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+-1 DOUGLAS R. HOFSTA DTER
+
+godel,escher,bach
+
+I ANETERNALGOLDEN BRAID
+
+■): i
+
+A METAPHORIC'AL FUGL E ON MIN DS A N E> M AC HIN ES
+, IN THE SPIRIT OF LEWIS CARROLL
+
+
+
+Contents
+
+
+Overview viii
+
+List of Illustrations xiv
+
+Words of Thanks xix
+
+Part I: GEB
+
+Introduction: A Musico-Logical Offering 3
+
+Three-Part Invention 29
+
+Chapter I: The MU-puzzle 33
+
+Two-Part Invention 43
+
+Chapter II: Meaning and Form in Mathematics 46
+
+Sonata for Unaccompanied Achilles 61
+
+Chapter III: Figure and Ground 64
+
+Contracrostipunctus 75
+
+Chapter IV: Consistency, Completeness, and Geometry 82
+
+Little Harmonic Labyrinth 103
+
+Chapter V: Recursive Structures and Processes 127
+
+Canon by Intervallic Augmentation 153
+
+Chapter VI: The Location of Meaning 158
+
+Chromatic Fantasy, And Feud 177
+
+Chapter VII: The Propositional Calculus 181
+
+Crab Canon 199
+
+Chapter VIII: Typographical Number Theory 204
+
+A Mu Offering 231
+
+Chapter IX: Mum on and Godel 246
+
+
+Contents
+
+
+VI
+
+
+
+Part II EGB
+
+
+Prelude ... 275
+
+Chapter X: Levels of Description, and Computer Systems 285
+
+Ant Fugue 311
+
+Chapter XI: Brains and Thoughts 337
+
+English French German Suit 366
+
+Chapter XII: Minds and Thoughts 369
+
+Aria with Diverse Variations 391
+
+Chapter XIII: BlooP and FlooP and GlooP 406
+
+Air on G's String 431
+
+Chapter XIV: On Formally Undecidable Propositions of TNT
+
+and Related Systems 438
+
+B irthday Cantatatata ... 461
+
+Chapter XV: Jumping out of the System 465
+
+Edifying Thoughts of a Tobacco Smoker 480
+
+Chapter XVI: Self-Ref and Self-Rep 495
+
+The Magn fierab, Indeed 549
+
+Chapter XVII: Church, Turing, Tarski, and Others 559
+
+SHRDFU, Toy of Man's Designing 586
+
+Chapter XVIII: Artificial Intelligence: Retrospects 594
+
+Contraf actus 633
+
+Chapter XIX: Artificial Intelligence: Prospects 641
+
+Sloth Canon 681
+
+Chapter XX: Strange Foops, Or Tangled Hierarchies 684
+
+Six-Part Ricercar 720
+
+Notes 743
+
+Bibliography 746
+
+Credits 757
+
+Index 759
+
+
+Contents
+
+
+VII
+
+
+
+Overview
+Part I: GEB
+
+Introduction: A Musico-Logical Offering. The book opens with the story of Bach's Musical
+Offering. Bach made an impromptu visit to King Frederick the Great of Prussia, and was
+requested to improvise upon a theme presented by the King. His improvisations formed the basis
+of that great work. The Musical Offering and its story form a theme upon which I "improvise"
+throughout the book, thus making a sort of "Metamusical Offering". Self-reference and the
+interplay between different levels in Bach are discussed: this leads to a discussion of parallel
+ideas in Escher's drawings and then Godel’s Theorem. A brief presentation of the history of logic
+and paradoxes is given as background for Godel’s Theorem. This leads to mechanical reasoning
+and computers, and the debate about whether Artificial Intelligence is possible. I close with an
+explanation of the origins of the book-particularly the why and wherefore of the Dialogues.
+
+Three-Part Invention. Bach wrote fifteen three-part inventions. In this three-part Dialogue, the
+Tortoise and Achilles-the main fictional protagonists in the Dialogues-are "invented" by Zeno (as
+in fact they were, to illustrate Zeno's paradoxes of motion). Very short, it simply gives the flavor
+of the Dialogues to come.
+
+Chapter I: The MU-puzzle. A simple formal system (the MIL'-system) is presented, and the reader
+is urged to work out a puzzle to gain familiarity with formal systems in general. A number of
+fundamental notions are introduced: string, theorem, axiom, rule of inference, derivation, formal
+system, decision procedure, working inside/outside the system.
+
+Two-Part Invention. Bach also wrote fifteen two-part inventions. This two-part Dialogue was written
+not by me, but by Lewis Carroll in 1895. Carroll borrowed Achilles and the Tortoise from Zeno,
+and I in turn borrowed them from Carroll. The topic is the relation between reasoning, reasoning
+about reasoning, reasoning about reasoning about reasoning, and so on. It parallels, in a way,
+Zeno's paradoxes about the impossibility of motion, seeming to show, by using infinite regress,
+that reasoning is impossible. It is a beautiful paradox, and is referred to several times later in the
+book.
+
+Chapter II: Meaning and Form in Mathematics. A new formal system (the pq-system) is
+presented, even simpler than the MlU-system of Chapter I. Apparently meaningless at first, its
+symbols are suddenly revealed to possess meaning by virtue of the form of the theorems they
+appear in. This revelation is the first important insight into meaning: its deep connection to
+isomorphism. Various issues related to meaning are then discussed, such as truth, proof, symbol
+manipulation, and the elusive concept, "form".
+
+Sonata for Unaccompanied Achilles. A Dialogue which imitates the Bach Sonatas for
+unaccompanied violin. In particular, Achilles is the only speaker, since it is a transcript of one
+end of a telephone call, at the far end of which is the Tortoise. Their conversation concerns the
+concepts of "figure" and "ground" in various
+
+
+Overview
+
+
+VIII
+
+
+
+contexts- e.g., Escher's art. The Dialogue itself forms an example of the distinction, since
+Achilles' lines form a "figure", and the Tortoise's lines-implicit in Achilles' lines-form a "ground".
+
+Chapter III: Figure and Ground. The distinction between figure and ground in art is compared to
+the distinction between theorems and nontheorems in formal systems. The question "Does a
+figure necessarily contain the same information as its ground%" leads to the distinction between
+recursively enumerable sets and recursive sets.
+
+Contracrostipunctus. This Dialogue is central to the book, for it contains a set of paraphrases of
+Godel’s self-referential construction and of his Incompleteness Theorem. One of the paraphrases
+of the Theorem says, "For each record player there is a record which it cannot play." The
+Dialogue's title is a cross between the word "acrostic" and the word "contrapunctus", a Latin word
+which Bach used to denote the many fugues and canons making up his Art of the Fugue. Some
+explicit references to the Art of the Fugue are made. The Dialogue itself conceals some acrostic
+tricks.
+
+Chapter IV: Consistency, Completeness, and Geometry. The preceding Dialogue is explicated to
+the extent it is possible at this stage. This leads back to the question of how and when symbols in
+a formal system acquire meaning. The history of Euclidean and non-Euclidean geometry is given,
+as an illustration of the elusive notion of "undefined terms". This leads to ideas about the
+consistency of different and possibly "rival" geometries. Through this discussion the notion of
+undefined terms is clarified, and the relation of undefined terms to perception and thought
+processes is considered.
+
+Little Harmonic Labyrinth. This is based on the Bach organ piece by the same name. It is a playful
+introduction to the notion of recursive-i.e., nested structures. It contains stories within stories. The
+frame story, instead of finishing as expected, is left open, so the reader is left dangling without
+resolution. One nested story concerns modulation in music-particularly an organ piece which
+ends in the wrong key, leaving the listener dangling without resolution.
+
+Chapter V: Recursive Structures and Processes. The idea of recursion is presented in many
+different contexts: musical patterns, linguistic patterns, geometric structures, mathematical
+functions, physical theories, computer programs, and others.
+
+Canon by Intervallic Augmentation. Achilles and the Tortoise try to resolve the question, "Which
+contains more information-a record, or the phonograph which plays it This odd question arises
+when the Tortoise describes a single record which, when played on a set of different
+phonographs, produces two quite different melodies: B-A-C-H and C-A-G-E. It turns out,
+however, that these melodies are "the same", in a peculiar sense.
+
+Chapter VI: The Location of Meaning. A broad discussion of how meaning is split among coded
+message, decoder, and receiver. Examples presented include strands of DNA, undeciphered
+inscriptions on ancient tablets, and phonograph records sailing out in space. The relationship of
+intelligence to "absolute" meaning is postulated.
+
+Chromatic Fantasy, And Feud. A short Dialogue bearing hardly any resemblance, except in title, to
+Bach's Chromatic Fantasy and Fugue. It concerns the proper way to manipulate sentences so as
+to preserve truth-and in particular the question
+
+
+Overview
+
+
+IX
+
+
+
+of whether there exist rules for the usage of the word "arid". This Dialogue has much in common
+with the Dialogue by Lewis Carroll.
+
+Chapter VII: The Propositional Calculus. It is suggested how words such as ,,and" can be
+governed by formal rules. Once again, the ideas of isomorphism and automatic acquisition of
+meaning by symbols in such a system are brought up. All the examples in this Chapter,
+incidentally, are "Zentences"-sentences taken from Zen koans. This is purposefully done,
+somewhat tongue-in-cheek, since Zen koans are deliberately illogical stories.
+
+Crab Canon. A Dialogue based on a piece by the same name from the Musical Offering. Both are so
+named because crabs (supposedly) walk backwards. The Crab makes his first appearance in this
+Dialogue. It is perhaps the densest Dialogue in the book in terms of formal trickery and level-
+play. Godel, Escher, and Bach are deeply intertwined in this very short Dialogue.
+
+Chapter VIII: Typographical Number Theory. An extension of the Propositional Calculus called
+"TNT" is presented. In TNT, number-theoretical reasoning can be done by rigid symbol
+manipulation. Differences between formal reasoning and human thought are considered.
+
+A Mu Offering. This Dialogue foreshadows several new topics in the book. Ostensibly concerned
+with Zen Buddhism and koans, it is actually a thinly veiled discussion of theoremhood and
+nontheoremhood, truth and falsity, of strings in number theory. There are fleeting references to
+molecular biology-particular) the Genetic Code. There is no close affinity to the Musical
+Offering, other than in the title and the playing of self-referential games.
+
+Chapter IX: Mumon and Godel. An attempt is made to talk about the strange ideas of Zen
+Buddhism. The Zen monk Mumon, who gave well known commentaries on many koans, is a
+central figure. In a way, Zen ideas bear a metaphorical resemblance to some contemporary ideas
+in the philosophy of mathematics. After this "Zennery", Godel’s fundamental idea of Godel-
+numbering is introduced, and a first pass through Godel’s Theorem is made.
+
+Part II: EGB
+
+Prelude ... This Dialogue attaches to the next one. They are based on preludes and fugues from
+Bach's Well-Tempered Clavier. Achilles and the Tortoise bring a present to the Crab, who has a
+guest: the Anteater. The present turns out to be a recording of the W.T.C.; it is immediately put
+on. As they listen to a prelude, they discuss the structure of preludes and fugues, which leads
+Achilles to ask how to hear a fugue: as a whole, or as a sum of parts? This is the debate between
+holism and reductionism, which is soon taken up in the Ant Fugue.
+
+Chapter X: Levels of Description, and Computer Systems. Various levels of seeing pictures,
+chessboards, and computer systems are discussed. The last of these is then examined in detail.
+This involves describing machine languages, assembly languages, compiler languages, operating
+systems, and so forth. Then the discussion turns to composite systems of other types, such as
+sports teams, nuclei, atoms, the weather, and so forth. The question arises as to how man
+intermediate levels exist-or indeed whether any exist.
+
+
+Overview
+
+
+X
+
+
+
+...Ant Fugue. An imitation of a musical fugue: each voice enters with the same statement. The
+theme-holism versus reductionism-is introduced in a recursive picture composed of words
+composed of smaller words, etc. The words which appear on the four levels of this strange picture
+are "HOLISM", "REDLCTIONIsM", and "ML". The discussion veers off to a friend of the
+Anteater's Aunt Hillary, a conscious ant colony. The various levels of her thought processes are
+the topic of discussion. Many fugal tricks are ensconced in the Dialogue. As a hint to the reader,
+references are made to parallel tricks occurring in the fugue on the record to which the foursome
+is listening. At the end of the Ant Fugue, themes from the Prelude return, transformed
+considerably.
+
+Chapter XI: Brains and Thoughts. "How can thoughts he supported by the hardware of the brain is
+the topic of the Chapter. An overview of the large scale and small-scale structure of the brain is
+first given. Then the relation between concepts and neural activity is speculatively discussed in
+some detail.
+
+English French German Suite. An interlude consisting of Lewis Carroll's nonsense poem
+"Jabberwocky' 1 together with two translations: one into French and one into German, both done
+last century.
+
+Chapter XII: Minds and Thoughts. The preceding poems bring up in a forceful way the question
+of whether languages, or indeed minds, can be "mapped" onto each other. How is communication
+possible between two separate physical brains: What do all human brains have in common? A
+geographical analogy is used to suggest an answer. The question arises, "Can a brain be
+understood, in some objective sense, by an outsider?"
+
+Aria with Diverse Variations. A Dialogue whose form is based on Bach's Goldberg Variations, and
+whose content is related to number-theoretical problems such as the Goldbach conjecture. This
+hybrid has as its main purpose to show how number theory's subtlety stems from the fact that
+there are many diverse variations on the theme of searching through an infinite space. Some of
+them lead to infinite searches, some of them lead to finite searches, while some others hover in
+between.
+
+Chapter XIII: BIooP and FlooP and GIooP. These are the names of three computer languages.
+BIooP programs can carry out only predictably finite searches, while FlooP programs can carry
+out unpredictable or even infinite searches. The purpose of this Chapter is to give an intuition for
+the notions of primitive recursive and general recursive functions in number theory, for they are
+essential in Godel’s proof.
+
+Air on G's String. A Dialogue in which Godel’s self-referential construction is mirrored in words.
+The idea is due to W. V. O. Quine. This Dialogue serves as a prototype for the next Chapter.
+
+Chapter XIV: On Formally Undeeidable Propositions of TNT and Related Systems. This
+Chapter's title is an adaptation of the title of Godel’s 1931 article, in which his Incompleteness
+Theorem was first published. The two major parts of Godel’s proof are gone through carefully. It
+is shown how the assumption of consistency of TNT forces one to conclude that TNT (or any
+similar system) is incomplete. Relations to Euclidean and non-Euclidean geometry are discussed.
+Implications for the philosophy of mathematics are gone into with some care.
+
+
+Overview
+
+
+XI
+
+
+
+Birthday Cantatatata ... In which Achilles cannot convince the wily and skeptical Tortoise that today
+is his (Achilles') birthday. His repeated but unsuccessful tries to do so foreshadow the
+repeatability of the Godel argument.
+
+Chapter XV: Jumping out of the System. The repeatability of Godel’s argument is shown, with
+the implication that TNT is not only incomplete, but "essentially incomplete The fairly notorious
+argument by J. R. Lucas, to the effect that Godel’s Theorem demonstrates that human thought
+cannot in any sense be "mechanical", is analyzed and found to be wanting.
+
+Edifying Thoughts of a Tobacco Smoker. A Dialogue treating of many topics, with the thrust being
+problems connected with self-replication and self-reference. Television cameras filming
+television screens, and viruses and other subcellular entities which assemble themselves, are
+among the examples used. The title comes from a poem by J. S. Bach himself, which enters in a
+peculiar way.
+
+Chapter XVI: Self-Ref and Self-Rep. This Chapter is about the connection between self-reference
+in its various guises, and self-reproducing entities e.g., computer programs or DNA molecules).
+The relations between a self-reproducing entity and the mechanisms external to it which aid it in
+reproducing itself (e.g., a computer or proteins) are discussed-particularly the fuzziness of the
+distinction. How information travels between various levels of such systems is the central topic of
+this Chapter.
+
+The Magnificrab, Indeed. The title is a pun on Bach's Magnifacat in D. The tale is about the Crab,
+who gives the appearance of having a magical power of distinguishing between true and false
+statements of number theory by reading them as musical pieces, playing them on his flute, and
+determining whether they are "beautiful" or not.
+
+Chapter XVII: Church, Turing, Tarski, and Others. The fictional Crab of the preceding Dialogue
+is replaced by various real people with amazing mathematical abilities. The Church-Turing
+Thesis, which relates mental activity to computation, is presented in several versions of differing
+strengths. All are analyzed, particularly in terms of their implications for simulating human
+thought mechanically, or programming into a machine an ability to sense or create beauty. The
+connection between brain activity and computation brings up some other topics: the halting
+problem of Turing, and Tarski's Truth Theorem.
+
+SHRDLU, Toy of Man's Designing. This Dialogue is lifted out of an article by Terry Winograd on
+his program SHRDLU: only a few names have been changed. In it. a program communicates
+with a person about the so-called "blocks world" in rather impressive English. The computer
+program appears to exhibit some real understanding-in its limited world. The Dialogue's title is
+based on Jesu, joy of Mans Desiring , one movement of Bach's Cantata 147.
+
+Chapter XVIII: Artificial Intelligence: Retrospects, This Chapter opens with a discussion of the
+famous "Turing test"-a proposal by the computer pioneer Alan Turing for a way to detect the
+presence or absence of "thought" in a machine. From there, we go on to an abridged history of
+Artificial Intelligence. This covers programs that can-to some degree-play games, prove
+theorems, solve problems, compose music, do mathematics, and use "natural language" (e.g.,
+English).
+
+
+Overview
+
+
+XII
+
+
+
+Contrafactus. About how we unconsciously organize our thoughts so that we can imagine
+hypothetical variants on the real world all the time. Also about aberrant variants of this ability-
+such as possessed by the new character, the Sloth, an avid lover of French fries, and rabid hater of
+counterfactuals.
+
+Chapter XIX: Artificial Intelligence: Prospects. The preceding Dialogue triggers a discussion of
+how knowledge is represented in layers of contexts. This leads to the modern A1 idea of "frames".
+A frame-like way of handling a set of visual pattern puzzles is presented, for the purpose of
+concreteness. Then the deep issue of the interaction of concepts in general is discussed, which
+leads into some speculations on creativity. The Chapter concludes with a set of personal
+"Questions and Speculations" on A1 and minds in general.
+
+Sloth Canon. A canon which imitates a Bach canon in which one voice plays the same melody as
+another, only upside down and twice as slowly, while a third voice is free. Here, the Sloth utters
+the same lines as the Tortoise does, only negated (in a liberal sense of the term) and twice as
+slowly, while Achilles is free.
+
+Chapter XX: Strange Loops, Or Tangled Hierarchies. A grand windup of many of the ideas
+about hierarchical systems and self-reference. It is concerned with the snarls which arise when
+systems turn back on themselves-for example, science probing science, government investigating
+governmental wrongdoing, art violating the rules of art, and finally, humans thinking about their
+own brains and minds. Does Godel’s Theorem have anything to say about this last "snarl"? Are
+free will and the sensation of consciousness connected to Godel’s Theorem? The Chapter ends by
+tying Godel, Escher, and Bach together once again.
+
+Six-Part Ricercar. This Dialogue is an exuberant game played with many of the ideas which have
+permeated the book. It is a reenactment of the story of the Musical Offering, which began the
+book; it is simultaneously a "translation" into words of the most complex piece in the Musical
+Offering: the Six-Part Ricercar. This duality imbues the Dialogue with more levels of meaning
+than any other in the book. Frederick the Great is replaced by the Crab, pianos by computers, and
+so on. Many surprises arise. The Dialogue's content concerns problems of mind, consciousness,
+free will. Artificial Intelligence, the Turing test, and so forth, which have been introduced earlier.
+It concludes with an implicit reference to the beginning of the book, thus making the book into
+one big self-referential loop, symbolizing at once Bach's music, Escher's drawings, and Godel’s
+Theorem.
+
+
+Overview
+
+
+XIII
+
+
+
+
+FIGURE I. Johann StbtMmn Bach. in 174H. From a panning bj Elias Golllieb
+Haussmann.
+
+
+FIGURE I. Johann Sebastian Bach, in 1748. From a painting by Elias Gottlieb
+Hanssmann.
+
+
+Introduction: A Musico-Logical Offering
+
+
+2
+
+
+Introduction:
+
+
+A Musico-Logical Offering
+
+Author:
+
+FREDERICK THE GREAT, King of Prussia, came to power in 1740. Although he is
+remembered in history books mostly for his military astuteness, he was also devoted to
+the life of the mind and the spirit. His court in Potsdam was one of the great centers of
+intellectual activity in Europe in the eighteenth century. The celebrated mathematician
+Leonhard Euler spent twenty-five years there. Many other mathematicians and scientists
+came, as well as philosophers-including Voltaire and La Mettrie, who wrote some of their
+most influential works while there.
+
+But music was Frederick's real love. He was an avid flutist and composer. Some of his
+compositions are occasionally performed even to this day. Frederick was one of the first
+patrons of the arts to recognize the virtues of the newly developed "piano-forte" ("soft-
+loud"). The piano had been developed in the first half of the eighteenth century as a
+modification of the harpsichord. The problem with the harpsichord was that pieces could
+only be played at a rather uniform loudness-there was no way to strike one note more
+loudly than its neighbors. The "soft-loud", as its name implies, provided a remedy to this
+problem. From Italy, where Bartolommeo Cristofori had made the first one, the soft-loud
+idea had spread widely. Gottfried Silbermann, the foremost German organ builder of the
+day, was endeavoring to make a "perfect" piano-forte. Undoubtedly King Frederick was
+the greatest supporter of his efforts-it is said that the King owned as many as fifteen
+Silbermann pianos!
+
+
+Bach
+
+Frederick was an admirer not only of pianos, but also of an organist and composer by the
+name of J. S. Bach. This Bach's compositions were somewhat notorious. Some called
+them "turgid and confused", while others claimed they were incomparable masterpieces.
+But no one disputed Bach's ability to improvise on the organ. In those days, being an
+organist not only meant being able to play, but also to extemporize, and Bach was known
+far and wide for his remarkable extemporizations. (For some delightful anecdotes about
+Bach's extemporization, see The Bach Reader, by H. T. David and A. Mendel.)
+
+In 1747, Bach was sixty-two, and his fame, as well as one of his sons, had reached
+Potsdam: in fact, Carl Philipp Emanuel Bach was the Capellmeister (choirmaster) at the
+court of King Frederick. For years the King had let it be known, through gentle hints to
+Philipp Emanuel, how
+
+
+Introduction: A Musico-Logical Offering
+
+
+3
+
+
+
+pleased he would be to have the elder Bach come and pay him a visit; but this wish had
+never been realized. Frederick was particularly eager for Bach to try out his new
+Silbermann pianos, which lie (Frederick) correctly foresaw as the great new wave in
+music.
+
+It was Frederick's custom to have evening concerts of chamber music in his court.
+Often he himself would be the soloist in a concerto for flute Here we have reproduced a
+painting of such an evening by the German painter Adolph von Menzel, who, in the
+1800's, made a series of paintings illustrating the life of Frederick the Great. At the
+cembalo is C. P. E. Bach, and the figure furthest to the right is Joachim Quantz, the
+King's flute master-and the only person allowed to find fault with the King's flute
+playing. One May evening in 1747, an unexpected guest showed up. Johann Nikolaus
+Forkel, one of Bach's earliest biographers, tells the story
+as follows:
+
+One evening, just as lie was getting his flute ready, and his musicians were ssembled,
+an officer brought him a list of the strangers who had arrived. With his flute in his hand
+he ran ever the list, but immediately turned to the assembled musicians, and said, with a
+kind of agitation, "Gentlemen, old Bach is come." The Hute was now laid aside, and old
+Bach, who had alighted at his son's lodgings, was immediately summoned to the Palace.
+Wilhelm Friedemann, who accompanied his father, told me this story, and I must say
+that 1 still think with pleasure on the manner in which lie related it. At that time it was
+the fashion to make rather prolix compliments. The first appearance of J. S. Bach before
+se great a King, who did not even give him time to change his traveling dress for a
+black chanter's gown, must necessarily be attended with many apologies. I will net here
+dwell en these apologies, but merely observe, that in Wilhelm Friedemann's mouth they
+made a formal Dialogue between the King and the Apologist.
+
+But what is mere important than this is that the King gave up his Concert for this
+evening, and invited Bach, then already called the Old Bach, to try his fortepianos,
+made by Silbermann, which steed in several rooms of the palace. [Forkel here inserts
+this footnote: "The pianofortes manufactured by Silbermann, of Frevberg, pleased the
+King se much, that he resolved to buy them all up. He collected fifteen. I hear that they
+all now stand unfit for use in various corners of the Royal Palace."] The musicians went
+with him from room to room, and Bach was invited everywhere to try them and to play
+unpremeditated compositions. After he had gene en for some time, he asked the King to
+give him a subject for a Fugue, in order to execute it immediately without any
+preparation. The King admired the learned manner in which his subject was thus
+executed extempore: and, probably to see hew far such artt could be carried, expressed
+a wish to hear a Fugue with six Obligato parts. But as it is not every subject that is fit
+for such full harmony, Bach chose one himself, and immediately executed it to the
+astonishment of all present in the same magnificent and learned manner as he had done
+that of the King. His Majesty desired also to hear his performance en the organ. The
+next day therefore Bach was taken to all the organs in Potsdam, as lie had before been
+to Silbermann's fortepianos. After his return to Leipzig, he composed the subject, which
+he had received from the King, in three and six parts, added several artificial passages
+in strict canon to it, and had it engraved, under the title of "Musikalisches Opfer"
+[Musical Offering], and dedicated it to the Inventor.'
+
+
+Introduction: A Musico-Logical Offering
+
+
+4
+
+
+
+
+Introduction: A Musico-Logical Offering
+
+
+5
+
+
+FIGURE 2. Flute Concert in Sanssouci, by Adolph von Menu! (1852).
+
+
+
+
+
+
+FIGURE 3. The Royal Theme.
+
+When Bach sent a copy of his Musical Offering to the King, he included a dedicatory
+letter, which is of interest for its prose style if nothing else rather submissive and
+flattersome. From a modern perspective it seems comical. Also, it probably gives
+something of the flavor of Bach's apology for his appearance.2
+
+MOST GRACIOUS KING!
+
+
+In deepest humility I dedicate herewith to Your Majesty a musical offering, the
+noblest part of which derives from Your Majesty's own august hand. With awesome
+pleasure I still remember the very special Royal grace when, some time ago, during
+my visit in Potsdam, Your Majesty's Self deigned to play to me a theme for a fugue
+upon the clavier, and at the same time charged me most graciously to carry it out in
+Your Majesty's most august presence. To obey Your Majesty's command was my most
+humble dim. I noticed very soon, however, that, for lack of necessary preparation, the
+execution of the task did not fare as well as such an excellent theme demanded. I
+resoled therefore and promptly pledged myself to work out this right Royal theme
+more fully, and then make it known to the world. This resolve has now been carried
+out as well as possible, and it has none other than this irreproachable intent, to glorify,
+if only in a small point, the fame of a monarch whose greatness and power, as in all
+the sciences of war and peace, so especially in music, everyone must admire and
+revere. I make bold to add this most humble request: may Your Majesty deign to
+dignify the present modest labor with a gracious acceptance, and continue to grant
+Your Majesty’s most august Royal grace to
+
+
+Leipzig, July 7 1747
+
+
+Your Majesty's
+
+most humble and obedient servant,
+THE AUTHOR
+
+
+Some twenty-seven years later, when Bach had been dead for twentyfour years, a Baron
+named Gottfried van Swieten-to whom, incidentally, Forkel dedicated his biography of
+Bach, and Beethoven dedicated his First Symphony-had a conversation with King
+Frederick, which he reported as follows:
+
+
+He [Frederick] spoke to me, among other things, of music, and of a great organist
+named Bach, who has been for a while in Berlin. This aitist [Wilhelm Friedemann
+Bach] is endowed with a talent superior, in depth of harmonic knowledge and power
+of execution, to any I have heard or can imagine, while those who knew his father
+claim that he, in turn, was even greater. The King
+
+
+Introduction: A Musico-Logical Offering
+
+
+6
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+is of this opinion, and to prove it to me he sang aloud a chromatic fugue subject which
+he had given this old Bach, who on the spot had made of it a fugue in four paits, then
+in five parts, and finally in eight parts.'
+
+Of course there is no way of knowing whether it was King Frederick or Baron van
+Swieten who magnified the story into larger-than-life proportions. But it shows how
+powerful Bach's legend had become by that time. To give an idea of how extraordinary a
+six-part fugue is, in the entire Well-Tempered Clavier by Bach, containing forty-eight
+Preludes and Fugues, only two have as many as five parts, and nowhere is there a six-pait
+fugue! One could probably liken the task of improvising a six-part fugue to the playing of
+sixty simultaneous blindfold games of chess, and winning them all. To improvise an
+eight-part fugue is really beyond human capability.
+
+In the copy which Bach sent to King Frederick, on the page preceding the first sheet of
+music, was the following inscription:
+
+
+dTVcyis lam (untie El Jvl icjua Gnoi lica A rtc ^Rdolula .
+
+FIGURE4.
+
+("At the King's Command, the Song and the Remainder Resolved with Canonic Art.")
+Here Bach is punning on the word "canonic", since it means not only "with canons" but
+also "in the best possible way". The initials of this inscription are
+
+RICERCAR
+
+-an Italian word, meaning "to seek". And certainly there is a great deal to seek in the
+Musical Offering. It consists of one three-part fugue, one six-part fugue, ten canons, and a
+trio sonata. Musical scholars have concluded that the three-part fugue must be, in
+essence, identical with the one which Bach improvised for King Frederick. The six-pait
+fugue is one of Bach's most complex creations, and its theme is, of course, the Royal
+Theme. That theme, shown in Figure 3, is a very complex one, rhythmically irregular and
+highly chromatic (that is, filled with tones which do not belong to the key it is in). To
+write a decent fugue of even two voices based on it would not be easy for the average
+musician!
+
+Both of the fugues are inscribed "Ricercar", rather than "Fuga". This is another
+meaning of the word; "ricercar" was, in fact, the original name for the musical form now
+known as "fugue". By Bach's time, the word "fugue" (or fuga, in Latin and Italian) had
+become standard, but the term "ricercar" had survived, and now designated an erudite
+kind of fugue, perhaps too austerely intellectual for the common ear. A similar usage
+survives in English today: the word "recherche" means, literally, "sought out", but carries
+the same kind of implication, namely of esoteric or highbrow cleverness.
+
+The trio sonata forms a delightful relief from the austerity of the fugues and canons,
+because it is very melodious and sweet, almost dance-
+
+
+introduction: A Musico-Logical Offering
+
+
+7
+
+
+able. Nevertheless, it too is based largely on the King's theme, chromatic and austere as it
+is. It is rather miraculous that Bach could use such a theme to make so pleasing an
+interlude.
+
+The ten canons in the Musical Offering are among the most sophisticated canons Bach
+ever wrote. However, curiously enough, Bach himself never wrote them out in full. This
+was deliberate. They were posed as puzzles to King Frederick. It was a familiar musical
+game of the day to give a single theme, together with some more or less tricky hints, and
+to let the canon based on that theme be "discovered" by someone else. In order to know
+how this is possible, you must understand a few facts about canons.
+
+Canons and Fugues
+
+The idea of a canon is that one single theme is played against itself. This is done by
+having "copies" of the theme played by the various participating voices. But there are
+means' ways to do this. The most straightforward of all canons is the round, such as
+"Three Blind Mice", "Row, Row, Row Your Boat", or " Frere Jacques". Here, the theme
+enters in the first voice and, after a fixed time-delay, a "copy" of it enters, in precisely the
+same key. After the same fixed time-delay in the second voice, the third voice enters
+carrying the theme, and so on. Most themes will not harmonize with themselves in this
+way. In order for a theme to work as a canon theme, each of its notes must be able to
+serve in a dual (or triple, or quadruple) role: it must firstly be part of a melody, and
+secondly it must be part of a harmonization of the same melody. When there are three
+canonical voices, for instance, each note of the theme must act in two distinct harmonic
+ways, as well as melodically. Thus, each note in a canon has more than one musical
+meaning; the listener's ear and brain automatically figure out the appropriate meaning, by
+referring to context.
+
+There are more complicated sorts of canons, of course. The first escalation in
+complexity comes when the "copies" of the theme are staggered not only in time, but also
+in pitch; thus, the first voice might sing the theme starting on C, and the second voice,
+overlapping with the first voice, might sing the identical theme starting five notes higher,
+on G. A third voice, starting on the D yet five notes higher, might overlap with the first
+two, and so on. The next escalation in complexity comes when the speeds of the different
+voices are not equal; thus, the second voice might sing twice as quickly, or twice as
+slowly, as the first voice. The former is called diminution, the latter augmentation (since
+the theme seems to shrink or to expand).
+
+We are not yet done! The next stage of complexity in canon construction is to invert the
+theme, which means to make a melody which jumps down wherever the original theme
+jumps up, and by exactly the same number of semitones. This is a rather weird melodic
+transformation, but when one has heard many themes inverted, it begins to seem quite
+natural. Bach was especially fond of inversions, and they show up often in his work-and
+the Musical Offering is no exception. (For a simple example of
+
+
+Introduction: A Musico-Logical Offering
+
+
+8
+
+
+
+inversion, try the tune "Good King Wenceslas". When the original and its inversion are
+sung together, starting an octave apart and staggered with a time-delay of two beats, a
+pleasing canon results.) Finally, the most esoteric of "copies" is the retrograde copy-
+where the theme is played backwards in time. A canon which uses this trick is
+affectionately known as a crab canon, because of the peculiarities of crab locomotion.
+Bach included a crab canon in the Musical Offering, needless to say. Notice that every
+type of "copy" preserves all the information in the original theme, in the sense that the
+theme is fully recoverable from any of the copies. Such an information preserving
+transformation is often called an isomorphism, and we will have much traffic with
+isomorphisms in this book.
+
+Sometimes it is desirable to relax the tightness of the canon form. One way is to allow
+slight departures from perfect copying, for the sake of more fluid harmony. Also, some
+canons have "free" voices-voices which do not employ the canon's theme, but which
+simply harmonize agreeably with the voices that are in canon with each other.
+
+Each of the canons in the Musical Offering has for its theme a different variant of the
+King's Theme, and all the devices described above for making canons intricate are
+exploited to the hilt; in fact, they are occasionally combined. Thus, one three-voice canon
+is labeled "Canon per Augmentationem, contrario Motu"; its middle voice is free (in fact,
+it sings the Royal Theme), while the other two dance canonically above and below it,
+using the devices of augmentation and inversion. Another bears simply the cryptic label
+"Quaerendo invenietis" ("By seeking, you will discover"). All of the canon puzzles have
+been solved. The canonical solutions were given by one of Bach's pupils, Johann Philipp
+Kirnberger. But one might still wonder whether there are more solutions to seek!
+
+I should also explain briefly what a fugue is. A fugue is like a canon, in that it is
+usually based on one theme which gets played in different voices and different keys, and
+occasionally at different speeds or upside down or backwards. However, the notion of
+fugue is much less rigid than that of canon, and consequently it allows for more
+emotional and artistic expression. The telltale sign of a fugue is the way it begins: with a
+single voice singing its theme. When it is done, then a second voice enters, either five
+scale-notes up, or four down. Meanwhile the first voice goes on, singing the
+"countersubject": a secondary theme, chosen to provide rhythmic, harmonic, and melodic
+contrasts to the subject. Each of the voices enters in turn, singing the theme, often to the
+accompaniment of the countersubject in some other voice, with the remaining voices
+doing whatever fanciful things entered the composer's mind. When all the voices have
+"arrived", then there are no rules. There are, to be sure, standard kinds of things to do-but
+not so standard that one can merely compose a fugue by formula. The two fugues in the
+Musical Offering are outstanding examples of fugues that could never have been
+"composed by formula". There is something much deeper in them than mere fugality.
+
+All in all, the Musical Offering represents one of Bach's supreme accomplishments in
+counterpoint. It is itself one large intellectual fugue, in
+
+
+Introduction: A Musico-Logical Offering
+
+
+9
+
+
+
+which many ideas and forms have been woven together, and in which playful double
+meanings and subtle allusions are commonplace. And it is a very beautiful creation of the
+human intellect which we can appreciate forever. (The entire work is wonderfully
+described in the book f. S. Bach's Musical Offering, by H. T. David.)
+
+An Endlessly Rising Canon
+
+There is one canon in the Musical Offering which is particularly unusual. Labeled simply
+"Canon per Tonos", it has three voices. The uppermost voice sings a variant of the Royal
+Theme, while underneath it, two voices provide a canonic harmonization based on a
+second theme. The lower of this pair sings its theme in C minor (which is the key of the
+canon as a whole), and the upper of the pair sings the same theme displaced upwards in
+pitch by an interval of a fifth. What makes this canon different from any other, however,
+is that when it concludes-or, rather, seems to conclude-it is no longer in the key of C
+minor, but now is in D minor. Somehow Bach has contrived to modulate (change keys)
+right under the listener's nose. And it is so constructed that this "ending" ties smoothly
+onto the beginning again; thus one can repeat the process and return in the key of E, only
+to join again to the beginning. These successive modulations lead the ear to increasingly
+remote provinces of tonality, so that after several of them, one would expect to be
+hopelessly far away from the starting key. And yet magically, after exactly six such
+modulations, the original key of C minor has been restored! All the voices are exactly one
+octave higher than they were at the beginning, and here the piece may be broken off in a
+musically agreeable way. Such, one imagines, was Bach's intention; but Bach indubitably
+also relished the implication that this process could go on ad infinitum, which is perhaps
+why he wrote in the margin "As the modulation rises, so may the King's Glory." To
+emphasize its potentially infinite aspect, I like to call this the "Endlessly Rising Canon".
+
+In this canon, Bach has given us our first example of the notion of Strange Loops. The
+"Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards)
+through the levels of some hierarchical system, we unexpectedly find ourselves right
+back where we started. (Here, the system is that of musical keys.) Sometimes I use the
+term Tangled Hierarchy to describe a system in which a Strange Loop occurs. As we go
+on, the theme of Strange Loops will recur again and again. Sometimes it will be hidden,
+other times it will be out in the open; sometimes it will be right side up, other times it will
+be upside down, or backwards. "Quaerendo invenietis" is my advice to the reader.
+
+
+Escher
+
+To my mind, the most beautiful and powerful visual realizations of this notion of Strange
+Loops exist in the work of the Dutch graphic artist M. C. Escher, who lived from 1902 to
+1972. Escher was the creator of some of the
+
+
+Introduction: A Musico-Logical Offering
+
+
+10
+
+
+
+
+FIGURE 5. Waterfall, by M. C. Escher (lithograph, 1961).
+
+
+most intellectually stimulating drawings of all time. Many of them have their origin in
+paradox, illusion, or double-meaning. Mathematicians were among the first admirers of
+Escher's drawings, and this is understandable because they often are based on
+mathematical principles of symmetry or pattern ... But there is much more to a typical
+Escher drawing than just symmetry or pattern; there is often an underlying idea, realized
+in artistic form. And in pailicular, the Strange Loop is one of the most recurrent themes in
+Escher's work. Look, for example, at the lithograph Waterfall (Fig. 5), and compare its
+six-step endlessly falling loop with the six-step endlessly rising loop of the "Canon per
+Tonos". The similarity of vision is
+
+
+Introduction: A Musico-Logical Offering
+
+
+11
+
+
+
+
+
+FIGURE 6. Ascending and Descending, by M. C. Escher (lithograph, 1960).
+
+
+Introduction: A Musico-Logical Offering
+
+
+12
+
+
+
+
+remarkable. Bach and Escher are playing one single theme in two different "keys": music
+and art.
+
+Escher realized Strange Loops in several different ways, and they can be arranged
+according to the tightness of the loop. The lithograph Ascending and Descending (Fig. 6),
+in which monks trudge forever in loops, is the loosest version, since it involves so many
+steps before the starting point is regained. A tighter loop is contained in Waterfall, which,
+as we already observed, involves only six discrete steps. You may be thinking that there
+is some ambiguity in the notion of a single "step"-for instance, couldn't Ascending and
+Descending be seen just as easily as having four levels (staircases) as forty-five levels
+(stairs)% It is indeed true that there is an inherent
+
+
+
+FIGURE 7. Hand with Reflecting Globe. Self-portrait In, M. C. Escher (lithograph,
+
+1935).
+
+
+Introduction: A Musico-Logical Offering
+
+
+13
+
+
+
+
+
+Introduction: A Musico-Logical Offering
+
+
+14
+
+
+FIGURE 8. Metamorphosis 11, b\ M. C. F.seller (woodrut. 19.5 cm. x 400 cm., 1959- 40).
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+haziness in level-counting, not only in Escher pictures, but in hierarchical, many-level
+systems. We will sharpen our understanding of this haziness later on. But let us not get
+too distracted now' As we tighten our loop, we come to the remarkable Drawing Hands
+(Fig. 135), in which each of two hands draws the other: a two-step Strange Loop. And
+finally, the tightest of all Strange Loops is realized in Print Gallery (Fig. 142): a picture
+of a picture which contains itself. Or is it a picture of a gallery which contains itself? Or
+of a town which contains itself? Or a young man who contains himself? (Incidentally, the
+illusion underlying Ascending and Descending and Waterfall was not invented by Escher,
+but by Roger Penrose, a British mathematician, in 1958. However, the theme of the
+Strange Loop was already present in Escher's work in 1948, the year he drew Drawing
+Hands. Print Gallery dates from 1956.)
+
+Implicit in the concept of Strange Loops is the concept of infinity, since what else is a
+loop but a way of representing an endless process in a finite way? And infinity plays a
+large role n many of Escher's drawings. Copies of one single theme often fit into each'
+other, forming visual analogues to the canons of Bach. Several such patterns can be seen
+in Escher's famous print Metamorphosis (Fig. 8). It is a little like the "Endlessly Rising
+Canon": wandering further and further from its starting point, it suddenly is back. In the
+tiled planes of Metamorphosis and other pictures, there are already suggestions of
+infinity. But wilder visions of infinity appear in other drawings by Escher. In some of his
+drawings, one single theme can appear on different levels of reality. For instance, one
+level in a drawing might clearly be recognizable as representing fantasy or imagination;
+another level would be recognizable as reality. These two levels might be the only
+explicitly portrayed levels. But the mere presence of these two levels invites the viewer to
+look upon himself as part of yet another level; and by taking that step, the viewer cannot
+help getting caught up in Escher's implied chain of levels, in which, for any one level,
+there is always another level above it of greater "reality", and likewise, there is always a
+level below, "more imaginary" than it is. This can be mind-boggling in itself. However,
+what happens if the chain of levels is not linear, but forms a loop? What is real, then, and
+what is fantasy? The genius of Escher was that he could not only concoct, but actually
+portray, dozens of half-real, half-mythical worlds, worlds filled with Strange Loops,
+which he seems to be inviting his viewers to enter.
+
+Godel
+
+In the examples we have seen of Strange Loops by Bach and Escher, there is a conflict
+between the finite and the infinite, and hence a strong sense of paradox. Intuition senses
+that there is something mathematical involved here. And indeed in our own century a
+mathematical counterpart was discovered, with the most enormous repercussions. And,
+just as the Bach and Escher loops appeal to very simple and ancient intuitions-a musical
+scale, a staircase-so this discovery, by K. Godel, of a Strange Loop in
+
+
+Introduction: A Musico-Logical Offering
+
+
+15
+
+
+
+
+FIGURE 9. Kurl Godel.
+
+
+InlroducUon: A Musico-Logical Offering
+
+
+16
+
+
+
+
+mathematical systems has its origins in simple and ancient intuitions. In its absolutely
+barest form, Godel's discovery involves the translation of an ancient paradox in
+philosophy into mathematical terms. That paradox is the so-called Epimenides paradox,
+or liar paradox. Epimenides was a Cretan who made one immortal statement: "All
+Cretans are liars." A sharper version of the statement is simply "I am lying"; or, "This
+statement is false". It is that last version which I will usually mean when I speak of the
+Epimenides paradox. It is a statement which rudely violates the usually assumed
+dichotomy of statements into true and false, because if you tentatively think it is true,
+then it immediately backfires on you and makes you think it is false. But once you've
+decided it is false, a similar backfiring returns you to the idea that it must be true. Try it!
+
+The Epimenides paradox is a one-step Strange Loop, like Escher's Print Gallery. But
+how does it have to do with mathematics? That is what Godel discovered. His idea was to
+use mathematical reasoning in exploring mathematical reasoning itself. This notion of
+making mathematics "introspective" proved to be enormously powerful, and perhaps its
+richest implication was the one Godel found: Godel's Incompleteness Theorem. What the
+Theorem states and how it is proved are two different things. We shall discuss both in
+quite some detail in this book. The Theorem can De likened to a pearl, and the method of
+proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex
+living beast whose innards give rise to this mysteriously simple gem.
+
+Godel's Theorem appears as Proposition VI in his 1931 paper "On Formally
+Undecidable Propositions in Principia Mathematica and Related Systems I." It states:
+
+To every w-consistent recursive class K of formulae there correspond recursive
+class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Fig (K) (where v
+is the free variable of r).
+
+Actually, it was in German, and perhaps you feel that it might as well be in German
+anyway. So here is a paraphrase in more normal English:
+
+All consistent axiomatic formulations of number theory
+include undecidable propositions.
+
+This is the pearl.
+
+In this pearl it is hard to see a Strange Loop. That is because the Strange Loop is buried
+in the oyster-the proof. The proof of Godel's Incompleteness Theorem hinges upon the
+writing of a self-referential mathematical statement, in the same way as the Epimenides
+paradox is a self-referential statement of language. But whereas it is very simple to talk
+about language in language, it is not at all easy to see how a statement about numbers can
+talk about itself. In fact, it took genius merely to connect the idea of self-referential
+statements with number theory. Once Godel had the intuition that such a statement could
+be created, he was over the major hurdle. The actual creation of the statement was the
+working out of this one beautiful spark of intuition.
+
+
+Introduction: A Musico-Logical Offering
+
+
+17
+
+
+
+We shall examine the Godel construction quite carefully in Chapters to come, but so that
+you are not left completely in the dark, I will sketch here, in a few strokes, the core of the
+idea, hoping that what you see will trigger ideas in your mind. First of all, the difficulty
+should be made absolutely clear. Mathematical statements-let us concentrate on number-
+theoretical ones-are about properties of whole numbers. Whole numbers are not
+statements, nor are their properties. A statement of number theory is not about a.
+statement of number theory; it just is a statement of number theory. This is the problem;
+but Godel realized that there was more here than meets the eye.
+
+Godel had the insight that a statement of number theory could be about a statement of
+number theory (possibly even itself), if only numbers could somehow stand for
+statements. The idea of a code, in other words, is at the heart of his construction. In the
+Godel Code, usually called "Godel-numbering", numbers are made to stand for symbols
+and sequences of symbols. That way, each statement of number theory, being a sequence
+of specialized symbols, acquires a Godel number, something like a telephone number or a
+license plate, by which it can be referred to. And this coding trick enables statements of
+number theory to be understood on two different levels: as statements of number theory,
+and also as statements about statements of number theory.
+
+Once Godel had invented this coding scheme, he had to work out in detail a way of
+transporting the Epimenides paradox into a numbertheoretical formalism. His final
+transplant of Epimenides did not say, "This statement of number theory is false", but
+rather, "This statement of number theory does not have any proof". A great deal of
+confusion can be caused by this, because people generally understand the notion of
+"proof" rather vaguely. In fact, Godel's work was just part of a long attempt by
+mathematicians to explicate for themselves what proofs are. The important thing to keep
+in mind is that proofs are demonstrations within fixed systems of propositions. In the case
+of Godel's work, the fixed system of numbertheoretical reasoning to which the word
+"proof" refers is that of Principia Mathematica (P.M.), a giant opus by Bertrand Russell
+and Alfred North Whitehead, published between 1910 and 1913. Therefore, the Godel
+sentence G should more properly be written in English as:
+
+This statement of number theory does not have any proof in the system of Principia
+
+Mathematica.
+
+Incidentally, this Godel sentence G is not Godel's Theorem-no more than the Epimenides
+sentence is the observation that "The Epimenides sentence is a paradox." We can now
+state what the effect of discovering G is. Whereas the Epimenides statement creates a
+paradox since it is neither true nor false, the Godel sentence G is unprovable (inside
+P.M.) but true. The grand conclusion% That the system of Principia Mathematica is
+"incomplete"-there are true statements of number theory which its methods of proof are
+too weak to demonstrate.
+
+
+Introduction: A Musico-Logical Offering
+
+
+18
+
+
+
+But if Principia Mathematica was the first victim of this stroke, it was certainly not the
+last! The phrase "and Related Systems" in the title of Godel's article is a telling one: for if
+Godel's result had merely pointed out a defect in the work of Russell and Whitehead, then
+others could have been inspired to improve upon P.M. and to outwit Godel's Theorem.
+But this was not possible: Godel's proof pertained to any axiomatic system which
+purported to achieve the aims which Whitehead and Russell had set for themselves. And
+for each different system, one basic method did the trick. In short, Godel showed that
+provability is a weaker notion than truth, no matter what axiomatic system is involved.
+
+Therefore Godel's Theorem had an electrifying effect upon logicians, mathematicians,
+and philosophers interested in the foundations of mathematics, for it showed that no fixed
+system, no matter how complicated, could represent the complexity of the whole
+numbers: 0, 1, 2, 3, ... Modern readers may not be as nonplussed by this as readers of
+1931 were, since in the interim our culture has absorbed Godel's Theorem, along with the
+conceptual revolutions of relativity and quantum mechanics, and their philosophically
+disorienting messages have reached the public, even if cushioned by several layers of
+translation (and usually obfuscation). There is a general mood of expectation, these days,
+of "limitative" results-but back in 1931, this came as a bolt from the blue.
+
+Mathematical Logic: A Synopsis
+
+A proper appreciation of Godel's Theorem requires a setting of context. Therefore, I will
+now attempt to summarize in a short space the history of mathematical logic prior to
+1931-an impossible task. (See DeLong, Kneebone, or Nagel and Newman, for good
+presentations of history.) It all began with the attempts to mechanize the thought
+processes of reasoning. Now our ability to reason has often been claimed to be what
+distinguishes us from other species; so it seems somewhat paradoxical, on first thought,
+to mechanize that which is most human. Yet even the ancient Greeks knew that reasoning
+is a patterned process, and is at least partially governed by statable laws. Aristotle
+codified syllogisms, and Euclid codified geometry; but thereafter, many centuries had to
+pass before progress in the study of axiomatic reasoning would take place again.
+
+One of the significant discoveries of nineteenth-century mathematics was that there are
+different, and equally valid, geometries-where by "a geometry" is meant a theory of
+properties of abstract points and lines. It had long been assumed that geometry was what
+Euclid had codified, and that, although there might be small flaws in Euclid's
+presentation, they were unimportant and any real progress in geometry would be
+achieved by extending Euclid. This idea was shattered by the roughly simultaneous
+discovery of non-Euclidean geometry by several people-a discovery that shocked the
+mathematics community, because it deeply challenged the idea that mathematics studies
+the real world. How could there be many differ
+
+
+Introduction: A Musico-Logical Offering
+
+
+19
+
+
+
+ent kinds of "points" and "lines" in one single reality? Today, the solution to the dilemma
+may be apparent, even to some nonmathematicians-but at the time, the dilemma created
+havoc in mathematical circles.
+
+Later in the nineteenth century, the English logicians George Boole and Augustus De
+Morgan went considerably further than Aristotle in codifying strictly deductive reasoning
+patterns. Boole even called his book "The Laws of Thought"-surely an exaggeration, but
+it was an important contribution. Lewis Carroll was fascinated by these mechanized
+reasoning methods, and invented many puzzles which could be solved with them. Gottlob
+Lrege in Jena and Giuseppe Peano in Turin worked on combining formal reasoning with
+the study of sets and numbers. David Hilbert in Gottingen worked on stricter
+formalizations of geometry than Euclid's. All of these efforts were directed towards
+clarifying what one means by "proof".
+
+In the meantime, interesting developments were taking place in classical mathematics.
+A theory of different types of infinities, known as the theory of sets, was developed by
+Georg Cantor in the 1880's. The theory was powerful and beautiful, but intuition-defying.
+Before long, a variety of set-theoretical paradoxes had been unearthed. The situation was
+very disturbing, because just as mathematics seemed to be recovering from one set of
+paradoxes-those related to the theory of limits, in the calculusalong came a whole new
+set, which looked worse!
+
+The most famous is Russell's paradox. Most sets, it would seem, are not members of
+themselves-for example, the set of walruses is not a walrus, the set containing only Joan
+of Arc is not Joan of Arc (a set is not a person)-and so on. In this respect, most sets are
+rather "run-of-the-mill". However, some "self-swallowing" sets do contain themselves as
+members, such as the set of all sets, or the set of all things except Joan of Arc, and so on.
+Clearly, every set is either run-of-the-mill or self-swallowing, and no set can be both.
+Now nothing prevents us from inventing R: the set of all run-o,-the-mill sets. At first, R
+might seem a rather run-of-the-mill invention-but that opinion must be revised when you
+ask yourself, "Is R itself "a run-of-the-mill set or a self-swallowing set?" You will find
+that the answer is: "R is neither run-of-the-mill nor self-swallowing, for either choice
+leads to paradox." Try it!
+
+But if R is neither run-of-the-mill nor self-swallowing, then what is it? At the very
+least, pathological. But no one was satisfied with evasive answers of that sort. And so
+people began to dig more deeply into the foundations of set theory. The crucial questions
+seemed to be: "What is wrong with our intuitive concept of 'set'? Can we make a rigorous
+theory of sets which corresponds closely with our intuitions, but which skirts the
+paradoxes?" Here, as in number theory and geometry, the problem is in trying to line up
+intuition with formalized, or axiomatized, reasoning systems.
+
+A startling variant of Russell's paradox, called "Grelling's paradox", can be made using
+adjectives instead of sets. Divide the adjectives in English into two categories: those
+which are self-descriptive, such as "pentasyllabic", "awkwardnessful", and "recherche",
+and those which are not, such
+
+
+Introduction: A Musico-Logical Offering
+
+
+20
+
+
+
+as "edible", "incomplete", and "bisyllabic". Now if we admit "non-selfdescriptive" as an
+adjective, to which class does it belong? If it seems questionable to include hyphenated
+words, we can use two terms invented specially for this paradox: autological (= "self-
+descriptive"), and heterological (= "non-self-descriptive"). The question then becomes:
+"Is 'heterological' heterological?" Try it!
+
+There seems to he one common culprit in these paradoxes, namely self-reference, or
+"Strange Loopiness". So if the goal is to ban all paradoxes, why not try banning self¬
+reference and anything that allows it to arise? This is not so easy as it might seem,
+because it can be hard to figure out just where self-reference is occurring. It may be
+spread out over a whole Strange Loop with several steps, as in this "expanded" version of
+Epimenides, reminiscent of Drawing Hands:
+
+The following sentence is false.
+
+The preceding sentence is true.
+
+Taken together, these sentences have the same effect as the original Epimenides paradox:
+yet separately, they are harmless and even potentially useful sentences. The "blame" for
+this Strange Loop can't he pinned on either sentence-only on the way they "point" at each
+other. In the same way, each local region of Ascending and Descending is quite
+legitimate; it is only the way they are globally put together that creates an impossibility.
+Since there are indirect as well as direct ways of achieving self-reference, one must figure
+out how to ban both types at once-if one sees self reference as the root of all evil.
+Banishing Strange Loops
+
+Russell and Whitehead did subscribe to this view, and accordingly, Principia
+Mathematica was a mammoth exercise in exorcising Strange Loops from logic, set
+theory, and number theory. The idea of their system was basically this. A set of the
+lowest "type" could contain only "objects" as membersnot sets. A set of the next type up
+could only contain objects, or sets of the lowest type. In general, a set of a given type
+could only contain sets of lower type, or objects. Every set would belong to a specific
+type. Clearly, no set could contain itself because it would have to belong to a type higher
+than its own type. Only "run-of-the-mill" sets exist in such a system; furthermore, old R-
+the set of all run-of-the-mill sets-no longer is considered a set at all, because it does not
+belong to any finite type. To all appearances, then, this theory of types, which we might
+also call the "theory of the abolition of Strange Loops", successfully rids set theory of its
+paradoxes, but only at the cost of introducing an artificial-seeming hierarchy, and of
+disallowing the formation of certain kinds of sets-such as the set of all run-of-the-mill
+sets. Intuitively, this is not the way we imagine sets.
+
+The theory of types handled Russell's paradox, but it did nothing about the Epimenides
+paradox or Grelling's paradox. For people whose
+
+
+Introduction: A Musico-Logical Offering
+
+
+21
+
+
+
+interest went no further than set theory, this was quite adequate-but for people interested
+in the elimination of paradoxes generally, some similar "hierarchization" seemed
+necessary, to forbid looping back inside language. At the bottom of such a hierarchy
+would be an object language. Here, reference could be made only to a specific domain-
+not to aspects of the object language itself (such as its grammatical rules, or specific
+sentences in it). For that purpose there would be a metalanguage. This experience of two
+linguistic levels is familiar to all learners of foreign languages. Then there would be a
+metametalanguage for discussing the metalanguage, and so on. It would be required that
+every sentence should belong to some precise level of the hierarchy. Therefore, if one
+could find no level in which a given utterance fit, then the utterance would be deemed
+meaningless, and forgotten.
+
+An analysis can be attempted on the two-step Epimenides loop given above. The first
+sentence, since it speaks of the second, must be on a higher level than the second. But by
+the same token, the second sentence must be on a higher level than the first. Since this is
+impossible, the two sentences are "meaningless". More precisely, such sentences simply
+cannot be formulated at all in a system based on a strict hierarchy of languages. This
+prevents all versions of the Epimenides paradox as well as Grelling's paradox. (To what
+language level could "heterological" belong?)
+
+Now in set theory, which deals with abstractions that we don't use all the time, a
+stratification like the theory of types seems acceptable, even if a little strange-but when it
+comes to language, an all-pervading part of life, such stratification appears absurd. We
+don't think of ourselves as jumping up and down a hierarchy of languages when we speak
+about various things. A rather matter-of-fact sentence such as, "In this book, I criticize
+the theory of types" would be doubly forbidden in the system we are discussing. Firstly, it
+mentions "this book", which should only be mentionable in a
+
+metabook"-and secondly, it mentions me-a person whom I should not be allowed to
+speak of at all! This example points out how silly the theory of types seems, when you
+import it into a familiar context. The remedy it adopts for paradoxes-total banishment of
+self-reference in any form-is a real case of overkill, branding many perfectly good
+constructions as meaningless. The adjective "meaningless", by the way, would have to
+apply to all discussions of the theory of linguistic types (such as that of this very
+paragraph) for they clearly could not occur on any of the levels-neither object language,
+nor metalanguage, nor metametalanguage, etc. So the very act of discussing the theory
+would be the most blatant possible violation of it!
+
+Now one could defend such theories by saying that they were only intended to deal
+with formal languages-not with ordinary, informal language. This may be so, but then it
+shows that such theories are extremely academic and have little to say about paradoxes
+except when they crop up in special tailor-made systems. Besides, the drive to eliminate
+paradoxes at any cost, especially when it requires the creation of highly artificial
+formalisms, puts too much stress on bland consistency, and too little on the
+
+
+Introduction: A Musico-Logical Offering
+
+
+22
+
+
+
+quirky and bizarre, which make life and mathematics interesting. It is of course important
+to try to maintain consistency, but when this effort forces you into a stupendously ugly
+theory, you know something is wrong.
+
+These types of issues in the foundations of mathematics were responsible for the high
+interest in codifying human reasoning methods which was present in the early part of this
+century. Mathematicians and philosophers had begun to have serious doubts about
+whether even the most concrete of theories, such as the study of whole numbers (number
+theory), were built on solid foundations. If paradoxes could pop up so easily in set
+theory-a theory whose basic concept, that of a set, is surely very intuitively appealing-
+then might they not also exist in other branches of mathematics? Another related worry
+was that the paradoxes of logic, such as the Epimenides paradox, might turn out to be
+internal to mathematics, and thereby cast in doubt all of mathematics. This was especially
+worrisome to those-and there were a good number-who firmly believed that mathematics
+is simply a branch of logic (or conversely, that logic is simply a branch of mathematics).
+In fact, this very question-"Are mathematics and logic distinct, or separate%"-was the
+source of much controversy.
+
+This study of mathematics itself became known as metamathematics-or occasionally,
+metalogic, since mathematics and logic are so intertwined. The most urgent priority of
+metamathematicians was to determine the true nature of mathematical reasoning. What is
+a legal method of procedure, and what is an illegal one? Since mathematical reasoning
+had always been done in "natural language" (e.g., French or Latin or some language for
+normal communication), there was always a lot of possible ambiguity. Words had
+different meanings to different people, conjured up different images, and so forth. It
+seemed reasonable and even important to establish a single uniform notation in which all
+mathematical work could be done, and with the aid of which any two mathematicians
+could resolve disputes over whether a suggested proof was valid or not. This would
+require a complete codification of the universally acceptable modes of human reasoning,
+at least as far as they applied to mathematics.
+
+Consistency, Completeness, Hilbert's Program
+
+This was the goal of Principia Mathematica, which purported to derive all of mathematics
+from logic, and, to be sure, without contradictions! It was widely admired, but no one
+was sure if (1) all of mathematics really was contained in the methods delineated by
+Russell and Whitehead, or (2) the methods given were even self-consistent. Was it
+absolutely clear that contradictory results could never be derived, by any mathematicians
+whatsoever, following the methods of Russell and Whitehead?
+
+This question particularly bothered the distinguished German mathematician (and
+metamathematician) David Hilbert, who set before the world community of
+mathematicians (and metamathematicians) this chal
+
+
+Introduction: A Musico-Logical Offering
+
+
+23
+
+
+
+lenge: to demonstrate rigorously-perhaps following the very methods outlined by Russell
+and Whitehead-that the system defined in Principia Mathematica was both consistent
+(contradiction-free), and complete (i.e., that every true statement of, number theory could
+be derived within the framework drawn up in P.M.). This was a tall order, and one could
+criticize it on the grounds that it was somewhat circular: how can you justify your
+methods of reasoning on the basis of those same methods of reasoning? It is like lifting
+yourself up by your own bootstraps. (We just don't seem to be able to get away from
+these Strange Loops!)
+
+Hilbert was fully aware of this dilemma, of course, and therefore expressed the hope
+that a demonstration of consistency or completeness could be found which depended only
+on "finitistic" modes of reasoning, "these were a small set of reasoning methods usually
+accepted by mathematicians. In this way, Hilbert hoped that mathematicians could
+partially lift themselves by their own bootstraps: the sum total of mathematical methods
+might be proved sound, by invoking only a smaller set of methods. This goal may sound
+rather esoteric, but it occupied the minds of many of the greatest mathematicians in the
+world during the first thirty years of this century.
+
+In the thirty-first year, however, Godel published his paper, which in some ways
+utterly demolished Hilbert's program. This paper revealed not only that there were
+irreparable "holes" in the axiomatic system proposed by Russell and Whitehead, but more
+generally, that no axiomatic system whatsoever could produce all number-theoretical
+truths, unless it were an inconsistent system! And finally, the hope of proving the
+consistency of a system such as that presented in P.M. was shown to be vain: if such a
+proof could be found using only methods inside P.M., then-and this is one of the most
+mystifying consequences of Godel's work-P.M. itself would be inconsistent!
+
+The final irony of it all is that the proof of Gi del's Incompleteness Theorem involved
+importing the Epimenides paradox right into the heart ofPrincipia Mathematica, a bastion
+supposedly invulnerable to the attacks of Strange Loops! Although Godel's Strange Loop
+did not destroy Principia Mathematica, it made it far less interesting to mathematicians,
+for it showed that Russell and Whitehead's original aims were illusory.
+
+Babbage, Computers, Artificial Intelligence ...
+
+When Godel's paper came out, the world was on the brink of developing electronic digital
+computers. Now the idea of mechanical calculating engines had been around for a while.
+In the seventeenth century, Pascal and Leibniz designed machines to perform fixed
+operations (addition and multiplication). These machines had no memory, however, and
+were not, in modern parlance, programmable.
+
+The first human to conceive of the immense computing potential of machinery was the
+Londoner Charles Babbage (1792-1871). A character who could almost have stepped out
+of the pages of the Pickwick Papers,
+
+
+Introduction: A Musico-Logical Offering
+
+
+24
+
+
+
+Babbage was most famous during his lifetime for his vigorous campaign to rid London
+of "street nuisances"-organ grinders above all. These pests, loving to get his goat, would
+come and serenade him at any time of day or night, and he would furiously chase them
+down the street. Today, we recognize in Babbage a man a hundred years ahead of his
+time: not only inventor of the basic principles of modern computers, he was also one of
+the first to battle noise pollution.
+
+His first machine, the "Difference Engine", could generate mathematical tables of
+many kinds by the "method of differences". But before any model of the "D.E." had been
+built, Babbage became obsessed with a much more revolutionary idea: his "Analytical
+Engine". Rather immodestly, he wrote, "The course through which I arrived at it was the
+most entangled and perplexed which probably ever occupied the human mind.'" Unlike
+any previously designed machine, the A.E. was to possess both a "store" (memory) and a
+"mill" (calculating and decision-making unit). These units were to be built of thousands
+of intricate geared cylinders interlocked in incredibly complex ways. Babbage had a
+vision of numbers swirling in and out of the mill tinder control of a program contained in
+punched cards-an idea inspired by the jacquard loom, a card-controlled loom that wove
+amazingly complex patterns. Babbage's brilliant but ill-fated Countess friend, Lady Ada
+Lovelace (daughter of Lord Byron), poetically commented that "the Analytical Engine
+weaves algebraic patterns just as the Jacquard-loom weaves flowers and leaves."
+Unfortunately, her use of the present tense was misleading, for no A.E. was ever built,
+and Babbage died a bitterly disappointed man.
+
+Lady Lovelace, no less than Babbage, was profoundly aware that with the invention of
+the Analytical Engine, mankind was flirting with mechanized intelligence-particularly if
+the Engine were capable of "eating its own tail" (the way Babbage described the Strange
+Loop created when a machine reaches in and alters its own stored program). In an 1842
+memoir,5 she wrote that the A.E. "might act upon other things besides number". While
+Babbage dreamt of creating_ a chess or tic-tac-toe automaton, she suggested that his
+Engine, with pitches and harmonies coded into its spinning cylinders, "might compose
+elaborate and scientific pieces of music of any degree of complexity or extent." In nearly
+the same breath, however, she cautions that "The Analytical Engine has no pretensions
+whatever to originate anything. It can do whatever we know how to order it to perform."
+Though she well understood the power of artificial computation, Lady Lovelace was
+skeptical about the artificial creation of intelligence. However, could her keen insight
+allow her to dream of the potential that would be opened up with the taming of
+electricity?
+
+In our century the time was ripe for computers-computers beyond the wildest dreams of
+Pascal, Leibniz, Babbage, or Lady Lovelace. In the 1930's and 1940's, the first "giant
+electronic brains" were designed and built. They catalyzed the convergence of three
+previously disparate areas: the theory of axiomatic reasoning, the study of mechanical
+computation, and the psychology of intelligence.
+
+These same years saw the theory of computers develop by leaps and
+
+
+Introduction: A Musico-Logical Offering
+
+
+25
+
+
+
+bounds. This theory was tightly linked to metamathematics. In fact, Godel's Theorem has
+a counterpart in the theory of computation, discovered by Alan Turing, which reveals the
+existence of inelucPable "holes" in even the most powerful computer imaginable.
+Ironically, just as these somewhat eerie limits were being mapped out, real computers
+were being built whose powers seemed to grow and grow beyond their makers' power of
+prophecy. Babbage, who once declared he would gladly give up the rest of his life if he
+could come back in five hundred years and have a three-day guided scientific tour of the
+new age, would probably have been thrilled speechless a mere century after his death-
+both by the new machines, and by their unexpected limitations.
+
+By the early 1950's, mechanized intelligence seemed a mere stone's throw away; and
+yet, for each barrier crossed, there always cropped up some new barrier to the actual
+creation of a genuine thinking machine. Was there some deep reason for this goal's
+mysterious recession?
+
+No one knows where the borderline between non-intelligent behavior and intelligent
+behavior lies; in fact, to suggest that a sharp borderline exists is probably silly. But
+essential abilities for intelligence are certainly:
+
+to respond to situations very flexibly;
+
+to take advantage of fortuitous circumstances;
+
+to make sense out of ambiguous or contradictory messages;
+
+to recognize the relative importance of different elements of a
+
+situation;
+
+to find similarities between situations despite differences which may separate them;
+to draw distinctions between situations despite similarities may link them;
+to synthesize new concepts by taking old them together in new ways; to come up
+with ideas which are novel.
+
+Here one runs up against a seeming paradox. Computers by their very nature are the
+most inflexible, desireless, rule-following of beasts. Fast though they may be, they are
+nonetheless the epitome of unconsciousness. How, then, can intelligent behavior be
+programmed? Isn't this the most blatant of contradictions in terms? One of the major
+theses of this book is that it is not a contradiction at all. One of the major purposes of this
+book is to urge each reader to confront the apparent contradiction head on, to savor it, to
+turn it over, to take it apart, to wallow in it, so that in the end the reader might emerge
+with new insights into the seemingly unbreathable gulf between the formal and the
+informal, the animate and the inanimate, the flexible and the inflexible.
+
+This is what Artificial Intelligence (Al) research is all about. And the strange flavor of
+AI work is that people try to put together long sets of rules in strict formalisms which tell
+inflexible machines how to be flexible.
+
+What sorts of "rules" could possibly capture all of what we think of as intelligent
+behavior, however? Certainly there must be rules on all sorts of
+
+
+Introduction: A Musico-Logical Offering
+
+
+26
+
+
+
+different levels. There must be many "just plain" rules. There must be "metarules" to
+modify the "just plain" rules; then "metametarules" to modify the metarules, and so on.
+The flexibility of intelligence comes from the enormous number of different rules, and
+levels of rules. The reason that so many rules on so many different levels must exist is
+that in life, a creature is faced with millions of situations of completely different types. In
+some situations, there are stereotyped responses which require "just plain" rules. Some
+situations are mixtures of stereotyped situations-thus they require rules for deciding
+which of the just plain" rules to apply. Some situations cannot be classified-thus there
+must exist rules for inventing new rules ... and on and on. Without doubt, Strange Loops
+involving rules that change themselves, directly or indirectly, are at the core of
+intelligence. Sometimes the complexity of our minds seems so overwhelming that one
+feels that there can be no solution to the problem of understanding intelligence-that it is
+wrong to think that rules of any sort govern a creature's behavior, even if one takes "rule"
+in the multilevel sense described above.
+
+
+...and Bach
+
+In the year 1754, four years after the death of J. S. Bach, the Leipzig theologian Johann
+Michael Schmidt wrote, in a treatise on music and the soul, the following noteworthy
+passage:
+
+Not many years ago it was reported from France that a man had made a statue that
+could play various pieces on the Fleuttraversiere, placed the flute to its lips and took it
+down again, rolled its eyes, etc. But no one has yet invented an image that thinks, or
+wills, or composes, or even does anything at all similar. Let anyone who wishes to be
+convinced look carefully at the last fugal work of the above-praised Bach, which has
+appeared in copper engraving, but which was left unfinished because his blindness
+intervened, and let him observe the art that is contained therein; or what must strike
+him as even more wonderful, the Chorale which he dictated in his blindness to the pen
+of another: Wenn wir in hochsten Nothen seen. I am sure that he will soon need his
+soul if he wishes to observe all the beauties contained therein, let alone wishes to play
+it to himself or to form a judgment of the author. Everything that the
+champions of Materialism put forward must fall to the ground in view of this
+single example.6
+
+Quite likely, the foremost of the "champions of Materialism" here alluded to was none
+other than Julien Offroy de la Mettrie-philosopher at the court of Frederick the Great,
+author of L'homme machine ("Man, the Machine"), and Materialist Par Excellence. It is
+now more than 200 years later, and the battle is still raging between those who agree with
+Johann Michael Schmidt, and those who agree with Julien Offroy de la Mettrie. I hope in
+this book to give some perspective on the battle.
+
+"Godel, Escher, Bach"
+
+The book is structured in an unusual way: as a counterpoint between Dialogues and
+Chapters. The purpose of this structure is to allow me to
+
+
+Introduction: A Musico-Logical Offering
+
+
+27
+
+
+
+present new concepts twice: almost every new concept is first presented metaphorically
+in a Dialogue, yielding a set of concrete, visual images; then these serve, during the
+reading of the following'Chapter, as an intuitive background for a more serious and
+abstract presentation of the same concept. In many of the Dialogues I appear to be talking
+about one idea on the surface, but in reality I am talking about some other idea, in a thinly
+disguised way.
+
+Originally, the only characters in my Dialogues were Achilles and the Tortoise, who
+came to me from Zeno of Elea, by way of Lewis Carroll. Zeno of Elea, inventor of
+paradoxes, lived in the fifth century B.C. One of his paradoxes was an allegory, with
+Achilles and the Tortoise as protagonists. Zeno's invention of the happy pair is told in my
+first Dialogue, Three-Part Invention. In 1895, Lewis Carroll reincarnated Achilles and the
+Tortoise for the purpose of illustrating his own new paradox of infinity. Carroll's paradox,
+which deserves to be far better known than it is, plays a significant role in this book.
+Originally titled "What the Tortoise Said to Achilles", it is reprinted here as Two-Part
+Invention.
+
+When I began writing Dialogues, somehow I connected them up with musical forms. I
+don't remember the moment it happened; I just remember one day writing "Fugue" above
+an early Dialogue, and from then on the idea stuck. Eventually I decided to pattern each
+Dialogue in one way or another on a different piece by Bach. This was not so
+inappropriate. Old Bach himself used to remind his pupils that the separate parts in their
+compositions should behave like "persons who conversed together as if in a select
+company". I have taken that suggestion perhaps rather more literally than Bach intended
+it; nevertheless I hope the result is faithful to the meaning. I have been particularly
+inspired by aspects of Bach's compositions which have struck me over and over, and
+which are so well described by David and Mendel in The Bach Reader:
+
+His form in general was based on relations between separate sections. These relations
+ranged from complete identity of passages on the one hand to the
+
+return of a single principle of elaboration or a mere thematic allusion on the other. The
+resulting patterns were often symmetrical, but by no means
+
+necessarily so. Sometimes the relations between the various sections make up a maze of
+interwoven threads that only detailed analysis can unravel. Usually,
+however, a few dominant features afford proper orientation at first sight or hearing, and
+while in the course of study one may discover unending sub
+
+tleties, one is never at a loss to grasp the unity that holds together every single creation by
+Bach.'
+
+I have sought to weave an Eternal Golden Braid out of these three strands: Godel,
+Escher, Bach. I began, intending to write an essay at the core of which would be Godel's
+Theorem. I imagined it would be a mere pamphlet. But my ideas expanded like a sphere,
+and soon touched Bach and Escher. It took some time for me to think of making this
+connection explicit, instead of just letting it be a private motivating force. But finally 1
+realized that to me, Godel and Escher and Bach were only shadows cast in different
+directions by some central solid essence. I tried to reconstruct the central object, and
+came up with this book.
+
+
+Introduction: A Musico-Logical Offering
+
+
+28
+
+
+
+Three-Part Invention
+
+
+Achilles (a Greek warrior, the fleetest of foot of all mortals) and a Tortoise are
+standing together on a dusty runway in the hot sun. Far down the runway, on a
+tall flagpole, there hangs a large rectangular flag. The flag is sold red, except
+where a thin ring-shaped holes has been cut out of it, through which one can see
+the sky.
+
+ACHILLES: What is that strange flag down at the other end of the track? It reminds me
+somehow of a print by my favourite artists M.C. Escher.
+
+TORTOISE: That is Zeno’s flag
+
+ACHILLES: Could it be that the hole in it resembles the holes in a Mobian strip Escher once
+drew? Something is wrong about the flag, I can tell.
+
+TORTOISE: The ring which has been cut from it has the shape of the numeral for zero, which
+is Zeno's favourite number.
+
+ACHILLES: The ring which hasn't been invented yet! It will only be invented by a Hindu
+mathematician some millennia hence. And thus, Mr. T, mt argument proves that such a
+flag is impossible.
+
+TORTOISE: Your argument is persuasive, Achilles, and I must agree that such a flag is indeed
+impossible. But it is beautiful anyway, is it not?
+
+ACHILLES: Oh, yes, there is no doubt of its beauty.
+
+TORTOISE: I wonder if it's beauty is related to it's impossibility. I don't know, I've never had
+the time to analyze Beauty. It's a Capitalized Essence, and I never seem to have time for
+Capitalized Essences.
+
+ACHILLES: Speaking of Capitalized Essences, Mr. T, have you ever wondered about the
+Purpose of Life?
+
+TORTOISE: Oh, heavens, no;
+
+ACHILLES: Haven’t you ever wondered why we are here, or who invented us?
+
+TORTOISE: Oh, that is quite another matter. We are inventions of Zeno (as you will shortly
+see) and the reason we are here is to have a footrace.
+
+ACHILLES::: A footrace? How outrageous! Me, the fleetest of foot of all mortals, versus you,
+the ploddingest of the plodders! There can be no point to such a race.
+
+TORTOISE: You might give me a head start.
+
+ACHILLES: It would have to be a huge one.
+
+TORTOISE: I don’t object.
+
+ACHILLES: But I will catch you, sooner or later - most likely sooner.
+
+TORTOISE: Not if things go according to Zeno's paradox, you won’t. Zeno is hoping to use
+our footrace to show that motion is impossible, you see. It is only in the mind that motion
+seems possible, according to Zeno. In truth, Motion Is Inherently Impossible. He proves
+it quite elegantly.
+
+
+Three-Part Invention
+
+
+29
+
+
+
+
+
+
+
+Figure 10. Mobius strip by M.C.Escher (wood-engraving printed from four blocks, 1961)
+
+ACHILLES: Oh, yes, it comes back to me now: the famous Zen koan about Zen
+
+Master Zeno. As you say it is very simple indeed.
+
+TORTOISE: Zen Koan? Zen Master? What do you mean?
+
+ACHILLES: It goes like this: Two monks were arguing about a flag. One said, “The
+
+flag is moving.” The other said, “The wind is moving.” The sixth patriarch, Zeno,
+happened to be passing by. He told them, “Not the wind, not the flag, mind is
+moving.”
+
+TORTOISE: I am afraid you are a little befuddled, Achilles. Zeno is no Zen master, far
+
+from it. He is in fact, a Greek philosopher from the town of Elea (which lies halfway
+between points A and B). Centuries hence, he will be celebrated for his paradoxes of
+motion. In one of those paradoxes, this very footrace between you and me will play a
+central role.
+
+ACHILLES: I’m all confused. I remember vividly how I used to repeat over and over
+
+the names of the six patriarchs of Zen, and I always said, “The sixth patriarch is Zeno,
+The sixth patriarch is Zeno...” (Suddenly a soft warm breeze picks up.) Oh, look Mr.
+Tortoise - look at the flag waving! How I love to watch the ripples shimmer through
+it’s soft fabric. And the ring cut out of it is waving, too!
+
+
+Three-Part Invention
+
+
+30
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+TORTOISE: Don't be silly. The flag is impossible, hence it can’t be waving. The wind is
+waving.
+
+(At this moment, Zeno happens by.)
+
+Zeno: Hallo! Hulloo! What’s up? What’s new?
+
+ACHILLES: The flag is moving.
+
+TORTOISE: The wind is moving.
+
+Zeno: O friends, Friends! Cease your argumentation! Arrest your vitriolics! Abandon your
+discord! For I shall resolve the issue for you forthwith. Ho! And on such a fine day.
+
+ACHILLES: This fellow must be playing the fool.
+
+TORTOISE: No, wait, Achilles. Let us hear what he has to say. Oh Unknown Sir, do impart to
+us your thoughts on this matter.
+
+Zeno: Most willingly. Not thw ind, not the flag - neither one is moving, nor is anything moving
+at all. For I have discovered a great Theorem, which states; “Motion Is Inherently
+Impossible.” And from this Theorem follows an even greater Theorem - Zeno’s
+Theorem: “Motion Unexists.”
+
+ACHILLES: “Zeno’s Theorem”? Are you, sir, by any chance, the philosopher Zeno of Elea?
+
+Zeno: I am indeed, Achilles.
+
+ACHILLES: (scratching his head in puzzlement). Now how did he know my name?
+
+Zeno: Could I possibly persuade you two to hear me out as to why this is the case? I’ve come
+all the way to Elea from point A this afternoon, just trying to find someone who’ll pay
+some attention to my closely honed argument. But they’re all hurrying hither and thither,
+and they don’t have time. You’ve no idea how disheartening it is to meet with refusal
+after refusal. Oh, I’m sorry to burden you with my troubles, I’d just like to ask you one
+thing: Would the two of you humour a sill old philosopher for a few moments - only a
+few, I promise you - in his eccentric theories.
+
+ACHILLES: Oh, by all means! Please do illuminate us! I know I speak for both of us, since my
+companion, Mr. Tortoise, was only moments ago speaking of you with great veneration -
+and he mentioned especially your paradoxes.
+
+Zeno: Thank you. You see, my Master, the fifth patriarch, taught me that reality is one,
+immutable, and unchanging, all plurality, change, and motion are mere illusions of the
+sense. Some have mocked his views; but I will show the absurdity of their mockery. My
+argument is quite simple. I will illustrate it with two characters of my own Invention:
+Achilles )a Greek warrior, the fleetest of foot of all mortals), and a Tortoise. In my tale,
+they are persuaded by a passerby to run a footrace down a runway towards a distant flag
+waving in the breeze. Let us assume that, since the Tortoise is a much slowerrunner, he
+gets a head start of, say, ten rods. Now the race begins. In a few bounds Achilles has
+reached the spot where the Tortoise started.
+
+
+Three-Part Invention
+
+
+31
+
+
+
+
+ACHILLES: Hah!
+
+Zeno: And now the Tortoise is but a single rod ahead of Achilles. Within only a moment,
+Achilles has attained that spot.
+
+ACHILLES: Ho ho!
+
+Zeno: Yet, in that short moment, the Tortoise has managed to advance a slight amount. In a
+flash, Achilles covers that distance too.
+
+ACHILLES: Heeheehee!
+
+Zeno: But in that very short flash, the Tortoise has managed to inch ahead by ever so little, and
+so Achilles is still behind. Now you see that in order for Achilles to catch the Tortoise,
+this game of “try-to-catch-me” will have to be played an INFINITE number of times -
+and therefore Achilles can NEVER catch up with the Tortoise.
+
+TORTOISE: Heh heh heh heh!
+
+ACHILLES: Hmm... Hmm... Hmm... Hmm... Hmm...That argument sounds wrong to me.
+And yes, I can’t quite make out what’s wrong with it
+
+Zeno: Isn’t it a teaser? It’s my favourite paradox.
+
+TORTOISE: Excuse me, Zeno, but I believe your tale illustrates the wrong principle, doe sit
+not? You have just told us what will come to known, centuries hence, as Zeno’s “Achilles
+paradox” , which shows (ahem!) that Achilles will never catch the Tortoise; but the proof
+that Motion Is Inherently Impossible (and thence that Motion Unexists) is your
+“dichotomy paradox”, isn’t that so?
+
+Zeno: Oh, shame on me. Of course, you’re right. That’s the new one about how, in going from
+A to B, one has to go halfway first - and of that stretch one also has to go halfway, and so
+on and so forth. But you see, both those paradoxes really have the same flavour. Frankly,
+I’ve only had one Great Idea - I just exploit it in different ways.
+
+ACHILLES: I swear, these arguments contain a flaw. I don’t quite see where, but they cannot
+be correct.
+
+Zeno: You doubt the validity of my paradox? Why not just try it outl? You see that red flag
+waving down here, at the far end of the runway?
+
+ACHILLES: The impossible one, based on an Escher print?
+
+Zeno: Exactly. What do you say to you and Mr. Tortoise racing for it, allowing Mr. T a fair
+head start of, well, I don’t know -
+
+TORTOISE: How about ten rods?
+
+Zeno: Very good - ten rods.
+
+ACHILLES: Any time.
+
+Zeno: Excellent! How exciting! An empirical test of my rigorously proven Theorem! Mr.
+Tortoise, will you position yourself ten rods upwind?
+
+(The Tortoise moves ten rods closer to the flag)
+
+Tortoise and Achlles: Ready!
+
+Zeno: On your mark! Get set! Go!
+
+
+Three-Part Invention
+
+
+32
+
+
+
+Chapter 1
+
+
+The MU-puzzle
+
+
+Formal Systems
+
+ONE OF THE most central notions in this book is that of a formal system. The type of
+formal system I use was invented by the American logician Emil Post in the 1920's, and
+is often called a "Post production system". This Chapter introduces you to a formal
+system and moreover, it is my hope that you will want to explore this formal system at
+least a little; so to provoke your curiosity, I have posed a little puzzle.
+
+"Can you produce MU?" is the puzzle. To begin with, you will be supplied with a
+string (which means a string of letters).* Not to keep you in suspense, that string will be
+MI. Then you will be told some rules, with which you can change one string into another.
+If one of those rules is applicable at some point, and you want to use it, you may, but-
+there is nothing that will dictate which rule you should use, in case there are several
+applicable rules. That is left up to you-and of course, that is where playing the game of
+any formal system can become something of an art. The major point, which almost
+doesn't need stating, is that you must not do anything which is outside the rules. We
+might call this restriction the "Requirement of Formality". In the present Chapter, it
+probably won't need to be stressed at all. Strange though it may sound, though, I predict
+that when you play around with some of the formal systems of Chapters to come, you
+will find yourself violating the Requirement of Formality over and over again, unless you
+have worked with formal systems before.
+
+The first thing to say about our formal system-the MlU-system-is that it utilizes
+only three letters of the alphabet: M, I, U. That means that the only strings of the MIU-
+system are strings which are composed of those three letters. Below are some strings of
+the MlU-system:
+
+
+MU
+
+UIM
+
+MUUMUU
+
+UIIUMIUUIMUIIUMIUUIMUIIU
+
+* In this book, we shall employ the following conventions when we refer to strings. When the
+string is in the same typeface as the text, then it will be enclosed in single or double quotes.
+Punctuation which belongs to the sentence and not to the string under discussion will go outside
+of the quotes, as logic dictates. For example, the first letter of this sentence is 'F', while the first
+letter of 'this ‘sentence’.is 'f. When the string is in Quadrata Roman, however, quotes will
+usually be left off, unless clarity demands them. For example, the first letter of Quadrata is Q.
+
+
+The MU-puzzle
+
+
+33
+
+
+
+But although all of these are legitimate strings, they are not strings which are "in your
+possession". In fact, the only string in your possession so far is MI. Only by using the
+rules, about to be introduced, can you enlarge your private collection. Here is the first
+rule:
+
+RULE I: If you possess a string whose last letter is I, you can add on a U at the end.
+
+By the way, if up to this point you had not guessed it, a fact about the meaning of "string"
+is that the letters are in a fixed order. For example, MI and IM are two different strings.
+A string of symbols is not just a "bag" of symbols, in which the order doesn't make any
+difference.
+
+Here is the second rule:
+
+RULE II: Suppose you have Mx. Then you may add Mxx to your collection.
+
+What I mean by this is shown below, in a few examples.
+
+From MIU, you may get MIUIU.
+
+From MUM, you may get MUMUM.
+
+From MU, you may get MUU.
+
+So the letter 'x' in the rule simply stands for any string; but once you have decided which
+string it stands for, you have to stick with your choice (until you use the rule again, at
+which point you may make a new choice). Notice the third example above. It shows how,
+once you possess MU, you can add another string to your collection; but you have to get
+MU first! I want to add one last comment about the letter 'x': it is not part of the formal
+system in the same way as the three letters 'M', 'I', and 'U' are. It is useful for us,
+though, to have some way to talk in general about strings of the system, symbolically-and
+that is the function of the 'x': to stand for an arbitrary string. If you ever add a string
+containing an Y to your "collection", you have done something wrong, because strings of
+the MlU-system never contain "x" “s”!
+
+Here is the third rule:
+
+RULE III: If III occurs in one of the strings in your collection, you may make a new
+string with U in place of III.
+
+Examples:
+
+From UMIIIMU, you could make UMUMU.
+
+From MII11, you could make MIU (also MUI).
+
+From IIMII, you can't get anywhere using this rule.
+
+(The three I's have to be consecutive.)
+
+From Mill, make MU.
+
+Don't, under any circumstances, think you can run this rule backwards, as in the
+following example:
+
+
+The MU-puzzle
+
+
+34
+
+
+
+From MU, make Mill
+
+
+<- This is wrong.
+
+
+Rules are one-way.
+
+Here is the final rule.
+
+RULE IV: If UU occurs inside one of your strings, you can drop it.
+
+From UUU, get U.
+
+From MUUUIII, get MUIII.
+
+There you have it. Now you may begin trying to make MU. Don't worry you don't get it.
+Just try it out a bit-the main thing is for you to get the flavor of this MU-puzzle. Have
+fun.
+
+Theorems, Axioms, Rules
+
+The answer to the MU-puzzle appears later in the book. For now, what important is not
+finding the answer, but looking for it. You probably hay made some attempts to produce
+MU. In so doing, you have built up your own private collection of strings. Such strings,
+producible by the rules, are called theorems. The term "theorem" has, of course, a
+common usage mathematics which is quite different from this one. It means some
+statement in ordinary language which has been proven to be true by a rigorous argument,
+such as Zeno's Theorem about the "unexistence" of motion, c Euclid's Theorem about the
+infinitude of primes. But in formal system theorems need not be thought of as statements-
+they are merely strings c symbols. And instead of being proven , theorems are merely
+produced , as if F machine, according to certain typographical rules. To emphasize this
+important distinction in meanings for the word "theorem", I will adopt the following
+convention in this book: when "theorem" is capitalized, its meaning will be the everyday
+one-a Theorem is a statement in ordinary language which somebody once proved to be
+true by some sort of logic argument. When uncapitalized, "theorem" will have its
+technical meaning a string producible in some formal system. In these terms, the MU-
+puzzle asks whether MU is a theorem of the MlU-system.
+
+I gave you a theorem for free at the beginning, namely MI. Such "free" theorem is called
+an axiom -the technical meaning again being qui different from the usual meaning. A
+formal system may have zero, or several, or even infinitely many axioms. Examples of all
+these types v appear in the book.
+
+Every formal system has symbol-shunting rules, such as the four rules of the MIU-
+system. These rules are called either rules of production or rules of inference. I will use
+both terms.
+
+The last term which I wish to introduce at this point is derivation. Shown below is a
+derivation of the theorem MUIIU:
+
+(1) MI axiom
+
+(2) Mil from (1) by rule II
+
+
+The MU-puzzle
+
+
+35
+
+
+
+(3) Mill
+
+(4) MIIIIU
+
+(5) MUIU
+
+(6) MUIUUIU
+
+(7) MUIIU
+
+
+from (2) by rule II
+from (3) by rule I
+from (4) by rule III
+from (5) by rule II
+from (6) by rule IV
+
+
+A derivation of a theorem is an explicit, line-by-line demonstration of how to produce
+that theorem according to the rules of the formal system. The concept of derivation is
+modeled on that of proof, but a derivation is an austere cousin of a proof. It would sound
+strange to say that you had proven MUIIU, but it does not sound so strange to say you
+have derived MUIIU.
+
+
+Inside and Outside the System
+
+Most people go about the MU-puzzle by deriving a number of theorems, quite at random,
+just to see what kind of thing turns up. Pretty soon, they begin to notice some properties
+of the theorems they have made; that is where human intelligence enters the picture. For
+instance, it was probably not obvious to you that all theorems would begin with M, until
+you had tried a few. Then, the pattern emerged, and not only could you see the pattern,
+but you could understand it by looking at the rules, which have the property that they
+make each new theorem inherit its first letter from an earlier theorem; ultimately, then, all
+theorems' first letters can be traced back to the first letter of the sole axiom Ml-and that is
+a proof that theorems of the MlU-system must all begin with M.
+
+There is something very significant about what has happened here. It shows one
+difference between people and machines. It would certainly be possible-in fact it would
+be very easy-to program a computer to generate theorem after theorem of the MIU-
+system; and we could include in the program a command to stop only upon generating U.
+You now know that a computer so programmed would never stop. And this does not
+amaze you. But what if you asked a friend to try to generate U? It would not surprise you
+if he came back after a while, complaining that he can't get rid of the initial M, and
+therefore it is a wild goose chase. Even if a person is not very bright, he still cannot help
+making some observations about what he is doing, and these observations give him good
+insight into the task-insight which the computer program, as we have described it, lacks.
+
+Now let me be very explicit about what I meant by saying this shows a difference
+between people and machines. I meant that it is possible to program a machine to do a
+routine task in such a way that the machine will never notice even the most obvious facts
+about what it is doing; but it is inherent in human consciousness to notice some facts
+about the things one is doing. But you knew this all along. If you punch "1" into an
+adding machine, and then add 1 to it, and then add 1 again, and again, and again, and
+continue doing so for hours and hours, the machine will never learn to anticipate you, and
+do it itself, although any person would pick up the
+
+
+The MU-puzzle
+
+
+36
+
+
+
+pick up the idea, no matter how much or how well it is driven, that it i supposed to avoid
+other cars and obstacles on the road; and it will never learn even the most frequently
+traveled routes of its owner.
+
+The difference, then, is that it is possible for a machine to act unobservant; it is
+impossible for a human to act unobservant. Notice I am not saying that all machines are
+necessarily incapable of making sophisticated observations; just that some machines are.
+Nor am I saying that all people are always making sophisticated observations; people, in
+fact, are often very unobservant. But machines can be made to be totally unobservant;
+any people cannot. And in fact, most machines made so far are pretty close ti being
+totally unobservant. Probably for this reason, the property of being; unobservant seems to
+be the characteristic feature of machines, to most people. For example, if somebody says
+that some task is "mechanical", i does not mean that people are incapable of doing the
+task; it implies though, that only a machine could do it over and over without eve
+complaining, or feeling bored.
+
+Jumping out of the System
+
+It is an inherent property of intelligence that it can jump out of the tas which it is
+performing, and survey what it has done; it is always looking for and often finding,
+patterns. Now I said that an intelligence can jump out o its task, but that does not mean
+that it always will. However, a little prompting will often suffice. For example, a human
+being who is reading a boo may grow sleepy. Instead of continuing to read until the book
+is finished he is just as likely to put the book aside and turn off the light. He ha stepped
+"out of the system" and yet it seems the most natural thing in the world to us. Or, suppose
+person A is watching television when person B comes in the room, and shows evident
+displeasure with the situation Person A may think he understands the problem, and try to
+remedy it b exiting the present system (that television program), and flipping the channel
+knob, looking for a better show. Person B may have a more radio concept of what it is to
+"exit the system"-namely to turn the television oft Of course, there are cases where only a
+rare individual will have the vision to perceive a system which governs many peoples
+lives, a system which ha never before even been recognized as a system; then such people
+often devote their lives to convincing other people that the system really is there and that
+it ought to be exited from!
+
+How well have computers been taught to jump out of the system? I w cite one
+example which surprised some observers. In a computer chess: tournament not long ago
+in Canada, one program-the weakest of all the competing ones-had the unusual feature of
+quitting long before the game was over. It was not a very good chess player, but it at least
+had the redeeming quality of being able to spot a hopeless position, and to resign then
+and there, instead of waiting for the other program to go through the
+
+
+The MU-puzzle
+
+
+37
+
+
+
+boring ritual of checkmating. Although it lost every game it played, it did it in style. A lot
+of local chess experts were impressed. Thus, if you define "the system" as "making
+moves in a chess game", it is clear that this program had a sophisticated, preprogrammed
+ability to exit from the system. On the other hand, if you think of "the system" as being
+"whatever the computer had been programmed to do", then there is no doubt that the
+computer had no ability whatsoever to exit from that system.
+
+It is very important when studying formal systems to distinguish working within
+the system from making statements or observations about the system. I assume that you
+began the MU-puzzle, as do most people, by working within the system; and that you
+then gradually started getting anxious, and this anxiety finally built up to the point where
+without any need for further consideration, you exited from the system, trying to take
+stock of what you had produced, and wondering why it was that you had not succeeded in
+producing MU. Perhaps you found a reason why you could not produce MU; that is
+thinking about the system. Perhaps you produced MIU somewhere along the way; that is
+working within the system. Now I do not want to make it sound as if the two modes are
+entirely incompatible; I am sure that every human being is capable to some extent of
+working inside a system and simultaneously thinking about what he is doing. Actually, in
+human affairs, it is often next to impossible to break things neatly up into "inside the
+system" and "outside the system"; life is composed of so many interlocking and
+interwoven and often inconsistent "systems" that it may seem simplistic to think of things
+in those terms. But it is often important to formulate simple ideas very clearly so that one
+can use them as models in thinking about more complex ideas. And that is why I am
+showing you formal systems; and it is about time we went back to discussing the MIU-
+system.
+
+
+M-Mode, I-Mode, U-Mode
+
+The MU-puzzle was stated in such a way that it encouraged some amount of exploration
+within the MlU-system-deriving theorems. But it was also stated in a way so as not to
+imply that staying inside the system would necessarily yield fruit. Therefore it
+encouraged some oscillation between the two modes of work. One way to separate these
+two modes would be to have two sheets of paper; on one sheet, you work "in your
+capacity as a machine", thus filling it with nothing but M's, I's, and U's; on the second
+sheet, you work "in your capacity as a thinking being", and are allowed to do whatever
+your intelligence suggests-which might involve using English, sketching ideas, working
+backwards, using shorthand (such as the letter 'x'), compressing several steps into one,
+modifying the rules of the system to see what that gives, or whatever else you might
+dream up. One thing you might do is notice that the numbers 3 and 2 play an important
+role, since I's are gotten rid of in three's, and U's in two's-and doubling of length (except
+for the M) is allowed by rule II. So the second sheet might
+
+
+The MU-puzzle
+
+
+38
+
+
+
+also have some figuring on it. We will occasionally refer back to these two modes of
+dealing with a formal system, and we will call them the Mechanic mode (M-mode ) and
+the Intelligent mode (I-mode). To round out our mode with one for each letter of the
+MlU-system, I will also mention a fin mode-the Un-mode (U-mode ), which is the Zen
+way of approaching thing. More about this in a few Chapters.
+
+
+Decision Procedures
+
+An observation about this puzzle is that it involves rules of two opposite tendencies-the
+lengthening rules and the shortening rules. Two rules (I and II) allow you to increase the
+size of strings (but only in very rigid, pr scribed ways, of course); and two others allow
+you to shrink strings somewhat (again in very rigid ways). There seems to be an endless
+variety to the order in which these different types of rules might be applied, and this gives
+hope that one way or another, MU could be produced. It might involve lengthening the
+string to some gigantic size, and then extracting piece after piece until only two symbols
+are left; or, worse yet, it might involve successive stages of lengthening and then
+shortening and then lengthening and then shortening, and so on. But there is no guarantee
+it. As a matter of fact, we already observed that U cannot be produced at all and it will
+make no difference if you lengthen and shorten till kingdom come.
+
+Still, the case of U and the case of MU seem quite different. It is by very
+superficial feature of U that we recognize the impossibility of producing it: it doesn't
+begin with an M (whereas all theorems must). It is very convenient to have such a simple
+way to detect nontheorems. However who says that that test will detect all nontheorems?
+There may be lots strings which begin with M but are not producible. Maybe MU is one
+of them. That would mean that the "first-letter test" is of limited usefulness able only to
+detect a portion of the nontheorems, but missing others. B there remains the possibility of
+some more elaborate test which discriminates perfectly between those strings which can
+be produced by the rules and those which cannot. Here we have to face the question,
+"What do mean by a test?" It may not be obvious why that question makes sense, of
+important, in this context. But I will give an example of a "test" which somehow seems to
+violate the spirit of the word.
+
+Imagine a genie who has all the time in the world, and who enjoys using it to
+produce theorems of the MlU-system, in a rather methodical way. Here, for instance, is a
+possible way the genie might go about it
+
+Step 1: Apply every applicable rule to the axiom MI. This yields two new theorems
+MIU, MIL
+
+Step 2: Apply every applicable rule to the theorems produced in step 1. This yields
+three new theorems: MIIU, MIUIU, MIIII.
+
+
+The MU-puzzle
+
+
+39
+
+
+
+Step 3: Apply every applicable rule to the theorems produced in step 2. This yields
+five new theorems: MIIIIU, MIIUIIU, MIUIUIUIU, MIIIIIIII, MUI.
+
+
+This method produces eveiy single theorem sooner or later, because the rules are applied
+in every conceivable order. (See Fig. 11.) All of the lengthening-shortening alternations
+which we mentioned above eventually get canned out. However, it is not clear how long
+to wait for a given string
+
+
+Ml
+
+
+
+MIUIUIUIU MIIUIIU MIIIIU MIIIIIIII MUI MIU
+
+/ / ^t \ 1\ \
+
+MU
+
+FIGURE II. A systematically constructed "tree” of all the theorems of the MIU-system.
+The N th level down contains those theorems whose derivations contain exactly N steps.
+The encircled numbers tell which rule was employed. Is MU anywhere in this tree?
+
+
+to appear on this list, since theorems are listed according to the shortness of their
+derivations. This is not a very useful order, if you are interested in a specific string (such
+as MU), and you don't even know if it has any derivation, much less how long that
+derivation might be.
+
+Now we state the proposed "theoremhood-test”:
+
+Wait until the string in question is produced; when that happens, you know it
+is a theorem-and if it never happens, you know that it is not a theorem.
+
+This seems ridiculous, because it presupposes that we don’t mind waiting around literally
+an infinite length of time for our answer. This gets to the crux of the matter of what
+should count as a "test". Of prime importance is a guarantee that we will get our answer
+in a finite length of time. If there is a test for theoremhood, a test which does always
+terminate in a finite
+
+
+The MU-puzzle
+
+
+40
+
+
+amount of time, then that test is called a decision procedure for the given formal system.
+
+When you have a decision procedure, then you have a very concrete
+characterization of the nature of all theorems in the system. Offhand, it might seem that
+the rules and axioms of the formal system provide no less complete a characterization of
+the theorems of the system than a decision procedure would. The tricky word here is
+"characterization". Certainly the rules of inference and the axioms of the MlU-system do
+characterize, implicitly, those strings that are theorems. Even more implicitly, they
+characterize those strings that are not theorems. But implicit characterization is not
+enough, for many purposes. If someone claims to have a characterization of all theorems,
+but it takes him infinitely long to deduce that some particular string is not a theorem, you
+would probably tend to say that there is something lacking in that characterization-it is
+not quite concrete enough. And that is why discovering that a decision procedure exists is
+a very important step. What the discovery means, in effect, is that you can perform a test
+for theoremhood of a string, and that, even if the test is complicated, it is guaranteed to
+terminate. In principle, the test is just as easy, just as mechanical, just as finite, just as full
+of certitude, as checking whether the first letter of the string is M. A decision procedure
+is a "litmus test" for theoremhood!
+
+Incidentally, one requirement on formal systems is that the set of axioms must be
+characterized by a decision procedure-there must be a litmus test for axiomhood. This
+ensures that there is no problem in getting off the ground at the beginning, at least. That
+is the difference between the set of axioms and the set of theorems: the former always has
+a decision procedure, but the latter may not.
+
+I am sure you will agree that when you looked at the MlU-system for the first
+time, you had to face this problem exactly. The lone axiom was known, the rules of
+inference were simple, so the theorems had been implicitly characterized-and yet it was
+still quite unclear what the consequences of that characterization were. In particular, it
+was still totally unclear whether MU is, or is not, a theorem.
+
+
+The MU-puzzle
+
+
+41
+
+
+
+
+FIGURE 12. Sky Castle, by M. C.: Escher (woodcut, 1928).
+
+
+The MU-puzzle
+
+
+42
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Two-Part Invention
+
+
+or,
+
+What the Tortoise Said to Achilles
+
+by Lewis Carroll'
+
+Achilles had overtaken the Tortoise, and had seated himself comfortably on its back.
+
+"So you've got to the end of our race-course?" said the Tortoise. "Even though it
+DOES consist of an infinite series of distances? I thought some wiseacre or other had
+proved that the thing couldn't be done?"
+
+"It CAN be done," said Achilles. "It HAS been done! Solvitur ambulando. You see the
+distances were constantly DIMINISHING; and so-"
+
+"But if they had been constantly INCREASING?" the Tortoise interrupted. "How
+then?"
+
+"Then I shouldn't be here," Achilles modestly replied; "and You would have got
+several times round the world, by this time!"
+
+"You flatter me-FLATTEN, I mean," said the Tortoise; "for you ARE a heavy weight,
+and NO mistake! Well now, would you like to hear of a race-course, that most people
+fancy they can get to the end of in two or three steps, while it REALLY consists of an
+infinite number of distances, each one longer than the previous one?"
+
+"Very much indeed!" said the Grecian warrior, as he drew from his helmet (few
+Grecian warriors possessed POCKETS in those days) an enormous note-book and pencil.
+"Proceed! And speak SLOWLY, please! SHORTHAND isn't invented yet!"
+
+"That beautiful First Proposition by Euclid!" the Tortoise murmured dreamily. "You
+admire Euclid?"
+
+"Passionately! So far, at least, as one CAN admire a treatise that won't be published
+for some centuries to come!"
+
+"Well, now, let's take a little bit of the argument in that First Proposition just TWO
+steps, and the conclusion drawn from them. Kindly enter them in your note-book. And in
+order to refer to them conveniently, let's call them A, B, and Z:
+
+(A) Things that are equal to the same are equal to each other.
+
+(B) The two sides of this Triangle are things that are equal to the same.
+
+(Z) The two sides of this Triangle are equal to each other.
+
+Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that
+any one who accepts A and B as true, MUST accept Z as true?"
+
+"Undoubtedly! The youngest child in a High School-as soon as High
+
+
+Two-Part Invention
+
+
+43
+
+
+
+Schools are invented, which will not be till some two thousand years later-will grant
+THAT."
+
+"And if some reader had NOT yet accepted A and B as true, he might still accept the
+SEQUENCE as a VALID one, I suppose?"
+
+"No doubt such a reader might exist. He might say, 'I accept as true the Hypothetical
+Proposition that, IF A and B be true, Z must be true; but I DON'T accept A and B as true.'
+Such a reader would do wisely in abandoning Euclid, and taking to football."
+
+"And might there not ALSO be some reader who would say 'I accept A and B as true,
+but I DON'T accept the Hypothetical'?"
+
+"Certainly there might. HE, also, had better take to football."
+
+"And NEITHER of these readers," the Tortoise continued, "is AS YET under any
+logical necessity to accept Z as true?"
+
+"Quite so," Achilles assented.
+
+"Well, now, I want you to consider ME as a reader of the SECOND kind, and to force
+me, logically, to accept Z as true."
+
+"A tortoise playing football would be-" Achilles was beginning.
+
+'-an anomaly, of course," the Tortoise hastily interrupted. "Don't wander from the
+point. Let's have Z first, and football afterwards!"
+
+"I'm to force you to accept Z, am I?" Achilles said musingly. "And your present
+position is that you accept A and B, but you DON'T accept the Hypothetical-"
+
+"Let's call it C," said the Tortoise.
+
+"-but you DON'T accept
+
+(C) If A and B are true, Z must be true."
+
+"That is my present position," said the Tortoise.
+
+"Then I must ask you to accept C."
+
+"I'll do so," said the Tortoise, "as soon as you've entered it in that notebook of yours.
+What else have you got in it?"
+
+"Only a few memoranda," said Achilles, nervously fluttering the leaves: "a few
+memoranda of-of the battles in which I have distinguished myself!"
+
+"Plenty of blank leaves, I see!" the Tortoise cheerily remarked. "We shall need them
+ALL!" (Achilles shuddered.) "Now write as I dictate:
+
+(A) Things that are equal to the same are equal to each other.
+
+(B) The two sides of this Triangle are things that are equal to the same.
+
+(C) If A and B are true, Z must be true.
+
+(Z) The two sides of this Triangle are equal to each other."
+
+"You should call it D, not Z," said Achilles. "It comes NEXT to the other three. If you
+accept A and B and C, you MUST accept Z.
+
+
+Two-Part Invention
+
+
+44
+
+
+
+“And why must I?”
+
+"Because it follows LOGICALLY from them. If A and B and C are true, Z MUST be
+true. You can't dispute THAT, I imagine?"
+
+"If A and B and C are true, Z MUST be true," the Tortoise thoughtfully repeated.
+"That's ANOTHER Hypothetical, isn't it? And, if I failed to see its truth, I might accept A
+and B and C, and STILL not accept Z, mightn't I?"
+
+"You might," the candid hero admitted; "though such obtuseness would certainly be
+phenomenal. Still, the event is POSSIBLE. So I must ask you to grant ONE more
+Hypothetical."
+
+"Very good, I'm quite willing to grant it, as soon as you've written it down. We will
+call it
+
+(D) If A and B and C are true, Z must be true.
+
+Have you entered that in your note-book?"
+
+"I HAVE!" Achilles joyfully exclaimed, as he ran the pencil into its sheath. "And at
+last we've got to the end of this ideal race-course! Now that you accept A and B and C
+and D, OF COURSE you accept Z."
+
+"Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and
+C and D. Suppose I STILL refused to accept Z?"
+
+"Then Logic would take you by the throat, and FORCE you to do it!" Achilles
+triumphantly replied. "Logic would tell you, 'You can't help yourself. Now that you've
+accepted A and B and C and D, you MUST accept Z!' So you've no choice, you see.",
+
+"Whatever LOGIC is good enough to tell me is worth WRITING DOWN," said the
+Tortoise. "So enter it in your book, please. We will call it
+
+(E) If A and B and C and D are true, Z must be true.
+
+Until I've granted THAT, of course I needn't grant Z. So it's quite a NECESSARY
+step, you see?"
+
+"I see," said Achilles; and there was a touch of sadness in his tone.
+
+Here the narrator, having pressing business at the Bank, was obliged to leave the
+happy pair, and did not again pass the spot until some months afterwards. When he did
+so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing
+in his notebook, which appeared to be nearly full. The Tortoise was saying, "Have you
+got that last step written down? Unless I've lost count, that makes a thousand and one.
+There are several millions more to come. And WOULD you mind, as a personal favour,
+considering what a lot of instruction this colloquy of ours will provide for the Logicians
+of the Nineteenth Century-WOULD you mind adopting a pun that my cousin the Mock-
+Turtle will then make, and allowing yourself to be renamed TAUGHT-US?"
+
+"As you please," replied the weary warrior, in the hollow tones of despair, as he buried
+his face in his hands. "Provided that YOU, for YOUR part, will adopt a pun the Mock-
+Turtle never made, and allow yourself to be re-named A KILL-EASE!"
+
+
+Two-Part Invention
+
+
+45
+
+
+
+CHAPTER 11
+
+
+Meaning and Form
+in Mathematics.
+
+
+THIS Two-Part Invention was the inspiration for my two characters. Just as Lewis
+Carroll took liberties with Zeno's Tortoise and Achilles, so have I taken liberties with
+Lewis Carroll's Tortoise and Achilles. In Carroll's dialogue, the same events take place
+over and over again, only each time on a higher and higher level; it is a wonderful
+analogue to Bach's Ever-Rising Canon. The Carrollian Dialogue, with its wit subtracted
+out, still leaves a deep philosophical problem: Do words and thoughts follow formal
+rules, or do they notl That problem is the problem of this book.
+
+In this Chapter and the next, we will look at several new formal systems. This will give
+us a much wider perspective on the concept of formal system. By the end of these two
+Chapters, you should have quite a good idea of the power of formal systems, and why
+they are of interest to mathematicians and logicians.
+
+The pq-System
+
+The formal system of this Chapter is called the pq-system. It is not important to
+mathematicians or logicians-in fact, it is just a simple invention of mine. Its importance
+lies only in the fact that it provides an excellent example of many ideas that play a large
+role in this book. There are three distinct symbols of the pq-system:
+
+p q -
+
+
+-the letters p, q, and the hyphen.
+
+The pq-system has an infinite number of axioms. Since we can't write them all down, we
+have to have some other way of describing what they are. Actually, we want more than
+just a description of the axioms; we want a way to tell whether some given string is an
+axiom or not. A mere description of axioms might characterize them fully and yet
+weakly-which was the problem with the way theorems in the MlU-system were
+characterized. We don't want to have to struggle for an indeterminate-possibly infinite
+length of time, just to find out if some string is an axiom or not. Therefore, we will define
+axioms in such a way that there is an obvious decision procedure for axiomhood of a
+string composed of p's, q's, and hyphens.
+
+
+Meaning and Form in Mathematics
+
+
+46
+
+
+
+DEFINITION: xp-qx is an axiom, whenever x is composed of hyphens only.
+
+Note that 'x' must stand for the same string of hyphens in both occurrences For example, -
+-p-q—is an axiom. The literal expression 'xp-qx-' i„ not an axiom, of course (because 'x'
+does not belong to the pq-system); it is more like a mold in which all axioms are cast-and
+it is called an axiom schema.
+
+The pq-system has only one rule of production:
+
+RUFE: Suppose x, y, and z all stand for particular strings containing only hyphens. And
+suppose that x py qz is known to be a theorem. The' xpy-qz- is a theorem.
+
+For example, take x to be 1 —', y to be'—', and z to be'-'. The rule tells us:
+
+If —p—q- turns out to be a theorem, then so will —p—q—.
+
+
+As is typical of rules of production, the statement establishes a causal connection between
+the theoremhood of two strings, but without asserting theoremhood for either one on its
+own.
+
+A most useful exercise for you is to find a decision procedure for the theorems of
+the pq-system. It is not hard; if you play around for a while you will probably pick it up.
+Try it.
+
+
+The Decision Procedure
+
+I presume you have tried it. First of all, though it may seem too obvious to mention, I
+would like to point out that every theorem of the pq-system has three separate groups of
+hyphens, and the separating elements are one p, and one q, in that order. (This can be
+shown by an argument based on "heredity", just the way one could show that all MIU-
+system theorems had to begin with M.) This means that we can rule out, from its form
+alone, o string such as — p-p-p—q .
+
+Now, stressing the phrase "from its form alone" may seem silly; what else is there to a
+string except its form? What else could possibly play a roll in determining its properties?
+Clearly nothing could. But bear this in mint as the discussion of formal systems goes on;
+the notion of "form" will star to get rather more complicated and abstract, and we will
+have to think more about the meaning of the word "form". In any case, let us give the
+name well formed string to any string which begins with a hyphen-group, then ha one p,
+then has a second hyphen-group, then a q, and then a final hyphen-group.
+
+Back to the decision procedure ... The criterion for theoremhood is that the first two
+hyphen-groups should add up, in length, to the third
+
+
+Meaning and Form in Mathematics
+
+
+47
+
+
+
+hyphen-group, for instance, —p--q - is a theorem, since 2 plus 2 equals 4, whereas — p-q-
+is not, since 2 plus 2 is not 1. To see why this is the proper criterion, look first at the
+axiom schema. Obviously, it only manufactures axioms which satisfy the addition
+criterion. Second, look at the rule of production. If the first string satisfies the addition
+criterion, so must the second one-and conversely, if the first string does not satisfy the
+addition criterion, then neither does the second string. The rule makes the addition
+criterion into a hereditary property of theorems: any theorem passes the property on to its
+offspring. This shows why the addition criterion is correct.
+
+There is, incidentally, a fact about the pq-system which would enable us to say
+with confidence that it has a decision procedure, even before finding the addition
+criterion. That fact is that the pq-system is not complicated by the opposing currents of
+lengthening and shortening rules; it has only lengthening rules. Any formal system which
+tells you how to make longer theorems from shorter ones, but never the reverse, has got
+to have a decision procedure for its theorems. For suppose you are given a string. First
+check whether it's an axiom or not (I am assuming that there is a decision procedure for
+axiomhood-otherwise, things are hopeless). If it is an axiom, then it is by definition a
+theorem, and the test is over. So suppose instead that it's not an axiom. Then, to be a
+theorem, it must have come from a shorter string, via one of the rules. By going over the
+various rules one by one, you can pinpoint not only the rules that could conceivably
+produce that string, but also exactly which shorter strings could be its forebears on the
+"family tree". In this way, you "reduce" the problem to determining whether any of
+several new but shorter strings is a theorem. Each of them can in turn be subjected to the
+same test. The worst that can happen is a proliferation of more and more, but shorter and
+shorter, strings to test. As you continue inching your way backwards in this fashion, you
+must be getting closer to the source of all theorems-the axiom schemata. You just can't
+get shorter and shorter indefinitely; therefore, eventually either you will find that one of
+your short strings is an axiom, or you'll come to a point where you're stuck, in that none
+of your short strings is an axiom, and none of them can be further shortened by running
+some rule or other backwards. This points out that there really is not much deep interest
+in formal systems with lengthening rules only; it is the interplay of lengthening and
+shortening rules that gives formal systems a certain fascination..
+
+Bottom-up vs. Top-down
+
+The method above might be called a top-down decision procedure, to be contrasted with a
+bottom-up decision procedure, which I give now. It is very reminiscent of the genie's
+systematic theorem-generating method for the MlU-system, but is complicated by the
+presence of an axiom schema. We are going to form a "bucket" into which we throw
+theorems as they are generated. Here is how it is done:
+
+
+Meaning and Form in Mathematics
+
+
+48
+
+
+
+(la) Throw the simplest possible axiom (-p-q— ) into the bucket.
+
+(I b) Apply the rule of inference to the item in the bucket, and put the result into the
+bucket.
+
+(2a) Throw the second-simplest axiom into the bucket.
+
+(2b) Apply the rule to each item in the bucket, and throw all results into the bucket.
+
+(3a) Throw the third-simplest axiom into the bucket.
+
+(3b) Apply the rule to each item in the bucket, and throw all results into the bucket,
+etc., etc.
+
+A moment's reflection will show that you can't fail to produce every theorem of the pq-
+system this way. Moreover, the bucket is getting filled with longer and longer theorems,
+as time goes on. It is again a consequence of that lack of shortening rules. So if you have
+a particular string, such as —p—q—, which you want to test for theoremhood, just
+follow the numbered steps, checking all the while for the string in question. If it turns up-
+theorem! If at some point everything that goes into the bucket is longer than the string in
+question, forget it-it is not a theorem. This decision procedure is bottom=up because it is
+working its way up from the basics, which is to say the axioms. The previous decision
+procedure is top-down because it does precisely the reverse: it works its way back down
+towards the basics.
+
+
+Isomorphisms Induce Meaning
+
+Now we come to a central issue of this Chapter-indeed of the book. Perhaps you have
+already thought to yourself that the pq-theorems are like additions. The string —p—q— is
+a theorem because 2 plus 3 equals 5. It could even occur to you that the theorem — p—q—
+is a statement, written in an odd notation, whose meaning is that 2 plus 3 is 5. Is this a
+reasonable way to look at things? Well, I deliberately chose 'p' to remind you of 'plus',
+and 'q' to remind you of'equals'. . . So, does the string --p — q — actually mean "2 plus
+3 equals 5"?
+
+What would make us feel that way? My answer would be that we have perceived
+an isomorphism between pq-theorems and additions. In the Introduction, the word
+"isomorphism" was defined as an information preserving transformation. We can now go
+into that notion a little more deeply, and see it from another perspective. The word
+"isomorphism 1 applies when two complex structures can be mapped onto each other, in
+such a way that to each part of one structure there is a corresponding part in the other
+structure, where "corresponding" means that the two part play similar roles in their
+respective structures. This usage of the word "isomorphism" is derived from a more
+precise notion in mathematics.
+
+
+Meaning and Form in Mathematics
+
+
+49
+
+
+
+It is cause for joy when a mathematician discovers an isomorphism between two
+structures which he knows. It is often a "bolt from the blue", and a source of
+wonderment. The perception of an isomorphism between two known structures is a
+significant advance in knowledge-and I claim that it is such perceptions of isomorphism
+which create meanings in the minds of people. A final word on the perception of
+isomorphisms: since they come in many shapes and sizes, figuratively speaking, it is not
+always totally clear when you really have found an isomorphism. Thus, "isomorphism" is
+a word with all the usual vagueness of words-which is a defect but an advantage as well.
+
+In this case, we have an excellent prototype for the concept of isomorphism.
+There is a "lower level" of our isomorphism-that is, a mapping between the parts of the
+two structures:
+
+
+p <= => plus
+q <= => equals
+
+- <= => one
+
+— <= => two
+— <= => three
+
+etc.
+
+This symbol-word correspondence has a name: interpretation.
+
+Secondly, on a higher level, there is the correspondence between true statements
+and theorems. But-note carefully-this higher-level correspondence could not be perceived
+without the prior choice of an interpretation for the symbols. Thus it would be more
+accurate to describe it as a correspondence between true statements and interpreted
+theorems. In any case we have displayed a two-tiered correspondence, which is typical of
+all isomorphisms.
+
+When you confront a formal system you know nothing of, and if you hope to
+discover some hidden meaning in it, your problem is how to assign interpretations to its
+symbols in a meaningful way-that is, in such a way that a higher-level correspondence
+emerges between true statements and theorems. You may make several tentative stabs in
+the dark before finding a good set of words to associate with the symbols. It is very
+similar to attempts to crack a code, or to decipher inscriptions in an unknown language
+like Linear B of Crete: the only way to proceed is by trial and error, based on educated
+guesses. When you hit a good choice, a "meaningful" choice, all of a sudden things just
+feel right, and work speeds up enormously. Pretty soon everything falls into place. The
+excitement of such an experience is captured in The Decipherment of Linear B by John
+Chadwick.
+
+But it is uncommon, to say the least, for someone to be in the position of
+"decoding" a formal system turned up in the excavations of a ruined civilization!
+Mathematicians (and more recently, linguists, philosophers, and some others) are the only
+users of formal systems, and they invariably have an interpretation in mind for the formal
+systems which they use and publish. The idea of these people is to set up a formal system
+whose
+
+
+Meaning and Form in Mathematics
+
+
+50
+
+
+
+Theorems reflect some portion of reality isomorphically. In such a case, the choice of
+symbols is a highly motivated one, as is the choice of typographical rules of production.
+When I devised the pq-system, I was in position. You see why I chose the symbols I
+chose. It is no accident theorems are isomorphic to additions; it happened because I
+deliberately sought out a way to reflect additions typographically.
+
+Meaningless and Meaningful Interpretations
+
+You can choose interpretations other than the one I chose. You need make every
+theorem come out true. But there would be very little reason make an interpretation in
+which, say, all theorems came out false, certainly even less reason to make an
+interpretation under which there is no correlation at all, positive or negative, between
+theoremhood and tri Let us therefore make a distinction between two types of
+interpretations a formal system. First, we can have a meaningless interpretation, one un
+which we fail to see any isomorphic connection between theorems of system, and reality.
+Such interpretations abound-any random choice a will do. For instance, take this one:
+
+p <= => horse
+q <= => happy
+- <= => apple
+
+
+Now -p-q-- acquires a new interpretation: "apple horse apple hat apple apple"-a
+poetic sentiment, which might appeal to horses, and mi! even lead them to favor this
+mode of interpreting pq-strings! However, t interpretation has very little
+"meaningfulness"; under interpretative, theorems don't sound any truer, or any better,
+than nontheorems. A ho might enjoy "happy happy happy apple horse" (mapped onto q q
+q) just as much as any interpreted theorem.
+
+The other kind of interpretation will be called meaningful. Under si an
+interpretation, theorems and truths correspond-that is, an isomorphism exists between
+theorems and some portion of reality. That is why it is good to distinguish between
+interpretations and meanings. Any old word can be used as an interpretation for 'p', but
+'plus' is the only meaningful choice we've come up with. In summary, the meaning of 'p'
+seems to be 'plus’ though it can have a million different interpretations.
+
+Active vs. Passive Meanings
+
+Probably the most significant fact of this Chapter, if understood deeply this: the pq-
+system seems to force us into recognizing that symbols of a formal system, though
+initially without meaning, cannot avoid taking on "meaning" of sorts at least if an
+isomorphism is found. The difference between meaning it formal system and in a
+language is a very important one, however. It is this:
+
+
+Meaning and Form in Mathematics
+
+
+51
+
+
+
+in a language, when we have learned a meaning for a word, we then mar-c new
+statements based on the meaning of the word. In a sense the meaning becomes active,
+since it brings into being a new rule for creating sentences. This means that our command
+of language is not like a finished product: the rules for making sentences increase when
+we learn new meanings. On the other hand, in a formal system, the theorems are
+predefined, by the rules of production. We can choose "meanings" based on an
+isomorphism (if we can find one) between theorems and true statements. But this does
+not give us the license to go out and add new theorems to the established theorems. That
+is what the Requirement of Formality in Chapter I was warning you of.
+
+In the MlU-system, of course, there was no temptation to go beyond the four
+rules, because no interpretation was sought or found. But here, in our new system, one
+might be seduced by the newly found "meaning" of each symbol into thinking that the
+string
+
+
+-p-p-p-q
+
+is a theorem. At least, one might wish that this string were a theorem. But wishing doesn't
+change the fact that it isn’t. And it would be a serious mistake to think that it "must" be a
+theorem, just because 2 plus 2 plus 2 plus 2 equals 8. It would even be misleading to
+attribute it any meaning at all, since it is not well-formed, and our meaningful
+interpretation is entirely derived from looking at well-formed strings.
+
+In a formal system, the meaning must remain passive', we can read each string
+according to the meanings of its constituent symbols, but we do not have the right to
+create new theorems purely on the basis of the meanings we've assigned the symbols.
+Interpreted formal systems straddle the line between systems without meaning, and
+systems with meaning. Their strings can be thought of as "expressing" things, but this
+must come only as a consequence of the formal properties of the system.
+
+Double-Entendre!
+
+And now, I want to destroy any illusion about having found the meanings for the symbols
+of the pq-system. Consider the following association:
+
+p <= => equals
+q <= => taken from
+
+- <= => one
+
+— <= => two
+
+etc.
+
+Now, —p—q— has a new interpretation: "2 equals 3 taken from 5". Of course it is a true
+statement. All theorems will come out true under this new interpretation. It is just as
+meaningful as the old one. Obviously, it is silly to ask, "But which one is the meaning of
+the string?" An interpreta
+
+
+Meaning and Form in Mathematics
+
+
+52
+
+
+
+tion will me meaningful to the extent that it accurately reflects some isomorphism to the
+real world. When different aspects of the real world a isomorphic to each other (in this
+case, additions and subtractions), or single formal system can be isomorphic to both, and
+therefore can take (two passive meanings. This kind of double-valuedness of symbols at
+strings is an extremely important phenomenon. Here it seems trivial curious, annoying.
+But it will come back in deeper contexts and bring with it a great richness of ideas.
+
+Here is a summary of our observations about the pq-system. Und either of the two
+meaningful interpretations given, every well-form( string has a grammatical assertion for
+its counterpart-some are true, son false. The idea of well formed strings in any formal
+system is that they a those strings which, when interpreted symbol for symbol, yield
+grammatical sentences. (Of course, it depends on the interpretation, but usually, there one
+in mind.) Among the well-formed strings occur the theorems. The: are defined by an
+axiom schema, and a rule of production. My goal in inventing the pq-system was to
+imitate additions: I wanted every theorem] to express a true addition under interpretation;
+conversely, I wanted every true addition of precisely two positive integers to be
+translatable into a string, which would be a theorem. That goal was achieved. Notice,
+then fore, that all false additions, such as "2 plus 3 equals 6", are mapped into strings
+which are well-formed, but which are not theorems.
+
+
+Formal Systems and Reality
+
+This is our first example of 'a case where a formal system is based upon portion of
+reality, and seems to mimic it perfectly, in that its theorems a] isomorphic to truths about
+that part of reality. However, reality and tt formal system are independent. Nobody need
+be aware that there is a isomorphism between the two. Each side stands by itself-one plus
+or equals two, whether or not we know that -p-q— is a theorem; and -p-q— is still a
+theorem whether or not we connect it with addition.
+
+You might wonder whether making this formal system, or any form system, sheds
+new light on truths in the domain of its interpretation. Hat we learned any new additions
+by producing pq-theorems? Certainly not but we have learned something about the nature
+of addition as process-namely, that it is easily mimicked by a typographical rule
+governing meaningless symbols. This still should not be a big surprise sing addition is
+such a simple concept. It is a commonplace that addition can I captured in the spinning
+gears of a device like a cash register.
+
+But it is clear that we have hardly scratched the surface, as far formal systems go;
+it is natural to wonder about what portion of reality co be imitated in its behavior by a set
+of meaningless symbols governed I formal rules. Can all of reality be turned into a formal
+system? In a very broad sense, the answer might appear to be yes. One could suggest, for
+instance, that reality is itself nothing but one very complicated formal
+
+
+Meaning and Form in Mathematics
+
+
+53
+
+
+
+system. Its symbols do not move around on paper, but rather in a three-dimensional
+vacuum (space); they are the elementary particles of which everything is composed.
+(Tacit assumption: that there is an end to the descending chain of matter, so that the
+expression "elementary particles" makes sense.) The "typographical rules" are the laws of
+physics, which tell how, given the positions and velocities of all particles at a given
+instant, to modify them, resulting in a new set of positions and velocities belonging to the
+"next" instant. So the theorems of this grand formal system are the possible
+configurations of particles at different times in the history of the universe. The sole axiom
+is (or perhaps, was) the original configuration of all the particles at the "beginning of
+time". This is so grandiose a conception, however, that it has only the most theoretical
+interest; and besides, quantum mechanics (and other parts of physics) casts at least some
+doubt on even the theoretical worth of this idea. Basically, we are asking if the universe
+operates deterministically, which is an open question.
+
+Mathematics and Symbol Manipulation
+
+Instead of dealing with such a big picture, let's limit ourselves to mathematics as
+our "real world". Here, a serious question arises: How can we be sure, if we've tried to
+model a formal system on some part of mathematics, that we've done the job accurately-
+especially if we're not one hundred per cent familiar with that portion of mathematics
+already? Suppose the goal of the formal system is to bring us new knowledge in that
+discipline. How will we know that the interpretation of every theorem is true, unless
+we've proven that the isomorphism is perfect? And how will we prove that the
+isomorphism is perfect, if we don't already know all about the truths in the discipline to
+begin with?
+
+Suppose that in an excavation somewhere, we actually did discover some
+mysterious formal system. We would try out various interpretations and perhaps
+eventually hit upon one which seemed to make every theorem come out true, and every
+nontheorem come out false. But this is something which we could only check directly in
+a finite number of cases. The set of theorems is most likely infinite. How will we know
+that all theorems express truths under this interpretation, unless we know everything there
+is to know about both the formal system and the corresponding domain of interpretation?
+
+It is in somewhat this odd position that we will find ourselves when we attempt to
+match the reality of natural numbers (i.e., the nonnegative integers: 0, 1,2, ...) with the
+typographical symbols of a formal system. We will try to understand the relationship
+between what we call "truth" in number theory and what we can get at by symbol
+manipulation.
+
+So let us briefly look at the basis for calling some statements of number theory
+true, and others false. How much is 12 times 12? Everyone knows it is 144. But how
+many of the people who give that answer have actually at
+
+
+Meaning and Form in Mathematics
+
+
+54
+
+
+
+any time in their lives drawn a 12 by 12 rectangle, and then counted the little squares in
+it? Most people would regard the drawing and counting unnecessary. They would instead
+offer as proof a few marks on paper, such as are shown below:
+
+
+12
+X 12
+
+
+24
+
+12
+
+
+144
+
+
+And that would be the "proof". Nearly everyone believes that if you counted the
+squares, you would get 144 of them; few people feel that outcome is in doubt.
+
+The conflict between the two points of view comes into sharper focus when you
+consider the problem of determining the value 987654321 x 123456789. First of all, it is
+virtually impossible to construct the appropriate rectangle; and what is worse, even if it
+were constructed and huge armies of people spent centuries counting the little squares, o
+a very gullible person would be willing to believe their final answer. It is just too likely
+that somewhere, somehow, somebody bobbled just a little bit. So is it ever possible to
+know what the answer is? If you trust the symbolic process which involves manipulating
+digits according to certain simple rules, yes. That process is presented to children as a
+device which gets right answer; lost in the shuffle, for many children, are the rhyme
+reason of that process. The digit-shunting laws for multiplication are based mostly on a
+few properties of addition and multiplication which are assumed to hold for all numbers.
+
+The Basic Laws of Arithmetic
+
+The kind of assumption I mean is illustrated below. Suppose that you down a few sticks:
+
+/////// / /
+
+Now you count them. At the same time, somebody else counts them, starting from the
+other end. Is it clear that the two of you will get the s: answer? The result of a counting
+process is independent of the way in which it is done. This is really an assumption about
+what counting i would be senseless to try to prove it, because it is so basic; either you s or
+you don't-but in the latter case, a proof won't help you a bit.
+
+From this kind of assumption, one can get to the commutativity and associativity
+of addition (i.e., first that b + c = c + b always, and second that b + (c + d) = (b + c) + d
+always). The same assumption can also you to the commutativity and associativity of
+multiplication; just think of
+
+
+Meaning and Form in Mathematics
+
+
+55
+
+
+
+
+
+many cubes assembled to form a large rectangular solid. Multiplicative commutativity
+and associativity are just the assumptions that when you rotate the solid in various ways,
+the number of cubes will not change. Now these assumptions are not verifiable in all
+possible cases, because the number of such cases is infinite. We take them for granted;
+we believe them (if we ever think about them) as deeply as we could believe anything.
+The amount of money in our pocket will not change as we walk down the street, jostling
+it up and down; the number of books we have will not change if we pack them up in a
+box, load them into our car, drive one hundred miles, unload the box, unpack it, and place
+the books in a new shelf. All of this is part of what we mean by number.
+
+There are certain types of people who, as soon as some undeniable fact is written
+down, find it amusing to show why that "fact" is false after all. I am such a person, and as
+soon as I had written down the examples above involving sticks, money, and books, I
+invented situations in which they were wrong. You may have done the same. It goes to
+show that numbers as abstractions are really quite different from the everyday numbers
+which we use.
+
+People enjoy inventing slogans which violate basic arithmetic but which illustrate
+"deeper" truths, such as "1 and 1 make 1" (for lovers), or "1 plus 1 plus 1 equals 1" (the
+Trinity). You can easily pick holes in those slogans, showing why, for instance, using the
+plus-sign is inappropriate in both cases. But such cases proliferate. Two raindrops
+running down a windowpane merge; does one plus one make one? A cloud breaks up into
+two clouds-more evidence for the same? It is not at all easy to draw a sharp line between
+cases where what is happening could be called "addition", and where some other word is
+wanted. If you think about the question, you will probably come up with some criterion
+involving separation of the objects in space, and making sure each one is clearly
+distinguishable from all the others. But then how could one count ideas? Or the number
+of gases comprising the atmosphere? Somewhere, if you try to look it up, you can
+probably find a statement such as, "There are 17 languages in India, and 462 dialects."
+There is something strange about precise statements like that, when the concepts
+"language" and "dialect" are themselves fuzzy.
+
+Ideal Numbers
+
+Numbers as realities misbehave. However, there is an ancient and innate sense in
+people that numbers ought not to misbehave. There is something clean and pure in the
+abstract notion of number, removed from counting beads, dialects, or clouds; and there
+ought to be a way of talking about numbers without always having the silliness of reality
+come in and intrude. The hard-edged rules that govern "ideal" numbers constitute
+arithmetic, and their more advanced consequences constitute number theory. There is
+only one relevant question to be asked, in making the transition from numbers as
+practical things to numbers as formal things. Once you have
+
+
+Meaning and Form in Mathematics
+
+
+56
+
+
+
+
+FIGURE 13. Liberation, by M.C. Escher (lithograph, 1955).
+
+
+Meaning and Form in Mathematics
+
+
+57
+
+
+
+decided to try to capsulize all of number theory in an ideal system, is it really possible to
+do the job completely? Are numbers so clean and crystalline and regular that their nature
+can be completely captured in the rules of a formal system? The picture Liberation (Fig.
+13), one of Escher's most beautiful, is a marvelous contrast between the formal and the
+informal, with a fascinating transition region. Are numbers really as free as birds? Do
+they suffer as much from being crystallized into a rule-obeying system? Is there a
+magical transition region between numbers in reality and numbers on paper?
+
+When I speak of the properties of natural numbers, I don't just mean properties
+such as the sum of a particular pair of integers. That can be found out by counting, and
+anybody who has grown up in this century cannot doubt the mechanizability of such
+processes as counting, adding, multiplying, and so on. I mean the kinds of properties
+which mathematicians are interested in exploring, questions for which no counting-
+process is sufficient to provide the answer-not even theoretically sufficient. Let us take a
+classic example of such a property of natural numbers. The statement is: "There are
+infinitely many prime numbers." First of all, there is no counting process which will ever
+be able to confirm, or refute, this assertion. The best we could do would be to count
+primes for a while and concede that there are "a lot". But no amount of counting alone
+would ever resolve the question of whether the number of primes is finite or infinite.
+There could always be more. The statement-and it is called "Euclid's Theorem" (notice
+the capital "T")-is quite unobvious. It may seem reasonable, or appealing, but it is not
+obvious. However, mathematicians since Euclid have always called it true. What is the
+reason?
+
+
+Euclid's Proof
+
+The reason is that reasoning tells them it is so. Let us follow the reasoning involved. We
+will look at a variant of Euclid's proof. This proof works by showing that whatever
+number you pick, there is a prime larger than it. Pick a number-N. Multiply all the
+positive integers starting with 1 and ending with N; in other words, form the factorial of
+N, written "N!". What you get is divisible by every number up to N. When you add 1 to
+N!, the result
+
+can't be a multiple of 2 (because it leaves 1 over, when you divide
+
+by 2);
+
+can't be a multiple of 3 (because it leaves I over, when you divide
+
+by 3);
+
+can't be a multiple of 4 (because it leaves 1 over, when you divide
+
+by 4);
+
+
+Meaning and Form in Mathematics
+
+
+58
+
+
+
+can't be a multiple of N (because it leaves 1 over, when you
+
+divide by N);
+
+In other words, N\ + 1, if it is divisible at all (other than by 1 and itself only is
+divisible by numbers greater than N. So either it is itself prime, or prime divisors are
+greater than N. But in either case we've shown the must exist a prime above N. The
+process holds no matter what number is. Whatever N is, there is a prime greater than N.
+And thus ends the demonstration of the infinitude of the primes.
+
+This last step, incidentally, is called generalization , and we will meet again later
+in a more formal context. It is where we phrase an argument terms of a single number
+(AT), and then point out that N was unspecified and therefore the argument is a general
+one.
+
+Euclid's proof is typical of what constitutes "real mathematics". It simple,
+compelling, and beautiful. It illustrates that by taking several rash short steps one can get
+a long way from one's starting point. In our case, t starting points are basic ideas about
+multiplication and division and forth. The short steps are the steps of reasoning. And
+though eve individual step of the reasoning seems obvious, the end result is not obvious.
+We can never check directly whether the statement is true or not; } we believe it, because
+we believe in reasoning. If you accept reasoning there seems to be no escape route; once
+you agree to hear Euclid out, you’ll have to agree with his conclusion. That's most
+fortunate-because it mea that mathematicians will always agree on what statements to
+label "true and what statements to label "false".
+
+This proof exemplifies an orderly thought process. Each statement related to
+previous ones in an irresistible way. This is why it is called "proof rather than just "good
+evidence". In mathematics the goal always to give an ironclad proof for some unobvious
+statement. The very fact of the steps being linked together in an ironclad way suggests ti
+there may be a patterned structure binding these statements together. TI structure can
+best be exposed by finding a new vocabulary-a stylized vocabulary, consisting of
+symbols-suitable only for expressing statements about numbers. Then we can look at the
+proof as it exists in its translated version. It will be a set of statements which are related,
+line by line, in some detectable way. But the statements, since they're represented by
+means a small and stylized set of symbols, take on the aspect of patterns. In other words,
+though when read aloud, they seem to be statements about numb and their properties, still
+when looked at on paper, they seem to be abstract patterns-and the line-by-line structure
+of the proof may start to look like slow transformation of patterns according to some few
+typographical rules.
+
+
+Getting Around Infinity
+
+Although Euclid's proof is a proof that all numbers have a certain property it avoids
+treating each of the infinitely many cases separately. It gets around
+
+
+Meaning and Form in Mathematics
+
+
+59
+
+
+
+it by using phrases like "whatever N is", or "no matter what number N is". We could also
+phrase-the proof over again, so that it uses the phrase "all N". By knowing the appropriate
+context and correct ways of using such phrases, we never have to deal with infinitely
+many statements. We deal with just two or three concepts, such as the word "all"-which,
+though themselves finite, embody an infinitude; and by using them, we sidestep the
+apparent problem that there are an infinite number of facts we want to prove.
+
+We use the word "all" in a few ways which are defined by the thought processes
+of reasoning. That is, there are rules which our usage of "all" obeys. We may be
+unconscious of them, and tend to claim we operate on the basis of the meaning of the
+word; but that, after all, is only a circumlocution for saying that we are guided by rules
+which we never make explicit. We have used words all our lives in certain patterns, and
+instead of calling the patterns "rules", we attribute the courses of our thought processes to
+the "meanings" of words. That discovery was a crucial recognition in the long path
+towards the formalization of number theory.
+
+If we were to delve into Euclid's proof more and more carefully, we would see
+that it is composed of many, many small-almost infinitesimal steps. If all those steps were
+written out line after line, the proof would appear incredibly complicated. To our minds it
+is clearest when several steps are telescoped together, to form one single sentence. If we
+tried to look at the proof in slow motion, we would begin to discern individual frames. In
+other words, the dissection can go only so far, and then we hit the "atomic" nature of
+reasoning processes. A proof can be broken down into a series of tiny but discontinuous
+jumps which seem to flow smoothly when perceived from a higher vantage point. In
+Chapter VIII, I will show one way of breaking the proof into atomic units, and you will
+see how incredibly many steps are involved. Perhaps it should not surprise you, though.
+The operations in Euclid's brain when he invented the proof must have involved millions
+of neurons (nerve cells), many of which fired several hundred times in a single second.
+The mere utterance of a sentence involves hundreds of thousands of neurons. If Euclid's
+thoughts were that complicated, it makes sense for his proof to contain a huge number of
+steps! (There may be little direct connection between the neural actions in his brain, and a
+proof in our formal system, but the complexities of the two are comparable. It is as if
+nature wants the complexity of the proof of the infinitude of primes to be conserved, even
+when the systems involved are very different from each other.)
+
+In Chapters to come, we will lay out a formal system that (1) includes a stylized
+vocabulary in which all statements about natural numbers can be expressed, and (2) has
+rules corresponding to all the types of reasoning which seem necessary. A very important
+question will be whether the rules for symbol manipulation which we have then
+formulated are really of equal power (as far as number theory is concerned) to our usual
+mental reasoning abilities-or, more generally, whether it is theoretically possible to attain
+the level of our thinking abilities, by using some formal system.
+
+
+Meaning and Form in Mathematics
+
+
+60
+
+
+
+Sonata
+
+for Unaccompanied Achilles
+
+
+The telephone rings; Achilles picks it up.
+
+
+Achilles: Hello, this is Achilles.
+
+Achilles: Oh, hello, Mr. T. How are you?
+
+Achilles: A torticollis? Oh, I'm sorry to hear it. Do you have any idea what caused it?
+Achilles: How long did you hold it in that position?
+
+Achilles: Well, no wonder it's stiff, then. What on earth induced you keep your neck
+twisted that way for so long?
+
+Achilles: Wondrous many of them, eh? What kinds, for example? Achilles: What do you
+mean, "phantasmagorical beasts"?
+
+
+FIGURE 14. Mosaic II, by M. C. Escher (lithograph, 1957).
+
+
+
+Sonata for Unaccompanied Achilles
+
+
+61
+
+
+
+
+Achilles: Wasn't it terrifying to see so many of them at the same time? Achilles: A
+guitar!? Of all things to be in the midst of all those weird creatures. Say, don't you
+play the guitar?
+
+Achilles: Oh, well, it's all the same to me.
+
+Achilles: You're right; I wonder why I never noticed that difference between fiddles and
+guitars before. Speaking of fiddling, how would you like to come over and listen
+to one of the sonatas for unaccompanied violin by your favorite composer, J. S.
+Bach? I just bought a marvelous recording of them. I still can't get over the way
+Bach uses a single violin to create a piece with such interest.
+
+Achilles: A headache too? That's a shame. Perhaps you should just go to bed.
+
+Achilles: I see. Have you tried counting sheep?
+
+Achilles: Oh, oh, I see. Yes, I fully know what you mean. Well, if it's THAT distracting,
+perhaps you'd better tell it to me, and let me try to work on it, too.
+
+Achilles: A word with the letters 'A', 'D\ 'A', 'C' consecutively inside it ... Hmm ...
+What about "abracadabra"?
+
+Achilles: True, "ADAC" occurs backwards, not forwards, in that word. Achilles: Hours
+and hours? It sounds like I'm in for a long puzzle, then. Where did you hear this
+infernal riddle?
+
+Achilles: You mean he looked like he was meditating on esoteric Buddhist matters, but in
+reality he was just trying to think up complex word puzzles?
+
+Achilles: Aha!-the snail knew what this fellow was up to. But how did you come to talk
+to the snail?
+
+Achilles: Say, I once heard a word puzzle a little bit like this one. Do you want to hear it?
+Or would it just drive you further into distraction? Achilles: I agree-can't do any
+harm. Here it is: What's a word that begins with the letters "HE" and also ends
+with "HE"?
+
+Achilles: Very ingenious-but that's almost cheating. It's certainly not what I meant!
+
+Achilles: Of course you're right-it fulfills the conditions, but it's a sort of "degenerate"
+solution. There's another solution which I had in mind. Achilles: That's exactly it!
+How did you come up with it so fast? Achilles: So here's a case where having a
+headache actually might have helped you, rather than hindering you. Excellent!
+But I'm still in the dark on your "ADAC" puzzle.
+
+Achilles: Congratulations! Now maybe you'll be able to get to sleep! So tell me, what is
+the solution?
+
+Achilles: Well, normally I don't like hints, but all right. What's your hint? Achilles: I
+don't know what you mean by "figure" and "ground" in this case.
+
+Achilles: Certainly I know Mosaic II! I know ALL of Escher's works. After all, he's my
+favorite artist. In any case, I've got a print of Mosaic II hanging on my wall, in
+plain view from here.
+
+
+Sonata for Unaccompanied Achilles
+
+
+62
+
+
+
+Achilles:: Yes, t see all the black animals.
+
+Achilles: Yes, I also see how their "negative space" — what's left out— defines the white
+animals.
+
+Achilles: So THAT'S what you mean by "figure" and "ground". But what does that have
+to do with the "ADAC" puzzle?
+
+Achilles: Oh, this is too tricky for me. I think I'M starting to get a headache
+
+Achilles: You want to come over now? But I thought ~
+
+Achilles: Very well. Perhaps by then I'll have thought of the right answer to YOUR
+puzzle, using your figure-ground hint, relating it to MY puzzle
+
+Achilles: I'd love to play them for you.
+
+Achilles: You've invented a theory about them?
+
+Achilles: Accompanied by what instrument?
+
+Achilles: Well, if that's the case, it seems a little strange that he would have written out
+the harpsichord part, then, and had it published a s well.
+
+Achilles: I see — sort of an optional feature. One could listen to them either way — with
+or without accompaniment. But how would one know what the accompaniment is
+supposed to sound like?
+
+Achilles: Ah, yes, I guess that it is best, after all, to leave it to the listener’s imagination.
+And perhaps, as you said, Bach never even had accompaniment in mind at all.
+Those sonatas seem to work very indeed as they are.
+
+Achilles: Right. Well, I'll see you shortly.
+
+Achilles: Good-bye, Mr. T.
+
+
+Sonata for Unaccompanied Achilles
+
+
+63
+
+
+
+CHAPTER III
+
+
+Figure and Ground
+
+Primes vs. Composites
+
+THERE IS A strangeness to the idea that concepts can be captured by simple
+typographical manipulations. The one concept so far captured is that of addition, and it
+may not have appeared very strange. But suppose the goal were to create a formal system
+with theorems of the form Px, the letter 'x' standing for a hyphen-string, and where the
+only such theorems would be ones in which the hyphen-string contained exactly a prime
+number of hyphens. Thus, P-— would be a theorem, but P— would not. How could this
+be done typographically? First, it is important to specify clearly what is meant by
+typographical operations. The complete repertoire has been presented in the MlU-system
+and the pq-system, so we really only need to make a list of the kinds of things we have
+permitted:
+
+(1) reading and recognizing any of a finite set of symbols;
+
+(2) writing down any symbol belonging to that set;
+
+(3) copying any of those symbols from one place to another;
+
+(4) erasing any of those symbols;
+
+(5) checking to see whether one symbol is the same as another;
+
+(6) keeping and using a list of previously generated theorems.
+
+The list is a little redundant, but no matter. What is important is that it clearly involves
+only trivial abilities, each of them far less than the ability to distinguish primes from
+nonprimes. How, then, could we compound some of these operations to make a formal
+system in which primes are distinguished from composite numbers?
+
+The tq-System
+
+A first step might be to try to solve a simpler, but related, problem. We could try to make
+a system similar to the pq-system, except that it represents multiplication, instead of
+addition. Let's call it the tq-system, Y for times'. More specifically, suppose X, Y, and Z
+are, respectively, the numbers of hyphens in the hyphen-strings x, y, and z. (Notice I am
+taking special pains to distinguish between a string and the number of hyphens it
+contains.) Then we wish the string x ty q z to be a theorem if and only if X times Y
+
+equals Z. For instance, — t — q -should be a theorem because 2 times 3 equals 6, but —
+
+t—q— should not be a theorem. The tq-system can be characterized just about as easily as
+the pq-system namely, by using just one axiom schema and one rule of inference:
+
+
+Figure and Ground
+
+
+64
+
+
+
+AXIOM SCHEMA: xt-qx is an axiom, whenever x is a hyphen string.
+
+RULE OF INFERENCE: Suppose that x, y, and z are all hyphen-strings. An suppose that
+x ty qz is an old theorem. Then, xty-qzx is a ne' theorem.
+
+Below is the derivation of the theorem — t—q-
+
+
+(1) —t-q— (axiom)
+
+(2) —t—q— (by rule of inference, using line (1) as the old theorem)
+
+(3) —t—q - (by rule of inference, using line (2) as the old theorem)
+
+
+Notice how the middle hyphen-string grows by one hyphen each time the rule of
+inference is applied; so it is predictable that if you want a theorem with ten hyphens in
+the middle, you apply the rule of inference nine times in a row.
+
+Capturing Compositeness
+
+Multiplication, a slightly trickier concept than addition, has now bee] "captured"
+typographically, like the birds in Escher's Liberation. What about primeness? Here's a
+plan that might seem smart: using the tq-system define a new set of theorems of the form
+Cx, which characterize compost, numbers, as follows:
+
+RULE: Suppose x, y, and z are hyphen-strings. If x-ty-qz is a theorem then C z is a
+theorem.
+
+This works by saying that Z (the number of hyphens in z) is composite a long as it is the
+product of two numbers greater than 1-namely, X + (the number of hyphens in x-), and Y
++ l (the number of hyphens in y I am defending this new rule by giving you some
+"Intelligent mode justifications for it. That is because you are a human being, and want t,
+know why there is such a rule. If you were operating exclusively in the "Mechanical
+mode", you would not need any justification, since M-mod. workers just follow the rules
+mechanically and happily, never questioning; them!
+
+Because you work in the I-mode, you will tend to blur in your mind the
+distinction between strings and their interpretations. You see, things Cal become quite
+confusing as soon as you perceive "meaning" in the symbol which you are manipulating.
+You have to fight your own self to keep from thinking that the string'—' is the number 3.
+The Requirement of Formality, which in Chapter I probably seemed puzzling (because it
+seemed so obvious), here becomes tricky, and crucial. It is the essential thing which
+keeps you from mixing up the I-mode with the M-mode; or said another way, it keeps
+you from mixing up arithmetical facts with typographical theorems.
+
+
+Figure and Ground
+
+
+65
+
+
+
+
+Illegally Characterizing Primes
+
+
+It is very tempting to jump from the C-type theorems directly to P-type theorems, by
+proposing a rule of the following kind:
+
+PROPOSED RULE: Suppose x is a hyphen-string. If Cx is not a theorem, then Px is a
+theorem.
+
+The fatal flaw here is that checking whether Cx is not a theorem is not an explicitly
+typographical operation. To know for sure that MU is not a theorem of the MlU-system,
+you have to go outside of the system ... and so it is with this Proposed Rule. It is a rule
+which violates the whole idea of formal systems, in that it asks you to operate informally-
+that is, outside the system. Typographical operation (6) allows you to look into the
+stockpile of previously found theorems, but this Proposed Rule is asking you to look into
+a hypothetical "Table of Nontheorems". But in order to generate such a table, you would
+have to do some reasoning outside the system -reason ing which shows why various strings
+cannot be generated inside the system. Now it may well be that there is another formal
+system which can generate the "Table of Nontheorems", by purely typographical means.
+In fact, our aim is to find just such a system. But the Proposed Rule is not a typographical
+rule, and must be dropped.
+
+This is such an important point that we might dwell on it a bit more. In our C-
+system (which includes the tq-system and the rule which defines C-type theorems), we
+have theorems of the form Cx, with 'x' standing, as usual, for a hyphen-string. There are
+also nontheorems of the form Cx. (These are what I mean when I refer to "nontheorems",
+although of course tt-Cqq and other ill-formed messes are also nontheorems.) The
+difference is that theorems have a composite number of hyphens, nontheorems have a
+prime number of hyphens. Now the theorems all have a common "form", that is, originate
+from a common set of typographical rules. Do all nontheorems also have a common
+"form", in the same sense? Below is a list of C-type theorems, shown without their
+derivations. The parenthesized numbers following them simply count the hyphens in
+them.
+
+
+C— (4)
+
+
+C-(6)
+
+C -(8)
+
+C -(9)
+
+C -(10)
+
+C -(12)
+
+C -(14)
+
+C -(15)
+
+C ---(16)
+
+C -(18)
+
+
+Figure and Ground
+
+
+66
+
+
+
+
+
+
+
+
+
+
+
+
+I he "holes" in this list are the nontheorems. I o repeat the earlier quest Do the holes also
+have some "form" in common? Would it be reasonable say that merely by virtue of being
+the holes in this list, they share a common form? Yes and no. That they share some
+typographical quality is and able, but whether we want to call it "form" is unclear. The
+reason hesitating is that the holes are only negatively defined-they are the things that are
+left out of a list which is positively defined.
+
+Figure and Ground
+
+This recalls the famous artistic distinction between figure and ground. When a figure or
+"positive space" (e.g., a human form, or a letter, or a still life is drawn inside a frame, an
+unavoidable consequence is that its complementary shape-also called the "ground”, or
+"background", or "negative space"-has also been drawn. In most drawings, however, this
+fig ground relationship plays little role. The artist is much less interested in ground than
+in the figure. But sometimes, an artist will take interest in ground as well.
+
+There are beautiful alphabets which play with this figure-ground distinction. A
+message written in such an alphabet is shown below. At fir looks like a collection of
+somewhat random blobs, but if you step back ways and stare at it for a while, all of a
+sudden, you will see seven letters appear in this ..
+
+FIGURE 15.
+
+
+For a similar effect, take a look at my drawing Smoke Signal (Fig. 139). Along these
+lines, you might consider this puzzle: can you somehow create a drawing containing
+words in both the figure and the ground?
+
+Let us now officially distinguish between two kinds of figures: cursive!y
+drawable ones, and recursive ones (by the way, these are my own terms are not in
+common usage). A cursively drawable figure is one whose ground is merely an
+accidental by-product of the drawing act. A recursive figure is one whose ground can be
+seen as a figure in its own right. Usually this is quite deliberate on the part of the artist.
+The "re" in "recursive" represents the fact that both foreground and background are
+cursively drawable - the figure is "twice-cursive". Each figure-ground boundary in a
+recursive figure is a double-edged sword. M. C. Escher was a master at drawing recursive
+figures-see, for instance, his beautiful recursive drawing of birds (Fig. 16).
+
+
+Figure and Ground
+
+
+67
+
+
+
+
+FIGURE 16. Tiling of the plane using birds, by M. C. Escher (from a 1942 notebook).
+
+Our distinction is not as rigorous as one in mathematics, for who can definitively
+say that a particular ground is not a figure? Once pointed out, almost any ground has
+interest of its own. In that sense, every figure is recursive. But that is not what I intended
+by the term. There is a natural and intuitive notion of recognizable forms. Are both the
+foreground and background recognizable forms? If so, then the drawing is recursive. If
+you look at the grounds of most line drawings, you will find them rather unrecognizable.
+This demonstrates that
+
+There exist recognizable forms whose negative space is not any recognizable form.
+In more "technical" terminology, this becomes:
+
+There exist cursively drawable figures which are not recursive.
+
+Scott Kim's solution to the above puzzle, which I call his "FIGURE-FIGURE
+Figure", is shown in Figure 17. If you read both black and white.
+
+
+Figure and Ground
+
+
+68
+
+
+
+
+
+FIGURE 17. FIGURE-FIGURE Figure, by Scott E. Kim (1975).
+
+
+Figure and Ground
+
+
+69
+
+
+you will see "FIGURE" everywhere, but "GROUND" nowhere! It is a paragon of
+recursive figures. In this clever drawing, there are two nonequivalent ways of
+characterizing the black regions:
+
+
+(1) as the negative space to the white regions;
+
+(2) as altered copies of the white regions (produced by coloring and shifting each
+
+white region).
+
+
+(In the special case of the FIGURE-FIGURE Figure, the two characterizations are
+equivalent-but in most black-and-white pictures, they would not be.) Now in Chapter
+VIII, when we create our Typographical Number Theory (TNT), it will be our hope that
+the set of all false statements of number theory can be characterized in two analogous
+ways:
+
+
+(1) as the negative space to the set of all TNT-theorems;
+
+(2) as altered copies of the set of all TNT-theorems (produced by negating each
+
+TNT-theorem).
+
+But this hope will be dashed, because:
+
+(1) inside the set of all nontheorems are found some truths
+
+(2) outside the set of all negated theorems are found some falsehoods
+
+You will see why and how this happens, in Chapter XIV. Meanwhile, ponder over a
+pictorial representation of the situation (Fig. 18).
+
+Figure and Ground in Music
+
+One may also look for figures and grounds in music. One analogue is the distinction
+between melody and accompaniment-for the melody is always in the forefront of our
+attention, and the accompaniment is subsidiary, in some sense. Therefore it is surprising
+when we find, in the lower lines of a piece of music, recognizable melodies. This does
+not happen too often in post-baroque music. Usually the harmonies are not thought of as
+foreground. But in baroque music-in Bach above all-the distinct lines, whether high or
+low or in between, all act as "figures". In this sense, pieces by Bach can be called
+"recursive".
+
+Another figure-ground distinction exists in music: that between on-beat and off¬
+beat. If you count notes in a measure "one-and, two-and, three-and, four-and", most
+melody-notes will come on numbers, not on "and'"s. But sometimes, a melody will be
+deliberately pushed onto the "and" 's, for the sheer effect of it. This occurs in several
+etudes for the piano by Chopin, for instance. It also occurs in Bach-particularly in his
+Sonatas and Partitas for unaccompanied violin, and his Suites for unaccompanied cello.
+There, Bach manages to get two or more musical lines going simultaneously. Sometimes
+he does this by having the solo instrument play "double stops"-two notes at once. Other
+times, however, he
+
+
+Figure and Ground
+
+
+70
+
+
+
+
+FIGURE 18. Considerable visual symbolism is featured in this diagram of the relation
+between various classes of TNT strings. The biggest box represents the set of all TNT
+strings The next-biggest box represents the set of all well-formed TNT strings. Within it is
+found- set of all sentences of TNT. Now things begin to get interesting. The set of
+theorems pictured as a tree growing out of a trunk (representing the set of axioms). The
+tree-symbol chosen because of the recursive growth pattern which it exhibits: new
+branches (theorems constantly sprouting from old ones. The fingerlike branches probe
+into the corners of constraining region (the set of truths), yet can never fully occupy it.
+The boundary beta the set of truths and the set of falsities is meant to suggest a randomly
+meandering coastline which, no matter how closely you examine it, always has finer
+levels of structure, an consequently impossible to describe exactly in any finite way. (See
+B. Mandelbrot's book Fractals.) The reflected tree represents the set of negations of
+theorems: all of them false yet unable collectively to span the space of false statements.
+IDrawing by the author.J
+
+puts one voice on the on-beats, and the other voice on the off-beats, so ear separates them
+and hears two distinct melodies weaving in and out, - harmonizing with each other.
+Needless to say, Bach didn't stop at this level of complexity...
+
+Recursively Enumerable Sets vs. Recursive Sets
+
+Now let us carry back the notions of figure and ground to the domain formal
+systems. In our example, the role of positive space is played by C-type theorems, and the
+role of negative space is played by strings with a
+
+
+Figure and Ground
+
+
+71
+
+
+
+
+
+
+
+
+
+
+prime number of hyphens. So far, the only way we have found to represent prime
+numbers typographically is as a negative space. Is there, however, some way-I don't care
+how complicated-of representing the primes as a positive space-that is, as a set of
+theorems of some formal system?
+
+Different people's intuitions give different answers here. I remember quite vividly
+how puzzled and intrigued I was upon realizing the difference between a positive
+characterization and a negative characterization. I was quite convinced that not only the
+primes, but any set of numbers which could be represented negatively, could also be
+represented positively. The intuition underlying my belief is represented by the question:
+"How could a figure and its ground not carry exactly the same information ?" They
+seemed to me to embody the same information, just coded in two complementary ways.
+What seems right to you?
+
+It turns out I was right about the primes, but wrong in general. This astonished
+me, and continues to astonish me even today. It is a fact that:
+
+There exist formal systems whose negative space (set of nontheorems) is not
+the positive space (set of theorems) of any formal system.
+
+This result, it turns out, is of depth equal to Godel’s Theorem-so it is not
+surprising that my intuition was upset. I, just like the mathematicians of the early
+twentieth century, expected the world of formal systems and natural numbers to be more
+predictable than it is. In more technical terminology, this becomes:
+
+There exist recursively enumerable sets which are not recursive.
+
+The phrase recursively enumerable (often abbreviated "r.e.") is the mathematical
+counterpart to our artistic notion of "cursively drawable"-and recursive is the counterpart
+of "recursive". For a set of strings to be "r.e." means that it can be generated according to
+typographical rules-for example, the set of C-type theorems, the set of theorems of the
+MlU-system-indeed, the set of theorems of any formal system. This could be compared
+with the conception of a "figure" as "a set of lines which can be generated according to
+artistic rules" (whatever that might mean!). And a "recursive set" is like a figure whose
+ground is also a figure-not only is it r.e., but its complement is also r.e.
+
+It follows from the above result that:
+
+There exist formal systems for which there is no typographical decision
+procedure.
+
+How does this follow? Very simply. A typographical decision procedure is a
+method which tells theorems from nontheorems. The existence of such a test allows us to
+generate all nontheorems systematically, simply by going down a list of all strings and
+performing the test on them one at a time, discarding ill-formed strings and theorems
+along the way. This amounts to
+
+
+Figure and Ground
+
+
+72
+
+
+
+a typographical method for generating the set of nontheorems. But according to the
+earlier statement (which we here accept on faith), for some systems this is not possible.
+So we must conclude that typographical decision procedures do not exist for all formal
+systems.
+
+Suppose we found a set F of natural numbers ('F' for 'Figure') whi4 we could
+generate in some formal way-like the composite numbers. Suppose its complement is the
+set G (for 'Ground')-like the primes. Together F and G make up all the natural numbers,
+and we know a rule for making all the numbers in set F, but we know no such rule for
+making all tl numbers in set G. It is important to understand that if the members of were
+always generated in order of increasing size, then we could always characterize G. The
+problem is that many r.e. sets are generated I methods which throw in elements in an
+arbitrary order, so you never know if a number which has been skipped over for a long
+time will get included you just wait a little longer.
+
+We answered no to the artistic question, "Are all figures recursive We have now
+seen that we must likewise answer no to the analogous question in mathematics: "Are all
+sets recursive?" With this perspective, 1 us now come back to the elusive word "form".
+Let us take our figure-set and our ground-set G again. We can agree that all the numbers
+in set have some common "form"-but can the same be said about numbers in s G? It is a
+strange question. When we are dealing with an infinite set to sta with-the natural
+numbers-the holes created by removing some subs may be very hard to define in any
+explicit way. And so it may be that th< are not connected by any common attribute or
+"form". In the last analysis it is a matter of taste whether you want to use the word
+"form"-but just thinking about it is provocative. Perhaps it is best not to define "form", bi
+to leave it with some intuitive fluidity.
+
+Here is a puzzle to think about in connection with the above matter Can you
+characterize the following set of integers (or its negative space)
+
+1 3 7 12 18 26 35 45 56 69...
+
+How is this sequence like the FIGURE-FIGURE Figure?
+
+
+Primes as Figure Rather than Ground
+
+Finally, what about a formal system for generating primes? How is it don< The trick is to
+skip right over multiplication, and to go directly to nondivisibility as the thing to
+represent positively. Here are an axiom schema and rule for producing theorems which
+represent the notion that one number does not divide (D N D) another number exactly:
+
+AXIOM SCHEMA: xy D N Dx where x and y are hyphen-strings.
+
+For example —D N D—, where x has been replaced by'—'and y by
+
+
+Figure and Ground
+
+
+73
+
+
+
+
+RULE: If x D N Dy is a theorem, then so is x D N Dx y.
+
+If you use the rule twice, you can generate this theorem:
+
+.—D N D.
+
+which is interpreted as "5 does not divide 12". But —D N D.is not a theorem.
+
+What goes wrong if you try to produce it?
+
+Now in order to determine that a given number is prime, we have to build up
+some knowledge about its nondivisibility properties. In particular, we want to know that
+it is not divisible by 2 or 3 or 4, etc., all the way up to 1 less than the number itself. But
+we can't be so vague in formal systems as to say "et cetera". We must spell things out.
+We would like to have a way of saying, in the language of the system, "the number Z is
+divisor free up to X", meaning that no number between 2 and X divides Z. This can be
+done, but there is a trick to it. Think about it if you want.
+
+Here is the solution:
+
+RULE: If —D N D z is a theorem, so is z D F—.
+
+RULE: If z D Fx is a theorem and also x-D N Dz is a theorem, z D Fx- is a theorem.
+
+These two rules capture the notion of divisor freeness. All we need to do is to say that
+primes are numbers which are divisor-free up to 1 less than themselves:
+
+RULE: If z-DFz is a theorem, then Pz- is a theorem.
+
+Oh-let’s not forget that 2 is prime!
+
+
+Axiom: P—.
+
+
+And there you have it. The principle of representing primality formally is that there is a
+test for divisibility which can be done without any backtracking. You march steadily
+upward, testing first for divisibility by 2, then by 3, and so on. It is this "monotonicity" or
+unidirectionality-this absence of cross-play between lengthening and shortening,
+increasing and decreasing-that allows primality to be captured. And it is this potential
+complexity of formal systems to involve arbitrary amounts of backwards-forwards
+interference that is responsible for such limitative results as Godel’s Theorem, Turing's
+Halting Problem, and the fact that not all recursively enumerable sets are recursive.
+
+
+Ligure and Ground
+
+
+74
+
+
+
+
+
+
+
+
+
+Contracrostipunctus
+
+Achilles has come to visit his friend and jogging companion, the
+Tortoise, at his home
+
+Achilles: Heavens, you certainly have an admirable boomerang collection
+
+Tortoise: Oh, pshaw. No better than that of any other Tortoise. And now would you like
+to step into the parlor?
+
+Achilles: Fine. (Walks to the corner of the room.) I see you also have a large collection of
+records. What sort of music do you enjoy?
+
+Tortoise: Sebastian Bach isn't so bad, in my opinion. But these days, I must say, I am
+developing more and more of an interest in a rather specialized sort of music.
+
+Achilles: Tell me, what kind of music is that?
+
+Tortoise: A type of music which you are most unlikely to have heard of. call it "music to
+break phonographs by".
+
+Achilles: Did you say "to break phonographs by"? That is a curious concept. I can just
+see you, sledgehammer in hand, whacking on phonograph after another to pieces,
+to the strains of Beethoven's heroic masterpiece Wellington's Victory.
+
+Tortoise: That's not quite what this music is about. However, you might find its true
+nature just as intriguing. Perhaps I should give you a brief description of it?
+
+Achilles: Exactly what I was thinking.
+
+Tortoise: Relatively few people are acquainted with it. It all began whet my friend the
+Crab-have you met him, by the way?-paid m» a visit.
+
+Achilles: ' twould be a pleasure to make his acquaintance, I'm sure Though I've heard so
+much about him, I've never met him
+
+Tortoise: Sooner or later I'll get the two of you together. You'd hit it of splendidly.
+Perhaps we could meet at random in the park on day ...
+
+Achilles: Capital suggestion! I'll be looking forward to it. But you were going to tell me
+about your weird "music to smash phone graphs by", weren't you?
+
+Tortoise: Oh, yes. Well, you see, the Crab came over to visit one day. You must
+understand that he's always had a weakness for fang gadgets, and at that time he
+was quite an aficionado for, of al things, record players. He had just bought his
+first record player, and being somewhat gullible, believed every word the
+salesman had told him about it-in particular, that it was capable of reproducing
+any and all sounds. In short, he was convinced that it was a Perfect phonograph.
+
+
+Contracrostipunctus
+
+
+75
+
+
+
+Achilles: Naturally, I suppose you disagreed.
+
+Tortoise: True, but he would hear nothing of my arguments. He staunchly maintained that
+any sound whatever was reproducible on his machine. Since I couldn't convince
+him of the contrary, I left it at that. But not long after that, I returned the visit,
+taking with me a record of a song which I had myself composed. The song was
+called "I Cannot Be Played on Record Player 1".
+
+Achilles: Rather unusual. Was it a present for the Crab?
+
+Tortoise: Absolutely. I suggested that we listen to it on his new phonograph, and he was
+very glad to oblige me. So he put it on. But unfortunately, after only a few notes,
+the record player began vibrating rather severely, and then with a loud "pop",
+broke into a large number of fairly small pieces, scattered all about the room. The
+record was utterly destroyed also, needless to say.
+
+Achilles: Calamitous blow for the poor fellow, I'd say. What was the matter with his
+record player?
+
+Tortoise: Really, there was nothing the matter, nothing at all. It simply couldn't reproduce
+the sounds on the record which I had brought him, because they were sounds that
+would make it vibrate and break.
+
+Achilles: Odd, isn't it? I mean, I thought it was a Perfect phonograph. That's what the
+salesman had told him, after all.
+
+Tortoise: Surely, Achilles, you don't believe everything that salesmen tell you! Are you
+as naive as the Crab was?
+
+Achilles: The Crab was naiver by far! I know that salesmen are notorious prevaricators. I
+wasn't born yesterday!
+
+Tortoise: In that case, maybe you can imagine that this particular salesman had somewhat
+exaggerated the quality of the Crab's piece of equipment ... perhaps it was indeed
+less than Perfect, and could not reproduce every possible sound.
+
+Achilles: Perhaps that is an explanation. But there's no explanation for the amazing
+coincidence that your record had those very sounds on it...
+
+Tortoise: Unless they got put there deliberately. You see, before returning the Crab's
+visit, I went to the store where the Crab had bought his machine, and inquired as
+to the make. Having ascertained that, I sent off to the manufacturers for a
+description of its design. After receiving that by return mail, I analyzed the entire
+construction of the phonograph and discovered a certain set of sounds which, if
+they were produced anywhere in the vicinity, would set the device to shaking and
+eventually to falling apart.
+
+Achilles: Nasty fellow! You needn't spell out for me the last details: that you recorded
+those sounds yourself, and offered the dastardly item as a gift...
+
+
+Contracrostipunctus
+
+
+76
+
+
+
+Tortoise: Clever devil! You jumped ahead of the story! But that wasn't t end of the
+adventure, by any means, for the Crab did r believe that his record player was at
+fault. He was quite stubborn. So he went out and bought a new record player, this
+o even more expensive, and this time the salesman promised give him double his
+money back in case the Crab found a soul which it could not reproduce exactly.
+So the Crab told r excitedly about his new model, and I promised to come over
+and see it.
+
+Achilles: Tell me if I'm wrong-I bet that before you did so, you on again wrote the
+manufacturer, and composed and recorded new song called "I Cannot Be Played
+on Record Player based on the construction of the new model.
+
+Tortoise: Utterly brilliant deduction, Achilles. You've quite got the spirit.
+
+Achilles: So what happened this time?
+
+Tortoise: As you might expect, precisely the same thing. The phonograph fell into
+innumerable pieces, and the record was shattered. Achilles: Consequently, the
+Crab finally became convinced that there could be no such thing as a Perfect
+record player.
+
+Tortoise: Rather surprisingly, that's not quite what happened. He was sure that the next
+model up would fill the bill, and having twice the money, h e—
+
+Achilles: Oho-I have an idea! He could have easily outwitted you, I obtaining a LOW-
+fidelity phonograph-one that was not capable of reproducing the sounds which
+would destroy it. In that way, he would avoid your trick.
+
+Tortoise: Surely, but that would defeat the-original purpose-namely, to have a
+phonograph which could reproduce any sound whatsoever, even its own self¬
+breaking sound, which is of coup impossible.
+
+Achilles: That's true. I see the dilemma now. If any record player-si
+
+Record Player X-is sufficiently high-fidelity, then when attempts to play the song "I
+Cannot Be Played on Record Player X", it will create just those vibrations which
+will cause to break. .. So it fails to be Perfect. And yet, the only way to g, around
+that trickery, namely for Record Player X to be c lower fidelity, even more
+directly ensures that it is not Perfect It seems that every record player is
+vulnerable to one or the other of these frailties, and hence all record players are
+defective.
+
+Tortoise: I don't see why you call them "defective". It is simply an inherent fact about
+record players that they can't do all that you might wish them to be able to do. But
+if there is a defect anywhere, is not in THEM, but in your expectations of what
+they should b able to do! And the Crab was just full of such unrealistic
+expectations.
+
+
+Contracrostipunctus
+
+
+77
+
+
+
+Achilles: Compassion for the Crab overwhelms me. High fidelity or low fidelity, he loses
+either way.
+
+Tortoise: And so, our little game went on like *_his for a few more rounds, and
+eventually our friend tried to become very smart. He got wind of the principle
+upon which I was basing my own records, and decided to try to outfox me. He
+wrote to the phonograph makers, and described a device of his own invention,
+which they built to specification. He called it "Record Player Omega". It was
+considerably more sophisticated than an ordinary record player.
+
+Achilles: Let me guess how: Did it have no of cotton? Or-
+
+Tortoise: Let me tell you, instead. That will save some time. In the first place, Record
+Player Omega incorporated a television camera whose purpose it was to scan any
+record before playing it. This camera was hooked up to a small built-in computer,
+which would determine exactly the nature of the sounds, by looking at the groove-
+patterns.
+
+Achilles: Yes, so far so good. But what could Record Player Omega do with this
+information?
+
+Tortoise: By elaborate calculations, its little computer figured out what effects the sounds
+would have upon its phonograph. If it deduced that the sounds were such that they
+would cause the machine in its present configuration to break, then it did
+something very clever. Old Omega contained a device which could disassemble
+large parts of its phonograph subunit, and rebuild them in new ways, so that it
+could, in effect, change its own structure. If the sounds were "dangerous", a new
+configuration was chosen, one to which the sounds would pose no threat, and this
+new configuration would then be built by the rebuilding subunit, under direction
+of the little computer. Only after this rebuilding operation would Record Player
+Omega attempt to play the record.
+
+Achilles: Aha! That must have spelled the end of your tricks. I bet you were a little
+disappointed.
+
+Tortoise: Curious that you should think so ... I don't suppose that you know Godel's
+Incompleteness Theorem backwards and forwards, do you?
+
+Achilles: Know WHOSE Theorem backwards and forwards? I've
+
+heard of anything that sounds like that. I'm sure it's fascinating, but I'd rather hear more
+about "music to break records by". It's an amusing little story. Actually, I guess I
+can fill in the end. Obviously, there was no point in going on, and so you
+sheepishly admitted defeat, and that was that. Isn't that exactly it?
+
+Tortoise: What! It's almost midnight! I'm afraid it's my bedtime. I'd love to talk some
+more, but really I am growing quite sleepy.
+
+
+Contracrostipunctus
+
+
+78
+
+
+
+Achilles: As am 1. Well, 1 u be on my way. (As he reaches the door, he suddenly stops,
+and turns around.) Oh, how silly of me! I almost forgo brought you a little
+present. Here. (Hands the Tortoise a small neatly wrapped package.)
+
+Tortoise: Really, you shouldn't have! Why, thank you very much indeed think I'll open it
+now. ( Eagerly tears open the package, and ins discovers a glass goblet.) Oh, what
+an exquisite goblet! Did y know that I am quite an aficionado for, of all things, gl
+goblets?
+
+Achilles: Didn't have the foggiest. What an agreeable coincidence!
+
+Tortoise: Say, if you can keep a secret, I'll let you in on something: I trying to find a
+Perfect goblet: one having no defects of a sort in its shape. Wouldn't it be
+something if this goblet-h call it "G"-were the one? Tell me, where did you come
+across Goblet G?
+
+Achilles: Sorry, but that's MY little secret. But you might like to know w its maker is.
+
+Tortoise: Pray tell, who is it?
+
+Achilles: Ever hear of the famous glassblower Johann Sebastian Bach? Well, he wasn't
+exactly famous for glassblowing-but he dabbled at the art as a hobby, though
+hardly a soul knows it-a: this goblet is the last piece he blew.
+
+Tortoise: Literally his last one? My gracious. If it truly was made by Bach its value is
+inestimable. But how are you sure of its maker
+
+Achilles: Look at the inscription on the inside-do you see where tletters 'B', 'A', 'C', 'H'
+have been etched?
+
+Tortoise: Sure enough! What an extraordinary thing. ( Gently sets Goblet G down on a
+shelf.) By the way, did you know that each of the four letters in\Bach's name is
+the name of a musical note?
+
+Achilles:' tisn't possible, is it? After all, musical notes only go from ‘A’ through 'G'.
+
+Tortoise: Just so; in most countries, that's the case. But in Germany, Bach’s own
+homeland, the convention has always been similar, except that what we call 'B',
+they call 'H', and what we call 'B-flat', they call 'B'. Lor instance, we talk about
+Bach's "Mass in B Minor whereas they talk about his "H-moll Messe". Is that
+clear?
+
+Achilles: ... hmm ... I guess so. It's a little confusing: H is B, and B B-flat. I suppose
+
+his name actually constitutes a melody, then
+
+Tortoise: Strange but true. In fact, he worked that melody subtly into or of his most
+elaborate musical pieces-namely, the final Contrapunctus in his Art of the Fugue.
+It was the last fugue Bach ever wrote. When I heard it for the first time, I had no
+idea how would end. Suddenly, without warning, it broke off. And the ... dead
+silence. I realized immediately that was where Bach died. It is an indescribably
+sad moment, and the effect it had o me was-shattering. In any case, B-A-C-H is
+the last theme c that fugue. It is hidden inside the piece. Bach didn't point it out
+
+
+Contracrostipunctus
+
+
+79
+
+
+
+
+FIGURE 19. The last page of Bach's Art of the Fugue. In the original manuscript, in the
+handwriting of Bach's son Carl Philipp Emanuel, is written: "N.B. In the course of this
+fugue, at the point where the name B.A.C.H. was brought in as countersubject, the
+composer died.” (B-A-C-H in box.) I have let this final page of Bach's last fugue serve as
+an epitaph.
+
+[Music Printed by Donald Byrd's program "SMUT”, developed at Indiana University]
+
+
+Contracrostipunctus
+
+
+80
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+
+
+Explicitly, but if you know about it, you can find it without much trouble. Ah, me-there
+are so many clever ways of hiding things in music .. .
+
+Achilles: . . or in poems. Poets used to do very similar things, you know (though it's
+rather out of style these days). For instance, Lewis Carroll often hid words and
+names in the first letters (or characters) of the successive lines in poems he wrote.
+Poems which conceal messages that way are called "acrostics".
+
+Tortoise: Bach, too, occasionally wrote acrostics, which isn't surprising. After all,
+counterpoint and acrostics, with their levels of hidden meaning, have quite a bit in
+common. Most acrostics, however, have only one hidden level-but there is no
+reason that one couldn't make a double-decker-an acrostic on top of an acrostic.
+Or one could make a "contracrostic"-where the initial letters, taken in reverse
+order, form a message. Heavens! There's no end to the possibilities inherent in the
+form. Moreover, it's not limited to poets; anyone could write acrostics-even a
+dialogician.
+
+Achilles: A dial-a-logician? That's a new one on me.
+
+Tortoise: Correction: I said "dialogician", by which I meant a writer of dialogues. Hmm
+... something just occurred to me. In the unlikely event that a dialogician should
+write a contrapuntal acrostic in homage to J. S. Bach, do you suppose it would be
+more proper for him to acrostically embed his OWN name-or that of Bach? Oh,
+well, why worry about such frivolous matters? Anybody who wanted to write
+such a piece could make up his own mind. Now getting back to Bach's melodic
+name, did you know that the melody B-A-C-H, if played upside down and
+backwards, is exactly the same as the original?
+
+Achilles: How can anything be played upside down? Backwards, I can see-you get H-C-
+A-B-but upside down? You must be pulling my leg.
+
+Tortoise: ' pon my word, you're quite a skeptic, aren't you? Well, I guess I'll have to give
+you a demonstration. Let me just go and fetch my fiddle- (Walks into the next
+room, and returns in a jiffy with an ancient-looking violin.) -and play it for you
+forwards and backwards and every which way. Let's see, now ... (Places his copy
+of the Art of the Fugue on his music stand and opens it to the last page.) ... here's
+the last Contrapunctus, and here's the last theme ...
+
+The Tortoise begins to play: B-A-C- - but as he bows the final H, suddenly,
+without warning, a shattering sound rudely interrupts his performance. Both
+he and Achilles spin around, just in time to catch a glimpse of myriad
+fragments of glass tinkling to the floor from the shelf where Goblet G had
+stood, only moments before. And then ... dead silence.
+
+
+Contracrostipunctus
+
+
+81
+
+
+
+Chapter IV
+
+
+Consistency, Completeness,
+and Geometry
+
+Implicit and Explicit Meaning
+
+IN CHAPTER II, we saw how meaning-at least in the relatively simple context of formal
+systems-arises when there is an isomorphism between rule-governed symbols, and things
+in the real world. The more complex the isomorphism, in general, the more "equipment"-
+both hardware and software-is required to extract the meaning from the symbols. If an
+isomorphism is very simple (or very familiar), we are tempted to say that the meaning
+which it allows us to see is explicit. We see the meaning without seeing the isomorphism.
+The most blatant example is human language, where people often attribute meaning to
+words in themselves, without being in the slightest aware of the very complex
+"isomorphism" that imbues them with meanings. This is an easy enough error to make. It
+attributes all the meaning to the object (the word), rather than to the link between that
+object and the real world. You might compare it to the naive belief that noise is a
+necessary side effect of any collision of two objects. This is a false belief; if two objects
+collide in a vacuum, there will be no noise at all. Here again, the error stems from
+attributing the noise exclusively to the collision, and not recognizing the role of the
+medium, which carries it from the objects to the ear.
+
+Above, I used the word "isomorphism" in quotes to indicate that it must be taken
+with a grain of salt. The symbolic processes which underlie the understanding of human
+language are so much more complex than the symbolic processes in typical formal
+systems, that, if we want to continue thinking of meaning as mediated by isomorphisms,
+we shall have to adopt a far more flexible conception of what isomorphisms can be than
+we have up till now. In my opinion, in fact, the key element in answering the question
+"What is consciousness?" will be the unraveling of the nature of the "isomorphism"
+which underlies meaning.
+
+Explicit Meaning of the Contracrostipunctus
+
+All this is by way of preparation for a discussion of the Contracrostipunctus-a study in
+levels of meaning. The Dialogue has both explicit and implicit meanings. Its most
+explicit meaning is simply the story
+
+
+Consistency, Completeness, and Geometry
+
+
+82
+
+
+
+Which was related. This “explicit meaning is, strictly speaking extremely implicit , in the
+sense that the brain processes required to understand the events in the story, given only
+the black marks on paper, are incredibly complex. Nevertheless, we shall consider the
+events in the story to be the explicit meaning of the Dialogue, and assume that every
+reader of English uses more or less the same "isomorphism" in sucking that meaning
+from the marks on the paper.
+
+Even so, I'd like to be a little more explicit about the explicit meaning of the story.
+First I'll talk about the record players and the records. The main point is that there are two
+levels of meaning for the grooves in the records. Level One is that of music. Now what is
+"music"-a sequence of vibrations in the air, or a succession of emotional responses in a
+brain? It is both. But before there can be emotional responses, there have to be vibrations.
+Now the vibrations get "pulled" out of the grooves by a record player, a relatively
+straightforward device; in fact you can do it with a pin, just pulling it down the grooves.
+After this stage, the ear converts the vibrations into firings of auditory neurons in the
+brain. Then ensue a number of stages in the brain, which gradually transform the linear
+sequence of vibrations into a complex pattern of interacting emotional responses-far too
+complex for us to go into here, much though I would like to. Let us therefore content
+ourselves with thinking of the sounds in the air as the "Level One" meaning of the
+grooves.
+
+What is the Level Two meaning of the grooves? It is the sequence of vibrations induced
+in the record player. This meaning can only arise after the Level One meaning has been
+pulled out of the grooves, since the vibrations in the air cause the vibrations in the
+phonograph. Therefore, the Level Two meaning depends upon a chain of two
+isomorphisms:
+
+(1) Isomorphism between arbitrary groove patterns and air
+vibrations;
+
+(2) Isomorphism between graph vibrations, arbitrary air
+vibrations and phonograph vibrations
+
+This chain of two isomorphisms is depicted in Figure 20. Notice that isomorphism I is the
+one which gives rise to the Level One meaning. The Level Two meaning is more implicit
+than the Level One meaning, because it is mediated by the chain of two isomorphisms. It
+is the Level Two meaning which "backfires", causing the record player to break apart.
+What is of interest is that the production of the Level One meaning forces the production
+of the Level Two meaning simultaneously-there is no way to have Level One without
+Level Two. So it was the implicit meaning of the record which turned back on it, and
+destroyed it.
+
+Similar comments apply to the goblet. One difference is that the mapping from
+letters of the alphabet to musical notes is one more level of isomorphism, which we could
+call "transcription". That is followed by "translation"-conversion of musical notes into
+musical sounds. Thereafter, the vibrations act back on the goblet just as they did on the
+escalating series of phonographs.
+
+
+Consistency, Completeness, and Geometry
+
+
+83
+
+
+
+
+FIGURE 20. Visual rendition of the principle underlying Gide/'s Theorem: two back-to-
+back mappings which have an unexpected boomer anging effect. The first is from groove
+patterns to sounds, carried out by a phonograph. The second-familiar, but usually
+ignored — is from sounds to vibrations of the phonograph. Note that the second mapping
+exists independently of the first one, for any sound in the vicinity, not just ones produced
+by the phonograph itself, will cause such vibrations. The paraphrase of Gidel’s Theorem
+says that for any record player, there are records which it cannot play because they will
+cause its indirect self-destruction. I Drawing by the author.
+
+
+Implicit Meanings of the Contracrostipunctus
+
+What about implicit meanings of the Dialogue? (Yes, it has more than one of these.) The
+simplest of these has already been pointed out in the paragraphs above-namely, that the
+events in the two halves of the dialogue are roughly isomorphic to each other: the
+phonograph becomes a violin, the Tortoise becomes Achilles, the Crab becomes the
+Tortoise, the grooves become the etched autograph, etc. Once you notice this simple
+isomorphism, you can go a little further. Observe that in the first half of the story, the
+Tortoise is the perpetrator of all the mischief, while in the second half, he is the victim.
+What do you know, but his own method has turned around and backfired on him!
+Reminiscent of the backfiring of the records' muusic-or the goblet's inscription-or perhaps
+of the Tortoise's boomerang collection? Yes, indeed. The story is about backf uing on two
+levels, as follows ...
+
+Level One: Goblets and records which backfire;
+
+Level Two: The Tortoise’s devilish method of exploiting implicit meaning to
+cause backfires-which backfires.
+
+Therefore we can even make an isomorphism between the two levels of the story,
+in which we equate the way in which the records and goblet boomerang back to destroy
+themselves, with the way in which the Tortoise's own fiendish method boomerangs back
+to get him in the end. Seen this
+
+
+Consistency, Completeness, and Geometry
+
+
+84
+
+
+
+
+
+
+
+
+
+
+
+
+way, the story itself is an example of the backfirings which it discusses. So we can think
+of the Contracrostipunctus as referring to itself indirectly that its own structure is
+isomorphic to the events it portrays. (Exactly goblet and records refer implicitly to
+themselves via the back-to-back morphisms of playing and vibration-causing.) One may
+read the Dialogue without perceiving this fact, of course-but it is there all the time.
+
+Mapping Between the Contracrostipunctus
+and Godel’s Theorem
+
+Now you may feel a little dizzy-but the best is yet to come. (Actually, levels of
+implicit meaning will not even be discussed here-they will 1 for you to ferret out.) The
+deepest reason for writing this Dialogue illustrate Godel’s Theorem, which, as I said in
+the Introduction, heavily on two different levels of meaning of statements of number tl
+Each of the two halves of the Contracrostipunctus is an "isomorphic co Godel’s Theorem.
+Because this mapping is the central idea of the Dialogue and is rather elaborate, I have
+carefully charted it out below.
+
+Phonograph <= =>axiomatic system for number theory
+
+low-fidelity phonograph <= =>"weak" axiomatic system
+
+high-fidelity phonograph <= =>"strong" axiomatic system
+
+"Perfect" phonograph" <= => complete system for number theory'
+
+Blueprint" of phonograph <= => axioms and rules of formal system
+
+record <= => string of the formal system
+
+playable record<= => theorem of the axiomatic system
+
+unplayable record <= =>nontheorem of the axiomatic system
+
+sound <= =>true statement of number theory
+
+reproducible sound <= => 'interpreted theorem of the system
+
+unreproducible sound <= => true statement which isn't a theorem:
+
+song title <= =>implicit meaning of Godel’s string:
+
+"I Cannot Be Played "I Cannot Be Derived
+
+on Record Player X" in Formal System X"
+
+This is not the full extent of the isomorphism between Godel’s theorem and the
+Contracrostipunctus, but it is the core of it. You need not if you don't fully grasp Godel’s
+Theorem by now-there are still Chapters to go before we reach it! Nevertheless, having
+read this Dialogue you have already tasted some of the flavor of Godel’s Theorem
+without necessarily being aware of it. I now leave you to look for any other types of
+implicit meaning in the Contracrostipunctus. "Quaerendo invenietis!"
+
+
+Consistency, Completeness, and Geometry
+
+
+85
+
+
+
+The Art of the Fugue
+
+
+A few words on the Art of the Fugue ... Composed in the last year of Bach's life, it is a
+collection of eighteen fugues all based on one theme. Apparently, writing the Musical
+Offering was an inspiration to Bach. He decided to compose another set of fugues on a
+much simpler theme, to demonstrate the full range of possibilities inherent in the form. In
+the Art of the Fugue, Bach uses a very simple theme in the most complex possible ways.
+The whole work is in a single key. Most of the fugues have four voices, and they
+gradually increase in complexity and depth of expression. Toward the end, they soar to
+such heights of intricacy that one suspects he can no longer maintain them. Yet he does . .
+. until the last Contrapunctus.
+
+The circumstances which caused the break-off of the Art of the Fugue (which is to
+say, of Bach's life) are these: his eyesight having troubled him for years, Bach wished to
+have an operation. It was done; however, it came out quite poorly, and as a consequence,
+he lost his sight for the better part of the last year of his life. This did not keep him from
+vigorous work on his monumental project, however. His aim was to construct a complete
+exposition of fugal writing, and usage of multiple themes was one important facet of it. In
+what he planned as the next-to-last fugue, he inserted his own name coded into notes as
+the third theme. However, upon this very act, his health became so precarious that he was
+forced to abandon work on his cherished project. In his illness, he managed to dictate to
+his son-in-law a final chorale prelude, of which Bach's biographer Forkel wrote, "The
+expression of pious resignation and devotion in it has always affected me whenever I
+have played it; so that I can hardly say which I would rather miss-this Chorale, or the end
+of the last fugue."
+
+One day, without warning, Bach regained his vision. But a few hours later, he
+suffered a stroke; and ten days later, he died, leaving it for others to speculate on the
+incompleteness of the Art of the Fugue. Could it have been caused by Bach's attainment
+of self-reference?
+
+
+Problems Caused by Godel’s Result
+
+The Tortoise says that no sufficiently powerful record player can be perfect, in the
+sense of being able to reproduce every possible sound from a record. Godel says that no
+sufficiently powerful formal system can be perfect, in the sense of reproducing every
+single true statement as a theorem. But as the Tortoise pointed out with respect to
+phonographs, this fact only seems like a defect if you have unrealistic expectations of
+what formal systems should be able to do. Nevertheless, mathematicians began this
+century with just such unrealistic expectations, thinking that axiomatic reasoning was the
+cure to all ills. They found out otherwise in 1931. The fact that truth transcends
+theoremhood, in any given formal system, is called "incompleteness" of that system.
+
+A most puzzling fact about Godel’s method of proof is that he uses
+
+
+Consistency, Completeness, and Geometry
+
+
+86
+
+
+
+reasoning methods which seemingly cannot be "encapsulated"-they re being incorporated
+into any formal system. Thus, at first sight, it seems that Godel has unearthed a hitherto
+unknown, but deeply significant, difference between human reasoning and mechanical
+reasoning. This mysterious discrepancy in the power of living and nonliving systems is
+mirrored in the discrepancy between the notion of truth, and that of theoremhood or at
+least that is a "romantic" way to view the situation.
+
+The Modified pq-System and Inconsistency
+
+In order to see the situation more realistically, it is necessary to see in, depth why
+and how meaning is mediated, in formal systems, by isomorphisms. And I believe that
+this leads to a more romantic way to view i situation. So we now will proceed to
+investigate some further aspects of 1 relation between meaning and form. Our first step is
+to make a new formal system by modifying our old friend, the pq-system, very slightly.
+We a one more axiom schema (retaining the original one, as well as the sin rule of
+inference):
+
+Axiom SCHEMA II: If x is a hyphen-string, then xp-qx is an axiom.
+
+Clearly, then, —p-q— is a theorem in the new system, and so —p—q—. And yet, their
+interpretations are, respectively, "2 plus; equals 2", and "2 plus 2 equals 3". It can be seen
+that our new system contain a lot of false statements (if you consider strings to be
+statement Thus, our new system is inconsistent with the external world.
+
+As if this weren't bad enough, we also have internal problems with < new system,
+since it contains statements which disagree with one another such as -p-q- (an old
+axiom) and -p-q- (a new axiom). So our system is inconsistent in a second sense:
+internally.
+
+Would, therefore, the only reasonable thing to do at this point be drop the new
+system entirely? Hardly. I have deliberately presented the "inconsistencies" in a wool¬
+pulling manner: that is, I have tried to press fuzzy-headed arguments as strongly as
+possible, with the purpose of n leading. In fact, you may well have detected the fallacies
+in what I hi said. The crucial fallacy came when I unquestioningly adopted the very same
+interpreting words for the new system as I had for the old of Remember that there was
+only one reason for adopting those words in I last Chapter, and that reason was that the
+symbols acted isomorphically to concepts which they were matched with, by the
+interpretation. But when y modify the rules governing the system, you are bound to
+damage t isomorphism. It just cannot be helped. Thus all the problems which we
+lamented over in preceding paragraphs were bogus problems; they can made to vanish in
+no time, by suitably reinterpreting some of the symbols of system. Notice that I said
+"some"; not necessarily all symbols will have to mapped onto new notions. Some may
+very well retain their "meaning while others change.
+
+
+Consistency, Completeness, and Geometry
+
+
+87
+
+
+
+Suppose, for instance, that we reinterpret just the symbol q, leaving all the others
+constant; in particular, interpret q by the phrase "is greater than or equal to". Now, our
+"contradictory" theorems -p-q-and -p-q—come out harmlessly as: "1 plus 1 is greater than
+or equal to 1", and "1 plus 1 is greater than or equal to 2". We have simultaneously gotten
+rid of (1) the inconsistency with the external world, and (2) the internal inconsistency.
+And our new interpretation is a meaningful interpretation; of course the original one is
+meaningless. That is, it is meaningless for the new system ; for the original pq-system, it is
+fine. But it now seems as pointless and arbitrary to apply it to the new pq-system as it
+was to apply the "horse-apple-happy" interpretation to the old pq-system.
+
+The History of Euclidean Geometry
+
+Although I have tried to catch you off guard and surprise you a little, this lesson
+about how to interpret symbols by words may not seem terribly difficult once you have
+the hang of it. In fact, it is not. And yet it is one of the deepest lessons of all of nineteenth
+century mathematics! It all begins with Euclid, who, around 300 B.C., compiled and
+systematized all of what was known about plane and solid geometry in his day. The
+resulting work, Euclid's Elements, was so solid that it was virtually a bible of geometry
+for over two thousand years-one of the most enduring works of all time. Why was this
+so?
+
+The principal reason was that Euclid was the founder of rigor in mathematics. The
+Elements began with very simple concepts, definitions, and so forth, and gradually built
+up a vast body of results organized in such a way that any given result depended only on
+foregoing results. Thus, there was a definite plan to the work, an architecture which made
+it strong and sturdy.
+
+Nevertheless, the architecture was of a different type from that of, say, a
+skyscraper. (See Fig. 21.) In the latter, that it is standing is proof enough that its structural
+elements are holding it up. But in a book on geometry, when each proposition is claimed
+to follow logically from earlier propositions, there will be no visible crash if one of the
+proofs is invalid. The girders and struts are not physical, but abstract. In fact, in Euclid's
+Elements, the stuff out of which proofs were constructed was human language-that
+elusive, tricky medium of communication with so many hidden pitfalls. What, then, of
+the architectural strength of the Elements? Is it certain that it is held up by solid structural
+elements, or could it have structural weaknesses?
+
+Every word which we use has a meaning to us, which guides us in our use of it.
+The more common the word, the more associations we have with it, and the more deeply
+rooted is its meaning. Therefore, when someone gives a definition for a common word in
+the hopes that we will abide by that
+
+
+Consistency, Completeness, and Geometry
+
+
+88
+
+
+
+
+FIGURE 21. Tower of Babel, by M. C. Escher (woodcut, 1928).
+
+
+Consistency, Completeness, and Geometry
+
+
+89
+
+
+
+
+
+
+definition, it is a foregone conclusion that we will not do so but will instead be guided,
+largely unconsciously, by what our minds find in their associative stores. I mention this
+because it is the sort of problem which Euclid created in his Elements, by attempting to
+give definitions of ordinary, common words such as "point", "straight line", "circle", and
+so forth. How can you define something of which everyone already has a clear concept?
+The only way is if you can make it clear that your word is supposed to be a technical
+term, and is not to be confused with the everyday word with the same spelling. You have
+to stress that the connection with the everyday word is only suggestive. Well, Euclid did
+not do this, because he felt that the points and lines of his Elements were indeed the
+points and lines of the real world. So by not making sure that all associations were
+dispelled, Euclid was inviting readers to let their powers of association run free ...
+
+This sounds almost anarchic, and is a little unfair to Euclid. He did set down
+axioms, or postulates, which were supposed to be used in the proofs of propositions. In
+fact, nothing other than those axioms and postulates was supposed to be used. But this is
+where he slipped up, for an inevitable consequence of his using ordinary words was that
+some of the images conjured up by those words crept into the proofs which he created.
+However, if you read proofs in the Elements, do not by any means expect to find glaring
+"jumps" in the reasoning. On the contrary, they are very subtle, for Euclid was a
+penetrating thinker, and would not have made any simpleminded errors. Nonetheless,
+gaps are there, creating slight imperfections in a classic work. But this is not to be
+complained about. One should merely gain an appreciation for the difference between
+absolute rigor and relative rigor. In the long run, Euclid's lack of absolute rigor was the
+cause of some of the most fertile path-breaking in mathematics, over two thousand years
+after he wrote his work.
+
+Euclid gave five postulates to be used as the "ground story" of the infinite
+skyscraper of geometry, of which his Elements constituted only the first several hundred
+stories. The first four postulates are rather terse and elegant:
+
+(1) A straight line segment can be drawn joining any two points.
+
+(2) Any straight line segment can be extended indefinitely in a straight line.
+
+(3) Given any straight line segment, a circle can be drawn having the segment as
+
+radius and one end point as center.
+
+(4) All right angles are congruent.
+
+The fifth, however, did not share their grace:
+
+(5) If two lines are drawn which intersect a third in such a way that the sum of the
+
+inner angles on one side is less than two right angles, then the two lines
+inevitably must intersect each other on that side if extended far enough
+
+
+Consistency, Completeness, and Geometry
+
+
+90
+
+
+
+Though he never explicitly said so, Euclid considered this postulate to be somehow
+inferior to the others, since he managed to avoid using it in t proofs of the first twenty-
+eight propositions. Thus, the first twenty-eight propositions belong to what might be
+called "four-postulate geometry" that part of geometry which can be derived on the basis
+of the first to postulates of the Elements, without the help of the fifth postulate. (It is al
+often called absolute geometry.) Certainly Euclid would have found it 1 preferable to
+prove this ugly duckling, rather than to have to assume it. B he found no proof, and
+therefore adopted it.
+
+But the disciples of Euclid were no happier about having to assume this fifth
+postulate. Over the centuries, untold numbers of people ga untold years of their lives in
+attempting to prove that the fifth postulate s itself part of four-postulate geometry. By
+1763, at least twenty-eight deficient proofs had been published-all erroneous! (They were
+all criticized the dissertation of one G. S. Klugel.) All of these erroneous proofs involve a
+confusion between everyday intuition and strictly formal properties. It safe to say that
+today, hardly any of these "proofs" holds any mathematic or historical interest-but there
+are certain exceptions.
+
+
+The Many Faces of Noneuclid
+
+Girolamo Saccheri (1667-1733) lived around Bach's time. He had t ambition to
+free Euclid of every flaw. Based on some earlier work he h; done in logic, he decided to
+try a novel approach to the proof of the famous fifth: suppose you assume its opposite;
+then work with that as your fif postulate ... Surely after a while you will create a
+contradiction. Since i mathematical system can support a contradiction, you will have
+shown t unsoundness of your own fifth postulate, and therefore the soundness Euclid's
+fifth postulate. We need not go into details here. Suffice it to s that with great skill,
+Saccheri worked out proposition after proposition "Saccherian geometry" and eventually
+became tired of it. At one point, decided he had reached a proposition which was
+"repugnant to the nature of the straight line". That was what he had been hoping for-to his
+mind was the long-sought contradiction. At that point, he published his work under the
+title Euclid Freed of Every Flaw, and then expired.
+
+But in so doing, he robbed himself of much posthumous glory, sir he had
+unwittingly discovered what came later to be known as "hyperbolic geometry". Fifty
+years after Saccheri, J. H. Lambert repeated the "near miss", this time coming even
+closer, if possible. Finally, forty years after Lambert, and ninety years after Saccheri,
+non-Euclidean geometry was recognized for what it was-an authentic new brand of
+geometry, a bifurcation the hitherto single stream of mathematics. In 1823, non-
+Euclidean geometry was discovered simultaneously, in one of those inexplicable
+coincidences, by a Hungarian mathematician, Janos (or Johann) Bolyai, age twenty-one,
+and a Russian mathematician, Nikolay Lobachevskiy, ag thirty. And, ironically, in that
+same year, the great French mathematician
+
+
+Consistency, Completeness, and Geometry
+
+
+91
+
+
+
+Adrien-Marie Legendre came up with what he was sure was a proof of Euclid's fifth
+postulate, very much along the lines of Saccheri.
+
+Incidentally, Bolyai's father, Farkas (or Wolfgang) Bolyai, a close friend of the
+great Gauss, invested much effort in trying to prove Euclid's fifth postulate. In a letter to
+his son Janos, he tried to dissuade him from thinking about such matters:
+
+You must not attempt this approach to parallels. I know this way to its very end. I have
+traversed this bottomless night, which extinguished all light and joy of my life. I entreat
+you, leave the science of parallels alone.... I thought 1 would sacrifice myself for the sake
+of the truth. I was ready to become a martyr who would remove the flaw from geometry
+and return it purified to mankind. I accomplished monstrous, enormous labors; my
+creations are far better than those of others and yet I have not achieved complete
+satisfaction. For here it is true that si paullum a summo discessit, vergit ad imum. I turned
+back when I saw that no man can reach the bottom of this night. I turned back nnconsoled,
+pitying myself and all mankind.... I have traveled past all reefs of this infernal Dead Sea
+and have always come back with broken mast and tom sail. The ruin of my disposition and
+my fall date back to this time. I thoughtlessly risked my life and happiness sut Caesar aut
+nihil.'
+
+But later, when convinced his son really "had something", he urged him to
+publish it, anticipating correctly the simultaneity which is so frequent in scientific
+discovery:
+
+When the time is ripe for certain things, these things appear in different places in
+the manner of violets coming to light in early spring.
+
+How true this was in the case of non-Euclidean geometry! In Germany, Gauss
+himself and a few others had more or less independently hit upon non-Euclidean ideas.
+These included a lawyer, F. K. Schweikart, who in 1818 sent a page describing a new
+"astral" geometry to Gauss; Schweikart's nephew, F. A. Taurinus, who did non-Euclidean
+trigonometry; and F. L. Wachter, a student of Gauss, who died in 1817, aged twenty-five,
+having found several deep results in non-Euclidean geometry.
+
+The clue to non-Euclidean geometry was "thinking straight" about the
+propositions which emerge in geometries like Saccheri's and Lambert's. The Saccherian
+propositions are only "repugnant to the nature of the straight line" if you cannot free
+yourself of preconceived notions of what "straight line" must mean. If, however, you can
+divest yourself of those preconceived images, and merely let a "straight line" be
+something which satisfies the new propositions, then you have achieved a radically new
+viewpoint.
+
+
+Undefined Terms
+
+This should begin to sound familiar. In particular, it harks back to the pq-system, and its
+variant, in which the symbols acquired passive meanings by virtue of their roles in
+theorems. The symbol q is especially interesting,
+
+
+Consistency, Completeness, and Geometry
+
+
+92
+
+
+
+since its "meaning" changed when a new axiom schema was added. In the very same
+way, one can let the meanings of "point", "line", and so on I determined by the set of
+theorems (or propositions) in which they occur. This was th great realization of the
+discoverers of non-Euclidean geometry. The found different sorts of non-Euclidean
+geometries by denying Euclid's fifth postulate in different ways and following out the
+consequences. Strict] speaking, they (and Saccheri) did not deny the fifth postulate
+directly, but rather, they denied an equivalent postulate, called the parallel postulate,
+which runs as follows:
+
+Given any straight line, and a point not on it, there exists one, and only one, straight
+line which passes through that point and never intersects the first line, no matter
+how far they are extended.
+
+The second straight line is then said to be parallel to the first. If you assert that no
+such line exists, then you reach elliptical geometry, if you assert that, at east two such
+lines exist, you reach hyperbolic geometry. Incidentally, tf reason that such variations are
+still called "geometries" is that the cot element-absolute, or four-postulate, geometry-is
+embedded in them, is the presence of this minimal core which makes it sensible to think
+of the] as describing properties of some sort of geometrical space, even if the spa( is not
+as intuitive as ordinary space.
+
+Actually, elliptical geometry is easily visualized. All "points", "lines and so forth
+are to be parts of the surface of an ordinary sphere. Let t write "POINT" when the
+technical term is meant, and "point" when tl everyday sense is desired. Then, we can say
+that a POINT consists of a pa of diametrically opposed points of the sphere's surface. A
+LINE is a great circle on the sphere (a circle which, like the equator, has its center at tl
+center of the sphere). Under these interpretations, the propositions ( elliptical geometry,
+though they contain words like "POINT" and "LINE speak of the goings-on on a sphere,
+not a plane. Notice that two LINT always intersect in exactly two antipodal points of the
+sphere's surface that is, in exactly one single POINT! And just as two LINES determine
+POINT, so two POINTS determine a LINE.
+
+By treating words such as "POINT" and "LINE" as if they had only tt meaning
+instilled in them by the propositions in which they occur, we take step towards complete
+formalization of geometry. This semiformal version still uses a lot of words in English
+with their usual meanings (words such "the", ' if ", "and", "join", "have"), although the
+everyday meaning has bee drained out of special words like "POINT" and "LINE", which
+are consequently called undefined terms. Undefined terms, like the p and q of th pq-
+system, do get defined in a sense: implicitly-by the totality of all propos dons in which
+they occur, rather than explicitly, in a definition.
+
+One could maintain that a full definition of the undefined tern resides in the
+postulates alone, since the propositions which follow from them are implicit in the
+postulates already. This view would say that the postulates are implicit definitions of all
+the undefined terms, all of the undefined terms being defined in terms of the others.
+
+
+Consistency, Completeness, and Geometry
+
+
+93
+
+
+
+The Possibility of Multiple Interpretations
+
+
+A full formalization of geometry would take the drastic step of making every term
+undefined-that is, turning every term into a "meaningless" symbol of a formal system. I
+put quotes around "meaningless" because, as you know, the symbols automatically pick
+up passive meanings in accordance with the theorems they occur in. It is another
+question, though, whether people discover those meanings, for to do so requires finding a
+set of concepts which can be linked by an isomorphism to the symbols in the formal
+system. If one begins with the aim of formalizing geometry, presumably one has an
+intended interpretation for each symbol, so that the passive meanings are built into the
+system. That is what I did for p and q when I first created the pq-system.
+
+But there may be other passive meanings which are potentially perceptible, which
+no one has yet noticed. For instance, there were the surprise interpretations of p as
+"equals" and q as "taken from", in the original pq-system. Although this is rather a trivial
+example, it contains the essence of the idea that symbols may have many meaningful
+interpretations-it is up to the observer to look for them.
+
+We can summarize our observations so far in terms of the word "consistency".
+We began our discussion by manufacturing what appeared to be an inconsistent formal
+system-one which was internally inconsistent, as well as inconsistent with the external
+world. But a moment later we took it all back, when we realized our error: that we had
+chosen unfortunate interpretations for the symbols. By changing the interpretations, we
+regained consistency! It now becomes clear that consistency is not a property of a formal
+system per se, but depends on the interpretation which is proposed for it. By the same
+token, inconsistency is not an intrinsic property of any formal system.
+
+Varieties of Consistency
+
+We have been speaking of "consistency" and "inconsistency" all along, without
+defining them. We have just relied on good old everyday notions. But now let us say
+exactly what is meant by consistency of a formal system (together with an interpretation):
+that every theorem, when interpreted, becomes a true statement. And we will say that
+inconsistency occurs when there is at least one false statement among the interpreted
+theorems.
+
+This definition appears to be talking about inconsistency with the external world-
+what about internal inconsistencies? Presumably, a system would be internally
+inconsistent if it contained two or more theorems whose interpretations were
+incompatible with one another, and internally consistent if all interpreted theorems were
+compatible with one another. Consider, for example, a formal system which has only the
+following three theorems: TbZ, ZbE, and EbT. If T is interpreted as "the Tortoise", Z as
+"Zeno", E as "Egbert", and x by as "x beats y in chess always", then we have the
+following interpreted theorems:
+
+
+Consistency, Completeness, and Geometry
+
+
+94
+
+
+
+The Tortoise always beats Zeno at chess
+
+Zeno always beats Egbert at chess.
+
+Egbert always beats the Tortoise at chess.
+
+The statements are not incompatible, although they describe a rather bizarre circle of
+chess players. Hence, under this interpretation, the form; system in which those three
+strings are theorems is internally consistent although, in point of fact, none of the three
+statements is true! Intern< consistency does not require all theorems to come out true, but
+merely that they come out compatible with one another.
+
+Now suppose instead that x by is to be interpreted as "x was invented by y". Then
+we would have:
+
+The Tortoise was invented by Zeno.
+
+Zeno was invented by Egbert.
+
+Egbert was invented by the Tortoise.
+
+In this case, it doesn't matter whether the individual statements are true c false-and
+perhaps there is no way to know which ones are true, and which are not. What is
+nevertheless certain is that not all three can be true at one Thus, the interpretation makes
+the system internally inconsistent. The internal inconsistency depends not on the
+interpretations of the three capital letters, but only on that of b, and on the fact that the
+three capita are cyclically permuted around the occurrences of b. Thus, one can have
+internal inconsistency without having interpreted all of the symbols of the formal system.
+(In this case it sufficed to interpret a single symbol.) By tl time sufficiently many symbols
+have been given interpretations, it may t clear that there is no way that the rest of them
+can be interpreted so that a theorems will come out true. But it is not just a question of
+truth-it is question of possibility. All three theorems would come out false if the capitals
+were interpreted as the names of real people-but that is not why we would call the system
+internally inconsistent; our grounds for doing s would be the circularity, combined with
+the interpretation of the letter I (By the way, you'll find more on this "authorship triangle"
+in Chapter XX.;
+
+
+Hypothetical Worlds and Consistency
+
+We have given two ways of looking at consistency: the first says that system-
+plus-interpretation is consistent with the external world if every theorem comes out true
+when interpreted; the second says that a system-plus: interpretation is internally
+consistent if all theorems come out mutually compatible when interpreted. Now there is a
+close relationship between these two types of consistency. In order to determine whether
+several statements at mutually compatible, you try to imagine a world in which all of
+them could be simultaneously true. Therefore, internal consistency depends upon
+consistency with the external world-only now, "the external world" allowed to be any
+imaginable world , instead of the one we live in. But this is
+
+
+Consistency, Completeness, and Geometry
+
+
+95
+
+
+
+an extremely vague, unsatisfactory conclusion. What constitutes an “imaginable" world?
+After all, it is possible to imagine a world in which three characters invent each other
+cyclically. Or is it? Is it possible to imagine a world in which there are square circles? Is a
+world imaginable in which Newton's laws, and not relativity, hold? Is it possible to
+imagine a world in which something can be simultaneously green and not green? Or a
+world in which animals exist which are not made of cells? In which Bach improvised an
+eight-part fugue on a theme of King Frederick the Great? In which mosquitoes are more
+intelligent than people? In which tortoises can play football-or talk? A tortoise talking
+football would be an anomaly, of course.
+
+Some of these worlds seem more imaginable than others, since some seem to
+embody logical contradictions-for example, green and not green-while some of them
+seem, for want of a better word, "plausible" — such as Bach improvising an eight-part
+fugue, or animals which are not made of cells. Or even, come to think of it, a world in
+which the laws of physics are different... Roughly, then, it should be possible to establish
+different brands of consistency. For instance, the most lenient would be "logical
+consistency", putting no restraints on things at all, except those of logic. More
+specifically, a system-plus-interpretation would be logically consistent just as long as no
+two of its theorems, when interpreted as statements, directly contradict each other; and
+mathematically consistent just as long as interpreted theorems do not violate
+mathematics; and physically consistent just as long as all its interpreted theorems are
+compatible with physical law; then comes biological consistency, and so on. In a
+biologically consistent system, there could be a theorem whose interpretation is the
+statement "Shakespeare wrote an opera", but no theorem whose interpretation is the
+statement "Cell-less animals exist". Generally speaking, these fancier kinds of
+inconsistency are not studied, for the reason that they are very hard to disentangle from
+one another. What kind of inconsistency, for example, should one say is involved in the
+problem of the three characters who invent each other cyclically? Logical? Physical?
+Biological? Literary?
+
+Usually, the borderline between uninteresting and interesting is drawn between
+physical consistency and mathematical consistency. (Of course, it is the mathematicians
+and logicians who do the drawing-hardly an impartial crew . . .) This means that the kinds
+of inconsistency which "count", for formal systems, are just the logical and mathematical
+kinds. According to this convention, then, we haven't yet found an interpretation which
+makes the trio of theorems TbZ, ZbE, EbT inconsistent. We can do so by interpreting b
+as "is bigger than". What about T and Z and E? They can be interpreted as natural
+numbers-for example, Z as 0, T as 2, and E as 11. Notice that two theorems come out
+true this way, one false. If, instead, we had interpreted Z as 3, there would have been two
+falsehoods and only one truth. But either way, we'd have had inconsistency. In fact, the
+values assigned to T, Z, and E are irrelevant, as long as it is understood that they are
+restricted to natural numbers. Once again we see a case where only some of the
+interpretation is needed, in order to recognize internal inconsistency.
+
+
+Consistency, Completeness, and Geometry
+
+
+96
+
+
+
+Embedding of One Formal System In Another
+
+
+The preceding example, in which some symbols could have interpretations while others
+didn't, is reminiscent of doing geometry in natural languag4 using some words as
+undefined terms. In such a case, words are divide into two classes: those whose meaning
+is fixed and immutable, and, those whose meaning is to be adjusted until the system is
+consistent (these are th undefined terms). Doing geometry in this way requires that
+meanings have already been established for words in the first class, somewhere outside c
+geometry. Those words form a rigid skeleton, giving an underlying structure to the
+system; filling in that skeleton comes other material, which ca vary (Euclidean or non-
+Euclidean geometry).
+
+Formal systems are often built up in just this type of sequential, c hierarchical,
+manner. For example, Formal System I may be devised, wit rules and axioms that give
+certain intended passive meanings to its symbol Then Formal System I is incorporated
+fully into a larger system with more symbols-Formal System II. Since Formal System I's
+axioms and rules at part of Formal System II, the passive meanings of Formal System I
+symbols remain valid; they form an immutable skeleton which then plays large role in the
+determination of the passive meanings of the new symbols of Formal System II. The
+second system may in turn play the role of skeleton with respect to a third system, and so
+on. It is also possible-an geometry is a good example of this-to have a system (e.g.,
+absolute geometry) which partly pins down the passive meanings of its undefined terms,
+and which can be supplemented by extra rules or axioms, which then further restrict the
+passive meanings of the undefined terms. This the case with Euclidean versus non-
+Euclidean geometry.
+
+
+Layers of Stability in Visual Perception
+
+In a similar, hierarchical way, we acquire new knowledge, new vocabulary or
+perceive unfamiliar objects. It is particularly interesting in the case understanding
+drawings by Escher, such as Relativity (Fig. 22), in which there occur blatantly
+impossible images. You might think that we won seek to reinterpret the picture over and
+over again until we came to interpretation of its parts which was free of contradictions-
+but we dot do that at all. We sit there amused and puzzled by staircases which go eve
+which way, and by people going in inconsistent directions on a sing staircase. Those
+staircases are "islands of certainty" upon which we base of interpretation of the overall
+picture. Having once identified them, we try extend our understanding, by seeking to
+establish the relationship which they bear to one another. At that stage, we encounter
+trouble. But if i attempted to backtrack-that is, to question the "islands of certainty"-s
+would also encounter trouble, of another sort. There's no way of backtracking and
+"undeciding" that they are staircases. They are not fishes, or whip or hands-they are just
+staircases. (There is, actually, one other on t-i leave all the lines of the picture totally
+uninterpreted, like the "meaningless
+
+
+Consistency, Completeness, and Geometry
+
+
+97
+
+
+
+
+FIGURE 22. Relativity, by M. C. Escher (lithograph, 1953).
+
+symbols" of a formal system. This ultimate escape route is an example of a "U-mode"
+response-a Zen attitude towards symbolism.)
+
+So we are forced, by the hierarchical nature of our perceptive processes, to see
+either a crazy world or just a bunch of pointless lines. A similar analysis could be made
+of dozens of Escher pictures, which rely heavily upon the recognition of certain basic
+forms, which are then put together in nonstandard ways; and by the time the observer
+sees the paradox on a high level, it is too late-he can't go back and change his mind about
+how to interpret the lower-level objects. The difference between an Escher drawing and
+non-Euclidean geometry is that in the latter, comprehensible interpretations can be found
+for the undefined terms, resulting in a com
+
+
+Consistency, Completeness, and Geometry
+
+
+98
+
+
+
+
+
+
+
+
+
+
+
+
+
+prehensible total system, whereas for the former, the end result is not reconcilable with
+one's conception of the world, no matter how long or stares at the pictures. Of course, one
+can still manufacture hypothetic worlds, in which Escherian events can happen ... but in
+such worlds, tl laws of biology, physics, mathematics, or even logic will be violated on
+or level, while simultaneously being obeyed on another, which makes the: extremely
+weird worlds. (An example of this is in Waterfall (Fig. 5), whet normal gravitation
+applies to the moving water, but where the nature space violates the laws of physics.)
+
+Is Mathematics the Same in Every Conceivable World?
+
+We have stressed the fact, above, that internal consistency of a form; system (together
+with an interpretation) requires that there be some imaginable world-that is, a world
+whose only restriction is that in it, mathematics and logic should be the same as in our
+world-in which all the interpreted theorems come out true. External consistency, however
+consistency with the external world-requires that all theorems come of true in the real
+world. Now in the special case where one wishes to create consistent formal system
+whose theorems are to be interpreted as statements of mathematics, it would seem that
+the difference between the two types of consistency should fade away, since, according to
+what we sat above, all imaginable worlds have the same mathematics as the real world.
+Thus, i every conceivable world, 1 plus 1 would have to be 2; likewise, there would have
+to be infinitely many prime numbers; furthermore, in every conceivable world, all right
+angles would have to be congruent; and of cours4 through any point not on a given line
+there would have to be exactly on parallel line ...
+
+But wait a minute! That's the parallel postulate-and to assert i universality would
+be a mistake, in light of what's just been said. If in all conceivable worlds the parallel
+postulate-is obeyed, then we are asserting that non-Euclidean geometry is inconceivable,
+which puts us back in the same mental state as Saccheri and Lambert-surely an unwise
+move. But what, then, if not all of mathematics, must all conceivable worlds share ?
+Could it I as little as logic itself? Or is even logic suspect? Could there be worlds where
+contradictions are normal parts of existence-worlds where contradictious are not
+contradictions?
+
+Well, in some sense, by merely inventing the concept, we have shoe that such
+worlds are indeed conceivable; but in a deeper sense, they are al: quite inconceivable.
+(This in itself is a little contradiction.) Quite serious] however, it seems that if we want to
+be able to communicate at all, we ha, to adopt some common base, and it pretty well has
+to include logic. (The are belief systems which reject this point of view-it is too logical,
+particular, Zen embraces contradictions and non-contradictions with equ eagerness. This
+may seem inconsistent, but then being inconsistent is pa of Zen, and so ... what can one
+say?)
+
+
+Consistency, Completeness, and Geometry
+
+
+99
+
+
+
+Is Number Theory the Same In All Conceivable Worlds?
+
+
+If we assume that logic is part of every conceivable world (and note that we have
+not defined logic, but we will in Chapters to come), is that all? Is it really conceivable
+that, in some worlds, there are not infinitely many primes? Would it not seem necessary
+that numbers should obey the same laws in all conceivable worlds? Or ... is the concept
+"natural number" better thought of as an undefined term, like "POINT" or "LINE"? In
+that case, number theory would be a bifurcated theory, like geometry: there would be
+standard and nonstandard number theories. But there would have to be some counterpart
+to absolute geometry: a "core" theory, an invariant ingredient of all number theories
+which identified them as number theories rather than, say, theories about cocoa or rubber
+or bananas. It seems to be the consensus of most modern mathematicians and
+philosophers that there is such a core number theory, which ought to be included, along
+with logic, in what we consider to be "conceivable worlds". This core of number theory,
+the counterpart to absolute geometry-is called Peano arithmetic, and we shall formalize it
+in Chapter VIII. Also, it is now well established-as a matter of fact as a direct
+consequence of Godel’s Theorem-that number theory is a bifurcated theory, with
+standard and nonstandard versions. Unlike the situation in geometry, however, the
+number of "brands" of number theory is infinite, which makes the situation of number
+theory considerably more complex.
+
+For practical purposes, all number theories are the same. In other words, if bridge
+building depended on number theory (which in a sense it does), the fact that there are
+different number theories would not matter, since in the aspects relevant to the real world,
+all number theories overlap. The same cannot be said of different geometries; for
+example, the sum of the angles in a triangle is 180 degrees only in Euclidean geometry; it
+is greater in elliptic geometry, less in hyperbolic. There is a story that Gauss once
+attempted to measure the sum of the angles in a large triangle defined by three mountain
+peaks, in order to determine, once and for all, which kind of geometry really rules our
+universe. It was a hundred years later that Einstein gave a theory (general relativity)
+which said that the geometry of the universe is determined by its content of matter, so
+that no one geometry is intrinsic to space itself. Thus to the question, "Which geometry is
+true?" nature gives an ambiguous answer not only in mathematics, but also in physics. As
+for the corresponding question, "Which number theory is true?", we shall have more to
+say on it after going through Godel’s Theorem in detail.
+
+Completenes
+
+If consistency is the minimal condition under which symbols acquire passive meanings,
+then its complementary notion, completeness, is the maximal confirmation of those
+passive meanings. Where consistency is the property
+
+
+Consistency, Completeness, and Geometry
+
+
+100
+
+
+
+way round: "Every true statement is produced by the system". Now I refine the notion
+slightly. We can't mean every true statement in th world-we mean only those which
+belong to the domain which we at attempting to represent in the system. Therefore,
+completeness mean! "Every true statement which can be expressed in the notation of the
+system is a theorem."
+
+Consistency: when every theorem, upon interpretation, comes out true (in some
+imaginable world).
+
+Completeness: when all statements which are true (in some imaginable world), and
+which can be expressed as well-formed strings of the system, are
+theorems.
+
+An example of a formal system which is complete on its own mode level is the
+original pq-system, with the original interpretation. All true additions of two positive
+integers are represented by theorems of th system. We might say this another way: "All
+true additions of two positive integers are provable within the system." (Warning: When
+we start using th term "provable statements" instead of "theorems", it shows that we at
+beginning to blur the distinction between formal systems and their interpretations. This is
+all right, provided we are very conscious of th blurring that is taking place, and provided
+that we remember that multiple interpretations are sometimes possible.) The pq-system
+with the origin interpretation is complete ; it is also consistent, since no false statement is-,
+use our new phrase-provable within the system.
+
+Someone might argue that the system is incomplete, on the grounds that additions
+of three positive integers (such as 2 + 3 + 4 =9) are not represented by theorems of the
+pq-system, despite being translatable into the notation of the system (e.g., — p—p — q—
+
+.). However, this string is not well-formed, and hence should be considered to I just
+
+as devoid of meaning as is p q p — q p q. Triple additions are simply not expressible in
+the notation of the system-so the completeness of the system is preserved.
+
+Despite the completeness of the pq-system under this interpretation, certainly falls
+far short of capturing the full notion of truth in numb theory. For example, there is no
+way that the pq-system tells us how mat prime numbers there are. Godel’s
+Incompleteness Theorem says that any system which is "sufficiently powerful" is, by
+virtue of its power, incomplete, in the sense that there are well-formed strings which
+express tr statements of number theory, but which are not theorems. (There a truths
+belonging to number theory which are not provable within the system.) Systems like the
+pq-system, which are complete but not very powerful, are more like low-fidelity
+phonographs; they are so poor to beg with that it is obvious that they cannot do what we
+would wish them do-namely tell us everything about number theory.
+
+
+Consistency, Completeness, and Geometry
+
+
+101
+
+
+
+
+How an Interpretation May Make or Break Completeness
+
+
+What does it mean to say, as I did above, that "completeness is the maximal confirmation
+of passive meanings"? It means that if a system is consistent but incomplete, there is a
+mismatch between the symbols and their interpretations. The system does not have the
+power to justify being interpreted that way. Sometimes, if the interpretations are
+"trimmed" a little, the system can become complete. To illustrate this idea, let's look at
+the modified pq-system (including Axiom Schema II) and the interpretation we used for
+it.
+
+After modifying the pq-system, we modified the interpretation for q from "equals"
+to "is greater than or equal to". We saw that the modified pq-system was consistent under
+this interpretation; yet something about the new interpretation is not very satisfying. The
+problem is simple: there are now many expressible truths which are not theorems. For
+instance, "2 plus 3 is greater than or equal to 1" is expressed by the nontheorem —p—q-.
+The interpretation is just too sloppy! It doesn't accurately reflect what the theorems in the
+system do. Under this sloppy interpretation, the pq-system is not complete. We could
+repair the situation either by (1) adding new rules to the system, making it more
+powerful, or by (2) tightening up the interpretation. In this case, the sensible alternative
+seems to be to tighten the interpretation. Instead of interpreting q as "is greater than or
+equal to", we should say "equals or exceeds by 1". Now the modified pq-system becomes
+both consistent and complete. And the completeness confirms the appropriateness of the
+interpretation.
+
+
+Incompleteness of Formalized Number Theory
+
+In number theory, we will encounter incompleteness again; but there, to remedy the
+situation, we will be pulled in the other direction-towards adding new rules, to make the
+system more powerful. The irony is that we think, each time we add a new rule, that we
+surely have made the system complete nowl The nature of the dilemma can be illustrated'
+by the following allegory ...
+
+We have a record player, and we also have a record tentatively labeled "Canon on
+B-A-C-H". However, when we play the record on the record player, the feedback-
+induced vibrations (as caused by the Tortoise's records) interfere so much that we do not
+even recognize the tune. We conclude that something is defective-either our record, or
+our record player. In order to test our record, we would have to play it on friends' record
+players, and listen to its quality. In order to test our phonograph, we would have to play
+friends' records on it, and see if the music we hear agrees with the labels. If our record
+player passes its test, then we will say the record was defective; contrariwise, if the
+record passes its test, then we will say our record player was defective. What, however,
+can we conclude when we find out that both pass their respective tests? That is the
+moment to remember the chain of two isomorphisms (Fig. 20), and think carefully!
+
+
+Consistency, Completeness, and Geometry
+
+
+102
+
+
+
+Little Harmonic Labyrinth
+
+
+The Tortoise and Achilles are spending a day at Coney Island After buying a
+couple of cotton candies, they decide to take a ride on the Ferris wheel.
+
+Tortoise: This is my favorite ride. One seems to move so far, and
+reality one gets nowhere.
+
+Achilles: I can see why it would appeal to you. Are you all strapped in?
+
+Tortoise: Yes, I think I've got this buckle done. Well, here we go.
+
+Achilles: You certainly are exuberant today.
+
+Tortoise: I have good reason to be. My aunt, who is a fortune-teller me that a stroke of
+Good Fortune would befall me today. So I am tingling with anticipation.
+
+Achilles: Don't tell me you believe in fortune-telling!
+
+Tortoise: No ... but they say it works even if you don't believe ii
+Achilles: Well, that's fortunate indeed.
+
+Tortoise: Ah, what a view of the beach, the crowd, the ocean, the city. . .
+
+Achilles: Yes, it certainly is splendid. Say, look at that helicopter there. It seems to be
+flying our way. In fact it's almost directly above us now.
+
+Tortoise: Strange-there's a cable dangling down from it, which is very close to us. It's
+coming so close we could practically grab it
+Achilles: Look! At the end of the line there's a giant hook, with a note
+
+(He reaches out and snatches the note. They pass by and are on their z down.)
+
+Tortoise: Can you make out what the note says?
+
+Achilles: Yes-it reads, "Howdy, friends. Grab a hold of the hook time around, for an
+Unexpected Surprise."
+
+Tortoise: The note's a little corny but who knows where it might lead, Perhaps it's got
+something to do with that bit of Good Fortune due me. By all means, let's try it!
+Achilles: Let's!
+
+(On the trip up they unbuckle their buckles, and at the crest of the ride, grab for the
+giant hook. All of a sudden they are whooshed up by the ca which quickly reels
+them skyward into the hovering helicopter. A It strong hand helps them in.)
+
+Voice: Welcome aboard-Suckers.
+
+Achilles: Wh-who are you?
+
+
+Little Harmonic Labyrinth
+
+
+103
+
+
+
+Voce: Allow me to introduce myself. I am Hexachlorophene J. Goodforttune, Kidnapper
+At-Large, and Devourer of Tortoises par Excellence, at your service.
+
+Tortoise: Gulp!
+
+Achilles ( whispering to his friend): Uh-oh-I think that this "Goodfortune" is not exactly
+what we'd anticipated. (To Goodfortune) Ah-if I may be so bold-where are you
+spiriting us off to?
+
+Goodfortune: Ho ho! To my all-electric kitchen-in-the-sky, where I will prepare THIS
+tasty morsel -{leering at the Tortoise as he says this)- in a delicious pie-in-the-sky!
+And make no mistake-it's all just for my gobbling pleasure! Ho ho ho!
+
+Achilles: All I can say is you've got a pretty fiendish laugh.
+
+Goodfortune {laughing fiendishly): Ho ho ho! For that remark, my friend, you will pay
+dearly. Ho ho!
+
+Achilles: Good grief-I wonder what he means by that!
+
+Goodfortune: Very simple-I've got a Sinister Fate in store for both of you! Just you wait!
+Ho ho ho! Ho ho ho!
+
+Achilles: Yikes!
+
+Goodfortune: Well, we have arrived. Disembark, my friends, into my fabulous all-electric
+kitchen-in-the-sky.
+
+(They walk inside.)
+
+Fet me show you around, before I prepare your fates. Here is my bedroom. Here is
+my study. Please wait here for me for a moment. I've got to go sharpen my knives.
+While you're waiting, help yourselves to some popcorn. Ho ho ho! Tortoise pie!
+Tortoise pie! My favorite kind of pie! (Exit.)
+
+Achilles: Oh, boy-popcorn! I'm going to munch my head off!
+
+Tortoise: Achilles! You just stuffed yourself with cotton candy! Besides, how can you
+think about food at a time like this?
+
+Achilles: Good gravy-oh, pardon me-I shouldn't use that turn of phrase, should I? I mean
+in these dire circumstances ... Tortoise: I'm afraid our goose is cooked.
+
+Achilles: Say-take a gander at all these books old Goodfortune has in his study. Quite a
+collection of esoterica: Birdbrains I Have Known; Chess and Umbrella-Twirling
+Made Easy; Concerto for Tapdancer and Orchestra ... Hmmm.
+
+Tortoise: What's that small volume lying open over there on the desk, next to the
+dodecahedron and the open drawing pad?
+
+Achilles: This one? Why, its title is Provocative Adventures of Achilles and the Tortoise
+Taking Place in Sundry Spots of the Globe. Tortoise: A moderately provocative
+title.
+
+Achilles: Indeed-and the adventure it's opened to looks provocative. It's called "Djinn and
+Tonic".
+
+Tortoise: Hmm ... I wonder why. Shall we try reading it? I could take the Tortoise's part,
+and you could take that of Achilles.
+
+
+Fittle Harmonic Fabyrinth
+
+
+104
+
+
+
+Achilles: I’m game. Here goes nothing . . .
+
+(They begin reading "Djinn and Tonic".)
+
+(Achilles has invited the Tortoise over to see his collection of prints by
+his favorite artist, M. C. Escher.)
+
+Tortoise: These are wonderful prints, Achilles.
+
+Achilles: I knew you would enjoy seeing them. Do you have any particular
+favorite?
+
+Tortoise: One of my favorites is Convex and Concave, where two internally
+consistent worlds, when juxtaposed, make a completely inconsistent
+composite world. Inconsistent worlds are always fun places to visit,
+but I wouldn't want to live there.
+
+Achilles: What do you mean, "fun to visit"? Inconsistent worlds don't EXIST,
+so how can you visit one?
+
+Tortoise: I beg your pardon, but weren't we just agreeing that in
+
+this Escher picture, an inconsistent world is portrayed?
+
+Achilles: Yes, but that's just a two-dimensional world-a fictitious world-a
+picture. You can't visit that world.
+
+Tortoise: I have my ways ...
+
+Achilles: How could you propel yourself into a flat picture-universe?
+
+Tortoise: By drinking a little glass of PUSHING-POTION. That does the
+trick.
+
+Achilles: What on earth is pushing-potion?
+
+Tortoise: It's a liquid that comes in small ceramic phials, and which, when
+drunk by someone looking at a picture, "pushes" him right into the
+world of that picture. People who aren't aware of the powers of
+pushing-potion often are pretty surprised by the situations they wind
+up in.
+
+Achilles: Is there no antidote? Once pushed, is one irretrievably lost?
+
+Tortoise: In certain cases, that's not so bad a fate. But there is, in fact, another
+potion-well, not a potion, actually, but an elixir-no, not an elixir, but
+a-a
+
+Tortoise: He probably means "tonic".
+
+Achilles: Tonic?
+
+Tortoise: That's the word I was looking for! "POPPING-TONIC" iu what it's
+called, and if you remember to carry a bottle of it in your right hand as
+you swallow the pushing-potion, it too will be pushed into the picture;
+then, whenever you get a hanker ing to "pop" back out into real life,
+you need only take a swallow of popping-tonic, and presto! You're
+back in the rea. world, exactly where you were before you pushed
+yourself in.
+
+Achilles: That sounds very interesting. What would happen it you took some
+popping-tonic without having previously pushed yourself into a
+picture?
+
+
+Little Harmonic Labyrinth
+
+
+105
+
+
+
+Tortoise: I don’t precisely know, Achilles, but I would be rather wary of
+horsing around with these strange pushing and popping liquids. Once I
+had a friend, a Weasel, who did precisely what you suggested-and no
+one has heard from him since.
+
+Achilles: That's unfortunate. Can you also carry along the bottle of pushing-
+potion with you?
+
+Tortoise: Oh, certainly. Just hold it in your left hand, and it too will get
+pushed right along with you into the picture you're looking at.
+
+Achilles: What happens if you then find a picture inside the picture which you
+have already entered, and take another swig of pushing-potion?
+
+Tortoise: Just what you would expect: you wind up inside that picture-in-a-
+picture.
+
+Achilles: I suppose that you have to pop twice, then, in order to extricate
+yourself from the nested pictures, and re-emerge back in real life.
+
+Tortoise: That's right. You have to pop once for each push, since a push takes
+you down inside a picture, and a pop undoes that.
+
+Achilles: You know, this all sounds pretty fishy to me . . . Are you sure you're
+not just testing the limits of my gullibility?
+
+Tortoise: I swear! Look-here are two phials, right here in my pocket.
+(.Reaches into his lapel pocket, and pulls out two rather large
+unlabeled phials, in one of which one can hear a red liquid sloshing
+around, and in the other of which one can hear a blue liquid sloshing
+around.) If you're willing, we can try them. What do you say?
+
+Achilles: Well, I guess, ahm, maybe, ahm ...
+
+Tortoise: Good! I knew you'd want to try it out. Shall we push ourselves into
+the world of Escher's Convex and Concave ?
+
+Achilles: Well, ah, .. .
+
+Tortoise: Then it's decided. Now we've got to remember to take along this
+flask of tonic, so that we can pop back out. Do you want to take that
+heavy responsibility, Achilles?
+
+Achilles: If it's all the same to you, I'm a little nervous, and I'd prefer letting
+you, with your experience, manage the operation.
+
+Tortoise: Very well, then.
+
+(So saying, the Tortoise pours two small portions of pushing-potion. Then
+he picks up the flask of tonic and grasps it firmly in his right hand, and
+both he and Achilles lift their glasses to their lips.)
+
+Tortoise: Bottoms up!
+
+(They swallow.)
+
+
+Little Harmonic Labyrinth
+
+
+106
+
+
+
+FIGURE 23. Convex and Concave, by M. C. Escher (lithograph, 1955).
+
+
+Achilles: That’s an exceedingly strange taste.
+
+Tortoise: One gets used to it.
+
+Achilles: Does taking the tonic feel this strange? Tortoise: Oh, that's quite
+another sensation. Whenever you taste the tonic, you feel a deep sense
+of satisfaction, as if you'd been waiting to taste it all your life.
+Achilles: Oh, I'm looking forward to that. Tortoise: Well, Achilles,
+where are we?
+
+Achilles (taking cognizance of his surroundings): We're in a little gondola,
+gliding down a canal! I want to get out. Mr.Gondolier, please let us
+out here.
+
+(The gondolier pays no attention to this request.)
+
+Tortoise: He doesn't speak English. If we want to get out here, we'd better just
+clamber out quickly before he
+
+
+Little Harmonic Labyrinth
+
+
+107
+
+
+
+Enters the sinister “Tunnel of Love”; just ahead of us.
+
+(Achilles, his face a little pale scrambles out in a split second and then
+pulls his slower friend out.)
+
+Achilles: I didn't like the sound of that place, somehow. I'm glad we got out
+here. Say, how do you know so much about this place, anyway? Have
+you been here before?
+
+Tortoise: Many times, although I always came in from other Escher pictures.
+They're all connected behind the frames, you know. Once you're in
+one, you can get to any other one.
+
+Achilles: Amazing! Were I not here, seeing these things with my own eyes,
+I'm not sure I'd believe you. (They wander out through a little arch.)
+Oh, look at those two cute lizards!
+
+Tortoise: Cute? They aren't cute-it makes me shudder just to think of them!
+They are the vicious guardians of that magic copper lamp hanging
+from the ceiling over there. A mere touch of their tongues, and any
+mortal turns to a pickle.
+
+Achilles: Dill, or sweet?
+
+Tortoise: Dill.
+
+Achilles: Oh, what a sour fate! But if the lamp has magical powers, I would
+like to try for it.
+
+Tortoise: It's a foolhardy venture, my friend. I wouldn't risk it.
+
+Achilles: I'm going to try just once.
+
+(He stealthily approaches the lamp, making sure not to awaken the
+sleeping lad nearby. But suddenly, he slips on a strange shell-like
+indentation in the floor, and lunges out into space. Lurching crazily, he
+reaches for anything, and manages somehow to grab onto the lamp with
+one hand. Swinging wildly, with both lizards hissing and thrusting their
+tongues violently out at him, he is left dangling helplessly out in the middle
+of space.)
+
+Achilles: He-e-e-elp!
+
+(His cry attracts the attention of a woman who rushes downstairs and
+awakens the sleeping boy. He takes stock of the situation, and, with a
+kindly smile on his face, gestures to Achilles that all will be well. He shouts
+something in a strange guttural tongue to a pair of trumpeters high up in
+windows, and immediately,
+
+
+Little Harmonic Labyrinth
+
+
+108
+
+
+
+Weird tones begin ringing out and making beats each other. The sleepy
+young lad points at the lizards, and Achilles sees that the music is having a
+strong soporific effect on them. Soon, they are completely unconscious.
+Then the helpful lad shouts to two companions climbing up ladders. They
+both pull their ladders up and then extend them out into space just
+underneath the stranded Achilles, forming a sort of bridge. Their gestures
+make it clear that Achilles should hurry and climb on. But before he does
+so, Achilles carefully unlinks the top link of the chain holding the lamp, and
+detaches the lamp. Then he climbs onto the ladder-bridge and the three
+young lads pull him in to safety. Achilles throws his arms around them and
+hugs them gratefully.)
+
+Achilles: Oh, Mr. T, how can I repay them?
+
+Tortoise: I happen to know that these valiant lads just love coffee, and down
+in the town below, there's a place where they make an incomparable
+cup of espresso. Invite them for a cup of espresso! Achilles: That
+would hit the spot.
+
+(And so, by a rather comical series of gestures, smiles, and words, Achilles
+manages to convey his invitation to the young lads, and the party of five
+walks out and down a steep staircase descending into the town. They reach
+a charming small cafe, sit down outside, and order five espressos. ,4.v they
+sip their drinks, Achilles remembers he has the lamp with him.)
+
+Achilles: I forgot, Mr. Tortoise-I've got this ma; lamp with me! But-what's
+magic about it? Tortoise: Oh, you know, just the usual-a genie.
+
+Achilles: What? You mean a genie comes out when you rub it, and grants you
+wishes?
+
+Tortoise: Right. What did you expect? Pennies fry heaven?
+
+Achilles: Well, this is fantastic! I can have any wish want, eh? I've always
+wished this would happen to me ...
+
+(And so Achilles gently rubs the large letter 'L' which is etched on the
+lamp's copper surface ... Suddenly a huge puff of smoke appears, and in the
+forms of the smoke the five friends can make out a weird, ghostly figure
+towering above them.)
+
+
+Little Harmonic Labyrinth
+
+
+109
+
+
+
+I
+
+Genie: Hello, my friends - and thanks ever so much for rescuing my Lamp
+from the evil Lizard-Duo.
+
+(And so saying, the Genie picks up the Lamp, and stuffs it into a pocket
+concealed among the folds of his long ghostly robe which swirls out of the
+Lamp.)
+
+As a sign of gratitude for your heroic deed, I would like to offer you, on the
+part of my Lamp, the opportunity to have any three of your wishes
+realized.
+
+Achilles: How stupefying! Don't you think so, Mr. T?
+
+Tortoise: I surely do. Go ahead, Achilles, take the first wish.
+
+Achilles: Wow! But what should I wish? Oh, I know! It's what I thought of
+the first time I read the Arabian Nights (that collection of silly (and
+nested) tales)-I wish that I had a HUNDRED wishes, instead of just
+three! Pretty clever, eh, Mr. T? I bet YOU never would have thought
+of that trick. I always wondered why those dopey people in the stories
+never tried it themselves.
+
+Tortoise: Maybe now you'll find out the answer.
+
+Genie: I am sorry, Achilles, but I don't grant metawishes.
+
+Achilles: I wish you'd tell me what a "meta-wish" is!
+
+Genie: But THAT is a meta-meta-wish, Achilles-and I don't grant them,
+either. Achilles: Whaaat? I don't follow you at all.
+
+Tortoise: Why don't you rephrase your last request, Achilles?
+
+Achilles: What do you mean? Why should I?
+
+Tortoise: Well, you began by saying "I wish". Since you're just asking for
+information, why don't you just ask a question?
+
+Achilles: All right, though I don't see why. Tell me, Mr. Genie-what is a
+meta-wish? Genie: It is simply a wish about wishes. I am not allowed
+to grant meta-wishes. It is only within my purview to grant plain
+ordinary wishes, such as wishing for ten bottles of beer, to have Helen
+of Troy on a blanket, or to have an all-expenses-paid weekend for two
+at the Copacabana. You know-simple things like that. But meta¬
+wishes I cannot grant. GOD won't permit me to.
+
+Achilles: GOD? Who is GOD? And why won't he let you grant meta-wishes?
+That seems like such a puny thing compared to the others you
+mentioned.
+
+
+Little Harmonic Labyrinth
+
+
+110
+
+
+
+Genie: Well, it’s a complicated matter, you see. Why don’t you just go ahead
+and make your three wishes? Or at least make one of them. I don't
+have all I time in the world, you know ...
+
+Achilles: Oh, I feel so rotten. I was REALLY HOPING wish for a hundred
+wishes ...
+
+Genie: Gee, I hate to see anybody so disappointed that. And besides, meta¬
+wishes are my favorite k of wish. Let me just see if there isn't anything
+I do about this. This'll just take one moment
+
+(The Genie removes from the wispy folds of his robe an object which looks
+just like the copper Lamp he had put away, except that this one is made of
+silver; and where the previous one had 'L' etched on it, this one has 'ML' in
+smaller letters, so as to cover the same area.)
+
+I
+
+Achilles: And what is that?
+
+Genie: This is my Meta-Lamp ...
+
+(He rubs the Meta-Lamp, and a huge puff of smoke appears. In the billows
+of smoke, they can all make out a ghostly form towering above them.)
+
+Meta-Genie: I am the Meta-Genie. You summoned me, 0 Genie? What is
+your wish?
+
+Genie: I have a special wish to make of you, 0 Djinn and of GOD. I wish for
+permission for tempos suspension of all type-restrictions on wishes,
+for duration of one Typeless Wish. Could you ph grant this wish for
+me?
+
+Meta-Genie: I'll have to send it through Channels, of course. One half a
+moment, please
+
+(And, twice as quickly as the Genie did, this Meta-Genie removes from the
+wispy folds of her robe an object which looks just like the silver Meta-
+Lamp, except that it is made of gold; and where the previous one had 'ML'
+etched on it, this one has 'MML' in smaller letters, so as to cover the same
+area.)
+
+Achilles (his voice an octave higher than before)-. And what is that? Meta-
+Genie: This is my Meta-Meta-Lamp. . .
+
+(She rubs the Meta-Meta-Lamp, and a hugs puff of smoke appears. In the
+billows o smoke, they can all make out a ghostly fore towering above
+them.)
+
+
+Little Harmonic Labyrinth
+
+
+111
+
+
+
+Meta-Meta-Genie: I am the MetaMeta-Genie. You summoned me,
+
+0 Meta-Genie? What is your wish?
+
+Meta-Genie: I have a special wish to make of you, 0 Djinn, and of GOD. I
+wish for permission for temporary suspension of all type-restrictions
+on wishes, for the duration of one Typeless Wish. Could you please
+grant this wish for me?
+
+Meta-Meta-Genie: I'll have to send it through Channels, of course.
+
+One quarter of a moment, please.
+
+(And, twice as quickly as the Meta-Genie did, this MetaMeta-
+Genie removes from the folds of his robe an object which looks
+just like the gold MetaLamp, except that it is made of ...)
+
+
+. .{GOD}
+
+
+( ... swirls back into the MetaMeta-Meta-Lamp, which the Meta-
+Meta-Genie then folds back into his robe, half as quickly as the
+Meta-Meta-Meta-Genie did.)
+
+Your wish is granted, 0 MetaGenie.
+
+Meta-Genie: Thank you, 0 Djinn, and GOD.
+
+(And the Meta-Meta-Genie, as all the higher ones before him, swirls
+back into the Meta-Meta-Lamp, which the Meta-Genie then folds back
+into her robe, half as quickly as the Meta-Meta-Genie did.)
+
+Your wish is granted, 0 Genie.
+
+Genie: Thank you, 0 Djinn, and GOD.
+
+(And the Meta-Genie, as all the higher ones before her,
+
+
+Little Harmonic Labyrinth
+
+
+112
+
+
+
+swirls back into the Meta-Lamp, which the Genie folds back into his robe,
+
+half as quickly as the M Genie did.)
+
+Your wish is granted, Achilles.
+
+(And one precise moment has elapsed since he "This will just take one
+moment.")
+
+Achilles: Thank you, 0 Djinn, and GOD.
+
+Genie: I am pleased to report, Achilles, that you r have exactly one (1)
+Typeless Wish-that is to sa wish, or a meta-wish, or a meta-meta-wish,
+as many "meta"'s as you wish-even infinitely many (if wish).
+
+Achilles: Oh, thank you so very much, Genie. But curiosity is provoked.
+Before I make my wish, would you mind telling me who-or what-
+GOD is?
+
+Genie: Not at all. "GOD" is an acronym which stands "GOD Over Djinn".
+The word "Djinn" is used designate Genies, Meta-Genies, Meta-Meta-
+Gen etc. It is a Typeless word.
+
+Achilles: But-but-how can "GOD" be a word in own acronym? That doesn't
+make any sense!
+
+Genie: Oh, aren't you acquainted with recursive acronyms? I thought
+everybody knew about them. \ see, "GOD" stands for "GOD Over
+Djinn"-which can be expanded as "GOD Over Djinn, O, Djinn"-and
+that can, in turn, be expanded to "G( Over Djinn, Over Djinn, Over
+Djinn"-which can its turn, be further expanded ... You can go as as
+you like.
+
+Achilles: But I'll never finish!
+
+Genie: Of course not. You can never totally expand GOD.
+
+Achilles: Hmm ... That's puzzling. What did you me when you said to the
+Meta-Genie, "I have a sped wish to make of you, 0 Djinn, and of
+GOD"?
+
+Genie: I wanted not only to make a request of Meta-Genie, but also of all the
+Djinns over her. 'I recursive acronym method accomplishes this qL
+naturally. You see, when the Meta-Genie received my request, she
+then had to pass it upwards to I GOD. So she forwarded a similar
+message to I Meta-Meta-Genie, who then did likewise to t Meta-Meta-
+Meta-Genie ... Ascending the chain this way transmits the message to
+GOD.
+
+
+Little Harmonic Labyrinth
+
+
+113
+
+
+
+Achilles: I see. You mean GOD sits up at the top of the ladder of djinns?
+
+Genie: No, no, no! There is nothing "at the top", for there is no top. That is
+why GOD is a recursive acronym. GOD is not some ultimate djinn;
+GOD is the tower of djinns above any given djinn.
+
+Tortoise: It seems to me that each and every djinn would have a different
+concept of what GOD is, then, since to any djinn, GOD is the set of
+djinns above him or her, and no two djinns share that set.
+
+Genie: You're absolutely right-and since I am the lowest djinn of all, my
+notion of GOD is the most exalted one. I pity the higher djinns, who
+fancy themselves somehow closer to GOD. What blasphemy!
+
+Achilles: By gum, it must have taken genies to invent GOD.
+
+Tortoise: Do you really believe all this stuff about GOD, Achilles?
+
+Achilles: Why certainly, I do. Are you atheistic, Mr. T? Or are you agnostic?
+
+Tortoise: I don't think I'm agnostic. Maybe I'm metaagnostic.
+
+Achilles: Whaaat? I don't follow you at all.
+
+Tortoise: Let's see ... If I were meta-agnostic, I'd be confused over whether
+I'm agnostic or not-but I'm not quite sure if I feel THAT way; hence I
+must be meta-meta-agnostic (I guess). Oh, well. Tell me, Genie, does
+any djinn ever make a mistake, and garble up a message moving up or
+down the chain?
+
+Genie: This does happen; it is the most common cause for Typeless Wishes
+not being granted. You see, the chances are infinitesimal, that a
+garbling will occur at any PARTICULAR link in the chain-but when
+you put an infinite number of them in a row, it becomes virtually
+certain that a garbling will occur SOMEWHERE. In fact, strange as it
+seems, an infinite number of garblings usually occur, although they
+are very sparsely distributed in the chain.
+
+Achilles: Then it seems a miracle that any Typeless Wish ever gets carried
+out.
+
+Genie: Not really. Most garblings are inconsequential, and many garblings
+tend to cancel each other out. But occasionally-in fact, rather seldom-
+the nonfulfillment of a Typeless Wish can be traced back to a single
+unfortunate djinn's garbling. When this happens, the guilty djinn is
+forced to run an infinite
+
+
+Little Harmonic Labyrinth
+
+
+114
+
+
+
+Gauntlet and get paddled on his or her rump, by GOD. It's good fun for the
+paddlers, and q harmless for the paddlee. You might be amused by the
+sight.
+
+Achilles: I would love to see that! But it only happens when a Typeless Wish
+goes ungranted?
+
+Genie: That's right.
+
+Achilles: Hmm ... That gives me an idea for my w Tortoise: Oh, really? What
+is it? Achilles: I wish my wish would not be granted!
+
+(At that moment, an event-or is "event" the word for it? -takes place which
+cannot be described, and hence no attempt will be made to describe it.)
+
+Achilles: What on earth does that cryptic comment mean?
+
+Tortoise: It refers to the Typeless Wish Achilles made.
+
+Achilles: But he hadn't yet made it.
+
+Tortoise: Yes, he had. He said, "I wish my wish would not be
+granted", and the Genie took THAT to be his wish.
+
+(At that moment, some footsteps are heard coming down the hallway in
+their direction.)
+
+Achilles: Oh, my! That sounds ominous.
+
+(The footsteps stop; then they turn around and fade away.)
+
+Tortoise: Whew!
+
+Achilles: But does the story go on, let's see. or is that the end? Turn the page
+and let’s see.
+
+(The Tortoise turns the page of "Djinn and Tonic", where they find that the
+story goes on...)
+
+Achilles: Hey! What happened? Where is my Genie: lamp? My cup of
+espresso? What happened to young friends from the Convex and
+Concave worlds? What are all those little lizards doing hi
+Tortoise: I'm afraid our context got restored incorrectly Achilles.
+
+Achilles: What on earth does that cryptic comment mean?
+
+Tortoise: I refer to the Typeless Wish you made.
+
+Achilles: But I hadn't yet made it.
+
+Tortoise: Yes, you had. You said, "I wish my wish would not be
+granted", and the Genie took THAT to be your wish.
+
+Achilles: Oh, my! That sounds ominous.
+
+Tortoise; It spells PARADOX. For that Typeless wish to be
+
+
+Little Harmonic Labyrinth
+
+
+115
+
+
+
+granted, it had to be denied - yet not to grant it would be to grant
+it.
+
+Achilles: So what happened? Did the earth come to a standstill? Did the
+universe cave in?
+
+Tortoise: No. The System crashed. Achilles: What does that mean?
+
+Tortoise: It means that you and I, Achilles, were suddenly and
+instantaneously transported to Tumbolia. Achilles: To where?
+
+Tortoise: Tumbolia: the land of dead hiccups and extinguished light
+bulbs. It's a sort of waiting room, where dormant software waits
+for its host hardware to come back up. No telling how long the
+System was down, and we were in Tumbolia. It could have been
+moments, hours, days-even years.
+
+Achilles: I don't know what software is, and I don't know what hardware
+is. But I do know that I didn't get to make my wishes! I want my
+Genie back!
+
+Tortoise: I'm sorry, Achilles-you blew it. You crashed the System, and
+you should thank your lucky stars that we're back at all. Things
+could have come out a lot worse. But I have no idea where we
+are.
+
+Achilles: I recognize it now-we're inside another of Escher's pictures.
+This time it's Reptiles.
+
+Tortoise: Aha! The System tried to save as much of our context as it
+could before it crashed, and it got as far as recording that it was
+an Escher picture with lizards before it went down. That's
+commendable.
+
+Achilles: And look-isn't that our phial of poppingtonic over there on the
+table, next to the cycle of lizards?
+
+Tortoise: It certainly is, Achilles. I must say, we are very lucky indeed.
+The System was very kind to us, in giving us back our popping-
+tonic-it's precious stuff!
+
+Achilles: I'll say! Now we can pop back out of the Escher world, into my
+house.
+
+Tortoise: There are a couple of books on the desk, next to the tonic. I
+wonder what they are. {He picks up the smaller one, which is
+open to a random page.) This looks like a moderately
+provocative book.
+
+Achilles: Oh, really? What is its title?
+
+Tortoise: Provocative Adventures of the Tortoise and Achilles Taking
+Place in Sundry Parts of the Globe. It sounds like an interesting
+book to read out of.
+
+
+Little Harmonic Labyrinth
+
+
+116
+
+
+
+
+FIGURE 24. Reptiles, by M. C. Escher (lithograph, 1943).
+
+Achilles: Well, You can read it if you want, but as for I’m not going to
+take any chances with t popping-tonic-one of the lizards might
+knock it off the table, so I'm going to get it right now!
+
+(He dashes over to the table and reaches for the popping-tonic, but in
+his haste he somehow bumps the flask of tonic, and it tumbles off the
+desk and begins rolling.)
+
+Oh, no! Mr. T-Iook! I accidentally knocked tonic onto the floor, and it’s
+rolling toward towards-the stairwell! ^uick-before it falls!
+
+(The Tortoise, however, is completely wrapped up in the thin volume
+which he has in his hands.) Achilles: Well, You can read it if you
+want, but as for I'm not going to take any chances with t popping-
+tonic-one of the lizards might knock it off the table, so I’m going
+to get it right
+
+
+Little Harmonic Labyrinth
+
+
+117
+
+
+
+Tortoise ( muttering ): Eh? This story looks fascinating.
+
+Achilles: Mr. T, Mr. T, help! Help catch the tonic-flask!
+
+Tortoise: What's all the fuss about?
+
+Achilles: The tonic-flask-I knocked it down from the desk, and now it's
+rolling and
+
+(At that instant it reaches the brink of the stairwell, and plummets
+over... )
+
+Oh no! What can we do? Mr. Tortoise-aren't you alarmed? We're
+losing our tonic! It's just fallen down the stairwell! There's only one
+thing to do! We'll have to go down one story!
+
+Tortoise: Go down one story? My pleasure. Won't you join me?
+
+(He begins to read aloud, and Achilles, pulled in two directions at
+once, finally stays, taking the role of the Tortoise.)
+
+Achilles: It's very dark here, Mr. T. I can't see a thing. Oof! I bumped
+into a wall. Watch out!
+
+Tortoise: Here-I have a couple of walking sticks. Why don't you take one
+of them? You can hold it out in front of you so that you don't
+bang into things.
+
+Achilles: Good idea. (He takes the stick.) Do you get the sense that this
+path is curving gently to the left as we walk? Tortoise: Very
+slightly, yes.
+
+Achilles: I wonder where we are. And whether we'll ever see the light of
+day again. I wish I'd never listened to you, when you suggested I
+swallow some of that "DRINK ME" stuff.
+
+Tortoise: I assure you, it's quite harmless. I've done it scads of times, and
+not a once have I ever regretted it. Relax and enjoy being small.
+
+Achilles: Being small? What is it you've done to me, Mr. T?
+
+Tortoise: Now don't go blaming me. You did it of your own free will.
+Achilles: Have you made me shrink? So that this labyrinth we're
+in is actually some teeny thing that someone could STEP on?
+
+
+Little Harmonic Labyrinth
+
+
+118
+
+
+
+
+FIGURE 25. Cretan Labyrinth (Italian engraving; School of
+Finiguerra). [From N Matthews', Mazes' and Labyrinths: Their
+History and Development (New York: Dover Publications, 1970).
+
+Tortoise: Labyrinth? Labyrinth? Could it Are we in the notorious Little
+Harmonic Labyrinth of the dreaded Majotaur?
+
+Achilles: Yiikes! What is that?
+
+Tortoise: They say-although I person never believed it myself-that an I
+Majotaur has created a tiny labyrinth sits in a pit in the middle of
+it, waiting innocent victims to get lost in its fears complexity.
+Then, when they wander and dazed into the center, he laughs and
+laughs at them-so hard, that he laughs them to death!
+
+Achilles: Oh, no!
+
+Tortoise: But it's only a myth. Courage, Achilles.
+
+(And the dauntless pair trudge on.)
+
+Achilles: Feel these walls. They're like o gated tin sheets, or something.
+But the corrugations have different sizes.
+
+
+Little Harmonic Labyrinth
+
+
+119
+
+
+
+
+
+
+
+
+
+(To emphasize his point, he sticks out his walking stick against the
+wall surface as he walks. Av the stick bounces back and forth against
+the corrugations, strange noises echo up and down the long curved
+corridor they are in.)
+
+Tortoise (alarmed): What was THAT?
+
+Achilles: Oh, just me, rubbing my walking stick against the wall.
+
+Tortoise: Whew! I thought for a moment it was the bellowing of the
+ferocious Majotaur! Achilles: I thought you said it was all a
+myth.
+
+Tortoise: Of course it is. Nothing to be afraid of.
+
+(Achilles puts his walking stick back against the wall, and continues
+walking. As he does so, some musical sounds are heard, coming from
+the point where his stick is scraping the wall.)
+
+Tortoise: Uh-oh. I have a bad feeling, Achilles.
+
+That Labyrinth may not be a myth, after all. Achilles: Wait a minute.
+What makes you change your mind all of a sudden? Tortoise: Do
+you hear that music?
+
+(To hear more clearly, Achilles lowers the stick, and the strains of
+melody cease.)
+
+Hey! Put that back! I want to hear the end of this piece!
+
+(Confused, Achilles obeys, and the music resumes.)
+
+Thank you. Now as I was about to say, I have just figured out where
+
+we are.
+
+Achilles: Really? Where are we?
+
+Tortoise: We are walking down a spiral groove of a record in its jacket.
+Your stick scraping against the strange shapes in the wall acts
+like a needle running down the groove, allowing us to hear the
+music.
+
+Achilles: Oh, no, oh, no ...
+
+Tortoise: What? Aren't you overjoyed? Have you ever had the chance to
+be in such intimate contact with music before?
+
+
+Little Harmonic Labyrinth
+
+
+120
+
+
+
+Achzltes: How am I ever going to win footraces against full-sized people
+when I am smaller than a flea, Mr. Tortoise?
+
+Tortoise: Oh, is that all that's bothering you That's nothing to fret abopt,
+Achilles.
+
+Achilles: The way you talk, I get the impression that you never worry at
+all.
+
+Tortoise: I don't know. But one thing for certain is that I don't worry
+about being small. Especially not when faced with the awful
+danger of the dreaded Majotaur!
+
+Achilles: Horrors! Are you telling me
+
+Tortoise: I'm afraid so, Achilles. The music gave it away.
+
+Achilles: How could it do that?
+
+Tortoise: Very simple. When I heard melody B-A-C-H in the top voice,
+I immediately realized that the grooves we're walking through
+could only be Little Harmonic Labyrinth, one of Bach's er known
+organ pieces. It is so named cause of its dizzyingly frequent
+modulations.
+
+Achilles: Wh-what are they?
+
+Tortoise: Well, you know that most music pieces are written in a key, or
+tonality, as C major, which is the key of this o;
+
+Achilles: I had heard the term before. Do that mean that C is the note
+you want to on?
+
+Tortoise: Yes, C acts like a home base, in a Actually, the usual word is
+"tonic".
+
+Achilles: Does one then stray away from tonic with the aim of eventually
+returning
+
+Tortoise: That's right. As the piece develops ambiguous chords and
+melodies are t which lead away from the tonic. Little by little,
+tension builds up-you feel at creasing desire to return home, to
+hear the tonic.
+
+Achilles: Is that why, at the end of a pie always feel so satisfied, as if I
+had waiting my whole life to hear the ton
+
+Tortoise: Exactly. The composer has uses knowledge of harmonic
+progressions to
+
+
+Little Harmonic Labyrinth
+
+
+121
+
+
+
+manipulate your emotions, and to build up hopes in you to hear
+that tonic.
+
+Achilles: But you were going to tell me about modulations.
+
+Tortoise: Oh, yes. One very important thing a composer can do is to
+"modulate" partway through a piece, which means that he sets up
+a temporary goal other than resolution into the tonic.
+
+Achilles: I see ... I think. Do you mean that some sequence of chords
+shifts the harmonic tension somehow so that I actually desire to
+resolve in a new key?
+
+Tortoise: Right. This makes the situation more complex, for although in
+the short term you want to resolve in the new key, all the while at
+the back of your mind you retain the longing to hit that original
+goal-in this case, C major. And when the subsidiary goal is
+reached, there is
+
+Achilles ( suddenly gesturing enthusiastically ): Oh, listen to the gorgeous
+upward-swooping chords which mark the end of this Little
+Harmonic Labyrinth!
+
+Tortoise: No, Achilles, this isn't the end. It's merely
+
+Achilles: Sure it is! Wow! What a powerful, strong ending! What a sense
+of relief! That's some resolution! Gee!
+
+(And sure enough, at that moment the music stops, as they emerge into
+an open area with no walls.)
+
+You see, it Is over. What did I tell you? Tortoise: Something is very
+wrong. This record
+
+is a disgrace to the world of music. Achilles: What do you mean?
+
+Tortoise: It was exactly what I was telling you about. Here Bach had
+modulated from C into G, setting up a secondary goal of hearing
+G. This means that you experience two tensions at once-waiting
+for resolution into G, but also keeping in mind that ultimate
+desire-to resolve triumphantly into C Major.
+
+Achilles: Why should you have to keep any
+
+
+Little Harmonic Labyrinth
+
+
+122
+
+
+
+thing in mind when listening to a piece of music? Is music only an
+intellectual exercise?
+
+Tortoise: No, of course not. Some music is highly intellectual, but most
+music is not. And most of the time your ear or br the
+"calculation" for you, and lets your emotions know what they
+want to hear, don't have to think about it consciously in this
+piece, Bach was playing tricks hoping to lead you astray. And in
+your case Achilles, he succeeded.
+
+Achilles: Are you telling me that I responded to a resolution in a
+subsidiary key?
+
+Tortoise: That's right.
+
+Achilles: It still sounded like an ending to me
+
+Tortoise: Bach intentionally made it sot way. You just fell into his trap.
+It was deliberately contrived to sound like an ending but if you
+follow the harmonic progression carefully, you will see that it is
+in the wrong key. Apparently not just you but this miserable
+record company fell for the same trick-and they truncated the
+piece early.
+
+Achilles: What a dirty trick Bach played
+
+Tortoise: That is his whole game-to m lose your way in his Labyrinth! '1
+Majotaur is in cahoots with Bach, And if you don't watch out, he
+i laugh you to death-and perhaps n with you!
+
+Achilles: Oh, let us hurry up and get here! Quick! Let's run backwards
+grooves, and escape on the outside record before the Evil
+Majotaur finds us.
+
+Tortoise: Heavens, no! My sensibility is delicate to handle the bizarre the
+gressions which occur when time versed.
+
+Achilles: Oh, Mr. T, how will we ever get out of here, if we can't just
+retrace our steps
+
+Tortoise: That's a very good question.
+
+(A little desperately, Achilles starts runt about aimlessly in the dark.
+
+Suddenly t is a slight gasp, and then a "thud".)
+
+
+Little Harmonic Labyrinth
+
+
+123
+
+
+
+Achilles-are you all right?
+
+Achilles: Just a bit shaken up but otherwise fine. I fell into some big
+hole.
+
+Tortoise: You've fallen into the pit of the Evil Majotaur! Here, I'll come
+help you out. We've got to move fast!
+
+Achilles: Careful, Mr. T-I don't want You to fall in here, too ...
+
+Tortoise: Don't fret, Achilles. Everything will be all —
+
+(Suddenly, there is a slight gasp, and then a "thud".)
+
+Achilles: Mr. T-you fell in, too! Are you all right?
+
+Tortoise: Only my pride is hurt-otherwise I'm fine.
+
+Achilles: Now we're in a pretty pickle, aren't we?
+
+(Suddenly, a giant, booming laugh is heard, alarmingly close to
+them.)
+
+Tortoise: Watch out, Achilles! This is no laughing matter.
+
+Majotaur: Hee hee hee! Ho ho! Haw haw haw!
+
+Achilles: I'm starting to feel weak, Mr. T ...
+
+Tortoise: Try to pay no attention to his laugh,
+
+Achilles. That's your only hope.
+
+Achilles: I'll do my best. If only my stomach weren't empty!
+
+Tortoise: Say, am I smelling things, or is there a bowl of hot buttered
+popcorn around here? Achilles: I smell it, too. Where is it coming
+from?
+
+Tortoise: Over here, I think. Oh! I just ran into a big bowl of the stuff.
+
+Yes, indeed-it seems to be a bowl of popcorn!
+
+Achilles: Oh, boy-popcorn! I'm going to munch my head off!
+
+Tortoise: Let's just hope it isn't pushcorn! Pushcorn and popcorn are
+extraordinarily difficult to tell apart.
+
+Achilles: What's this about Pushkin?
+
+Tortoise: I didn't say a thing. You must be hearing things.
+
+Achilles: Go-golly! I hope not. Well, let's dig in!
+
+
+Little Harmonic Labyrinth
+
+
+124
+
+
+
+(And the two Jriends begin muncnai popcorn (or pushcorn?)-and t
+once POP! I guess it was popcorn; all.)
+
+Tortoise: What an amusing story. Did you enjoy it?
+
+Achilles: Mildly. Only I wonder whether the' out of that Evil Majotaur’s
+pit or r Achilles-he wanted to be full-sized again
+
+Tortoise: Don't worry-they're out, and he is again. That's what the "POP"
+was all abo
+
+Achilles: Oh, I couldn't tell. Well, now I REAL: find that bottle of tonic.
+For some reason, burning. And nothing would taste bett drink of
+popping-tonic.
+
+Tortoise: That stuff is renowned for its thirst powers. Why, in some
+places people very crazy over it. At the turn of the century the
+Schonberg food factory stopped ma] and started making cereal
+instead. You cai the uproar that caused.
+
+Achilles: I have an inkling. But let's go look fo Hey just a moment.
+Those lizards on the you see anything funny about them?
+
+Tortoise: Umm ... not particularly. What do you see of such great
+interest?
+
+Achilles: Don't you see it? They're emerging flat picture without
+drinking any pop] How are they able to do that?
+
+Tortoise: Oh, didn't I tell you? You can ge picture by moving
+perpendicularly to it you have no popping-tonic. The little li
+learned to climb UP when they want to ge two-dimensional
+sketchbook world.
+
+Achilles: Could we do the same thing to get Escher picture we're in?
+
+Tortoise: Of course! We just need to go UP one story, you want to try it?
+
+Achilles: Anything to get back to my house! I all these provocative
+adventures.
+
+Tortoise: Follow me, then, up this way.
+
+(And they go up one story.)
+
+Achilles: It's good to be back. But something seems wrong. This isn't my
+house! This is YOUR house, Mr. Tortoise
+
+Tortoise: Well, so it is-and am I glad for that! I wasn’t looking
+
+
+Little Harmonic Labyrinth
+
+
+125
+
+
+
+forward one whit to the long walk back from your house. I am bushed,
+and doubt if I could have made it.
+
+Achilles: I don't mind walking home, so I guess it's lucky we ended up
+here, after all.
+
+Tortoise: I'll say! This certainly is a piece of Good Fortune!
+
+
+Little Harmonic Labyrinth
+
+
+126
+
+
+
+Recursive Structures
+and Processes
+
+
+What Is Recursion?
+
+
+WHAT IS RECURSION? It is what was illustrated in the Dialogue Little Harmonic
+Labyrinth : nesting, and variations on nesting. The concept is very general. (Stories inside
+stories, movies inside movies, paintings inside paintings, Russian dolls inside Russian
+dolls (even parenthetical comments in. side parenthetical comments!)-these are just a few
+of the charms of recursion.) However, you should he aware that the meaning of
+"recursive 1 in this Chapter is only faintly related to its meaning in Chapter 111. The
+relation should be clear by the end of this Chapter.
+
+Sometimes recursion seems to brush paradox very closely. For example, there are
+recursive definitions. Such a definition may give the casual viewer the impression that
+something is being defined in terms of itself. That would be circular and lead to infinite
+regress, if not to paradox proper. Actually, a recursive definition (when properly
+formulated) never leads to infinite regress or paradox. This is because a recursive
+definition never defines something in terms of itself, but always in terms of simpler
+versions of itself. What I mean by this will become clearer shortly, when ' show some
+examples of recursive definitions.
+
+One of the most common ways in which recursion appears in daily life is when
+you postpone completing a task in favor of a simpler task, often o the same type. Here is
+a good example. An executive has a fancy telephone and receives many calls on it. He is
+talking to A when B calls. To A he say,, "Would you mind holding for a moment?" Of
+course he doesn't really car if A minds; he just pushes a button, and switches to B. Now C
+calls. The same deferment happens to B. This could go on indefinitely, but let us not get
+too bogged down in our enthusiasm. So let's say the call with C terminates. Then our
+executive "pops" back up to B, and continues. Meanwhile A is sitting at the other end of
+the line, drumming his fingernails again some table, and listening to some horrible
+Muzak piped through the phone lines to placate him ... Now the easiest case is if the call
+with B simply terminates, and the executive returns to A finally. But it could happen that
+after the conversation with B is resumed, a new caller-D-calls. B is once again pushed
+onto the stack of waiting callers, and D is taken care of. Aft D is done, back to B, then
+back to A. This executive is hopelessly mechanical, to be sure-but we are illustrating
+recursion in its most precise form
+
+
+Recursive Structures and Processes
+
+
+127
+
+
+
+Pushing, Popping, and Stacks
+
+
+In the preceding example, I have introduced some basic terminology of recursion-at least
+as seen through the eyes of computer scientists. The terms are push, pop , and stack (or
+push-down stack, to be precise) and they are all related. They were introduced in the late
+1950's as part of IPL, one of the first languages for Artificial Intelligence. You have
+already encountered "push" and "pop" in the Dialogue. But I will spell things out
+anyway. To push means to suspend operations on the task you're currently working on,
+without forgetting where you are-and to take up a new task. The new task is usually said
+to be "on a lower level" than the earlier task. To pop is the reverse-it means to close
+operations on one level, and to resume operations exactly where you left off, one level
+higher.
+
+But how do you remember exactly where you were on each different level? The
+answer is, you store the relevant information in a stack. So a stack is just a table telling
+you such things as (1) where you were in each unfinished task (jargon: the "return
+address"), (2) what the relevant facts to know were at the points of interruption (jargon:
+the "variable bindings"). When you pop back up to resume some task, it is the stack
+which restores your context, so you don't feel lost. In the telephone-call example, the
+stack tells you who is waiting on each different level, and where you were in the
+conversation when it was interrupted.
+
+By the way, the terms "push", "pop", and "stack" all come from the visual image
+of cafeteria trays in a stack. There is usually some sort of spring underneath which tends
+to keep the topmost tray at a constant height, more or less. So when you push a tray onto
+the stack, it sinks a little-and when you remove a tray from the stack, the stack pops up a
+little.
+
+One more example from daily life. When you listen to a news report on the radio,
+oftentimes it happens that they switch you to some foreign correspondent. "We now
+switch you to Sally Swumpley in Peafog, England." Now Sally has got a tape of some
+local reporter interviewing someone, so after giving a bit of background, she plays it. "I'm
+Nigel Cadwallader, here on scene just outside of Peafog, where the great robbery took
+place, and I'm talking with ..." Now you are three levels down. It may turn out that the
+interviewee also plays a tape of some conversation. It is not too uncommon to go down
+three levels in real news reports, and surprisingly enough, we scarcely have any
+awareness of the suspension. It is all kept track of quite easily by our subconscious mind.
+Probably the reason it is so easy is that each level is extremely different in flavor from
+each other level. If they were all similar, we would get confused in no time flat.
+
+An example of a more complex recursion is, of course, our Dialogue. There,
+Achilles and the Tortoise appeared on all the different levels. Sometimes they were
+reading a story in which they appeared as characters. That is when your mind may get a
+little hazy on what's going on, and you have to concentrate carefully to get things straight.
+"Let's see, the real Achilles and Tortoise are still up there in Goodfortune's helicopter, but
+the
+
+
+Recursive Structures and Processes
+
+
+128
+
+
+
+secondary ones are in some Escher picture-and then they found this book and are reading
+in it, so it's the tertiary Achilles and Tortoise who wandering around inside the grooves
+of the Little Harmonic Labyrinth, wait a minute-I left out one level somewhere ..." You
+have to ha conscious mental stack like this in order to keep track of the recursion the
+Dialogue. (See Fig. 26.)
+
+
+»»*v l/iuiu^uv. y. »v.v. i i^. I.W.;
+
+
+
+FIGURE 26. Diagram of the structure of the Dialogue Little Harmonic Labyrinth
+Vertical descents are "pushes"; rises ore "pops". Notice the similarity of this diagram to
+indentation pattern of the Dialogue. From the diagram it is clear that the initial tension
+Goodfortitne's threat-never was resolved; Achilles and the Tortoise were just left
+dangling the sky. Some readers might agonize over this unpopped push, while others
+might not ba eyelash. In the story, Bach's musical labyrinth likewise was cut off too soon-
+but Achilles d even notice anything funny. Only the Tortoise was aware of the more
+global dangling tension
+
+
+Stacks in Music
+
+While we’re talking about the Little Harmonic Labyrinth, we should discuss
+something which is hinted at, if not stated explicitly in the Dialogue: that hear music
+recursively-in particular, that we maintain a mental stack of keys, and that each new
+modulation pushes a new key onto the stack, implication is further that we want to hear
+that sequence of keys retrace reverse order-popping the pushed keys off the stack, one by
+one, until the tonic is reached. This is an exaggeration. There is a grain of truth to it
+however.
+
+Any reasonably musical person automatically maintains a shallow with two keys.
+In that "short stack”, the true tonic key is held and also most immediate "pseudotonic"
+(the key the composer is pretending t in). In other words, the most global key and the
+most local key. That the listener knows when the true tonic is regained, and feels a strong
+s of "relief". The listener can also distinguish (unlike Achilles) between a local easing of
+tension-for example a resolution into the pseudotonic —
+
+
+Recursive Structures and Processes
+
+
+129
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+and a global resolution. In fact, a pseudoresolution should heighten the global tension,
+not relieve it, because it is a piece of irony-just like Achilles' rescue from his perilous
+perch on the swinging lamp, when all the while you know he and the Tortoise are really
+awaiting their dire fates at the knife of Monsieur Goodfortune.
+
+Since tension and resolution are the heart and soul of music, there are many, many
+examples. But let us just look at a couple in Bach. Bach wrote many pieces in an
+"AABB" form-that is, where there are two halves, and each one is repeated. Let's take the
+gigue from the French Suite no. 5, which is quite typical of the form. Its tonic key is G,
+and we hear a gay dancing melody which establishes the key of G strongly. Soon,
+however, a modulation in the A-section leads to the closely related key of D (the
+dominant). When the A-section ends, we are in the key of D. In fact, it sounds as if the
+piece has ended in the key of D! (Or at least it might sound that way to Achilles.) But
+then a strange thing happens-we abruptly jump back to the beginning, back to G, and
+rehear the same transition into D. But then a strange thing happens-we abruptly jump
+back to the beginning, back to G, and rehear the same transition into D.
+
+Then comes the B-section. With the inversion of the theme for our melody, we
+begin in D as if that had always been the tonic-but we modulate back to G after all, which
+means that we pop back into the tonic, and the B-section ends properly. Then that funny
+repetition takes place, jerking us without warning back into D, and letting us return to G
+once more. Then that funny repetition takes place, jerking us without warning
+
+back into D, and letting us return to G once more.
+
+The psychological effect of all this key shifting-some jerky, some smooth-is very
+difficult to describe. It is part of the magic of music that we can automatically make sense
+of these shifts. Or perhaps it is the magic of Bach that he can write pieces with this kind
+of structure which have such a natural grace to them that we are not aware of exactly
+what is happening.
+
+The original Little Harmonic Labyrinth is a piece by Bach in which he tries to
+lose you in a labyrinth of quick key changes. Pretty soon you are so disoriented that you
+don't have any sense of direction left-you don't know where the true tonic is, unless you
+have perfect pitch, or like Theseus, have a friend like Ariadne who gives you a thread that
+allows you to retrace your steps. In this case, the thread would be a written score. This
+piece-another example is the Endlessly Rising Canon-goes to show that, as music
+listeners, we don't have very reliable deep stacks.
+
+
+Recursion in Language
+
+Our mental stacking power is perhaps slightly stronger in language. The grammatical
+structure of all languages involves setting up quite elaborate push-down stacks, though, to
+be sure, the difficulty of understanding a sentence increases sharply with the number of
+pushes onto the stack. The proverbial German phenomenon of the "verb-at-the-end",
+about which
+
+
+Recursive Structures and Processes
+
+
+130
+
+
+
+Droll tales of absentminded professors who would begin a sentence, ramble on for
+an entire lecture, and then finish up by rattling off a string of verbs by which their
+audience, for whom the stack had long since lost its coherence, would be totally
+nonplussed, are told, is an excellent example of linguistic pushing and popping. The
+confusion among the audience out-of-order popping from the stack onto which the
+professor's verbs been pushed, is amusing to imagine, could engender. But in normal ken
+German, such deep stacks almost never occur-in fact, native speaker of German often
+unconsciously violate certain conventions which force verb to go to the end, in order to
+avoid the mental effort of keeping track of the stack. Every language has constructions
+which involve stacks, though usually of a less spectacular nature than German. But there
+are always of rephrasing sentences so that the depth of stacking is minimal.
+
+Recursive Transition Networks
+
+The syntactical structure of sentences affords a good place to present a of describing
+recursive structures and processes: the Recursive Transition Network (RTN). An RTN is
+a diagram showing various paths which can be followed to accomplish a particular task.
+Each path consists of a number of nodes , or little boxes with words in them, joined by
+arcs , or lines with arrows. The overall name for the RTN is written separately at the left,
+and the and last nodes have the words begin and end in them. All the other nodes contain
+either very short explicit directions to perform, or else name other RTN's. Each time you
+hit a node, you are to carry out the direct inside it, or to jump to the RTN named inside it,
+and carry it out.
+
+Let's take a sample RTN, called ORNATE NOUN, which tells how to construct
+a certain type of English noun phrase. (See Fig. 27a.) If traverse ORNATE NOUN
+purely horizontally, we begin', then we create ARTICLE, an ADJECTIVE, and a
+NOUN, then we end. For instance, "the shampoo" or "a thankless brunch". But the arcs
+show other possibilities such as skipping the article, or repeating the adjective. Thus we
+co construct "milk", or "big red blue green sneezes", etc.
+
+When you hit the node NOUN, you are asking the unknown black I called NOUN
+to fetch any noun for you from its storehouse of nouns. This is known as a procedure
+call, in computer science terminology. It means you temporarily give control to a
+procedure (here, NOUN) which (1) does thing (produces a noun) and then (2) hands
+control back to you. In above RTN, there are calls on three such procedures: ARTICLE,
+ADJECTIVE and NOUN. Now the RTN ORNATE NOUN could itself be called from
+so other RTN-for instance an RTN called SENTENCE. In this case, ORNATE NOUN
+would produce a phrase such as "the silly shampoo" and d return to the place inside
+SENTENCE from which it had been called. I quite reminiscent of the way in which you
+resume where you left off nested telephone calls or nested news reports.
+
+However, despite calling this a "recursive transition network", we have
+
+
+Recursive Structures and Processes
+
+
+131
+
+
+
+
+
+
+FIGURE 27. Recursive Transition Networks for ORNATE NOUN and FANCY NOUN.
+
+not exhibited any true recursion so far. Things get recursive-and seemingly circular-when
+you go to an RTN such as the one in Figure 27b, for FANCY NOUN. As you can see,
+every possible pathway in FANCY NOUN involves a call on ORNATE NOUN, so there
+is no way to avoid getting a noun of some sort or other. And it is possible to be no more
+ornate than that, coming out merely with "milk" or "big red blue green sneezes". But
+three of the pathways involve recursive calls on FANCY NOUN itself. It certainly looks
+as if something is being defined in terms of itself. Is that what is happening, or not?
+
+The answer is "yes, but benignly". Suppose that, in the procedure SENTENCE,
+there is a node which calls FANCY NOUN, and we hit that node. This means that we
+commit to memory (viz., the stack) the location of that node inside SENTENCE, so we'll
+know where to return to-then we transfer our attention to the procedure FANCY NOUN.
+Now we must choose a pathway to take, in order to generate a FANCY NOUN. Suppose
+we choose the lower of the upper pathways-the one whose calling sequence goes:
+
+ORNATE NOUN; RELATIVE PRONOUN; FANCY NOUN; VERB.
+
+
+Recursive Structures and Processes
+
+
+132
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+So we spit out an ORNATE NOUN: " the strange bagels a RELATIVE NOUN:
+"that"; and now we are suddenly asked for a FANCY NOUN. B are in the middle of
+FANCY NOUN! Yes, but remember our executive was in the middle of one phone call
+when he got another one. He n stored the old phone call's status on a stack, and began the
+new one nothing were unusual. So we shall do the same.
+
+We first write down in our stack the node we are at in the outer call on FANCY
+NOUN, so that we have a "return address"; then we jump t beginning of FANCY NOUN
+as if nothing were unusual. Now we h~ choose a pathway again. For variety's sake, let's
+choose the lower pat] ORNATE NOUN; PREPOSITION; FANCY NOUN. That
+means we produce an ORNATE NOUN (say "the purple cow"), then a PREPOSITION
+(say “without”), and once again, we hit the recursion. So we hang onto our hats descend
+one more level. To avoid complexity, let's assume that this the pathway we take is the
+direct one just ORNATE NOUN. For example: we might get "horns". We hit the node
+END in this call on FANCY NOUN which amounts to popping out, and so we go to our
+stack to find the return address. It tells us that we were in the middle of executing
+FANCY NOUN one level up-and so we resume there. This yields " the purple cow
+without horns". On this level, too, we hit END, and so we pop up once more, this finding
+ourselves in need of a VERB-so let's choose "gobbled". This ends highest-level call on
+FANCY NOUN, with the result that the phrase
+
+"the strange bagels that the purple cow without horns gobbled"
+
+will get passed upwards to the patient SENTENCE, as we pop for the last time.
+
+As you see, we didn't get into any infinite regress. The reason is tl least one
+pathway inside the RTN FANCY NOUN does not involve recursive calls on FANCY
+NOUN itself. Of course, we could have perversely insisted on always choosing the
+bottom pathway inside FANCY NOUN then we would never have gotten finished, just
+as the acronym "GOD” never got fully expanded. But if the pathways are chosen at
+random, an infinite regress of that sort will not happen.
+
+"Bottoming Out" and Heterarchies
+
+This is the crucial fact which distinguishes recursive definitions from circular
+ones. There is always some part of the definition which avoids reference, so that the
+action of constructing an object which satisfies the definition will eventually "bottom
+out".
+
+Now there are more oblique ways of achieving recursivity in RTNs than by self¬
+calling. There is the analogue of Escher's Drawing (Fig. 135), where each of two
+procedures calls the other, but not itself. For example, we could have an RTN named
+CLAUSE, which calls FANCY NOUN whenever it needs an object for a transitive verb,
+and conversely, the u path of FANCY NOUN could call RELATIVE PRONOUN and
+then CLAUSE
+
+
+Recursive Structures and Processes
+
+
+133
+
+
+
+whenever it wants a relative clause. This is an example of indirect recursion. It is
+reminiscent also of the two-step version of the Epimenides paradox.
+
+Needless to say, there can be a trio of procedures which call one another,
+cyclically-and so on. There can be a whole family of RTN's which are all tangled up,
+calling each other and themselves like crazy. A program which has such a structure in
+which there is no single "highest level", or "monitor", is called a heterarchy (as
+distinguished from a hierarchy). The term is due, I believe, to Warren McCulloch, one of
+the first cyberneticists, and a reverent student of brains and minds.
+
+Expanding Nodes
+
+One graphic way of thinking about RTN’s is this. Whenever you are moving along some
+pathway and you hit a node which calls on an RTN, you "expand" that node, which
+means to replace it by a very small copy of the RTN it calls (see Fig. 2S). Then you
+proceed into the very small RTN,
+
+
+
+FIGURE 28. The FANCY NOUN RTN with one node recursively expanded
+
+When you pop out of it, you are automatically in the right place in the big one. While in
+the small one, you may wind up constructing even more miniature RTN’s. But by
+expanding nodes only when you come across them, you avoid the need to make an
+infinite diagram, even when an RTN calls itself.
+
+Expanding a node is a little like replacing a letter in an acronym by the word it
+stands for. The "GOD" acronym is recursive but has the defect or advantage-that you
+must repeatedly expand the 'G'; thus it never bottoms out. When an RTN is implemented
+as a real computer program, however, it always has at least one pathway which avoids
+recursivity (direct or indirect) so that infinite regress is not created. Even the most
+heterarchical program structure bottoms out-otherwise it couldn't run! It would just be
+constantly expanding node after node, but never performing any action.
+
+
+Recursive Structures and Processes
+
+
+134
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Diagram G and Recursive Sequences
+
+
+Infinite geometrical structures can be defined in just this way-that is by expanding
+node after node. For example, let us define an infinite diagram called "Diagram G”. To
+do so, vve shall use an implicit representation. In two nodes, we shall write merely the
+letter 'G', which, however, will stand for an entire copy of Diagram G. In Figure 29a,
+Diagram G is portrayed implicitly. Now if we wish to see Diagram G more explicitly, we
+expand each of the two G's-that is, we replace them by the same diagram , only reduced
+in scale (see Fig. 29b). This "second-order" version of Diagram gives us an inkling of
+what the final, impossible-to-realize Diagram G really looks like. In Figure 30 is shown a
+larger portion of Diagram G, where all the nodes have been numbered from the bottom
+up, and from left to right. Two extra nodes-numbers — 1 and 2— have been inserted at
+the bottom
+
+This infinite tree has some very curious mathematical properties Running up its
+right-hand edge is the famous sequence of Fibonacci numbers.
+
+I, I, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
+
+discovered around the year 12t2 by Leonardo of Pisa, son of Bonaccio, ergo "Filius
+Bonacci", or "Fibonacci" for short. These number are best
+
+FIGURE 29. (a) Diagram G, unexpanded. (c) Diagram H, unexpanded
+
+(b) Diagram G, expanded once. (d) Diagram H, expanded once
+
+
+
+Recursive Structures and Processes
+
+
+135
+
+
+
+
+
+
+
+
+
+
+
+FIGURE 30. Diagram G, further expanded and with numbered nodes.
+defined recursively by the pair of formulas
+FIBO(n) = FIBO(n- 1) + FIBO(n 2) for n > 2
+
+FIBO(I) = FIBO(2) = 1
+
+Notice how new Fibonacci numbers are defined in terms of previous Fibonacci numbers.
+We could represent this pair of formulas in an RTN (see Fig. 31).
+
+
+
+FIGURE 31. An RTN for Fibonacci numbers.
+
+
+Thus you can calculate FIBO(15) by a sequence of recursive calls on the procedure
+defined by the RTN above. This recursive definition bottoms out when you hit FIBO(I)
+or FIBO(2) (which are given explicitly) after you have worked your way backwards
+through descending values of n. It is slightly awkward to work your way backwards,
+when you could just as well work your way forwards, starting with FIBO(I) and FIBO(2)
+and always adding the most recent two values, until you reach FIBO(15). That way you
+don't need to keep track of a stack.
+
+Now Diagram G has some even more surprising properties than this. Its entire
+structure can be coded up in a single recursive definition, as follows:
+
+
+Recursive Structures and Processes
+
+
+136
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+G(n) = n - G(G(n- 1)) for n > 0
+
+
+G(O) = 0
+
+How does this function G(n) code for the tree-structure? Quite simply you construct a
+tree by placing G(n) below n, for all values of n, you recreate Diagram G. In fact, that is
+how I discovered Diagram G in the place. I was investigating the function G, and in
+trying to calculate its values quickly, I conceived of displaying the values I already knew
+in a tree. T surprise, the tree turned out to have this extremely orderly recursive
+geometrical description.
+
+What is more wonderful is that if you make the analogous tree function H(n)
+defined with one more nesting than G—
+
+H(n) = n - H(H(H(n - 1))) for n > 0
+H(0) = 0
+
+
+—then the associated "Diagram H" is defined implicitly as shown in Figure 29c. The
+right-hand trunk contains one more node; that is the difference. The first recursive
+expansion of Diagram H is shown in Figure 29d. And so it goes, for any degree of
+nesting. There is a beautiful regularity to the recursive geometrical structures, which
+corresponds precisely to the recursive algebraic definitions.
+
+A problem for curious readers is: suppose you flip Diagram G around as if in a
+mirror, and label the nodes of the new tree so they increase left to right. Can you find a
+recursive algebraic definition for this "flip-tree. What about for the "flip" of the H-tree?
+Etc.?
+
+Another pleasing problem involves a pair of recursively intertwined functions
+F(n) and M(n) — "married" functions, you might say — defined this way:
+
+
+F(n) = n - M(F(n- 1))
+M(n) = n - F(M(n- 1))
+
+
+For n > 0
+
+
+F(0) = 1, and M(0) = 0
+
+
+The RTN's for these two functions call each other and themselves as well. The
+problem is simply to discover the recursive structures of Diagram F; and Diagram M.
+They are quite elegant and simple.
+
+
+A Chaotic Sequence
+
+One last example of recursion in number theory leads to a small my Consider the
+following recursive definition of a function:
+
+Q(n) = Q(n - Q(n- 1)) + Q(n - Q(n-2)) for n > 2
+Q(l) = Q(2) = 1.
+
+
+Recursive Structures and Processes
+
+
+137
+
+
+
+It is reminiscent of the Fibonacci definition in that each new value is a sum of two
+previous values-but not of the immediately previous two values. Instead, the two
+immediately previous values tell how far to count back to obtain the numbers to be added
+to make the new value! The first 17 Q-numbers run as follows:
+
+1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, ... .
+
+5 + 6 = 11 how far to move to the left
+
+New term
+
+To obtain the next one, move leftwards (from the three dots) respectively 10 and 9 terms;
+you will hit a 5 and a 6, shown by the arrows. Their sum-1 1-yields the new value: Q(18).
+This is the strange process by which the list of known Q-numbers is used to extend itself.
+The resulting sequence is, to put it mildly, erratic. The further out you go, the less sense it
+seems to make. This is one of those very peculiar cases where what seems to be a
+somewhat natural definition leads to extremely puzzling behavior: chaos produced in a
+very orderly manner. One is naturally led to wonder whether the apparent chaos conceals
+some subtle regularity. Of course, by definition, there is regularity, but what is of interest
+is whether there is another way of characterizing this sequence-and with luck, a
+nonrecursive way.
+
+
+Two Striking Recursive Graphs
+
+The marvels of recursion in mathematics are innumerable, and it is not my purpose to
+present them all. However, there are a couple of particularly striking examples from my
+own experience which I feel are worth presenting. They are both graphs. One came up in
+the course of some number-theoretical investigations. The other came up in the course of
+my Ph.D. thesis work, in solid state physics. What is truly fascinating is that the graphs
+are closely related.
+
+The first one (Fig. 32) is a graph of a function which I call INT(x). It is plotted
+here for x between 0 and 1. For x between any other pair of integers n and n + 1, you just
+find INT(x-n), then add n back. The structure of the plot is quite jumpy, as you can see. It
+consists of an infinite number of curved pieces, which get smaller and smaller towards
+the corners-and incidentally, less and less curved. Now if you look closely at each such
+piece, you will find that it is actually a copy of the full graph, merely curved! The
+implications are wild. One of them is that the graph of INT consists of nothing but copies
+of itself, nested down infinitely deeply. If you pick up any piece of the graph, no matter
+how small, you are holding a complete copy of the whole graph-in fact, infinitely many
+copies of it!
+
+The fact that INT consists of nothing but copies of itself might make you think it is too
+ephemeral to exist. Its definition sounds too circular.
+
+
+Recursive Structures and Processes
+
+
+138
+
+
+
+
+FIGURE 32. Graph of the function INT(x). There is a jump discontinuity at every rat
+value of x.
+
+
+How does it ever get off the ground? That is a very interesting matter, main thing to
+notice is that, to describe INT to someone who hasn't see it will not suffice merely to say,
+"It consists of copies of itself." The o half of the story-the nonrecursive half-tells where
+those copies lie in the square, and how they have been deformed, relative to the full
+graph. Only the combination of these two aspects of INT will specify structure of INT. It
+is exactly as in the definition of Fibonacci number where you need two lines-one to
+define the recursion, the other to de the bottom (i.e., the values at the beginning). To be
+very concrete, if make one of the bottom values 3 instead of 1, you will produce a
+completely different sequence, known as the Lucas sequence:
+
+1, 3, 4, 7, II, 18, 29, 47, 76, 123,...
+
+the "bottom" 29 + 47 = 76
+
+same recursive rule
+
+as for the Fibonacci numbers
+
+
+Recursive Structures and Processes
+
+
+139
+
+
+
+
+What corresponds to the bottom in the definition of INT is a picture (Fig. 33a)
+composed of many boxes, showing where the copies go, and how they are distorted. I call
+it the "skeleton" of INT. To construct INT from its skeleton, you do the following. First,
+for each box of the skeleton, you do two operations: (1) put a small curved copy of the
+skeleton inside the box, using the curved line inside it as a guide; (2) erase the containing
+box and its curved line. Once this has been done for each box of the original skeleton,
+you are left with many "baby" skeletons in place of one big one. Next you repeat the
+process one level down, with all the baby skeletons. Then again, again, and again ... What
+you approach in the limit is an exact graph of INT, though you never get there. By
+nesting the skeleton inside itself over and over again, you gradually construct the graph
+of INT "from out of nothing". But in fact the "nothing" was not nothing-it was a picture.
+
+To see this even more dramatically, imagine keeping the recursive part of the
+definition of INT, but changing the initial picture, the skeleton. A variant skeleton is
+shown in Figure 33b, again with boxes which get smaller and smaller as they trail off to
+the four corners. If you nest this second skeleton inside itself over and over again, you
+will create the key graph from my Ph.D. thesis, which I call Gplot (Fig. 34). (In fact,
+some complicated distortion of each copy is needed as well-but nesting is the basic idea.).
+
+Gplot is thus a member of the INT-family. It is a distant relative, because its
+skeleton is quite different from-and considerably more complex than-that of INT.
+However, the recursive part of the definition is identical, and therein lies the family tie.
+
+I should not keep you too much in the dark about the origin of these beautiful graphs.
+INT-standing for "interchange"-comes from a problem involving "Eta-sequences", which
+are related to continued fractions. The basic idea behind INT is that plus and minus signs
+are interchanged in a certain kind of continued fraction. As a consequence, INT(INT(x))
+= x. INT has the property that if x is rational, so is INT(x); if x is quadratic, so is INT(x).
+I do not know if this trend holds for higher algebraic degrees. Another lovely feature of
+INT is that at all rational values of x, it has a jump discontinuity, but at all irrational
+values of x, it is continuous.
+
+Gplot comes from a highly idealized version of the question, "What are the
+allowed energies of electrons in a crystal in a magnetic field?" This problem is interesting
+because it is a cross between two very simple and fundamental physical situations: an
+electron in a perfect crystal, and an electron in a homogeneous magnetic field. These two
+simpler problems are both well understood, and their characteristic solutions seem almost
+incompatible with each other. Therefore, it is of quite some interest to see how nature
+manages to reconcile the two. As it happens, the crystal without-magnetic-field situation
+and the magnetic-field-without-crystal situation do have one feature in common: in each
+of them, the electron behaves periodically in time. It turns out that when the two
+situations are combined, the ratio of their two time periods is the key parameter. In fact,
+that ratio holds all the information about the distribution of allowed electron energies-but
+it only gives up its secret upon being expanded into a continued fraction.
+
+
+Recursive Structures and Processes
+
+
+140
+
+
+
+(a)
+
+
+
+FIGURE 33(a) The skeleton from which 1ST can be constructed by recursive substitutions.
+
+(b) The skeleton from which Gplot can be constructed by recursive substitutions.
+
+
+
+Recursive Structures and Processes
+
+
+141
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Gplot shows that distribution. The horizontal axis represents energy, and the
+vertical axis represents the above-mentioned ratio of time periods, which we can call "a".
+At the bottom, a is zero, and at the top a is unity. When a is zero, there is no magnetic
+field. Each of the line segments making up Gplot is an "energy band"-that is, it represents
+allowed values of energy. The empty swaths traversing Gplot on all different size scales
+are therefore regions of forbidden energy. One of the most startling properties of Gplot is
+that when a is rational (say p/q in lowest terms), there are exactly q such bands (though
+when q is even, two of them "kiss" in the middle). And when a is irrational, the bands
+shrink to points, of which there are infinitely many, very sparsely distributed in a so-
+called "Cantor set" — another recursively defined entity which springs up in topology.
+
+You might well wonder whether such an intricate structure would ever show up in
+an experiment. Frankly, I would be the most surprised person in the world if Gplot came
+out of any experiment. The physicality of Gplot lies in the fact that it points the way to
+the proper mathematical treatment of less idealized problems of this sort. In other words,
+Gplot is purely a contribution to theoretical physics, not a hint to experimentalists as to
+what to expect to see! An agnostic friend of mine once was so struck by Gplot's infinitely
+many infinities that he called it "a picture of God", which I don't think is blasphemous at
+all.
+
+
+Recursion at the Lowest Level of Matter
+
+We have seen recursion in the grammars of languages, we have seen recursive
+geometrical trees which grow upwards forever, and we have seen one way in which
+recursion enters the theory of solid state physics. Now we are going to see yet another
+way in which the whole world is built out of recursion. This has to do with the structure
+of elementary particles: electrons, protons, neutrons, and the tiny quanta of
+electromagnetic radiation called "photons". We are going to see that particles are-in a
+certain sense which can only be defined rigorously in relativistic quantum mechanics —
+nested inside each other in a way which can be described recursively, perhaps even by
+some sort of "grammar".
+
+We begin with the observation that if particles didn't interact with each other,
+things would be incredibly simple. Physicists would like such a world because then they
+could calculate the behavior of all particles easily (if physicists in such a world existed,
+which is a doubtful proposition). Particles without interactions are called bare particles,
+and they are purely hypothetical creations; they don't exist.
+
+Now when you "turn on" the interactions, then particles get tangled up together in
+the way that functions F and M are tangled together, or married people are tangled
+together. These real particles are said to be renormalized -an ugly but intriguing term.
+What happens is that no particle can even be defined without referring to all other
+particles, whose definitions in turn depend on the first particles, etc. Round and round, in
+a never-ending loop.
+
+
+Recursive Structures and Processes
+
+
+142
+
+
+
+
+Figure 34. Gplot; a recursive graph, showing energy hands for electrons in an idealized
+crystal in a magnetic field, a representing magnetic field strength, runs vertically from 0
+to I. Energy runs horizontally. The horizontal line segments *re bands of allowed
+electron energies.
+
+
+Recursive Structures and Processes
+
+
+143
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Let us be a little more concrete, now. Let’s limit ourselves to only two kinds of
+particles: electrons and photons. We’ll also have to throw in the electron's antiparticle, the
+positron. (Photons are their own antipailicles.) Imagine first a dull world where a bare
+electron wishes to propagate from point A to point B, as Zeno did in my Three-Part
+Invention. A physicist would draw a picture like this:
+
+
+A 9 -►-« B
+
+
+There is a mathematical expression which corresponds to this line and its endpoints, and
+it is easy to write down. With it, a physicist can understand the behavior of the bare
+electron in this trajectory.
+
+Now let us "turn on" the electromagnetic interaction, whereby electrons and
+photons interact. Although there are no photons in the scene, there will nevertheless be
+profound consequences even for this simple trajectory. In particular, our electron now
+becomes capable of emitting and then reabsorbing virtual photons -photons which flicker
+in and out of existence before they can be seen. Let us show one such process:
+
+
+
+Now as our electron propagates, it may emit and reabsorb one photon after another, or it
+may even nest them, as shown below:
+
+
+
+The mathematical expressions corresponding to these diagrams-called "Feynman
+diagrams"-are easy to write down, but they are harder to calculate than that for the bare
+electron. But what really complicates matters is that a photon (real or virtual) can decay
+for a brief moment into an electron-positron pair. Then these two annihilate each other,
+and, as if by magic, the original photon reappears. This sort of process is shown below:
+
+
+
+The electron has a right-pointing arrow, while the positron's arrow points leftwards.
+
+
+Recursive Structures and Processes
+
+
+144
+
+
+
+
+
+
+
+
+As you might have anticipated, these virtual processes can be inside each other to
+arbitrary depth. This can give rise to some complicated-looking drawings, such as the one
+in Figure 35. In that man diagram, a single electron enters on the left at A, does some an
+acrobatics, and then a single electron emerges on the right at B. outsider who can't see the
+inner mess, it looks as if one electron peacefully sailed from A to B. In the diagram, you
+can see how el lines can get arbitrarily embellished, and so can the photon lines diagram
+would be ferociously hard to calculate.
+
+
+
+FIGURE 35. A Feynman diagram showing the propagation of a renormalized electron
+from A to B. In this diagram, time increases to the right. Therefore, in the segments
+where the electron's arrow points leftwards, it is moving "backwards in time”. A more
+intuitive way to say this is that an antielectron (positron) is moving forwards in time.
+Photons are their own antiparticles; hence their lines have no need of arrows.
+
+There is a soil of "grammar" to these diagrams, that only certain pictures to be
+realized in nature. For instance, the one be impossible:
+
+
+
+You might say it is not a "well-formed” Feynman diagram. The gram a result of basic
+laws of physics, such as conservation of energy, conservation of electric charge, and so
+on. And, like the grammars of I - languages, this grammar has a recursive structure, in
+that it allow' nestings of structures inside each other. It would be possible to drat set of
+recursive transition networks defining the "grammar” of the electromagnetic interaction.
+
+When bare electrons and bare photons are allowed to interact ii arbitrarily tangled
+ways, the result is renormalized electrons and ph Thus, to understand how a real, physical
+electron propagates from A to B,
+
+
+Recursive Structures and Processes
+
+
+145
+
+
+
+
+
+
+the physicist has to be able to take a sort of average of all the infinitely many different
+possible drawings which involve virtual particles. This is Zeno with a vengeance!
+
+Thus the point is that a physical particle-a renormalized particle involves (1) a
+bare particle and (2) a huge tangle of virtual particles, inextricably wound together in a
+recursive mess. Every real particle's existence therefore involves the existence of
+infinitely many other particles, contained in a virtual "cloud" which surrounds it as it
+propagates. And each of the virtual particles in the cloud, of course, also drags along its
+own virtual cloud, and so on ad infinitum.
+
+Particle physicists have found that this complexity is too much to handle, and in
+order to understand the behavior of electrons and photons, they use approximations
+which neglect all but fairly simple Feynman diagrams. Fortunately, the more complex a
+diagram, the less important its contribution. There is no known way of summing up all of
+the infinitely many possible diagrams, to get an expression for the behavior of a fully
+renormalized, physical electron. But by considering roughly the simplest hundred
+diagrams for certain processes, physicists have been able to predict one value (the so-
+called g-factor of the muon) to nine decimal places — correctly!
+
+Renormalization takes place not only among electrons and photons. Whenever
+any types of particle interact together, physicists use the ideas of renormalization to
+understand the phenomena. Thus protons and neutrons, neutrinos, pi-mesons, quarks-all
+the beasts in the subnuclear zoo they all have bare and renormalized versions in physical
+theories. And from billions of these bubbles within bubbles are all the beasts and baubles
+of the world composed.
+
+
+Copies and Sameness
+
+Fet us now consider Gplot once again. You will remember that in the
+Introduction, we spoke of different varieties of canons. Each type of canon exploited
+some manner of taking an original theme and copying it by an isomorphism, or
+information-preserving transformation. Sometimes the copies were upside down,
+sometimes backwards, sometimes shrunken or expanded ... In Gplot we have all those
+types of transformation, and more. The mappings between the full Gplot and the "copies"
+of itself inside itself involve size changes, skewings, reflections, and more. And yet there
+remains a sort of skeletal identity, which the eye can pick up with a bit of effort,
+particularly after it has practiced with INT.
+
+Escher took the idea of an object's parts being copies of the object itself and made
+it into a print: his woodcut Fishes and Scales (Fig. 36). Of course these fishes and scales
+are the same only when seen on a sufficiently abstract plane. Now everyone knows that a
+fish's scales aren't really small copies of the fish; and a fish's cells aren't small copies of
+the fish; however, a fish's DNA, sitting inside each and every one of the fish's cells, is a
+very convo-
+
+
+Recursive Structures and Processes
+
+
+146
+
+
+
+
+FIGURE 36. Fish and Scales, by M. C. Escher (woodcut, 1959).
+
+luted "copy" of the entire fish-and so there is more than a grain of truth to the Escher
+picture.
+
+What is there that is the "same" about all butterflies? The mapping from one
+butterfly to another does not map cell onto cell; rather, it m; functional part onto
+functional part, and this may be partially on a macroscopic scale, partially on a
+microscopic scale. The exact proportions of pa are not preserved; just the functional
+relationships between parts. This is the type of isomorphism which links all butterflies in
+Escher’s wood engraving Butterflies (Fig. 37) to each other. The same goes for the more
+abstract butterflies of Gplot, which are all linked to each other by mathematical mappings
+that carry functional part onto functional part, but totally ignore exact line proportions,
+angles, and so on.
+
+Taking this exploration of sameness to a yet higher plane of abstraction, we might
+well ask, "What is there that is the 'same' about all Esc I drawings?" It would be quite
+ludicrous to attempt to map them piece by piece onto each other. The amazing thing is
+that even a tiny section of an
+
+
+Recursive Structures and Processes
+
+
+147
+
+
+
+FIGURE 37. Butterflies, by M. C. Escher (wood-engraving, 1950).
+
+Escher drawing or a Bach piece gives it away. Just as a fish's DNA is contained inside
+evety tiny bit of the fish, so a creator's "signature" is contained inside every tiny section
+of his creations. We don't know what to call it but "style" — a vague and elusive word.
+
+We keep on running up against "sameness-in-differentness", and the question
+
+When are two things the same?
+
+It will recur over and over again in this book. We shall come at it from all sorts of skew
+angles, and in the end, we shall see how deeply this simple question is connected with the
+nature of intelligence.
+
+That this issue arose in the Chapter on recursion is no accident, for recursion is a
+domain where "sameness-in-differentness" plays a central role. Recursion is based on the
+"same" thing happening on several differ-
+
+
+Recursive Structures and Processes
+
+
+148
+
+
+
+
+ent levels at once. But the events on different levels aren't exactly same-rather, we find
+some invariant feature in them, despite many s in which they differ. For example, in the
+Little Harmonic Labyrinth, all stories on different levels are quite unrelated-their
+"sameness" reside only two facts: (1) they are stories, and (2) they involve the Tortoise
+and Achilles. Other than that, they are radically different from each other.
+
+Programming and Recursion: Modularity, Loops, Procedures
+
+One of the essential skills in computer programming is to perceive wl two processes are
+the same in this extended sense, for that leads modularization-the breaking-up of a task
+into natural subtasks. For stance, one might want a sequence of many similar operations
+to be cart out one after another. Instead of writing them all out, one can write a h which
+tells the computer to perform a fixed set of operations and then loop back and perform
+them again, over and over, until some condition is satisfied. Now the body of the loop-the
+fixed set of instructions to repeated-need not actually be completely fixed. It may vary in
+so predictable way.
+
+An example is the most simple-minded test for the primality o natural number N,
+in which you begin by trying to divide N by 2, then 3, 4, 5, etc. until N - 1. If N has
+survived all these tests without be divisible, it's prime. Notice that each step in the loop is
+similar to, but i the same as, each other step. Notice also that the number of steps varies
+with N-hence a loop of fixed length could never work as a general test primality. There
+are two criteria for "aborting" the loop: (1) if so number divides N exactly, quit with
+answer "NO"; (2) if N - 1 is react as a test divisor and N survives, quit with answer
+"YES".
+
+The general idea of loops, then, is this: perform some series of related steps over
+and over, and abort the process when specific conditions are n Now sometimes, the
+maximum number of steps in a loop will be known advance; other times, you just begin,
+and wait until it is aborted. The second type of loop — which I call a free loop — is
+dangerous, because criterion for abortion may never occur, leaving the computer in a so-
+cal "infinite loop". This distinction between bounded loops and free loops is one the most
+important concepts in all of computer science, and we shall dev an entire Chapter to it:
+"BlooP and FlooP and GlooP".
+
+Now loops may be nested inside each other. For instance, suppose t we wish to
+test all the numbers between 1 and 5000 for primality. We c write a second loop which
+uses the above-described test over and over starting with N = I and finishing with N =
+5000. So our program i have a "loop-the-loop" structure. Such program structures are
+typical - in fact they are deemed to be good programming style. This kind of nest loop
+also occurs in assembly instructions for commonplace items, and such activities as
+knitting or crocheting-in which very small loops are
+
+
+Recursive Structures and Processes
+
+
+149
+
+
+
+repeated several times in larger loops, which in turn are carried out repeatedly ... While
+the result of a low-level loop might be no more than couple of stitches, the result of a
+high-level loop might be a substantial portion of a piece of clothing.
+
+In music, too, nested loops often occur-as, for instance, when a scale (a small
+loop) is played several times in a row, perhaps displaced in pitch each new time. For
+example, the last movements of both the Prokofiev fifth piano concerto and the
+Rachmaninoff second symphony contain extended passages in which fast, medium, and
+slow scale-loops are played simultaneously by different groups of instruments, to great
+effect. The Prokofiev scales go up; the Rachmaninoff-scales, down. Take your pick.
+
+A more general notion than loop is that of subroutine, or procedure, which we
+have already discussed somewhat. The basic idea here is that a group of operations are
+lumped together and considered a single unit with a name-such as the procedure
+ORNATE NOUN. As we saw in RTN's, procedures can call each other by name, and
+thereby express very concisely sequences of operations which are to be carried out. This
+is the essence of modularity in programming. Modularity exists, of course, in hi-fi
+systems, furniture, living cells, human society-wherever there is hierarchical
+organization.
+
+More often than not, one wants a procedure which will act variably, according to
+context. Such a procedure can either be given a way of peering out at what is stored in
+memory and selecting its actions accordingly, or it can be explicitly fed a list of
+parameters which guide its choice of what actions to take. Sometimes both of these
+methods are used. In RTN terminology, choosing the sequence of actions to carry out
+amounts to choosing which pathway to follow. An RTN which has been souped up with
+parameters and conditions that control the choice of pathways inside it is called an
+Augmented Transition Network (ATN). A place where you might prefer ATN's to RTN's
+is in producing sensible-as distinguished from nonsensical-English sentences out of raw
+words, according to a grammar represented in a set of ATN's. The parameters and
+conditions would allow you to insert various semantic constraints, so that random
+juxtapositions like "a thankless brunch" would be prohibited. More on this in Chapter
+XVIII, however.
+
+
+Recursion in Chess Programs
+
+A classic example of a recursive procedure with parameters is one for choosing the "best"
+move in chess. The best move would seem to be the one which leaves your opponent in
+the toughest situation. Therefore, a test for goodness of a move is simply this: pretend
+you've made the move, and now evaluate the board from the point of view of your
+opponent. But how does your opponent evaluate the position? Well, he looks for his best
+move. That is, he mentally runs through all possible moves and evaluates them from what
+he thinks is your point of view, hoping they will look bad to you. But
+
+
+Recursive Structures and Processes
+
+
+150
+
+
+
+notice that we have now defined "best move" recursively, simply maxim that what is best
+for one side is worst for the other. The procedure which looks for the best move operates
+by trying a move and then colling on itself in the rote of opponent ! As such, it tries
+another n calls on itself in the role of its opponent’s opponent-that is, its
+
+This recursion can go several levels deep-but it’s got to bottom out somewhere!
+How do you evaluate a board position wit hoot looking There are a number of useful
+criteria for this purpose, such as si number of pieces on each side, the number and type of
+pieces undo the control of the center, and so on. By using this kind of evaluation at the
+bottom, the recursive move-generator can pop back upwards an( evaluation at the top
+level of each different move. One of the parameters in the self-calling, then, must tell
+how many moves to look ahead. TI most call on the procedure will use some externally
+set value parameter. Thereafter, each time the procedure recursively calls must decrease
+this look-ahead parameter by 1. That way, w parameter reaches zero, the procedure will
+follow the alternate pathway -the non-recursive evaluation.
+
+In this kind of game-playing program, each move investigate the generation of a
+so-called "look-ahead tree", with the move trunk, responses as main branches, counter¬
+responses as subsidiary branches, and so on. In Figure 38 I have shown a simple look¬
+ahead tree depicting the start of a tic-tar-toe game. There is an art to figuring to avoid
+exploring every branch of a look-ahead tree out to its tip. trees, people-not computers-
+seem to excel at this art; it is known that top-level players look ahead relatively little,
+compared to most chess programs - yet the people are far better! In the early days of
+compute people used to estimate that it would be ten years until a computer (or
+
+FIGURE 38. The branching tree of moves and countermoves at the start of c tic-tac-toe.
+
+
+
+
+Recursive Structures and Processes
+
+
+151
+
+
+
+
+
+program) was world champion. But after ten years had passed, it seemed that the day a
+computer would become world champion was still more than ten years away ... This is
+just one more piece of evidence for the rather recursive
+
+Hofstadter's Law. It always takes longer than you expect, even when you take into
+account Hofstadter's Law.
+
+Recursion and Unpredictability
+
+Now what is the connection between the recursive processes of this Chapter, and the
+recursive sets of the preceding Chapter? The answer involves the notion of a recursively
+enumerable set. For a set to be r.e. means that it can be generated from a set of starting
+points (axioms), by the repeated application of rules of inference. Thus, the set grows and
+grows, each new element being compounded somehow out of previous elements, in a sort
+of "mathematical snowball". But this is the essence of recursion-something being defined
+in terms of simpler versions of itself, instead of explicitly. The Fibonacci numbers and
+the Lucas numbers are perfect examples of r.e. sets-snowballing from two elements by a
+recursive rule into infinite sets. It is just a matter of convention to call an r.e. set whose
+complement is also r.e. "recursive".
+
+Recursive enumeration is a process in which new things emerge from old things
+by fixed rules. There seem to be many surprises in such processes-for example the
+unpredictability of the Q-sequence. It might seem that recursively defined sequences of
+that type possess some sort of inherently increasing complexity of behavior, so that the
+further out you go, the less predictable they get. This kind of thought carried a little
+further suggests that suitably complicated recursive systems might be strong enough to
+break out of any predetermined patterns. And isn't this one of the defining properties of
+intelligence? Instead of just considering programs composed of procedures which can
+recursively call themselves, why not get really sophisticated, and invent programs which
+can modify themselves-programs which can act on programs, extending them, improving
+them, generalizing them, fixing them, and so on? This kind of "tangled recursion"
+probably lies at the heart of intelligence.
+
+
+Recursive Structures and Processes
+
+
+152
+
+
+
+Canon
+
+by Intervallic Augmentation
+
+
+Achilles and the Tortoise have just finished a delicious Chinese banquet for
+two, at the best Chinese restaurant in town.
+
+
+Achilles: You wield a mean chopstick, Mr. T.
+
+Tortoise: I ought to. Ever since my youth, I have had a fondness for Oriental cuisine. And you-
+did you enjoy your meal, Achilles? Achilles: Immensely. I'd not eaten Chinese food
+before. This meal was a splendid introduction. And now, are you in a hurry to go, or shall
+we just sit here and talk a little while?
+
+Tortoise: I'd love to talk while we drink our tea. Waiter!
+
+(A waiter comes up.)
+
+Could we have our bill, please, and some more tea?
+
+(The waiter rushes off.)
+
+Achilles: You may know more about Chinese cuisine than I do, Mr.T, I'll bet I know more about
+Japanese poetry than you do. Have you ever read any haiku?
+
+Tortoise: I'm afraid not. What is a haiku?
+
+Achilles: A haiku is a Japanese seventeen-syllable poem-or minipoem rather, which is evocative
+in the same way, perhaps, as a fragrant petal is, or a lily pond in a light drizzle. It
+generally consists of groups of: of five, then seven, then five syllables.
+
+Tortoise: Such compressed poems with seventeen syllables can't much meaning ...
+
+Achilles: Meaning lies as much in the mind of the reader as i haiku.
+
+Tortoise: Hmm ... That's an evocative statement.
+
+(The waiter arrives with their bill, another pot of tea, and two fortune cookies.)
+
+Thank you, waiter. Care for more tea, Achilles?
+
+Achilles: Please. Those little cookies look delicious. ( Picks one up, bites I into it and begins to
+chew.) Hey! What's this funny thing inside? A piece of paper?
+
+Tortoise: That's your fortune, Achilles. Many Chinese restaurants give out fortune cookies with
+their bills, as a way of softening the blow. I frequent Chinese restaurants, you come to
+think of fortune cookies
+
+
+
+less as cookies than as message bearers Unfortunately you seem to have swallowed some
+of your fortune. What does the rest say?
+
+Achilles: It's a little strange, for all the letters are run together, with no spaces in between.
+Perhaps it needs decoding in some way? Oh, now I see. If you put the spaces back in
+where they belong, it says, "ONE WAR TWO EAR EWE". I can't quite make head or tail
+of that. Maybe it was a haiku-like poem, of which I ate the majority of syllables.
+
+Tortoise: In that case, your fortune is now a mere 5/17-haiku. And a curious image it evokes. If
+5/17-haiku is a new art form, then I'd say woe, 0, woe are we ... May I look at it?
+
+Achilles ( handing the Tortoise the small slip of paper): Certainly.
+
+Tortoise: Why, when I "decode" it, Achilles, it comes out completely different! It's not a 5/17-
+haiku at all. It is a six-syllable message which says, "0 NEW ART WOE ARE WE". That
+sounds like an insightful commentary on the new art form of 5/17-haiku.
+
+Achilles: You're right. Isn't it astonishing that the poem contains its own commentary!
+
+Tortoise: All I did was to shift the reading frame by one unit-that is, shift all the spaces one unit
+to the right.
+
+Achilles: Let's see what your fortune says, Mr. Tortoise.
+
+Tortoise ( deftly splitting open his cookie, reads): "Fortune lies as much in the hand of the eater as
+in the cookie."
+
+Achilles: Your fortune is also a haiku, Mr. Tortoise-at least it's got seventeen syllables in the 5-7-
+5 form.
+
+Tortoise: Glory be! I would never have noticed that, Achilles. It's the kind of thing only you
+would have noticed. What struck me more is what it says-which, of course, is open to
+interpretation.
+
+Achilles: I guess it just shows that each of us has his own characteristic way of interpreting
+messages which we run across ...
+
+(Idly, Achilles gazes at the tea leaves on the bottom of his empty teacup.)
+
+Tortoise: More tea, Achilles?
+
+Achilles: Yes, thank you. By the way, how is your friend the Crab? I have been thinking about
+him a lot since you told me of your peculiar phonograph-battle.
+
+Tortoise: I have told him about you, too, and he is quite eager to meet you. He is getting along
+just fine. In fact, he recently made a new acquisition in the record player line: a rare type
+of jukebox.
+
+Achilles: Oh, would you tell me about it? I find jukeboxes, with their flashing colored lights and
+silly songs, so quaint and reminiscent of bygone eras.
+
+Tortoise: This jukebox is too large to fit in his house, so he had a shed specially built in back for
+it.
+
+Achilles: I can't imagine why it would be so large, unless it has an unusually large selection of
+records. Is that it?
+
+Tortoise: As a matter of fact, it has exactly one record.
+
+
+
+Achilles: What? A jukebox with only one record? That's a contradiction in terms. Why is the
+jukebox so big, then? Is its single record gigantic — twenty feet in diameter?
+
+Tortoise: No, it's just a regular jukebox-style record.
+
+Achilles: Now, Mr. Tortoise, you must be joshing me. After all, what I
+
+of a jukebox is it that has only a single song?
+
+Tortoise: Who said anything about a single song, Achilles?
+
+Achilles: Every- jukebox I've ever run into obeyed the fundamental jukebox-axiom: "One record,
+one song".
+
+Tortoise: This jukebox is different, Achilles. The one record sits vertically, suspended, and
+behind it there is a small but elaborate network of overhead rails, from which hang
+various record players. When push a pair of buttons, such as B-l, that selects one of the
+record players. This triggers an automatic mechanism that starts the record player
+squeakily rolling along the rusty tracks. It gets shunted alongside the record-then it clicks
+into playing position.
+
+Achilles: And then the record begins spinning and music comes out — right?
+
+Tortoise: Not quite. The record stands still-it's the record player which rotates.
+
+Achilles: I might have known. But how, if you have but one record to play can you get more than
+one song out of this crazy contraption?
+
+Tortoise: I myself asked the Crab that question. He merely suggested I try it out. So I fished a
+quarter from my pocket (you get three plays for a quarter), stuffed it in the slot, and hit
+buttons B-l, then C-3 then B-10-all just at random.
+
+Achilles: So phonograph B-l came sliding down the rail, I suppose, plugged itself into the
+vertical record, and began spinning?
+
+Tortoise: Exactly. The music that came out was quite agreeable, based the famous old tune B-A-
+C-H, which I believe you remember.
+
+
+
+
+m r +
+
+
+
+- ~ .
+
+
+Achilles: Could 1 ever forget it?
+
+Tortoise: This was record player B-l. Then it finished, and was s rolled back into its hanging
+position, so that C-3 could be slid into position.
+
+Achilles: Now don’t tell me that C-3 played another song?
+
+Tortoise: It did just that.
+
+Achilles: Ah, I understand. It played the flip side of the first song, or another band on the same
+side.
+
+Tortoise: No, the record has grooves only on one side, and has only a single band.
+
+
+
+
+Achilles: I don't understand that at all. You CAN'T pull different songs out of the same record!
+
+Tortoise: That's what I thought until I saw Mr. Crab's jukebox. Achilles: How did the second
+song go?
+
+Tortoise: That's the interesting thing ... It was a song based on the melody C-A-G-E.
+
+Achilles: That's a totally different melody!
+
+Tortoise: True.
+
+Achilles: And isn't John Cage a composer of modern music? I seem to remember reading about
+him in one of my books on haiku.
+
+Tortoise: Exactly. He has composed many celebrated pieces, such as 4'33", a three-movement
+piece consisting of silences of different lengths. It's wonderfully expressive-if you like
+that sort of thing.
+
+Achilles: I can see where if I were in a loud and brash cafe I might gladly pay to hear Cage's
+4'33" on a jukebox. It might afford some relief!
+
+Tortoise: Right-who wants to hear the racket of clinking dishes and jangling silverware? By the
+way, another place where 4'33" would come in handy is the Hall of Big Cats, at feeding
+time.
+
+Achilles: Are you suggesting that Cage belongs in the zoo? Well, I guess that makes some sense.
+But about the Crab's jukebox ... I am baffled. How could both "BACH" and "CAGE" be
+coded inside a single record at once?
+
+Tortoise: You may notice that there is some relation between the two, Achilles, if you inspect
+them carefully. Let me point the way. What do you get if you list the successive intervals
+in the melody B-A-C-H?
+
+Achilles: Let me see. Lirst it goes down one semitone, from B to A (where B is taken the
+German way); then it rises three semitones to C; and finally it falls one semitone, to H.
+That yields the pattern:
+
+
+-1, +3,-1.
+
+Tortoise: Precisely. What about C-A-G-E, now?
+
+Achilles: Well, in this case, it begins by falling three semitones, then ten semitones (nearly an
+octave), and finally falls three more semitones. That means the pattern is:
+
+-3,+10, -3.
+
+
+It's very much like the other one, isn't it?
+
+Tortoise: Indeed it is. They have exactly the same "skeleton", in a certain sense. You can make
+C-A-G-E out of B-A-C-H by multiplying all the intervals by 31/3, and taking the nearest
+whole number.
+
+Achilles: Well, blow me down and pick me up! So does that mean that only
+
+
+
+some sort of skeletal code is present in the grooves, and that the various record players
+add their own interpretations to that code?
+
+Tortoise: I don’t know, for sure. The cagey Crab wouldn't fill me in on the details. But I did get
+to hear a third song, when record player B-l swiveled into place.
+
+Achilles: How did it go?
+
+Tortoise: The melody consisted of enormously wide intervals, and we B-C-A-H.
+
+
+
+The interval pattern in semitones was:
+
+-10, +33, -10.
+
+It can be gotten from the CAGE pattern by yet another multiplication by 3%3, and
+rounding to whole numbers.
+
+Achilles: Is there a name for this kind of interval multiplication?
+
+Tortoise: One could call it "intervallic augmentation". It is similar to tl canonic device of
+temporal augmentation, where all the time values notes in a melody get multiplied by
+some constant. There, the effect just to slow the melody down. Here, the effect is to
+expand the melodic range in a curious way.
+
+Achilles: Amazing. So all three melodies you tried were intervallic augmentations of one single
+underlying groove-pattern in the record:
+
+Tortoise: That’s what I concluded.
+
+Achilles: I find it curious that when you augment BACH you get CAGE and when you augment
+CAGE over again, you get BACH back, except jumbled up inside, as if BACH had an
+upset stomach after passing through the intermediate stage of CAGE.
+
+Toiloise: That sounds like an insightful commentary on the new art form of Cage.
+
+
+
+
+
+
+
+
+CHAPTER VI
+
+
+The Location of Meaning
+
+When Is One Thing Not Always the Same?
+
+LAST CHAPTER, WE came upon the question, "When are two things the same?" In this
+Chapter, we will deal with the flip side of that question: "When is one thing not always
+the same?" The issue we are broaching is whether meaning can be said to be inherent in a
+message, or whether meaning is always manufactured by the interaction of a mind or a
+mechanism with a message-as in the preceding Dialogue. In the latter case, meaning
+could not said to be located in any single place, nor could it be said that a message has
+any universal, or objective, meaning, since each observer could bring its own meaning to
+each message. But in the former case, meaning would have both location and
+universality. In this Chapter, I want to present the case for the universality of at least
+some messages, without, to be sure, claiming it for all messages. The idea of an
+"objective meaning" of a message will turn out to be related, in an interesting way, to the
+simplicity with which intelligence can be described.
+
+Information-Bearers and Information- Revealers
+
+I'll begin with my favorite example: the relationship between records, music, and record
+players. We feel quite comfortable with the idea that a record contains the same
+information as a piece of music, because of the existence of record players, which can
+"read" records and convert the groove-patterns into sounds. In other words, there is an
+isomorphism between groove-patterns and sounds, and the record player is a mechanism
+which physically realizes that isomorphism. It is natural, then, to think of the record as an
+information-bearer, and the record-player as an information-revealer. A second example
+of these notions is given by the pq-system. There, the "information-bearers" are the
+theorems, and the "information-revealer" is the interpretation, which is so transparent that
+we don't need any electrical machine to help us extract the information from pq-
+theorems.
+
+One gets the impression from these two examples that isomorphisms and
+decoding mechanisms (i.e., information-revealers) simply reveal information which is
+intrinsically inside the structures, waiting to be "pulled out". This leads to the idea that
+for each structure, there are certain pieces of information which can be pulled out of it,
+while there are other pieces of information which cannot be pulled out of it. But what
+does this phrase
+
+
+The Location of Meaning
+
+
+158
+
+
+
+"pull out" really mean? How hard are you allowed to pull? There are c where by
+investing sufficient effort, you can pull very recondite piece of information out of certain
+structures. In fact, the pulling-out may inv such complicated operations that it makes you
+feel you are putting in n information than you are pulling out.
+
+Genotype and Phenotype
+
+Take the case of the genetic information commonly said to reside in double helix of
+deoxyribonucleic acid (DNA). A molecule of DNA - a genotype-is converted into a
+physical organism-a phenotype -by a complex process, involving the manufacture of
+proteins, the replication the DNA, the replication of cells, the gradual differentiation of
+cell types and so on. Incidentally, this unrolling of phenotype from genotype epigenesis-
+is the most tangled of tangled recursions, and in Chapter we shall devote our full attention
+to it. Epigenesis is guided by a se enormously complex cycles of chemical reactions and
+feedback loops the time the full organism has been constructed, there is not even remotest
+similarity between its physical characteristics and its genotype.
+
+And yet, it is standard practice to attribute the physical structure of organism to
+the structure of its DNA, and to that alone. The first evidence for this point of view came
+from experiments conducted by Oswald A, in 1946, and overwhelming corroborative
+evidence has since been amassed Avery's experiments showed that, of all the biological
+molecules, only E transmits hereditary properties. One can modify other molecules it
+organism, such as proteins, but such modifications will not be transmitted to later
+generations. However, when DNA is modified, all successive generations inherit the
+modified DNA. Such experiments show that the only of changing the instructions for
+building a new organism is to change DNA-and this, in turn, implies that those
+instructions must be cc somehow in the structure of the DNA.
+
+Exotic and Prosaic Isomorphisms
+
+Therefore one seems forced into accepting the idea that the DNA's structure contains the
+information of the phenotype's structure, which is to the two are isomorphic. However,
+the isomorphism is an exotic one, by w] I mean that it is highly nontrivial to divide the
+phenotype and genotype into "parts" which can be mapped onto each other. Prosaic
+isomorphic by contrast, would be ones in which the parts of one structure are easily
+mappable onto the parts of the other. An example is the isomorphism between a record
+and a piece of music, where one knows that to any so in the piece there exists an exact
+"image" in the patterns etched into grooves, and one could pinpoint it arbitrarily
+accurately, if the need arose Another prosaic isomorphism is that between Gplot and any
+of its internal butterflies.
+
+
+The Location of Meaning
+
+
+159
+
+
+
+The isomorphism between DNA structure and phenotype structure is anything but
+prosaic, and the mechanism which carries it out physically is awesomely complicated.
+For instance, if you wanted to find some piece of your DNA which accounts for the shape
+of your nose or the shape of your fingerprint, you would have a very hard time. It would
+be a little like trying to pin down the note in a piece of music which is the carrier of the
+emotional meaning of the piece. Of course there is no such note, because the emotional
+meaning is carried on a very high level, by large "chunks" of the piece, not by single
+notes. Incidentally, such "chunks" are not necessarily sets of contiguous notes; there may
+be disconnected sections which, taken together, carry some emotional meaning.
+
+Similarly, "genetic meaning"-that is, information about phenotype structure-is
+spread all through the small parts of a molecule of DNA, although nobody understands
+the language yet. (Warning: Understanding this "language" would not at all be the same
+as cracking the Genetic Code, something which took place in the early 1960's. The
+Genetic Code tells how to translate short portions of DNA into various amino acids.
+Thus, cracking the Genetic Code is comparable to figuring out the phonetic values of the
+letters of a foreign alphabet, without figuring out the grammar of the language or the
+meanings of any of its words. The cracking of the Genetic Code was a vital step on the
+way to extracting the meaning of DNA strands, but it was only the first on a long path
+which is yet to be trodden.)
+
+
+Jukeboxes and Triggers
+
+The genetic meaning contained in DNA is one of the best possible examples of implicit
+meaning. In order to convert genotype into phenotype, a set of mechanisms far more
+complex than the genotype must operate on the genotype. The various parts of the
+genotype serve as triggers for those mechanisms. A jukebox-the ordinary type, not the
+Crab type!-provides a useful analogy here: a pair of buttons specifies a very complex
+action to be taken by the mechanism, so that the pair of buttons could well be described
+as "triggering" the song which is played. In the process which converts genotype into
+phenotype, cellular jukeboxes-if you will pardon the notion!-accept "button-pushings"
+from short excerpts from a long strand of DNA, and the "songs" which they play are
+often prime ingredients in the creation of further "jukeboxes". It is as if the output of real
+jukeboxes, instead of being love ballads, were songs whose lyrics told how to build more
+complex jukeboxes ... Portions of the DNA trigger the manufacture of proteins; those
+proteins trigger hundreds of new reactions; they in turn trigger the replicating-operation
+which, in several steps, copies the DNA-and on and on ... This gives a sense of how
+recursive the whole process is. The final result of these many-triggered triggerings is the
+phenotype-the individual. And one says that the phenotype is the revelation-the "pulling-
+out"-of the information that was present in the DNA to start with, latently. (The term
+"revelation" in this context is due to
+
+
+The Location of Meaning
+
+
+160
+
+
+
+Jacques Monod, one of the deepest and most original of twentieth-century molecular
+biologists.)
+
+Now no one would say that a song coming out of the loudspeaker of jukebox
+constitutes a "revelation" of information inherent in the pair buttons which were pressed,
+for the pair of buttons seem to be mere triggers, whose purpose is to activate information¬
+bearing portions of the jukebox mechanism. On the other hand, it seems perfectly
+reasonable to call t extraction of music from a record a "revelation" of information
+inherent the record, for several reasons:
+
+(1) the music does not seem to be concealed in the mechanism of the record player;
+
+(2) it is possible to match pieces of the input (the record) with pieces of the output
+(the music) to an arbitrary degree of accuracy;
+
+(3) it is possible to play other records on the same record player and get other
+sounds out;
+
+(4) the record and the record player are easily separated from one another.
+
+It is another question altogether whether the fragments of a smashed record contain
+intrinsic meaning. The edges of the separate pieces together and in that way allow the
+information to be reconstituted-t something much more complex is going on here. Then
+there is the question of the intrinsic meaning of a scrambled telephone call ... There is a
+vast spectrum of degrees of inherency of meaning. It is interesting to try place epigenesis
+in this spectrum. As development of an organism takes place, can it be said that the
+information is being "pulled out" of its DNA? Is that where all of the information about
+the organism's structure reside;
+
+DNA and the Necessity of Chemical Context
+
+In one sense, the answer seems to be yes, thanks to experiments li Avery's. But in another
+sense, the answer seems to be no, because so much of the pulling-out process depends on
+extraordinarily complicated cellular chemical processes, which are not coded for in the
+DNA itself. The DNA relies on the fact that they will happen, but does not seem to
+contain a code which brings them about. Thus we have two conflicting views on the
+nature of the information in a genotype. One view says that so much of t information is
+outside the DNA that it is not reasonable to look upon the DNA as anything more than a
+very intricate set of triggers, like a sequence of buttons to be pushed on a jukebox;
+another view says that the information is all there, but in a very implicit form.
+
+Now it might seem that these are just two ways of saying the same thing, but that
+is not necessarily so. One view says that the DNA is quite meaningless out of context; the
+other says that even if it were taken out context, a molecule of DNA from a living being
+has such a compelling inner
+
+
+The Location of Meaning
+
+
+161
+
+
+
+logic to its structure that its message could be deduced anyway. To put it as succinctly as
+possible, one view says that in order for DNA to have meaning, chemical context is
+necessary; the other view says that only intelligence is necessary to reveal the "intrinsic
+meaning" of a strand of DNA.
+
+
+An Unlikely UFO
+
+We can get some perspective on this issue by considering a strange hypothetical
+event. A record of David Oistrakh and Lev Oborin playing Bach's sonata in F Minor for
+violin and clavier is sent up in a satellite. From the satellite it is then launched on a course
+which will carry it outside of the solar system, perhaps out of the entire galaxy just a thin
+plastic platter with a hole in the middle, swirling its way through intergalactic space. It
+has certainly lost its context. How much meaning does it carry?
+
+If an alien civilization were to encounter it, they would almost certainly be struck
+by its shape, and would probably be very interested in it. Thus immediately its shape,
+acting as a trigger, has given them some information: that it is an artifact, perhaps an
+information-bearing artifact. This idea-communicated, or triggered, by the record itself-
+now creates a new context in which the record will henceforth be perceived. The next
+steps in the decoding might take considerably longer-but that is very hard for us to assess.
+We can imagine that if such a record had arrived on earth in Bach's time, no one would
+have known what to make of it, and very likely it would not have gotten deciphered. But
+that does not diminish our conviction that the information was in principle there; we just
+know that human knowledge in those times was not very sophisticated with respect to the
+possibilities of storage, transformation, and revelation of information.
+
+Levels of Understanding of a Message
+
+Nowadays, the idea of decoding is extremely widespread; it is a significant part of the
+activity of astronomers, linguists, archaeologists, military specialists, and so on. It is
+often suggested that we may be floating in a sea of radio messages from other
+civilizations, messages which we do not yet know how to decipher. And much serious
+thought has been given to the techniques of deciphering such a message. One of the main
+problems perhaps the deepest problem-is the question, "How will we recognize the fact
+that there is a message at all? How to identify a frame?" The sending of a record seems to
+be a simple solution-its gross physical structure is very attention-drawing, and it is at
+least plausible to us that it would trigger, in any sufficiently great intelligence, the idea of
+looking for information hidden in it. However, for technological reasons, sending of solid
+objects to other star systems seems to be out of the question. Still, that does not prevent
+our thinking about the idea.
+
+Now suppose that an alien civilization hit upon the idea that the appropriate
+mechanism for translation of the record is a machine which
+
+
+The Location of Meaning
+
+
+162
+
+
+
+converts the groove-patterns into sounds. This would still be a far cry from a true
+deciphering. What, indeed, would constitute a successful deciphering of such a record?
+Evidently, the civilization would have to be able to ma sense out of the sounds. Mere
+production of sounds is in itself hart worthwhile, unless they have the desired triggering
+effect in the brains that is the word) of the alien creatures. And what is that desired
+effect? would be to activate structures in their brains which create emotional effects in
+them which are analogous to the emotional effects which experience in hearing the piece.
+In fact, the production of sounds cot even be bypassed, provided that they used the record
+in some other way get at the appropriate structures in their brains. (If we humans had a w
+of triggering the appropriate structures in our brains in sequential order, as music does,
+we might be quite content to bypass the sounds-but it see] extraordinarily unlikely that
+there is any way to do that, other than via o ears. Deaf composers-Beethoven, Dvofak,
+Faure-or musicians who can "hear" music by looking at a score, do not give the lie to this
+assertion, for such abilities are founded upon preceding decades of direct auditory
+experiences.)
+
+Here is where things become very unclear. Will beings of an alien civilization
+have emotions? Will their emotions-supposing they have some-be mappable, in any
+sense, onto ours? If they do have emotions somewhat like ours, do the emotions cluster
+together in somewhat the same way as ours do? Will they understand such amalgams as
+tragic beauty courageous suffering? If it turns out that beings throughout the universe do
+share cognitive structures with us to the extent that even emotions overlap, then in some
+sense, the record can never be out of its natural context; that context is part of the scheme
+of things, in nature. And if such is the case, then it is likely that a meandering record, if
+not destroyed en route, would eventually get picked up by a being or group of beings, at
+get deciphered in a way which we would consider successful.
+
+
+"Imaginary Spacescape"
+
+In asking about the meaning of a molecule of DNA above, I used t phrase
+"compelling inner logic"; and I think this is a key notion. To illustrate this, let us slightly
+modify our hypothetical record-into-spa event by substituting John Cage's "Imaginary
+Landscape no. 4" for the Bach. This piece is a classic of aleatoric , or chance, music-
+music who structure is chosen by various random processes, rather than by an attempt to
+convey a personal emotion. In this case, twenty-four performers attar themselves to the
+twenty-four knobs on twelve radios. For the duration the piece they twiddle their knobs in
+aleatoric ways so that each radio randomly gets louder and softer, switching stations all
+the while. The tot sound produced is the piece of music. Cage's attitude is expressed in 14
+own words: "to let sounds be themselves, rather than vehicles for man made theories or
+expressions of human sentiments."
+
+
+The Location of Meaning
+
+
+163
+
+
+
+Now imagine that this is the piece on the record sent out into space. It would be
+extraordinarily unlikely-if not downright impossible-for an alien civilization to
+understand the nature of the artifact. They would probably be very puzzled by the
+contradiction between the frame message ("I am a message; decode me"), and the chaos
+of the inner structure. There are few "chunks" to seize onto in this Cage piece, few
+patterns which could guide a decipherer. On the other hand, there seems to be, in a Bach
+piece, much to seize onto-patterns, patterns of patterns, and so on. We have no way of
+knowing whether such patterns are universally appealing. We do not know enough about
+the nature of intelligence, emotions, or music to say whether the inner logic of a piece by
+Bach is so universally compelling that its meaning could span galaxies.
+
+However, whether Bach in particular has enough inner logic is not the issue here;
+the issue is whether any message has, per se, enough compelling inner logic that its
+context will be restored automatically whenever intelligence of a high enough level
+comes in contact with it. If some message did have that context-restoring property, then it
+would seem reasonable to consider the meaning of the message as an inherent property of
+the message.
+
+
+The Heroic Decipherers
+
+Another illuminating example of these ideas is the decipherment of ancient texts written
+in unknown languages and unknown alphabets. The intuition feels that there is
+information inherent in such texts, whether or not we succeed in revealing it. It is as
+strong a feeling as the belief that there is meaning inherent in a newspaper written in
+Chinese, even if we are completely ignorant of Chinese. Once the script or language of a
+text has been broken, then no one questions where the meaning resides: clearly it resides
+in the text, not in the method of decipherment just as music resides in a record, not inside
+a record player! One of the ways that we identify decoding mechanisms is by the fact that
+they do not add any meaning to the signs or objects which they take as input; they merely
+reveal the intrinsic meaning of those signs or objects. A jukebox is not a decoding
+mechanism, for it does not reveal any meaning belonging to its input symbols; on the
+contrary, it supplies meaning concealed inside itself.
+
+Now the decipherment of an ancient text may have involved decades of labor by
+several rival teams of scholars, drawing on knowledge stored in libraries all over the
+world ... Doesn't this process add information, too? Just how intrinsic is the meaning of a
+text, when such mammoth efforts are required in order to find the decoding rules? Has
+one put meaning into the text, or was that meaning already there? My intuition says that
+the meaning was always there, and that despite the arduousness of the pulling-out
+process, no meaning was pulled out that wasn't in the text to start with. This intuition
+comes mainly from one fact: I feel that the result was inevitable; that, had the text not
+been deciphered by this group at this time, it would have been deciphered by that group
+at that time-and it would have come
+
+
+The Location of Meaning
+
+
+164
+
+
+
+
+a*«^f i -wU f
+
+^S^^SS^i^E^lISS^,
+
+av GnBS&m tr. *
+
+
+*W*^s ««3»sj55i
+
+
+FIGURE 39. The Rosetta Stone [courtesy of the British Museum.
+
+out the same way. That is why the meaning is part of the text itself; it acts upon
+intelligence in a predictable way. Generally, we can say: meaning is part of an object to
+the extent that it acts upon intelligence in a predictable way.
+
+In Figure 39 is shown the Rosetta stone, one of the most precious of all historic
+discoveries. It was the key to the decipherment of Egyptian hieroglyphics, for it contains
+parallel text in three ancient scripts: hieroglyphic demotic characters, and Greek. The
+inscription on this basalt stele was firs deciphered in 1821 by Jean Francois Champollion,
+the "father of Egyptology"; it is a decree of priests assembled at Memphis in favor of
+Ptolemy Epiphanes.
+
+
+The Location of Meaning
+
+
+165
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Three Layers of Any Message
+
+In these examples of decipherment of out-of-context messages, we can separate
+out fairly clearly three levels of information: (1) the frame message; (2) the outer
+message; (3) the inner message. The one we are most familiar with is (3), the inner
+message; it is the message which is supposed to be transmitted: the emotional
+experiences in music, the phenotype in genetics, the royalty and rites of ancient
+civilizations in tablets, etc.
+
+To understand the inner message is to have extracted the meaning intended by the
+
+sender..
+
+The frame message is the message "I am a message; decode me if you can!"; and
+it is implicitly conveyed by the gross structural aspects of any information-bearer.
+
+To understand the frame message is to recognize the need for a decoding-
+
+mechanism.
+
+If the frame message is recognized as such, then attention is switched to level (2),
+the outer message. This is information, implicitly carried by symbol-patterns and
+structures in the message, which tells how to decode the inner message.
+
+To understand the outer message is to build, or know how to build, the correct
+decoding mechanism for the inner message.
+
+This outer level is perforce an implicit message, in the sense that the sender cannot ensure
+that it will be understood. It would be a vain effort to send instructions which tell how to
+decode the outer message, for they would have to be part of the inner message, which can
+only be understood once the decoding mechanism has been found. For this reason, the
+outer message is necessarily a set of triggers, rather than a message which can be
+revealed by a known decoder.
+
+The formulation of these three "layers" is only a rather crude beginning at
+analyzing how meaning is contained in messages. There may be layers and layers of
+outer and inner messages, rather than just one of each. Think, for instance, of how
+intricately tangled are the inner and outer messages of the Rosetta stone. To decode a
+message fully, one would have to reconstruct the entire semantic structure which
+underlay its creation and thus to understand the sender in every deep way. Hence one
+could throw away the inner message, because if one truly understood all the finesses of
+the outer message, the inner message would be reconstructible.
+
+The book After Babel , by George Steiner, is a long discussion of the interaction
+between inner and outer messages (though he never uses that terminology). The tone of
+his book is given by this quote:
+
+We normally use a shorthand beneath which there lies a wealth of subconscious,
+deliberately concealed or declared associations so extensive and intri-
+
+
+The Location of Meaning
+
+
+166
+
+
+
+cate that they probably equal the sum and uniqueness of our status as an individual person.'
+
+Thoughts along the same lines are expressed by Leonard B. Meyer, in h book Music, the
+Arts, and Ideas:
+
+The way of listening to a composition by Elliott Carter is radically different from the way
+of listening appropriate to a work by John Cage. Similarly, a novel by Beckett must in a
+significant sense be read differently from one by Bellow. A painting by Willem de
+Kooning and one by Andy Warhol require different perceptional-cognitive attitudes.'
+
+Perhaps works of art are trying to convey their style more than an thing else. In
+that case, if you could ever plumb a style to its very bottom you could dispense with the
+creations in that style. "Style", "outer message "decoding technique "-all ways of
+expressing the same basic idea.
+
+Schrodinger's Aperiodic Crystals
+
+What makes us see a frame message in certain objects, but none in other; Why
+should an alien civilization suspect, if they intercept an errant record that a message lurks
+within? What would make a record any different from a meteorite? Clearly its geometric
+shape is the first clue that "something funny is going on". The next clue is that, on a more
+microscopic scale, consists of a very long aperiodic sequence of patterns, arranged in a
+spiral If we were to unwrap the spiral, we would have one huge linear sequence (around
+2000 feet long) of minuscule symbols. This is not so different from a DNA molecule,
+whose symbols, drawn from a meager "alphabet" of four different chemical bases, are
+arrayed in a one-dimensional sequence, an then coiled up into a helix. Before Avery had
+established the connection between genes and DNA, the physicist Erwin Schrodinger
+predicted, o purely theoretical grounds, that genetic information would have to be stored
+in "aperiodic crystals", in his influential book What Is Lifel In fact books themselves are
+aperiodic crystals contained inside neat geometric forms. These examples suggest that,
+where an aperiodic crystal is found "packaged" inside a very regular geometric structure,
+there may lurk a inner message. (I don't claim this is a complete characterization of frame
+messages; however, it is a fact that many common messages have frame messages of this
+description. See Figure 40 for some good examples.)
+
+Languages for the Three Levels
+
+The three levels are very clear in the case of a message found in a bottle washed up on a
+beach. The first level, the frame message, is found when one picks up the bottle and sees
+that it is sealed, and contains a dry piece c paper. Even without seeing writing, one
+recognizes this type of artifact an information-bearer, and at this point it would take an
+extraordinary almost inhuman-lack of curiosity, to drop the bottle and not look further.
+
+
+The Location of Meaning
+
+
+167
+
+
+
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+
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+
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+
+
+The Location of Meaning
+
+
+168
+
+
+
+
+
+
+
+
+
+
+
+Next, one opens the bottle and examines the marks on the paper. Perhaps, they are in
+Japanese; this can be discovered without any of the inner message being understood-it
+merely comes from a recognition of 1 characters. The outer message can be stated as an
+English sentence: "I in Japanese." Once this has been discovered, then one can proceed
+the inner message, which may be a call for help, a haiku poem, a lover’s lament...
+
+It would be of no use to include in the inner message a translation the sentence
+"This message is in Japanese", since it would take someone who knew Japanese to read
+it. And before reading it, he would have recognize the fact that, as it is in Japanese, he
+can read it. You might try wriggle out of this by including translations of the statement
+"This mess2 is in Japanese" into many different languages. That would help it practical
+sense, but in a theoretical sense the same difficulty is there. . English-speaking person
+still has to recognize the "Englishness" of the message; otherwise it does no good. Thus
+one cannot avoid the problem that one has to find out how to decipher the inner message
+from the outside the inner message itself may provide clues and confirmations, but those ;
+at best triggers acting upon the bottle finder (or upon the people whom enlists to help).
+
+Similar kinds of problem confront the shortwave radio listener. First he has to
+decide whether the sounds he hears actually constitute a message or are just static. The
+sounds in themselves do not give the answer, not e% in the unlikely case that the inner
+message is in the listener's own native language, and is saying, "These sounds actually
+constitute a message a are not just static!" If the listener recognizes a frame message in
+the soup then he tries to identify the language the broadcast is in-and clearly, he is still on
+the outside; he accepts triggers from the radio, but they cam explicitly tell him the
+answer.
+
+It is in the nature of outer messages that they are not conveyed in any
+
+FIGURE 40. A collage of scripts. Uppermost on the left is an inscription in the un ciphered
+boustrophedonic writing system from Easter Island, in which every second lin upside down. The
+characters are chiseled on a wooden tablet, 4 inches by 35 inches. Mov clockwise, we encounter
+vertically written Mongolian: above, present-day Mongolian, below, a document dating from
+1314. Then we come to a poem in Bengali by Rabindran Tagore in the bottom righthand corner.
+Next to it is a newspaper headline in Malay a lam (II Kerala, southern India), above which is the
+elegant curvilinear language Tamil (E Kerala). The smallest entry is part of a folk tale in
+Buginese (Celebes Island, Indonesia). In center of the collage is a paragraph in the Thai
+language, and above it a manuscript in Rn dating from the fourteenth century, containing a
+sample of the provincial law of Scania (so Sweden). Finally, wedged in on the left is a section of
+the laws of Hammurabi, written Assyrian cuneiform. As an outsider, I feel a deep sense of
+mystery as I wonder how meanin cloaked in the strange curves and angles of each of these
+beautiful aperiodic crystals. Info there is content. [From Ham Jensen, Sign, Symbol, and Script
+(New York: G. Putnam's S. 1969), pp. 89 (cuneiform), 356 (Easter Island), 386, 417 (Mongolian),
+552 (Runic); from Keno Katzner, The Languages of the World (New York: Funk & Wagnalls,
+1975), pp. 190 (Bengali),
+
+(Buginese); from 1. A. Richards and Christine Gibson, English Through Pictures (New Y
+Washington Square Press, 1960), pp. 73 (Tamil), 82 (Thai).
+
+
+The Location of Meaning
+
+
+169
+
+
+
+explicit language. To find an explicit language in which to convey outer messages would
+not be a breakthrough-it would be a contradiction in terms! It is always the listener's
+burden to understand the outer message. Success lets him break through into the inside, at
+which point the ratio of triggers to explicit meanings shifts drastically towards the latter.
+By comparison with the previous stages, understanding the inner message seems
+effortless. It is as if it just gets pumped in.
+
+The "Jukebox" Theory of Meaning.
+
+These examples may appear to be evidence for the viewpoint that no message has
+intrinsic meaning, for in order to understand any inner message, no matter how simple it
+is, one must first understand its frame message and its outer message, both of which are
+carried only by triggers (such as being written in the Japanese alphabet, or having
+spiraling grooves, etc.). It begins to seem, then, that one cannot get away from a
+"jukebox" theory of meaning-the doctrine that no message contains inherent meaning,
+because, before any message can be understood, it has to be used as the input to some
+"jukebox", which means that information contained in the "jukebox" must be added to the
+message before it acquires meaning.
+
+This argument is very similar to the trap which the Tortoise caught Achilles in, in
+Lewis Carroll's Dialogue. There, the trap was the idea that before you can use any rule,
+you have to have a rule which tells you how to use that rule; in other words, there is an
+infinite hierarchy of levels of rules, which prevents any rule from ever getting used. Here,
+the trap is the idea that before you can understand any message, you have to have a
+message which tells you how to understand that message; in other words, there is an
+infinite hierarchy of levels of messages, which prevents any message from ever getting
+understood. However, we all know that these paradoxes are invalid, for rules do get used,
+and messages do get understood. How come?
+
+Against the Jukebox Theory
+
+This happens because our intelligence is not disembodied, but is instantiated in physical
+objects: our brains. Their structure is due to the long process of evolution, and their
+operations are governed by the laws of physics. Since they are physical entities, our
+brains run without being told how to run. So it is at the level where thoughts are produced
+by physical law that Carroll's rule-paradox breaks down; and likewise, it is at the level
+where a brain interprets incoming data as a message that the message-paradox breaks
+down. It seems that brains come equipped with "hardware" for recognizing that certain
+things are messages, and for decoding those messages. This minimal inborn ability to
+extract inner meaning is what allows the highly recursive, snowballing process of
+language acquisition to take place. The inborn hardware is like a jukebox: it supplies the
+additional information which turns mere triggers into complete messages.
+
+
+The Location of Meaning
+
+
+170
+
+
+
+Meaning Is Intrinsic If Intelligence Is Natural
+
+
+Now if different people's "jukeboxes" had different "songs" in then responded to given
+triggers in completely idiosyncratic ways, the would have no inclination to attribute
+intrinsic meaning to those tri; However, human brains are so constructed that one brain
+responds in much the same way to a given trigger as does another brain, all other t being
+equal. This is why a baby can learn any language; it responds to triggers in the same way
+as any other baby. This uniformity of "human jukeboxes" establishes a uniform
+"language" in which frame message outer messages can be communicated. If,
+furthermore, we believe human intelligence is just one example of a general phenomena
+nature-the emergence of intelligent beings in widely varying contexts then presumably
+the "language" in which frame messages and outer sages are communicated among
+humans is a "dialect" of a universal gauge by which intelligences can communicate with
+each other. Thus, would be certain kinds of triggers which would have "universal
+triggering power", in that all intelligent beings would tend to respond to them i same way
+as we do.
+
+This would allow us to shift our description of where meaning located. We could
+ascribe the meanings (frame, outer, and inner) message to the message itself, because of
+the fact that deciphering mechanisms are themselves universal-that is, they are
+fundamental f of nature which arise in the same way in diverse contexts. To make it
+concrete, suppose that "A-5" triggered the same song in all jukeboxes suppose moreover
+that jukeboxes were not man-made artifacts, but w occurring natural objects, like galaxies
+or carbon atoms. Under such circumstances, we would probably feel justified in calling
+the universal triggering power of "A-5" its "inherent meaning"; also, "A-5" would merit:
+the name of "message", rather than "trigger", and the song would indeed "revelation" of
+the inherent, though implicit, meaning of "A-5".
+
+Earth Chauvinism
+
+This ascribing of meaning to a message comes from the invariance c processing of the
+message by intelligences distributed anywhere ii universe. In that sense, it bears some
+resemblance to the ascribing of to an object. To the ancients, it must have seemed that an
+object's weight was an intrinsic property of the object. But as gravity became understood,
+it was realized that weight varies with the gravitational field the object is immersed in.
+Nevertheless, there is a related quantity, the mass, which not vary according to the
+gravitational field; and from this invariance the conclusion that an object's mass was an
+intrinsic property of the object itself. If it turns out that mass is also variable, according to
+context, then will backtrack and revise our opinion that it is an intrinsic property of an
+object. In the same way, we might imagine that there could exist other
+
+
+The Location of Meaning
+
+
+171
+
+
+
+kinds of "jukeboxes"-intelligences-which communicate among each other via messages
+which we would never recognize as messages, and who also would never recognize our
+messages as messages. If that were the case, then the claim that meaning is an intrinsic
+property of a set of symbols would have to be reconsidered. On the other hand, how
+could we ever realize that such beings existed?
+
+It is interesting to compare this argument for the inherency of meaning with a
+parallel argument for the inherency of weight. Suppose one defined an object's weight as
+"the magnitude of the downward force which the object exerts when on the surface of the
+planet Earth". Under this definition, the downward force which an object exerts when on
+the surface of Mars would have to be given another name than "weight". This definition
+makes weight an inherent property, but at the cost of geocentricity" Earth chauvinism". It
+would be like "Greenwich chauvinism "-refusing to accept local time anywhere on the
+globe but in the GMT time zone. It is an unnatural way to think of time.
+
+Perhaps we are unknowingly burdened with a similar chauvinism with respect to
+intelligence, and consequently with respect to meaning. In our chauvinism, we would call
+any being with a brain sufficiently much like our own "intelligent", and refuse to
+recognize other types of objects as intelligent. To take an extreme example, consider a
+meteorite which, instead of deciphering the outer-space Bach record, punctures it with
+colossal indifference, and continues in its merry orbit. It has interacted with the record in
+a way which we feel disregards the record's meaning. Therefore, we might well feel
+tempted to call the meteorite "stupid". But perhaps we would thereby do the meteorite a
+disservice. Perhaps it has a "higher intelligence" which we in our Earth chauvinism
+cannot perceive, and its interaction with the record was a manifestation of that higher
+intelligence. Perhaps, then, the record has a "higher meaning"-totally different from that
+which we attribute to it; perhaps its meaning depends on the type of intelligence
+perceiving it. Perhaps.
+
+It would be nice if we could define intelligence in some other way than "that
+which gets the same meaning out of a sequence of symbols as we do". For if we can only
+define it this one way, then our argument that meaning is an intrinsic property is circular,
+hence content-free. We should try to formulate in some independent way a set of
+characteristics which deserve the name "intelligence". Such characteristics would
+constitute the uniform core of intelligence, shared by humans. At this point in history we
+do not yet have a well-defined list of those characteristics. However, it appears likely that
+within the next few decades there will be much progress made in elucidating what human
+intelligence is. In particular, perhaps cognitive psychologists, workers in Artificial
+Intelligence, and neuroscientists will be able to synthesize their understandings, and come
+up with a definition of intelligence. It may still be human-chauvinistic; there is no way
+around that. But to counterbalance that, there may be some elegant and beautiful-and
+perhaps even simple-abstract ways of characterizing the essence of intelligence. This
+would serve to lessen the feeling of having
+
+
+The Location of Meaning
+
+
+172
+
+
+
+formulated an anthropocentric concept. And of course, if contact were established with an
+alien civilization from another star system, we feel supported in our belief that our own
+type of intelligence is not just a fluke, but an example of a basic form which reappears in
+nature in contexts, like stars and uranium nuclei. This in turn would support the idea of
+meaning being an inherent property.
+
+To conclude this topic, let us consider some new and old ex; and discuss the
+degree of inherent meaning which they have, by ourselves, to the extent that we can, in
+the shoes of an alien civilization which intercepts a weird object...
+
+
+Two Plaques in Space
+
+Consider a rectangular plaque made of an indestructible metallic alloy which are
+engraved two dots, one immediately above the another preceding colon shows a picture.
+Though the overall form of the might suggest that it is an artifact, and therefore that it
+might conceal some message, two dots are simply not sufficient to convey anything. (Can
+before reading on, hypothesize what they are supposed to mean suppose that we made a
+second plaque, containing more dots, as follows.
+
+
+Now one of the most obvious things to do-so it might seer terrestrial intelligence
+at least-would be to count the dots in the successive rows. The sequence obtained is:
+
+1, 1, 2, 3, 5, 8, 13, 21, 34.
+
+
+Here there is evidence of a rule governing the progression from one the next. In fact, the
+recursive part of the definition of the Fib numbers can be inferred, with some confidence,
+from this list. Supp think of the initial pair of values (1,1) as a "genotype" from which the
+"phenotype"-the full Fibonacci sequence-is pulled out by a recursive rule. By sending the
+genotype alone-namely the first version plaque-we fail to send the information which
+allows reconstitution phenotype. Thus, the genotype does not contain the full
+specification of
+
+
+The Location of Meaning
+
+
+173
+
+
+
+
+
+
+
+
+the phenotype. On the other hand, if we consider the second version of the plaque to be
+the genotype, then there is much better cause to suppose that the phenotype could
+actually be reconstituted. This new version of the genotype-a "long genotype"-contains so
+much information that the mechanism by which phenotype is pulled out of genotype can
+be inferred by intelligence from the genotype alone.
+
+Once this mechanism is firmly established as the way to pull phenotype from
+genotype, then we can go back to using "short genotypes"-like the first plaque. For
+instance, the "short genotype" (1,3) would yield the phenotype
+
+1, 3, 4, 7, 11, 18, 29, 47,...
+
+-the Lucas sequence. And for every set of two initial values-that is, for every short
+genotype-there will be a corresponding phenotype. But the short genotypes, unlike the
+long ones, are only triggers-buttons to be pushed on the jukeboxes into which the
+recursive rule has been built. The long genotypes are informative enough that they
+trigger, in an intelligent being, the recognition of what kind of "jukebox" to build. In that
+sense, the long genotypes contain the information of the phenotype, whereas the short
+genotypes do not. In other words, the long genotype transmits not only an inner message,
+but also an outer message, which enables the inner message to be read. It seems that the
+clarity of the outer message resides in the sheer length of the message. This is not
+unexpected; it parallels precisely what happens in deciphering ancient texts. Clearly,
+one's likelihood of success depends crucially on the amount of text available.
+
+Bach us'. Cage Again
+
+But just having a long text may not be enough. Let us take up once more the difference
+between sending a record of Bach's music into space, and a record of John Cage's music.
+Incidentally, the latter, being a Composition of Aleatorically Generated Elements, might
+be handily called a "CAGE", whereas the former, being a Beautiful Aperiodic Crystal of
+Harmony, might aptly be dubbed a "BACH". Now let's consider what the meaning of a
+Cage piece is to ourselves. A Cage piece has to be taken in a large cultural setting-as a
+revolt against certain kinds of traditions. Thus, if we want to transmit that meaning, we
+must not only send the notes of the piece, but we must have earlier communicated an
+extensive history of Western culture. It is fair to say, then, that an isolated record of John
+Cage's music does not have an intrinsic meaning. However, for a listener who is
+sufficiently well versed in Western and Eastern cultures, particularly in the trends in
+Western music over the last few decades, it does carry meaning-but such a listener is like
+a jukebox, and the piece is like a pair of buttons. The meaning is mostly contained inside
+the listener to begin with; the music serves only to trigger it. And this "jukebox", unlike
+pure intelligence, is not at all universal; it is highly earthbound, depending on
+idiosyncratic se-
+
+
+The Location of Meaning
+
+
+174
+
+
+
+quences of events all over our globe for long period of time. Hoping that John Cage's
+music will be understood by another civilization is like hoping that your favorite tune, on
+a jukebox on the moon, will have the same buttons as in a saloon in Saskatoon.
+
+On the other hand, to appreciate Bach requires far less cultural k edge. This may
+seem like high irony, for Bach is so much more con and organized, and Cage is so devoid
+of intellectuality. But there strange reversal here: intelligence loves patterns and balks at
+randomness For most people, the randomness in Cage's music requires much explanation;
+and even after explanations, they may feel they are missing the message-whereas with
+much of Bach, words are superfluous. In sense, Bach's music is more self-contained than
+Cage's music. Still, it is clear how much of the human condition is presumed by Bach.
+
+For instance, music has three major dimensions of structure (me harmony,
+rhythm), each of which can be further divided into small intermediate, and overall
+aspects. Now in each of these dimensions, there is a certain amount of complexity which
+our minds can handle before boggling; clearly a composer takes this into account, mostly
+unconsciously when writing a piece. These "levels of tolerable complexity" along
+different dimensions are probably very dependent on the peculiar conditions of our
+evolution as a species, and another intelligent species might have developed music with
+totally different levels of tolerable complexity along these many dimensions. Thus a Bach
+piece might conceivably have to be accompanied, by a lot of information about the
+human species, which simply could not inferred from the music's structure alone. If we
+equate the Bach music a genotype, and the emotions which it is supposed to evoke with
+the phenotype, then what we are interested in is whether the genotype con all the
+information necessary for the revelation of the phenotype.
+
+How Universal Is DNA's Message?
+
+The general question which we are facing, and which is very similar t questions inspired
+by the two plaques, is this: "How much of the co necessary for its own understanding is a
+message capable of restoring? can now revert to the original biological meanings of
+"genotype" "phenotype"-DNA and a living organism-and ask similar quest Does DNA
+have universal triggering power? Or does it need a "biojukebox" to reveal its meaning?
+Can DNA evoke a phenotype without being embedded in the proper chemical context?
+To this question to answer is no-but a qualified no. Certainly a molecule of DNA in a
+vacuum will not create anything at all. However, if a molecule of DNA were set to seek
+its fortune in the universe, as we imagined the BACH and the CAGE were, it might be
+intercepted by an intelligent civilization. They might first of all recognize its frame
+message. Given that, they might to try to deduce from its chemical structure what kind of
+chemical environment it seemed to want, and then supply such an environment. Succes-
+
+
+The Location of Meaning
+
+
+175
+
+
+
+sively more refined attempts along these lines might eventually lead to a full restoration
+of the chemical context necessary for the revelation of DNA's phenotypical meaning.
+This may sound a little implausible, but if one allows many millions of years for the
+experiment, perhaps the DNA's meaning would finally emerge.
+
+On the other hand, if the sequence of bases which compose a strand of DNA were
+sent as abstract symbols (as in Fig. 41), not as a long helical molecule, the odds are
+virtually nil that this, as an outer message, would trigger the proper decoding mechanism
+which would enable the phenotype to be drawn out of the genotype. This would be a case
+of wrapping an inner message in such an abstract outer message that the context-restoring
+power of the outer message would be lost, and so in a very pragmatic sense, the set of
+symbols would have no intrinsic meaning. Lest you think this all sounds hopelessly
+abstract and philosophical, consider that the exact moment when phenotype can be said to
+be "available", or "implied", by genotype, is a highly charged issue in our day: it is the
+issue of abortion.
+
+
+FIGURE 41. This Giant Aperiodic Crystal is the base sequence for the chromosome of
+bacteriophage 0X174. It is the first complete genome ever mapped out for any organism.
+About 2,000 of these boustrophedonic pages would be needed to show the base sequence
+of a single E. Coli cell, and about one million pages to show the base sequence of the
+DNA of a single human cell. The book now in your hands contains roughly the same
+amount of information as a molecular blueprint for one measly E. Coli cell.
+
+
+CCGTCAGCArTCACACCCTCCCAATTCTATGTTTTCATGCCTCCAAATCrrCGAGCCTTTTrTATGGTTCGTTCTTArTACCCTTCTGAATCTCACCCTG\
+r ACGAATACCTTCGGrTCCTAACCCCTAACTCTTTCTCArCTTTACCGTCTTCCCAGTTATCGTCCAAATTCTCGGACCTATGCGACTTTCACTTTTATTA'
+»GATGCATAACCGCATCAAGCTCTTCGAAGAGATTCTCTCnTTCGTATGCAGCCCCTTGAGTTCGArAATGGTCATArGTATGTTCACCGCCArAAGGCT-\
+/ ACAATAATTATAGTTCAACCCCCTCGTCTAACATCCTAACACCGrTAAGTAGCTAATTCAAGACTCATTGTCTArCTTTOACTAGTGCnGCAOTCTTCC*
+'CTATACACCACCCCCCCCAACCCCACCAAAAATGCTTTnACAGAACCAGAACACGGTTACCCAGTTTTCCCGCAACCTCCCTCCTGAACCCCCTCTTAA 1
+f mcCCACATCCCCrATACAATCACCTCCCCACCTCCTTAGAACTTGTGAGrAGCAATTATGGAAAGAAAAACCCCATTAATATCACTAGCCCtTATAGC*
+
+* GCTArTCACCGTTTGATGAATGCAATCCGACACCCTCArCCTC AICCTTCCTTTATCCTrTTTCA CACT CTCACCTTGr.CXC ACC ACCCATTACACGCGT T
+/"CTCACCCCCACACTTACTAATCGGAACCCTCCGAGCCCTCCTTCTTCGTATGCTCCnATACTCCnTTATCAGTCCGTTTCCTAACCCTAATAGTATTT*
+^CTArCAGTATTTTrGTCrCCCTCACTATCCTACACCTAATGCCCCTCTTCATTTCCATCCCCTCCACTrTATCCGCACACTTCCTACAGCTACCGTTCAC}
+/■ CGCACArCGCTTCACOCTACCCGTATGACATTCCTATTCCCCTCCATAAAACGTTCCArAAATTGACCCCCGCTAACCCATAGCCTCCTGCTTTTAATCC*
+*ACCACCCrTmCACCrrcTCCTTCCrTCrCCCCTCTTCAICCTAAAGCrCACCCCCTTAAACCTACCAGTTATATCGCTGTTCCTTTCTATGrCCCTAA}
+
+(* caacccttcatccctgtcgaaccaaaaatcactcaacaacctaagaaatcgaccatctggaaatcctccttccacctatacactcaaaaacaattgcata*
+
+^AACCTCTrCACAATCACAATCACCCCCAACTTCCCCATCAAAATGCTCACAATGACAAATClGTCCAGCCACTCCrtAATCCAACTTACCAAGCrGGCTT}
+CACCCCCCCCTTTTCCACCCCATCTCArTCAAAAGGCTCGCACTTACAGTAGAGAGAAAAACCCAACACGAAGTTATACACCAACTTCCCCCACCGCAGCA*
+v ACCTCICACCACAAATCTCCTCAAATTTATCCCCGCTTCCATAAAAArGArTGCCCIATCCAACCTCCAGAGTTTTAICCCTTCCATGACGCAGAACTTAj
+X AAGCGGGTCCtCAGGTGAACCTAAATTAAGCATrTCTtCCTCATCArTAAGGACGAAATAGTTCTATTAAAAAGCTGACTACTCTTTArAGGCTTTCACA*
+
+* AAIGACAAAATTCGACCIATCCTICCCCACCTCGAGAAGCrCTTACTrrGCCACCirTCGCCATCAACTAACCATTCTGTC AAAAACTCACCCCTTGGAr
+£ AATTrTACACTrCTTCTCTTAGAGATGGrACTTGTTrTACACTGAGTATAGATTTGGTCAGCAACTCCrTGCACGCTTCCTATAATTCGCTGAAGAGGAC*
+
+* AAC AOCCTCGATTACTA rCTCACTC.CC ArGCTCTTCAACC ACT AAIACGTAACAAATCATGAGTCAAGTTACTCAAC.AATCCCTACGTTTCCACACCGCT t
+rCCACTCATtCrrACGTTTCAAACAATGAGCACTCTTTTACCnTACTAGAAGCCAATTTACGTTTTCCCCTCTTCCCACTTACTCCAATTATCTCCGGrT'
+'*CCTCTCCTCCTCCTCCCTCCCTTCACCCTTCCGTTTATCCTACCCTGCACTTTCTACCArACCCTCeCTTTCCTGCTCCTGTTGACTTTATTCCTCCCCT\
+f CCCTCCCACCTGCGGTAAXTATTACAAAAGCCATTTAACTCCCCGAAGCTACTACTCTGTCCCCCAAACTrACAACTGCCCTACTTGTATT ATTCCTTAC*
+
+CrTAAAGCCGCTCAATTCTTCGCCTTT ACCTTGCGTGTACCCCCACC AAACACTGACCTT CTT ACTGACCCACAAC AAA ACGTC CCTCA AAAATT ACGTG )
+C tcccggaaatcccaacggaaatcatggagcgttgccgacgcctcctgctccccctcgccctcttgcaaaaaatcgaaatctgtaatctactgaccaaggc'
+
+* CCTCTTTCCTATCTACCTGCrCAACAATTTTAATTCCAGGGGCTTCGCCCCCTTACTTCACGAT AAATTATCTCTAATAITCAAACTGGCGCCGACCGTAT
+r rAGACCTTCCTCACCCCTCCCTATTGCCCTCATCAAcrTTACCATTAnCTCCrGGTTACACTGGTCCTTCCTTCGCTTCTACCCTTTCCACTACCCCCT*
+
+CGAGCCCGTTGGCGCTCTCCCTCTTTCTCCArfCCCTCGTGCCCTTCCTATTCACTCTACTGTACACATTTTTACTTTTTA TCTCCCTCATCCTCACCTT}
+QGTTcnTTCCCCCIACCACTTATAtTGGTCAICACAATTCICACCCCTCTCCTCACCGTAAnCTCGTAGGAACTACrTCAATTACCTCACAACTCGTAT*
+
+
+ACTCAGGTrTCTCCCCAAATGACCACTTCTACCACATCTATTGACATTATGGGTCTGCAACCTGCTTATGGTAATTTCCATACTCACCAAGAACCTGAnT
+
+rGTCTCTAATCTCGCGTACTCTTCATTTCCTCCCAACAGTCGCACTATACTCCAAAATGCAGCTTTACTTCTTTATTCTACTACCATTGCGACCTACTTCA*
+
+GGCArCTGGCTATGArCTTCATOGAACTGACCAAACCTCCTTACCCCACTTTTCTGCTCGTCTTCAACAGACCTATAAACATTCTGTGCCGCCTTTCTtT\
+C rtCACT rrcGTCGAAATCCCAATTCCA fGACTTAGAGAAATCACCGTCATCCCCCTrTTGCTTGTTCC CCTTC TCATTTGTA TCACCCTACGAGTCCrTC*
+' ArACCGATATTGCTGGCCACCCTCTTTTCTATGCCAACTTCCCCCCCCCrCAAATTTCTATGAACCATCTTTTCCCTTCTGGTGArTCCTCTAAGAACTTT
+C XCTrCCGCCAACCACTTACTTACCCTTCCCAACTTCTTCCACTATTCGTCCTCTrTGTATGCTTCCCCCTArTGCTATGGTGACTGCGAGTCGTTAGAAT*
+•GCTCAITTCCAACAACGCCTACTTATTCCCAACCATCAnATCACCACTCTnCACTCGTTCAGTTCTTCCACTGCArACTCTTACCTCATGTCACCTTTT
+f CCCCcTrTTTAATTTTAAAAAICGCGAAGCCAATATTOCAGTCTGACrTAGAAAATACTGCTTCACTACTAACTTACCGCTCACCAGCCGTCTAACGCrA*
+'TCACCGCrrCACCAAGCC.WCCGCCCTACGTrrrCTCCrTAGCACTTTAATCATCTTTCACACTTTTATTTCTCCCCACAATTCAAACTTTrtTTCTCAr5
+G ATTGCCACT'TTCATACTrTTArArrGCAACTGCTACATCCAAATCCACAGACArTTTGTCCACCCCTTCTTCGACCTCATTCTCTTCACTCTTGCTCGAA*
+* ATCCTCGTAATCGTCCTTTTCrTCATTGCATTCACATCCATACATCTGTCAACGCCCCTAATCAGGTTGTTTCAGTTGGTGCTGATArTCCnTTGATCC }
+^TACCGCTGGTAGCTTTCCTArf TGTAGTATCCCTCAGCCCTCCCATCACCCTrGGCTTCrTCTCACTTTCCCTTCCTTTGTCCCTmrTAAATCCCAGC'
+GArGCTCGTTATTATACCCTCAAGGACTGTGTGACTAT TGA CGTCCTICCCCCTACGCCCCGCAATAACGTCTACGTTGGTTTCATCCnTIXTCTAACTj
+
+
+ArrCCTGCCCCTATTCCT T GIGCTCTTCCTGCTGGCGCCA fGTCTAAAtTGmCG.M~J_
+
+C rTTCATCCCCCCCGGACTACTCCCAAICCTTGTAATCTCGGAACTTACCGTCTAAArTATt/rrCG rACTCGGTACGGATGTCATAACAAT ACCCATCCTT
+' TCTTTCTGCTGCTATCCCT AAACCTCGTAAAGGACTTCTTGAAGGTACGTTGCAGCCTGGCACTT CTCCCCTTTCTGATAAGTTCCTTCATTTCCTTGGA a
+rCTACTCCTGCTCGTCCCACCGTTCCTAArrCCACTCCrTTACCtCCTCGTTCrArrACTCCTCArACGAAAGGAAATAGTCCCCCTCTCAACCCTCCrrC'
+’CrTCCTCTCCTGCrATCCnCACCCCCCAniGACAATCAAAAACACCrTACTAAAArGCAACTGCACAATCACAAACACATTCCCCACArCCAAAATCAT
+rGAAGACAACTArTCCTTCCTAGACTAAAACACGTATATGGACCAGAAAGCATAACACCCCACTTCAGCGCCTGACTTACGGTCCTTAGACAAAAACTCAG*
+CAGTCTACTCCTCGCCriCCCICTATTATGCAAAACACCAATCTTTCCAACCAACAGCACCTTTCCGAGATTATGCGCCAAAT CCTT ACTCAACrTCAAAT
+C ATCGCGACTAAGACGCAAACCACTACTTCATTCAGTTCCACTCCTCATTCCAACCCTCACTAAAGAAACTAAACCAGTAACCArTTTArCACTGCTCGGC^
+* TCCCTCrrCTCATArTCGCGCTACTCCAAAGGATArrTCTAATCTCGTCACTGArCC'rCCTTCTGGTCTGGTTGArATTTTTCATGCTAnGAT AAACCT 3
+AATAAACCATCTGTrTAArCTCCCTTATGGTACTCCAAATCCCACAAAGCTCTTTAACAAGCTTCATAGCCCTTG*
+
+
+The Location of Meaning
+
+
+176
+
+
+
+
+
+
+
+
+
+
+Chromatic Fantasy, And Feud .
+
+Having had a splendid dip in the pond, the Tortoise is just crawling out and
+shaking himself dry, when who but Achilles walks by.
+
+Tortoise: Ho there, Achilles. I was just thinking of you as I splash around in the pond.
+
+Achilles: Isn't that curious? I was just thinking of you, too, while I meandered through the
+meadows. They're so green at this time of year.
+
+Tortoise: You think so? It reminds me of a thought I was hoping to share with you.
+
+Would you like to hear it?
+
+Achilles: Oh, I would be delighted. That is, I would be delighted as long you're not going
+to try to snare me in one of your wicked traps of log Mr. T.
+
+Tortoise: Wicked traps? Oh, you do me wrong. Would I do anything wicked? I'm a
+peaceful soul, bothering nobody and leading a gent; herbivorous life. And my
+thoughts merely drift among the oddities and quirks of how things are (as I see
+them). I, humble observer phenomena, plod along and puff my silly words into
+the air rather unspectacularly, I am afraid. But to reassure you about my intention
+I was only planning to speak of my Tortoise-shell today, and as you know, those
+things have nothing-nothing whatsoever-to do with logic!
+
+Achilles: Your words Do reassure me, Mr. T. And, in fact, my curiosity quite piqued. I
+would certainly like to listen to what you have to say even if it is unspectacular.
+
+Tortoise: Let's see ... how shall I begin? Hmm ... What strikes you me about my shell,
+Achilles?
+
+Achilles: It looks wonderfully clean!
+
+Tortoise: Thank you. I just went swimming and washed off several layers of dirt which
+had accumulated last century. Now you can see ho green my shell is.
+
+Achilles: Such a good healthy green shell, it's nice to see it shining in sun.
+
+Tortoise: Green? It's not green.
+
+Achilles: Well, didn't you just tell me Tortoise: I did.
+
+Achilles: Then, we agree: it is green. Tortoise: No, it isn't green.
+
+Achilles: Oh, I understand your game. You're hinting to me that what you say isn't
+necessarily true; that Tortoises play with language; that your statements and
+reality don't necessarily match; that —
+
+
+
+Tortoise: I certainly am not. Tortoises treat words as sacred. Tortoises revere accuracy.
+
+Achilles: Well, then, why did you say that your shell is green, and that it is not green
+also?
+
+Tortoise: I never said such a thing; but I wish I had. Achilles: You would have liked to
+say that?
+
+Tortoise: Not a bit. I regret saying it, and disagree wholeheartedly with it. Achilles: That
+certainly contradicts what you said before!
+
+Tortoise: Contradicts? Contradicts? I never contradict myself. It's not part of Tortoise-
+nature.
+
+Achilles: Well, I've caught you this time, you slippery fellow, you. Caught you in a full-
+fledged contradiction.
+
+Tortoise: Yes, I guess you did.
+
+Achilles: There you go again! Now you're contradicting yourself more and more! You are
+so steeped in contradiction it's impossible to argue with you!
+
+Tortoise: Not really. I argue with myself without any trouble at all. Perhaps the problem
+is with you. I would venture a guess that maybe you're the one who's
+contradictory, but you're so trapped in your own tangled web that you can't see
+how inconsistent you're being.
+
+Achilles: What an insulting suggestion! I'm going to show you that you're the
+contradictory one, and there are no two ways about it.
+
+Tortoise: Well, if it's so, your task ought to be cut out for you. What could be easier than
+to point out a contradiction? Go ahead-try it out.
+
+Achilles: Hmm ... Now I hardly know where to begin. Oh ... I know. You first said that
+(1) your shell is green, and then you went on to say that (2) your shell is not
+green. What more can I say?
+
+Tortoise: Just kindly point out the contradiction. Quit beating around the bush.
+
+Achilles: But-but-but... Oh, now I begin to see. (Sometimes I am so slow-witted!) It must
+be that you and I differ as to what constitutes a contradiction. That's the trouble.
+Well, let me make myself very clear: a contradiction occurs when somebody says
+one thing and denies it at the same time.
+
+Tortoise: A neat trick. I'd like to see it done. Probably ventriloquists would excel at
+
+contradictions, speaking out of both sides of their mouth, as it were. But I'm not a
+ventriloquist.
+
+Achilles: Well, what I actually meant is just that somebody can say one thing and deny it
+all within one single sentence! It doesn't literally have to be in the same instant.
+
+Tortoise: Well, you didn't give ONE sentence. You gave TWO.
+
+Achilles: Yes-two sentences that contradict each other!
+
+Tortoise: I am sad to see the tangled structure of your thoughts becoming so exposed,
+Achilles. First you told me that a contradiction is some thing which occurs in a
+single sentence. Then you told me that you
+
+
+
+Found a contradiction in a pair of sentences I uttered. Frankly, it’s just as I said. Your
+own system of thought is so delusional that you manage to avoid seeing how
+inconsistent it is. From the outside, however plain as day.
+
+Achilles: Sometimes I get so confused by your diversionary tactics tl can't quite tell if
+we're arguing about something utterly petty, or something deep and profound!
+
+Tortoise: I assure you, Tortoises don't spend their time on the petty. Hence it's the latter.
+
+Achilles: I am very reassured. Thank you. Now I have had a moment to reflect, and I see
+the necessary logical step to convince you that you contradicted yourself.
+
+Tortoise: Good, good. I hope it's an easy step, an indisputable one.
+
+Achilles: It certainly is. Even you will agree with it. The idea is that you believed
+
+sentence 1 ("My shell is green"), AND you believed sentence 2 ("My shell is not
+green"), you would believe one compound( sentence in which both were
+combined, wouldn't you?
+
+Tortoise: Of course. It would only be reasonable ... providing just that the manner of
+combination is universally acceptable. But I'm sure we'll agree on that.
+
+Achilles: Yes, and then I'll have you! The combination I propose is
+
+Tortoise: But we must be careful in combining sentences. For instance you'd grant that
+"Politicians lie" is true, wouldn't you?
+
+Achilles: Who could deny it?
+
+Tortoise: Good. Likewise, "Cast-iron sinks" is a valid utterance, isn't it?
+
+Achilles: Indubitably.
+
+Tortoise: Then, putting them together, we get "Politicians lie in cast iron sinks". Now
+that's not the case, is it?
+
+Achilles: Now wait a minute ... "Politicians lie in cast-iron sinks?" N no, but
+
+Tortoise: So, you see, combining two true sentences in one is not a policy, is it?
+
+Achilles: But you-you combined the two-in such a silly way!
+
+Tortoise: Silly? What have you got to object to in the way I combined them Would you
+have me do otherwise?
+
+Achilles: You should have used the word "and", not "in".
+
+Tortoise: I should have? You mean, if YOU'D had YOUR way, I should h;
+
+Achilles: No-it's the LOGICAL thing to do. It's got nothing to do with personally.
+
+Tortoise: This is where you always lose me, when you resort to your L and its high-
+sounding Principles. None of that for me today, plea
+
+Achilles: Oh, Mr. Tortoise, don't put me through all this agony. You k very well that
+that's what "and" means! It's harmless to combine true sentences with "and"!
+
+Tortoise: "Harmless", my eye! What gall! This is certainly a pernicious plot
+
+
+
+to entrap a poor, innocent, bumbling Tortoise in a fatal contradiction. If it were so
+harmless, why would you be trying so bloody hard to get me to do it? Eh?
+
+Achilles: You've left me speechless. You make me feel like a villain, where I really had
+only the most innocent of motivations.
+
+Tortoise: That's what everyone believes of himself...
+
+Achilles: Shame on me-trying to outwit you, to use words to snare you in a self-
+contradiction. I feel so rotten.
+
+Tortoise: And well you should. I know what you were trying to set up. Your plan was to
+make me accept sentence 3, to wit: "My shell is green and my shell is not green".
+And such a blatant falsehood is repellent to the Tongue of a Tortoise.
+
+Achilles: Oh, I'm so sorry I started all this.
+
+Tortoise: You needn't be sorry. My feelings aren't hurt. After all, I'm used to the
+
+unreasonable ways of the folk about me. I enjoy your company, Achilles, even if
+your thinking lacks clarity.
+
+Achilles: Yes ... Well, I fear I am set in my ways, and will probably continue to err and
+err again, in my quest for Truth.
+
+Tortoise: Today's exchange may have served a little to right your course. Good day,
+Achilles.
+
+Achilles: Good day, Mr. T.
+
+
+
+CHAPTER VII
+
+
+The Propositional Calculus
+
+Words and Symbols
+
+THE PRECEDING DIALOGUE is reminiscent of the Two-Part Invention by Lewis
+Carroll. In both, the Tortoise refuses to use normal, ordinary in the normal, ordinary way-
+or at least he refuses to do so when it is his advantage to do so. A way to think about the
+Carroll paradox was given last Chapter. In this Chapter we are going to make symbols dc
+Achilles couldn't make the Tortoise do with his words. That is, we are to make a formal
+system one of whose symbols will do just what A wished the word 'and' would do, when
+spoken by the Tortoise, and ail of whose symbols will behave the way the words 'if... then
+. . .' ought to behave. There are only two other words which we will attempt to deal with
+'or' and 'not'. Reasoning which depends only on correct usage of these words is termed
+propositional reasoning.
+
+Alphabet and First Rule of the Propositional Calculus
+
+I will present this new formal system, called the Propositional Calculus, like a puzzle, not
+explaining everything at once, but letting you things out to some extent. We begin with
+the list of symbols:
+
+
+< >
+
+P Q R "
+
+A V Z>
+
+The first rule of this system that I will reveal is the following:
+
+RULE OF JOINING: If x and y are theorems of the system, then so is the string < xAy >.
+
+This rule takes two theorems and combines them into one. It s remind you of the
+Dialogue.
+
+Well-Formed Strings
+
+There will be several other rules of inference, and they will all be pres shortly-but first, it
+is important to define a subset of all strings, namely the
+
+
+The Propositional Calculus
+
+
+181
+
+
+
+well formed strings. They will be defined in a recursive way. We begin with the
+ATOMS: P, Q, and R are called atoms.. New atoms are formed by appending primes
+onto the right of old atoms-thus, R\ Q", P'", etc. This gives an endless supply of atoms.
+All atoms are well-formed.
+
+Then we have four recursive
+
+FORMATION RULES: If x and y are well-formed, then the following four strings are
+also well-formed:
+
+(1) ~x
+
+(2) < XAy>
+
+(3) < xvy>
+
+(4) < X3y>
+
+
+For example, all of the following are well-formed:
+
+P
+
+atom
+
+~P
+
+by (1)
+
+~~P
+
+by (1)
+
+Q'
+
+atom
+
+~Ql
+
+by (1)
+
+<Pa~Q' >
+
+by (2)
+
+~<Pa~Q' >
+
+by (1)
+
+~~<P=>~Q' >
+
+by (4)
+
+<~<Pa~Q' >v —<P=>~Q' »
+
+by (3)
+
+
+The last one may look quite formidable, but it is built up straightforwardly from two
+components-namely the two lines just above it. Each of them is in turn built up from
+previous lines ... and so on. Every well-formed string can in this way be traced back to its
+elementary constituents-that is, atoms. You simply run the formation rules backwards
+until you can no more. This process is guaranteed to terminate, since each formation rule
+(when run forwards) is a lengthening rule, so that running it backwards always drives you
+towards atoms.
+
+This method of decomposing strings thus serves as a check on the well-
+formedness of any string. It is a top-down decision procedure for wellformedness. You
+can test your understanding of this decision procedure by checking which of the
+following strings are well-formed:
+
+(1) <P>
+
+(2) (2) <~P>
+
+(3) <PaQaR>
+
+(4) <PaQ>
+
+(5) «PaQ>aQ~aP»
+
+(6) <Pa~P>
+
+(7) «Pv<Qz>R»a<~Pv~R»
+
+(8) <PaQ>a< QaP:
+
+
+The Propositional Calculus
+
+
+182
+
+
+
+(Answer: Those whose numbers are Fibonacci numbers are not formed. The rest are well-
+formed.)
+
+
+More Rules of Inference
+
+Now we come to the rest of the rules by which theorems of this system constructed. A
+few rules of inference follow. In all of them, the symbols 'x' and 'y' are always to be
+understood as restricted to well formed strings
+
+RULE OF SEPARATION: If < XAy> is a theorem, then both x and theorems.
+
+Incidentally, you should have a pretty good guess by now as to concept the symbol 'A'
+stands for. (Hint: it is the troublesome word the preceding Dialogue.) From the following
+rule, you should be a figure out what concept the tilde ('-') represents:
+
+DOUBLE-TILDE RULE: The string '—' can be deleted from any theorem. It can also be
+inserted into any theorem, provided that the rest string is itself well-formed.
+
+
+The Fantasy Rule
+
+Now a special feature of this system is that it has no axioms- only rule you think back to
+the previous formal systems we've seen, you may w( how there can be any theorems,
+then. How does everything get started? The answer is that there is one rule which
+manufactures theorems from out of thin air-it doesn't need an "old theorem" as input.
+(The rest of the do require input.) This special rule is called the fantasy rule. The reason I
+call it that is quite simple.
+
+To use the fantasy rule, the first thing you do is to write down an well-formed
+string x you like, and then "fantasize" by asking, "What if string x were an axiom, or a
+theorem?" And then, you let the system give an answer. That is, you go ahead and make a
+derivation with x ; opening line; let us suppose y is the last line. (Of course the derivation
+must strictly follow the rules of the system.) Everything from x to y (inclusive) is the
+fantasy; x is the premise of the fantasy, and y is its outcome. The next step is to jump out
+of the fantasy, having learned from it that out.
+
+If x were a theorem, y would be a theorem.
+
+Still, you might wonder, where is the real theorem? The real theorem is the string
+
+<x3y>
+
+Notice the resemblance of this string to the sentence printed above
+
+To signal the entry into, and emergence from, a fantasy, one uses the
+
+
+The Propositional Calculus
+
+
+183
+
+
+
+square brackets '[' and respectively. Thus, whenever you see a left square bracket, you
+know you are "pushing" into a fantasy, and the next line will contain the fantasy's
+premise. Whenever you see a right square bracket, you know you are "popping" back out,
+and the preceding line was the outcome. It is helpful (though not necessary) to indent
+those lines of a derivation which take place in fantasies.
+
+Here is an illustration of the fantasy rule, in which the string P is taken as a
+premise. (It so happens that P is not a theorem, but that is of no import; we are merely
+inquiring, "What if it were?") We make the following fantasy:
+
+[ push into fantasy
+
+P premise
+
+~~~P outcome (by double tilde rule)
+
+] pop out of fantasy
+
+The fantasy shows that:
+
+If P were a theorem, so would ~~P be one.
+
+
+We now "squeeze" this sentence of English (the metalanguage) into the formal
+notation (the object language): <P=)~~P>. This, our first theorem of the Propositional
+Calculus, should reveal to you the intended interpretation of the symbol 'z>'.
+
+Here is another derivation using the fantasy rule:
+
+
+[
+
+<PaQ>
+
+P
+
+Q
+
+<QaP>
+
+]
+
+«PaQ>=><QaP»
+
+
+push
+
+premise
+
+separation
+
+separation
+
+joining
+
+pop
+
+fantasy rule
+
+
+It is important to understand that only the last line is a genuine theorem, here-everything
+else is in the fantasy.
+
+
+Recursion and the Fantasy Rule
+
+As you might guess from the recursion terminology "push" and "pop", the fantasy rule
+can be used recursively-thus, there can be fantasies within fantasies, thrice-nested
+fantasies, and so on. This means that there are all sorts of "levels of reality", just as in
+nested stories or movies. When you pop out of a movie-within-a-movie, you feel for a
+moment as if you had reached the real world, though you are still one level away from the
+top. Similarly, when you pop out of a fantasy-within-a-fantasy, you are in a "realer"
+world than you had been, but you are still one level away from the top.
+
+Now a "No Smoking" sign inside a movie theater does not apply to the
+
+
+The Propositional Calculus
+
+
+184
+
+
+
+characters in the movie-there is no carry-over from the real world in fantasy world, in
+movies. But in the Propositional Calculus, then carry-over from the real world into the
+fantasies; there is even carry from a fantasy to fantasies inside it. This is formalized by
+the following rule:
+
+CARRY-OVER RULE: Inside a fantasy, any theorem from the "reality level higher can
+be brought in and used.
+
+It is as if a "No Smoking" sign in a theater applied not only to a moviegoers, but also to
+all the actors in the movie, and, by repetition of the same idea, to anyone inside multiply
+nested movies! (Warning: There carry-over in the reverse direction: theorems inside
+fantasies cannot be exported to the exterior! If it weren't for this fact, you could write any
+as the first line of a fantasy, and then lift it out into the real world as a theorem.)
+
+To show how carry-over works, and to show how the fantasy rule can be used
+recursively, we present the following derivation:
+
+
+[
+
+P
+
+[
+
+Q
+
+P
+
+<PaQ>
+
+]
+
+<Q=><PaQ»
+
+]
+
+<P=)<Q=><PaQ»>
+
+
+push
+
+premise of outer fantasy
+push again
+
+premise of inner fantasy
+carry-over of P into inner fantasy
+joining
+
+pop out of inner fantasy, regain outer fantasy
+fantasy rule
+
+pop out of outer fantasy, reach real world!
+fantasy rule
+
+
+Note that I’ve indented the outer fantasy once, and the inner fantasy twice, to
+emphasize the nature of these nested "levels of reality". One to look at the fantasy rule is
+to say that an observation made about the system is inserted into the system. Namely, the
+theorem < x^y> which gets produced can be thought of as a representation inside the
+system of the statement about the system "If x is a theorem, then y is too". To be specific,
+the intended interpretation for <P=>Q> is "if P, then Q equivalently, "P implies Q".
+
+
+The Converse of the Fantasy Rule
+
+Now Lewis Carroll's Dialogue was all about "if-then" statements. In particular, Achilles
+had a lot of trouble in persuading the Tortoise to accept the second clause of an "if-then"
+statement, even when the "if-then" state itself was accepted, as well as its first clause. The
+next rule allows y infer the second "clause" of a'z>'-string, provided that the 'zf-string it a
+theorem, and that its first "clause" is also a theorem.
+
+
+The Propositional Calculus
+
+
+185
+
+
+
+RULE OF DETACHMENT: If x and < xz>y> are both theorems, then y is a theorem.
+
+Incidentally, this rule is often called "Modus Ponens", and the fantasy rule is often called
+the "Deduction Theorem".
+
+The Intended Interpretation of the Symbols
+
+We might as well let the cat out of the bag at this point, and reveal the "meanings" of the
+rest of the symbols of our new system. In case it is not yet apparent, the symbol 'A' is
+meant to be acting isomorphically to the normal, everyday word 'and'. The symbol
+represents the word 'not'-it is a formal sort of negation. The angle brackets '<’ and '>' are
+groupers-their function being very similar to that of parentheses in ordinary algebra. The
+main difference is that in algebra, you have the freedom to insert parentheses or to leave
+them out, according to taste and style, whereas in a formal system, such anarchic freedom
+is not tolerated. The symbol V represents the word 'or' ('vel' is a Latin word for 'or'). The
+'or' that is meant is the so-called inclusive 'or', which means that the interpretation of
+<xvy> is "either x or y-or both".
+
+The only symbols we have not interpreted are the atoms. An atom has no single
+interpretation-it may be interpreted by any sentence of English (it must continue to be
+interpreted by the same sentence if it occurs multiply within a string or derivation). Thus,
+for example, the well-formed string <Pa~P> could be interpreted by the compound
+sentence
+
+This mind is Buddha, and this mind is not Buddha.
+
+Now let us look at each of the theorems so far derived, and interpret them. The first one
+was <Pz)~~P>. If we keep the same interpretation for P, we have the following
+interpretation:
+
+If this mind is Buddha,
+
+then it is not the case that this mind is not Buddha.
+
+Note how I rendered the double negation. It is awkward to repeat a negation in any
+natural language, so one gets around it by using two different ways of expressing
+negation. The second theorem we derived was «PaQ>zkQaP». If we let Q be
+interpreted by the sentence "This flax weighs three pounds", then our theorem reads as
+follows:
+
+If this mind is Buddha and this flax weighs three pounds,
+
+then this flax weighs three pounds and this mind is Buddha.
+
+The third theorem was <PzxQz><PaQ>». This one goes into the following nested "if-
+then" sentence:
+
+
+The Propositional Calculus
+
+
+186
+
+
+
+If this mind is Buddha,
+
+then, if this flax weighs three pounds,
+
+then this mind is Buddha and this flax weighs three pounds.
+
+You probably have noticed that each theorem, when interpreted, something
+absolutely trivial and self-evident. (Sometimes they are so s evident that they sound
+vacuous and-paradoxically enough-confusing or even wrong!) This may not be very
+impressive, but just remember there are plenty of falsities out there which could have
+been produced they weren't. This system-the Propositional Calculus-steps neatly ft truth
+to truth, carefully avoiding all falsities, just as a person who is concerned with staying dry
+will step carefully from one stepping-stone creek to the next, following the layout of
+stepping-stones no matter I twisted and tricky it might be. What is impressive is that-in
+the Propositional Calculus-the whole thing is done purely typographically. There is
+nobody down "in there", thinking about the meaning of the strings. It i! done
+mechanically, thoughtlessly, rigidly, even stupidly.
+
+Rounding Out the List of Rules
+
+We have not yet stated all the rules of the Propositional Calculus. The complete set of
+rules is listed below, including the three new ones.
+
+JOINING RULE: If x and y are theorems, then < XAy> is a theorem.
+
+SEPARATION RULE: If < XAy> is a theorem, then both x and y are theorems.
+
+DOUBLE-TILDE RULE: The string '—' can be deleted from any theorem can also be
+inserted into any theorem, provided that the result string is itself well-formed.
+
+FANTASY RULE: If y can be derived when x is assumed to be a theorem then < X3y> is
+a theorem.
+
+CARRY-OVER RULE: Inside a fantasy, any theorem from the "reality" c level higher
+can be brought in and used.
+
+RULE OF DETACHMENT: If x and < X3y> are both theorems, then y is a theorem.
+CONTRAPOSITIVE RULE: <x3y> and <~y=)~x> are interchangeable
+DE MORGAN'S RULE: <~XA~y> and ~< xvy> are interchangeable.
+
+SWITCHEROO RULE: <xvy> and <~xz>y> are interchangeable.
+
+(The Switcheroo rule is named after Q. q. Switcheroo, an Albanian railroad engineer who
+worked in logic on the siding.) By "interchangeable" in foregoing rules, the following is
+meant: If an expression of one form occurs as either a theorem or part of a theorem, the
+other form may be
+
+
+The Propositional Calculus
+
+
+187
+
+
+
+substituted, and the resulting string will also be a theorem. It must be kept in mind that
+the symbols ‘x’ and ‘y’ always stand for well-formed strings of the system.
+
+Justifying the Rules
+
+Before we see these rules used inside derivations, let us look at some very short
+justifications for them. You can probably justify them to yourself better than my
+examples - which is why I only give a couple.
+
+The contrapositive rule expresses explicitly a way of turning around conditional
+statements which we carry out unconsciously. For instance, the “Zentence”
+
+If you are studying it, then you are far from the Way
+Means the same thing as
+
+If you are close to the Way, then you are not studying it.
+
+De Morgan’s rule can be illustrated by our familiar sentence “The flag is not
+moving and the wind is not moving”. If P symbolizes “the flag is not moving”, and Q
+symbolizes “the wind is moving”, then the compound sentence is symbolized by
+<~Pa~Q>, which, according to Morgan’s law, is interchangeable with ~<PvQ>. whose
+interpretation would be “It is not true that either the flag or the wind is moving”. And no
+one could deny that it is a Zensible conclusion to draw.
+
+For the Switrcheroo rule, consider the sentence “Either a cloud is hanging over
+the mountain, or the moonlight is penetrating the waves of the lake,” which might be
+spoken, I suppose, by a wistful Zen master remembering a familiar lake which he can
+visualize mentally but cannot see. Now hang on to your seat, for the Swircheroo rule tells
+us that this is interchangeable with the thought “If a cloud is not hanging over the
+mountain, then the moonlight is penetrating the waves of the lake.” This may not be
+enlightenment, but it is the best the Propositional Calculus has to offer.
+
+Playing around with the system
+
+Now, let us apply these rules to a previous theorem, ands see what we get: For instance,
+take the theorem <Pid~~P>:
+
+
+<P=>—P>:
+<~~~P=d~P>:
+<~P=>~P>
+<Pv~P>
+
+
+old theorem
+contrapositive
+double-tilde
+switcheroo
+
+
+This new theorem, when interpreted, says:
+
+
+The Propositional Calculus
+
+
+188
+
+
+
+Either this mind is Buddha, or this mind is not Buddha
+
+
+Once again, the interpreted theorem, though perhaps less than mind boggling, is at least
+true.
+
+
+Semi-Interpretations
+
+It is natural, when one reads theorems of the Propositional Calculus out loud, to interpret
+everything but the atoms. I call this semi-interpreting. For example, the semi¬
+interpretation of <Pv~P>:: would be
+
+
+P or not P.
+
+Despite the fact that P is not a sentence, the above semisentence still sounds true, because
+you can very easily imagine sticking any sentence in for P - and the form of the semi-
+interpreted theorem assures you that however you make your choice, the resulting
+sentence will be true. And that is the key idea of the Propositional Calculus: it produces
+theorems which, when semi-interpreted, are seen to be “universally true semisaentences”,
+by which is meant that no matter how you complete the interpretation, the final result will
+be a true statement.
+
+
+Ganto’s Ax
+
+Now we can do a more advanced exercise, based on a Zen koan called “Ganto’s Ax”.
+Here is how it began.
+
+One day Tokusan told his student Ganto, “I have two monks who have been here
+for many years. Go and examine them.” Ganto picked up an ax and went to the hut
+where the two monks were meditating. He raised the ax, saying “If you say a word,
+
+I will cut off your heads; and if you do not say a word, I will also cut off your
+heads.” 1
+
+If you say a word I will cut off this koan, and if you do not say a word, I will also cut off
+this koan - because I want you to translate some of it into our notation. Let us symbolize
+“you say a word” by P and “I will cut off your heads” by Q. Then Ganto’s ax threat is
+symbolized by the string «Pz>Q>a<~'Pz>Q». What if this ax threat were an axiom?
+
+
+Here is a fantasy to answer that question.
+
+(1) [
+
+push
+
+(2)
+
+«P=>Q>a<~'P=>Q».
+
+Ganto’s axiom
+
+(3)
+
+<P=>Q>
+
+separation
+
+(4)
+
+<~Q=>~P>.
+
+contrapositive
+
+(5)
+
+<~P=>Q>
+
+separation
+
+(6)
+
+<~Q=>~~P>.
+
+contrapositive
+
+(7)
+
+]
+
+push again
+
+(8)
+
+~Q
+
+premise
+
+
+The Propositional Calculus
+
+
+189
+
+
+
+(9) <~Qz>~P>.
+
+(10) ~P
+
+(11) <~Qz)~~P>.
+
+(12) ~~P
+
+(13) <~Pa~~P>
+
+(14) <~Pv~~P>
+
+(15) ]
+
+(16) <~Qz>~<Pv~P».
+
+(17) <~Pv~P>z>Q>.
+
+(18) [
+
+(19) . ~P
+
+( 20 ) ]
+
+(21) <~P3~P>.
+
+(22) <Pv~P>.
+
+(23) Q
+
+(24) ]
+
+
+carry-over of line 4
+
+detachment
+
+carry-over of line 6
+
+detachment (lines 8 and 11)
+
+joining
+
+De Morgan
+
+pop once
+
+fantasy rule
+
+contrapositive
+
+push
+
+premise (also outcome)
+pop
+
+fantasy rule
+switcheroo
+
+detachment (lines 22 and 17)
+pop out
+
+
+The power of the Propositional Calculus is shown in this example. Why, in but two dozen
+steps, we have deduced Q: that the heads will be cut off! (Ominously, the rule last
+invoked was "detachment" ...) It might seem superfluous to continue the koan now, since
+we know what must ensue ... However, I shall drop my resolve to cut the koan off; it is a
+true Zen koan, after all. The rest of the incident is here related:
+
+Both monks continued their meditation as if he had not spoken. Ganto dropped the
+ax and said, "You are true Zen students." He returned to Tokusan and related the
+incident. "I see your side well," Tokusan agreed, "but tell me, how is their side?"
+"Tozan may admit them," replied Ganto, "but they should not be admitted under
+Tokusan. "2
+
+Do you see my side well? How is the Zen side?
+
+Is There a Decision Procedure for Theorems?
+
+The Propositional Calculus gives us a set of rules for producing statements which would
+be true in all conceivable worlds. That is why all of its theorems sound so simple-minded;
+it seems that they have absolutely no content! Looked at this way, the Propositional
+Calculus might seem to be a waste of time, since what it tells us is absolutely trivial. On
+the other hand, it does it by specifying the form of statements that are universally true,
+and this throws a new kind of light onto the core truths of the universe: they are not only
+fundamental, but also regular, they can be produced by one set of typographical rules. To
+put it another way, they are all "cut from the same cloth". You might consider whether
+the same could be said about Zen koans: could they all be produced by one set of
+typographical rules?
+
+It is quite relevant here to bring up the question of a decision procedure. That is,
+does there exist any mechanical method to tell nontheorems from theorems? If so, that
+would tell us that the set of theorems of the
+
+
+The Propositional Calculus
+
+
+190
+
+
+
+Propositional Calculus is not only r.e., but also recursive. It turns out that there is an
+interesting decision procedure-the method of truth u would take us a bit afield to present
+it here; you can find it in almost any standard book on logic. And what about Zen koans?
+Could there conceivably be a mechanical decision procedure which distinguishes genuine
+Zen koans from other things?
+
+Do We Know the System Is Consistent?
+
+Up till now, we have only presumed that all theorems, when interpreted as indicated, are
+true statements. But do we know that that is the case' we prove it to be? This is just
+another way of asking whether the intended interpretations ('and' for 'a', etc.) merit being
+called the "passive meanings” of the symbols. One can look at this issue from two very
+different points of view, which might be called the "prudent" and "imprudent" points I
+will now present those two sides as I see them, personifying their as "Prudence" and
+"Imprudence".
+
+Prudence: We will only KNOW that all theorems come out true un intended
+interpretation if we manage to PROVE it. That is the c: thoughtful way to proceed.
+Imprudence: On the contrary. It is OBVIOUS that all theorems will come out true. If you
+doubt me, look again at the rules of the system. You will find that each rule makes a
+symbol act exactly as the word it represents ought to be used. For instance, the joining
+rule makes the symbol ‘a’ act as 'and' ought to act; the rule of detachment makes '3'
+act as it ought to, if it is to stand for 'implies', or 'if-then'; and so on. Unless you are
+like the Tortoise, you will recognize in each rule a codification of a pattern you use in
+your own thought patterns. So if you trust your own thought patterns, then you HAVE
+to believe that all theorems come out true! That's the way I see it. I don't need any
+further proof. If you think that some theorem comes out false, then presumably you
+think that some rule must be wrong. Show me which one.
+
+Prudence: I'm not sure that there is any faulty rule, so I can't point one out to you. Still, I
+can imagine the following kind of scenario. You, following the rules, come up with a
+theorem - say x. Meanwhile I, also following the rules, come up with another
+theorem-it happens to be ~x. Can't you force yourself to conceive of that?
+Imprudence: All right; let's suppose it happened. Why would it bother you? Or let me put
+it another way. Suppose that in playing with the MlU-system, I came up with a
+theorem x, and you came up with xU Can you force yourself to conceive of that?
+Prudence: Of course-in fact both MI and MIU are theorems.
+
+Imprudence: Doesn't that bother you?
+
+Prudence: Of course not. Your example is ridiculous, because MI and MIU are not
+CONTRADICTORY, whereas two strings x and ~x in the Propositional Calculus
+ARE contradictory.
+
+
+The Propositional Calculus
+
+
+191
+
+
+
+Imprudence: Well, yes - provided you wish to interpretas 'not'. But what would lead
+you to think that'-' should be interpreted as 'not'?
+
+Prudence: The rules themselves. When you look at them, you realize that the only
+conceivable interpretation for '-' is 'not'-and likewise, the only conceivable
+interpretation for 'a' is 'and', etc.
+
+Imprudence: In other words, you are convinced that the rules capture the meanings of
+those words?
+
+Prudence: Precisely.
+
+Imprudence: And yet you are still willing to entertain the thought that both x and ~x
+could be theorems? Why not also entertain the notion that hedgehogs are frogs, or that
+1 equals 2, or that the moon is made of green cheese? I for one am not prepared even
+to consider whether such basic ingredients of my thought processes are wrong —
+because if I entertained that notion, then I would also have to consider whether my
+modes of analyzing the entire question are also wrong, and I would wind up in a total
+tangle.
+
+Prudence: Your arguments are forceful ... Yet I would still like to see a PROOF that all
+theorems come out true, or that x and ~x can never both be theorems.
+
+Imprudence: You want a proof. I guess that means that you want to be more convinced
+that the Propositional Calculus is consistent than you are convinced of your own
+sanity. Any proof I could think of would involve mental operations of a greater
+complexity than anything in the Propositional Calculus itself. So what would it prove?
+Your desire for a proof of consistency of the Propositional Calculus makes me think
+of someone who is learning English and insists on being given a dictionary which
+definers all the simple words in terms of complicated ones...
+
+The Carroll Dialogue Again
+
+This little debate shows the difficulty of trying to use logic and reasoning to defend
+themselves. At some point, you reach rock bottom, and there is no defense except loudly
+shouting, "I know I'm right!" Once again, we are up against the issue which Lewis
+Carroll so sharply set forth in his Dialogue: you can't go on defending your patterns of
+reasoning forever. There comes a point where faith takes over.
+
+A system of reasoning can be compared to an egg. An egg has a shell which
+protects its insides. If you want to ship an egg somewhere, though, you don't rely on the
+shell. You pack the egg in some sort of container, chosen according to how rough you
+expect the egg's voyage to be. To be extra careful, you may put the egg inside several
+nested boxes. However, no matter how many layers of boxes you pack your egg in, you
+can imagine some cataclysm which could break the egg. But that doesn't mean that you'll
+never risk transporting your egg. Similarly, one can never give an ultimate, absolute
+proof that a proof in some system is correct. Of course,
+
+
+The Propositional Calculus
+
+
+192
+
+
+
+one can give a proof of a proof, or a proof of a proof of a proof - but the validity of the
+outermost system always remains an unproven assumption, accepted on faith. One can
+always imagine that some unsuspected subtlety will invalidate every single level of proof
+down to the bottom, and tl "proven" result will be seen not to be correct after all. But that
+doesn’t mean that mathematicians and logicians are constantly worrying that the whole
+edifice of mathematics might be wrong. On the other hand, unorthodox proofs are
+proposed, or extremely lengthy proofs, or proofs generated by computers, then people do
+stop to think a bit about what they really mean by that quasi-sacred word "proven".
+
+An excellent exercise for you at this point would be to go back Carroll Dialogue,
+and code the various stages of the debate into our notation — beginning with the original
+bone of contention:
+
+Achilles: If you have «AaB>z>Z>, and you also have <AaB>, then surely you have Z.
+Tortoise: Oh! You mean: ««AaB>=>Z>a<AaB»z>Z>, : don't you?
+
+(Hint: Whatever Achilles considers a rule of inference, the Tortoise immediately flattens
+into a mere string of the system. If you use or letters A, B, and Z, you will get a recursive
+pattern of longer and strings.)
+
+Shortcuts and Derived Rules
+
+When carrying out derivations in the Propositional Calculus, one quickly invents various
+types of shortcut, which are not strictly part of the system For instance, if the string
+<Qv~Q> were needed at some point, and <Pv~P> had been derived earlier, many people
+would proceed as if <Qv~Q> had been derived, since they know that its derivation is an
+exact parallel to that of <Pv~P>. The derived theorem is treated as a "theorem schema" —
+a mold for other theorems. This turns out to be a perfect valid procedure, in that it always
+leads you to new theorems, but it is not a rule of the Propositional Calculus as we
+presented it. It is, rather, a derived ride , It is part of the knowledge which we have about
+the system. That this rule keeps you within the space of theorems needs proof, of course -
+but such a proof is not like a derivation inside the system. It is a proof in the ordinary,
+intuitive sense — a chain of reasoning carried out in the I-mode. The theory about the
+Propositional Calculus is a "metatheory", and results in it can be called "metatheorems" -
+Theorems about theorems. (Incidentally, note the peculiar capitalization in the phrase
+"Theorems about theorems". It is a consequence of our convention: metatheorems are
+Theorems (proven results) concerning theorems (derivable strings).)
+
+In the Propositional Calculus, one could discover many metatheorems, or derived
+rules of inference. For instance, there is a De Morgan's Rule:
+
+
+The Propositional Calculus
+
+
+193
+
+
+
+<~xv~y> and ~<XAy> are interchangeable.
+
+
+If this were a rule of the system, it could speed up many derivations considerably. But if
+we prove that it is correct, isn't that good enough? Can't we use it just like a rule of
+inference, from then on?
+
+There is no reason to doubt the correctness of this particular derived rule. But
+once you start admitting derived rules as part of your procedure in the Propositional
+Calculus, you have lost the formality of the system, since derived rules are derived
+informally-outside the system. Now formal systems were proposed as a way to exhibit
+every step of a proof explicitly, within one single, rigid framework, so that any
+mathematician could check another's work mechanically. But if you are willing to step
+outside of that framework at the drop of a hat, you might as well never have created it at
+all. Therefore, there is a drawback to using such shortcuts.
+
+Formalizing Higher Levels
+
+On the other hand, there is an alternative way out. Why not formalize the metatheory,
+too? That way, derived rules (metatheorems) would be theorems of a larger formal
+system, and it would be legitimate to look for shortcuts and derive them as theorems-that
+is, theorems of the formalized metatheory-which could then be used to speed up the
+derivations of theorems of the Propositional Calculus. This is an interesting idea, but as
+soon as it is suggested, one jumps ahead to think of metametatheories, and so on. It is
+clear that no matter how many levels you formalize, someone will eventually want to
+make shortcuts in the top level.
+
+It might even be suggested that a theory of reasoning could be identical to its own
+metatheory, if it were worked out carefully. Then, it might seem, all levels would
+collapse into one, and thinking about the system would be just one way of working in the
+system! But it is not that easy. Even if a system can "think about itself", it still is not
+outside itself. You, outside the system, perceive it differently from the way it perceives
+itself. So there still is a metatheory-a view from outside-even for a theory which can
+"think about itself" inside itself. We will find that there are theories which can "think
+about themselves". In fact, we will soon see a system in which this happens completely
+accidentally, without our even intending it! And we will see what kinds of effects this
+produces. But for our study of the Propositional Calculus, we will stick with the simplest
+ideas-no mixing of levels.
+
+Fallacies can result if you fail to distinguish carefully between working in the
+system (the M-mode) and thinking about the system (the I-mode). For example, it might
+seem perfectly reasonable to assume that, since <Pv~P> (whose semi-interpretation is
+"either P or not P") is a theorem, either P or ~P must be a theorem. But this is dead
+wrong: neither one of the latter pair is a theorem. In general, it is a dangerous practice to
+assume that symbols can be slipped back and forth between different levels-here, the
+language of the formal system and its metalanguage (English).
+
+
+The Propositional Calculus
+
+
+194
+
+
+
+Reflections on the Strengths and Weaknesses of the System
+
+
+You have now seen one example of a system with a purpose-to re part of the architecture
+of logical thought. The concepts which this handles are very few in number, and they are
+very simple, precise co But the simplicity and precision of the Propositional Calculus are
+the kinds of features which make it appealing to mathematicians. There are two reasons
+for this. (1) It can be studied for its own properties, ex geometry studies simple, rigid
+shapes. Variants can be made on it, employing different symbols, rules of inference,
+axioms or axiom schemata on. (Incidentally, the version of the Propositional Calculus
+here pr is related to one invented by G. Gentzen in the early 1930's. The other versions in
+which only one rule of inference is used-detachment usually-and in which there are
+several axioms, or axiom schemata study of ways to carry out propositional reasoning in
+elegant formal systems is an appealing branch of pure mathematics. (2) The Propositional
+Calculus can easily be extended to include other fundamental aspects of reasoning. Some
+of this will be shown in the next Chapter, where the Propositional Calculus is
+incorporated lock, stock and barrel into a much larger and deeper system in which
+sophisticated number-theoretical reasoning can be done.
+
+Proofs vs. Derivations
+
+The Propositional Calculus is very much like reasoning in some w one should not equate
+its rules with the rules of human thought. A proof is something informal, or in other
+words a product of normal thought written in a human language, for human consumption.
+All sorts of complex features of thought may be used in proofs, and, though they may
+“feel right", one may wonder if they can be defended logically. That is really what
+formalization is for. A derivation is an artificial counterpart of and its purpose is to reach
+the same goal but via a logical structure whose methods are not only all explicit, but also
+very simple.
+
+If - and this is usually the case -it happens that a formal derivation is extremely
+lengthy compared with the corresponding "natural" proof that is just too bad. It is the
+price one pays for making each step so simple. What often happens is that a derivation
+and a proof are "simple" in complementary senses of the word. The proof is simple in
+that each step sounds right", even though one may not know just why; the derivation is
+simple in that each of its myriad steps is considered so trivial that it is beyond reproach,
+and since the whole derivation consists just of such trivial steps it is supposedly error-
+free. Each type of simplicity, however, brings along a characteristic type of complexity.
+In the case of proofs, it is the complexity of the underlying system on which they rest —
+namely, human language — and in the case of derivations, it is their astronomical size,
+which makes them almost impossible to grasp.
+
+Thus, the Propositional Calculus should be thought of as part of a
+
+
+The Propositional Calculus
+
+
+195
+
+
+
+general method for synthesizing artificial proof-like structures. It does not, however, have
+much flexibility or generality. It is intended only for use in connection with mathematical
+concepts-which are themselves quite rigid. As a rather interesting example of this, let us
+make a derivation in which a very peculiar string is taken as a premise in a fantasy:
+<Pa~P>. At least its semi-interpretation is peculiar. The Propositional Calculus,
+however, does not think about semi-interpretations; it just manipulates strings
+typographically-and typographically, there is really nothing peculiar about this string.
+Here is a fantasy with this string as its premise:
+
+
+(1)
+
+[
+
+push
+
+(2)
+
+<Pa~P>
+
+premise
+
+(3)
+
+P
+
+separation
+
+(4)
+
+~P
+
+separation
+
+(5)
+
+[
+
+push
+
+(6)
+
+~Q
+
+premise
+
+(7)
+
+P
+
+carry-over line 3
+
+(8)
+
+~~P
+
+double-tilde
+
+(9)
+
+]
+
+pop
+
+(10)
+
+<~Q=>~~P>
+
+fantasy
+
+(ID
+
+<~P=>Q>
+
+contrapositive
+
+(12)
+
+Q
+
+detachment (Lines 4,11)
+
+(13)
+
+]
+
+pop
+
+(14)
+
+«Pa~P >=)Q>
+
+fantasy
+
+
+Now this theorem has a very strange semi-interpretation:
+
+P and not P together imply Q
+
+Since Q is interpretable by any statement, we can loosely take the theorem to say that
+"From a contradiction, anything follows"! Thus, in systems based on the Propositional
+Calculus, contradictions cannot be contained; they infect the whole system like an
+instantaneous global cancer.
+
+The Handling of Contradictions
+
+This does not sound much like human thought. If you found a contradiction in your own
+thoughts, it's very unlikely that your whole mentality would break down. Instead, you
+would probably begin to question the beliefs or modes of reasoning which you felt had
+led to the contradictory thoughts. In other words, to the extent you could, you would step
+out of the systems inside you which you felt were responsible for the contradiction, and
+try to repair them. One of the least likely things for you to do would be to throw up your
+arms and cry, "Well, I guess that shows that I believe everything now!" As a joke, yes-
+but not seriously.
+
+Indeed, contradiction is a major source of clarification and progress in all domains
+of life-and mathematics is no exception. When in times past, a
+
+
+The Propositional Calculus
+
+
+196
+
+
+
+contradiction in mathematics was found, mathematicians would immediately seek to
+pinpoint the system responsible for it, to jump out of it, to reason about it, and to amend
+it. Rather than weakening mathematics, the discovery and repair of a contradiction would
+strengthen it. This might take time and a number of false starts, but in the end it would
+yield fmit. For instance, in the Middle Ages, the value of the infinite series
+
+
+1 - 1 + 1 - 1 + 1 -. ..
+
+
+was hotly disputed. It was "proven" to equal 0, 1, Vi, and perhaps other values. Out of
+such controversial findings came a fuller, deeper about infinite series.
+
+A more relevant example is the contradiction right now confronting us-namely the
+discrepancy between the way we really think, and t the Propositional Calculus imitates
+us. This has been a source of discomfort for many logicians, and much creative effort has
+gone into trying to patch up the Propositional Calculus so that it would not act so stupidly
+and inflexibly. One attempt, put forth in the book Entailment by A. R. Anderson and N.
+Belnap,' involves "relevant implication", which tries to make the symbol for "if-then"
+reflect genuine causality, or at least connect meanings. Consider the following theorems
+of the Propositional Calculus
+
+
+<P=><Q=>P»
+
+<P=><Qv~P»
+
+«Pa~P>z)Q>
+
+«P=)Q>v<Q=>P»
+
+
+They, and many others like them, all show that there need be no relationship at all
+between the first and second clauses of an if-then statement for it to be provable within
+the Propositional Calculus. In protest, "relevant implication" puts certain restrictions on
+the contexts in which the rules of inference can be applied. Intuitively, it says that
+"something can only be derived from something else if they have to do with each other”.
+For example, line 10 in the derivation given above would not be allowed in such a
+system, and that would block the derivation of the «Pa~P >z>Q>
+
+More radical attempts abandon completely the quest for completeness or
+consistency, and try to mimic human reasoning with all its inconsistencies. Such research
+no longer has as its goal to provide a solid underpinning for mathematics, but purely to
+study human thought processes.
+
+Despite its quirks, the Propositional Calculus has some feat recommend itself. If
+one embeds it into a larger system (as we will do next Chapter), and if one is sure that the
+larger system contains no contradictions (and we will be), then the Propositional Calculus
+does all that one could hope: it provides valid propositional inferences - all that can be
+made. So if ever an incompleteness or an inconsistency is uncovered, can be sure that it
+will be the fault of the larger system, and not of its subsystem which is the Propositional
+Calculus.
+
+
+The Propositional Calculus
+
+
+197
+
+
+
+
+FIGURE 42. “Crab Canon”, by M. C. Escher(~1965)
+
+
+Crab Canon
+
+
+198
+
+
+
+Crab Canon
+
+
+Achilles and the Tortoise happen upon each other
+in the park one day while strolling.
+
+
+Tortoise: Good day, Mr. A.
+
+Achilles: Why, same to you. Tortoise: So nice to run into you. Achilles: That echoes my
+thoughts.
+
+Tortoise: And it's a perfect day for a walk. I think I'll be walking home soon.
+
+Achilles: Oh, really? I guess there's nothing better for you than w Tortoise: Incidentally,
+you're looking in very fine fettle these days, I must say.
+
+Achilles: Thank you very much.
+
+Tortoise: Not at all. Here, care for one of my cigars?
+
+Achilles: Oh, you are such a philistine. In this area, the Dutch contributions are of
+markedly inferior taste, don't you think?
+
+Tortoise: I disagree, in this case. But speaking of taste, I finally saw that Crab Canon by
+your favorite artist, M. C. Escher, in a gallery the other day, and I fully appreciate the
+beauty and ingenuity with which he made one single theme mesh with itself going
+both backwards and forwards. But I am afraid I will always feel Bach is superior to
+Escher.
+
+
+Crab Canon
+
+
+199
+
+
+
+Achilles: I don't know. But one thing for certain is that I don't worry about arguments of
+taste. De gustibus non est disputandum.
+
+Tortoise: Tell me, what's it like to be your age? Is it true that one has no worries at all?
+
+Achilles: To be precise, one has no frets.
+
+Tortoise: Oh, well, it's all the same to me.
+
+Achilles: Fiddle. It makes a big difference, you know. Tortoise: Say, don't you play the
+guitar?
+
+Achilles: That's my good friend. He often plays, the fool. But I myself wouldn't touch a
+guitar with a ten-foot pole!
+
+(Suddenly, the Crab, appearing from out of nowhere, wanders up excitedly,
+pointing to a rather prominent black eye.)
+
+Crab: Hallo! Hulloo! What's up? What's new? You see this bump, this lump? Given to
+me by a grump. Ho! And on such a fine day. You see, I was just idly loafing about the
+park when up lumbers this giant fellow from Warsaw-a colossal bear of a man¬
+playing a lute. He was three meters tall, if I'm a day. I mosey on up to the chap, reach
+skyward and manage to tap him on the knee, saying, "Pardon me, sir, but you are
+Pole-luting our park with your mazurkas." But wow! he had no sense of humor-not a
+bit, not a wit-and POW!-he lets loose and belts me one, smack in the eye! Were it in
+my nature, I would crab up a storm, but in the time-honored tradition of my species, I
+backed off. After all, when we walk forwards, we move backwards. It's in our genes,
+you know, turning round and round. That reminds me-I've always wondered, "Which
+came first-the Crab, or the Gene?" That is to say, "Which came last the Gene, or the
+Crab?" I'm always turning things round and round, you know. It's in our genes, after
+all. When we walk backwards, we move forwards. Ah me, oh my! I must lope along
+on my merry way-so off I go on such a fine day. Sing "ho!" for the life of a Crab!
+TATA! iOle!
+
+(And he disappears as suddenly as he arrived.)
+
+Tortoise: That's my good friend. He often plays the fool. But I myself wouldn't touch a
+ten-foot Pole with a guitar!
+
+Achilles: Say, don't you play the guitar? Tortoise: Fiddle. It makes a big difference,
+Achilles: Oh, well, it's all the same to me.
+
+Tortoise: To be precise, one has no frets.
+
+Achilles: Tell me, what's it like to be your age? Is it true that one has no worries at all?
+
+Tortoise: I don't know. But one thing for certain is that I don't worry about arguments of
+taste. Disputandum non est de gustibus.
+
+
+Crab Canon
+
+
+200
+
+
+
+
+FIGURE 43. Here is a short section one of the Crab's
+Genes, turning round and round. When the two DNA
+strands are raveled and laid out side by side, they
+read this way:
+
+XXTTTTXTTCGAAAAAAAAA
+.... A A A A A A A AGCTTTTTTTTTT
+
+Notice that they are the same, only one forwards while
+the other goes backwards This is the defining
+property of the form called "crab canon” in music. It
+is reminiscent of, though a little different from
+palindrome, which is a sentence that reads the same
+backwards and forwards ,In molecular biology,
+such segments of DNA are called "palindromes "-a
+slight misnomer, since "crab canon" would be more
+accurate. Not only is this DNA segment crab-
+canonical-but moreover its base sequence codes for
+the Dialogue's structure Look care fully!
+
+
+Achilles: I disagree, in this case. But speaking of taste, I finally heard that Crab Canon
+by your favorite composer, J. S Bach, in a concert other day, and I fully appreciate
+the beauty and ingenuity with which he made one single theme mesh with itself going
+both backwards and forwards. But I'm afraid I will always feel Escher is superior to
+Bach
+
+Tortoise: Oh, you are such a philistine. In this area, the Dutch contributions are of
+markedly inferior taste, don’t you think?
+
+Achilles: Not at all. Here, care for one of my cigars?
+
+Tortoise: Thank you very much.
+
+Achilles: Incidentally, you’re looking in very fine fettle these days, I must say.
+
+
+Crab Canon
+
+
+201
+
+
+
+
+
+.$*_« j j-*r xxxstxxs
+
+
+i -i-=
+
+
+
+
+-J. 7
+
+X&JTJ}
+
+
+CRJ\S> CAl XJOM ja>
+
+
+^>1 j r* 1
+
+*.. 3,
+
+¥-- J—-
+
+s®HH5
+
+J-
+
+JS&OT
+
+T~~3~—
+
+
+
+«l wooyo a/\jo
+
+
+
+
+^ ^ j g V J U w * • »i < J * g *
+
+
+
+
+•’ * 4
+
+
+FKiL'Ri .' 44 Crjb (jnon /twb A# Mu«ma! Ottering. h J. S. Both (.llmar fr m ttd h l lmm U By<i ^w fi '.OUT ]
+
+
+Crab Canon
+
+
+202
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Tortoise: Oh, really? I guess there's nothing better for you than walking.
+Achilles: And it's a perfect day for a walk. I think I'll be walking home soon.
+Tortoise: That echoes my thoughts.
+
+Achilles: So nice to run into you.
+
+Tortoise: Why, same to you.
+
+Achilles: Good day, Mr. T
+
+
+Crab Canon
+
+
+203
+
+
+
+CHAPTER VIII
+
+
+Typographical Number Theory
+
+
+The Crab Canon and Indirect Self-Reference
+
+THREE EXAMPLES OF indirect self-reference are found in the Crab Canon. Achilles
+and the Tortoise both describe artistic creations they know-and, quite accidentally, those
+creations happen to have the same structure as the Dialogue they're in. (Imagine my
+surprise, when I, the author, noticed this!) Also, the Crab describes a biological structure
+and that, too, has the same property. Of course, one could read the Dialogue and
+understand it and somehow fail to notice that it, too, has the form of a crab canon. This
+would be understanding it on one level, but not on another. To see the self-reference, one
+has to look at the form, as well as the content, of the Dialogue.
+
+Godel’s construction depends on describing the form, as well as the content, of
+strings of the formal system we shall define in this Chapter — Typographical Number
+Theory (TNT). The unexpected twist is that, because of the subtle mapping which Godel
+discovered, the form of strings can be described in the formal system itself. Let us
+acquaint ourselves with this strange system with the capacity for wrapping around.
+
+What We Want to Be Able to Express in TNT
+
+We'll begin by citing some typical sentences belonging to number theory; then we will
+try to find a set of basic notions in terms of which all our sentences can be rephrased.
+Those notions will then be given individual symbols. Incidentally, it should be stated at
+the outset that the term "number theory" will refer only to properties of positive integers
+and zero (and sets of such integers). These numbers are called the natural numbers.
+Negative numbers play no role in this theory. Thus the word "number", when used, will
+mean exclusively a natural number. And it is important — vital-for you to keep separate in
+your mind the formal system (TNT) and the rather ill-defined but comfortable old branch
+of mathematics that is number theory itself; this I shall call "N".
+
+Some typical sentences of N-number theory-are:
+
+(1) 5 is prime.
+
+(2) 2 is not a square.
+
+(3) 1729 is a sum of two cubes.
+
+(4) No sum of two positive cubes is itself a cube.
+
+(5) There are infinitely many prime numbers.
+
+(6) 6 is even.
+
+
+Typographical Number Theory
+
+
+204
+
+
+
+Now it may seem that we will need a symbol for each notion such as "prime” or "cube"
+or "positive" — but those notions are really not primitive. Primeness, for instance, has to
+do with the factors which a number has, which in turn has to do with multiplication.
+Cubeness as well is defined in terms multiplication. Let us rephrase the sentences, then,
+in terms of what seem to be more elementary notions.
+
+(1) There do not exist numbers a and b, both greater than 1. such that 5 equals a
+times b.
+
+(2) There does not exist a number b, such that b times b equals 2.
+
+(3) There exist numbers b and c such that b times b times b, plus c times c times c,
+equals 1729.
+
+(4') For all numbers b and c, greater than 0, there is no number a such that a times a
+times a equals b times b times b plus c times c times c.
+
+(5) For each number a, there exists a number b, greater than a, with the property
+that there do not exist numbers c and d, both greater than 1 , such that b equals c
+times d.
+
+(6') There exists a number e such that 2 times e equals 6.
+
+This analysis has gotten us a long ways towards the basic elements of language of
+number theory. It is clear that a few phrases reappear over a over:
+
+for all numbers b
+
+there exists a number b, such that
+
+greater than
+
+equals
+
+times
+
+plus
+
+0 , 1 , 2 ,..
+
+Most of these will be granted individual symbols. An exception is "greater than", which
+can be further reduced. In fact, the sentence "a is greater than b" becomes
+
+there exists a number c, not equal to 0, such that a equals b plus c.
+
+Numerals
+
+We will not have a distinct symbol for each natural number. Instead, we have a very
+simple, uniform way of giving a compound symbol to e natural number - very much as
+we did in the pq-system. Here is notation for natural numbers:
+
+
+Typographical Number Theory
+
+
+205
+
+
+
+etc.
+
+
+zero:
+
+one:
+
+two:
+
+three:
+
+
+0
+
+SO
+
+sso
+
+ssso
+
+
+The symbol S has an interpretation-"the successor of". Hence, the interpretation of SSO
+is literally "the successor of the successor of zero". Strings of this form are called
+numerals.
+
+
+Variables and Terms
+
+Clearly, we need a way of referring to unspecified, or variable, numbers. For that, we will
+use the letters a, b, c, d, e. But five will not be enough. We need an unlimited supply of
+them, just as we had of atoms in the Propositional Calculus. We will use a similar method
+for making more variables: tacking on any number of primes. (Note: Of course the
+symbol "'-read "prime"-is not to be confused with prime numbers!) For instance:
+
+e
+
+d 1
+
+c"
+
+b'"...
+
+a""
+
+
+are all variables.
+
+In a way it is a luxury to use the first five letters of the alphabet when we could
+get away with just a and the prime. Later on, I will actually drop b, c, d, and e, which will
+result in a sort of "austere" version of TNT-austere in the sense that it is a little harder to
+decipher complex formulas. But for now well be luxurious.
+
+Now what about addition and multiplication? Very simple: we will use the
+ordinary symbols '+' and However, we will also introduce a parenthesizing
+requirement (we are now slowly slipping into the rules which define well-formed strings
+of TNT). To write "b plus c" and "b times c", for instance, we use the strings
+
+(b+c)
+
+(b • c)
+
+There is no laxness about such parentheses; to violate the convention is to produce a non¬
+well-formed formula. ("Formula"? I use the term instead of "string" because it is
+conventional to do so. A formula is no more and no less than a string of TNT.)
+
+Incidentally, addition and multiplication are always to be thought of as binary operations-
+that is, they unite precisely two numbers, never three or more. Hence, if you wish to
+translate "1 plus 2 plus 3", you have to decide which of the following two expressions
+you want:
+
+
+Typographical Number Theory
+
+
+206
+
+
+
+(SO+(SSO+SSSO))
+
+((SO+SSO)+SSSO)
+
+
+The next notion we'll symbolize is equals. That is very simple: we use '=2The advantage
+of taking over the standard symbol used N — nonformal number theory — iis obvious:
+easy legibility. The disadvantage is very much like the disadvantage of using the words
+"point" a "line" in a formal treatment of geometry: unless one is very conscious a careful,
+one may blur the distinction between the familiar meaning and strictly rule-governed
+behavior of the formal symbol. In discuss geometry, I distinguished between the
+everyday word and the formal to by capitalizing the formal term: thus, in elliptical
+geometry, a POINT was 1 union of two ordinary points. Here, there is no such
+distinction; hen mental effort is needed not to confuse a symbol with all of the association
+is laden with. As I said earlier, with reference to the pq-system: the string — is not the
+number 3, but it acts isomorphically to 3, at least in the context of additions. Similar
+remarks go for the string SSSO.
+
+Atoms and Propositional Symbols
+
+All the symbols of the Propositional Calculus except the letters used making atoms (P, Q,
+and R) will be used in TNT, and they retain their interpretations. The role of atoms will
+be played by strings which, when interpreted, are statements of equality, such as
+so=sso or (SO • SO) Now, we have the equipment to do a fair amount of translation of
+simple sentences into the notation of TNT:
+
+2 plus 3 equals 4: (SSO+SSSO)=SSSSO
+
+2 plus 2 is not equal to 3: ~(SSO+SSO)=SSSO
+
+If 1 equals 0, then 0 equals 1: <SO=OJO=SO>
+
+The first of these strings is an atom; the rest are compound formulas (Warning: The 'and'
+in the phrase "I and 1 make 2" is just another word for 'plus', and must be represented by
+'+' (and the requisite parentheses).)
+
+Free Variables and Quantifiers
+
+All the well-formed formulas above have the property that their interpretations are
+sentences which are either true or false. There are, however, well-formed formulas which
+do-not have that property, such as this one
+
+(b+SO)=SSO
+
+Its interpretation is "b plus 1 equals 2". Since b is unspecified, there is way to assign a
+truth value to the statement. It is like an out-of-context statement with a pronoun, such as
+"she is clumsy". It is neither true nor false; it is waiting for you to put it into a context.
+Because it is neither true nor false, such a formula is called open , and the variable b is
+called a free variable.
+
+
+Typographical Number Theory
+
+
+207
+
+
+
+One way of changing an open formula into a closed formula, or sentence, is by
+prefixing it with a quantifier-e ither the phrase "there exists a number b such that , or the
+phrase "for all numbers b". In the first instance, you get the sentence
+
+There exists a number b such that b plus 1 equals 2.
+
+Clearly this is true. In the second instance, you get the sentence
+
+For all numbers b, b plus 1 equals 2.
+
+Clearly this is false. We now introduce symbols for both of these quantifiers. These
+sentences are translated into TNT-notation as follows:
+
+3b:(b+SO)=SSO ('3' stands for 'exists'.)
+
+¥b:(b+SO)=SSO (’¥' stands for 'all'.)
+
+It is very important to note that these statements are no longer about unspecified
+numbers; the first one is an assertion of existence, and the second one is a universal
+assertion. They would mean the same thing, even if written with c instead of b:
+
+3c:(c+SO)=SSO'
+
+¥c:(c+SO)=SSO
+
+A variable which is under the dominion of a quantifier is called a quantified variable. The
+following two formulas illustrate the difference between free variables and quantified
+variables:
+
+
+(b.b)=SSO (open)
+
+—-3b:(b*b)=SSO (closed; a sentence of TNT)
+
+The first one expresses a property which might be possessed by some natural number. Of
+course, no natural number has that property. And that is precisely what is expressed by
+the second one. It is very crucial to understand this difference between a string with a free
+variable, which expresses a property, and a string where the variable is quantified, which
+expresses a truth or falsity. The English translation of a formula with at least one free
+variable-an open formula-is called a predicate. It is a sentence without a subject (or a
+sentence whose subject is an out-of-context pronoun). For instance,
+
+"is a sentence without a subject"
+
+"would be an anomaly"
+
+"runs backwards and forwards simultaneously"
+
+"improvised a six-part fugue on demand"
+
+are nonarithmetical predicates. They express properties which specific entities might or
+might not possess. One could as well stick on a "dummy
+
+
+Typographical Number Theory
+
+
+208
+
+
+
+subject", such as "so-and-so". A string with free variables is like a predicate with "so-
+and-so" as its subject. For instance,
+
+
+(SO+SO)=b
+
+is like saying "1 plus 1 equals so-and-so". This is a predicate in the variable b. It
+expresses a property which the number b might have. If one wet substitute various
+numerals for b, one would get a succession of forms most of which would express
+falsehoods. Here is another example of difference between open formulas and sentences :
+
+'¥b:'¥c:(b+c)=(c+b)
+
+The above formula is a sentence representing, of course, the commutativity of addition.
+On the other hand,
+
+
+'¥c:(b+c)=(c+b)
+
+is an open formula, since b is free. It expresses a property which unspecified number b
+might or might not have - namely of commuting with all numbers c.
+
+Translating Our Sample Sentences
+
+This completes the vocabulary with which we will express all num theoretical statements!
+It takes considerable practice to get the hang of expressing complicated statements of N
+in this notation, and converse] figuring out the meaning of well-formed formulas. For this
+reason return to the six sample sentences given at the beginning, and work their
+translations into TNT. By the way, don't think that the translations given below are
+unique-far from it. There are many - infinitely many - ways to express each one.
+
+Let us begin with the last one: "6 is even". This we rephrased in to of more
+primitive notions as "There exists a number e such that 2 times e equals 6". This one is
+easy
+
+
+3e:(SSO. e)=SSSSSSO
+
+Note the necessity of the quantifier; it simply would not do to write
+
+(SSO . e)=SSSSSSO
+
+alone. This string's interpretation is of course neither true nor false; it expresses a
+property which the number e might have.
+
+It is curious that, since we know multiplication is commutative might easily have
+
+written
+
+
+3e:(e - SSO)=SSSSSSO
+
+instead. Or, knowing that equality is a symmetrical relation, we might 1 chosen to write
+the sides of the equation in the opposite order:
+
+
+Typographical Number Theory
+
+
+209
+
+
+
+3e:SSSSSSO=(SSO • e)
+
+
+Now these three translations of "6 is even" are quite different strings, and it is by no
+means obvious that theoremhood of any one of them is tied to theoremhood of any of the
+others. (Similarly, the fact that —p-q— was a theorem had very little to do with the fact
+that its "equivalent" string -p—q— was a theorem. The equivalence lies in our minds,
+since, as humans, we almost automatically think about interpretations, not structural
+properties of formulas.)
+
+We can dispense with sentence 2: "2 is not a square", almost immediately:
+
+-3b:(b • b)=SSO
+
+However, once again, we find an ambiguity. What if we had chosen to write it this way?
+
+Vb: -(b • b) =SSO
+
+The first way says, "It is not the case that there exists a number b with the property that
+b's square is 2", while the second way says, "For all numbers b, it is not the case that b's
+square is 2." Once again, to us, they are conceptually equivalent-but to TNT, they are
+distinct strings.
+
+Let us proceed to sentence 3: "1729 is a sum of two cubes." This one will involve
+two existential quantifiers, one after the other, as follows:
+
+3b:3c:SSSSSS.SSSSSO=(((b • b) • b)+((c • c) • c))
+
+1729 of them
+
+There are alternatives galore. Reverse the order of the quantifiers; switch the sides of the
+equation; change the variables to d and e; reverse the addition; write the multiplications
+differently; etc., etc. However, I prefer the following two translations of the sentence:
+
+3b:3c:(((SSSSSSSSSSO.SSSSSSSSSSO).SSSSSSSSSSO)+
+((SSSSSSSSSO • SSSSSSSSSO) • SSSSSSSSSO))=(((b • b) • b)+((c • c) • c))
+
+and
+
+3b:3c:(((SSSSSSSSSSSSO.SSSSSSSSSSSSO). SSSSSSSSSSSSO)+
+
+((SO 'SO) • SO))=(((b *b) »b)+((c • c) «c))
+
+
+Do you see why?
+
+
+Tricks of the Trade
+
+Now let us tackle the related sentence 4: "No sum of two positive cubes is itself a cube".
+Suppose that we wished merely to state that 7 is not a sum of two positive cubes. The
+easiest way to do this is by negating the formula
+
+
+Typographical Number Theory
+
+
+210
+
+
+
+
+which asserts that 7 is a sum of two positive cubes. This will be just like the preceding
+sentence involving 1729, except that we have to add in the proviso of the cubes being
+positive. We can do this with a trick: prefix variables with the symbol S, as follows:
+
+3b:3c:SSSSSSSO=(((Sb • Sb) • Sb)+((Sc • Sc) -Sc))
+
+You see, we are cubing not b and c, but their successors, which must be positive, since
+the smallest value which either b or c can take on is zero. Hence the right-hand side
+represents a sum of two positive cubes. In( tally, notice that the phrase "there exist
+
+numbers b and c such that.”) when translated, does not involve the symbol 'ii' which
+
+stands for ‘and’. That symbol is used for connecting entire well-formed strings, not for
+joining two quantifiers.
+
+Now that we have translated "7 is a sum of two positive cubes", we wish to negate
+it. That simply involves prefixing the whole thing by a single (Note: you should not
+negate each quantifier, even though the desired phrase runs "There do not exist numbers
+b and c such that...".) Thus we get:
+
+-3b:3c:SSSSSSSO=(((Sb • Sb) • Sb)+((Sc -Sc) -Sc))
+
+Now our original goal was to assert this property not of the number of all cubes.
+Therefore, let us replace the numeral SSSSSSSO by the ((a-a)-a), which is the translation
+of "a cubed":
+
+
+3b:3c:((a *a) •a)=(((Sb *Sb) • Sb)+((Sc -Sc) -Sc))
+
+At this stage, we are in possession of an open formula, since a is still free. This formula
+expresses a property which a number a might or might not have-and it is our purpose to
+assert that all numbers do have that property. That is simple — just prefix the whole thing
+with a universal quantifier
+
+¥a:-3b:3c:((a -a) • a)=(((Sb • Sb) • Sb) +((Sc -Sc) -Sc))
+
+An equally good translation would be this:
+
+--3a:3b:3c:((a-a) a)=(((Sb , Sb)»Sb)+((Sc , Sc) , Sc))
+
+In austere TNT, we could use a' instead of b, and a" instead of c, and the formula would
+become:
+
+
+--3a: 3a': 3a":((a • a) • a) =(((Sa' • Sa') • Sa') +((Sa" • Sa") • Sa"))
+
+What about sentence 1: "5 is prime"? We had reworded it in this way "There do not exist
+numbers a and b, both greater than 1, such equals a times b". We can slightly modify it,
+as follows: "There do not exist numbers a and b such that 5 equals a plus 2, times b plus
+2". This is another trick-since a and b are restricted to natural number values, this is an
+adequate way to say the same thing. Now "b plus 2" could be translated into
+
+
+Typographical Number Theory
+
+
+211
+
+
+
+
+(b+SSO), but there is a shorter way to write it — namely, SSb. Likewise, "c plus 2" can
+be written SSc. Now, our translation is extremely concise:
+
+3b: 3c:SSSSSO=(SSb • SSc)
+
+Without the initial tilde, it would be an assertion that two natural numbers do exist,
+which, when augmented by 2, have a product equal to 5. With the tilde in front, that
+whole statement is denied, resulting in an assertion that 5 is prime.
+
+If we wanted to assert that d plus e plus 1, rather than 5, is prime, the most
+economical way would be to replace the numeral for 5 by the string (d+Se):
+
+3b: 3c:(d+Se)=(SSb SSc)
+
+Once again, an open formula, one whose interpretation is neither a true nor a false
+sentence, but just an assertion about two unspecified numbers, d and e. Notice that the
+number represented by the string (d+Se) is necessarily greater than d, since one has
+added to d an unspecified but definitely positive amount. Therefore, if we existentially
+quantify over the variable e, we will have a formula which asserts that:
+
+There exists a number which is greater than d and which is prime.
+
+3e:- 3b:3c:(d+Se)=(SSb • SSc)
+
+Well, all we have left to do now is to assert that this property actually obtains, no matter
+what d is. The way to do that is to universally quantify over the variable d:
+
+Vd:3e:-3b:3c:(d+Se)=(SSb -SSc)
+
+That's the translation of sentence 5!
+
+
+Translation Puzzles for You
+
+This completes the exercise of translating all six typical number-theoretical sentences.
+However, it does not necessarily make you an expert in the notation of TNT. There are
+still some tricky issues to be mastered. The following six well-formed formulas will test
+your understanding of TNT notation. What do they mean? Which ones are true (under
+interpretation, of course), and which ones are false? (Hint: the way to tackle this exercise
+is to move leftwards. First, translate the atom; next, figure out what adding a single
+quantifier or a tilde does; then move leftwards, adding another quantifier or tilde; then
+move leftwards again, and do the same.)
+
+-¥c: 3b:(SSO • b)=c
+
+¥c:- 3b:(SSO • b)=c
+
+
+Typographical Number Theory
+
+
+212
+
+
+
+¥c: 3b:—(SSO • b)=c
+~3b:¥c:(SSO • b)=c
+3b:~¥c:(SSO • b)=c
+3b:-¥c:-(SSO • b)=c
+
+(Second hint: Either four of them are true and two false, or four false and two true.)
+
+How to Distinguish True from False?
+
+At this juncture, it is worthwhile pausing for breath and contempt what it would mean to
+have a formal system that could sift out the true from the false ones. This system would
+treat all these strings-which look like statements-as designs having form, but no content.
+An( system would be like a sieve through which could pass only designs v special style-
+the "style of truth". If you yourself have gone through ti formulas above, and have
+separated the true from the false by this about meaning, you will appreciate the subtlety
+that any system would to have, that could do the same thing-but typographically! The
+bout separating the set of true statements from the set of false statements written in the
+TNT-notation) is anything but straight; it is a boundary with many treacherous curves
+(recall Fig. 18), a boundary of which mathematicians have delineated stretches, here and
+there, working over hundreds years. Just think what a coup it would be to have a
+typographical m( which was guaranteed to place any formula on the proper side o border!
+
+The Rules of Well-Formedness
+
+It is useful to have a table of Rules of Formation for well-formed formulas This is
+provided below. There are some preliminary stages, defining numerals, variables, and
+terms. Those three classes of strings are ingredients of well-formed formulas, but are not
+in themselves well-formed. The smallest well-formed formulas are the atoms ; then there
+are ways of compounding atoms. Many of these rules are recursive lengthening rules, in
+that they take as input an item of a given class and produce a longer item of the class. In
+this table, I use 'x' and 'y' to stand for well-formed formulas, and 's', 't', and 'u' to stand
+for other kinds of TNT-strings. Needless to say, none of these five symbols is itself a
+symbol of TNT.
+
+NUMERAFS.
+
+0 is a numeral.
+
+A numeral preceded by S is also a numeral.
+
+Examples : 0 SO S50 SSSO SSSSO SSSSSO
+
+
+Typographical Number Theory
+
+
+213
+
+
+
+VARIABLES.
+
+a is a variable. If we're not being austere, so are b, c, d and e. A variable followed
+by a prime is also a variable.
+
+Examples: a b' c" d"' a""
+
+
+TERMS.
+
+All numerals and variables are terms.
+
+A term preceded by S is also a term.
+
+If s and t are terms, then so are (,v+ t) and (s • t).
+
+Examples-. 0 b SSa' (SO • (SSO+c)) S(Sa • (Sb • Sc))
+
+TERMS may be divided into two categories:
+
+(1) DEFINITE terms. These contain no variables.
+
+Examples: 0 (SO+SO) SS((SSO.SSO)+(SO.SO))
+
+(2) INDEFINITE terms. These contain variables.
+
+Examples: b Sa (b+SO) (((SO+SO)+SO)+e)
+
+The above rules tell how to make parts of well-formed formulas; the remaining
+rules tell how to make complete well-formed formulas.
+
+ATOMS.
+
+I f .v and t are terms, then s = t is an atom.
+
+Examples: SO=0 (SS0+SS0)=5SSS0 5(b+c)=((c«d).e)
+
+If an atom contains a variable u, then u is free in it. Thus there are
+four free variables in the last example.
+
+NEGATIONS.
+
+A well-formed formula preceded by a tilde is well-formed.
+
+Examples: ~S0=0 ~3b:(b+b)=SO -<O=0z>S0=O> ~b=SO
+
+The quantification status of a variable (which says whether the variable is
+free or quantified) does not change under negation.
+
+COMPOUNDS.
+
+If x and y are well-formed formulas, and provided that no variable which is free in
+one is quantified in the other, then the following are all well-formed formulas:
+
+< xa y>, < xv y>, < xd y>.
+
+Examples: <O=Oa~-0=0> <b=bv~3c:c=b>
+
+<S0=0=>¥c:~3b:(b+b)=c>
+
+The quantification status of a variable doesn't change here.
+
+QUANTI FI CATIONS.
+
+If u is a variable, and x is a well-formed formula in which u is free then the
+following strings are well-formed formulas:
+
+3u: x and ¥u: x.
+
+Examples: ¥b:<b=bv~3c:c=b> vc:~3b:(b+b)=c ~3c:Sc=d
+
+OPEN FORMULAS contain at least one free variable.
+
+Examples: --c=c b=b <¥b:b=bn—c=c>
+
+CLOSED FORMULAS (SENTENCES) contain no free variables.
+
+Examples: 50=0 ~Vd:d=0 3c:<Vb:b=bA~c=c>
+
+
+Typographical Number Theory
+
+
+214
+
+
+
+This completes the table of Rules of Formation for the well-formed formulas of TNT.
+
+A Few More Translation Exercises
+
+
+And now, a few practice exercises for you, to test your understanding of the notation of
+TNT. Try to translate the first four of the following N-sentences into TNT-sentences, and
+the last one into an open formed formula.
+
+All natural numbers are equal to 4.
+
+There is no natural number which equals its own square.
+
+Different natural numbers have different successors.
+
+If 1 equals 0, then every number is odd.
+b is a power of 2.
+
+The last one you may find a little tricky. But it is nothing, compared to this one:
+b is a power of 10.
+
+Strangely, this one takes great cleverness to render in our notation. I would caution you to
+try it only if you are willing to spend hours and hours on it — and if you know quite a bit
+of number theory!
+
+
+A Non typographical System
+
+This concludes the exposition of the notation of TNT; however, we still left with the
+problem of making TNT into the ambitious system which we have described. Success
+would justify the interpretations which we given to the various symbols. Until we have
+done that, however, particular interpretations are no more justified than the "horse-apple
+happy" interpretations were for the pq-system's symbols.
+
+Someone might suggest the following way of constructing TNT: (II) Do not have
+any rules of inference; they are unnecessary, because (2) We take as axioms all true
+statements of number theory (as written in TNT-notation). What a simple prescription!
+Unfortunately it is as empty as instantaneous reaction says it is. Part (2) is, of course, not
+a typographical description of strings. The whole purpose of TNT is to figure out if and
+how it is possible to- characterize the true strings typographically.
+
+The Five Axioms and First Rules of TNT
+
+Thus we will follow a more difficult route than the suggestion above; we will have
+axioms and rules of inference. Firstly, as was promised, all of the rules of the
+Propositional Calculus are taken over into TNT. Therefore one theorem of TNT will be
+this one:
+
+
+<S0=0v~S0=0>
+
+
+Typographical Number Theory
+
+
+215
+
+
+
+which can be derived in the same way as <Pv-P> was derived.
+
+Before we give more rules, let us give the five axioms of TNT:
+
+Axiom 1: ¥a:~Sa=0
+
+Axiom 2: ¥a:(a+0)=a
+
+Axiom 3: ¥a:¥b:(a+Sb)=S(a+b)
+
+Axiom 4: ¥a:(a-0)=0
+
+Axiom 5: ¥a:¥b:(a-Sb)=((a-b)+a)
+
+(In the austere versions, use a' instead of b.) All of them are very simple to understand.
+Axiom 1 states a special fact about the number 0; Axioms 2 and 3 are concerned with the
+nature of addition; Axioms 4 and 5 are concerned with the nature of multiplication, and in
+particular with its relation to addition.
+
+The Five Peano Postulates
+
+By the way, the interpretation of Axiom l-"Zero is not the successor of any natural
+number"-is one of five famous properties of natural numbers first explicitly recognized
+by the mathematician and logician Giuseppe Peano, in 1889. In setting out his postulates,
+Peano was following the path of Euclid in this way: he made no attempt to formalize the
+principles of reasoning, but tried to give a small set of properties of natural numbers from
+which everything else could be derived by reasoning. Peano's attempt might thus be
+considered "semiformal". Peano's work had a significant influence, and thus it would be
+good to show Peano's five postulates. Since the notion of "natural number" is the one
+which Peano was attempting to define, we will not use the familiar term "natural
+number", which is laden with connotation. We will replace it with the undefined term
+djinn, a word which comes fresh and free of connotations to our mind. Then Peano's five
+postulates place five restrictions on djinns. There are two other undefined terms: Genie ,
+and meta. I will let you figure out for yourself what usual concept each of them is
+supposed to represent. The five Peano postulates:
+
+(1) Genie is a djinn.
+
+(2) Every djinn has a mesa (which is also a djinn).
+
+(3) Genie is not the mesa of any djinn. (4) Different djinns have different metas.
+
+(5) If Genie has X, and each djinn relays X to its mesa, then all djinns get X.
+
+In light of the lamps of the Little Harmonic Labyrinth, we should name the set of all
+djinns "GOD". This harks back to a celebrated statement by the German mathematician
+and logician Leopold Kronecker, archenemy of Georg Cantor: "God made the natural
+numbers; all the rest is the work of man."
+
+
+Typographical Number Theory
+
+
+216
+
+
+
+You may recognize Peano's fifth postulate as the principle of mathematical
+induction-another term for a hereditary argument. Peano he that his five restrictions on
+the concepts "Genie", "djinn", and "mesa" so strong that if two different people formed
+images in their minds o concepts, the two images would have completely isomorphic
+structures, example, everybody's image would include an infinite number of distinct
+djinns. And presumably everybody would agree that no djinn coins with its own meta, or
+its meta's meta, etc.
+
+Peano hoped to have pinned down the essence of natural numbers in his five
+postulates. Mathematicians generally grant that he succeeded that does not lessen the
+importance of the question, "How is a true statement about natural numbers to be
+distinguished from a false one?" At answer this question, mathematicians turned to totally
+formal systems, as TNT. However, you will see the influence of Peano in TNT, because
+all of his postulates are incorporated in TNT in one way or another.
+
+New Rules of TNT: Specification and Generalization
+
+Now we come to the new rules of TNT. Many of these rules will allow reach in and
+change the internal structure of the atoms of TNT. In sense they deal with more
+"microscopic" properties of strings than the of the Propositional Calculus, which treat
+atoms as indivisible units, example, it would be nice if we could extract the string -S0=0
+from the first axiom. To do this we would need a rule which permits us to di universal
+quantifier, and at the same time to change the internal strut of the string which remains, if
+we wish. Here is such a rule:
+
+RULE OF SPECIFICATION: Suppose u is a variable which occurs inside string x. If the
+string Yu:x is a theorem, then so is x, and so an strings made from x by replacing u,
+wherever it occurs, by one the same term.
+
+0 Restriction : The term which replaces u must not contain any vat that is quantified
+in x.)
+
+The rule of specification allows the desired string to be extracted Axiom 1. It is a one-
+step derivation:
+
+
+¥a -Sa=0 axiom 1
+
+~S0=0 specification
+
+Notice that the rule of specification will allow some formulas which co: free variables
+(i.e., open formulas) to become theorems. For example following strings could also be
+derived from Axiom 1, by specification:
+
+
+Sa=0
+
+~S(c+SSO)=0
+
+There is another rule, the rule of generalization, which allows us to put
+
+
+Typographical Number Theory
+
+
+217
+
+
+
+back the universal quantifier on theorems which contain variables that became free as a
+result of usage of specification. Acting on the lower string, for example, generalization
+would give:
+
+
+¥c:~S(c+SS0)=0
+
+Generalization undoes the action of specification, and vice versa. Usually, generalization
+is applied after several intermediate steps have transformed the open formula in various
+ways. Here is the exact statement of the rule:
+
+RULE OF GENERALIZATION: Suppose x is a theorem in which u, a variable, occurs
+free. Then ¥u:x is a theorem.
+
+( Restriction'. No generalization is allowed in a fantasy on any variable which
+appeared free in the fantasy's premise.)
+
+The need for restrictions on these two rules will shortly be demonstrated explicitly.
+Incidentally, this generalization is the same generalization as was mentioned in Chapter
+II, in Euclid's proof about the infinitude of primes. Already we can see how the symbol-
+manipulating rules are starting to approximate the kind of reasoning which a
+mathematician uses.
+
+
+The Existential Quantifier
+
+These past two rules told how to take off universal quantifiers and put them back on; the
+next two rules tell how to handle existential quantifiers.
+
+RULE OF INTERCHANGE: Suppose u is a variable. Then the strings Vu:- and -3u: are
+interchangeable anywhere inside any theorem.
+
+For example, let us apply this rule to Axiom 1:
+
+
+¥a:-Sa=0 axiom 1
+~3a:Sa=0 interchange
+
+By the way, you might notice that both these strings are perfectly natural renditions, in
+TNT, of the sentence "Zero is not the successor of any natural number". Therefore it is
+good that they can be turned into each other with ease.
+
+The next rule is, if anything, even more intuitive. It corresponds to the very
+simple kind of inference we make when we go from "2 is prime" to "There exists a
+prime". The name of this rule is self-explanatory:
+
+RULE OF EXISTENCE: Suppose a term (which may contain variables as long as they
+are free) appears once, or multiply, in a theorem. Then any (or several, or all) of the
+appearances of the term may be replaced by a variable which otherwise does not occur in
+the theorem, and the corresponding existential quantifier must be placed in front.
+
+Let us apply the rule to —as usual—Axiom 1:
+
+
+Typographical Number Theory
+
+
+218
+
+
+
+¥a:-Sa=0 axiom 1
+3b:¥a:-Sa=b existence
+
+
+You might now try to shunt symbols, according to rules so far giver produce the theorem
+
+~¥b: 3a:Sa=b.
+
+
+Rules of Equality and Successorship
+
+We have given rules for manipulating quantifiers, but so far none for symbols '=' and 'S'.
+We rectify that situation now. In what follows, r, s, t all stand for arbitrary terms.
+
+RULES OF EQUALITY:
+
+SYMMETRY: If r = s is a theorem, then so is s = r.
+
+TRANSITIVITY: If r = s and s = t are theorems, then so is r = t.
+
+RULFS OF SUCCESSORSHIP:
+
+ADD S: If r = t is a theorem, then Sr = St is a theorem.
+
+DROP S: If Sr = St is a theorem, then r = t is a theorem.
+
+Now we are equipped with rules that can give us a fantastic variet theorems. For
+example, the following derivations yield theorems which pretty fundamental:
+
+
+(1)
+
+¥a:-¥b:(a+Sb)=S(a+b)
+
+axiom 3
+
+(2)
+
+¥b:(SO+Sb)=S(SO+b)
+
+specification (SO for a)
+
+(3)
+
+(S0+S0)=S(S0+0)
+
+specification (0 for b)
+
+(4)
+
+¥a:(a+0)=a
+
+axiom 2
+
+(5)
+
+(S0+0)=S0
+
+specification (SO for a)
+
+(6)
+
+S(S0+0)=SS0
+
+addS
+
+(7)
+
+(SO+SO)=SSO
+
+* * *
+
+transitivity (lines 3,6)
+
+* *
+
+(1)
+
+¥a:-¥b:(a-Sb)=((a-b)+a)
+
+axiom 5
+
+(2)
+
+¥b:(SO , Sb)=((SO»b)+SO)
+
+specification (SO for a)
+
+(3)
+
+(SO.SO)=((SO.O)+SO)
+
+specification (0 for b)
+
+(4)
+
+¥a:-¥b:(a+Sb)=S(a+b)
+
+axiom 3
+
+(5)
+
+¥b:((SO.O)+Sb)=S((50 0)+b)
+
+specification ((SO-O) for a)
+
+(6)
+
+((SO .0)+SO)=S((SO.0)+0)
+
+specification (0 for b)
+
+(7)
+
+¥a:(a+0)=a
+
+axiom 2
+
+(8)
+
+((SO.O)+0)=(SO.O)
+
+specification ((S0.0) for a)
+
+(9)
+
+¥a:(a.0)=0
+
+axiom 4
+
+(10)
+
+(S0-0)=0
+
+specification (SO for a)
+
+(ID
+
+((S0.0)+0)=0
+
+transitivity (lines 8,10)
+
+(12)
+
+S((SO.O)+0)=SO
+
+addS
+
+(13)
+
+((SO -0)+SO)=SO
+
+transitivity (lines 6,12)
+
+(14)
+
+(SO.SO)=SO
+
+transitivity (lines 3,13)
+
+
+Typographical Number Theory
+
+
+219
+
+
+
+Illegal Shortcuts
+
+
+Now here is an interesting question: "How can we make a derivation for the string 0=0?"
+It seems that the obvious route to go would be first to derive the string ¥a:a=a, and then
+to use specification. So, what about the following "derivation" of ¥a:a=a ... What is
+wrong with it? Can you fix it up?
+
+
+(1)
+
+¥a:(a+0)=a
+
+axiom 2
+
+(2)
+
+¥a:a=(a+0)
+
+symmetry
+
+(3)
+
+¥a:a=a
+
+transitivity (lines 2,1)
+
+
+I gave this mini-exercise to point out one simple fact: that one should not jump too fast in
+manipulating symbols (such as '=') which are familiar. One must follow the rules, and not
+one's knowledge of the passive meanings of the symbols. Of course, this latter type of
+knowledge is invaluable in guiding the route of a derivation.
+
+Why Specification and Generalization Are Restricted
+
+Now let us see why there are restrictions necessary on both specification and
+generalization. Here are two derivations. In each of them, one of the restrictions is
+violated. Look at the disastrous results they produce:
+
+
+(1)
+
+[
+
+push
+
+(2)
+
+a=0
+
+premise
+
+(3)
+
+¥a:a=0
+
+generalization ( Wrong ])
+
+(4)
+
+Sa=()
+
+specification
+
+(5)
+
+]
+
+pop
+
+(6)
+
+<a=0=>Sa=0>
+
+fantasy rule
+
+(7)
+
+¥a:<a=0=>Sa=0>
+
+generalization
+
+(8)
+
+<o=o^so=o>
+
+specification
+
+(9)
+
+0=0
+
+previous theorem
+
+(10)
+
+S0=0
+
+detachment (lines 9,8)
+
+
+This is the first disaster. The other one is via faulty specification.
+
+
+(1)
+
+¥a:a=a
+
+previous theorem
+
+(2)
+
+Sa=Sa
+
+specification
+
+(3)
+
+3b:b=Sa
+
+existence
+
+(4)
+
+¥a: 3b:b=Sa
+
+generalization
+
+(5)
+
+3b:b=Sb
+
+specification ( Wrongl )
+
+
+So now you can see why those restrictions are needed.
+
+Here is a simple puzzle: translate (if you have not already done so) Peano's fourth
+postulate into TNT-notation, and then derive that string as a theorem.
+
+
+Typographical Number Theory
+
+
+220
+
+
+
+Something Is Missing
+
+
+Now if you experiment around for a while with the rules and axioms of TNT so far
+presented, you will find that you can produce the following pyramidal family of theorems
+(a set of strings all cast from an identical mold, differing from one another only in that the
+numerals 0, SO, SSO, and s have been stuffed in):
+
+( 0 + 0)=0
+
+(O+SO)=S0
+
+(0+SS0)=SS0
+
+(0+SSS0)=SSS0
+
+(0+SSSS0)=SSSS0
+
+etc.
+
+As a matter of fact, each of the theorems in this family can be derived the one directly
+above it, in only a couple of lines. Thus it is a so "cascade" of theorems, each one
+triggering the next. (These theorem very reminiscent of the pq-theorems, where the
+middle and right-] groups of hyphens grew simultaneously.)
+
+Now there is one string which we can easily write down, and v summarizes the
+passive meaning of them all, taken together. That un sally quantified summarizing string
+is this:
+
+
+¥a:(0+a)=a
+
+Yet with the rules so far given, this string eludes production. Ti produce it yourself if you
+don't believe me.
+
+You may think that we should immediately remedy the situation the following
+
+(PROPOSED) RULE OF ALL: If all the strings in a pyramidal family are theorems, then
+so is the universally quantified string which summarizes them.
+
+The problem with this rule is that it cannot be used in the M-mode. people who are
+thinking about the system can ever know that an infinite set of strings are all theorems.
+Thus this is not a rule that can be stuck i any formal system.
+
+co-Incomplete Systems and Undecidable Strings
+
+So we find ourselves in a strange situation, in which we can typographically produce
+theorems about the addition of any specific numbers, but even a simple string as the one
+above, which expresses a property of addition in general , is not a theorem. You might
+think that is not all that strange, we were in precisely that situation with the pq-system.
+However, the pq-system had no pretensions about what it ought to be able to do; and ii
+fact
+
+
+Typographical Number Theory
+
+
+221
+
+
+
+there was no way to express general statements about addition in its symbolism, let alone
+prove them. The equipment simply was not there, and it did not even occur to us to think
+that the system was defective. Here, however, the expressive capability is far stronger,
+and we have correspondingly higher expectations of TNT than of the pq-system. If the
+string above is not a theorem, then we will have good reason to consider TNT to be
+defective. As a matter of fact, there is a name for systems with this kind of defect-they
+are called co -incomplete. (The prefix 'oo'-'omega'- comes from the fact that the totality of
+natural numbers is sometimes denoted by 'oo'.) Here is the exact definition:
+
+A system is oo-incomplete if all the strings in a pyramidal family are theorems, but
+the universally quantified summarizing string is not a theorem.
+
+Incidentally, the negation of the above summarizing string
+
+~¥a:(0+a)=a
+
+-is also a nontheorem of TNT. This means that the original string is undecidable within
+the system. If one or the other were a theorem, then we would say that it was decidable.
+Although it may sound like a mystical term, there is nothing mystical about
+undecidability within a given system. It is only a sign that the system could be extended.
+For example, within absolute geometry, Euclid's fifth postulate is undecidable. It has to
+be added as an extra postulate of geometry, to yield Euclidean geometry; or conversely,
+its negation can be added, to yield non-Euclidean geometry. If you think back to
+geometry, you will remember why this curious thing happens. It is because the four
+postulates of absolute geometry simply do not pin down the meanings of the terms
+"point" and "line", and there is room for different extensions of the notions. The points
+and lines of Euclidean geometry provide one kind of extension of the notions of "point"
+and "line"; the POINTS and LINES of non-Euclidean geometry, another. However, using
+the pre-flavored words "point" and "line" tended, for two millennia, to make people
+believe that those words were necessarily univalent, capable of only one meaning.
+
+Non-Euclidean TNT
+
+We are now faced with a similar situation, involving TNT. We have adopted a notation
+which prejudices us in certain ways. For instance, usage of the symbol '+'tends to make
+us think that every theorem with a plus sign in it ought to say something known and
+familiar and "sensible" about the known and familiar operation we call "addition".
+Therefore it would run against the grain to propose adding the following "sixth axiom":
+
+~¥a:(0+a)=a
+
+
+Typographical Number Theory
+
+
+222
+
+
+
+It doesn't jibe with what we believe about addition. But it is one possible extension of
+TNT, as we have so far formulated TNT. The system which uses this as its sixth axiom is
+a consistent system, in the sense of not has, two theorems of the form x and - x. However,
+when you juxtapose this "sixth axiom" with the pyramidal family of theorems shown
+above, you will probably be bothered by a seeming inconsistency between the family and
+the new axiom. But this kind of inconsistency is riot so damaging as the other kind
+(where x and x are both theorems). In fact, it is not a true inconsistency, because there is
+a way of interpreting the symbols so that everything comes out all right.
+
+©-Inconsistency Is Not the Same as Inconsistency
+
+This kind of inconsistency, created by the opposition of (1) a pyramidal family of
+theorems which collectively assert that all natural numbers have some property, and (2) a
+single theorem which seems to assert that not all numbers have it, is given the name of w-
+inconsistency. An w-inconsistent system is more like the at-the-outset-distasteful-but-in-
+the-end-accept non-Euclidean geometry. In order to form a mental model of what is
+going on, you have to imagine that there are some "extra", unsuspected numbers—let us
+not call them "natural", but supernatural numbers-which have no numerals. Therefore,
+facts about them cannot be represented in the pyramidal family. (This is a little bit like
+Achilles' conception GOD-as a sort of "superdjinn", a being greater than any of the djinn
+This was scoffed at by the Genie, but it is a reasonable image, and may I you to imagine
+supernatural numbers.)
+
+What this tells us is that the axioms and rules of TNT, as so presented, do not
+fully pin down the interpretations for the symbol TNT. There is still room for variation in
+one's mental model of the notions they stand for. Each of the various possible extensions
+would pin d, some of the notions further; but in different ways. Which symbols we begin
+to take on "distasteful" passive meanings, if we added the "s axiom" given above? Would
+all of the symbols become tainted, or we some of them still mean what we want them to
+mean? I will let you tt about that. We will encounter a similar question in Chapter XIV,
+discuss the matter then. In any case, we will not follow this extension r but instead go on
+to try to repair the w-incompleteness of TNT.
+
+The Last Rule
+
+The problem with the "Rule of All" was that it required knowing that all lines of an
+infinite pyramidal family are theorems — too much for a finite being. But suppose that
+each line of the pyramid can be derived from its predecessor in a patterned way. Then
+there would be a. finite reason accounting for the fact that all the strings in the pyramid
+are theorems. The trick then, is to find the pattern that causes the cascade, and show that
+
+
+Typographical Number Theory
+
+
+223
+
+
+
+pattern is a theorem in itself. That is like proving that each djinn passes a message to its
+meta, as in the children's game of "Telephone". The other thing left to show is that Genie
+starts the cascading message-that is, to establish that the first line of the pyramid is a
+theorem. Then you know that GOD will get the message!
+
+In the particular pyramid we were looking at, there is a pattern, captured by lines
+4-9 of the derivation below.
+
+
+(1)
+
+¥a:¥b:(a+Sb)=S(a+b)
+
+axiom 3
+
+(2)
+
+¥b:(0+Sb)=S(0+b)
+
+specification
+
+(3)
+
+(0+Sb)=S(0+b)
+
+specification
+
+(4)
+
+[
+
+push
+
+(5)
+
+(0+b)=b
+
+premise
+
+(6)
+
+S(0+b)=Sb
+
+addS
+
+(7)
+
+(0+Sb)=S(0+b)
+
+carry over line 3
+
+(8)
+
+(0+Sb)=Sb
+
+transitivity
+
+(9)
+
+]
+
+pop
+
+
+The premise is (0+b)=b; the outcome is (0+Sb)=Sb.
+
+The first line of the pyramid is also a theorem; it follows directly from Axiom 2.
+All we need now is a rule which lets us deduce that the string which summarizes the
+entire pyramid is itself a theorem. Such a rule will he a formalized statement of the fifth
+Peano postulate.
+
+To express that rule, we need a little notation. Let us abbreviate a well-formed
+formula in which the variable a is free by the following notation:
+
+X{a}
+
+(There may be other free variables, too, but that is irrelevant.) Then the notation X{Sa/a}
+will stand for that string but with every occurrence of a replaced by Sa. Likewise, X{0/a}
+would stand for the same string, with each appearance of a replaced by 0.
+
+A specific example would be to let X{a} stand for the string in question: (0+a)=a.
+Then X{Sa/a} would represent the string (0+Sa)=Sa, and X{0/a} would represent
+(0+0)=0. (Warning: This notation is not part of TNT; it is for our convenience in talking
+about TNT.)
+
+With this new notation, we can state the last rule of TNT quite precisely:
+
+RULE OF INDUCTION: Suppose u is a variable, and X{u} is a well-formed formula in
+which u occurs free. If both ¥u:< X{u}=3 X{Su/u}> and X{0/u} are theorems,
+then ¥u: X{u} is also a theorem.
+
+This is about as close as we can come to putting Peano's fifth postulate into TNT. Now
+let us use it to show that Va:(0+a)=a is indeed a theorem in TNT. Emerging from the
+fantasy in our derivation above, we can apply the fantasy rule, to give us
+
+(10) <(0+b)=b=>(0+Sb)=Sb> fantasy rule
+
+(11) ¥b:<(0+b)=bz>(0+Sb)=Sb> generalization
+
+
+Typographical Number Theory
+
+
+224
+
+
+
+This is the first of the two input theorems required by the induction The other
+requirement is the first line of the pyramid, which we have. Therefore, we can apply the
+rule of induction, to deduce what we wanted.
+
+
+¥b:(0+b)=b
+
+Specification and generalization will allow us to change the variable from b to a; thus
+¥a:(0+a)=a is no longer an undecidable string of TNT..
+
+A Long Derivation
+
+Now I wish to present one longer derivation in TNT, so that you ca what one is like, and
+also because it proves a significant, if simple, fact of number theory.
+
+
+(1) ¥a:-¥b:(a+Sb)=S(a+b)
+
+axiom 3
+
+(2) ¥b:(d+Sb)=S(d+b)
+
+specification
+
+(3) (d+SSc)=S(d+Sc)
+
+specificatic
+
+(4) b:(Sd+Sb)=S(Sd+b)
+
+specification (line 1)
+
+(5) (Sd+Sc)-S(Sd+c)
+
+specification
+
+6) S(Sd+c)=(Sd+Sc)
+
+symmetry
+
+(7)[
+
+push
+
+(8) ¥d:(d+Sc)=(Sd+c)
+
+premise
+
+(9) (d+Sc)=(Sd+c)
+
+specification
+
+(10) S(d+Sc)=S(Sd+c)
+
+addS
+
+(11) (d+SSc)=S(d+Sc)
+
+carry over 3
+
+(12) (d+SSc)=S(Sd+c)
+
+transitivity
+
+(13) S(Sd+c)=(Sd+Sc)
+
+carry over 6
+
+(14) (d+SSc)=(Sd+Sc)
+
+transitivity
+
+(15) ¥d:(d+SSc)=(Sd+Sc)
+
+generalization
+
+(16)]
+
+pop
+
+(17) <¥d:(d+5c)=(Sd+c)z>¥d:(d+SSc)=(Sd+Sc)>
+
+fantasy rule
+
+(18) ¥c:<-¥d:(d+Sc)=(Sd+c) 3¥d:(d+SSc)=(Sd+Sc)>
+
+generalization
+
+sfc ^ ^ ^ 5J:
+
+
+(19) (d+S0)=5(d+0)
+
+specification (line 2)
+
+(20) ¥a:(a+0)=a
+
+axiom 1
+
+(21) (d+0)=d
+
+specification
+
+(22) S(d+0)=Sd
+
+addS
+
+(23) (d+SO)=Sd
+
+transitivity (lines 19,2)
+
+(24) (Sd+0)=Sd
+
+specification (line 20)
+
+(25) Sd=(Sd+0)
+
+symmetry
+
+
+Typographical Number Theory
+
+
+225
+
+
+
+(26) (d+SO)=(Sd+o)
+
+(27) ¥d:(d+50)=(Sd+0)
+
+
+transitivity (lines 23,25)
+generalization
+
+
+^ ^
+
+
+(28) ¥c:-¥d:(d+Sc)=(Sd+c)
+
+induction (lines 18,27)
+
+[S can be slipped back and forth
+
+^ 5ji jf: sj: 5J:
+
+in an addition]
+
+(29) ¥b:(c+Sb)=S(c+b)
+
+specification (line 1)
+
+(30) (c+Sd)=S(c+d)
+
+specification
+
+(31) ¥b:(d+Sb)=S(d+b)
+
+specification (line 1)
+
+(32) (d+Sc)=S(d+c)
+
+specification
+
+(33) S(d+c)=(d+Sc)
+
+symmetry
+
+(34) bed:(d+Sc)=(Sd+c)
+
+specification (line 28)
+
+(35) (d+Sc)=(Sd+c)
+
+specification
+
+(36) [
+
+push
+
+(37) ¥c:(c+d)=(d+c)
+
+premise
+
+(38) (c+d)=(d+c)
+
+specification
+
+(39) S(c+d)=S(d+c)
+
+addS
+
+(40) (c+Sd)=S(c+d)
+
+carry over 30
+
+(41) (c+Sd)=S(d+c)
+
+transitivity
+
+(42) S(d+c)=(d+Sc)
+
+carry over 33
+
+(43) (c+Sd)=(d+Sc)
+
+transitivity
+
+(44) (d+Sc)=(Sd+c)
+
+carry over 35
+
+(45) (c+Sd)=(Sd+c)
+
+transitivity
+
+(46) ¥c:(c+Sd)=(Sd+c)
+
+generalization
+
+(47) ]
+
+pop
+
+(48) <¥e:(c+d)=(d+c) 3¥c:(c+Sd)=(Sd+c)>
+
+fantasy rule
+
+(49) ¥d:<-¥c:(c+d)=(d+c) 3¥c:(c+Sd)=(Sd+c)>
+
+generalization
+
+
+[If d commutes with every c, then Sd does too.
+
+sfc ;Ji jJ: ^ ;Ji
+
+
+(50) (c+0)=c
+
+(51) ¥a:(0+a)=a
+
+(52) (0+c)=c
+
+(53) c=(0+c)
+
+
+specification (line 20)
+previous theorem
+specification
+symmetry
+
+
+Typographical Number Theory
+
+
+226
+
+
+
+(54) (c+0)=(0+c)
+
+(55) ¥c:(c+0)=(O+c)
+
+
+transitivity (lines 50,53)
+generalization
+
+
+[0 commutes with every c.]
+
+
+(56) ¥d:-¥c:(c+d)=(d+c) induction (lines 49,55)
+
+[Therefore, every d commutes with every c.]
+
+Tension and Resolution in TNT
+
+TNT has proven the commutativity of addition. Even if you do not follow this derivation
+in detail, it is important to realize that, like a piece of music, it has its own natural
+"rhythm". It is not just a random walk that happens to have landed on the desired last
+line. I have inserted "breathing marks” to show some of the "phrasing" of this derivation.
+Line 28 in particular turning point in the derivation, something like the halfway point it
+AABB type of piece, where you resolve momentarily, even if not in the t key. Such
+important intermediate stages are often called "lemmas".
+
+It is easy to imagine a reader starting at line 1 of this derivation ignorant of where
+it is to end up, and getting a sense of where it is going as he sees each new line. This
+would set up an inner tension, very much the tension in a piece of music caused by chord
+progressions that let know what the tonality is, without resolving. Arrival at line 28 w,
+confirm the reader's intuition and give him a momentary feeling of satisfaction while at
+the same time strengthening his drive to progress tow what he presumes is the true goal.
+
+Now line 49 is a critically important tension-increaser, because of "almost-there"
+feeling which it induces. It would be extremely unsatisfactory to leave off there! From
+there on, it is almost predictable how things must go. But you wouldn't want a piece of
+music to quit on you just when had made the mode of resolution apparent. You don't
+want to imagine ending-you want to hear the ending. Likewise here, we have to c things
+through. Line 55 is inevitable, and sets up all the final tension which are resolved by Line
+56.
+
+This is typical of the structure not only of formal derivations, but of informal
+proofs. The mathematician's sense of tension is intimately related to his sense of beauty,
+and is what makes mathematics worthy doing. Notice, however, that in TNT itself, there
+seems to be no reflection of these tensions. In other words, TNT doesn't formalize the
+notions of tension and resolution, goal and subgoal, "naturalness" and "inevitable any
+more than a piece of music is a book about harmony and rhythm. Could one devise a
+much fancier typographical system which is aware of the tensions and goals inside
+derivations?
+
+
+Typographical Number Theory
+
+
+227
+
+
+
+Formal Reasoning vs. Informal Reasoning
+
+
+I would have preferred to show how to derive Euclid's Theorem (the infinitude of
+primes) in TNT, but it would probably have doubled the length of the book. Now after
+this theorem, the natural direction to go would be to prove the associativity of addition,
+the commutativity and associativity of multiplication and the distributivity of
+multiplication over addition. These would give a powerful base to work from.
+
+As it is now formulated, TNT has reached "critical mass" (perhaps a strange
+metaphor to apply to something called "TNT"). It is of the same strength as the system of
+Principia Mathematica; in TNT one can now prove every theorem which you would find
+in a standard treatise on number theory. Of course, no one would claim that deriving
+theorems in TNT is the best way to do number theory. Anybody who felt that way would
+fall in the same class of people as those who think that the best way to know what 1000 x
+1000 is, is to draw a 1000 by 1000 grid, and count all the squares in it ... No; after total
+formalization, the only way to go is towards relaxation of the formal system. Otherwise,
+it is so enormously unwieldy as to be, for all practical purposes, useless. Thus, it is
+important to embed TNT within a wider context, a context which enables new rules of
+inference to be derived, so that derivations can be speeded up. This would require
+formalization of the language in which rules of inference are expressed-that is, the
+metalanguage. And one could go considerably further. However, none of these speeding-
+up tricks would make TNT any more powerful ; they would simply make it more usable.
+The simple fact is that we have put into TNT every mode of thought that number
+theorists rely on. Embedding it in ever larger contexts will not enlarge the space of
+theorems; it will just make working in TNT-or in each "new, improved version"-look
+more like doing conventional number theory.
+
+Number Theorists Go out of Business
+
+Suppose that you didn't have advance knowledge that TNT will turn out to be
+incomplete, but rather, expected that it is complete-that is, that every true statement
+expressible in the TNT-notation is a theorem. In that case, you could make a decision
+procedure for all of number theory. The method would be easy: if you want to know if N-
+statement X is true or false, code it into TNT-sentence x. Now if X is true, completeness
+says that x is a theorem; and conversely, if not-X is true, then completeness says that ~x
+is a theorem. So either x or ~x must be a theorem, since either X or not-X is true. Now
+begin systematically enumerating all the theorems of TNT, in the way we did for the
+MlU-system and pq-system. You must come to x or ~x after a while; and whichever one
+you hit tells you which of X and not-X is true. (Did you follow this argument? It crucially
+depends on your being able to hold separate in your mind the formal system TNT and its
+informal counterpart N. Make sure you understand it.) Thus, in prince -
+
+
+Typographical Number Theory
+
+
+228
+
+
+
+pie, if TNT were complete, number theorists would be put out of business any question
+in their field could be resolved, with sufficient time, in a purely mechanical way. As it
+turns out, this is impossible, which, depending on your point of view, is a cause either for
+rejoicing, or for mourning.
+
+
+Hilbert's Program
+
+The final question which we will take up in this Chapter is whether should have
+as much faith in the consistency of TNT as we did consistency of the Propositional
+Calculus; and, if we don't, whether possible to increase our faith in TNT, by proving it to
+be consistent could make the same opening statement on the "obviousness" of TNT’s
+consistency as Imprudence did in regard to the Propositional Calculus namely, that each
+rule embodies a reasoning principle which we believe in, and therefore to question the
+consistency of TNT is to question our own sanity. To some extent, this argument still
+carries weight-but not quite so much weight as before. There are just too many rules of
+inference and some of them just might be slightly "off ". Furthermore, how do we know
+that this mental model we have of some abstract entities called "natural numbers" is
+actually a coherent construct? Perhaps our own thought processes, those informal
+processes which we have tried to capture in the formal rules of the system, are themselves
+inconsistent! It is of course not the kind of thing we expect, but it gets more and more
+conceivable that our thoughts might lead us astray, the more complex the subject matter
+gets-and natural numbers are by no means a trivial subject matter. Prudence's cry for a
+proof of consistency has to be taken more seriously in this case. It's not that we seriously
+doubt that TNT could be inconsistent but there is a little doubt, a flicker, a glimmer of a
+doubt in our minds, and a proof would help to dispel that doubt.
+
+But what means of proof would we like to see used? Once again, faced with the
+recurrent question of circularity. If we use all the equipment in a proof about our system
+as we have inserted into it, what will we have accomplished? If we could manage to
+convince ourselves consistency of TNT, but by using a weaker system of reasoning than
+we will have beaten the circularity objection! Think of the way a heavy rope is passed
+between ships (or so I read when I was a kid): first a light arrow is fired across the gap,
+pulling behind it a thin rope. Once a connection has been established between the two
+ships this way, then the heavy rope pulled across the gap. If we can use a "light" system
+to show that a system is consistent, then we shall have really accomplished something.
+
+Now on first sight one might think there is a thin rope. Our goal is to prove that
+TNT has a certain typographical property (consistency): that no theorems of the form x
+and ,~x ever occur. This is similar to trying to show that MU is not a theorem of the
+MlU-system. Both are statements about typographical properties of symbol-
+manipulation systems. The visions of a thin rope are based on the presumption that facts
+about number theory won't be
+
+
+Typographical Number Theory
+
+
+229
+
+
+
+needed in proving that such a typographical property holds. In other words, if properties
+of integers are not used-or if only a few extremely simple ones are used-then we could
+achieve the goal of proving TNT consistent, by using means which are weaker than its
+own internal modes of reasoning.
+
+This is the hope which was held by an important school of mathematicians and
+logicians in the early part of this century, led by David Hilbert. The goal was to prove the
+consistency of formalizations of number theory similar to TNT by employing a very
+restricted set of principles of reasoning called "finitistic" methods of reasoning. These
+would be the thin rope. Included among finitistic methods are all of propositional
+reasoning, as embodied in the Propositional Calculus, and additionally some kinds of
+numerical reasoning. But Godel’s work showed that any effort to pull the heavy rope of
+TNT's consistency across the gap by using the thin rope of finitistic methods is doomed
+to failure. Godel showed that in order to pull the heavy rope across the gap, you can't use
+a lighter rope; there just isn't a strong enough one. Less metaphorically, we can say: Any
+system that is strong enough to prove TNT's consistency is at least as strong as TNT
+itself. And so circularity is inevitable.
+
+
+Typographical Number Theory
+
+
+230
+
+
+
+A Mu Offering
+
+The Tortoise and Achilles have just been to hear a lecture on the origins of the
+Genetic Code, and are now drinking some tea at Achilles' home.
+
+Achilles: I have something terrible to confess, Mr. T.
+
+Tortoise: What is it, Achilles?
+
+Achilles: Despite the fascinating subject matter of that lecture, I drifter to sleep a time or
+two. But in my drowsy state, I still was semi-awake aware of the words coming into
+my ears. One strange image that floated up from my lower levels was that 'A' and 'T',
+instead of standing "adenine" and "thymine", stood for my name and yours-and
+double-strands of DNA had tiny copies of me and you along backbones, always
+paired up, just as adenine and thymine always Isn't that a strange symbolic image?
+
+Tortoise: Phooey! Who believes in that silly kind of stuff? Anyway, about 'C' and 'G'?
+
+Achilles: Well, I suppose 'C' could stand for Mr. Crab, instead o cytosine. I'm not sure
+about 'G', but I'm sure one could thin something. Anyway, it was amusing to imagine
+my DNA being with minuscule copies of you-as well as tiny copies of myself, for
+matter. Just think of the infinite regress THAT leads to!
+
+Tortoise: I can see you were not paying too much attention to the lecture.
+
+Achilles: No, you're wrong. I was doing my best, only I had a hard keeping fancy
+separated from fact After all, it is such a strange netherworld that those molecular
+biologists are exploring.
+
+Tortoise: How do you mean?
+
+Achilles: Molecular biology is filled with peculiar convoluted loops which I can't quite
+understand, such as the way that folded proteins, which are coded for in DNA, can
+loop back and manipulate the DNA which came from, possibly even destroying it.
+Such strange loops always confuse the daylights out of me. They're eerie, in a way.
+
+Tortoise: I find them quite appealing.
+
+Achilles: You would, of course-they're just down your alley. But me, sometimes I like to
+retreat from all this analytic thought any meditate a little, as an antidote. It clears my
+mind of all those conf loops and incredible complexities which we were hearing about
+tonight.
+
+Tortoise: Fancy that. I wouldn't have guessed that you were a meditator
+
+Achilles: Did I never tell you that I am studying Zen Buddhism?
+
+Tortoise: Heavens, how did you come upon that?
+
+Achilles: I have always had a yen for the yin and yang, you know - the
+
+
+A Mu Offering
+
+
+231
+
+
+
+whole Oriental mysticism trip, with the I Ching, gurus, and whatnot. So one day I'm
+thinking to myself, "Why not Zen too?" And that's how it all began.
+
+Tortoise: Oh, splendid. Then perhaps I can finally become enlightened. Achilles: Whoa,
+now. Enlightenment is not the first step on the road to Zen; if anything, it', the last
+one! Enlightenment is not for novices like you, Mr. T!
+
+Tortoise: I see we have had a misunderstanding. By "enlightenment", I hardly meant
+something so weighty as is meant in Zen. All I meant is that I can perhaps become
+enlightened as to what Zen is all about. Achilles: For Pete's sake, why didn't you say
+so? Well, I'd be only too happy to tell you what I know of Zen. Perhaps you might
+even be tempted to become a student of it, like me.
+
+Tortoise: Well, nothing's impossible.
+
+Achilles: You could study with me under my master, Okanisama-the seventh patriarch.
+
+Tortoise: Now what in the world does that mean?
+
+Achilles: You have to know the history of Zen to understand that.
+
+Tortoise: Would you tell me a little of the history of Zen, then?
+
+Achilles: An excellent idea. Zen is a kind of Buddhism which was founded by a monk
+named Bodhidharma, who left India and went to China around the sixth century.
+Bodhidharma was the first patriarch. The sixth one was Eno. (I've finally got it
+straight now!)
+
+Tortoise: The sixth patriarch was Zeno, eh? I find it strange that he, of all people, would
+get mixed up in this business.
+
+Achilles: I daresay you underestimate the value of Zen. Listen just a little more, and
+maybe you'll come to appreciate it. As I was saying, about five hundred years later,
+Zen was brought to Japan, and it took hold very well there. Since that time it has been
+one of the principal religions in Japan.
+
+Tortoise: Who is this Okanisama, the "seventh patriarch"?
+
+Achilles: He is my master, and his teachings descend directly from those of the sixth
+patriarch. He has taught me that reality is one, immutable, and unchanging; all
+plurality, change, and motion are mere illusions of the senses.
+
+Tortoise: Sure enough, that's Zeno, a mile away. But how ever did he come to be tangled
+up in Zen? Poor fellow!
+
+Achilles: Whaaat? I wouldn't put it that way. If ANYONE is tangled up, it's ... But that's
+another matter. Anyway, I don't know the answer to your question. Instead, let me tell
+you something of the teachings of my master. I have learned that in Zen, one seeks
+enlightenment, or SATORI-the state of "No-mind". In this state, one does not think
+about the world-one just is. I have also learned that a student of Zen is not supposed
+to "attach" to any object or thought or person-which is to say, he must not believe in,
+or depend on, any absolute-not even this philosophy of nonattachment.
+
+
+A Mu Offering
+
+
+232
+
+
+
+Tortoise: Hmm ... Now THERE'S something I could like about Achilles: I had a hunch
+you'd get attached to it.
+
+Tortoise: But tell me: if Zen rejects intellectual activity, does it make sense to
+intellectualize about Zen, to study it rigorously?
+
+Achilles: That matter has troubled me quite a bit. But I think I have finally worked out an
+answer. It seems to me that you may begin approaching Zen through any path you
+know-even if it is completely antithetical to Zen. As you approach it, you gradually
+learn to stray from that path. The more you stray from the path, the closer you get to
+Zen.
+
+Tortoise: Oh, it all begins to sound so clear now.
+
+Achilles: My favorite path to Zen is through the short, fascinating and weird Zen parables
+called "koans".
+
+Tortoise: What is a koan?
+
+Achilles: A koan is a story about Zen masters and their student times it is like a riddle;
+other times like a fable; and other ti nothing you've ever heard before.
+
+Tortoise: Sounds rather intriguing. Would you say that to read al koans is to practice
+Zen?
+
+Achilles: I doubt it. However, in my opinion, a delight in koans million times closer to
+real Zen than reading volume after about Zen, written in heavy philosophical jargon.
+
+Tortoise: I would like to hear a koan or two.
+
+Achilles: And I would like to tell you one-or a few. Perhaps begin with the most famous
+one of all. Many centuries ago, the Zen master named Joshu, who lived to be 119
+years old.
+
+Tortoise: A mere youngster!
+
+Achilles: By your standards, yes. Now one day while Joshu and monk were standing
+together in the monastery, a dog wand The monk asked Joshu, "Does a dog have
+Buddha-nature,
+
+Tortoise: Whatever that is. So tell me-what did Joshu reply?
+
+Achilles: 'MU'.
+
+Tortoise: 'MU? What is this 'MU'? What about the dog? What about Buddha-nature?
+What's the answer?
+
+Achilles: Oh, but 'MU' is Joshu's answer. By saying 'MU', Joshu let the other monk know
+that only by not asking such questions can one know the answer to them.
+
+Tortoise: Joshu "unasked" the question.
+
+Achilles: Exactly!
+
+Tortoise: 'MU' sounds like a handy thing to have around. I'd like unask a question or two,
+sometimes. I guess I'm beginning to get the hang of Zen. Do you know any other
+koans, Achilles? I would like to hear some more.
+
+Achilles: My pleasure. I can tell you a pair of koans which go together
+Only ...
+
+Tortoise: What's the matter?
+
+
+A Mu Offering
+
+
+233
+
+
+
+Achilles: Well, there is one problem. Although both are widely told koans, my master has
+cautioned me that only one of them is genuine. And what is more, he does not know
+which one is genuine, and which one is a fraud.
+
+Tortoise: Crazy! Why don't you tell them both to me and we can speculate to our hearts'
+content!
+
+Achilles: All right. One of the alleged koans goes like this:
+
+A monk asked Baso: "What is Buddha?"
+
+Baso said: "This mind is Buddha."
+
+Tortoise: Hmm ... "This mind is Buddha"? Sometimes I don't quite understand what these
+Zen people are getting at. Achilles: You might prefer the other alleged koan then.
+
+Tortoise: How does it run? Achilles: Like this:
+
+A monk asked Baso: "What is Buddha?"
+
+Baso said: "This mind is not Buddha."
+
+Tortoise: My, my! If my shell isn't green and not green! I like that! Achilles: Now, Mr. T-
+you're not supposed to just "like" koans.
+
+Tortoise: Very well, then-I don't like it.
+
+Achilles: That's better. Now as I was saying, my master believes only one of the two is
+genuine.
+
+Tortoise: I can't imagine what led him to such a belief. But anyway, I suppose it's all
+academic, since there's no way to know if a koan is genuine or phony.
+
+Achilles: Oh, but there you are mistaken. My master has shown us how to do it.
+
+Tortoise: Is that so? A decision procedure for genuineness of koans? I should very much
+like to hear about THAT.
+
+Achilles: It is a fairly complex ritual, involving two stages. In the first stage, you must
+TRANSLATE the koan in question into a piece of string, folded all around in three
+dimensions.
+
+Tortoise: That's a curious thing to do. And what is the second stage?
+
+Achilles: Oh, that's easy-all you need to do is determine whether the string has Buddha-
+nature, or not! If it does, then the koan is genuine-if not, the koan is a fraud.
+
+Tortoise: Hmm ... It sounds as if all you've done is transfer the need for a decision
+procedure to another domain. Now it's a decision procedure for Buddha-nature that
+you need. What next? After all, if you can't even tell whether a Do(; has Buddha-
+nature or not, how can you expect to do so for every possible folded string?
+
+Achilles: Well, my master explained to me that shifting between domains can help. It's
+like switching your point of view. Things sometimes look complicated from one
+angle, but simple from another. He gave the example of an orchard, in which from
+one direction no order is
+
+
+A Mu Offering
+
+
+234
+
+
+
+
+FIGURE 45. La Mezquita, by M. C. Escher (black and white chalk, 1936
+
+apparent, but from special angles, beautiful regularity em, You’ve reordered the same
+information by changing your way of looking at it.
+
+Tortoise: I see. So perhaps the genuineness of a koan is concealed how very deeply inside
+it, but if you translate it into a string it ma in some way to float to the surface?
+Achilles: That's what my master has discovered.
+
+Tortoise: Then I would very much like to learn about the techniqu first, tell me: how can
+you turn a koan (a sequence of words) folded string (a three-dimensional object)?
+They are rather dif kinds of entities.
+
+Achilles: That is one of the most mysterious things I have learned i There are two steps:
+"transcription" and "translation". TRANSCF a koan involves writing it in a phonetic
+alphabet, which contain four geometric symbols. This phonetic rendition of the koan
+is called the MESSENGER.
+
+Tortoise: What do the geometric symbols look like?
+
+Achilles: They are made of hexagons and pentagons. Here is what they
+
+
+A Mu Offering
+
+
+235
+
+
+
+
+look like {picks up a nearby napkin, and draws for the Tortoise these four figures ):
+
+
+O O OO CO
+
+Tortoise: They are mysterious-looking.
+
+Achilles: Only to the uninitiated. Now once you have made the messenger, you rub your
+hands in some ribo, and
+
+Toitoise: Some ribo? Is that a kind of ritual anointment?
+
+Achilles: Not exactly. It is a special sticky preparation which makes the string hold its
+shape, when folded up. Tortoise: What is it made of?
+
+Achilles: I don't know, exactly. But it feels sort of gluey, and it works exceedingly well.
+Anyway, once you have some ribo on your hands, you can TRANSLATE the
+sequence of symbols in the messenger into certain kinds of folds in the string. It's as
+simple as that. Tortoise: Hold on! Not so fast! How do you do that?
+
+Achilles: You begin with the string entirely straight. Then you go to one end and start
+making folds of various types, according to the geometric symbols in the messenger.
+
+Tortoise: So each of those geometric symbols stands for a different way to curl the string
+up?
+
+Achilles: Not in isolation. You take them three at a time, instead of one at a time. You
+begin at one end of the string, and one end of the messenger. What to do with the first
+inch of the string is determined by the first three geometric symbols. The next three
+symbols tell you how to fold the second inch of string. And so you inch your way
+along the string and simultaneously along the messenger, folding each little segment
+of string until you have exhausted the messenger. If you have properly applied some
+ribo, the string will keep its folded shape, and what you thereby produce is the
+translation of the koan into a string.
+
+Toitoise: The procedure has a certain elegance to it. You must get some wild-looking
+strings that way.
+
+Achilles: That's for sure. The longer koans translate into quite bizarre shapes.
+
+Tortoise: I can imagine. But in order to carry out the translation of the messenger into the
+string, you need to know what kind of fold each triplet of geometric symbols in the
+messenger stands for. How do you know this? Do you have a dictionai 7 ?
+
+Achilles: Yes-there is a venerated book which lists the "Geometric Code”. If you don't
+have a copy of this book, of course, you can't translate a koan into a string.
+
+
+A Mu Offering
+
+
+236
+
+
+Tortoise: Evidently not. What is the origin of the Geometric Code Achilles: It came from
+an ancient master known as "Great Tutor" who my master says is the only one ever to
+attain the Enlightenment' Enlightenment.
+
+Tortoise: Good gravy! As if one level of the stuff weren't enough. But then there are
+gluttons of every sort-why not gluttons for enlighten] Achilles: Do you suppose that
+"Enlightenment 'Yond Enlighten] stands for "EYE"?
+
+Tortoise: In my opinion, it's rather doubtful that it stands for you, Ac More likely, it
+stands for "Meta-Enlightenment"-"ME", that is
+
+Achilles: For you? Why would it stand for you? You haven't even re; the FIRST stage of
+enlightenment, let alone the
+
+Tortoise: You never know, Achilles. Perhaps those who have learn( lowdown on
+enlightenment return to their state before enlighten I've always held that "twice
+enlightened is unenlightened." But le back to the Grand Tortue-uh, I mean the Great
+Tutor.
+
+Achilles: Little is known of him, except that he also invented the Art of Zen Strings.
+
+Tortoise: What is that?
+
+Achilles: It is an art on which the decision procedure for Buddha-nature is based. I shall
+tell you about it.
+
+Tortoise: I would be fascinated. There is so much for novices like absorb!
+
+Achilles: There is even reputed to be a koan which tells how the Art Strings began. But
+unfortunately, all this has long since been lost sands of time, and is no doubt gone
+forever. Which may be just a for otherwise there would be imitators who would take
+on the m~ name, and copy him in other ways.
+
+Tortoise: But wouldn't it be a good thing if all students of Zen copied that most
+enlightened master of all, the Great Tutor?
+
+Achilles: Let me tell you a koan about an imitator.
+
+Zen master Gutei raised his finger whenever he was asked a question about Zen. A
+young novice began to irritate him in this way. When Gut was told about the
+novice's imitation, he sent for him and asked him if were true. The novice
+admitted it was so. Gutei asked him if he understood. In reply the novice held up
+his index finger. Gutei promptly cut off. The novice ran from the room, howling in
+pain. As he reached it threshold, Gutei called, "Boy!" When the novice turned,
+Gutei raised h index finger. At that instant the novice vas enlightened.
+
+Tortoise: Well, what do you know! Just when I thought Zen was all about Joshu and his
+shenanigans, now I find out that Gutei is in on the merriment too. He seems to have
+quite a sense of humor.
+
+Achilles: That koan is very serious. I don't know how you got the idea that it is
+humorous.
+
+Tortoise: Perhaps Zen is instructive because it is humorous. I would guess
+
+
+A Mu Offering
+
+
+237
+
+
+
+that if you took all such stories entirely seriously, you would miss the point as often as
+you would get it.
+
+Achilles: Maybe there's something to your Tortoise-Zen.
+
+Tortoise: Can you answer just one question for me? I would like to know this: Why did
+Bodhidharma come from India into China?
+
+Achilles: Oho! Shall I tell you what Joshu said when he was asked that very question?
+
+Tortoise: Please do.
+
+Achilles: He replied, "That oak tree in the garden."
+
+Tortoise: Of course; that's just what I would have said. Except that I would have said it in
+answer to a different question-namely, "Where can I find some shade from the
+midday sun?"
+
+Achilles: Without knowing it, you have inadvertently hit upon one of the basic questions
+of all Zen. That question, innocent though it sounds, actually means, "What is the
+basic principle of Zen?"
+
+Tortoise: How extraordinary. I hadn't the slightest idea that the central aim of Zen was to
+find some shade.
+
+Achilles: Oh, no-you've misunderstood me entirely. I wasn't referring to THAT question.
+I meant your question about why Bodhidharma came from India into China.
+
+Tortoise: I see. Well, I had no idea that I was getting into such deep waters. But let's
+come back to this curious mapping. I gather that any koan can be turned into a folded
+string by following the method you outlined. Now what about the reverse process?
+Can any folded string be read in such a way as to yield a koan?
+
+Achilles: Well, in a way. However .. .
+
+Tortoise: What's wrong?
+
+Achilles: You're just not supposed to do it that way 'round. It would violate the Central
+Dogma of Zen strings, you see, which is contained in this picture (picks up a napkin
+and draws)'.
+
+
+koan => messenger folded string
+transcription translation
+
+You're not supposed to go against the arrows-especially not the second one.
+
+Tortoise: Tell me, does this Dogma have Buddha-nature, or not? Come to think of it, I
+think I'll unask the question. Is that all right?
+
+Achilles: I am glad you unasked the question. But-I'll let you in on a secret. Promise you
+won't tell anyone?
+
+Tortoise: Tortoise's honor.
+
+Achilles: Well, once in a while, I actually do go against the arrows. I get sort of an illicit
+thrill out of it, I guess.
+
+Tortoise: Why, Achilles! I had no idea you would do something so irreverent!
+
+Achilles: I've never confessed it to anyone before-not even Okanisama.
+
+
+A Mu Offering
+
+
+238
+
+
+
+Tortoise: So tell me, what happens when you go against the arrows i Central Dogma?
+Does that mean you begin with a string and m koan?
+
+Achilles: Sometimes-but some weirder things can happen.
+
+Tortoise: Weirder than producing koans?
+
+Achilles: Yes ... When you untranslate and untranscribe, you get THING, but not always
+a koan. Some strings, when read out Ion way, only give nonsense.
+
+Tortoise: Isn't that just another name for koans?
+
+Achilles: You clearly don't have the true spirit of Zen yet.
+
+Tortoise: Do you always get stories, at least?
+
+Achilles: Not always-sometimes you get nonsense syllables, other you get ungrammatical
+sentences. But once in a while you get seems to be a koan.
+
+Tortoise: It only SEEMS to be one?
+
+Achilles: Well, it might be fraudulent, you see.
+
+Tortoise: Oh, of course.
+
+Achilles: I call those strings which yield apparent koans "well-foi strings.
+
+Tortoise: Why don't you tell me about the decision procedure which allows you to
+distinguish phony koans from the genuine article?
+
+Achilles: That's what I was heading towards. Given the koan, or non* as the case may be,
+the first thing is to translate it into the dimensional string. All that’s left is to find out
+if the strip Buddha-nature or not.
+
+Tortoise: But how do you do THAT?
+
+Achilles: Well, my master has said that the Great Tutor was able, I glancing at a string, to
+tell if it had Buddha-nature or not.
+
+Tortoise: But what if you have not reached the stage of the Enlightenment: 'Yond
+Enlightenment? Is there no other way to tell if a string hasi Buddha-nature?
+
+Achilles: Yes, there is. And this is where the Art of Zen Strings come is a technique for
+making innumerably many strings, all of whit Buddha-nature.
+
+Tortoise: You don't say! And is there a corresponding way of n strings which DON'T
+have Buddha-nature?
+
+Achilles: Why would you want to do that?
+
+Tortoise: Oh, I just thought it might be useful.
+
+Achilles: You have the strangest taste. Imagine! Being more intere things that DON'T
+have Buddha-nature than things that DO!
+
+Tortoise: Just chalk it up to my unenlightened state. But go on. T how to make a string
+which DOES have Buddha-nature.
+
+Achilles: Well, you must begin by draping a loop of string over your in one of five legal
+starting positions, such as this one ... (Picks up a string and drapes it in a simple loop
+between a finger on each hand.:)
+
+
+A Mu Offering
+
+
+239
+
+
+
+Tortoise: What are the other four legal starting positions?
+
+Achilles: Each one is a position considered to be a SELF-EVIDENT manner of picking
+up a string. Even novices often pick up strings in those positions. And these five
+strings all have Buddha-nature. Tortoise: Of course.
+
+Achilles: Then there are some String Manipulation Rules, by which you can make more
+complex string figures. In particular, you are allowed to modify your string by doing
+certain basic motions of the hands. For instance, you can reach across like this-and
+pull like this-and twist like this. With each operation you are changing the overall
+configuration of the string draped over your hands.
+
+Tortoise: Why, it looks just like making cat's-cradles and such string figures!
+
+Achilles: That's right. Now as you watch, you'll see that some of these rules make the
+string more complex; some simplify it. But whichever way you go, as long as you
+follow the String Manipulation Rules, every string you produce will have Buddha-
+nature.
+
+Tortoise: That is truly marvelous. Now what about the koan concealed inside this string
+you've just made? Would it be genuine?
+
+Achilles: Why, according to what I've learned, it must. Since I made it according to the
+Rules, and began in one of the five self-evident positions, the string must have
+Buddha-nature, and consequently it must correspond to a genuine koan.
+
+Tortoise: Do you know what the koan is?
+
+Achilles: Are you asking me to violate the Central Dogma? Oh, you naughty fellow!
+
+(And with furrowed brow and code book in hand, Achilles points along the string
+inch by inch, recording each fold by a triplet of geometric symbols of the strange
+phonetic alphabet for koan, until he has nearly a napkinful.)
+
+Done!
+
+Tortoise: Terrific. Now let's hear it.
+
+Achilles: All right.
+
+A traveling monk asked an old woman the road to Taizan, a popular temple
+supposed to give wisdom to the one who worships there. The old woman said:
+"Go straight ahead." After the monk had proceeded a few steps, she said to herself,
+"He also is a common church-goer." Someone told this incident to Joshu, who
+said: "Wait until I investigate." The next day he went and asked the same question,
+and the old woman gave the same answer. Joshu remarked: "I have investigated
+that old woman."
+
+Tortoise: Why, with his flair for investigations, it's a shame that Joshu
+
+never was hired by the FBI. Now tell me-what you did, I could also
+
+do, if I followed the Rules from the Art of Zen Strings, right?
+
+Achilles: Right.
+
+Tortoise: Now would I have to perform the operations in just the same ORDER as you
+did?
+
+
+A Mu Offering
+
+
+240
+
+
+
+.Achilles: No, any old order will do.
+
+Tortoise: Of course, then I would get a different string, and consequently a different
+koan. Now would I have to perform the same NUMBER of steps as you did?
+
+Achilles: By no means. Any number of steps is fine.
+
+Tortoise: Well, then there are an infinite number of strings with Buddha nature-and
+consequently an infinite number of genuine koans Howdo you know there is any
+string which CAN "I- be made by your Achilles: Oh, yes-back to things which lack
+Buddha-nature. It just so happens that once you know how to make strings WITH
+Buddha nature, you can also make strings WITHOUT Buddha-nature. That is
+something which my master drilled into me right at the beg Tortoise: Wonderful!
+How does it work?
+
+Achilles: Easy. Here, for example-M make a string which lacks Buddha-nature .. .
+
+(He picks up the string out of which the preceding koan was "pulled", ties a little
+teeny knot at one end of it, pulling it tight with his thumb forefinger.)
+
+This is it — no Buddha-nature here.
+
+Tortoise: Very illuminating. All it takes is adding a knot? How know that the new string
+lacks Buddha-nature?
+
+Achilles: Because of this fundamental property of Buddha-nature; when two well-formed
+strings are identical but for a knot at one end, then only ONE of them can have
+Buddha-nature. It's a rule of thumb which my master taught me.
+
+Tortoise: I'm just wondering about something. Are there some strings with Buddha-
+nature which you CAN'T reach by following the Rules of Zen Strings, no matter in
+what order?
+
+Achilles: I hate to admit it, but I am a little confused on this point myself. At first my
+master gave the strongest impression that Buddha in a string was DEFINED by
+starting in one of the five legal positions, and then developing the string according to
+the Rules. But then later, he said something about somebody-o "Theorem". I never
+got it straight. Maybe I even misheard said. But whatever he said, it put some doubt in
+my mind as to this method hits ALL strings with Buddha-nature. To the be
+knowledge, at least, it does. But Buddha-nature is a pretty elusive thing, you know.
+
+Tortoise: I gathered as much, from Joshu's 'MU'. I wonder ...
+
+Achilles: What is it?
+
+Tortoise: I was just wondering about those two koans-I mean t and its un-koan-the ones
+which say "This mind is Buddha" at mind is not Buddha"-what do they look like,
+when turned int via the Geometric Code?
+
+Achilles: I'd be glad to show you.
+
+
+A Mu Offering
+
+
+241
+
+
+
+(He writes down the phonetic transcriptions, and then pulls from his pocket a
+couple of pieces of string, which he carefully folds inch by inch, following the
+triplets of symbols written in the strange alphabet. Then he places the finished
+strings side by side.)
+
+You see, here is the difference.
+
+Tortoise: They are very similar, indeed. Why, I do believe there is only one difference
+between them: it's that one of them has a little knot on its end!
+
+Achilles: By Joshu, you're right.
+
+Tortoise: Aha! Now I understand why your master is suspicious.
+
+Achilles: You do?
+
+Tortoise: According to your rule of thumb, AT MOST ONE of such a pair can have
+Buddha-nature, so you know right away that one of the koans must be phony.
+
+Achilles: But that doesn't tell which one is phony. I've worked, and so has my master, at
+trying to produce these two strings by following the String Manipulation Rules, but to
+no avail. Neither one ever turns up. It's quite frustrating. Sometimes you begin to
+wonder ...
+
+Tortoise: You mean, to wonder if either one has Buddha-nature? Perhaps neither of them
+has Buddha-nature-and neither koan is genuine!
+
+Achilles: I never carried my thoughts as far as that-but you're right-it's possible, I guess.
+But I think you should not ask so many questions about Buddha-nature. The Zen
+master Mumon always warned his pupils of the danger of too many questions.
+
+Tortoise: All right-no more questions. Instead, I have a sort of hankering to make a string
+myself. It would be amusing to see if what I come up with is well-formed or not.
+
+Achilles: That could be interesting. Here's a piece of string. (He passes one to the
+Tortoise.)
+
+Tortoise: Now you realize that I don't have the slightest idea what to do.
+
+We'll just have to take potluck with my awkward production, which will follow no rules
+and will probably wind up being completely undecipherable. (Grasps the loop
+between his feet and, with a few simple manipulations, creates a complex string which
+he proffers wordlessly to Achilles. At that moment, Achilles' face lights up.)
+
+Achilles: Jeepers creepers! I'll have to try out your method myself. I have never seen a
+string like this!
+
+Tortoise: I hope it is well-formed. Achilles: I see it's got a knot at one end.
+
+Tortoise: Oh just a moment! May I have it back? I want to do one thing to it.
+
+Achilles: Why, certainly. Here you are.
+
+(Hands it back to the Tortoise, who ties another knot at the same end. Then the
+Tortoise gives a sharp tug, and suddenly both knots disappear!)
+
+
+A Mu Offering
+
+
+242
+
+
+
+Achilles: What happened?
+
+Tortoise: I wanted to get rid of that knot.
+
+Achilles: But instead of untying it, you tied another one, and then BOTH disappeared!
+Where did they go?
+
+Tortoise: Tumbolia, of course. That's the Law of Double Nodulation
+
+(Suddenly, the two knots reappear from out of nowhere-that is to say, Tumbolia.)
+
+Achilles: Amazing. They must lie in a fairly accessible layer of Tumbol they can pop into
+it and out of it so easily. Or is all of Tumbolia equally inaccessible?
+
+Tortoise: I couldn't say. However, it does occur to me that burning string would make it
+quite improbable for the knots to come back such a case, you could think of them as
+being trapped in a deeper la of Tumbolia. Perhaps there are layers and layers of
+Tumbolia. that's neither here nor there. What I would like to know is how string
+sounds, if you turn it back into phonetic symbols. (As he hauls it back, once again, the
+knots pop into oblivion.)
+
+Achilles: I always feel so guilty about violating the Central Dogma ( Takes out his pen
+and code book, and carefully jots down the many sym triplets which correspond to the
+curvy involutions of the Tortoise's string; when he is finished, he clears his voice.)
+Ahem. Are you ready to hear w you have wrought?
+
+Tortoise: I'm willing if you're willing.
+
+Achilles: All right. It goes like this:
+
+A certain monk had a habit of pestering the Grand Tortue (the only one who had ever
+reached the Enlightenment 'Yond Enlightenment), by asking whether various objects
+had Buddha-nature or not. To such questions Tortue invariably sat silent. The monk
+had already asked about a bean, a lake, and a moonlit night. One day, he brought to
+Tortue a piece of string, and asked the same question. In reply, the Grand Tortue
+grasped the loop between his feet and
+
+Tortoise: Between his feet? How odd! Achilles: Why should you find that odd?
+
+Tortoise: Well, ah ... you've got a point there. But please go on!
+
+Achilles: All right.
+
+The Grand Tortue grasped the loop between his feet and, with a few simple
+manipulations, created a complex string which he proffered wordlessly to the
+monk. At that moment, the monk was enlightened.
+
+Tortoise: I'd rather be twice-enlightened, personally.
+
+Achilles: Then it tells how to make the Grand Tortue's string, if you be, with a string
+draped over your feet. I'll skip those boring details concludes this way:
+
+From then on, the monk did not bother Tortue. Instead, he made string after string
+by Tortue's method; and he passed the method on to his own disciples, who passed
+it on to theirs.
+
+
+A Mu Offering
+
+
+243
+
+
+
+Tortoise: Quite a yarn. It's hard to believe it was really hidden inside my string.
+
+Achilles: Yet it was. Astonishingly, you seem to have created a well-formed string right
+off the bat.
+
+Tortoise: But what did the Grand Tortue's string look like? That's the main point of this
+koan, I'd suppose.
+
+Achilles: I doubt it. One shouldn't "attach" to small details like that inside koans. It's the
+spirit of the whole koan that counts, not little parts of it. Say, do you know what I just
+realized? I think, crazy though it sounds, that you may have hit upon that long-lost
+koan which describes the very origin of the Art of Zen Strings!
+
+Tortoise: Oh, that would almost be too good to have Buddha-nature.
+
+Achilles: But that means that the great master-the only one who ever reached the mystical
+state of the Enlightenment 'Yond Enlightenment-was named "Tortue", not "Tutor".
+What a droll name!
+
+Tortoise: I don't agree. I think it's a handsome name. I still want to know how Tortue's
+string looked. Can you possibly recreate it from the description given in the koan?
+
+Achilles: I could try ... Of course, I'll have to use my feet, too, since it's described in
+terms of foot motions. That's pretty unusual. But I think I can manage it. Let me give
+it a go. (He picks up the koan and a piece of string, and for a few minutes twists and
+bends the string in arcane ways until he has the finished product.) Well, here it is.
+Odd, how familiar it looks.
+
+Tortoise: Yes, isn't that so? I wonder where I saw it before? Achilles: I know! Why, this
+is YOUR string, Mr. T! Or is Tortoise: Certainly not.
+
+Achilles: Of course not-it's the string which you first handed to me, before you took it
+back to tie an extra knot in it.
+
+Tortoise: Oh, yes-indeed it is. Fancy that. I wonder what that implies.
+
+Achilles: It's strange, to say the least.
+
+Tortoise: Do you suppose my koan is genuine?
+
+Achilles: Wait just a moment ...
+
+Tortoise: Or that my string has Buddha-nature?
+
+Achilles: Something about your string is beginning, to trouble me, Mr.Tortoise .
+
+Tortoise ( looking most pleased with himself and paying no attention to Achilles)-. And
+what about Tortue's string? Does it have Buddha nature? There are a host of questions
+to ask!
+
+Achilles: I would be scared to ask such questions, Mr. T. There is something mighty
+funny going on here, and I'm not sure I like it. Tortoise: I'm sorry to hear it. I can't
+imagine what's troubling you. Achilles: Well, the best way I know to explain it is to
+quote the words of another old Zen master, Kyogen.
+
+
+A Mu Offering
+
+
+244
+
+
+
+Kyogen said: Zen is like a man hanging in a tree by his teeth over a precipice. His
+har grasp no branch, his feet rest on no limb, and under the tree anotl person asks
+him: "Why did Bodhidharma come to China from India?" the man in the tree does
+not answer, he fails; and if he does answer, falls and loses his life. Now what shall
+he do?
+
+Tortoise: That's clear; he should give up Zen, and take up molecular biology.
+
+
+A Mu Offering
+
+
+245
+
+
+
+CHAPTER IX
+
+
+Mumon and Godel
+
+
+What Is Zen?
+
+I'M NOT SURE I know what Zen is. In a way, I think I understand it very well; but in a
+way, I also think I can never understand it at all. Ever since my freshman English teacher
+in college read Joshu's MU out loud to our class, I have struggled with Zen aspects of
+life, and probably I will never cease doing so. To me, Zen is intellectual quicksand-
+anarchy, darkness, meaninglessness, chaos. It is tantalizing and infuriating. And yet it is
+humorous, refreshing, enticing. Zen has its own special kind of meaning, brightness, and
+clarity. I hope that in this Chapter, I can get some of this cluster of reactions across to
+you. And then, strange though it may seem, that will lead us directly to Godelian matters.
+
+One of the basic tenets of Zen Buddhism is that there is no way to characterize
+what Zen is. No matter what verbal space you try to enclose Zen in, it resists, and spills
+over. It might seem, then, that all efforts to explain Zen are complete wastes of time. But
+that is not the attitude of Zen masters and students. For instance, Zen koans are a central
+part of Zen study, verbal though they are. Koans are supposed to be "triggers" which,
+though they do not contain enough information in themselves to impart enlightenment,
+may possibly be sufficient to unlock the mechanisms inside one's mind that lead to
+enlightenment. But in general, the Zen attitude is that words and truth are incompatible,
+or at least that no words can capture truth.
+
+Zen Master Mumon
+
+Possibly in order to point this out in an extreme way, the monk Mumon ("No-gate"), in
+the thirteenth century, compiled forty-eight koans, following each with a commentary
+and a small "poem". This work is called "The Gateless Gate" or the Mumonkan ("No¬
+gate barrier"). It is interesting to note that the lives of Mumon and Fibonacci coincided
+almost exactly: Mumon living from 1183 to 1260 in China, Fibonacci from 1180 to 1250
+in Italy. To those who would look to the Mumonkan in hopes of making sense of, or
+"understanding", the koans, the Mumonkan may come as a rude shock, for the comments
+and poems are entirely as opaque as the koans which they are supposed to clarify. Take
+this, for example:' -
+
+
+Mumon and Godel
+
+
+246
+
+
+
+
+FIGURE 46. Three Worlds by M. C. Escher (lithograph, 1955)
+
+
+Mumon and Godel
+
+
+247
+
+
+
+
+
+
+Koan:
+
+
+Hogen of Seiryo monastery was about to lecture before dinner when he noticed that the bamboo screen,
+lowered for meditation, had not been rolled up. He pointed to it. Two monks arose wordlessly from the
+audience and rolled it up. Hogen, observing the physical moment, said, "The state of the first monk is good,
+not that of the second."
+
+Mumon's Commentary:
+
+I want to ask you: which of those two monks gained and which lost? If any of you has one eye, he will see
+the failure on the teacher's part. However, I am not discussing gain and loss.
+
+Mumon's Poem:
+
+When the screen is rolled up the great sky opens.
+
+Yet the sky is not attuned to Zen.
+
+It is best to forget the great sky
+And to retire from every wind.
+
+Or then again, there is this one
+Koan:
+
+Goso said: "When a buffalo goes out of his enclosure to the edge of the abyss, his horns and his
+head and his hoofs all pass through, but why can’t the tail also pass?"
+
+Mumon's Commentary:
+
+If anyone can open one eye at this point and say a word of Zen, he is qualified to repay
+the four gratifications, and, not only that, he can save all sentient beings under him. But if
+he cannot say such a word of Zen, he should turn back to his tail.
+
+Mumon's Poem:
+
+If the buffalo runs, he will fall into the trench;
+
+If he returns, he will be butchered.
+
+That little tail
+
+Is a very strange thing.
+
+I think you will have to admit that Mumon does not exactly clear everything up. One
+might say that the metalanguage (in which Mumon writes) is not very different from the
+object language (the language of the koan). According to some, Mumon's comments are
+intentionally idiotic, perhaps meant to show how useless it is to spend one's time in
+chattering about Zen. How ever, Mumon's comments can be taken on more than one
+level. For instance, consider this :3
+
+Koan:
+
+A monk asked Nansen: "Is diere a teaching no master ever taught before?"
+
+Nansen said: "Yes, there is."
+
+"What is it?" asked the monk.
+
+Nansen replied: "It is not mind, it is not Buddha, it is not things."
+
+
+Mumon and Godel
+
+
+248
+
+
+
+FIGURE 47. Dewdrop, by M. C. Escher (mezzotint, 1948).
+
+
+Mumon’s Commentary:
+
+Old Nansen gave away his treasure-words. He must have been greatly upset
+Mumon’s Poem:
+
+Nansen was too kind and lost his treasure.
+
+Truly, words have no power.
+
+Even though the mountain becomes the sea.
+
+Words cannot open another's mind.
+
+In this poem Mumon seems to be saying something very central to Zen, not making
+idiotic statements. Curiously, however, the poem is referential, and thus it is a comment
+not only on Nansen’s words, but on its own ineffectiveness. This type of paradox is quite
+characteristic of Zen It is an attempt to "break the mind of logic”. You see this paradox
+quality in the koan, as well. Concerning Mumon's commentary, do think that Nansen was
+really so sure of his answer? Or did the "correctness of his answer matter at all? Or does
+correctness play any role in Zen? is the difference between correctness and truth, or is
+there any? What if Nansen had said, "No, there is not any such teaching"? Would it have
+r any difference? Would his remark have been immortalized in a koan?
+
+
+Mumon and Godel
+
+
+249
+
+
+
+
+FIGURE 48. Another World, by M. C. Escher (wood-engraving, 1947).
+
+Here is another koan which aims to break the mind of logic:'
+
+The student Doko came to a Zen master, and said: "I am seeking the truth. In what state
+of mind should I train myself, so as to find it?"
+
+Said the master, "There is no mind, so you cannot put it in any state. There is no
+truth, so you cannot train yourself for it."
+
+"If there is no mind to train, and no truths to find, why do you have these monks
+gather before you every day to study- Zen and train themselves for this study?"
+
+"But I haven't an inch of room here,” said the master, "so how could the monks
+gather? I have no tongue, so how could I call them together or teach them?"
+
+
+Mumon and Godel
+
+
+250
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+"Oh, how can you lie like this?" asked Doko.
+
+"But if I have no tongue to talk to others, how can I lie to you?" asked the master.
+Then Doko said sadly, "I cannot follow you. I cannot understand you
+"I cannot understand myself," said the master.
+
+If any koan serves to bewilder, this one does. And most likely, can bewilderment is its
+precise purpose, for when one is in a bewildered s one's mind does begin to operate
+nonlogically, to some extent. Only by stepping outside of logic, so the theory goes, can
+one make the lea enlightenment. But what is so bad about logic? Why does it prevent the
+to enlightenment?
+
+
+Zen's Struggle Against Dualism
+
+To answer that, one needs to understand something about what enlightenment is. Perhaps
+the most concise summary of enlightenment w be: transcending dualism. Now what is
+dualism? Dualism is the conceptual division of the world into categories. Is it possible to
+transcend this natural tendency? By prefixing the word "division" by the word
+"conceptual", I may have made it seem that this is an intellectual or cons effort, and
+perhaps thereby given the impression that dualism could overcome simply by suppressing
+thought (as if to suppress thinking act were simple!). But the breaking of the world into
+categories takes plat below the upper strata of thought; in fact, dualism is just as a
+perceptual division of the world into categories as it is a conceptual division In other
+words, human perception is by nature a dualistic phenomenon which makes the quest for
+enlightenment an uphill struggle, to say the least.
+
+At the core of dualism, according to Zen, are words just plain w The use of words
+is inherently dualistic, since each word represents, obviously, a conceptual category.
+Therefore, a major part of Zen is the against reliance on words. To combat the use of
+words, one of the devices is the koan, where words are so deeply abused that one's mi
+practically left reeling, if one takes the koans seriously. Therefore perhaps wrong to say
+that the enemy of enlightenment is logic; rather dualistic, verbal thinking. In fact, it is
+even more basic than that: perception. As soon as you perceive an object, you draw a line
+between it and the rest of the world; you divide the world, artificially, into parts you
+thereby miss the Way.
+
+Here is a koan which demonstrates the struggle against words:
+
+Koan:
+
+Shuzan held out his short staff and said: "If you call this a short staff, you oppose its
+reality. If you do not call it a short staff, you ignore the fact. N, what do you wish to call
+this?"
+
+
+Mumon and Godel
+
+
+251
+
+
+
+FIGURE 49. Day and Night, by M. C. Escher (woodcut, 1938).
+
+
+Mumon’s Commentary:
+
+If you call this a short staff, you oppose its reality. If you do not call it a short staff, you
+ignore the fact. It cannot be expressed with words and it cannot be expressed without
+words. Now say quickly what it is.
+
+Mumon's Poem:
+
+Holding out the short staff.
+
+He gave an order of life or death.
+
+Positive and negative interwoven,
+
+Even Buddhas and patriarchs cannot escape this attack.
+
+("Patriarchs" refers to six venerated founders of Zen Buddhism, of whom Bodhidharma is
+the first, and Eno is the sixth.)
+
+Why is calling it a short staff opposing its reality? Probably because such a
+categorization gives the appearance of capturing reality, whereas the surface has not even
+been scratched by such a statement. It could be compared to saying "5 is a prime
+number". There is so much more-an infinity of facts-that has been omitted. On the other
+hand, not to call it a staff is, indeed, to ignore that particular fact, minuscule as it may be.
+Thus words lead to some truth-some falsehood, perhaps, as well-but certainly not to all
+truth. Relying on words to lead you to the truth is like relying on an incomplete formal
+system to lead you to the truth. A formal system will give you some truths, but as we
+shall soon see, a formal system-no matter how powerful-cannot lead to all truths. The
+dilemma of mathematicians is: what else is there to rely on, but formal systems? And the
+dilemma of
+
+
+Mumon and Godel
+
+
+252
+
+
+Zen people is, what else is there to rely on, but words? Mumon states t dilemma very
+clearly: "It cannot be expressed with words and it cannot
+expressed without words."
+
+Here is Nansen, once again:'
+
+Joshu asked the teacher Nansen, "What is the true Way?"
+
+Nansen answered, "Everyday way is the true Way.’Joshu asked, "Can I study it?"
+Nansen answered, "The more you study, the further from the Way." Joshu asked, "If I
+don't study it, how can I know it?"
+
+Nansen answered, "The Way does not belong to things seen: nor to thing: unseen. It
+does not belong to things known: nor to things unknown. Do not seek it, study it, or
+name it. To find yourself on it, open yourself wide as the sky." [See Fig. 50.]
+
+
+FIGURE 50. Rind, by M. C. Escher (wood-engraving, 1955).
+
+
+
+Mumon and Godel
+
+
+253
+
+
+This curious statement seems to abound with paradox. It is a little reminiscent of
+this surefire cure for hiccups: "Run around the house three times without thinking of the
+word 'wolf." Zen is a philosophy which seems to have embraced the notion that the road
+to ultimate truth, like the only surefire cure for hiccups, may bristle with paradoxes.
+
+Ism, The Un-Mode, and Unmon
+
+If words are bad, and thinking is bad, what is good? Of course, to ask this is already
+horribly dualistic, but we are making no pretense of being faithful to Zen in discussing
+Zen-so we can try to answer the question seriously. I have a name for what Zen strives
+for: ism. Ism is an antiphilosophy, a way of being without thinking. The masters of ism
+are rocks, trees, clams; but it is the fate of higher animal species to have to strive for ism,
+without ever being able to attain it fully. Still, one is occasionally granted glimpses of
+ism. Perhaps the following koan offers such a glimpse 7
+
+Hyakujo wished to send a monk to open a new monastery. He told his pupils that
+whoever answered a question most ably would be appointed. Placing a water vase on
+the ground, he asked: "Who can say what this is without calling its name?"
+
+The chief monk said: "No one can call it a wooden shoe."
+
+Isan, the cooking monk, tipped over the vase with his foot and went out. Hyakujo
+smiled and said: "The chief monk loses." And Isan became the
+master of the new monastery.
+
+To suppress perception, to suppress logical, verbal, dualistic thinking-this is the essence
+of Zen, the essence of ism. This is the Unmode-not Intelligent, not Mechanical, just "Un".
+Joshu was in the Unmode, and that is why his 'MU' unasks the question. The Un-mode
+came naturally to Zen Master Unmon: 8
+
+One day Unmon said to his disciples, "This staff of mine has transformed itself into a
+dragon and has swallowed up the universe! Oh, where are the rivers and mountains
+and the great earth?"
+
+Zen is holism, carried to its logical extreme. If holism claims that things can only be
+understood as wholes, not as sums of their parts, Zen goes one further, in maintaining that
+the world cannot be broken into parts at all. To divide the world into parts is to be
+deluded, and to miss enlightenment.
+
+A master was asked the question, "What is the Way?" by a curious monk. "
+
+It is right before your eyes," said the master. "Why do I not see it for myself?"
+"Because you are thinking of yourself."
+
+"What about you: do you see it?"
+
+"So long as you see double, saying 'I don't 1 , and 'you do', and so on, your
+eyes are clouded," said the master.
+
+"When there is neither 'I' nor 'You', can one see it?"
+
+"When there is neither T nor 'You', who is the one that wants to see it?" 9
+
+
+Mumon and Godel
+
+
+254
+
+
+
+Apparently the master wants to get across the idea that an enlighte state is one
+where the borderlines between the self and the rest of universe are dissolved. This would
+truly be the end of dualism, for a says, there is no system left which has any desire for
+perception. But what is that state, if not death? How can a live human being dissolve the
+borderlines between himself and the outside world?
+
+Zen and Tumbolia
+
+The Zen monk Bassui wrote a letter to one of his disciples who was about to die, and in it
+he said: "Your end which is endless is as a snowflake dissolving in the pure air." The
+snowflake, which was once very much a discernible subsystem of the universe, now
+dissolves into the larger system which 4 held it. Though it is no longer present as a
+distinct subsystem, its essence somehow still present, and will remain so. It floats in
+Tumbolia, along hiccups that are not being hiccupped and characters in stories that are
+being read . . . That is how I understand Bassui's message.
+
+Zen recognizes its own limitations, just as mathematicians have lea: to recognize
+the limitations of the axiomatic method as a method attaining truth. This does not mean
+that Zen has an answer to what beyond Zen any more than mathematicians have a clear
+understanding the forms of valid reasoning which lie outside of formalization. One ol
+clearest Zen statements about the borderlines of Zen is given in the fol ing strange koan,
+very much in the spirit of Nansen: 10
+
+Tozan said to his monks, "You monks should know there is an even high
+understanding in Buddhism." A monk stepped forward and asked, "What the higher
+Buddhism?" Tozan answered, "It is not Buddha."
+
+There is always further to go; enlightenment is not the end-all of And there is no recipe
+which tells how to transcend Zen; the only thing can rely on for sure is that Buddha is not
+the way. Zen is a system cannot be its own metasystem; there is always something
+outside of which cannot be fully understood or described within Zen.
+
+Escher and Zen
+
+In questioning perception and posing absurd answerless riddles, Zen company, in the
+person of M. C. Escher. Consider Day and Night (Fig. 4 masterpiece of "positive and
+negative interwoven" (in the words of Mumoni). One might ask, "Are those really birds,
+or are they really field it really night, or day?" Yet we all know there is no point to such
+questions The picture, like a Zen koan, is trying to break the mind of logic. Es4 also
+delights in setting up contradictory pictures, such as Another World
+
+
+Mumon and Godel
+
+
+255
+
+
+
+FIGURE 51. Puddle, by M. C. Escher (woodcut, 1952).
+
+
+(Fig. 4S)-pictures that play with reality and unreality the same way as Zen plays with
+reality and unreality. Should one take Escher seriously? Should one take Zen seriously?
+
+There is a delicate, haiku-like study of reflections in Dewclrop (Fig. 47); and then
+there are two tranquil images of the moon reflected in still waters: Puddle (Fig. 51), and
+Rippled Surface (Fig. 52). The reflected moon is a theme which recurs in various koans.
+Here is an example:'
+
+Chiyono studied Zen for many years under Bukko of Engaku. Still, she could not
+attain the fruits of meditation. At last one moonlit night she was carrying water in an
+old wooden pail girded with bamboo. The bamboo broke, and the bottom fell out of
+the pail. At that moment, she was set free. Chiyono said, "No more water in the pail,
+no more moon in the water."
+
+Three Worlds : an Escher picture (Fig. 46), and the subject of a Zen koan: 12
+
+A monk asked Ganto, "When the three worlds threaten me, what shall I do?" Ganto
+answered, "Sit down." "I do not understand," said the monk. Canto said, "Pick up the
+mountain and bring it to me. Then I will tell you."
+
+
+Mumon and Godel
+
+
+256
+
+
+
+Hemiolia and Escher
+
+
+In Verbum (Fig. 149), oppositions are made into unities on several I Going around we see
+gradual transitions from black birds to white birds to black fish to white fish to black
+frogs to white frogs to black birds ... six steps, back where we started! Is this a
+reconciliation of the dichotomy of black and white? Or of the trichotomy of birds, fish,
+and frogs? Or sixfold unity made from the opposition of the evenness of 2 an oddness of
+3? In music, six notes of equal time value create a rhythmic ambiguity-are they 2 groups
+of 3, or 3 groups of 2? This ambiguity has a name: hemiolia. Chopin was a master of
+hemiolia: see his Waltz op. his Etude op. 25, no. 2. In Bach, there is the Tempo di
+Menuetto from the keyboard Partita no. 5, or the incredible Finale of the first Sonata
+unaccompanied violin, in G Minor.
+
+As one glides inward toward the center of Verbum, the distinctions gradually blur,
+so that in the end there remains not three, not two, but one single essence: "VERBUM",
+which glows with brilliancy-perhaps a symbol of enlightenment. Ironically, ' verbum"
+not only is a word, but "word"-not exactly the most compatible notion with Zen. On the
+hand, "verbum" is the only word in the picture. And Zen master I once said, "The
+complete Tripitaka can be expressed in one character ("Tripitaka", meaning "three
+baskets", refers to the complete texts c original Buddhist writings.) What kind of
+decoding-mechanism, I wonder would it take to suck the three baskets out of one
+character? Perhaps one with two hemispheres.
+
+
+FIGURE 52. Rippled Surface, by M. C. Escher (lino-cut, 1950).
+
+
+
+Mumon and Godel
+
+
+257
+
+
+
+FIGURE 53. Three Spheres II, by M. C. Escher (lithograph, 1946),
+
+Indra's Net
+
+
+Finally, consider Three Spheres 11 (Fig. 53), in which every part of the world seems to
+contain, and be contained in, every other part: the writing table reflects the spheres on top
+of it, the spheres reflect each other, as well as the writing table, the drawing of them, and
+the artist drawing it. The endless connections which all things have to each other is only
+hinted at here, yet the hint is enough. The Buddhist allegory of "Indra's Net" tells of an
+endless net of threads throughout the universe, the horizontal threads running through
+space, the vertical ones through time. At every crossing of threads is an individual, and
+every individual is a crystal bead. The great light of "Absolute Being" illuminates and
+penetrates every crystal bead; moreover, every crystal bead reflects not only the light
+from every other crystal in the net-but also every reflection of every reflection throughout
+the universe.
+
+To my mind, this brings forth an image of renormalized particles: in every
+electron, there are virtual photons, positrons, neutrinos, muons ... ; in every photon, there
+are virtual electrons, protons, neutrons, pions ... ; in every pion, there are ...
+
+But then another image rises: that of people, each one reflected in the minds of
+many others, who in turn are mirrored in yet others, and so on.
+
+Both of these images could be represented in a concise, elegant way by using
+Augmented Transition Networks. In the case of particles, there would be one network for
+each category of particle; in the case of people.
+
+
+Mumon and Godel
+
+
+258
+
+
+
+one for each person. Each one would contain calls to many others, t creating a virtual
+cloud of ATN's around each ATN. Calling one we create calls on others, and this process
+might cascade arbitrarily far, un~ bottomed out.
+
+Mumon on MU
+
+Let us conclude this brief excursion into Zen by returning to Mumon. H is his comment
+on Joshu's MU 13
+
+To realize Zen one has to pass through the barrier of the patriarchs. Enlightenment
+always comes after the road of thinking is blocked. If you do nc pass the barrier of the
+patriarchs or if your thinking road is not blocked whatever you think, whatever you
+do, is like a tangling ghost. You may ask "What is a barrier of a patriarch?" This one
+word, 'MU', is it.
+
+This is the barrier of Zen. If you pass through it, you will see Joshu face t face.
+Then you can work hand in hand with the whole line of patriarchs. I this not a pleasant
+thing to do?
+
+If you want to pass this barrier, you must work through every bone in you body,
+through every pore of your skin, filled with this question: "What 'MU'?" and carry it
+day and night. Do not believe it is the common negative symbol meaning nothing. It is
+not nothingness, the opposite of existence. I you really want to pass this barrier, you
+should feel like drinking a hot iro ball that you can neither swallow nor spit out.
+
+Then your previous lesser knowledge disappears. As a fruit ripening i season,
+your subjectivity and objectivity naturally become one. It is like dumb man who has
+had a dream. He knows about it but he cannot tell i
+
+When he enters this condition his ego-shell is crushed and he can shake th heaven
+and move the earth. He is like a great warrior with a sharp sword. If Buddha stands in
+his way, he will cut him down; if a patriarch offers him an obstacle, he will kill him;
+and he will be free in his way of birth and death. H can enter any world as if it were
+his own playground. I will tell you how to d this with this koan:
+
+Just concentrate your whole energy into this MU, and do not allow an
+discontinuation. When you enter this MU and there is no discontinuation - your
+attainment will be as a candle burning and illuminating the who] universe.
+
+From Mumon to the MU-puzzIe
+
+From the ethereal heights of Joshu's MU, we now descend to the private lowlinesses of
+Hofstadter's MU ... I know that you have already concentrated your whole energy into
+this MU (when you read Chapter 1). So n wish to answer the question which was posed
+there:
+
+
+Has MU theorem-nature, or not?
+
+The answer to this question is not an evasive MU; rather, it is a resounding NO. In order
+to show this, we will take advantage of dualistic, logical thinking.
+
+
+Mumon and Godel
+
+
+259
+
+
+
+We made two crucial observations in Chapter I:
+
+(1) that the MU-puzzle has depth largely because it involves the interplay of
+lengthening and shortening rules;
+
+(2) that hope nevertheless exists for cracking the problem by employing a tool which
+
+is in some sense of adequate depth to handle matters of that complexity: the
+theory of numbers.
+
+We did not analyze the MU-puzzle in those terms very carefully in Chapter I; we shall do
+so now. And we will see how the second observation (when generalized beyond the
+insignificant MlU-system) is one of the most fruitful realizations of all mathematics, and
+how it changed mathematicians' view of their own discipline.
+
+For your ease of reference, here is a recapitulation of the MlU-system:
+
+SYMBOLS: M, I, U
+
+Axiom: MI
+
+RULES:
+
+I. If xl is a theorem, so is xIU.
+
+II. If Mx is a theorem, so is Mxx.
+
+III. In any theorem, III can be replaced by U.
+
+IV. UU can be dropped from any theorem.
+
+Mumon Shows Us How to Solve the MU-puzzle
+
+According to the observations above, then, the MU-puzzle is merely a puzzle about
+natural numbers in typographical disguise. If we could only find a way to transfer it to the
+domain of number theory, we might be able to solve it. Let us ponder the words of
+Mumon, who said, "If any of you has one eye, he will see the failure on the teacher's
+part." But why should it matter to have one eye?
+
+If you try counting the number of l's contained in theorems, you will soon notice
+that it seems never to be 0. In other words, it seems that no matter how much lengthening
+and shortening is involved, we can never work in such a way that all l's are eliminated.
+Let us call the number of l's in any string the I-count of that string. Note that the I-count
+of the axiom MI is 1. We can do more than show that the I-count can't be 0-we can show
+that the I-count can never be any multiple of 3.
+
+To begin with, notice that rules I and IV leave the I-count totally undisturbed.
+Therefore we need only think about rules II and III. As far as rule III is concerned, it
+diminishes the I-count by exactly 3. After an application of this rule, the I-count of the
+output might conceivably be a multiple of 3-but only if the I-count of the input was also.
+Rule III, in short, never creates a multiple of 3 from scratch. It can only create one when
+it began with one. The same holds for rule II, which doubles the
+
+
+Mumon and Godel
+
+
+260
+
+
+
+I-count. The reason is that if 3 divides 2n, then-because 3 does not dig 2-it must divide n
+(a simple fact from the theory of numbers). Neither rule II nor rule III can create a
+multiple of 3 from scratch.
+
+But this is the key to the MU-puzzle! Here is what we know:
+
+(1) The I-count begins at 1 (not a multiple of 3);
+
+(2) Two of the rules do not affect the I-count at all; (3)
+
+(3) The two remaining rules which do affect the I-count do so in such a way as never
+
+to create a multiple of 3 unless given one initially.
+
+The conclusion-and a typically hereditary one it is, too-is that I-count can never become
+any multiple of 3. In particular, 0 is a forbid value of the I-count. Hence, MU is not a
+
+theorem of the MlU-system.
+
+Notice that, even as a puzzle about I-counts, this problem was plagued by the
+crossfire of lengthening and shortening rules. Zero became the goal; I-counts could
+increase (rule II), could decrease (rule III). 1 we analyzed the situation, we might have
+thought that, with enough switching back and forth between the rules, we might
+eventually hit 0. IS thanks to a simple number-theoretical argument, we know that the
+impossible.
+
+
+Godel-Numbering the MlU-System
+
+Not all problems of the the type which the MU-puzzle symbolizes at easy to solve as this
+one. But we have seen that at least one such pr could be embedded within, and solved
+within, number theory. We are going to see that there is a way to embed all problems
+about any for system, in number theory. This can happen thanks to the discovery Godel,
+of a special kind of isomorphism. To illustrate it, I will use MlU-system.
+
+We begin by considering the notation of the MlU-system. We map each symbol onto a
+new symbol:
+
+
+M <==>3
+I <= => 1
+U <= => 0
+
+The correspondence was chosen arbitrarily; the only rhyme or reason is that each symbol
+looks a little like the one it is mapped onto. I number is called the Godel number of the
+corresponding letter. Now I sure you can guess what the Godel number of a multiletter
+string will be:
+
+
+MU <==>30
+MIIU <= =>3110
+Etc.
+
+
+Mumon and Godel
+
+
+261
+
+
+
+It is easy. Clearly this mapping between notations is an information preserving
+transformation; it is like playing the same melody on two different instruments.
+
+Let us now take a look at a typical derivation in the MlU-system, written
+simultaneously in both notations:
+
+
+(1)
+
+MI
+
+axiom
+
+31
+
+(2)
+
+Mil
+
+rule 2
+
+311
+
+(3)
+
+miiii
+
+rule 2
+
+31111
+
+(4)
+
+MUI
+
+rule 3
+
+301
+
+(5)
+
+MUIU
+
+rule 1
+
+3010
+
+(6)
+
+MUIUUIU
+
+rule 2
+
+3010010
+
+(7)
+
+MUIIU
+
+rule 4
+
+30110
+
+
+The left-hand column is obtained by applying our four familiar typographical rules. The
+right-hand column, too, could be thought of as having been generated by a similar set of
+typographical rules. Yet the right-hand column has a dual nature. Let me explain what
+this means.
+
+Seeing Things Both Typographically and Arithmetically
+
+We could say of the fifth string ('3010') that it was made from the fourth, by appending a
+'O' on the right; on the other hand we could equally well view the transition as caused by
+an arithmetical operation-multiplication by 10, to be exact. When natural numbers are
+written in the decimal system, multiplication by 10 and putting a 'O' on the right are
+indistinguishable operations. We can take advantage of this to write an arithmetical rule
+which corresponds to typographical rule I:
+
+ARITHMETICAL RULE la: A number whose decimal expansion ends on the right in '1'
+can be multiplied by 10.
+
+We can eliminate the reference to the symbols in the decimal expansion by arithmetically
+describing the rightmost digit:
+
+ARITHMETICAL RULE lb: A number whose remainder when divided by 10 is 1, can
+be multiplied by 10.
+
+Now we could have stuck with a purely typographical rule, such as the following one:
+
+TYPOGRAPHICAL RULE I: From any theorem whose rightmost symbol is ' 1' a new
+theorem can be made, by appending 'O' to the right of that 1'.
+
+They would have the same effect. This is why the right-hand column has a "dual nature":
+it can be viewed either as a series of typographical opera-
+
+
+Mumon and Godel
+
+
+262
+
+
+
+tions changing one pattern of symbols into another, or as a series arithmetical operations
+changing one magnitude into another. But the are powerful reasons for being more
+interested in the arithmetical version Stepping out of one purely typographical system
+into another isomorphic typographical system is not a very exciting thing to do; whereas
+stepping clear out of the typographical domain into an isomorphic part of number theory
+has some kind of unexplored potential. It is as if somebody h known musical scores all
+his life, but purely visually-and then, all o: sudden, someone introduced him to the
+mapping between sounds a musical scores. What a rich, new world! Then again, it is as if
+somebody h been familiar with string figures all his life, but purely as string figur devoid
+of meaning-and then, all of a sudden, someone introduced him the mapping between
+stories and strings. What a revelation! The discovery of Godel-numbering has been
+likened to the discovery, by Descartes, of t isomorphism between curves in a plane and
+equations in two variables; incredibly simple, once you see it-and opening onto a vast
+new world
+
+Before we jump to conclusions, though, perhaps you would like to a more
+complete rendering of this higher level of the isomorphism. It i very good exercise. The
+idea is to give an arithmetical rule whose action is indistinguishable from that of each
+typographical rule of the MlU-system:
+
+A solution is given below. In the rules, m and k are arbitrary natural numbers, and n is
+any natural number which is less than 10 m
+
+RULE 1: If we have made 10m + 1, then we can make 10 x (10m + 1)
+
+Example: Going from line 4 to line 5. Here, m = 30.
+
+RULE 2: If we have made 3 x 10" + n, then we can make 10' X X (3 x 10"'+n)+n.
+Example: Going from line 1 to line 2, where both m and n equal 1.
+
+RULE 3: If we have made k x 10 "'+ 111 x 10'+n, then we can make k x 10"+' + n.
+
+Example: Going from line 3 to line 4. Here, m and n are 1, and k is 3.
+
+RULE 4: If we have made k x lOrn+z + n, k x 10" +n. then we can make k x 10m + n
+
+Example: Going from line 6 to line 7. Here, m = 2, n = 10, and k = 301.
+
+Let us not forget our axiom! Without it we can go nowhere. Therefore, let us postulate
+that:
+
+
+We can make 31.
+
+Now the right-hand column can be seen as a full-fledged arithmetic process, in a new
+arithmetical system which we might call the 310-system
+
+
+Mumon and Godel
+
+
+263
+
+
+
+I
+
+
+(1)
+
+31
+
+given
+
+(2)
+
+311
+
+rule 2 (m=l, n=l)
+
+(3)
+
+31111
+
+rule 2 (m=2, n=ll)
+
+(4)
+
+301
+
+rule 3 (m=l, n=l, k=3)
+
+(5)
+
+3010
+
+rule 1 (m=30)
+
+(6)
+
+3010010
+
+rule 2 (m=3, n=10)
+
+(7)
+
+30110
+
+rule 4 (m=2, n=10, k=301)
+
+
+Notice once again that the lengthening and shortening rules are ever with us in this "310-
+system"; they have merely been transposed into the domain of numbers, so that the Godel
+numbers go up and down. If you look carefully at what is going on, you will discover that
+the rules are based on nothing more profound than the idea that shifting digits to left and
+right in decimal representations of integers is related to multiplications and divisions by
+powers of 10. This simple observation finds its generalization in the following
+
+CENTRAL PROPOSITION: If there is a typographical rule which tells how
+certain digits are to be shifted, changed, dropped, or inserted in any number
+represented decimally, then this rule can be represented equally well by an
+arithmetical counterpart which involves arithmetical operations with powers of 10
+as well as additions, subtractions, and so forth.
+
+More briefly:
+
+Typographical rules for manipulating numerals are actually arithmetical rules for
+operating on numbers.
+
+This simple observation is at the heart of Godel’s method, and it will have an absolutely
+shattering effect. It tells us that once we have a Godel numbering for any formal system,
+we can straightaway form a set of arithmetical rules which complete the Godel
+isomorphism. The upshot is that we can transfer the study of any formal system-in fact
+the study of all formal systems-into number theory.
+
+MIU-Producible Numbers
+
+Just as any set of typographical rules generates a set of theorems, a corresponding set of
+natural numbers will be generated by repeated applications of arithmetical rules. These
+producible numbers play the same role inside number theory as theorems do inside any
+formal system. Of course, different numbers will be producible, depending on which
+rules are adopted. "Producible numbers" are only producible relative to a system of
+arithmetical rules. For example, such numbers as 31, 3010010, 3111, and so forth could
+be called MIU -producible numbers-an ungainly name, which might be shortened to
+MIU -numbers, symbolizing the fact that those numbers are the ones that result when you
+transcribe the MlU-system into number theory, via Godel-numbering. If we were to
+Godel-number the pq-system
+
+
+Mumon and Godel
+
+
+264
+
+
+
+and then "arithmetize" its rules, we could call the producible numbers "pq-numbers"-and
+so on.
+
+Note that the producible numbers (in any given system) are defined by a recursive
+method: given numbers which are known to be producible, we have rules telling how to
+make more producible numbers. Thus, the class of numbers known to be producible is
+constantly extending itself, in much the same way that the list of Fibonacci numbers, or
+Q-numbers, does. The set of producible numbers of any system is a recursively
+enumerable set. What about its complement-the set of nonproducible numbers? Is that set
+always recursively enumerable? Do numbers which are nonproducible share some
+common arithmetical feature?
+
+This is the sort of issue which arises when you transpose the study of formal
+systems into number theory. For each system which is arithmetized, one can ask, "Can
+we characterize producible numbers in a simple way?" "Can we characterize
+nonproducible numbers in a recursively enumerable way?" These are difficult questions
+of number theory. Depending on the system which has been arithmetized, such questions
+might prove too hard for us to resolve. But if there is any hope for solving such problems,
+it would have to reside in the usual kind of step-by-step reasoning as it applies to natural
+numbers. And that, of course, was put in its quintessential form in the previous Chapter.
+TNT seemed, to all appearances, to have captured all valid mathematical thinking
+processes in one single, compact system.
+
+Answering Questions about Producible Numbers
+by Consulting TNT
+
+Could it be, therefore, that the means with which to answer any question about any
+formal system lies within just a single formal system-TNT? It seems plausible. Take, for
+instance, this question:
+
+
+Is MU a theorem of the MlU-system?
+
+Finding the answer is equivalent to determining whether 30 is a MIU number or not.
+Because it is a statement of number theory, we should expect that, with some hard work,
+we could figure out how to translate the sentence "30 is a MlU-number" into TNT-
+notation, in somewhat the same way as we figured out how to translate other number-
+theoretical sentences into TNT-notation. I should immediately caution the reader that
+such a translation, though it does exist, is immensely complex. If you recall, I pointed out
+in Chapter VIII that even such a simple arithmetical predicate as "b is a power of 10" is
+very tricky to code into TNT-notation-and the predicate "b is a MlU-number" is a lot
+more complicated than that! Still, it can be found; and the numeral
+SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSO can be substituted for every b. This will
+result in a MONstrous string of TNT, a string of TNT which speaks about the MU-
+puzzle. Let us therefore call that string "MUMON". Through MUMON and strings like
+it, TNT is now capable of speaking "in code" about the MlU-system.
+
+
+Mumon and Godel
+
+
+265
+
+
+
+The Dual Nature of MUMON
+
+
+In order to gain some benefit from this peculiar transformation of the original question,
+we would have to seek the answer to a new question:
+
+Is MUMON a theorem of TNT?
+
+All we have done is replace one relatively short string (MU) by another (the monstrous
+MUMON), and a simple formal system (the MlU-system) by a complicated one (TNT).
+It isn't likely that the answer will be any more forthcoming even though the question has
+been reshaped. In fact, TNT has a full complement of both lengthening and shortening
+rules, and the reformulation of the question is likely to be far harder than the original.
+One might even say that looking at MU via MUMON is an intentionally idiotic way of
+doing things. However, MUMON can be looked at on more than one level.
+
+In fact, this is an intriguing point: MUMON has two different passive meanings.
+Firstly, it has the one which was given before:
+
+30 is a MlU-number.
+
+But secondly, we know that this statement is tied (via isomorphism) to the statement
+
+MU is a theorem of the MlU-system.
+
+So we can legitimately quote this latter as the second passive meaning of MUMON. It
+may seem very strange because, after all, MUMON contains nothing but plus signs,
+parentheses, and so forth-symbols of TNT. How can it possibly express any statement
+with other than arithmetical content?
+
+The fact is, it can. Just as a single musical line may serve as both harmony and
+melody in a single piece; just as "BACH" may be interpreted as both a name and a
+melody; just as a single sentence may be an accurate structural description of a picture by
+Escher, of a section of DNA, of a piece by Bach, and of the dialogue in which the
+sentence is embedded, so MUMON can be taken in (at least) two entirely different ways.
+This state of affairs comes about because of two facts:
+
+Fact 1. Statements such as "MU is a theorem" can be coded into number theory
+via Godel’s isomorphism.
+
+Fact 2. Statements of number theory can be translated into TNT.
+
+It could be said that MUMON is, by Fact 1, a coded message, where the symbols of the
+code are, by Fact 2, just symbols of TNT.
+
+
+Mumon and Godel
+
+
+266
+
+
+
+Codes and Implicit Meaning
+
+
+Now it could be objected here that a coded message, unlike an uncod message, does not
+express anything on its own-it requires knowledge the code. But in reality there is no
+such thing as an uncoded message. There are only messages written in more familiar
+codes, and message written in less familiar codes. If the meaning of a message is to be
+revealed it must be pulled out of the code by some sort of mechanism, or isomorphism. It
+may be difficult to discover the method by which the decoding should be done; but once
+that method has been discovered, the message becomes transparent as water. When a
+code is familiar enough, it ceases appearing like a code; one forgets that there is a
+decoding mechanism. The message is identified with its meaning.
+
+Here we have a case where the identification of message and meant is so strong
+that it is hard for us to conceive of an alternate meaning: residing in the same symbols.
+Namely, we are so prejudiced by the symbols of TNT towards seeing number-theoretical
+meaning (and only numb( theoretical meaning) in strings of TNT, that to conceive of
+certain string of TNT as statements about the MlU-system is quite difficult. But Godel’s
+isomorphism compels us to recognize this second level of meaning certain strings of
+TNT.
+
+
+Decoded in the more familiar way, MUMON bears the message:
+
+30 is a MlU-number.
+
+This is a statement of number theory, gotten by interpreting each sign the conventional
+way.
+
+But in discovering Godel-numbering and the whole isomorphism bu upon it, we
+have in a sense broken a code in which messages about the MlU-system are written in
+strings of TNT. Godel’s isomorphism is a n( information-revealer, just as the
+decipherments of ancient scripts we information-revealers. Decoded by this new and less
+familiar mechanism MUMON bears the message
+
+MU is a theorem of the MlU-system.
+
+The moral of the story is one we have heard before: that meaning is ; automatic by¬
+product of our recognition of any isomorphism; therefore there are at least two passive
+meanings of MUMON-maybe more!
+
+The Boomerang: Godel-Numbering TNT
+
+Of course things do not stop here. We have only begun realizing the: potential of Godel’s
+isomorphism. The natural trick would be to turn TNT's capability of mirroring other
+formal systems back on itself, as the Tortoise turned the Crab's phonographs against
+themselves, and as his Goblet G turned against itself, in destroying itself. In order to do
+this, we
+
+
+Mumon and Godel
+
+
+267
+
+
+
+will have to Godel-number TNT itself, just as we did the MlU-system, and then
+"arithmetize" its rules of inference. The Godel-numbering is easy to do. For instance, we
+could make the following correspondence:
+
+
+Symbol...
+
+Codon
+
+Mnemonic Justification
+
+0
+
+666
+
+Number of the Beast for the Mysterious Zero
+
+S
+
+123
+
+successorship: 1, 2, 3,
+
+=
+
+111
+
+visual resemblance, turned sideways
+
++
+
+112
+
+1+1=2
+
+• ....
+
+236
+
+2x3=6
+
+(
+
+362
+
+ends in 2 *
+
+)
+
+323
+
+ends in 3 *
+
+<
+
+212
+
+ends in 2 * these three pairs
+
+>
+
+213
+
+ends in 3 * form a pattern
+
+[
+
+312
+
+ends in 2 *
+
+]
+
+313
+
+ends in 3 *
+
+a
+
+262
+
+opposite to V (626)
+
+s
+
+163
+
+163 is prime
+
+A .
+
+161
+
+' a ' is a "graph" of the sequence 1-6-1
+
+V
+
+616
+
+V is a "graph" of the sequence 6-1-6
+
+3
+
+633
+
+' 6 "implies" 3 and 3, in some sense .
+
+~
+
+223
+
+. 2 + 2 is not 3
+
+3
+
+333
+
+'3’ looks like '3'
+
+¥
+
+626
+
+opposite to a; also a "graph" of 6-2-6
+
+
+636
+
+two dots, two sixes
+
+punc.
+
+611
+
+special number, as on Bell system (411, 911)
+
+
+Each symbol of TNT is matched up with a triplet composed of the digits 1, 2, 3,
+and 6, in a manner chosen for mnemonic value. I shall call each such triplet of digits a
+Godel codon, or codon for short. Notice that I have given no codon for b, c, d, or e; we
+are using austere TNT. There is a hidden motivation for this, which you will find out
+about in Chapter XVI. I will explain the bottom entry, "punctuation", in Chapter XIV.
+
+Now we can rewrite any string or rule of TNT in the new garb. Here, for instance,
+is Axiom 1 in the two notations, the old below the new:
+
+626,262,636,223,123,262,111,666
+
+V a : ~ S a = 0
+
+Conveniently, the standard convention of putting in a comma every third digit happens to
+coincide with our colons, setting them off for "easy" legibility.
+
+Here is the Rule of Detachment, in the new notation:
+
+RULE: If x and 212x6331213 are both theorems, then 1 is a theorem. Finally, here is an
+entire derivation taken from last Chapter, given in austere TNT and also transcribed into
+the new notation:
+
+
+Mumon and Godel
+
+
+268
+
+
+
+
+
+
+
+
+626,262,636,626262,163,636,362262,112,123,262,163,323,111,123,362,262,112,262,163,323 axiom:
+
+-¥ a :: ¥ a ' : (a + 5a' )=s (a+a ’)
+
+626,262.163,636,362,123,666,112,123,262,163,323,111,123,362,123,666,112,262,163,32 specification
+
+-¥a':(S0+sa') = S( so+a')
+
+
+362,123,666,112,123,666,323,1 11,123,362,123,666,112,666,323 specification
+
+( SO + 5 0 ) = S ( SO + 0 )
+
+626,262,636,362 262,112,666, 3 23,111,262 axiom
+
+¥• a : ( a + 0 ) = a
+
+362.123.666.112.666.323.111.123.666 specification
+
+( S 0 + 0 ) = S 0
+
+123,362,123,666,112,666,323,11 1,123,123,666 insert’12;
+
+S (S0+0 ) = sso
+
+362.123.666.112.123.666.323.111.123.123.666 transitivity
+
+(S0 + 50 ) = SSO
+
+
+Notice that I changed the name of the "Add S" rule to "Insert '123' ", since that is the
+typographical operation which it now legitimizes.
+
+This new notation has a pretty strange feel to it. You lose all sense o meaning; but
+if you had been brought up on it, you could read strings it this notation as easily as you do
+TNT. You would be able to look and, at glance, distinguish well-formed formulas from
+ill-formed ones. Naturally since it is so visual, you would think of this as a typographical
+operation but at the same time, picking out well-formed formulas in this notation i
+picking out a special class of integers , which have an arithmetical definition too.
+
+Now what about "arithmetizing" all the rules of inference? As matter stand, they
+are all still typographical rules. But wait! According to the Central Proposition, a
+typographical rule is really equivalent to al arithmetical rule. Inserting and moving digits
+in decimally represented numbers is an arithmetical operation, which can be carried out
+typographically. Just as appending a 'O' on the end is exactly the same as multiplying b,
+10, so each rule is a condensed way of describing a messy arithmetical operation.
+Therefore, in a sense, we do not even need to look for equivalent arithmetical rules,
+because all of the rules are already arithmetical!
+
+TNT-Numbers: A Recursively Enumerable Set of Numbers
+
+Looked at this way, the preceding derivation of the theorem
+"362,123,666,112,123,666,323,111,123,123,666" is a sequence of high] convoluted
+number-theoretical transformations, each of which acts on one or more input numbers,
+and yields an output number, which is, as before, called a producible number, or, to be
+more specific, a TNT -number. Some the arithmetical rules take an old TNT-number and
+increase it in a particular way, to yield a new TNT-number; some take an old TNT-
+number a and decrease it; other rules take two TNT-numbers, operate on each of them
+some odd way, and then combine the results into a new TNT-number
+and so on and so forth. And instead of starting with just one know: 'TNT-number, we
+have five initial TNT-numbers-one for each (austere axiom, of course. Arithmetized TNT
+is actually extremely similar to the
+
+
+Mumon and Godel
+
+
+269
+
+
+
+arithmetized MlU-system, only there are more rules and axioms, and to write out
+arithmetical equivalents explicitly would be a big bother-and quite unenlightening,
+incidentally. If you followed how it was done for the MlU-system, there ought to be no
+doubt on your part that it is quite analogous here.
+
+There is a new number-theoretical predicate brought into being by this
+"Godelization" of TNT: the predicate
+
+a is a TNT-number.
+
+For example, we know from the preceding derivation that
+362,123,666,112,123,666,323,111,123,123,666 is a TNT-number, while on the other
+hand, presumably 123,666,111,666 is not a TNT-number.
+
+Now it occurs to us that this new number-theoretical! predicate is expressible by
+some string of TNT with one free variable, say a. We could put a tilde in front, and that
+string would express the complementary notion
+
+a is not a TNT-number.
+
+Now if we replaced all the occurrences of a in this second string by the TNT-numeral for
+123,666,111,666-a numeral which would contain exactly 123,666,111,666 S's, much too
+long to write out-we would have a TNT-string which, just like MUMON, is capable of
+being interpreted on two levels. In the first place, that string would say
+
+123,666,111,666 is not a TNT-number.
+
+But because of the isomorphism which links TNT-numbers to theorems of TNT, there
+would be a second-level meaning of this string, which is:
+
+S0=0 is not a theorem of TNT.
+
+TNT Tries to Swallow Itself
+
+This unexpected double-entendre demonstrates that TNT contains strings which talk
+about other strings of TNT. In other words, the metalanguage in which we, on the
+outside, can speak about TNT, is at least partially imitated inside TNT itself. And this is
+not an accidental feature of TNT; it happens because the architecture of any formal
+system can be mirrored inside N (number theory). It is just as inevitable a feature of TNT
+as are the vibrations induced in a record player when it plays a record. It seems as if
+vibrations should come from the outside world-for instance, from jumping children or
+bouncing balls; but a side effect of producing sounds-and an unavoidable one-is that they
+wrap around and shake the very mechanism which produces them. It is no accident; it is a
+side effect which cannot be helped. It is in the nature of record players. And it is in the
+nature of any formalization of number theory that its metalanguage is embedded within it.
+
+
+Mumon and Godel
+
+
+270
+
+
+
+We can dignify this observation by calling it the Central Dogma of Mathematical
+Logic , and depicting it in a two-step diagram:
+
+TNT => N => meta-'TNT
+
+In words: a string of TNT has an interpretation in N; and a statement o may have a
+second meaning as a statement about TNT.
+
+G: A String Which Talks about Itself in Code
+
+This much is intriguing yet it is only half the story. The rest of the st involves an
+intensification of the self-reference. We are now at the st where the Tortoise was when he
+realized that a record could be m; which would make the phonograph playing it break-but
+now the quest is: "Given a record player, how do you actually figure out what to put the
+record?" That is a tricky matter.
+
+We want to find a string of TNT-which we'll call 'G'-which is ab itself, in the
+sense that one of its passive meanings is a sentence about G. particular the passive
+meaning will turn out to be
+
+
+"G is not a theorem of TNT."
+
+I should quickly add that G also has a passive meaning which is a statement of number
+theory; just like MUMON it is susceptible to being construed in least) two different
+ways. The important thing is that each passive mean is valid and useful and doesn't cast
+doubt on the other passive meaning in any way. (The fact that a phonograph playing a
+record can induce vibrations in itself and in the record does not diminish in any way the
+fact t those vibrations are musical sounds!)
+
+G's Existence Is What Causes TNT's Incompleteness
+
+The ingenious method of creating G, and some important concepts relating to TNT, will
+be developed in Chapters XIII and XIV; for now it is interesting to glance ahead, a bit
+superficially, at the consequences finding a self-referential piece of TNT. Who knows? It
+might blow up! In a sense it does. We focus down on the obvious question:
+
+Is G a theorem of TNT, or not?
+
+Let us be sure to form our own opinion on this matter, rather than rely G's opinion about
+itself. After all, G may not understand itself any be than a Zen master understands
+himself. Like MUMON, G may express a falsity. Like MU, G may be a nontheorem. We
+don't need to believe every possible string of TNT-only its theorems. Now let us use our
+power of reasoning to clarify the issue as best we can at this point.
+
+We will make our usual assumption: that TNT incorporates valid
+
+
+Mumon and Godel
+
+
+271
+
+
+
+methods of reasoning, and therefore that TNT never has falsities for theorems. In other
+words, anything which is a theorem of TNT expresses a truth. So if G were a theorem, it
+would express a truth, namely: "G is not a theorem". The full force of its self-reference
+hits us. By being a theorem, G would have to be a falsity. Relying on our assumption that
+TNT never has falsities for theorems, we'd be forced to conclude that G is not a theorem.
+This is all right; it leaves us, however, with a lesser problem. Knowing that G is not a
+theorem, we'd have to concede that G expresses a truth. Here is a situation in which TNT
+doesn't live up to our expectations-we have found a string which expresses a true
+statement yet the string is not a theorem. And in our amazement, we shouldn't lose track
+of the fact that G has an arithmetical interpretation, too-which allows us to summarize
+our findings this way:
+
+A string of TNT has been found; it expresses, unambiguously, a statement about
+certain arithmetical properties of natural numbers; moreover, by reasoning outside
+the system we can determine not only that the statement is a true one, but also that
+the string fails to be a theorem of TNT. And thus, if we ask TNT whether the
+statement is true, TNT says neither yes nor no.
+
+Is the Tortoise's string in the Mu Offering the analogue of G? Not quite. The
+analogue of the Tortoise's string is ~G. Why is this so? Well, let us think a moment about
+what -G says. It must say the opposite of what G says. G says, "G is not a theorem of
+TNT", so ~G must say "G is a theorem". We could rephrase both G and ~G this way:
+
+G: "I am not a theorem (of TNT)."
+
+~G: "My negation is a theorem (of TNT)."
+
+It is ~G which is parallel to the Tortoise's string, for that string spoke not about itself, but
+about the string which the Tortoise first proffered to Achilles — which had an extra knot
+on it (or one too few, however you want to look at it).
+
+Mumon Has the Last Word
+
+Mumon penetrated into the Mystery of the Undecidable anyone, in his concise poem on
+Joshu's MU:
+
+Has a dog Buddha-nature?
+
+This is the most serious question of all.
+
+If you say yes or no.
+
+You lose your own Buddha-nature.
+
+
+Mumon and Godel
+
+
+272
+
+
+
+Prelude
+
+
+Achilles and the Tortoise have come to the residence of their friend the Crab, to
+make the acquaintance of one of his friends, the Anteater. The introductions
+having been made, the four of them settle down to tea.
+
+Tortoise We have brought along a little something for you, Mr. Crab. Crab: That's most
+kind of you. But you shouldn't have.
+
+Tortoise: Just a token of our esteem. Achilles, would you like to give it to Mr. C?
+
+Achilles: Surely. Best wishes, Mr. Crab. I hope you enjoy it.
+
+(Achilles hands the Crab an elegantly wrapped present, square and very thin. The
+Crab begins unwrapping it.)
+
+Anteater: I wonder what it could be.
+
+Crab: We'll soon find out. (Completes the unwrapping, and pulls out the gif)t Two
+records! How exciting! But there's no label. Uh-oh-is this another of your "specials",
+Mr. T?
+
+Tortoise: If you mean a phonograph-breaker, not this time. But it is in fact a custom-
+recorded item, the only one of its kind in the entire world. In fact, it's never even been
+heard before-except, of course, when Bach played it.
+
+Crab: When Bach played it? What do you mean, exactly?
+
+Achilles: Oh, you are going to be fabulously excited, Mr. Crab, when Mr. T tells you
+what these records in fact are.
+
+Tortoise: Oh, you go ahead and tell him, Achilles.
+
+Achilles: May I? Oh, boy! I'd better consult my notes, then. (Pulls out a small filing card,
+and clears his voice.) Ahem. Would you be interested in hearing about the remarkable
+new result in mathematics, to which your records owe their existence?
+
+Crab: My records derive from some piece of mathematics? How curious Well, now that
+you've provoked my interest, I must hear about it.
+
+Achilles: Very well, then. (Pauses for a moment to sip his tea, then resumes) Have you
+heard of Fermat's infamous "Last Theorem"?
+
+Anteater: I'm not sure ... It sounds strangely familiar, and yet I can't qui place it.
+
+Achilles: It's a very simple idea. Pierre de Fermat, a lawyer by vocation b mathematician
+by avocation, had been reading in his copy of the class text Arithmetica by
+Diophantus, and came across a page containing the equation
+
+a 2 +b 2 =c 2
+
+
+Prelude
+
+
+275
+
+
+
+
+FIGURE 54. Mobius Strip II, by M. C. Escher (woodcut, 1963).
+
+
+Prelude
+
+
+276
+
+
+
+He immediately realized that this equation has infinitely many solutions a, b, c, and then
+wrote in the margin the following notorious comment:
+
+The equation
+
+
+a 11 +b n =c n
+
+has solutions in positive integers a, b, c, and n only when n = 2 (an then there are
+infinitely many triplets a, b, c which satisfy the equation); but there are no
+solutions for n > 2. I have discovered a truly marvelous proof of this statement,
+which, unfortunately, this margin is too small to contain.
+
+Ever since that day, some three hundred years ago, mathematicians have been vainly
+trying to do one of two things: either to I Fermat's claim, and thereby vindicate
+Fermat's reputation, whit though very high, has been somewhat tarnished by skeptics
+who he never really found the proof he claimed to have found-or e: refute the claim,
+by finding a counterexample: a set of four integers a, b, c, and n, with n > 2, which
+satisfy the equation. Until recently, every attempt in either direction had met with
+failure. 1 sure, the Theorem has been proven formally specific values of i particular,
+all n up to 125,000.
+
+Anteater: Shouldn't it be called a "Conjecture" rather than a "Theorem it's never been
+given a proper proof?
+
+Achilles: Strictly speaking, you're right, but tradition has kept it this i
+
+Crab: Has someone at last managed to resolve this celebrated quest Achilles: Indeed! In
+fact, Mr. Tortoise has done so, and as usual, by a wizardly stroke. He has not only
+found a PROOF of Fermat's Theorem (thus justifying its name as well as vindicating
+Fermat; also a COUNTEREXAMPFE, thus showing that the skeptics had good
+intuition!
+
+Crab: Oh my gracious! That is a revolutionary discovery.
+
+Anteater: But please don't leave us in suspense. What magical integer they, that satisfy
+Fermat's equation? I'm especially curious about the value of n.
+
+Achilles: Oh, horrors! I'm most embarrassed! Can you believe this? the values at home on
+a truly colossal piece of paper. Unfortunately was too huge to bring along. I wish I
+had them here to show to y( it's of any help to you, I do remember one thing-the value
+of n only positive integer which does not occur anywhere in the continued fraction for
+7T
+
+Crab: Oh, what a shame that you don't have them here. But there reason to doubt what
+you have told us.
+
+
+Prelude
+
+
+277
+
+
+
+FIGURE 55. Pierre de Fermat.
+
+
+
+Anteater: Anyway, who needs to see n written out decimally? Achilles has just told us
+how to find it. Well, Mr. T, please accept my hearty felicitations, on the occasion of
+your epoch-making discovery!
+
+Tortoise: Thank you. But what I feel is more important than the result itself is the
+practical use to which my result immediately led.
+
+Crab: I am dying to hear about it, since I always thought number theory was the Queen of
+Mathematics — the purest branch of mathematic — the one branch of mathematics
+which has No applications!
+
+Tortoise: You’re not the only one with that belief, but in fact it is quite impossible to
+make a blanket statement about when or how some branch-or even some individual
+Theorem-of pure mathematics will have important repercussions outside of
+mathematics. It is quite unpredictable-and this case is a perfect example of that
+phenomenon.
+
+Achilles: Mr. Tortoise's double-barreled result has created a breakthrough in the field of
+acoustico-retrieval!
+
+Anteater: What is acoustico-retrieval?
+
+Achilles: The name tells it all: it is the retrieval of acoustic information from extremely
+complex sources. A typical task of acoustico-retrieval is to reconstruct the sound
+which a rock made on plummeting into a lake from the ripples which spread out over
+the lake's surf ace.
+
+Crab: Why, that sounds next to impossible!
+
+Achilles: Not so. It is actually quite similar to what one’s brain does, when it reconstructs
+the sound made in the vocal cords of another person from the vibrations transmitted
+by the eardrum to the fibers in the cochlea.
+
+Crab: I see. But I still don’t see where number theory enters the picture, or what this all
+has to do with my new records.
+
+
+Prelude
+
+
+278
+
+
+
+
+Achilles: Well, in the mathematics of acoustico-retrieval, there arise rr questions which
+have to do with the number of solutions of cer Diophantine equations. Now Mr. T has
+been for years trying to fit way of reconstructing the sounds of Bach playing his
+harpsichord, which took place over two hundred years ago, from calculations in% ing
+the motions of all the molecules in the atmosphere at the pre time.
+
+Anteater: Surely that is impossible! They are irretrievably gone, g forever!
+
+Achilles: Thus think the nave ... But Mr. T has devoted many year this problem, and
+came to the realization that the whole thing hinged on the number of solutions to the
+equation
+
+
+a n +b n =c n
+
+
+in positive integers, with n > 2.
+
+Tortoise: I could explain, of course, just how this equation arises, but I’m sure it would
+bore you.
+
+Achilles: It turned out that acoustico-retrieval theory predicts that Bach sounds can be
+retrieved from the motion of all the molecule the atmosphere, provided that EITHER
+there exists at least one solution to the equation
+
+Crab: Amazing!
+
+Anteater: Fantastic!
+
+Tortoise: Who would have thought!
+
+Achilles: I was about to say, "provided that there exists EITHER such a solution OR a
+proof that there are tic) solutions!" And therefore, Mr. T, in careful fashion, set about
+working at both ends of the problem, simultaneously. As it turns out, the discovery of
+the counterexample was the key ingredient to finding the proof, so the one led directly
+to the other.
+
+Crab: How could that be?
+
+Tortoise: Well, you see, I had shown that the structural layout of any pr of Fermat's Last
+Theorem-if one existed-could be described by elegant formula, which, it so happened,
+depended on the values ( solution to a certain equation. When I found this second
+equation my surprise it turned out to be the Fermat equation. An amusing accidental
+relationship between form and content. So when I found the counterexample, all I
+needed to do was to use those numbers blueprint for constructing my proof that there
+were no solutions to equation. Remarkably simple, when you think about it. I can't
+imagine why no one had ever found the result before.
+
+Achilles: As a result of this unanticipatedly rich mathematical success, Mr. T was able to
+carry out the acoustico-retrieval which he had long dreamed of. And Mr. Crab's
+present here represents a palpable realization of all this abstract work.
+
+
+Prelude
+
+
+279
+
+
+
+Crab: Don't tell me it's a recording of Bach playing his own works for harpsichord!
+
+Achilles: I'm sorry, but I have to, for that is indeed just what it is! This is a set of two
+records of Johann Sebastian Bach playing all of his Well Tempered Clavier. Each
+record contains one of the two volumes of the Well-Tempered Clavier, that is to say,
+each record contains 24 preludes and fugues-one in each major and minor key.
+
+Crab: Well, we must absolutely put one of these priceless records on, immediately! And
+how can I ever thank the two of you?
+
+Tortoise: You have already thanked us plentifully, with this delicious tea which you have
+prepared.
+
+(The Crab slides one of the records out of its jacket, and puts it on. The sound of
+an incredibly masterful harpsichordist fills the room, in the highest imaginable
+fidelity. One even hears-or is it one's imagination?-the soft sounds of Bach singing
+to himself as he plays ...)
+
+Crab: Would any of you like to follow along in the score? I happen to have a unique
+edition of the Well-Tempered Clavier, specially illuminated by a teacher of mine who
+happens also to be an unusually fine calligrapher. Tortoise: I would very much enjoy
+that.
+
+(The Crab goes to his elegant glass-enclosed wooden bookcase, opens the doors, and
+draws out two large volumes.)
+
+Crab: Here you are, Mr. Tortoise. I've never really gotten to know all the beautiful
+illustrations in this edition. Perhaps your gift will provide the needed impetus for me
+to do so.
+
+Tortoise: I do hope so.
+
+Anteater: Have you ever noticed how in these pieces the prelude always sets the mood
+perfectly for the following fugue?
+
+Crab: Yes. Although it may be hard to put it into words, there is always some subtle
+relation between the two. Even if the prelude and fugue do not have a common
+melodic subject, there is nevertheless always some intangible abstract quality which
+underlies both of them, binding them together very strongly.
+
+Tortoise: And there is something very dramatic about the few moments of silent suspense
+hanging between prelude and fugue-that moment where the the theme of the fugue is
+about to ring out, in single tones, and then to join with itself in ever-increasingly
+complex levels of weird, exquisite harmony.
+
+Achilles: I know just what you mean. There are so many preludes and fugues which I
+haven't yet gotten to know, and for me that fleeting interlude of silence is very
+exciting; it's a time when I try to second-guess old Bach. For example, I always
+wonder what the fugue's tempo will be: allegro, or adagio? Will it be in 6/8, or 4/4?
+Will it have three voices, or five-or four? And then, the first voice starts ... Such an
+exquisite moment.
+
+
+Prelude
+
+
+280
+
+
+
+Crab: Ah, yes, well do I remember those long-gone days of my youth, days when I
+thrilled to each new prelude and fugue, filled with excitement of their novelty and
+beauty and the many unexpected' surprises which they conceal.
+
+Achilles: And now? Is that thrill all gone?
+
+Crab: It's been supplanted by familiarity, as thrills always will be. But that familiarity
+there is also a kind of depth, which has its own compensations. For instance, I find
+that there are always new surprises whit hadn't noticed before.
+
+Achilles: Occurrences of the theme which you had overlooked?
+
+Crab: Perhaps-especially when it is inverted and hidden among several other voices, or
+where it seems to come rushing up from the dept out of nowhere. But there are also
+amazing modulations which ii marvelous to listen to over and over again, and wonder
+how old B2 dreamt them up.
+
+Achilles: I am very glad to hear that there is something to look forward after I have been
+through the first flush of infatuation with the Well Tempered Clavier -although it also
+makes me sad that this stage cot not last forever and ever.
+
+Crab: Oh, you needn't fear that your infatuation will totally die. One the nice things about
+that sort of youthful thrill is that it can always resuscitated, just when you thought it
+was finally dead. It just takes the right kind of triggering from the outside.
+
+Achilles: Oh, really? Such as what?
+
+Crab: Such as hearing it through the ears, so to speak, of someone whom it is a totally
+new experience-someone such as you, Achilles. Somehow the excitement transmits
+itself, and I can feel thrilled again.
+
+Achilles: That is intriguing. The thrill has remained dormant somewhere inside you, but
+by yourself, you aren't able to fish it up out of your subconscious.
+
+Crab: Exactly. The potential of reliving the thrill is "coded", in sot unknown way, in the
+structure of my brain, but I don't have the power to summon it up at will; I have to
+wait for chance circumstance trigger it.
+
+Achilles: I have a question about fugues which I feel a little embarrass about asking, but
+as I am just a novice at fugue-listening, I was wondering if perhaps one of you
+seasoned fugue-listeners might help me learning .. .
+
+Tortoise: I'd certainly like to offer my own meager knowledge, if it might prove of some
+assistance.
+
+Achilles: Oh, thank you. Let me come at the question from an angle. Are you familiar
+with the print called Cube with Magic Ribbons , by M. Escher?
+
+Tortoise: In which there are circular bands having bubble-like distortions which, as soon
+as you've decided that they are bumps, seem to turn it dents-and vice versa?
+
+
+Prelude
+
+
+281
+
+
+
+
+FIGURE 56. Cube with Magic Ribbons, by M. C. Escher (lithograph, 1957).
+
+Achilles: Exactly.
+
+Crab: I remember that picture. Those little bubbles always seem to flip back and forth
+between being concave and convex, depending on the direction that you approach
+them from. There’s no way to see them simultaneously as concave AND convex-
+somehow one’s brain doesn't allow that. There are two mutually exclusive "modes" in
+which one can perceive the bubbles.
+
+Achilles: Just so. Well, I seem to have discovered two somewhat analogous modes in
+which I can listen to a fugue. The modes are these: either to follow one individual
+voice at a time, or to listen to the total effect of all of them together, without trying to
+disentangle one from another. I have tried out both of these modes, and, much to my
+frustration, each one of them shuts out the other. It’s simply not in my power to follow
+the paths of individual voices and at the same time to hear the whole effect. I find that
+I flip back and forth between one mode and the other, more or less spontaneously and
+involuntarily.
+
+
+Prelude
+
+
+282
+
+
+
+Anteater: Just as when you look at the magic bands, eh?
+
+Achilles: Yes. I was just wondering ... does my description of they modes of fugue¬
+listening brand me unmistakably as a naive, inexperienced listener, who couldn't even
+begin to grasp the deeper mo, perception which exist beyond his ken?
+
+Tortoise: No, not at all, Achilles. I can only speak for myself, but I to myself shifting
+back and forth from one mode to the other without exerting any conscious control
+over which mode should he dominant don't know if our other companions here have
+also experience( thing similar.
+
+Crab: Most definitely. It's quite a tantalizing phenomenon, since you feel that the essence
+of the fugue is flitting about you, and you can't grasp all of it, because you can't quite
+make yourself function ways at once.
+
+Anteater: Fugues have that interesting property, that each of their voices is a piece of
+music in itself; and thus a fugue might be thought o collection of several distinct
+pieces of music, all based on one theme, and all played simultaneously. And it is up to
+the listener subconscious) to decide whether it should be perceived as a unit, c
+collection of independent parts, all of which harmonize.
+
+Achilles: You say that the parts are "independent", yet that can't be literally true. There
+has to be some coordination between them, otherwise when they were put together
+one would just have an unsystematic clashing of tones-and that is as far from the truth
+as could b,
+
+Anteater: A better way to state it might be this: if you listened to each on its own, you
+would find that it seemed to make sense all by its could stand alone, and that is the
+sense in which I meant that it is independent. But you are quite right in pointing out
+that each of individually meaningful lines fuses with the others in a highly nonrandom
+way, to make a graceful totality. The art of writing a beautiful fugue lies precisely in
+this ability, to manufacture several diff lines, each one of which gives the illusion of
+having been written I own beauty, and yet which when taken together form a whole, ,
+does not feel forced in any way. Now, this dichotomy between he a fugue as a whole,
+and hearing its component voices, is a part: example of a very general dichotomy,
+which applies to many kit structures built up from lower levels.
+
+Achilles: Oh, really? You mean that my two "modes" may have some general type of
+applicability, in situations other than fugue-listening?
+
+Anteater: Absolutely.
+
+Achilles: I wonder how that could be. I guess it has to do with alternating between
+perceiving something as a whole, and perceiving it as a collection of parts. But the
+only place I have ever run into that dichotomy is in listening to fugues.
+
+Tortoise: Oh, my, look at this! I just turned the page while following the music, and came
+across this magnificent illustration facing the page of the fugue.
+
+
+Prelude
+
+
+283
+
+
+
+Crab: I have never seen that illustration before. Why don't you pass it 'round?
+
+(The Tortoise passes the book around. Each of the foursome looks at it in a
+characteristic way-this one from afar, that one from close up, everyone tipping his
+head this way and that in puzzlement. Finally it has made the rounds, and returns
+to the Tortoise, who peers at it rather intently.)
+
+Achilles: Well, I guess the prelude is just about over. I wonder if, as I listen to this fugue,
+I will gain any more insight into the question, "What is the right way to listen to a
+fugue: as a whole, or as the sum of its parts?"
+
+TTortoise: Listen carefully, and you will!
+
+(The prelude ends. There is a moment of silence; and ...
+
+
+[ATTACCA]
+
+
+Prelude
+
+
+284
+
+
+
+CHAPTER X
+
+
+Levels of Description,
+and Computer Systems
+
+
+Levels of Description
+
+
+GOD EL'S STRING G, and a Bach fugue: they both have the property that they can be
+understood on different levels. We are all familiar with this kind of thing; and yet in some
+cases it confuses us, while in others w handle it without any difficulty at all. For example,
+we all know that w human beings are composed of an enormous number of cells (around
+twenty-five trillion), and therefore that everything we do could in principle be described
+in terms of cells. Or it could even be described on the level c molecules. Most of us
+accept this in a rather matter-of-fact way; we go t the doctor, who looks at us on lower
+levels than we think of ourselves. W read about DNA and "genetic engineering" and sip
+our coffee. We seem t have reconciled these two inconceivably different pictures of
+ourselves simply by disconnecting them from each other. We have almost no way t relate
+a microscopic description of ourselves to that which we feel ourselves to be, and hence it
+is possible to store separate representations of ourselves in quite separate "compartments"
+of our minds. Seldom do we have to fir back and forth between these two concepts of
+ourselves, wondering "How can these two totally different things be the same me?"
+
+Or take a sequence of images on a television screen which show Shirley
+MacLaine laughing. When we watch that sequence, we know that we are actually looking
+not at a woman, but at sets of flickering dots on a flat surface. We know it, but it is the
+furthest thing from our mind. We have these two wildly opposing representations of what
+is on the screen, but that does not confuse us. We can just shut one out, and pay attention
+to th other-which is what all of us do. Which one is "more real"? It depends o; whether
+you're a human, a dog, a computer, or a television set.
+
+Chunking and Chess Skill
+
+One of the major problems of Artificial Intelligence research is to figure out how to
+bridge the gap between these two descriptions; how to construe a system which can
+accept one level of description, and produce the other One way in which this gap enters
+Artificial Intelligence is well illustrated b the progress in knowledge about how to
+program a computer to play goof chess. It used to be thought in the 1950's and on into the
+1960's-that the
+
+
+Levels of Description, and Computer Systems
+
+
+285
+
+
+
+trick to making a machine play well was to make the machine look further ahead into the
+branching network of possible sequences of play than any chess master can. However, as
+this goal gradually became attained, the level of computer chess did not have any sudden
+spurt, and surpass human experts. In fact, a human expert can quite soundly and
+confidently trounce the best chess programs of this day.
+
+The reason for this had actually been in print for many years. In the 1940's, the
+Dutch psychologist Adriaan de Groot made studies of how chess novices and chess
+masters perceive a chess situation. Put in their starkest terms, his results imply that chess
+masters perceive the distribution of pieces in chunks. There is a higher-level description
+of the board than the straightforward "white pawn on K5, black rook on Q6" type of
+description, and the master somehow produces such a mental image of the board. This
+was proven by the high speed with which a master could reproduce an actual position
+taken from a game, compared with the novice's plodding reconstruction of the position,
+after both of them had had five-second glances at the board. Highly revealing was the fact
+that masters' mistakes involved placing whole groups of pieces in the wrong place, which
+left the game strategically almost the same, but to a novice's eyes, not at all the same. The
+clincher was to do the same experiment but with pieces randomly assigned to the squares
+on the board, instead of copied from actual games. The masters were found to be simply
+no better than the novices in reconstructing such random boards.
+
+The conclusion is that in normal chess play, certain types of situation recur-
+certain patterns-and it is to those high-level patterns that the master is sensitive. He thinks
+on a different level from the novice; his set of concepts is different. Nearly everyone is
+surprised to find out that in actual play, a master rarely looks ahead any further than a
+novice does-and moreover, a master usually examines only a handful of possible moves!
+The trick is that his mode of perceiving the board is like a filter: he literally does not see
+bad moves when he looks at a chess situation-no more than chess amateurs see illegal
+moves when they look at a chess situation. Anyone who has played even a little chess has
+organized his perception so that diagonal rook-moves, forward captures by pawns, and so
+forth, are never brought to mind. Similarly, master-level players have built up higher
+levels of organization in the way they see the board; consequently, to them, bad moves
+are as unlikely to come to mind as illegal moves are, to most people. This might be called
+implicit pruning of the giant branching tree of possibilities. By contrast, explicit pruning
+would involve thinking of a move, and after superficial examination, deciding not to
+pursue examining it any further.
+
+The distinction can apply just as well to other intellectual activities — for instance,
+doing mathematics. A gifted mathematician doesn't usually think up and try out all sorts
+of false pathways to the desired theorem, as less gifted people might do; rather, he just
+"smells" the promising paths, and takes them immediately.
+
+Computer chess programs which rely on looking ahead have not been taught to
+think on a higher level; the strategy has just been to use brute
+
+
+Levels of Description, and Computer Systems
+
+
+286
+
+
+
+force look-ahead, hoping to crush all types of opposition. But it h worked. Perhaps
+someday, a look-ahead program with enough brute ,gill indeed overcome the best human
+players-but that will be a intellectual gain, compared to the revelation that intelligence de
+crucially on the ability to create high-level descriptions of complex such as chess boards,
+television screens, printed pages, or painting
+
+Similar Levels
+
+usually, we are not required to hold more than one level of understanding of a situation in
+our minds at once. Moreover, the different descriptions a single system are usually so
+conceptually distant from each other tl was mentioned earlier, there is no problem in
+maintaining them both are just maintained in separate mental compartments. What is
+confusing though, is when a single system admits of two or more descriptions different
+levels which nevertheless resemble each other in some way. we find it hard to avoid
+mixing levels when we think about the system can easily get totally lost.
+
+Undoubtedly this happens when we think about our psychology-for instance,
+when we try to understand people's motivations: for various actions. There are many
+levels in the human m structure-certainly it is a system which we do not understand very
+we But there are hundreds of rival theories which tell why people act the way they do,
+each theory based on some underlying assumptions about he down in this set of levels
+various kinds of psychological "forces" are f( Since at this time we use pretty much the
+same kind of language f mental levels, this makes for much level-mixing and most
+certain] hundreds of wrong theories. For instance, we talk of "drives"-for se power, for
+fame, for love, etc., etc.-without knowing where these drives come from in the human
+mental structure. Without belaboring the pc simply wish to say that our confusion about
+who we are is certainly r( to the fact that we consist of a large set of levels, and we use
+overlapping language to describe ourselves on all of those levels.
+
+Computer Systems
+
+There is another place where many levels of description coexist for a system, and where
+all the levels are conceptually quite close to one an( I am referring to computer systems.
+When a computer program is ping, it can be viewed on a number of levels. On each level,
+the description is given in the language of computer science, which makes all the de
+descriptions similar in some ways to each other-yet there are extremely imp( differences
+between the views one gets on the different levels. At the 1 level, the description can be
+so complicated that it is like the dot-description of a television picture. For some
+purposes, however, this is by far the important view. At the highest level, the description
+is greatly chunked and
+
+
+Levels of Description, and Computer Systems
+
+
+287
+
+
+
+takes on a completely different feel, despite the fact that many of the same concepts
+appear on the lowest and highest levels. The chunks on the high-level description are like
+the chess expert's chunks, and like the chunked description of the image on the screen:
+they summarize in capsule form a number of things which on lower levels are seen as
+separate. (See Fig. 57.) Now before things become too abstract, let us pass on to the
+
+
+
+FIGURE 57. The idea of "chunking": a group of items is reperceived as a single "chunk".
+The chunk's boundary is a little like a cell membrane or a national border: it establishes
+a separate identity for the cluster within. According to context, one may wish to ignore
+the chunk's internal structure or to take it into account.
+
+concrete facts about computers, beginning with a very quick skim of what a computer
+system is like on the lowest level. The lowest level? Well, not really, for I am not going
+to talk about elementary particles-but it is the lowest level which we wish to think about.
+
+At the conceptual rock-bottom of a computer, we find a memory, a central
+processing unit (CPU), and some input-output (I/O) devices. Let us first describe the
+memory. It is divided up into distinct physical pieces, called words. For the sake of
+concreteness, let us say there are 65,536 words of memory (a typical number, being 2 to
+the 16th power). A word is further divided into what we shall consider the atoms of
+computer science-bits. The number of bits in a typical word might be around thirty-six.
+Physically, a bit is just a magnetic "switch" that can be in either of two positions.
+
+
+
+□
+
+«»i \l<>
+
+0
+
+E
+
+FH
+
+0
+
+□
+
+
+
+
+E
+
+01
+
+E
+
+0
+
+n | \ 1
+
+E
+
+E
+
+7i]7>
+
+□
+
+0
+
+0
+
+
+- a word of 36 bits —
+
+
+Levels of Description, and Computer Systems
+
+
+288
+
+
+
+you could call the two positions "up" and "down", or "x" and "o", o and "0" ... The third
+is the usual convention. It is perfectly fine, but i the possibly misleading effect of making
+people think that a comp deep down, is storing numbers. This is not true. A set of thirty-
+six bits not have to be thought of as a number any more than two bits has i thought of as
+the price of an ice cream cone. Just as money can do va things depending on how you use
+it, so a word in memory can serve r functions. Sometimes, to be sure, those thirty-six bits
+will indeed repn a number in binary notation. Other times, they may represent thin dots
+on a television screen. And other times, they may represent a letters of text. How a word
+in memory is to be thought of depends eni on the role that this word plays in the program
+which uses it. It ma course, play more than one role-like a note in a canon.
+
+Instructions and Data
+
+There is one interpretation of a word which I haven't yet mentioned, that is as an
+instruction. The words of memory contain not only data t acted on, but also the program
+to act on the data. There exists a lin repertoire of operations which can be carried out by
+the central proce5 unit-the CPU-and part of a word, usually its first several bits-is it
+pretable as the name of the instruction-type which is to be carried What do the rest of the
+bits in a word-interpreted-as-instruction stand Most often, they tell which other words in
+memory are to be acted upoi other words, the remaining bits constitute a pointer to some
+other wor( words) in memory. Every word in memory has a distinct location, li house on
+a street; and its location is called its address. Memory may have "street", or many
+"streets"-they are called "pages". So a given wo addressed by its page number (if memory
+is paged) together wit position within the page. Hence the "pointer" part of an instruction
+i numerical address of some word(s) in memory. There are no restric on the pointer, so an
+instruction may even "point" at itself, so that whet executed, it causes a change in itself to
+be made.
+
+How does the computer know what instruction to execute at any € time? This is kept
+track of in the CPU. The CPU has a special pointer w points at (i.e., stores the address of)
+the next word which is to be inter ed as an instruction. The CPU fetches that word from
+memory, and c it electronically into a special word belonging to the CPU itself. (Wor the
+CPU are usually not called "words", but rather, registers.) Then the executes that
+instruction. Now the instruction may call for any of a number of types of operations to be
+carried out. Typical ones include:
+
+ADD the word pointed to in the instruction, to a register.
+
+(In this case, the word pointed to is obviously interpreted as number.)
+
+
+Levels of Description, and Computer Systems
+
+
+289
+
+
+
+PRINT the word pointed to in the instruction, as letters.
+
+(In this case, the word is obviously interpreted not as a number, but as a
+string of letters.)
+
+JUMP to the word pointed to in the instruction.
+
+(In this case, the CPU is being told to interpret that particular word as its
+next instruction.)
+
+Unless the instruction explicitly dictates otherwise, the CPU will pick up
+the very next word and interpret it as an instruction. In other words, the CPU
+assumes that it should move down the "street" sequentially, like a mailman,
+interpreting word after word as an instruction. But this sequential order can be
+broken by such instructions as the JUMP instruction, and others.
+
+Machine Language v.v. Assembly language
+
+This is a very brief sketch of machine language. In this language, the types of
+operations which exist constitute a finite repertoire which cannot be extended.
+Thus all programs, no matter how large and complex, must be made out of
+compounds of these types of instructions. Looking at a program written in
+machine language is vaguely comparable to looking at a DNA molecule atom by
+atom. If you glance back to Fig. 41, showing the nucleotide sequence of a DNA
+molecule—and then if you consider that each nucleotide contains two dozen atoms
+or so-and if you imagine trying to write the DNA, atom by atom, for a small virus
+(not to mention a human being !)-then you will get a feeling for what it is like to
+write a complex program in machine language, and what it is like to try to grasp
+what is going on in a program if you have access only to its machine language
+description. ,
+
+It must be mentioned, however, that computer programming was
+originally done on an even lower level, if possible, than that of machine language-
+-namely, connecting wires to each other, so that the proper operations were "hard¬
+wired" in. This is so amazingly primitive by modern standards that it is painful
+even to' imagine. Yet undoubtedly the people who first did it experienced as much
+exhilaration as the pioneers of modern computers ever do .. .
+
+We now wish to move to a higher level of the hierarchy of levels of
+description of programs. This is the assembly language level. There is not a
+gigantic spread between assembly language and machine language; indeed, the
+step is rather gentle. In essence, there is a one-to-one correspondence between
+assembly language instructions and machine language instructions. The idea of
+assembly language is to "chunk" the individual machine language instructions, so
+that instead of writing the sequence of bits "010111000" when you want an
+instruction which adds one number to another, you simply write ADD, and then
+instead of giving the address in binary representation, you can refer to the word in
+memory by a name.
+
+
+Levels of Description, and Computer Systems
+
+
+290
+
+
+
+Therefore, a program in assembly language is very much like a machine language
+program made legible to humans. You might compare the machine language
+version of a program to a TNT-derivation done in the obscure Godel-numbered
+notation, and the assembly language version to the isomorphic TNT-derivation,
+done in the original TNT-notation, which is much easier to understand. Or, going
+back to the DNA image, we can liken the difference between machine language
+and assembly language to the difference between painfully specifying each
+nucleotide, atom by atom, and specifying a nucleotide by simply giving its name
+(i.e., 'A', 'G', 'C', or 'T'). There is a tremendous saving of labor in this very
+simple "chunking" operation, although conceptually not much has been changed.
+
+Programs That Translate Programs
+
+Perhaps the central point about assembly language is not its differences from
+machine language, which are not that enormous, but just the key idea that
+programs could be written on a different level at all\ Just think about it: the
+hardware is built to "understand" machine language programs-sequences of bits-
+but not letters and decimal numbers. What happens when hardware is fed a
+program in assembly language% It is as if you tried to get a cell to accept a piece
+of paper with the nucleotide sequence written out in letters of the alphabet, instead
+of in chemicals. What can a cell do with a piece of paper? What can a computer
+do with an assembly language program?
+
+And here is the vital point: someone can write, in machine language, a
+translation program. This program, called an assembler , accepts mnemonic
+instruction names, decimal numbers, and other convenient abbreviations which a
+programmer can remember easily, and carries out the conversion into the
+monotonous but critical bit-sequences. After the assembly language program has
+been assembled (i.e., translated), it is run-ox rather, its machine language
+equivalent is run. But this is a matter of terminology. Which level program is
+running? You can never go wrong if you say that the machine language program
+is running, for hardware is always involved when any program runs-but it is also
+quite reasonable to think of the running program in terms of assembly language.
+For instance, you might very well say, "Right now, the CPU is executing a JUMP
+instruction", instead of saying, "Right now, the CPU is executing a ' 1 11010000'
+instruction". A pianist who plays the notes G-E-B E-G-B is also playing an
+arpeggio in the chord of E minor. There is no reason to be reluctant about
+describing things from a higher-level point of view. So one can think of the
+assembly language program running concurrently with the machine language
+program. We have two modes of describing what the CPU is doing.
+
+
+Levels of Description, and Computer Systems
+
+
+291
+
+
+
+Higher-Level Languages, Compilers, and Interpreters
+
+The next level of the hierarchy carries much further the extremely powerful idea
+of using the computer itself to translate programs from a high level into lower
+levels. After people had programmed in assembly language for a number of years,
+in the early 1950's, they realized that there were a number of characteristic
+structures which kept reappearing in program after program. There seemed to be,
+just as in chess, certain fundamental patterns which cropped up naturally when
+human beings tried to formulate algorithms—exact descriptions of processes they
+wanted carried out. In other words, algorithms seemed to have certain higher-
+level components, in terms of which they could be much more easily and
+esthetically specified than in the very restricted machine language, or assembly
+language. Typically, a high-level algorithm component consists not of one or two
+machine language instructions, but of a whole collection of them, not necessarily
+all contiguous in memory. Such a component could be represented in a higher-
+level language by a single item-a chunk.
+
+Aside from standard chunks-the newly discovered components out of
+which all algorithms can be built-people realized that almost all programs contain
+even larger chunks-superchunks, so to speak. These superchunks differ from
+program to program, depending on the kinds of high-level tasks the j program is
+supposed to carry out. We discussed superchunks in Chapter V, calling them by
+their usual names: "subroutines" and "procedures". It was clear that a most
+powerful addition to any programming language would be the ability to define
+new higher-level entities in terms of previously known ones, and then to call them
+by name. This would build the chunking operation right into the language. Instead
+of there being a determinate repertoire of instructions out of which all programs
+had to be explicitly assembled, the programmer could construct his own modules,
+each with its own name, each usable anywhere inside the program, just as if it had
+been a built-in feature of the language. Of course, there is no getting away from
+the fact that down below, on a machine language level, everything would still be
+composed of the same old machine language instructions, but that would not be
+explicitly visible to the highlevel programmer; it would be implicit.
+
+The new languages based on these ideas were called compiler languages.
+One of the earliest and most elegant was called "Algol", for "Algorithmic
+Language". Unlike the case with assembly language, there is no straightforward
+one-to-one correspondence between statements in Algol and machine language
+instructions. To be sure, there is still a type of mapping from Algol into machine
+language, but it is far more "scrambled" than that between assembly language and
+machine language. Roughly speaking, an Algol program is to its machine
+language translation as a word problem in an elementary algebra text is to the
+equation it translates into. (Actually, getting from a word problem to an equation
+is far more complex, but it gives some inkling of the types of "unscrambling" that
+have to be carried out in translating from a high-level language to a lower-level
+Ian
+
+
+Levels of Description, and Computer Systems
+
+
+292
+
+
+
+guage.) In the mid-1950's, successful programs called compilers were written
+whose function was to carry out the translation from compiler languages to
+machine language.
+
+Also, interpreters were invented. Like compilers, interpreters translate
+from high-level languages into machine language, but instead of translating all the
+statements first and then executing the machine code, they read one line and'
+execute it immediately. This has the advantage that a user need not have written a
+complete program to use an interpreter. He may invent his program line by line,
+and test it out as he goes along. Thus, an interpreter is to a compiler as a
+simultaneous interpreter is to a translator of a written speech. One of the most
+important and fascinating of all computer languages is LISP (standing for "List
+Processing"), which was invented by John McCarthy around the time Algol was
+invented. Subsequently, LISP has enjoyed great popularity with workers in
+Artificial Intelligence.
+
+There is one interesting difference between the way interpreters work and
+compilers work. A compiler takes input (a finished Algol program, for instance)
+and produces output (a long sequence of machine language instructions). At this
+point, the compiler has done its duty. The output is then given to the computer to
+run. By contrast, the interpreter is constantly running while the programmer types
+in one LISP statement after another, and each one gets executed then' and there.
+But this doesn't mean that each statement gets first translated, then executed, for
+then an interpreter would be nothing but a line-by-line compiler. Instead, in an
+interpreter, the operations of reading a new line, "understanding" it, and executing
+it are intertwined: they occur simultaneously.
+
+Here is the idea, expanded a little more. Each time a new line of LISP is
+typed in, the interpreter tries to process it. This means that the interpreter jolts into
+action, and certain (machine language) instructions inside it get executed.
+Precisely which ones get executed depends on the LISP statement itself, of
+course. There are many JUMP instructions inside the interpreter, so that the new
+line of LISP may cause control to move around in a complex way-forwards,
+backwards, then forwards again, etc.. Thus, each LISP statement gets converted
+into a "pathway" inside the interpreter, and the act of following that pathway
+achieves the desired effect.
+
+Sometimes it is helpful to think of the LISP statements as mere pieces of
+data which are fed sequentially to a constantly running machine language
+program (the LISP interpreter). When you think of things this way, you get a
+different image of the relation between a program written in a higher-level
+language and the machine which is executing it.
+
+Bootstrapping
+
+Of course a compiler, being itself a program, has to be written in some language.
+The first compilers were written in assembly language, rather than machine
+language, thus taking full advantage of the already ac-
+
+
+Levels of Description, and Computer Systems
+
+
+293
+
+
+
+omplished first step up from machine language. A summary of these rather tricky
+concepts is presented in Figure 58.
+
+
+
+FIGURE 58. Assemblers and
+compilers are both translators into
+machine language. This is indicated
+by the solid lines. Moreover, since
+they are themselves programs, they
+are originally written in a language
+also. The wavy lines indicate that aa
+compiler can be written in assembly
+language, and an assembler in
+machine language.
+
+
+Now as sophistication increased, people realized that a partially written compiler
+could be used to compile extensions of itself. In other words, once i certain
+minimal core of a compiler had been written, then that minimal compiler could
+translate bigger compilers into machine language-which n turn could translate yet
+bigger compilers, until the final, full-blown :compiler had been compiled. This
+process is affectionately known as 'bootstrapping"-for obvious reasons (at least if
+your native language is English it is obvious). It is not so different from the
+attainment by a child of a critical level of fluency in his native language, from
+which point on his 'vocabulary and fluency can grow by leaps and bounds, since
+he can use language to acquire new language.
+
+Levels on Which to Describe Running Programs
+
+Compiler languages typically do not reflect the structure of the machines which
+will run programs written in them. This is one of their chief advantages over the
+highly specialized assembly and machine languages. Of course, when a compiler
+language program is translated into machine language, the resulting program is
+machine-dependent. Therefore one can describe a program which is being
+executed in a machine-independent way or a machine-dependent way. It is like
+referring to a paragraph in a book by its subject matter (publisher-independent), or
+its page number and position on the page (publisher-dependent).
+
+As long as a program is running correctly, it hardly matters how you
+describe it or think of its functioning. It is when something goes wrong that
+
+
+Levels of Description, and Computer Systems
+
+
+294
+
+
+
+
+
+it is important to be able to think on different levels. If, for instance, the machine
+is instructed to divide by zero at some stage, it will come to a halt and let the user
+know of this problem, by telling where in the program the questionable event
+occurred. However, the specification is often given on a lower level than that in
+which the programmer wrote the program. Here are three parallel descriptions of
+a program grinding to a halt:
+
+Machine Language Level:
+
+"Execution of the program stopped in location 1110010101110111"
+Assembly Language Level*:
+
+"Execution of the program stopped when the DIV (divide) instruction was
+hit"
+
+Compiler Language Level:
+
+"Execution of the program stopped during evaluation of the algebraic
+expression '(A + B)/Z'
+
+One of the greatest problems for systems programmers (the people who write
+compilers, interpreters, assemblers, and other programs to be used by many
+people) is to figure out how to write error-detecting routines in such a way that
+the messages which they feed to the user whose program has a "bug" provide
+high-level, rather than low-level, descriptions of the problem. It is an interesting
+reversal that when something goes wrong in a genetic "program" (e.g., a
+mutation), the "bug" is manifest only to people on a high level-namely on the
+phenotype level, not the genotype level. Actually, modern biology uses mutations
+as one of its principal windows onto genetic processes, because of their multilevel
+traceability.
+
+
+Microprogramming and Operating Systems
+
+In modern computer systems, there are several other levels of the hierarchy. For
+instance, some systems-often the so-called "microcomputers" come with machine
+language instructions which are even more rudimentary than the instruction to add
+a number in memory to a number in a register. It is up to the user to decide what
+kinds of ordinary machine-level instructions he would like to be able to program
+in; he "microprograms" these instructions in terms of the "micro-instructions"
+which are available. Then the "higher-level machine language" instructions which
+he has designed may be burned into the circuitry and become hard-wired,
+although they need not be. Thus microprogramming allows the user to step a little
+below the conventional machine language level. One of the consequences is that a
+computer of one manufacturer can be hard-wired (via microprogramming) so as
+to have the same machine language instruction set as a computer of the same, or
+even another, manufacturer. The microprogrammed computer is said to be
+"emulating" the other computer. Then there is the level of the operating system,
+which fits between the
+
+
+Levels of Description, and Computer Systems
+
+
+295
+
+
+
+machine language program and whatever higher level the user is programming in.
+The operating system is itself a program which has the functions of shielding the
+bare machine from access by users (thus protecting the system), and also of
+insulating the programmer from the many extremely intricate and messy problems
+of reading the program, calling a translator, running the translated program,
+directing the output to the proper channels at the proper time, and passing control
+to the next user. If there are several users "talking" to the same CPU at once, then
+the operating system is the program that shifts attention from one to the other in
+some orderly fashion. The complexities of operating systems are formidable
+indeed, and I shall only hint at them by the following analogy.
+
+Consider the first telephone system. Alexander Graham Bell could phone
+his assistant in the next room: electronic transmission of a voice! Now that is like
+a bare computer minus operating system: electronic computation! Consider now a
+modern telephone system. You have a choice of other telephones to connect to.
+Not only that, but many different calls can be handled simultaneously. You can
+add a prefix and dial into different areas. You can call direct, through the
+operator, collect, by credit card, person-to-person, on a conference call. You can
+have a call rerouted or traced. You can get a busy signal. You can get a siren-like
+signal that says that the number you dialed isn't "well-formed", or that you have
+taken too in long in dialing. You can install a local switchboard so that a group of
+phones are all locally connected—etc., etc. The list is amazing, when you think of
+how much flexibility there is, particularly in comparison to the erstwhile miracle
+of a "bare" telephone. Now sophisticated operating systems carry out similar
+traffic-handling and level-switching operations with respect to users and their
+programs. It is virtually certain that there are somewhat parallel things which take
+place in the brain: handling of many stimuli at the same time; decisions of what
+should have priority over what and for how long; instantaneous "interrupts"
+caused by emergencies or other unexpected occurrences; and so on.
+
+Cushioning the User and Protecting the System
+
+The many levels in a complex computer system have the combined effect of
+"cushioning" the user, preventing him from having to think about the many lower-
+level goings-on which are most likely totally irrelevant to him anyway. A
+passenger in an airplane does not usually want to be aware of the levels of fuel in
+the tanks, or the wind speeds, or how many chicken dinners are to be served, or
+the status of the rest of the air traffic around the destination-this is all left to
+employees on different levels of the airlines hierarchy, and the passenger simply
+gets from one place to another. Here again, it is when something goes wrong-such
+as his baggage not arriving that the passenger is made aware of the confusing
+system of levels underneath him.
+
+
+Levels of Description, and Computer Systems
+
+
+296
+
+
+
+Are Computers Super-Flexible or Super-Rigid?
+
+
+One of the major goals of the drive to higher levels has always been to make as
+natural as possible the task of communicating to the computer what you want it to
+do. Certainly, the high-level constructs in compiler languages are closer to the
+concepts which humans naturally think in, than are lower-level constructs such as
+those in machine language. But in this drive towards ease of communication, one
+aspect of "naturalness" has been quite neglected. That is the fact that interhuman
+communication is far less rigidly constrained than human-machine
+communication. For instance, we often produce meaningless sentence fragments
+as we search for the best way to express something, we cough in the middle of
+sentences, we interrupt each other, we use ambiguous descriptions and "improper"
+syntax, we coin phrases and distort meanings-but our message still gets through,
+mostly. With programming languages, it has generally been the rule that there is a
+very strict syntax which has to be obeyed one hundred per cent of the time; there
+are no ambiguous words or constructions. Interestingly, the printed equivalent of
+coughing (i.e., a nonessential or irrelevant comment) is allowed, but only
+provided it is signaled in advance by a key word (e.g., COMMENT), and then
+terminated by another key word (e.g., a semicolon). This small gesture towards
+flexibility has its own little pitfall, ironically: if a semicolon (or whatever key
+word is used for terminating a comment) is used inside a comment, the translating
+program will interpret that semicolon as signaling the end of the comment, and
+havoc will ensue.
+
+If a procedure named INSIGHT has been defined and then called seventeen times
+in the program, and the eighteenth time it is misspelled as INSIHGT, woe to the
+programmer. The compiler will balk and print a rigidly unsympathetic error
+message, saying that it has never heard of INSIHGT. Often, when such an error
+is detected by a compiler, the compiler tries to continue, but because of its lack of
+insihgt, it has not understood what the programmer meant. In fact, it may very
+well suppose that something entirely different was meant, and proceed under that
+erroneous assumption. Then a long series of error messages will pepper the rest of
+the program, because the compiler-not the programmer-got confused. Imagine the
+chaos that would result if a simultaneous English-Russian interpreter, upon
+hearing one phrase of French in the English, began trying to interpret all the
+remaining English as French. Compilers often get lost in such pathetic ways. C'est
+la vie.
+
+Perhaps this sounds condemnatory of computers, but it is not meant to be. In some
+sense, things had to be that way. When you stop to think what most people use
+computers for, you realize that it is to carry out very definite and precise tasks,
+which are too complex for people to do. If the computer is to be reliable, then it is
+necessary that it should understand, without the slightest chance of ambiguity,
+what it is supposed to do. It is also necessary that it should do neither more nor
+less than it is explicitly instructed to do. If there is, in the cushion underneath the
+programmer, a program whose purpose is to "guess" what the programmer wants
+or
+
+
+Levels of Description, and Computer Systems
+
+
+297
+
+
+
+means, then it is quite conceivable that the programmer could try to communicate
+his task and be totally misunderstood. So it is important that the high-level
+program, while comfortable for the human, still should be unambiguous and
+precise.
+
+
+Second-Guessing the Programmer
+
+Now it is possible to devise a programming language-and a program which
+translates it into the lower levels-which allows some sorts of imprecision. One
+way of putting it would be to say that a translator for such a programming
+language tries to make sense of things which are done "outside of the rules of the
+language". But if a language allows certain "transgressions", then transgressions
+of that type are no longer true transgressions, because they have been included
+inside the rules' If a programmer is aware that he may make certain types of
+misspelling, then he may use this feature of the language deliberately, knowing
+that he is actually operating within the rigid rules of the language, despite
+appearances. In other words, if the user is aware of all the flexibilities
+programmed into the translator for his convenience, then he knows the bounds
+which he cannot overstep, and therefore, to him, the translator still appears rigid
+and inflexible, although it may allow him much more freedom than early versions
+of the language, which did not incorporate "automatic compensation for human
+error".
+
+With "rubbery" languages of that type, there would seem to be two
+alternatives: (1) the user is aware of the built-in flexibilities of the language and
+its translator; (2) the user is unaware of them. In the first case, the language is still
+usable for communicating programs precisely, because the programmer can
+predict how the computer will interpret the programs he writes in the language. In
+the second case, the "cushion" has hidden features which may do things that are
+unpredictable (from the vantage point of a user who doesn't know the inner
+workings of the translator). This may result in gross misinterpretations of
+programs, so such a language is unsuitable for purposes where computers are used
+mainly for their speed and reliability.
+
+Now there is actually a third alternative: (3) the user is aware of the built-
+in flexibilities of the language and its translator, but there are so many of them
+and they interact with each other in such a complex way that he cannot tell how
+his programs will be interpreted. This may well apply to the person who wrote the
+translating program; he certainly knows its insides as well as anyone could-but he
+still may not be able to anticipate how it will react to a given type of unusual
+construction.
+
+One of the major areas of research in Artificial Intelligence today is called
+automatic programming, which is concerned with the development of yet higher-
+level languages-languages whose translators are sophisticated, in that they can do
+at least some of the following impressive things: generalize from examples,
+correct some misprints or grammatical errors,
+
+
+Levels of Description, and Computer Systems
+
+
+298
+
+
+
+try to make sense of ambiguous descriptions, try to second-guess the user by
+having a primitive user model, ask questions when things are unclear, use English
+itself, etc. The hope is that one can walk the tightrope between reliability and
+flexibility.
+
+
+AI Advances Are Language Advances
+
+It is striking how tight the connection is between progress in computer science
+(particularly Artificial Intelligence) and the development of new languages. A
+clear trend has emerged in the last decade: the trend to consolidate new types of
+discoveries in new languages. One key for the understanding and creation of
+intelligence lies in the constant development and refinement of the languages in
+terms of which processes for symbol manipulation are describable. Today, there
+are probably three or four dozen experimental languages which have been
+developed exclusively for Artificial Intelligence research. It is important to realize
+that any program which can be written in one of these languages is in principle
+programmable in lower-level languages, but it would require a supreme effort for
+a human; and the resulting program would be so long that it would exceed the
+grasp of humans. It is not that each higher level extends the potential of the
+computer; the full potential of the computer already exists in its machine language
+instruction set. It is that the new concepts in a high-level language suggest
+directions and perspectives by their very nature.
+
+The "space" of all possible programs is so huge that no one can have a
+sense of what is possible. Each higher-level language is naturally suited for
+exploring certain regions of "program space"; thus the programmer, by using that
+language, is channeled into those areas of program space. He is not forced by the
+language into writing programs of any particular type, but the language makes it
+easy for him to do certain kinds of things. Proximity to a concept, and a gentle
+shove, are often all that is needed for a major discovery-and that is the reason for
+the drive towards languages of ever higher levels.
+
+Programming in different 'languages is like composing pieces in different
+keys, particularly if you work at the keyboard. If you have learned or written
+pieces in many keys, each key will have its own special emotional aura. Also,
+certain kinds of figurations "lie in the hand" in one key but are awkward in
+another. So you are channeled by your choice of key. In some ways, even
+enharmonic keys, such as C-sharp and D-flat, are quite distinct in feeling. This
+shows how a notational system can play a significant role in shaping the final
+product.
+
+A "stratified" picture of Al is shown in Figure 59, -with machine
+components such as transistors on the bottom, and "intelligent programs" on the
+top. The picture is taken from the book Artificial Intelligence by Patrick Henry
+Winston, and it represents a vision of Al shared by nearly all Al workers.
+Although I agree with the idea that Al must be stratified in some such way, I do
+not think that, with so few layers, intelligent programs
+
+
+Levels of Description, and Computer Systems
+
+
+299
+
+
+
+
+FIGURE 59. To create intelligent
+programs, one needs to build up a series
+of levels of hardware and software, so
+that one is spared the agonT of seeing
+everything only on the lowest level.
+Descriptions of a single process on
+different levels will sound verb different
+from each other, only the top one being
+sufficiently chunked that it is
+comprehensible to us. [Adapted from P.
+H. Winston, Artificial Intelligence
+(Reading, Mass.: Addison-ifele", 1977)]
+
+
+can he reached. Between the machine language level and the level where rue
+intelligence will be reached, I am convinced there will lie perhaps mother dozen
+(or even several dozen!) layers, each new layer building on and extending the
+flexibilities of the layer below. What they will be like we can hardly dream of
+now ...
+
+
+The Paranoid and the Operating System
+
+The similarity of all levels in a computer system can lead to some strange level¬
+mixing experiences. I once watched a couple of friends-both computer novices-
+playing with the program "PARRY” on a terminal. PARRY s a. rather infamous
+program which simulates a paranoid in an extremely rudimentary way, by spitting
+out canned phrases in English chosen from a vide repertoire; its plausibility is due
+to its ability to tell which of its stock phrases might sound reasonable in response
+to English sentences typed to t by a human.
+
+At one point, the response time got very sluggish-PARRY was taking very
+long to reply-and I explained to my friends that this was probably because of the
+heavy load on the time-sharing system. I told them they could find out how many
+users were logged on, by typing a special "control" character which would go
+directly to the operating system, and would )e unseen by PARRY. One of my
+friends pushed the control character. In a lash, some internal data about the
+operating system's status overwrote some of PARRY's words on the screen.
+PARRY knew nothing of this: it is a program with "knowledge" only of horse
+racing and bookies-not operating systems and terminals and special control
+characters. But to my friends, both PARRY and the operating system were just
+"the computer"-a mysterious, remote, amorphous entity that responded to them
+when they typed. And so it made perfect sense when one of them blithely typed,
+in 3nglish, "Why are you overtyping what's on the screen?" The idea that PARRY
+could know' nothing about the operating system it was running
+
+
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+
+
+300
+
+
+
+
+
+
+
+
+
+
+
+
+under was not clear to my friends. The idea that "you" know all about "yourself"
+is so familiar from interaction with people that it was natural to extend it to the
+computer-after all, it was intelligent enough that it could "talk" to them in
+English! Their question was not unlike asking a person, "Why are you making so
+few red blood cells today?" People do not know about that level-the "operating
+system level"-of their bodies.
+
+The main cause of this level-confusion was that communication with all
+levels of the computer system was taking place on a single screen, on a single
+terminal. Although my friends' naivete might seem rather extreme, even
+experienced computer people often make similar errors when several levels of a
+complex system are all present at once on the same screen. They forget "who"
+they are talking to, and type something which makes no sense at that level,
+although it would have made perfect sense on another level. It might seem
+desirable, therefore, to have the system itself sort out the levels-to interpret
+commands according to what "makes sense". Unfortunately, such interpretation
+would require the system to have a lot of common sense, as well as perfect
+knowledge of the programmer's overall intent-both of which would require more
+artificial intelligence than exists at the present time.
+
+The Border between Software and Hardware
+
+One can also be confused by the flexibility of some levels and the rigidity of
+others. For instance, on some computers there are marvelous text-editing systems
+which allow pieces of text to be "poured" from one format into another,
+practically as liquids can be poured from one vessel into another. A thin page can
+turn into a wide page, or vice versa. With such power, you might expect that it
+would be equally trivial to change from one font to another-say from roman to
+italics. Yet there may be only a single font available on the screen, so that such
+changes are impossible. Or it may be feasible on the screen but not printable by
+the printer-or the other way around. After dealing with computers for a long time,
+one gets spoiled, and thinks that everything should be programmable: no printer
+should be so rigid as to have only one character set, or even a finite repertoire of
+them-typefaces should be user-specifiable! But once that degree of flexibility has
+been attained, then one may be annoyed that the printer cannot print in different
+colors of ink, or that it cannot accept paper of all shapes and sizes, or that it does
+not fix itself when it breaks ...
+
+The trouble is that somewhere, all this flexibility has to "bottom out", to
+use the phrase from Chapter V. There must be a hardware level which underlies it
+all, and which is inflexible. It may lie deeply hidden, and there may be so much
+flexibility on levels above it that few users feel the hardware limitations-but it is
+inevitably there.
+
+What is this proverbial distinction between software and hardware? It is
+the distinction between programs and machines-between long complicated
+sequences of instructions, and the physical machines which carry
+
+
+Levels of Description, and Computer Systems
+
+
+301
+
+
+
+them out. I like to think of software as "anything which you could send over he
+telephone lines", and hardware as "anything else". A piano is hardware, gut
+printed music is software. A telephone set is hardware, but a telephone lumber is
+software. 'The distinction is a useful one, but not always so clear-cut.
+
+We humans also have "software" and "hardware" aspects, and the
+difference is second nature to us. We are used to the rigidity of our physiology:
+the fact that we cannot, at will, cure ourselves of diseases, or ;row hair of any
+color-to mention just a couple of simple examples. We an, however, "reprogram"
+our minds so that we operate in new conceptual frameworks. The amazing
+flexibility of our minds seems nearly irreconcilable with the notion that our brains
+must be made out of fixed-rule hardware, which cannot be reprogrammed. We
+cannot make our neurons ire faster or slower, we cannot rewire our brains, we
+cannot redesign the interior of a neuron, we cannot make anti choices about the
+hardware-and 'et, we can control how we think.
+
+But there are clearly aspects of thought which are beyond our control. We
+cannot make ourselves smarter by an act of will; we cannot learn a new language
+as fast as we want; we cannot make ourselves think faster than we lo; we cannot
+make ourselves think about several things at once; and so on. This is a kind of
+primordial self-knowledge which is so obvious that it is lard to see it at all; it is
+like being conscious that the air is there. We never really bother to think about
+what might cause these "defects" of our minds: lamely, the organization of our
+brains. To suggest ways of reconciling the software of mind with the hardware of
+brain is a main goal of this book.
+
+Intermediate Levels and the Weather
+
+We have seen that in computer systems, there are a number of rather sharply
+defined strata, in terms of any one of which the operation of a running program
+can be described. Thus there is not merely a single low bevel and a single high
+level-there are all degrees of lowness and highness, s the existence of intermediate
+levels a general feature of systems which lave low and high levels? Consider, for
+example, the system whose 'hardware" is the earth's atmosphere (not very hard,
+but no matter), and whose "software" is the weather. Keeping track of the motions
+of all of the molecules simultaneously would be a very low-level way of
+"understanding" he weather, rather like looking at a huge, complicated program
+on the machine language level. Obviously it is way beyond human
+comprehension. 3ut we still have our own peculiarly human ways of looking at,
+and describing, weather phenomena. Our chunked view of the weather is based >n
+very high-level phenomena, such as: rain, fog, snow, hurricanes, cold fronts,
+seasons, pressures, trade winds, the jet stream, cumulo-nimbus clouds,
+thunderstorms, inversion layers, and so on. All of these phenomena involve
+astronomical numbers of molecules, somehow behaving in concert o that large-
+scale trends emerge. This is a little like looking at the weather n a compiler
+language.
+
+
+Levels of Description, and Computer Systems
+
+
+302
+
+
+
+Is there something analogous to looking at the weather in an intermediate-
+level language, such as assembly language? For instance, are there very small
+local "mini-storms", something like the small whirlwinds which one occasionally
+sees, whipping up some dust in a swirling column a few feet wide, at most? Is a
+local gust of wind an intermediate-level chunk which plays a role in creating
+higher-level weather phenomena? Or is there just no practical way of combining
+knowledge of such kinds of phenomena to create a more comprehensive
+explanation of the weather?
+
+Two other questions come to my mind. The first is: "Could it be that the
+weather phenomena which we perceive on our scale-a tornado, a drought-are just
+intermediate-level phenomena: parts of vaster, slower phenomena?" If so, then
+true high-level weather phenomena would be global, and their time scale would
+be geological. The Ice Age would be a high-level weather event. The second
+question is: "Are there intermediate level weather phenomena which have so far
+escaped human perception, but which, if perceived, could give greater insight into
+why the weather is as it is?"
+
+
+From Tornados to Quarks
+
+This last suggestion may sound fanciful, but it is not all that far-fetched.
+We need only look to the hardest of the hard sciences-physics-to find peculiar
+examples of systems which are explained in terms of interacting "parts" which are
+themselves invisible. In physics, as in any other discipline, a system is a group of
+interacting parts. In most systems that we know, the parts retain their identities
+during the interaction, so that we still see the parts inside the system. For
+example, when a team of football players assembles, the individual players retain
+their separateness-they do not melt into some composite entity, in which their
+individuality is lost. Still-and this is important-some processes are going on in
+their brains which are evoked by the team-context, and which would not go on
+otherwise, so that in a minor way, the players change identity when they become
+part of the larger system, the team. This kind of system is called a nearly
+decomposable system (the term comes from H. A. Simon's article "The
+Architecture of Complexity"; see the Bibliography). Such a system consists of
+weakly interacting modules, each of which maintains its own private identity
+throughout the interaction but by becoming slightly different from how it is when
+outside of the system,, contributes to the cohesive behavior of the whole system.
+The systems studied in physics are usually of this type. For instance, an atom is
+seen as made of 'a nucleus whose positive charge captures a number of electrons
+in "orbits", or bound states. The bound electrons are very much like free electrons,
+despite their being internal to a composite object.
+
+Some systems studied in physics offer a contrast to the relatively
+straightforward atom. Such systems involve extremely strong interactions, as a
+result of which the parts are swallowed up into the larger system, and lose some
+or all of their individuality. An example of this is the nucleus of an atom, which is
+usually described as being "a collection of protons and
+
+
+Levels of Description, and Computer Systems
+
+
+303
+
+
+
+neutrons". But the forces which pull the component particles together are strong
+that the component particles do not survive to anything like their “free" form (the
+form they have when outside a nucleus). And in fact a nucleus acts in many ways
+as a single particle, rather than as a collection of interacting particles. When a
+nucleus is split, protons and neutrons are ten released, but also other particles,
+such as pi-mesons and gamma rays, are commonly produced. Are all those
+different particles physically present side a nucleus before it is split, or are then
+just "sparks" which fly off ten the nucleus is split- It is perhaps not meaningful to
+try to give an answer to such a question. On the level of particle physics, the
+difference between storing the potential to make "sparks" and storing actual sub
+particles is not so clear.
+
+A nucleus is thus one systems whose "parts!, even though they are not
+visible while on the inside, can be pulled out and made risible. However, ere are
+more pathological cases, such as the proton and neutron seen as stems themselves.
+Each of them has been hypothesized to be constituted from a trio of "quarks"-
+hypothetical particles which can be combined in twos or threes to make many
+known fundamental particles. However, the interaction between quarks is so
+strong that not only can they not he seen [side the proton and neutron, but they
+cannot even be pulled out at all', bus, although quarks help to give a theoretical
+understanding of certain properties of protons and neutrons, their own existence
+may perhaps ever be independently established. Here see have the antithesis of a
+nearly decomposable system"-it is a system which, if anything, is "nearly
+indecomposable". Yet what is curious is that a quark-based theory of rotors and
+neutrons (and other particles) has considerable explanatory power, in that many
+experimental results concerning the particles which narks supposedly compose
+can be accounted for quite well, quantitatively, by using the "quark model".
+
+Superconductivity: A "Paradox" of Renormalization
+
+In Chapter V we discussed how renormalized particles emerge from their bare
+cores, by recursively compounded interactions with virtual particles. A
+renormalized particle can be seen either as this complex mathematical construct,
+or as the single lump which it is, physically. One of the strangest rid most
+dramatic consequences of this way of describing particles is the explanation it
+provides for the famous phenomenon of superconductivity resistance-free flow of
+electrons in certain solids, at extremely low temperatures.
+
+It turns out that electrons in solids are renormalized by their interactions
+with strange quanta of vibration called phonons (themselves renormalized as
+well!). These renormalized electrons are called polarons. Calculation shows that
+at very low temperatures, two oppositely spinning polarons sill begin to attract
+each other, and can actually become bound together in i certain way. Under the
+proper conditions, all the current-carrying polar
+
+
+Levels of Description, and Computer Systems
+
+
+304
+
+
+
+ons will get paired up, forming Cooper pains. Ironically, this pairing comes about
+precisely because electrons-the hare cores of the paired polarons—repel each other
+electrically. In contrast to the electrons, each Cooper pair feels neither attracted to
+nor repelled by an other Cooper pair, and consequently it can slip freely through a
+metal as if the metal were a vacuum. If you convert the mathematical description
+of such a metal from one whose primitive units are polarons into one whose
+primitive units are Cooper pairs, you get a considerable- simplified set of
+equations. This mathematical simplicity is the physicist's way of knowing that
+"chunking" into Cooper pairs is the natural way to look at superconductivity.
+
+Here we have several levels of particle: the Cooper pair itself: the two
+oppositely-spinning polarons which compose it: the electrons and phonons which
+make up the polarons: and then, within the electrons, the virtual photons and
+positrons, etc. etc. We can look at each level and perceive phenomena there,
+which are explained by an understanding of the levels below.
+
+"Sealing-off"
+
+Similarly, and fortunately, one does not have to know all about quarks to
+understand many things about the particles which they may compose. Thus, a
+nuclear physicist can proceed with theories of nuclei that are based on protons and
+neutrons, and ignore quark theories and their rivals. The nuclear physicist has a
+chunked picture of protons and neutrons-a description derived from lower-level
+theories buf which does not require understanding the lower-level theories.
+Likewise, an atomic physicist has a chunked picture of an atomic nucleus derived
+from nuclear theory. Then a chemist has a chunked picture of the electrons and
+their orbits, and builds theories of small molecules, theories which can be taken
+over in a chunked way by the molecular biologist, who has an intuition for how
+small molecules hang together, but whose technical expertise is in the field of
+extremely large molecules and how they interact. Then the cell biologist has a
+chunked picture of the units which the molecular biologist pores over, and tries to
+use them to account for the ways that cells interact. The point is clear. Each level
+is, in some sense, "sealed off from the levels below it. This is another of Simon's
+vivid terms, recalling the way in which a submarine is built in compartments, so
+that if one part is damaged, and water begins pouring in, the trouble can be
+prevented from spreading, by closing the doors, thereby sealing off the damaged
+compartment from neighboring compartments.
+
+Although there is always some "leakage" between the hierarchical levels
+of science, so that a chemist cannot afford to ignore lower-level physics totally, or
+a biologist to ignore chemistry totally, there is almost no leakage from one level
+to a distant level. That is why people earl, have intuitive understandings of other
+people without necessarily understanding the quark model, the structure of nuclei,
+the nature of electron orbits.
+
+
+Levels of Description, and Computer Systems
+
+
+305
+
+
+
+the chemical bond, the structure of proteins, the organelles in a cell, the methods
+of intercellular communication, the physiology 'of the various organs of the
+human body, or the complex interactions among organs. All at a person needs is a
+chunked model of how the highest level acts; and as all know, such models are
+very realistic and successful.
+
+The Trade-off between Chunking and Determinism
+
+There is, however, perhaps one significant negative feature of a chunked model: it
+usually does not have exact predictive power. That is, we save ourselves from the
+impossible task of seeing people as collections of quarks (or whatever is at the
+lowest level) by using chunked models: but of course such models only give us
+probabilistic estimates of how other people feel, will react to what we say or do,
+and so on. In short, in using chunked high-level models, we sacrifice determinism
+for simplicity. Despite not being sure how people will react to a joke, we tell it
+with the expectation at they will do something such as laugh, or not laugh-rather
+than, say, climb the nearest flagpole. (Zen masters might well do the latter!) A
+chunked model defines a "space" within which behavior is expected to fall, and
+specifies probabilities of its falling in different parts of that space.
+
+"Computers Can Only Do What You Tell Them to Do"
+
+Now these ideas can be applied as well to computer programs as to
+compose physical systems. There is an old saw which says, "Computers can only
+what you tell them to do." This is right in one sense, but it misses the hint: you
+don't know in advance the consequences of what you tell a computer to do;
+therefore its behavior can be as baffling and surprising id unpredictable to you as
+that of a person. You generally know in advance the space in which the output
+will fall, but you don't know details of here it will fall. For instance, you might
+write a program to calculate the first million digits of 7r. Your program will spew
+forth digits of 7r much faster than you can-but there is no paradox in the fact that
+the computer outracing its programmer. You know in advance the space in which
+the output will lie-namely the space of digits between 0 and 9-which is to say, )u
+have a chunked model of the program's behavior; but if you'd known ie rest, you
+wouldn't have written the program.
+
+There is another sense in which this old saw is rusty. This involves the ct
+that as you program in ever higher-level languages, you know less and ss
+precisely what you've told the computer to do! Layers and layers of translation
+may separate the "front end" of a complex program from the actual machine
+language instructions. At the level you think and program, your statements may
+resemble declaratives and suggestions more than they resemble imperatives or
+commands. And all the internal rumbling provoked by the input of a high-level
+statement is invisible to you, generally, just as when you eat a sandwich, you are
+spared conscious awareness of the digestive processes it triggers
+
+
+Levels of Description, and Computer Systems
+
+
+306
+
+
+
+In any case, this notion that "computers can only do what they are told to
+do," first propounded by Lady Lovelace in her famous memoir, is so prevalent
+and so connected with the notion that "computers cannot think" that we shall
+return to it in later Chapters when our level of sophistication is greater.
+
+Two Types of System
+
+There is an important division between two types of system built up from many
+parts. There are those systems in which the behavior of some parts tends to cancel
+out the behavior of other parts, with the result that it does not matter too much
+what happens on the low level, because most anything will yield similar high-
+level behavior. An example of this kind of system is a container of gas, where all
+the molecules bump and bang against each other in very complex microscopic
+ways; but the total outcome, from a macroscopic point of view, is a very calm,
+stable system with a certain temperature, pressure, and volume. Then there are
+systems where the effect of a single low-level event may get magnified into an
+enormous high-level consequence. Such a system is a pinball machine, where the
+exact angle with which a ball strikes each post is crucial in determining the rest of
+its descending pathway.
+
+A computer is an elaborate combination of these two types of system. It
+contains subunits such as wires, which behave in a highly predictable fashion:
+they conduct electricity according to Ohm's law, a very precise, chunked law
+which resembles the laws governing gases in containers, since it depends on
+statistical effects in which billions of random effects cancel each other out,
+yielding a predictable overall behavior. A computer also contains macroscopic
+subunits, such as a printer, whose behavior is completely determined by delicate
+patterns of currents. What the printer prints is not by any means created by a
+myriad canceling microscopic effects. In fact, in the case of most computer
+programs, the value of every single bit in the program plays a critical role in the
+output that gets printed. If any bit were changed, the output would also change
+drastically.
+
+Systems which are made up of "reliable" subsystems only-that is,
+subsystems whose behavior can be reliably predicted from chunked descriptions-
+play inestimably important roles in our daily lives, because they are pillars of
+stability. We can rely on walls not to fall down, on sidewalks to go where they
+went yesterday, on the sun to shine, on clocks to tell the time correctly, and so on.
+Chunked models of such systems are virtually entirely deterministic. Of course,
+the other kind of system which plays a very large role in our lives is a system that
+has variable behavior which depends on some internal microscopic parameters-
+often a very large number of them, moreover-which we cannot directly observe.
+Our chunked model of such a system is necessarily in terms of the "space" of
+operation, and involves probabilistic estimates of landing in different regions of
+that space.
+
+A container of gas, which, as I already pointed out, is a reliable system
+
+
+Levels of Description, and Computer Systems
+
+
+307
+
+
+
+Because of many canceling effects, obeys precise, deterministic laws of physics.
+Such laws are chunked laws, in that they deal with the gas as a whole, nd ignore
+its constituents. Furthermore, the microscopic and macroscopic descriptions of a
+gas use entirely different terms. The former requires the pacification of the
+position and velocity of every single component molecule; the latter requires only
+the specification of three new quantities: temperature, pressure, and volume, the
+first two of which do not even have microscopic counterparts. The simple
+mathematical relationship which elates these three parameters- pV = cT, where c
+is a constant-is a law which depends on, yet is independent of, the lower-level
+phenomena. Less paradoxically, this law can be derived from the laws governing
+the molecular level; in that sense it depends on the lower level. On the other hand,
+it is law which allows you to ignore the lower level completely, if you wish: in hat
+sense it is independent of the lower level.
+
+It is important to realize that the high-level law cannot be stated in the
+vocabulary of the low-level description. "Pressure" and "temperature" are new
+terms which experience with the low level alone cannot convey. We humans
+perceive temperature and pressure directly; that is how we are guilt, so that it is
+not amazing that we should have found this law. But creatures which knew gases
+only as theoretical mathematical constructs would have to have an ability to
+synthesize new concepts, if they were to discover this law.
+
+Epiphenomena
+
+In drawing this Chapter to a close, I would like to relate a story about a complex
+system. I was talking one day with two systems programmers for he computer I
+was using. They mentioned that the operating system seemed to be able to handle
+up to about thirty-five users with great comfort, but at about thirty-five users or
+so, the response time all of a sudden hot up, getting so slow that you might as well
+log off and go home and wait until later. Jokingly I said, "Well, that's simple to fix
+just find the place in he operating system where the number '35' is stored, and
+change it to 60'!" Everyone laughed. The point is, of course, that there is no such
+place, where, then, does the critical number-35 users-come from? The answer is:
+It is a visible consequence of the overall system organization-an
+" epiphenometon,,.
+
+Similarly, you might ask about a sprinter, "Where is the '9.3' stored, hat
+makes him be able to run 100 yards in 9.3 seconds?" Obviously, it is not stored
+anywhere. His time is a result of how he is built, what his reaction time is, a
+million factors all interacting when he runs. The time is quite 'reproducible, but it
+is not stored in his body anywhere. It is spread around among all the cells of his
+body and only manifests itself in the act of the print itself.
+
+Epiphenomena abound. In the game of "Go", there is the feature that “two
+eyes live”. It is not built into the rules, but it is a consequence of the
+
+
+Levels of Description, and Computer Systems
+
+
+308
+
+
+
+rules. In the human brain, there is gullibility. How gullible are you? Is your
+gullibility located in some "gullibility center" in your brain? Could a
+neurosurgeon reach in and perform some delicate operation to lower your
+gullibility, otherwise leaving you alone? If you believe this, you are pretty
+gullible, and should perhaps consider such an operation.
+
+Mind vs. Brain
+
+In coming Chapters, where we discuss the brain, we shall examine whether the
+brain's top level-the mind-can be understood without understanding the lower
+levels on which it both depends and does not depend. Are there laws of thinking
+which are "sealed off" from the lower laws that govern the microscopic activity in
+the cells of the brain? Can mind be "skimmed" off of brain and transplanted into
+other systems? Or is it impossible to unravel thinking processes into neat and
+modular subsystems? Is the brain more like an atom, a renormalized electron, a
+nucleus, a neutron, or a quark? Is consciousness an epiphenomenon? To
+understand the mind, must one go all the way down to the level of nerve cells?
+
+
+Levels of Description, and Computer Systems
+
+
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+
+FIGURE 60. [Drawing by the author.)
+
+
+
+... Ant Fugue
+
+
+.... then, one by one, the four voices of the fugue chime in.)
+
+Achilles: I know the rest of you won't believe this, but the answer to the question is
+
+staring us all in the face, hidden in the picture. It is simply one word-but what an
+
+important one: "MU"!
+
+CCrab: I know the rest of you won't believe this, but the answer to the question is staring
+us all in the face, hidden in the picture. It is simply one word-but what an important
+one: "HOLISM"!
+
+Achilles: Now hold on a minute. You must be seeing things. It's plain as day that the
+message of this picture is "MU", not "HOLISM"!
+
+Crab: I beg your pardon, but my eyesight is extremely good. Please look again, and then
+tell me if the the picture doesn't say what I said it says!
+
+Anteater: I know the rest of you won't believe this, but the answer to the question is
+
+staring us all in the face, hidden in the picture. It is simply one word-but what an
+
+important one: "REDUCTIONISM"!
+
+Crab: Now hold on a minute. You must be seeing things. It's plain as day that the
+message of this picture is "HOLISM", not "REDUCTIONISM"!
+
+Achilles: Another deluded one! Not "HOLISM", not "REDUCTIONISM", but "MU" is
+the message of this picture, and that much is certain.
+
+Anteater: I beg your pardon, but my eyesight is extremely clear. Please look again, and
+then see if the picture doesn't say what I said it says.
+
+Achilles: Don't you see that the picture is composed of two pieces, and that each of them
+is a single letter?
+
+Crab: You are right about the two pieces, but you are wrong in your identification of
+what they are. The piece on the left is entirely composed of three copies of one word:
+"HOLISM"; and the piece on the right is composed of many copies, in smaller letters,
+of the same word. Why the letters are of different sizes in the two parts, I don't know,
+but I know what I see, and what I see is "HOLISM", plain as day. How you see
+anything else is beyond me.
+
+Anteater: You are right about the two pieces, but you are wrong in your identification of
+what they are. The piece on the left is entirely composed of many copies of one
+word: "REDUCTIONISM"; and the piece on the right is composed of one single
+copy, in larger letters, of the same word. Why the letters are of different sizes in the
+two parts, I don't know, but I know what I see, and what I see is
+"REDUCTIONISM", plain as day. How you see anything else is beyond me.
+
+Achilles: I know what is going on here. Each of you has seen letters which compose, or
+are composed of, other letters. In the left-hand piece,
+
+
+Ant Fugue
+
+
+311
+
+
+
+there are indeed three "HOLISM"'s, but each one of them is composed out of smaller
+copies of the word "REDUCTIONISM". And in complementary fashion, in the right-
+hand piece, there is indeed one "REDUCTIONISM", but it is composed out of
+smaller copies of the word "HOLISM". Now this is all fine and good, but in your
+silly squabble, the two of you have actually missed the forest for the trees. You see,
+what good is it to argue about whether "HOLISM" or "REDUCTIONISM" is right,
+when the proper way to understand the matter is to transcend the question, by
+answering "Mu",
+
+Crab: I now see the picture as you have described it, Achilles, but I have no idea of what
+you mean by the strange expression "transcending the question".
+
+Anteater: I now see the picture as you have described it, Achilles, but I have no idea of
+what you mean by the strange expression "Mu", .dies: I will be glad to indulge both
+of you, if you will first oblige me, by telling me the meaning of these strange
+expressions, "HOLISM" and "REDUCTIONISM".
+
+Crab: HOLISM is the most natural thing in the world to grasp. It's simply the belief that
+"the whole is greater than the sum of its parts". No one in his right mind could reject
+holism.
+
+Anteater: REDUCTIONISM is the most natural thing in the world to grasp. It's simply
+the belief that "a whole can be understood completely if you understand its parts, and
+the nature of their 'sum'". No one in her left brain could reject reductionism.
+
+Crab: I reject reductionism. I challenge you to tell me, for instance, how to understand a
+brain reductionistically. Any reductionistic explanation of a brain will inevitably fall
+far short of explaining where the consciousness experienced by a brain arises from.
+
+Anteater: I reject holism. I challenge you to tell me, for instance, how a holistic
+description of an ant colony sheds any more light on it than is shed by a description
+of the ants inside it, and their roles, and their interrelationships. Any holistic
+explanation of an ant colony will inevitably fall far short of explaining where the
+consciousness experienced by an ant colony arises from.
+
+Antilles: Oh, no! The last thing which I wanted to do was to provoke another argument.
+Anyway, now that I understand the controversy, I believe that my explanation of
+"Mu" will help greatly. You see, "Mu" is an ancient Zen answer which, when given
+to a question, UNASKS the question. Here, the question seems to be, "Should the
+world be understood via holism, or via reductionism?" And the answer of "Mu" here
+rejects the premises of the question, which are that one or the other must be chosen.
+By unasking the question, it reveals a wider truth: that there is a larger context into
+which both holistic and reductionistic explanations fit.
+
+Anteater: Absurd! Your "Mu" is as silly as a cow's moo. I'll have none of this Zen washy-
+wishiness.
+
+
+Ant Fugue
+
+
+312
+
+
+
+Crab: Ridiculous! Your "ML" is as silly as a kitten's mew. I’ll have none of this Zen
+washy-wishiness.
+
+Achilles: Oh, dear! We're getting nowhere fast. Why have you stayed so strangely silent,
+Mr. Tortoise? It makes me very uneasy. Surely you must somehow be capable of
+helping straighten out this mess?
+
+Tortoise: I know the rest of you won't believe this, but the answer to the question is
+staring us all in the face, hidden in the picture. It is simply one word-but what an
+important one: "Mu"!
+
+Gust as he says this, the fourth voice in the fugue being played enters, exactly one octave
+below the first entry.)
+
+Achilles: Oh, Mr. T, for once you have let me down. I was sure that you, who always see
+the most deeply into things, would be able to resolve this dilemma-but apparently,
+you have seen no further than I myself saw. Oh, well, I guess I should feel pleased to
+have seen as far as Mr. Tortoise, for once.
+
+Tortoise: I beg your pardon, but my eyesight is extremely fine. Please look again, and
+then tell me if the picture doesn't say what I said it says.
+
+Achilles: But of course it does! You have merely repeated my own original observation.
+
+Tortoise: Perhaps "Mu" exists in this picture on a deeper level than you imagine,
+Achilles-an octave lower (figuratively speaking). But for now I doubt that we can
+settle the dispute in the abstract. I would like to see both the holistic and
+reductionistic points of view laid out more explicitly; then there may be more of a
+basis for a decision. I would very much like to hear a reductionistic description of an
+ant colony, for instance.
+
+Crab: Perhaps Dr. Anteater will tell you something of his experiences in that regard.
+After all, he is by profession something of an expert on that subject.
+
+Tortoise: I am sure that we have much to learn from you, Dr. Anteater. Could you tell us
+more about ant colonies, from a reductionistic point of view?
+
+Anteater: Gladly. As Mr. Crab mentioned to you, my profession has led me quite a long
+way into the understanding of ant colonies.
+
+Achilles: I can imagine! The profession of anteater would seem to be synonymous with
+being an expert on ant colonies!
+
+Anteater: I beg your pardon. "Anteater" is not my profession; it is my species. By
+profession, I am a colony surgeon. I specialize in correcting nervous disorders of the
+colony by the technique of surgical removal.
+
+Achilles: Oh, I see. But what do you mean by "nervous disorders" of an ant colony?
+
+Anteater: Most of my clients suffer from some sort of speech impairment. You know,
+colonies which have to grope for words in everyday situations. It can be quite tragic.
+I attempt to remedy the situation by, uhh—removing the defective part of the colony.
+These operations
+
+
+Ant Fugue
+
+
+313
+
+
+
+are sometimes quite involved, and of course years of study are required before one
+can perform them.
+
+Achilles: But-isn't it true that, before one can suffer from speech impairment, one must
+have the faculty of speech?
+
+Anteater: Right.
+
+Achilles: Since ant colonies don’t have that faculty, I am a little confused. Crab: It's too
+bad, Achilles, that you weren't here last week, when Dr.
+
+Anteater and Aunt Hillary were my house guests. I should have thought of having you
+over then.
+
+Achilles: Is Aunt Hillary your aunt, Mr. Crab? Crab: Oh, no, she's not really anybody's
+aunt.
+
+Anteater: But the poor dear insists that everybody should call her that, even strangers. It's
+just one of her many endearing quirks.
+
+Crab: Yes, Aunt Hillary is quite eccentric, but such a merry old soul. It's a shame I didn't
+have you over to meet her last week.
+
+Anteater: She's certainly one of the best-educated ant colonies I have ever had the good
+fortune to know. The two of us have spent many a long evening in conversation on
+the widest range of topics.
+
+Achilles: I thought anteaters were devourers of ants, not patrons of antintellectualism!
+
+Anteater: Well, of course the two are not mutually inconsistent. I am on the best of terms
+with ant colonies. It's just ANTS that I eat, not colonies-and that is good for both
+parties: me, and the colony.
+
+Achilles: How is it possible that—
+
+Tortoise: How is it possible that—
+
+Achilles: -having its ants eaten can do an ant colony any good? Crab: How is it possible
+that
+
+Tortoise: -having a forest fire can do a forest any good? Anteater: How is it possible that
+
+Crab: -having its branches pruned can do a tree any good? Anteater: -having a haircut can
+do Achilles any good?
+
+Tortoise: Probably the rest of you were too engrossed in the discussion to notice the
+lovely stretto which just occurred in this Bach fugue.
+
+Achilles: What is a stretto?
+
+Tortoise: Oh, I'm sorry; I thought you knew the term. It is where one theme repeatedly
+enters in one voice after another, with very little delay between entries.
+
+Achilles: If I listen to enough fugues, soon I'll know all of these things and will be able to
+pick them out myself, without their having to be pointed out.
+
+Tortoise: Pardon me, my friends. I am sorry to have interrupted. Dr. Anteater was trying
+to explain how eating ants is perfectly consistent with being a friend of an ant colony.
+
+Achilles: Well, I can vaguely see how it might be possible for a limited and regulated
+amount of ant consumption to improve the overall health of
+
+
+Ant Fugue
+
+
+314
+
+
+
+a colony-but what is far more perplexing is all this talk about having conversations
+with ant colonies. That's impossible. An ant colony is simply a bunch of individual
+ants running around at random looking for food and making a nest.
+
+Anteater: You could put it that way if you want to insist on seeing the trees but missing
+the forest, Achilles. In fact, ant colonies, seen as wholes, are quite well-defined units,
+with their own qualities, at times including the mastery of language.
+
+Achilles: I find it hard to imagine myself shouting something out loud in the middle of
+the forest, and hearing an ant colony answer back.
+
+Anteater: Silly fellow! That's not the way it happens. Ant colonies don't converse out
+loud, but in writing. You know how ants form trails leading them hither and thither?
+
+Achilles: Oh, yes-usually straight through the kitchen sink and into my peach jam.
+
+Anteater: Actually, some trails contain information in coded form. If you know the
+system, you can read what they're saying just like a book. Achilles: Remarkable. And
+can you communicate back to them? Anteater: Without any trouble at all. That's how
+Aunt Hillary and I have conversations for hours. I take a stick and draw trails in the
+moist ground, and watch the ants follow my trails. Presently, a new trail starts getting
+formed somewhere. I greatly enjoy watching trails develop. As they are forming, I
+anticipate how they will continue (and more often I am wrong than right). When the
+trail is completed, I know what Aunt Hillary is thinking, and I in turn make my reply.
+
+Achilles: There must be some amazingly smart ants in that colony, I'll say that.
+
+Anteater: I think you are still having some difficulty realizing the difference in levels
+here. Just as you would never confuse an individual tree with a forest, so here you
+must not take an ant for the colony. You see, all the ants in Aunt Hillary are as dumb
+as can be. They couldn't converse to save their little thoraxes!
+
+Achilles: Well then, where does the ability to converse come from? It must reside
+somewhere inside the colony! I don't understand how the ants can all be unintelligent,
+if Aunt Hillary can entertain you for hours with witty banter.
+
+Tortoise: It seems to me that the situation is not unlike the composition of a human brain
+out of neurons. Certainly no one would insist that individual brain cells have to be
+intelligent beings on their own, in order to explain the fact that a person can have an
+intelligent conversation.
+
+Achilles: Oh, no, clearly not. With brain cells, I see your point completely. Only ... ants
+are a horse of another color. I mean, ants just roam about at will, completely
+randomly, chancing now and then upon a morsel of food ... They are free to do what
+they want to do, and with that freedom, I don’t see at all how their behaviour, seen as
+a whole, can
+
+
+Ant Fugue
+
+
+315
+
+
+
+amount to anything coherent-especially something so coherent as the brain behavior
+necessary for conversing.
+
+Crab: It seems to me that the ants are free only within certain constraints. For example,
+they are free to wander, to brush against each other, to pick up small items, to work
+on trails, and so on. But they never step out of that small world, that ant-system,
+which they are in. It would never occur to them, for they don't have the mentality to
+imagine anything of the kind. Thus the ants are very reliable components, in the
+sense that you can depend on them to perform certain kinds of tasks in certain ways.
+
+Achilles: But even so, within those limits they are still free, and they just act at random,
+running about incoherently without any regard for the thought mechanisms of a
+higher-level being which Dr. Anteater asserts they are merely components of.
+
+Anteater: Ah, but you fail to recognize one thing. Achilles-the regularity of statistics.
+
+Achilles: How is that?
+
+Anteater: For example, even though ants as individuals wander about in what seems a
+random way, there are nevertheless overall trends, involving large numbers of ants,
+which can emerge from that chaos.
+
+Achilles: Oh, I know what you mean. In fact, ant trails are a perfect example of such a
+phenomenon. There, you have really quite unpredictable motion on the part of any
+single ant-and yet, the trail itself seems to remain well-defined and stable. Certainly
+that must mean that the individual ants are not just running about totally at random.
+
+Anteater: Exactly, Achilles. There is some degree of communication among the ants, just
+enough to keep them from wandering off completely at random. By this minimal
+communication they can remind each other that they are not alone but are
+cooperating with teammates. It takes a large number of ants, all reinforcing each
+other this way, to sustain any activity-such as trail-building-for any length of time.
+Now my very hazy understanding of the operation of brains leads me to believe that
+something similar pertains to the firing of neurons. Isn't it true, Mr. Crab, that it takes
+a group of neurons firing in order to make another neuron fire?
+
+Crab: Definitely. Take the neurons in Achilles' brain, for example. Each neuron receives
+signals from neurons attached to its input lines, and if the sum total of inputs at any
+moment exceeds a critical threshold, then that neuron will fire and send its own
+output pulse rushing off to other neurons, which may in turn fire-and on down the
+line it goes. The neural flash swoops relentlessly in its Achillean path, in shapes
+stranger then the dash of a gnat-hungry swallow; every twist, every turn foreordained
+by the neural structure in Achilles' brain, until sensory input messages interfere.
+
+Achilles: Normally, I think that I'M in control of what I think-but the way you put it turns
+it all inside out, so that it sounds as though "I" am just
+
+
+Ant Fugue
+
+
+316
+
+
+
+what comes out of all this neural structure, and natural law. It makes what I consider
+my SELF sound at best like a by-product of an organism governed by natural law,
+and at worst, an artificial notion produced by my distorted perspective. In other
+words, you make me feel like I don't know who or what-I am, if anything.
+
+Tortoise: You'll come to understand much better as we go along. But Dr.
+
+Anteater-what do you make of this similarity?
+
+Anteater: I knew there was something parallel going on in the two very different systems.
+Now I understand it much better. It seems that group phenomena which have
+coherence-trail-building, for example-will take place only when a certain threshold
+number of ants get involved. If an effort is initiated, perhaps at random, by a few ants
+in some locale, one of two things can happen: either it will fizzle out after a brief
+sputtering start
+
+Achilles: When there aren't enough ants to keep the thing rolling?
+
+Anteater: Exactly. The other thing that can happen is that a critical mass of ants is
+present, and the thing will snowball, bringing more and more ants into the picture. In
+the latter case, a whole "team" is brought into being which works on a single project.
+That project might be trailmaking, or food-gathering, or it might involve nest¬
+keeping. Despite the extreme simplicity of this scheme on a small scale, it can give
+rise to very complex consequences on a larger scale.
+
+Achilles: I can grasp the general idea of order emerging from chaos, as you sketch it, but
+that still is a long way from the ability to converse. After all, order also emerges from
+chaos when molecules of a gas bounce against each other randomly-yet all that
+results there is an amorphous mass with but three parameters to characterize it:
+volume, pressure, and temperature. Now that's a far cry from the ability to understand
+the world, or to talk about it!
+
+Anteater: That highlights a very interesting difference between the explanation of the
+behavior of an ant colony and the explanation of the behavior of gas inside a
+container. One can explain the behavior of the gas simply by calculating the
+statistical properties of the motions of its molecules. There is no need to discuss any
+higher elements of structure than molecules, except the full gas itself. On the other
+hand, in an ant colony, you can't even begin to understand the activities of the colony
+unless you go through several layers of structure.
+
+Achilles: I see what you mean. In a gas, one jump takes you from the lowest level-
+molecules-to the highest level-the full gas. There are no intermediate levels of
+organization. Now how do intermediate levels of organized activity arise in an ant
+colony?
+
+Anteater: It has to do with the existence of several different varieties of ants inside any
+colony.
+
+Achilles: Oh, yes. I think I have heard about that. They are called "castes", aren't they?
+
+Anteater: That's correct. Aside from the queen, there are males, who do
+
+
+Ant Fugue
+
+
+317
+
+
+
+practically nothing towards, the upkeep of the nest, and then—
+
+Achilles: And of course there are soldiers-Glorious Fighters Against Communism!
+
+Crab: Hmm ... I hardly think that could be right, Achilles. An ant colony is quite
+communistic internally, so why would its soldiers fight against communism? Or am I
+right, Dr. Anteater? .
+
+Anteater: Yes, about colonies you are right, Mr. Crab; they are indeed based on
+somewhat communistic principles. But about soldiers Achilles is somewhat naive. In
+fact, the so-called "soldiers" are hardly adept at fighting at all. They are slow,
+ungainly ants with giant heads, who can snap with their strong jaws, but are hardly to
+be glorified. As in a true communistic state, it is rather the workers who are to be
+glorified. It is they who do most of the chores, such as food-gathering, hunting, and
+nursing of the young. It is even they who do most of the fighting.
+
+Achilles: Bah. That is an absurd state of affairs. Soldiers who won't fight!
+
+Anteater: Well, as I just said, they really aren't soldiers at all. It's the workers who are
+soldiers; the soldiers are just lazy fatheads.
+
+Achilles: Oh, how disgraceful! Why, if I were an ant, I'd put some discipline in their
+ranks! I'd knock some sense into those fatheads!
+
+Tortoise: If you were an ant? How could you be an ant? There is no way to map your
+brain onto an ant brain, so it seems to me to be a pretty fruitless question to worry
+over. More reasonable would be the proposition of mapping your brain onto an ant
+colony ... But let us not get sidetracked. Let Dr. Anteater continue with his most
+illuminating description of castes and their role in the higher levels of organization.
+
+Anteater: Very well. There are all sorts of tasks which must be accomplished in a colony,
+and individual ants develop specializations. Usually an ant's specialization changes as
+the ant ages. And of course it is also dependent on the ant's caste. At any one
+moment, in any small area of a colony, there are ants of all types present. Of course,
+one caste may be be very sparse in some places and very dense in others.
+
+Crab: Is the density of a given caste, or specialization, just a random thing? Or is there a
+reason why ants of one type might be more heavily concentrated in certain areas, and
+less heavily in others?
+
+Anteater: I'm glad you brought that up, since it is of crucial importance in understanding
+how a colony thinks. In fact, there evolves, over a long period of time, a very delicate
+distribution of castes inside a colony. And it is this distribution which allows the
+colony to have the complexity which underlies the ability to converse with me.
+
+Achilles: It would seem to me that the constant motion of ants to and fro would
+completely prevent the possibility of a very delicate distribution.Any delicate
+distribution would be quickly destroyed by all the random motions of ants, just as
+any delicate pattern among molecules in a gas would not survive for an instant, due
+to the random bombardment from all sides.
+
+Anteater: In an ant colony, the situation is quite the contrary. In fact, it is just exactly the
+to-ing and fro-ing of ants inside the colony
+
+
+Ant Fugue
+
+
+318
+
+
+
+which adapts the caste distribution to varying situations, and thereby preserves the
+delicate caste distribution. You see, the caste distribution cannot remain as one single
+rigid pattern; rather, it must constantly be changing so as to reflect, in some manner,
+the real-world situation with which the colony is dealing, and it is precisely the
+motion inside the colony which updates the caste distribution, so as to keep it in line
+with the present circumstances facing the colony.
+
+Tortoise: Could you give an example?
+
+Anteater: Gladly. When I, an anteater, arrive to pay a visit to Aunt Hillary, all the foolish
+ants, upon sniffing my odor, go into a panic-which means, of course, that they begin
+running around completely differently from the way they were before I arrived.
+
+Achilles: But that's understandable, since you're a dreaded enemy of the colony.
+
+Anteater: Oh, no. I must reiterate that, far from being an enemy of the colony, I am Aunt
+Hillary's favorite companion. And Aunt Hillary is my favorite aunt. I grant you, I'm
+quite feared by all the individual ants in the colony-but that's another matter entirely.
+In any case, you see that the ants' action in response to my arrival completely changes
+the internal distribution of ants.
+
+Achilles: That's clear.
+
+Anteater: And that sort of thing is the updating which I spoke of. The new distribution
+reflects my presence. One can describe the change from old state to new as having
+added a "piece of knowledge" to the colony.
+
+Achilles: How can you refer to the distribution of different types of ants inside a colony
+as a "piece of knowledge"?
+
+Anteater: Now there's a vital point. It requires some elaboration. You see, what it comes
+down to is how you choose to describe the caste distribution. If you continue to think
+in terms of the lower levels-individual ants-then you miss the forest for the trees.
+That's just too microscopic a level, and when you think microscopically, you're
+bound to miss some large-scale features. You've got to find the proper high-level
+framework in which to describe the caste distribution-only then will it make sense
+how the caste distribution can encode many pieces of knowledge.
+
+Achilles: Well, how DO you find the proper-sized units in which to describe the present
+state of the colony, then?
+
+Anteater: All right. Let's begin at the bottom. When ants need to get something done,
+they form little "teams", which stick together to perform a chore. As I mentioned
+earlier, small groups of ants are constantly forming and unforming. Those which
+actually exist for a while are the teams, and the reason they don't fall apart is that
+there really is something for them to do.
+
+Achilles: Earlier you said that a group will stick together if its size exceeds a certain
+threshold. Now you're saying that a group will stick together if there is something for
+it to do.
+
+Anteater: They are equivalent statements. For instance, in food-gathering.
+
+
+Ant Fugue
+
+
+319
+
+
+
+if there is an inconsequential amount of food somewhere which gets discovered by
+some wandering Ant who then attempts to communicate its enthusiasm to other ants,
+the number of ants who respond will be proportional to the size of the food sample-
+and an inconsequential amount will not attract enough ants to surpass the threshold.
+Which is exactly what I meant by saying there is nothing to do-too little food ought
+to be ignored.
+
+Achilles: I see. I assume that these "teams" are one of the levels of structure falling
+somewhere in between the single-ant level and the colony level.
+
+Anteater: Precisely. There exists a special kind of team, which I call a "signal"-and all the
+higher levels of structure are based on signals. In fact, all the higher entities are
+collections of signals acting in concert. There are teams on higher levels whose
+members are not ants, but teams on lower levels. Eventually you reach the lowest-
+level teams which is to say, signals-and below them, ants.
+
+Achilles: Why do signals deserve their suggestive name?
+
+Anteater: It comes from their function. The effect of signals is to transport ants of various
+specializations to appropriate parts of the colony. So the typical story of a signal is
+thus: it comes into existence by exceeding the threshold needed for survival, then it
+migrates for some distance through the colony, and at some point it more or less
+disintegrates into its individual members, leaving them on their own.
+
+Achilles: It sounds like a wave, carrying sand dollars and seaweed from afar, and leaving
+them strewn, high and dry, on the shore.
+
+4nteater: In a way that's analogous, since the team does indeed deposit something which
+it has carried from a distance, but whereas the water in the wave rolls back to the sea,
+there is no analogous carrier substance in the case of a signal, since the ants
+themselves compose it.
+
+Tortoise: And I suppose that a signal loses its coherency just at some spot in the colony
+where ants of that type were needed in the first place.
+
+Anteater: Naturally.
+
+Achilles: Naturally? It's not so obvious to ME that a signal should always go just where it
+is needed. And even if it goes in the right direction, how does it figure out where to
+decompose? How does it know it has arrived?
+
+Anteater: Those are extremely important matters, since they involve explaining the
+existence of purposeful behavior-or what seems to be purposeful behavior-on the part
+of signals. From the description, one would be inclined to characterize the signals'
+behavior as being oriented towards filling a need, and to call it "purposeful". But you
+can look at it otherwise.
+
+Achilles: Oh, wait. Either the behavior is purposeful, or it is NOT. I don't see how you
+can have it both ways.
+
+Anteater: Let me explain my way of seeing things, and then see if you agree. Once a
+signal is formed, there is no awareness on its part that it
+
+
+Ant Fugue
+
+
+320
+
+
+
+should head off in any particular direction. But here, the delicate caste distribution
+plays a crucial role. It is what determines the motion of signals through the colony,
+and also how long a signal will remain stable, and where it will "dissolve".
+
+Achilles: So everything depends on the caste distribution, eh?
+
+Anteater: Right. Let's say a signal is moving along. As it goes, the ants which compose it
+interact, either by direct contact or by exchange of scents, with ants of the local
+neighborhoods which it passes through. The contacts and scents provide information
+about local matters of urgency, such as nest-building, or nursing, or whatever. The
+signal will remain glued together as long as the local needs are different from what it
+can supply; but if it CAN contribute, it disintegrates, spilling a fresh team of usable
+ants onto the scene. Do you see now how the caste distribution acts as an overall
+guide of the teams inside the colony?
+
+Achilles: I do see that.
+
+Anteater: And do you see how this way of looking at things requires attributing no sense
+of purpose to the signal?
+
+Achilles: I think so. Actually, I'm beginning to see things from two different vantage
+points. From an ant's-eye point of view, a signal has NO purpose. The typical ant in a
+signal is just meandering around the colony, in search of nothing in particular, until it
+finds that it feels like stopping. Its teammates usually agree, and at that moment the
+team unloads itself by crumbling apart, leaving just its members but none of its
+coherency. No planning is required, no looking ahead; nor is any search required, to
+determine the proper direction. But from the COLONY'S point-of view, the team has
+just responded to a message which was written in the language of the caste
+distribution. Now from this perspective, it looks very much like purposeful activity.
+
+Crab: What would happen if the caste distribution were entirely random? Would signals
+still band and disband?
+
+Anteater: Certainly. But the colony would not last long, due to the meaninglessness of the
+caste distribution.
+
+Crab: -Precisely the point I wanted to make. Colonies survive because their caste
+distribution has meaning, and that meaning is a holistic aspect, invisible on lower
+levels. You lose explanatory power unless you take that higher level into account.
+
+Anteater: I see your side; but I believe you see things too narrowly.
+
+Crab: How so?
+
+Anteater: Ant colonies have been subjected to the rigors of evolution for billions of years.
+A few mechanisms were selected for, and most were selected against. The end result
+was a set of mechanisms which make ant colonies work as we have been describing.
+If you could watch the whole process in a movie-running a billion or so times faster
+than life, of course-the emergence of various mechanisms would be seen as natural
+responses to external pressures, just as bubbles in boiling water are natural responses
+to an external heat source. I don't suppose you
+
+
+Ant Fugue
+
+
+321
+
+
+
+see "meaning" and "purpose", in the bubbles in boiling water-or do you?
+
+Crab: No, but
+
+Anteater: Now that's MY point. No matter how big a bubble is, it owes its existence to
+processes on the molecular level, and you can forget about any "higher-level laws".
+The same goes for ant colonies and their teams. By looking at things from the vast
+perspective of evolution, you can drain the whole colony of meaning and purpose.
+They become superfluous notions.
+
+Achilles: Why, then, Dr. Anteater, did you tell me that you talked with Aunt Hillary? It
+now seems that you would deny that she can talk or think at all.
+
+Anteater: I am not being inconsistent, Achilles. You see, I have as much difficulty as
+anyone else in seeing things on such a grandiose time scale, so I find it much easier
+to change points of view. When I do so, forgetting about evolution and seeing things
+in the here and now, the vocabulary of teleology comes back: the MEANING of the
+caste distribution and the PURPOSEFULNESS of signals. This not only happens
+when I think of ant colonies, but also when I think about my own brain and other
+brains. However, with some effort I can always remember the other point of view if
+necessary, and drain all these systems of meaning, too.
+
+Crab: Evolution certainly works some miracles. You never know the next trick it will pull
+out of its sleeve. For instance, it wouldn't surprise me one bit if it were theoretically
+possible for two or more "signals" to pass through each other, each one unaware that
+the other one is also a signal; each one treating the other as if it were just part of the
+background population.
+
+Anteater: It is better than theoretically possible; in fact it happens routinely!
+
+Achilles: Hmm ... What a strange image that conjures up in my mind. I can just imagine
+ants moving in four different directions, some black, some white, criss-crossing,
+together forming an orderly pattern, almost like-like
+
+Tortoise: A fugue, perhaps?
+
+Achilles: Yes-that's it! An ant fugue!
+
+Crab: An interesting image, Achilles. By the way, all that talk of boiling water made me
+think of tea. Who would like some more? Achilles: I could do with another cup, Mr.
+C.
+
+Crab: Very good.
+
+Achilles: Do you suppose one could separate out the different visual "voices" of such an
+"ant fugue"? I know how hard it is for me-
+
+Tortoise: Not for me, thank you.
+
+Achilles: -to track a single voice-
+
+Anteater: I'd like some, too, Mr. Crab
+
+Achilles: - in a musical fugue—
+
+
+Ant Fugue
+
+
+322
+
+
+
+FIGURE 61. "Ant Fugue", by M. C. Escher (woodcut, 1953).
+
+
+Anteater: -if it isn't too much trouble
+Achilles: . -when all of them
+Crab: Not at all. Four cups of tea
+Tortoise: Three?
+
+Achilles: -are going at once.
+
+Crab: -coming right up!
+
+Anteater: That's an interesting thought, Achilles. But its unlikely that anyone could draw
+such a picture in a convincing way
+Achilles: That's too bad.
+
+Tortoise: Perhaps you could answer this, Dr. Anteater. Does a signal, from its creation
+until its dissolution, always consist of the same set of ants?
+
+Anteater: As a matter of fact, the individuals in a signal sometimes break off and get
+replaced by others of the same caste, if there are a few in the area. Most often, signals
+arrive at their disintegration points with nary an ant in common with their starting
+lineup.
+
+
+Ant Fugue
+
+
+323
+
+
+
+Crab: I can see that the signals are constantly affecting the caste distribution throughout
+the colony, and are doing so in response to the internal needs of the colony-which in
+turn reflect the external situation which the colony is faced with. Therefore the caste
+distribution, as you said, Dr. Anteater, gets continually updated in a way which
+ultimately reflects the outer world.
+
+Achilles: But what about those intermediate levels of structure? You were saying that the
+caste distribution should best be pictured not in terms of ants or signals, but in terms
+of teams whose members were other teams, whose members were other teams, and
+so on until you come down to the ant level. And you said that that was the key to
+understanding how it was possible to describe the caste distribution as encoding
+pieces of information about the world.
+
+Anteater: Yes, we are coming to all that. I prefer to give teams of a sufficiently high level
+the name of "symbols". Mind you, this sense of the word has some significant
+differences from the usual sense. My "symbols" are ACTIVE SUBSYSTEMS of a
+complex system, and they are composed of lower-level active subsystems ... They are
+therefore quite different from PASSIVE symbols, external to the system, such as
+letters of the alphabet or musical notes, which sit there immobile, waiting for an
+active system to process them.
+
+Achilles: Oh, this is rather complicated, isn't it? I just had no idea that ant colonies had
+such an abstract structure.
+
+Anteater: Yes, it's quite remarkable. But all these layers of structure are necessary for the
+storage of the kinds of knowledge which enable an organism to be "intelligent" in
+any reasonable sense of the word. Any system which has a mastery of language has
+essentially the same underlying sets of levels.
+
+Achilles: Now just a cotton-picking minute. Are you insinuating that my brain consists
+of, at bottom, just a bunch of ants running around?
+
+Anteater: Oh, hardly. You took me a little too literally. The lowest level may be utterly
+different. Indeed, the brains of anteaters, for instance, are not composed of ants. But
+when you go up a level or two in a brain, you reach a level whose elements have
+exact counterparts in other systems of equal intellectual strength-such as ant colonies.
+
+Tortoise: That is why it would be reasonable to think of mapping your brain, Achilles,
+onto an ant colony, but not onto the brain of a mere ant.
+
+Achilles: I appreciate the compliment. But how would such a mapping be carried out?
+For instance, what in my brain corresponds to the low level teams which you call
+signals?
+
+Anteater: Oh, I but dabble in brains, and therefore couldn't set up the map in its glorious
+detail. But-and correct me if I'm wrong, Mr. Crab-I would surmise that the brain
+counterpart to an ant colony's signal is the firing of a neuron; or perhaps it is a larger-
+scale event, such as a pattern of neural firings.
+
+
+Ant Fugue
+
+
+324
+
+
+
+Crab: I would tend to agree. But don't you think that, for the purposes of our discussion,
+delineating the exact counterpart is not in itself crucial, desirable though it might be?
+It seems to me that the main idea is that such a correspondence does exist, even if we
+don't know exactly how to define it right now. I would only question one point, Dr.
+Anteater, which you raised, and that concerns the level at which one can have faith
+that the correspondence begins. You seemed to think that a SIGNAL might have a
+direct counterpart in a brain; whereas I feel that it is only at the level of your
+ACTIVE SYMBOLS and above that it is likely that a correspondence must exist.
+
+Anteater: Your interpretation may very well be more accurate than mine, Mr. Crab.
+Thank you for bringing out that subtle point.
+
+Achilles: What does a symbol do that a signal couldn't do?
+
+Anteater: It is something like the difference between words and letters. Words, which are
+meaning-carrying entities, are composed of letters, which in themselves carry no
+meaning. This gives a good idea of the difference between symbols and signals. In
+fact it is a useful analogy, as long as you keep in mind the fact that words and letters
+are PASSIVE, symbols and signals are ACTIVE.
+
+Achilles: I'll do so, but I'm not sure I understand why it is so vital to stress the difference
+between active and passive entities.
+
+Anteater: The reason is that the meaning which you attribute to any passive symbol, such
+as a word on a page, actually derives from the meaning which is carried by
+corresponding active symbols in your brain. So that the meaning of passive symbols
+can only be properly understood when it is related to the meaning of active symbols.
+
+Achilles: All right. But what is it that endows a SYMBOL-an active one, to be sure-with
+meaning, when you say that a SIGNAL, which is a perfectly good entity in its own
+right, has none? Anteater: It all has to do with the way that symbols can cause other
+symbols to be triggered. When one symbol becomes active, it does not do so in
+isolation. It is floating about, indeed, in a medium, which is characterized by its caste
+distribution.
+
+Crab: Of course, in a brain there is no such thing as a caste distribution, but the
+counterpart is the "brain state". There, you describe the states of all the neurons, and
+all the interconnections, and the threshold for firing of each neuron.
+
+Anteater: Very well; let's lump "caste distribution" and "brain state" under a common
+heading, and call them just the "state". Now the state can be described on a low level
+or on a high level. A low-level description of the state of an ant colony would involve
+painfully specifying the location of each ant, its age and caste, and other similar
+items. A very detailed description, yielding practically no global insight as to WHY it
+is in that state. On the other hand, a description on a high level would involve
+specifying which symbols could be triggered by which combinations of other
+symbols, under what conditions, and so forth.
+
+
+Ant Fugue
+
+
+325
+
+
+
+Achilles: What about a description on the level of signals, or teams?
+
+Anteater: A description on that level would fall somewhere in between the low-level and
+symbol-level descriptions. It would contain a great deal of information about what is
+actually going on in specific locations throughout the colony, although certainly less
+than an ant-by-ant description, since teams consist of clumps of ants. A team-by-team
+description is like a summary of an ant-by-ant description. However, you have to add
+extra things which were not present in the ant-by-ant description-such as the
+relationships between teams, and the supply of various castes here and there. This
+extra complication is the price you pay for the right to summarize.
+
+Achilles: It is interesting to me to compare the merits of the descriptions at various levels.
+The highest-level description seems to carry the most explanatory power, in that it
+gives you the most intuitive picture of the ant colony, although strangely enough, it
+leaves out seemingly- the most important feature-the ants.
+
+Anteater: But you see, despite appearances, the ants are not the most important feature.
+Admittedly, were it not for them, the colony Wouldn't exist: but something
+equivalent-a brain-can exist, ant-free. So, at least from a high-level point of view, the
+ants are dispensable. .Achilles: I'm sure no ant would embrace your theory with
+eagerness.
+
+Anteater: Well, I never met an ant with a high-level point of view.
+
+Crab: What a counterintuitive picture you paint, Dr. Anteater. It seems that, if what you
+say is true, in order to grasp the whole structure, you have to describe it omitting any
+mention of its fundamental building blocks.
+
+Anteater: Perhaps I can make it a little clearer by an analogy. Imagine you have before
+you a Charles Dickens novel.
+
+Achilles: The Pickwick Papers-will that do?
+
+Anteater: Excellently! And now imagine trying the following game: you must find a way
+of mapping letters onto ideas, so that the entire Pickwick Papers makes sense when
+you read it letter by letter.
+
+Achilles: Hmm ... You mean that every time I hit a word such as "the", I have to think of
+three definite concepts, one after another, with no room for variation?
+
+Anteater: Exactly. They are the 't'-concept, the 'h'-concept, and the 'e'-concept-and every
+time, those concepts are as they were the preceding time.
+
+Achilles: Well, it sounds like that would turn the experience of "reading" The Pickwick
+Papers into an indescribably boring nightmare. It would be an exercise in
+meaninglessness, no matter what concept I associated with each letter.
+
+Anteater: Exactly. There is no natural mapping from the individual letters into the real
+world. The natural mapping occurs on a higher level between words, and parts of the
+real world. If you wanted to describe the book, therefore, you would make no
+mention of the letter level.
+
+
+Ant Fugue
+
+
+326
+
+
+
+Achilles: Of course not! I'd describe the plot and the characters, and so forth.
+
+Anteater: So there you are. You would omit all mention of the building blocks, even
+though the book exists thanks to them. They are the medium, but not the message.
+
+Achilles: All right-but what about ant colonies?
+
+Anteater: Here, there are active signals instead of passive letters, and active symbols
+instead of passive words-hut the idea carries over.
+
+Achilles: Do you mean I couldn't establish a mapping between signals and things in the
+real world?
+
+Anteater: You would find that you could not do it in such a way that the triggering of new
+signals would make am sense. Nor could you succeed on any lower level-for example
+the ant level. Only on the symbol level do the triggering patterns make sense.
+Imagine, for instance, that one day you were watching Aunt Hillary when I arrived to
+pay a call. You could watch as carefully as you wanted, and yet you would probably
+perceive nothing more than a rearrangement of ants.
+
+Achilles: I'm sure that's accurate.
+
+Anteater: And yet, as I watched, reading the higher level instead of the lower level, I
+would see several dormant symbols being awakened, those which translate into the
+thought, "Oh, here's that charming Dr. Anteater again-how pleasant!"-or words to
+that effect.
+
+Achilles: That sounds like what happened when the four of us all found different levels to
+read in the MU-picture-or at least THREE of us did .. .
+
+Tortoise: What an astonishing coincidence that there should be such a resemblance
+between that strange picture which I chanced upon in the Well-Tempered Clavier,
+and the trend of our conversation.
+
+Achilles: Do you think it's just coincidence?
+
+Tortoise: Of course.
+
+Anteater: Well, I hope you can grasp now how the thoughts in Aunt Hillary emerge from
+the manipulation of symbols composed of signals composed of teams composed of
+lower-level teams, all the way down to ants.
+
+Achilles: Why do you call it "symbol manipulation"? Who does the manipulating, if the
+symbols are themselves active? Who is the agent?
+
+Anteater: This gets back to the question which you earlier raised about purpose. You're
+right that symbols themselves are active, but the activities which they follow are
+nevertheless not absolutely free. The activities of all symbols are strictly determined
+by the state of the full system in which they reside. Therefore, the full system is
+responsible for how its symbols trigger each other, and so it is quite reasonable to
+speak of the full system as the "agent". As the symbols operate, the state of the
+system gets slowly transformed, or updated. But there are many features which
+remain over time. It is this partially constant, partially varying system which is the
+agent. One can give a name to the
+
+
+Ant Fugue
+
+
+327
+
+
+
+full system. For example, Aunt Hillary is the "who" who can be said to manipulate
+her symbols; and you are similar, Achilles.
+
+Achilles: That's quite a strange characterization of the notion of who I am. I'm not sure I
+can fully understand it, but I will give it some thought.
+
+Tortoise: It would be quite interesting to follow the symbols in your brain as you do that
+thinking about the symbols in your brain.
+
+Achilles: That's too complicated for me. I have trouble enough just trying to picture how
+it is possible to look at an ant colony and read it on the symbol level. I can certainly
+imagine perceiving it at the ant level; and with a little trouble, I can imagine what it
+must be like to perceive it at the signal level; but what in the world can it be like to
+perceive an ant colony at the symbol level?
+
+Anteater: One only learns through long practice. But when one is at my stage, one reads
+the top level of an ant colony as easily as you yourself read the "MU" in the MU-
+picture.
+
+Achilles: Really? That must be an amazing experience.
+
+Anteater: In a way-but it is also one which is quite familiar to you, Achilles.
+
+Achilles: Familiar to me? What do you mean? I have never looked at an ant colony on
+anything but the ant level.
+
+Anteater: Maybe not; but ant colonies are no different from brains in many respects.
+
+Achilles: I have never seen nor read any brain either, however.
+
+Anteater: What about your OWN brain? Aren't you aware of your own thoughts? Isn't
+that the essence of consciousness? What else are you doing but reading your own
+brain directly at the symbol level?
+
+Achilles: I never thought of it that way. You mean that I bypass all the lower levels, and
+only see the topmost level?
+
+Anteater: That's the way it is, with conscious systems. They perceive themselves on the
+symbol level only, and have no awareness of the lower levels, such as the signal
+levels.
+
+Achilles: Does it follow that in a brain, there are active symbols which are constantly
+updating themselves so that they reflect the overall state of the brain itself, always on
+the symbol level?
+
+Anteater: Certainly. In any conscious system there are symbols which represent the brain
+state, and they are themselves part of the very brain state which they symbolize. For
+consciousness requires a large degree of self-consciousness.
+
+Achilles: That is a weird notion. It means that although there is frantic activity occurring
+in my brain at all times, I am only capable of registering that activity in one way-on
+the symbol level; and I am completely insensitive to the lower levels. It is like being
+able to read a Dickens novel by direct visual perception, without ever having learned
+the letters of the alphabet. I can't imagine anything as weird as that really happening.
+
+Crab: But precisely that sort of thing can happen when you read “MU”,
+
+
+Ant Fugue
+
+
+328
+
+
+
+without perceiving the lower levels "HOLISM" and "REDUCTIONISM".
+
+Achilles: You're right-I bypassed the lower levels, and saw only the top. I wonder if I'm
+missing all sorts of meaning on lower levels of my brain as well, by reading only the
+symbol level. It's too bad that the top level doesn't contain all the information about
+the bottom level, so that by reading the top, one also learns what the bottom level
+says. But I guess it would be naive to hope that the top level encodes anything from
+the bottom level-it probably doesn't percolate up. The MU-picture is the most striking
+possible example of that: there, the topmost level says only "ML which bears no
+relation whatever to the lower levels!
+
+Crab: That's absolutely true. ( Picks up the MU-picture, to inspect it more closely.) Hmm
+... There's something strange about the smallest letters in this picture; they're very
+wiggly ...'
+
+Anteater: Let me take a look. ( Peers closely at the MU-picture.) I think there's yet
+another level, which all of us missed!
+
+Tortoise: Speak for yourself, Dr. Anteater.
+
+Achilles: Oh, no-that can't be! Let me see. ( Looks very carefully.) I know the rest of you
+won't believe this, but the message of this picture is staring us all in the face, hidden
+in its depths. It is simply one word, repeated over and over again, like a mantra-but
+what an important one: "Mu"! What do you know! It is the same as the top level!
+And none of us suspected it in the least.
+
+Crab: We would never have noticed it if it hadn't been for you, Achilles. Anteater: I
+wonder if the coincidence of the highest and lowest levels happened by chance? Or
+was it a purposeful act carried out by some creator?
+
+Crab: How could one ever decide that?
+
+Tortoise: I don't see any way to do so, since we have no idea why that particular picture is
+in the Crab's edition of the Well-Tempered Clavier. Anteater: Although we have been
+having a lively discussion, I have still managed to listen with a good fraction of an
+ear to this very long and complex four-voice fugue. It is extraordinarily beautiful.
+
+Tortoise: It certainly is. And now, in just a moment, comes an organ point.
+
+Achilles: Isn't an organ point what happens when a piece of music slows down slightly,
+settles for a moment or two on a single note or chord, and then resumes at normal
+speed after a short silence?
+
+Tortoise: No, you're thinking of a "fermata"-a sort of musical semicolon. Did you notice
+there was one of those in the prelude?
+
+Achilles: I guess I must have missed it.
+
+Tortoise: Well, you have another chance coming up to hear a fermata-in fact, there are a
+couple of them coming up, towards the end of this fugue.
+
+Achilles: Oh, good. You'll point them out in advance, won't you? Tortoise: If you like.
+
+Achilles: But do tell me, what is an organ point?
+
+Tortoise: An organ point is the sustaining of a single note by one of the
+
+
+Ant Fugue
+
+
+329
+
+
+
+voices in a polyphonic piece (often the lowest voice), while the other voices continue
+their own independent lines. This organ point is on the note of G. Listen carefully,
+and you'll hear it.
+
+Anteater:. There occurred an incident one day when I visited with Aunt Hillary which
+reminds me of your suggestion of observing the symbols in Achilles' brain as they
+create thoughts which are about themselves.
+
+Crab: Do tell us about it.
+
+Anteater: Aunt Hillary had been feeling very lonely, and was very happy to have
+someone to talk to that day. So she gratefully told me to help myself to the juiciest
+ants I could find. (She's always been most generous with her ants.)
+
+Achilles: Gee!
+
+Anteater: It just happened that I had been watching the symbols which were carrying out
+her thoughts, because in them were some particularly juicy-looking ants.
+
+Achilles: Gee!
+
+Anteater: So I helped myself to a few of the fattest ants which had been parts of the
+higher-level symbols which I had been reading. Specifically, the symbols which they
+were part of were the ones which had expressed the thought, "Help yourself to any of
+the ants which look appetizing."
+
+Achilles: Gee!
+
+Anteater: Unfortunately for them, but fortunately for me, the little bugs didn't have the
+slightest inkling of what they were collectively telling me, on the symbol level.
+
+Achilles: Gee! That is an amazing wraparound. They were completely unconscious of
+what they were participating in. Their acts could be seen as part of a pattern on a
+higher level, but of course they were completely unaware of that. Ah, what a pity-a
+supreme irony, in fact-that they missed it.
+
+Crab: You are right, Mr. T-that was a lovely organ point.
+
+Anteater: I had never heard one before, but that one was so conspicuous that no one could
+miss it. Very effective.
+
+Achilles: What? Has the organ point already occurred? How can I not have noticed it, if it
+was so blatant?
+
+Tortoise: Perhaps you were so wrapped up in what you were saying that you were
+completely unaware of it. Ah, what a pity-a supreme irony, in fact-that you missed it.
+
+Crab: Tell me, does Aunt Hillary live in an anthill?
+
+Anteater: Well, she owns a rather large piece of property. It used to belong to someone
+else, but that is rather a sad story. In any case, her estate is quite expansive. She lives
+rather sumptuously, compared to many other colonies.
+
+!chilies: How does that jibe with the communistic nature of ant colonies which you
+earlier described to us? It sounds quite inconsistent, to me, to preach communism and
+to live in a fancy estate.
+
+
+Ant Fugue
+
+
+330
+
+
+
+Anteater: The communism is on the ant level. In an ant colony all ants work for the
+common good, even to their own individual detriment at times. Now this is simply a
+built-in aspect of Aunt Hillary's structure, but for all I know, she may not even be
+aware of this internal communism. Most human beings are not aware of anything
+about their neurons; in fact they probably are quite content not to know anything
+about their brains, being somewhat squeamish creatures. Aunt Hillary is also
+somewhat squeamish; she gets rather antsy whenever she starts to think about ants at
+all. So she avoids thinking about them whenever possible. I truly doubt that she
+knows anything about the communistic society which is built into her very structure.
+She herself is a staunch believer in libertarianism-you know, laissez-faire and all that.
+So it makes perfect sense, to me at least, that she should live in a rather sumptuous
+manor.
+
+
+
+
+Tortoise: As I turned the page just now, while following along in this lovely edition of
+the Well-Tempered Clavier, I noticed that the first of the two fermatas is coming up
+soon-so you might listen for it, Achilles. Achilles: I will, I will.
+
+Tortoise: Also, there's a most curious picture facing this page. Crab: Another one? What
+next?
+
+Tortoise: See for yourself. (Passes the score over to the Crab.)
+
+Crab: Aha! It's just a few bunches of letters. Let's see-there are various numbers of the
+letters T, 'S', 'B', 'm', V, and't'. It's strange, how the first three letters grow, and then
+the last three letters shrink again. Anteater: May I see it?
+
+Crab: Why, certainly.
+
+Anteater: Oh, by concentrating on details, you have utterly missed the big picture. In
+reality, this group of letters is 'f, V, Y, 'A', "C, 'H', without any repetitions. First
+they get smaller, then they get bigger. Here, AchiUes-what do you make of it?
+
+Achilles: Let me see. Hmm. Well, I see it as a set of upper-case letters which grow as you
+move to the right.
+
+Tortoise: Do they spell anything?
+
+
+
+Ant Fugue
+
+
+331
+
+
+
+Achilles: Ah ... "J. S. BACH". Oh! I understand now. It's Bach's name!
+
+Tortoise: Strange that you should see it that way. I see it as a set of lower-case letters,
+shrinking as they move to the right, and ... spelling out... the name of... (Slows down
+slightly, especialh drawing out the last few words. Then there is a brief silence.
+Suddenly he resumes as if nothing unusual had happened.) -"fermat".
+
+Achilles: Oh, you've got Fermat on the brain, I do believe. You see Fermat's Last
+Theorem everywhere.
+
+Anteater: You were right, Mr. Tortoise-I just heard a charming little fermata in the fugue.
+
+Crab: So did I.
+
+Achilles: Do you mean everybody heard it but me? I'm beginning to feel stupid.
+
+Tortoise: There, there, Achilles-don't feel bad. I'm sure you won't miss Fugue's Last
+Fermata (which is coming up quite soon). But, to return to our previous topic. Dr.
+Anteater, what is the very sad story which you alluded to, concerning the former
+owner of Aunt Hillary's property
+
+Anteater: The former owner was an extraordinary individual, one of the most creative ant
+colonies who ever lived. His name was Johant Sebastiant Fermant, and he was a
+mathematiciant by vocation, but a musiciant by avocation.
+
+Achilles: How very versantile of him!
+
+Anteater: At the height of his creative powers, he met with a most untimely demise. One
+day, a very hot summer day, he was out soaking up the warmth, when a freak
+thundershower-the kind that hits only once every hundred years or so-appeared from
+out of the blue, and thoroughly drenched J. S F. Since the storm came utterly without
+warning, the ants got completely disoriented and confused. The intricate organization
+which had been so finely built up over decades, all went down the drain in a matter of
+minutes. It was tragic.
+
+Achilles: Do you mean that all the ants drowned, which obviously would spell the end of
+poor J. S. F.
+
+Anteater: Actually, no. The ants managed to survive, every last one of them, by crawling
+onto various sticks and logs which floated above the raging torrents. But when the
+waters receded and left the ants back on their home grounds, there was no
+organization left. The caste distribution was utterly destroyed, and the ants
+themselves had no ability to reconstruct what had once before been such a finely
+tuned organization. They were as helpless as the pieces of Humpty Dumpty in putting
+themselves back together again. I myself tried, like all the king's horses and all the
+king's men, to put poor Fermant together again. I faithfully put out sugar and cheese,
+hoping against hope that somehow Fermant would reappear ... (Pulls out a
+handkerchief and wipes his eyes.)
+
+Achilles: How valiant of you! I never knew Anteaters had such big hearts.
+
+Anteater: But it was all to no avail. He was Bone, beyond reconstitution.
+
+
+Ant Fugue
+
+
+332
+
+
+
+However, something very strange then began to take place: over the next few
+months, the ants which had been components of J. S. F. slowly regrouped, and built
+up a new organization. And thus was Aunt Hillary born.
+
+Crab: Remarkable! Aunt Hillary is composed of the very same ants as Fermant was
+
+Anteater: Well, originally she was, yes. By now, some of the older ants have died, and
+been replaced. But there are still many holdovers from the J. S. F.-days.
+
+Crab: And can't you recognize some of J. S. F.'s old traits coming to the fore, from time
+to time, in Aunt Hillary%
+
+Anteater: Not a one. They have nothing in common. And there is no reason they should,
+as I see it. There are, after all, often several distinct ways to rearrange a group of
+parts to form a "sum". And Aunt Hillary was just a new "sum" of the old parts. Not
+MORE than the sum, mind you just that particular KIND of sum.
+
+Tortoise: Speaking of sums, I am reminded of number theory, where occasionally one
+will be able to take apart a theorem into its component symbols, rearrange them in a
+new order, and come up with a new theorem.
+
+Anteater: I've never heard of such a phenomenon, although I confess to being a total
+ignoramus in the field.
+
+Achilles: Nor have I heard of it-and I am rather well versed in the field, If I don't say so
+myself. I suspect Mr. T is just setting up one of his elaborate spoofs. I know him
+pretty well by now. Anteater: Speaking of number theory, I am reminded of J. S. F.
+again, for number theory is one of the domains in which he excelled. In fact, he made
+some rather rema, ..able contributions to number theory. Aunt Hillary, on the other
+hand, is remarkably dull-witted in anything that has even the remotest connection
+with mathematics. Also, she has only a rather banal taste in music, whereas
+Sebastiant was extremely gifted in music.
+
+Achilles: I am very fond of number theory. Could you possibly relate to us something of
+the nature of Sebastiant's contributions,
+
+Anteater: Very well, then. (Pauses for a moment to sip his tea, then resumes.)
+
+Have you heard of Fourmi's infamous "Well-Tested Conjecture”?
+
+Achilles. I'm not sure ... It sounds strangely familiar, and yet I can't quite place it.
+
+Anteater: It's a very simple idea. Lierre de Fourmi, a mathematiciant by vocation but
+lawyer by avocation, had been reading in his copy-of the classic text Arithmetica by
+Di of Antus, and came across a page containing the equation
+
+2 a +2 b =2 c
+
+He immediately realized that this equation has infinitely many solutions a. b, c, and
+then wrote in the margin the following notorious comment:
+
+
+Ant Fugue
+
+
+333
+
+
+
+
+FIGURE 63. During emigrations arm' ants sometimes create living bridges of their
+own bodies. In this photograph of such a bridge (de Fourmi Lierre), the workers of
+an Eciton burchelli colony can be seen linking their legs and, along the top of the
+bridge, hooking their tarsal claws together to form irregular systems of chains. .A
+symbiotic silverfish, Trichatelura manni, is seen crossing the bridge in the center.
+I From E. O. Wilson, The Insect Societies 'Cambridge, Mass.: Harvard University
+Press, 1971), p. 62)
+
+
+The equation
+
+
+n a +n b =n
+
+
+C
+
+
+has solutions in positive integers a, b, c, and n only when n = 2 (and then there are
+infinitely many triplets a, b, c which satisfy the equation); but there are no solutions
+for n > 2. I have discovered a truly marvelous proof of this statement, which,
+unfortunately, is so small that it would be well-nigh invisible if written in the margin.
+Ever since that year, some three hundred days ago, mathematiciants have been vainly
+trying to do one of two things: either to prove Fourmi’s claim, and thereby vindicate
+Fourmi’s reputation, which, although very high, has been somewhat tarnished by
+skeptics who think he never really found the proof he claimed to have found-or else
+to refute the claim, by finding a counterexample: a set of four integers a, b, c, and n,
+with n > 2, which satisfy the equation. Until very recently, every attempt in either
+direction had met with failure. To be sure, the Conjecture has been verified for many
+specific values of n-in particular, all n up to 125,000. But no one had succeeded in
+proving it for ALL n-no one, that is, until Johant Sebastiant Fermant came upon the
+scene. It was he who found the proof that cleared Fourmi’s name.
+
+
+Ant Fugue
+
+
+334
+
+
+It now goes under the name "Johant Sebastiant's Well-Tested Conjecture".
+
+Achilles: Shouldn't it be called a "Theorem" rather than a "Conjecture", if it's finally been
+given a proper proof;
+
+Anteater: Strictly speaking, you're right, but tradition has kept it this way.
+
+Tortoise: What sort of music did Sebastiant do?
+
+Anteater: He had great gifts for composition. Unfortunately, his greatest work is shrouded
+in mystery, for he never reached the point of publishing it. Some believe that he had
+it all in his mind; others are more unkind, saying that he probably never worked it out
+at all, but merely blustered about it.
+
+Achilles: What was the nature of this magnum opus?
+
+Anteater: It was to be a giant prelude and fugue; the fugue was to have
+twenty-four voices, and to involve twenty-four distinct subjects, one in
+each of the major and minor keys.
+
+Achilles: It would certainly be hard to listen to a twenty-four-voice fugue
+as a whole!
+
+Crab: Not to mention composing one!
+
+Anteater: But all that we know of it is Sebastiant's description of it, which he wrote in the
+
+margin of his copy of Buxtehude's Preludes and Fugues for Organ. The last words which
+
+he wrote before his tragic demise were:
+
+I have composed a truly marvelous fugue. In it, I have added
+together the power of 24 keys, and the power of 24 themes; I
+came up with a fugue with the power of 24 voices. Unfortunately,
+this margin is too narrow to contain it.
+
+And the unrealized masterpiece simply goes by the name, "Fermant’s Last Fugue".
+
+Achilles: Oh, that is unbearably tragic.
+
+Tortoise: Speaking of fugues, this fugue which we have been listening to is nearly over.
+Towards the end, there occurs a strange new twist on its theme. (Flips the page in the
+Well-Tempered Clavier.) Well, what have we here? A new illustration-how
+appealing! (Shows it to the Crab.)
+
+
+
+
+I
+
+f
+
+t
+
+fc
+
+k
+
+(
+
+k
+
+i
+
+t
+
+t
+
+
+I
+
+1
+
+)
+
+1
+
+3
+
+l
+
+J
+
+J
+
+t
+
+J
+
+
+FIGURE 64. [l>rawi7ig by Iht author.]
+
+
+
+I
+
+1
+
+
+0o
+
+
+i
+
+il I
+
+u
+
+
+
+Ant Fugue
+
+
+335
+
+
+
+
+
+
+Crab: Well, what have we here? Oh, I see: It's HOLISMIONSIM”, written in large letters
+that first shrink and then grow back to their original size. But that doesn't make any
+sense, because it's not a word. Oh me, oh my! (Passes it to the Anteater.)
+
+Anteater: Well, what have we here? Oh, I see: it's "REDUCTHOLISM", written in small
+letters that first grow and then shrink back to their original size. But that doesn't make
+any sense, because it's not a word. Oh my, oh me! (Passes it to Achilles.)
+
+Achilles: I know the rest of you won't believe this, but in fact this picture consists of the
+word "HOLISM" written twice, with the letters continually shrinking as they proceed
+from left to right. (Returns it to the Tortoise.)
+
+Tortoise: I know the rest of you won't believe this, but in fact this picture consists of the
+word "REDUCTIONISM" written once, with the letters continually growing as they
+proceed from left to right.
+
+Achilles: At last-I heard the new twist on the theme this time! I am so glad that you
+pointed it out to me, Mr. Tortoise. Finally, I think I am beginning to grasp the art of
+listening to fugues.
+
+
+Ant Fugue
+
+
+336
+
+
+
+Brains and Thoughts
+
+New Perspectives on Thought
+
+IT WAS ONLY with the advent of computers that people actually tried to create
+"thinking" machines, and witnessed bizarre variations on the theme, of thought. Programs
+were devised whose "thinking" was to human thinking as a slinky flipping end over end
+down a staircase is to human locomotion. All of a sudden the idiosyncrasies, the
+weaknesses and powers, the vagaries and vicissitudes of human thought were hinted at by
+the newfound ability to experiment with alien, yet hand-tailored forms of thought-or
+approximations of thought. As a result, we have acquired, in the last twenty years or so, a
+new kind of perspective on what thought is, and what it is not. Meanwhile, brain
+researchers have found out much about the small-scale and large-scale hardware of the
+brain. This approach has not yet been able to shed much light on how the brain
+manipulates concepts, but it gives us some ideas about the biological mechanisms on
+which thought manipulation rests.
+
+In the coming two Chapters, then, we will try to unite some insights gleaned from
+attempts at computer intelligence with some of the facts learned from ingenious
+experiments on living animal brains, as well as with results from research on human
+thought processes done by cognitive psychologists. The stage has been set by the
+Prelude, Ant Fugue', now we develop the ideas more deeply.
+
+Intensionality and Extensionality
+
+Thought must depend on representing reality in the hardware of the brain. In the
+preceding Chapters, we have developed formal systems which represent domains of
+mathematical reality in their symbolisms. To what extent is it reasonable to use such
+formal systems as models for how the brain might manipulate ideas?
+
+We saw, in the pq-system and then in other more complicated systems, how meaning, in
+a limited sense of the term, arose as a result of an isomorphism which maps typographical
+symbols onto numbers, operations, and relations; and strings of typographical symbols
+onto statements. Now in the brain we don't have typographical symbols, but we have
+something even better: active elements which can store information and transmit it and
+receive it from other active elements. Thus we have active symbols, rather than passive
+typographical symbols. In the brain, the rules
+
+
+Brains and Thoughts
+
+
+337
+
+
+
+re mixed right in with the symbols themselves, whereas on paper, the symbols are static
+entities, and the rules are in our heads.
+
+It is important not to get the idea, from the rather strict nature of all ie formal systems we
+have seen, that the isomorphism between symbols and real things is a rigid, one-to-one
+mapping, like the strings which link a marionette and the hand guiding it. In TNT, the
+notion "fifty" can be expressed in different symbolic ways; for example,
+
+((SSSSSSSO.SSSSSSSO)+(SO-SO))
+
+((SSSSSO'SSSSSO)+(SSSSSO.SSSSSO))
+
+'hat these both represent the same number is not a priori clear. You can manipulate each
+expression independently, and at some point stumble cross a theorem which makes you
+exclaim, "Oh-it's that number!"
+
+In your mind, you can also have different mental descriptions for a single person; for
+example,
+
+The person whose book I sent to a friend in Poland a while back.
+
+The stranger who started talking with me and my friends tonight in this coffee
+house.
+
+:'hat they both represent the same person is not a priori clear. Both descriptions may sit in
+your mind, unconnected. At some point during the evening you may stumble across a
+topic of conversation which leads to the revelation that they designate the same person,
+making you exclaim, Oh-you're that person!"
+
+Not all descriptions of a person need be attached to some central symbol for that person,
+which stores the person's name. Descriptions can be manufactured and manipulated in
+themselves. We can invent nonexistent people by making descriptions of them; we can
+merge two descriptions 'hen we find they represent a single entity; we can split one
+description into two when we find it represents two things, not one-and so on. This
+calculus of descriptions" is at the heart of thinking. It is said to be intentional and not
+extensional, which means that descriptions can "float" without Being anchored down to
+specific, known objects. The intensionality of thought is connected to its flexibility; it
+gives us the ability to imagine hypothetical worlds, to amalgamate different descriptions
+or chop one description into separate pieces, and so on.
+
+Suppose a friend who has borrowed your car telephones you to say hat your car skidded
+off a wet mountain road, careened against a bank, .nd overturned, and she narrowly
+escaped death. You conjure up a series & images in your mind, which get progressively
+more vivid as she adds details, and in the end you "see it all in your mind's eye". Then
+she tells you hat it's all been an April Fool's joke, and both she and the car are fine! In
+many ways that is irrelevant. The story and the images lose nothing of their vividness,
+and the memory will stay with you for a long, long time. Later, you may even think of
+her as an unsafe driver because of the strength of
+
+
+Brains and Thoughts
+
+
+338
+
+
+
+the first impression, which should have been wiped out when you learned it was all
+untrue. Fantasy and fact intermingle very closely in our minds, and this is because
+thinking involves the manufacture and manipulation of complex descriptions, which need
+in no way be tied down to real events or things.
+
+A flexible, intensional representation of the world is what thinking is all about. Now how
+can a physiological system such as the brain support such a system?
+
+The Brain's "Ants"
+
+The most important cells in the brain are nerve cells, or neurons (see Fig. 65), of which
+there are about ten billion. (Curiously, outnumbering the neurons by about ten to one are
+the glial cells, or glia. Glia are believed to play more of a supporting role to the neurons'
+starring role, and therefore we will not discuss them.) Each neuron possesses a number of
+synapses ("entry ports") and one axon ("output channel"). The input and output are
+electrochemical flows: that is, moving ions. In between the entry ports of a neuron and its
+output channel is its cell body, where "decisions" are made.
+
+
+
+HOC RE 65.
+nsuroft. [/f daptx
+Maihinri j of
+AflCrtfw.Wr//, I9i
+
+
+FIGURE 65. Schematic drawing of a neuron. [Adapted From D. Wooldridge, The
+Machinery of the Brain (New York:"- McGraw-Hill, 1963), p. 6.
+
+
+Brains and Thoughts
+
+
+339
+
+
+
+The type of decision which a neuron faces-and this can take place up to a thousand times
+per second-is this: whether or not to fire-that is, to ease ions down its axon, which -
+eventually will cross over into the entry its of one or more other neurons, thus causing
+them to make the same sort of decision. The decision is made in a very simple manner: if
+the sum all inputs exceeds a certain threshold, yes; otherwise, no. Some of the inputs can
+be negative inputs, which cancel out positive inputs coming from somewhere else. In any
+case, it is simple addition which rules the lowest 'el of the mind. To paraphrase Descartes'
+famous remark, "I think, therefore I sum" (from the Latin Cogito, ergo am).
+
+Now although the manner of making the decision sounds very simple, here is one
+fact which complicates the issue: there may be as many as 200,000 separate entry ports to
+a neuron, which means that up to 200,000 Karate summands may be involved in
+determining the neuron's next ion. Once the decision has been made, a pulse of ions
+streaks down the on towards its terminal end. Before the ions reach the end, however, ey
+may encounter a bifurcation-or several. In such cases, the single output pulse splits up as
+it moves down the bifurcating axon, and by the tine it has reached the end, "it" has
+become "they"-and they may reach their destinations at separate times, since the axon
+branches along which they travel may be of different lengths and have different
+resistivities. The important thing, though, is that they all began as one single pulse,
+moving 'ay from the cell body. After a neuron fires, it needs a short recovery time fore
+firing again; characteristically this is measured in milliseconds, so at a neuron may fire up
+to about a thousand times per second.
+
+Larger Structures in the Brain
+
+Now we have described the brain's "ants". What about "teams", or "signals"? What about
+"symbols"? We make the following observation: despite e complexity of its input, a
+single neuron can respond only in a very primitive way-by firing, or not firing. This is a
+very small amount of Formation. Certainly for large amounts of information to be carried
+or processed, many neurons must be involved. And therefore one might guess at larger
+structures, composed from many neurons, would exist, which handle concepts on a
+higher level. This is undoubtedly true, but the most naive assumption-that there is a fixed
+group of neurons for each different concept-is almost certainly false.
+
+There are many anatomical portions of the brain which can be distinguished from
+each other, such as the cerebrum, the cerebellum, the hypothalamus (see Fig. 66). The
+cerebrum is the largest part of the human am, and is divided into a left hemisphere and a
+right hemisphere. The outer few millimeters of each cerebral hemisphere are coated with
+a layered "bark", or cerebral cortex. The amount of cerebral cortex is the major
+distinguishing feature, in terms of anatomy, between human brains and brains of less
+intelligent species. We will not describe any of the brain's suborgans in detail because, as
+it turns out, only the roughest mapping can
+
+
+Brains and Thoughts
+
+
+340
+
+
+
+Motor area
+
+
+Central sulcus
+
+
+
+FIGURE 66. The human brain, seen from the left side. It is strange that the visual area is
+in the back of the head. [From Steven Rose, The Conscious Brain, updated ed. (New
+York: Vintage, 1966), p. 50. ]
+
+at this time be made between such large-scale suborgans and the activities, mental or
+physical, which they are responsible for. For instance, it is known that language is
+primarily handled in one of the two cerebral hemispheres-in fact, usually the left
+hemisphere. Also, the cerebellum is the place where trains of impulses are sent off to
+muscles to control motor activity. But how these areas carry out their functions is still
+largely a mystery.
+
+
+Mappings between Brains
+
+Now an extremely important question comes up here. If thinking does take place in the
+brain, then how are two brains different from each other? How is my brain different from
+yours? Certainly you do not think exactly as I do, nor as anyone else does. But we all
+have the same anatomical divisions in our brains. How far does this identity of brains
+extend? Does it go to the neural level? Yes, if you look at animals on a low enough level
+of the thinking-hierarchy-the lowly earthworm, for instance. The following quote is from
+the neurophysiologist, David Hubei, speaking at a conference on communication with
+extraterrestrial intelligence:
+
+The number of nerve cells in an animal like a wonn would be measured, I
+suppose, in the thousands. One very interesting thing is that we may point to a
+particular individual cell in a particular earthworm, and then identify the same
+cell, the corresponding cell in another earthworm of the same species.'
+
+
+Brains and Thoughts
+
+
+341
+
+
+
+Earthworms have isomorphic brains! One could say, "There is only one earthworm."
+
+But such one-to-one mappability between individuals' brains disappears very soon as
+you ascend in the thinking-hierarchy and the number of neurons increases-confirming
+one's suspicions that there is not just one pan! Yet considerable physical similarity can be
+detected between different human brains when they are compared on a scale larger than a
+;le neuron but smaller than the major suborgans of the brain. What s this imply about how
+individual mental differences are represented in physical brain? If we looked at my
+neurons' interconnections, could we 1 various structures that could be identified as coding
+for specific things -tow, specific beliefs I have, specific hopes, fears, likes and dislikes I
+harbor? If mental experiences can be attributed to the brain, can knowledge and other
+aspects of mental life likewise be traced to specific locations de the brain, or to specific
+physical subsystems of the brain? This will be a central question to which we will often
+return in this Chapter and the next.
+
+Localization of Brain Processes: An Enigma
+
+In an attempt to answer this question, the neurologist Karl Lashley, in a series of
+experiments beginning around 1920 and running for many ,s, tried to discover where in
+its brain a rat stores its knowledge about :e running. In his book The Conscious Brain,
+Steven Rose describes Lashley's trials and tribulations this way:
+
+Lashley was attempting to identify the locus of memory within the cortex, and, to do so,
+first trained rats to run mazes, and then removed various cortical regions. He allowed the
+animals to recover and tested the retention of the maze-running skills. To his suiprise it
+was not possible to find a particular region corresponding to the ability to remember the
+way through a maze, instead all the rats which had had cortex regions removed suffered
+some kind f impairment, and the extent of the impairment was roughly proportional to the
+amount of cortex taken off. Removing cortex damaged the motor and sensory capacities
+of the animals, and they would limp, hop, roll, or stagger, but somehow they always
+managed to traverse the maze. So far as memory 'as concerned, the cortex appeared to be
+equipotential, that is, with all regions of equal possible utility. Indeed, Lashley concluded
+rather gloomily in is last paper "In Search of the Engram", which appeared in 1950, that
+the only conclusion was that memory was not possible at all.’
+
+Curiously, evidence for the opposite point of view was being developed :in Canada at
+roughly the same time that Lashley was doing his last work, in late 1940's. The
+neurosurgeon Wilder Penfield was examining the reactions of patients whose brains had
+been operated on, by inserting electrodes into various parts of their exposed brains, and
+then using small electrical pulses to stimulate the neuron or neurons to which the
+electrodes been attached. These pulses were similar to the pulses which come other
+neurons. What Penfield found was that stimulation of certain
+
+
+Brains and Thoughts
+
+
+342
+
+
+
+neurons would reliably create specific images or sensations in the patient. These
+artificially provoked impressions ranged from strange but indefinable fears to buzzes and
+colors, and, most impressively of all, to entire successions of events recalled from some
+earlier time of life, such as a childhood birthday party. The set of locations which could
+trigger such specific events was extremely small-basically centered upon a single neuron.
+Now these results of Penfield dramatically oppose the conclusions of Lashley, since they
+seem to imply that local areas are responsible for specific memories, after all.
+
+What can one make of this? One possible explanation could be that memories are coded
+locally, but over and over again in different areas of the cortex-a strategy perhaps
+developed in evolution as security against possible loss of cortex in fights, or in
+experiments conducted by neurophysiologists. Another explanation would be that
+memories can be reconstructed from dynamic processes spread over the whole brain, but
+can be triggered from local spots. This theory is based on the notion of modern telephone
+networks, where the routing of a long-distance call is not predictable in advance, for it is
+selected at the time the call is placed, and depends on the situation all over the whole
+country. Destroying any local part of the network would not block calls; it would just
+cause them to be routed around the damaged area. In this sense any call is potentially
+nonlocalizable. Yet any call just connects up two specific points; in this sense any call is
+localizable.
+
+
+Specificity in Visual Processing
+
+Some of the most interesting and significant work on localization of brain processes has
+been done in the last fifteen years by David Hubei and Torsten Wiesel, at Harvard. They
+have mapped out visual pathways in the brains of cats, starting with the neurons in the
+retina, following their connections towards the rear of the head, passing through the
+"relay station" of the lateral geniculate, and ending up in the visual cortex, at the very
+back of the brain. First of all, it is remarkable that there exist well defined neural
+pathways, in light of Lashley's results. But more remarkable are the properties of the
+neurons located at different stages along the pathway.
+
+It turns out that retinal neurons are primarily contrast sensors. More specifically, the
+way they act is this. Each retinal neuron is normally firing at a "cruising speed". When its
+portion of the retina is struck by light, it may either fire faster or slow down and even
+stop firing. However, it will do so only provided that the surrounding part of the retina is
+less illuminated. So this means that there are two types of neuron: ."on-center", and "off-
+center". The on-center neurons are those whose firing rate increases whenever, in the
+small circular retinal area to which they are sensitive, the center is bright but the outskirts
+are dark; the off-center neurons are those which fire faster when there is darkness in the
+center and brightness in the
+
+
+Brains and Thoughts
+
+
+343
+
+
+
+outer ring. If an on-center pattern is shown to an off-center neuron, the neuron will slow
+down in firing (and vice versa). Uniform illumination will .leave both types of retinal
+neuron unaffected; they will continue to fire at cruising speed.
+
+From the retina, signals from these neurons proceed via the optic nerve to the lateral
+geniculate, located somewhere towards the middle of the brain. There, one can find a
+direct mapping of the retinal surface in the .use that there are lateral-geniculate neurons
+which are triggered only by specific stimuli falling on specific areas of the retina. In that
+sense, the general geniculate is disappointing; it seems to be only a "relay station", and
+not a further processor (although to give it its due, the contrast sensitivity ,ms to be
+enhanced in the lateral geniculate). The retinal image is coded a straightforward way in
+the firing patterns of the neurons in the lateral geniculate, despite the fact that the neurons
+there are not arranged on a o-dimensional surface in the form of the retina, but in a three-
+dimensional block. So two dimensions get mapped onto three, yet the formation is
+preserved: an isomorphism. There is probably some deep meaning to the change in the
+dimensionality of the representation, which is not yet fully appreciated. In any case, there
+are so many further unexplained stages of vision that we should not be disappointed but
+pleased the fact that-to some extent-we have figured out this one stage!
+
+From the lateral geniculate, the signals proceed back to the visual cortex. Here, some
+new types of processing occur. The cells of the visual cortex are divided into three
+categories: simple, complex, and hyper complex. Simple cells act very much like retinal
+cells or lateral geniculate [Is: they respond to point-like light or dark spots with
+contrasting surrounds, in particular regions of the retina. Complex cells, by contrast,
+usually receive input from a hundred or more other cells, and they detect light dark bars
+oriented at specific angles on the retina (see Fig. 67). Hyper complex cells respond to
+corners, bars, or even "tongues" moving in specific directions (again see Fig. 67). These
+latter cells are so highly specialized at they are sometimes called "higher-order hyper
+complex cells".
+
+
+A "Grandmother Cell"?
+
+Because of the discovery of cells in the visual cortex which can be triggered stimuli of
+ever-increasing complexity, some people have wondered if things are not leading in the
+direction of "one cell, one concept"-for ample, you would have a "grandmother cell"
+which would fire if, and only if, your grandmother came into view. This somewhat
+humorous ample of a "superhypercomplex cell" is not taken very seriously. Rower, it is
+not obvious what alternative theory seems reasonable. One possibility is that larger neural
+networks are excited collectively by sufficiently complex visual stimuli. Of course, the
+triggering of these larger multineuron units would somehow have to come from
+integration of signals emanating from the many hyper complex cells. How this might be
+done nobody knows> Just when we seem to be approaching the threshold where
+
+
+Brains and Thoughts
+
+
+344
+
+
+
+(a) Complex cell (area 17)
+Stimulus, Response Stimulus; Response
+
+Strong
+
+
+Ulirasuperhyper-
+
+tb) H) percomplex cell complex tell
+(area 19) (until 1729)
+
+Stimulus; Response Stimulus: Response
+
+
+
+
+
+Weak
+
+
+H
+
+E
+
+E
+
+
+Zrro
+
+
+Zero
+
+
+
+Zero
+
+
+
+Strong
+
+
+I
+
+□
+
+
+Strong
+
+
+
+Stiong
+
+
+Zero
+
+
+□
+
+tl
+
+2]
+
+Efi
+
+d
+
+
+Weak
+
+
+Zero
+
+
+Strong
+
+
+c p ,
+
+■
+
+) Strong
+
+
+Medium
+
+
+. . . a Sl "
+
+% iS Weak
+
+
+| ^| Zrr »
+
+
+nr./rpr <7 I.. rsrlntn uimhl* nsurnrix
+
+
+FIGURE 67. Responses to patterns by certain sample neurons.
+
+(a) This edge-detecting neuron looks for vertical edges with light on the left
+and dark on the right. The first column shows how the orientation of an
+edge is relevant to this neuron. The second column shows how the position
+of the edge within the field is irrelevant, for this particular neuron, (b)
+Showing how a hyper complex cell responds more selectively: here, only
+when the descending tongue is in the middle of the field, (c) The responses
+of a hypothetical "grandmother cell" to various random stimuli; the reader
+may enjoy pondering how an "octopus cell" would respond to the same
+stimuli.
+
+
+"symbol" might emerge from "signal", the trail gets lost-a tantalizingly unfinished story.
+We will return to this story shortly, however, and try to fill in some of it.
+
+Earlier I mentioned the coarse-grained isomorphism between all human brains which
+exists on a large anatomical scale, and the very fine-grained, neural-level isomorphism
+which exists between earthworm brains. It is quite interesting that there is also an
+isomorphism between the visual processing apparatus of cat, monkey, and human, the
+"grain" of which is somewhere between coarse and fine. Here is how that isomorphism
+works. First of all, all three species have "dedicated" areas of cortex at the back of their
+brains where visual processing is done: the visual cortex. Secondly, in
+
+
+Brains and Thoughts
+
+
+345
+
+
+
+
+each of them, the visual cortex breaks up into three subregions, called areas 18, and 19 of
+the cortex. These areas are still universal, in the sense that y can be located in the brain of
+any normal individual in any of the three ties. Within each area you can go still further,
+reaching the "columnar" organization of the visual cortex. Perpendicular to the surface of
+the (ex, moving radially inwards towards the inner brain, visual neurons are inged in
+"columns"-that is, almost all connections move along the ial, columnar direction, and not
+between columns. And each column ps onto a small, specific retinal region. The number
+of columns is not same in each individual, so that one can't find "the same column", ally,
+within a column, there are layers in which simple neurons tend to found, and other layers
+in which complex neurons tend to be found, to hypercomplex neurons tend to be found in
+areas 18 and 19 predominately, while the simple and complex ones are found mostly in
+area 17.) appears that we run out of isomorphisms at this level of detail. From here down
+to the individual neuron level, each individual cat, monkey, or man has a completely
+unique pattern-somewhat like a fingerprint or a signature.
+
+One minor but perhaps telling difference between visual processing in ;'brains and
+monkeys' brains has to do with the stage at which informal from the two eyes is
+integrated to yield a single combined higher-level lal. It turns out that it takes place
+slightly later in the monkey than in the cat, which gives each separate eye's signal a
+slightly longer time to get processed by itself. This is not too surprising, since one would
+expect that higher a species lies in the intelligence hierarchy, the more complex will the
+problems which its visual system will be called upon to handle; and before signals ought
+to pass through more and more early processing ore receiving a final "label". This is quite
+dramatically confirmed by observations of the visual abilities of a newborn calf, which
+seems to be born with as much power of visual discrimination as it will ever have. It will
+shy away from people or dogs, but not from other cattle. Probably its entire visual system
+is "hard-wired" before birth, and involves relatively little optical processing. On the other
+hand, a human's visual system, so deeply ant on the cortex, takes several years to reach
+maturity.
+
+
+Funneling into Neural Modules
+
+A puzzling thing about the discoveries so far made about the organization the brain is
+that few direct correspondences have been found between large-scale hardware and high-
+level software. The visual cortex, for instance, is a large-scale piece of hardware, which is
+entirely dedicated to a it software purpose-the processing of visual information-yet all of
+processing so far discovered is still quite low-level. Nothing approaching recognition of
+objects has been localized in the visual cortex. This means that no one knows where or
+how the output from complex and hypercomplex cells gets transformed into conscious
+recognition of shapes.
+
+
+Brains and Thoughts
+
+
+346
+
+
+
+rooms, pictures, faces, and so on. People have looked for evidence of the "funneling" of
+many low-level neural responses into fewer and fewer higher-level ones, culminating in
+something such as the proverbial grandmother cell, or some kind of multineuron network,
+as mentioned above. It is evident that this will not be found in some gross anatomical
+division of the brain, but rather in a more microscopic analysis.
+
+One possible alternative to the the grandmother cell might be a fixed set of neurons, say
+a few dozen, at the thin end of the "funnel", all of which fire when Granny comes into
+view. And for each different recognizable object, there would be a unique network and a
+funneling process that would focus down onto that network. There are more complicated
+alternatives along similar lines, involving networks which can be excited in different
+manners, instead of in a fixed manner. Such networks would be the "symbols" in our
+brains.
+
+But is such funneling necessary? Perhaps an object being looked at is implicitly
+identified by its "signature" in the visual cortex-that is, the collected responses of simple,
+complex, and hypercomplex cells. Perhaps the brain does not need any further recognizer
+for a particular form. This theory, however, poses the following problem. Suppose you
+are looking at a scene. It registers its signature on your visual cortex; but then how do you
+get from that signature to a verbal description of the scene? For instance, the paintings of
+Edouard Vuillard, a French post-impressionist, often take a few seconds of scrutiny, and
+then suddenly a human figure will jump out at you. Presumably the signature gets
+imprinted on the visual cortex in the first fraction of a second-but the picture is only
+understood after a few seconds. This is but one example of what is actually a common
+phenomenon-a sensation of something "crystallizing" in your mind at the moment of
+recognition, which takes place not when the light rays hit your retina, but sometime later,
+after some part of your intelligence has had a chance to act on the retinal signals.
+
+The crystallization metaphor yields a pretty image derived from statistical mechanics,
+of a myriad microscopic and uncorrelated activities in a medium, slowly producing local
+regions of coherence which spread and enlarge; in the end, the myriad small events will
+have performed a complete structural revamping of their medium from the bottom up,
+changing' it from a chaotic assembly of independent elements into one large, coherent,
+fully linked structure. If one thinks of the early neural activities as independent, and of
+the end result of their many independent firings as the triggering of a well-defined large
+"module" of neurons, then the word "crystallization" seems quite apt.
+
+Another argument for funneling is based on the fact that there are a myriad distinct
+scenes which can cause you to feel you have perceived the same object-for example, your
+grandmother, who may be smiling or frowning, wearing a hat or not, in a bright garden or
+a dark train station, seen from near or far, from side or front, and so on. All these scenes
+produce extremely different signatures on the visual cortex; yet all of them could prompt
+you to say "Hello, Granny." So a funneling process must take
+
+
+Brains and Thoughts
+
+
+347
+
+
+
+place at some point after the reception of the visual signature and before e words are
+uttered. One could claim that this funneling is not part of the perception of Granny, but
+just part of verbalization. But it seems quite unnatural to partition the process that way,
+for you could internally use the formation that it is Granny without verbalizing it. It
+would be very it unwieldy to handle all of the information in the entire visual cortex,
+when much of it could be thrown away, since you don't care about where shadows fall or
+how many buttons there are on her blouse, etc.
+
+Another difficulty with a non-funneling theory is to explain how there in be different
+interpretations for a single signature-for example, the Escher picture Convex a4 Concave
+(Fig. 23). Just as it seems obvious to us tat we do not merely perceive dots on a television
+screen, but chunks, likewise it seems ridiculous to postulate that perception has taken
+place hen a giant dot-like "signature" has been created on the visual cortex, here must be
+some funneling, whose end result is to trigger some specific modules of neurons, each of
+which is associated with the concepts-the funks-in the scene.
+
+Modules Which Mediate Thought Processes
+
+Thus we are led to the conclusion that for each concept there is a fairly ell-defined
+module which can be triggered-a module that consists of a nail group of neurons-a
+"neural complex" of the type suggested earlier, problem with this theory-at least if it is
+taken naively-is that it would suggest that one should be able to locate such modules
+somewhere within to brain. This has not yet been done, and some evidence, such as the
+experiments by Lashley, points against localization. However, it is still too early to tell.
+There may be many copies of each module spread around, or modules may overlap
+physically; both of these effects would tend to obscure any division of neurons into
+"packets". Perhaps the complexes are like very thin pancakes packed in layers which
+occasionally pass through each other; perhaps they are like long snakes which curl
+around each other, here and there flattening out, like cobras' heads; perhaps they are like
+spiderwebs; or perhaps they are circuits in which signals travel round id round in shapes
+stranger than the dash of a gnat-hungry swallow, here is no telling. It is even possible that
+these modules are software, ether than hardware, phenomena-but this is something which
+we will discuss later
+
+There are many questions that come to mind concerning these hypothesized neural
+complexes. For instance:
+
+Do they extend into the lower regions of the brain, such as the
+midbrain, the hypothalamus, etc.?
+
+Can a single neuron belong to more than one such complex?
+
+To how many such complexes can a single neuron belong?
+
+By how many neurons can such complexes overlap?
+
+
+Brains and Thoughts
+
+
+348
+
+
+
+Are these complexes pretty much the same for everybody?
+
+Are corresponding ones found in corresponding places in different
+people's brains?
+
+Do they overlap in the same way in everybody's brain?
+
+Philosophically, the most important question of all is this: "hat would the existence of
+modules-for instance, a grandmother module-tell us? Would this give us any insight into
+the phenomenon of our own consciousness? Or would it still leave us as much in the dark
+about what consciousness is, as does knowledge that a brain is built out of neurons and
+glia? As you might guess from reading the Ant Fugue, my feeling is that it would go a
+long way towards giving us an understanding of the phenomenon of consciousness. The
+crucial step that needs to be taken is from a low-level-neuron-by-neuron-description of
+the state of a brain, to a high-level-module-by-module-description of the same state of the
+same brain. Or, to revert to the suggestive terminology of the Ant Fugue, we want to shift
+the description of the brain state from the signal level to the symbol, level.
+
+Active Symbols
+
+Let us from now on refer to these hypothetical neural complexes, neural modules,
+neural packets, neural networks, multineuron units-call them what you will, whether they
+come in the form of pancakes, garden rakes, rattlesnakes, snowflakes, or even ripples on
+lakes-as symbols. A description of a brain state in terms of symbols was alluded to in the
+Dialogue. What would such a description be like? What kinds of concepts is it reasonable
+to think actually might be "symbolized"? What kinds of interrelations would symbols
+have? And what insights would this whole picture provide into consciousness?
+
+The first thing to emphasize is that symbols can be either dormant, or awake (activated).
+An active symbol is one which has been triggered-that is, one in which a threshold
+number of neurons have been caused to fire by stimuli coming from outside. Since a
+symbol can be triggered in many different ways, it can act in many different ways when
+awakened. This suggests that we should think of a symbol not as a fixed entity, but as a
+variable entity. Therefore it would not suffice to describe a brain state by saying
+"Symbols A, B, ..., N are all active"; rather, we would have to supply in addition a set of
+parameters for each active symbol, characterizing some aspects of the symbol's internal
+workings. It is an interesting question whether in each symbol there are certain core
+neurons, which invariably fire when the symbol is activated. If such a core set of neurons
+exists, we might refer to it as the "invariant core" of the symbol. It is tempting to assume
+that each time you think of, say, a waterfall, some fixed neural process is repeated,
+without doubt embellished in different ways depending on the context, but reliably
+occurring. However, it is not clear that this must be so.
+
+
+Brains and Thoughts
+
+
+349
+
+
+
+Now what does a symbol do, when awakened? A low-level description would say,
+"Many of its neurons fire." But this no longer interests us. The high-level description
+should eliminate all reference to neurons, and concentrate exclusively on symbols. So a
+high-level description of what makes a symbol active, as distinguished from dormant,
+would be, "It sends out messages, or signals, whose purpose is to try to awaken, or
+trigger, other symbols." Of course these messages would be carried as streams of nerve
+impulses, by neurons-but to the extent that we can avoid such phraseology, we should,
+for it represents a low-level way of looking at things, and we hope that we can get along
+on purely a high level. In other words, we hope at thought processes can be thought of as
+being sealed off from neural events in the same way that the behavior of a clock is sealed
+off from the laws of quantum mechanics, or the biology of cells is sealed off from the
+laws of quarks.
+
+But what is the advantage of this high-level picture? Why is it better to say, "Symbols A
+and B triggered symbol C" than to say, "Neurons 183 through 612 excited neuron 75 and
+caused it to fire"? This question was answered in the Ant Fugue: It is better because
+symbols symbolize things, and neurons don't. Symbols are the hardware realizations of
+concepts. Whereas group of neurons triggering another neuron corresponds to no outer
+event, the triggering of some symbol by other symbols bears a relation to events in the
+real world-or in an imaginary world. Symbols are related to each other by the messages
+which they can send back and forth, in such a way that their triggering patterns are very
+much like the large-scale events rich do happen in our world, or could happen in a world
+similar to ours, essence, meaning arises here for the same reason as it did in the -system-
+isomorphism; only here, the isomorphism is infinitely more complex, subtle, delicate,
+versatile, and intensional.
+
+Incidentally, the requirement that symbols should be able to pass sophisticated
+messages to and fro is probably sufficient to exclude neurons themselves from playing
+the role of symbols. Since a neuron has only a single way of sending information out of
+itself, and has no way of selectively selecting a signal now in one direction, now in
+another, it simply does not have the kind of selective triggering power which a symbol
+must have to act e an object in the real world. I n his book The Insect Societies, E. O.
+Wilson makes a similar point about how messages propagate around inside ant colonies:
+
+[Mass communication] is defined as the transfer, among groups, of information that a
+
+single individual could not pass to another.'
+
+It is not such a bad image, the brain as an ant colony!
+
+The next question-and an extremely important one it is, too concerns the
+nature and "size" of the concepts which are represented in the tin by single
+symbols. About the nature of symbols there are questions like this: Would there be
+a symbol for the general notion of waterfalls, or would there be different symbols
+for various specific waterfalls? Or would both of these alternatives be realized?
+About the "size" of symbols, there are questions like this: Would there be a symbol
+for an entire story? Or for a
+
+
+Brains and Thoughts
+
+
+350
+
+
+
+melody? Or a joke? Or is it more likely that there would only be symbols for concepts
+roughly the size of words, and that larger ideas, such as phrases or sentences, would be
+represented by concurrent or sequential activation of various symbols?
+
+Let us consider the issue of the size of concepts represented by symbols. Most thoughts
+expressed in sentences are made up out of basic, quasi-atomic components which we do
+not usually analyze further. These are of word size, roughly-sometimes a little longer,
+sometimes a little shorter. For instance, the noun "waterfall", the proper noun "Niagara
+Falls", the past-tense suffix "-ed", the verb "to catch up with", and longer idiomatic
+phrases are all close to atomic. These are typical elementary brush strokes which we use
+in painting portraits of more complex concepts, such as the plot of a movie, the flavor of
+a city, the nature of consciousness, etc. Such complex ideas are not single brush strokes..
+It seems reasonable to think that the brush strokes of language are also brush strokes of
+thought, and therefore that symbols represent concepts of about this size. Thus a symbol
+would be roughly something for which you know a word or stock phrase, or with which
+you associate a proper name. And the representation in the brain of a more complex idea,
+such as a problem in a love affair, would be a very complicated sequence of activations
+of various symbols by other symbols.
+
+Classes and Instances
+
+There is a general distinction concerning thinking: that between categories and
+individuals, or classes and instances. (Two other terms sometimes used are "types" and
+"tokens".) It might seem at first sight that a given symbol would inherently be either a
+symbol for a class or a symbol for an instance-but that is an oversimplification. Actually,
+most symbols may play either role, depending on the context of their activation. For
+example, look at the list below:
+
+(1) a publication
+
+(2) a newspaper
+
+(3) The San Francisco Chronicle
+
+(4) the May 18 edition of the Chronicle
+
+(5) my copy of the May 18 edition of the Chronicle
+
+(6) my copy of the May 18 edition of the Chronicle as
+
+it was when I first picked it up (as contrasted with
+my copy as it was a few days later: in my fireplace,
+burning)
+
+Here, lines 2 to 5 all play both roles. Thus, line 4 is an instance of of the general class of
+line 3, and line 5 is an instance of line 4. Line 6 is a special kind of instance of a class: a
+manifestation. The successive stages of an object during its life history are its
+manifestations. It is interesting to wonder if the cows on a farm perceive the invariant
+individual underneath all the manifestations of the jolly farmer who feeds then hay.
+
+
+Brains and Thoughts
+
+
+351
+
+
+
+The Prototype Principle
+
+
+The list above seems to be a hierarchy of generality-the top being a very road conceptual
+category, the bottom some very humble particular thing located in space and time.
+However, the idea that a "class" must always be enormously broad and abstract is far too
+limited. The reason- is that our thought makes use of an ingenious principle, which might
+be called the prototype principle:
+
+The most specific event can serve as a general example
+of a class of events.
+
+Everyone knows that specific events have a vividness which imprints them i strongly on
+the memory that they can later be used as models for other vents which are like them in
+some way. Thus in each specific event, there is the germ of a whole class of similar
+events. This idea that there is generality in the specific is of far-reaching importance.
+
+Now it is natural to ask: Do the symbols in the brain represent classes, r instances? Are
+there certain symbols which represent only classes, while other symbols represent only
+instances? Or can a single symbol serve duty either as a class symbol or instance symbol,
+depending which parts of it are activated? The latter theory seems appealing; one might
+think that a "light" activation of a symbol might represent a class, and that a deeper, or
+more complex, activation would contain more detailed internal neural firing patterns, and
+hence would represent an instance. But on second thought, its is crazy: it would imply,
+for example, that by activating the symbol for publication" in a sufficiently complex way,
+you would get the very complex symbol which represents a specific newspaper burning
+in my fireplace. And very other possible manifestation of every other piece of printed
+matter would be represented internally by some manner of activating the single symbol
+for "publication". That seems much too heavy a burden to place on to single symbol
+"publication". One must conclude, therefore, that finance symbols can exist side by side
+with class symbols, and are not just lodes of activation of the latter.
+
+The Splitting-off of Instances from Classes
+
+On the other hand, instance symbols often inherit many of their properties from the
+classes to which those instances belong. If I tell you I went to see a Movie, you will begin
+"minting" a fresh new instance symbol for that particular movie; but in the absence of
+more information, the new instance symbol will have to lean rather heavily on your pre¬
+existing class symbol for movie". Unconsciously, you will rely on a host of
+presuppositions about at movie-for example, that it lasted between one and three hours,
+that it was shown in a local theater, that it told a story about some people, and so i. These
+are built into the class symbol as expected links to other symbols e., potential triggering
+relations), and are called de fault options. In any
+
+
+Brains and Thoughts
+
+
+352
+
+
+
+freshly minted instance symbol, the default options can easily be overridden, but unless
+this is explicitly done, they will remain in the instance symbol, inherited from its class
+symbol. Until they are overridden, they provide some preliminary basis for you to think
+about the new instance for example, the movie I went to see-by using the reasonable
+guesses which are supplied by the "stereotype", or class symbol.
+
+A fresh and simple instance is like a child without its own ideas or experiences-it relies
+entirely on its parents' experiences and opinions and just parrots them. But gradually, as it
+interacts more and more with the rest of the world, the child acquires its own
+idiosyncratic experiences and inevitably begins to split away from the parents.
+Eventually, the child becomes a full-fledged adult. In the same way, a fresh instance can
+split off from its parent class over a period of time, and become a class, or prototype, in
+its own right.
+
+For a graphic illustration of such a splitting-off process, suppose that some Saturday
+afternoon you turn on your car radio, and happen to tune in on a football game between
+two "random" teams. At first you do not know the names of the players on either team.
+All you register, when the announcer says, "Palindromi made the stop on the twenty-
+seven yard line, and that brings up fourth down and six to go," is that some player
+stopped some other player. Thus it is a case of activation of the class symbol "football
+player", with some sort of coordinated activation of the symbol for tackling. But then as
+Palindromi figures in a few more key plays, you begin building up a fresh instance
+symbol for him in particular, using his name, perhaps, as a focal point. This symbol is
+dependent, like a child, on the class symbol for "football player": most of your image of
+Palindromi is supplied by your stereotype of a football player as contained in the
+"football player" symbol. But gradually, as more information comes to you, the
+"Palindromi" symbol becomes more autonomous, and relies less and less on concurrent
+activation of its parent class symbol. This may happen in a few minutes, as Palindromi
+makes a few good plays and stands out. His teammates may still all be represented by
+activations of the class symbol, however. Eventually, perhaps after a few days, when you
+have read some articles in the sports section of your paper, the umbilical cord is broken,
+and Palindromi can stand on his own two feet. Now you know such things as his home
+town and his major in college; you recognize his face; and so on. At this point,
+Palindromi is no longer conceived of merely as a football player, but as a human being
+who happens also to be a football player. "Palindromi" is an instance symbol which can
+become active while its parent class symbol (football player) remains dormant.
+
+Once, the Palindromi symbol was a satellite orbiting around its mother symbol, like an
+artificial satellite circling the Earth, which is so much bigger and more massive. Then
+there came an intermediate stage, where one symbol was more important than the other,
+but they could be seen as orbiting around each other-something like the Earth and the
+Moon. Finally, the new symbol becomes quite autonomous; now it might easily serve as
+a class symbol around which could start rotating new satellites-
+
+
+Brains and Thoughts
+
+
+353
+
+
+
+symbols for other people who are less familiar but who have something in common with
+Palindromi, and for whom he can serve as a temporary stereotype, until you acquire more
+information, enabling the new symbols so to become autonomous.
+
+The Difficulty of Disentangling Symbols from Each Other
+
+These stages of growth and eventual detachment of an instance from a ass will be
+distinguishable from each other by the way in which the symbols involved are linked.
+Sometimes it will no doubt be very difficult to 11 just where one symbol leaves off and
+the other one begins. How "active" the one symbol, compared to the other? If one can be
+activated independently of the other, then it would be quite sensible to call them
+autonomous.
+
+We have used an astronomy metaphor above, and it is interesting that to problem of the
+motion of planets is an extremely complex one-in fact the general problem of three
+gravitationally interacting bodies (such as the Earth, Moon, and Sun) is far from solved,
+even after several centuries of work. One situation in which it is possible to obtain good
+approximate solutions, however, is when one body is much more massive than the other
+two (here, the Sun); then it makes sense to consider that body as stationary, with the other
+two rotating about it: on top of this can finally be added the interaction between the two
+satellites. But this approximation depends on breaking up the system into the Sun, and a
+"cluster": the Earth-Moon 'stem. This is an approximation, but it enables the system to be
+understood quite deeply. So to what extent is this cluster a part of reality, and to hat
+extent is it a mental fabrication, a human imposition of structure on me universe? This
+problem of the "reality" of boundaries drawn between hat are perceived to be autonomous
+or semi-autonomous clusters will create endless trouble when we relate it to symbols in
+the brain.
+
+One greatly puzzling question is the simple issue of plurals. How do we visualize, say,
+three dogs in a teacup? Or several people in an elevator? Do we begin with the class
+symbol for "dog" and then rub three "copies" off of it? That is, do we manufacture three
+fresh instance symbols using the class 'symbol "dog" as template? Or do we jointly
+activate the symbols "three" and log"? By adding more or less detail to the scene being
+imagined, either theory becomes hard to maintain. For instance, we certainly do not have
+a separate instance symbol for each nose, mustache, grain of salt, etc., that we have ever
+seen. We let class symbols take care of such numerous items, and when we pass people
+on the street who have mustaches, we somehow just activate the "mustache" class
+symbol, without minting fresh instance symbols, unless we scrutinize them carefully.
+
+On the other hand, once we begin to distinguish individuals, we cannot rely on a single
+class symbol (e.g., "person") to timeshare itself among all the different people. Clearly
+there must come into existence separate stance symbols for individual people. It would be
+ridiculous to imagine
+
+
+Brains and Thoughts
+
+
+354
+
+
+
+that this feat could be accomplished by 'juggling"-that is, by the single class symbol
+flitting back and forth between several different modes of activation (one for each
+person).
+
+Between the extremes, there must be room for many sorts of intermediate cases. There
+may be a whole hierarchy of ways of creating the class-instance distinction in the brain,
+giving rise to symbols-and symbol organizations-of varying degrees of specificity. The
+following different kinds of individual and joint activation of symbols might be
+responsible for mental images of various degrees of specificity:
+
+(1) various different modes or depths of activation of a single class symbol:
+
+(2) simultaneous activation of several class symbols in some in some coordinated
+
+manner:
+
+(3) activation of a single instance symbol:
+
+(4) activation of a single instance symbol in conjunction with activation of several
+
+class symbols:
+
+(5) simultaneous activation of several instance symbols and several class symbols
+
+in some coordinated manner.
+
+This brings us right hack to the question: "When is a symbol a distinguishable
+subsystem of the brain For instance, consider the second example-simultaneous
+activation of several class symbols in some coordinated manner. This could easily be
+what happens when "piano sonata" is the concept under consideration (the symbols for
+"piano" and "sonata" being at least two of the activated symbols). But if this pair of
+symbols gets activated in conjunction often enough, it is reasonable to assume that the
+link between them will become strong enough that they will act as a unit, when activated
+together in the proper way. So two or more symbols can act as one, under the proper
+conditions, which means that the problem of enumerating the number of symbols in the
+brain is trickier than one might guess.
+
+Sometimes conditions can arise where two previously unlinked symbols get activated
+simultaneously and in a coordinated fashion. They may fit together so well that it seems
+like an inevitable union, and a single new symbol is formed by the tight interaction of the
+two old symbols. If this happens, would it be fair to say that the new symbol "always had
+been there but never had been activated"-or should one say that it has been "created"?
+
+In case this sounds too abstract, let us take a concrete example: the Dialogue Crab
+Canon. In the invention of this Dialogue, two existing symbols-that for "musical crab
+canon", and that for "verbal dialogue “had to be activated simultaneously and in some
+way forced to interact. Once this was done, 'the rest was quite inevitable: a new symbol-a
+class symbol-was born from the interaction of these two, and from then on it was able to
+be activated on its own. Now had it always been a dormant symbol in my brain? If so,
+then it must have also been a dormant symbol in
+
+
+Brains and Thoughts
+
+
+355
+
+
+
+the brain of every human who ever had its component symbols, even if it never was
+awakened in them. This would mean that to enumerate the symbols in anyone's brain, one
+would have to count all dormant symbols-all possible combinations and permutations of
+all types of activations of all known symbols. This would even include those fantastic
+creatures of software that one's brain invents when one is asleep-the strange mixtures of
+ideas which wake up when their host goes to sleep ... The existence of these "potential
+symbols" shows that it is really a huge oversimplification to imagine that the brain is a
+well-defined collection of symbols in well-defined states of activation. It is much harder
+than that to pin down a brain state on the symbol level.
+
+Symbols -Software or Hardware?
+
+With the enormous and ever-growing repertoire of symbols that exist in each brain, you
+might wonder whether there eventually comes a point when the brain is saturated-when
+there is just no more room for a new symbol. This would come about, presumably, if
+symbols never overlapped each other-if a given neuron never served a double function, so
+that symbols would be like people getting into an elevator. "Warning: This brain has a
+maximum capacity of 350,275 symbols!"
+
+This is not a necessary feature of the symbol model of brain function, however. In fact,
+overlapping and completely tangled symbols are probably the rule, so that each neuron,
+far from being a member of a unique symbol, is probably a functioning part of hundreds
+of symbols. This gets a little disturbing, because if it is true, then might it not just as
+easily be the case that each neuron is part of every single symbol? If that were so, then
+there would be no localizability whatsoever of symbols-every symbol would be identified
+with the whole of the brain. This would account for results like Lashley's cortex removal
+in rats-but it would also mean abandonment of our original idea of breaking the brain up
+into physically distinct subsystems. Our earlier characterization of symbols as "hardware
+realizations of concepts" could at best be a great oversimplification. In fact, if every
+symbol were made up of the same component neurons as every other symbol, then what
+sense would it make to speak of distinct symbols at all? What would be the signature of a
+given symbol's activation-that is, how could the activation of symbol A be distinguished
+from the activation of symbol B? Wouldn't our whole theory go down the drain? And
+even if there is not a total overlap of symbols, is our theory not more and more difficult to
+maintain, the more that symbols do overlap? (One possible way of portraying
+overlapping symbols is shown in Figure 68.)
+
+There is a way to keep a theory based on symbols even if physically, they overlap
+considerably or totally. Consider the' surface of a pond, which can support many different
+types of waves or ripples. The hardware namely the water itself-is the same in all cases,
+but it possesses different possible modes of excitation. Such software excitations of the
+same
+
+
+Brains and Thoughts
+
+
+356
+
+
+
+
+••
+
+
+
+
+FIGURE 68. In this schematic diagram, neurons are imagined as laid out as dots in one plane.
+Two overlapping neural pathways are shown in different shades of gray. It may happen that two
+independent "neural flashes" simultaneously race down these two pathways, passing through one
+another like two ripples on a pond's surface (as in Fig. 52). This is illustrative of the idea of two
+"active symbols" which share neurons and which may even be simultaneously activated. [From
+John C. Eccles, Facing Reality (New York: Springer Verlag, 1970), p.21.]
+
+
+hardware can all be distinguished from each other. By this analogy, I do not mean to go
+so far as to suggest that all the different symbols are just different kinds of "waves"
+propagating through a uniform neural medium which admits of no meaningful division
+into physically distinct symbols. But it may be that in order to distinguish one symbol’s
+activation from that of another symbol, a process must be carried out which involves not
+only locating the neurons which are firing, but also identifying very precise details of the
+timing of the firing of those neurons. That is, which neuron preceded which other neuron,
+and by how much? How many times a second was a particular neuron firing? Thus
+perhaps several symbols can coexist in the same set of neurons by having different
+characteristic neural firing patterns. The difference between a theory having physically
+distinct symbols, and a theory having overlapping symbols which are distinguished from
+each other by modes of excitation, is that the former gives hardware realizations of
+concepts, while the latter gives partly hardware, partly software realizations of concepts.
+
+
+Brains and Thoughts
+
+
+357
+
+
+
+Liftability of Intelligence
+
+
+Thus we are left with two basic problems in the unraveling of thought processes, as they
+take place in the brain. One is to explain how the A,-level traffic of neuron firings gives
+rise to the high-level traffic of symbol activations. The other is to explain the high-level
+traffic of symbol activation in its own terms-to make a theory which does not talk about
+the ,v-level neural events. If this latter is possible-and it is a key assumption the basis of
+all present research into Artificial Intelligence-then intelligence can be realized in other
+types of hardware than brains. Then intelligence will have been shown to be a property
+that can be "lifted" right out of e hardware in which it resides-or in other words,
+intelligence will be a software property. This will mean that the phenomena of
+consciousness and intelligence are indeed high-level in the same sense as most other
+complex
+
+FIGURE 69. The construction of an arch by workers of the termite Macrotermes belosus. Each
+column is built up by the addition of pellets of soil and excrement. #n the outer part of the left
+column a worker is seen depositing a round fecal pellet. #ther workers, having carried pellets in
+their mandibles up the columns, are now placing them at the growing ends of ’ columns. When a
+column reaches a certain height the termites, evidently guided by odor, ;in to extend it at an angle
+in the direction of a neighboring column. A completed arch is shown in the background.
+[Drawing by Turid Holldobler; from E. 0. Wilson, The Insect Societies Cambridge, Mass.:
+Harvard University Press, 1971), p. 230]
+
+
+
+Brains and Thoughts
+
+
+358
+
+
+
+phenomena of nature: they have their own high-level laws which depend on, yet are
+"liftable" out of, the lower levels. If, on the other hand, there is absolutely no way to
+realize symbol-triggering patterns without having all the hardware of neurons (or
+simulated neurons), this will imply that intelligence is a brain-bound phenomenon, and
+much more difficult to unravel than one which owes its existence to a hierarchy of laws
+on several different levels.
+
+Here we come back to the mysterious collective behavior of ant colonies, which can build
+huge and intricate nests, despite the fact that the roughly 100,000 neurons of an ant brain
+almost certainly do not carry any. information about nest structure. How, then, does the
+nest get created? Where does the information reside? In particular, ponder where the
+information describing an arch such as is shown in Figure 69 can be found. Somehow, it
+must be spread about in the colony, in the caste distribution, the age distribution-and
+probably largely in the physical properties of the ant-body itself. That is, the interaction
+between ants is determined just as much by their six-leggedness and their size and so on,
+as by the information stored in their brain. Could there be an Artificial Ant Colony?
+
+Can One Symbol Be Isolated?
+
+Is it possible that one single symbol could be awakened in isolation from all others?
+Probably not. Just as objects in the world always exist in a context of other objects, so
+symbols are always connected to a constellation of other symbols. This does not
+necessarily mean that symbols can never be disentangled from each other. To make a
+rather simple analogy, males and females always arise in a species together: their roles
+are completely intertwined, and yet this does not mean that a male cannot be
+distinguished from a female. Each is reflected in the other, as the beads in Indra's net
+reflect each other. The recursive intertwining of the functions F(n) and M(n) in Chapter V
+does not prevent each function from having its own characteristics. The intertwining of F
+and M could be mirrored in a pair of RTN's which call each other. From this we can jump
+to a whole network of ATN's intertwined with each other-a heterarchy of interacting
+recursive procedures. Here, the meshing is so inherent that no one ATN could be
+activated in isolation; yet its activation may be completely distinctive, not confusable
+with that of any other of the ATN's. It is not such a bad image, the brain as an ATN-
+colony!
+
+Fikewise, symbols, with all their multiple links to each other, are meshed together and
+yet ought to be able to be teased apart. This might involve identifying a neural network, a
+network plus a mode of excitation-or possibly something of a completely different kind.
+In any case, if symbols are part of reality, presumably there exists a natural way to chart
+them out in a real brain. However, if some symbols were finally identified in a brain, this
+would not mean that any one of them could be awakened in isolation.
+
+
+Brains and Thoughts
+
+
+359
+
+
+
+The fact that a symbol cannot be awakened in isolation does not diminish the
+separate identity of the symbol; in fact, quite to the contrary: a symbol's identity lies
+precisely in its ways of being connected (via potential triggering links) to other symbols.
+The network by which symbols can potentially trigger each other constitutes the brain's
+working model of the real universe, as well as of the alternate universes which it
+considers (and which are every bit as important for the individual's survival in the real
+world as the real world is).
+
+
+The Symbols of Insects
+
+Our facility for making instances out of classes and classes out of instances lies at the
+basis of our intelligence, and it is one of the great differences between human thought and
+the thought processes of other animals. Not that I have ever belonged to another species
+and experienced at first hand how it feels to think their way-but from the outside it is
+apparent that no other species forms general concepts as we do, or imagines hypothetical
+worlds-variants on the world as it is, which aid in figuring out which future pathway to
+choose. For instance, consider the celebrated "language of the bees"-information-laden
+dances which are performed by worker bees returning to the hive, to inform other bees of
+the location of nectar. While there may be in each bee a set of rudimentary symbols
+which are activated by such a dance, there is no reason to believe that a bee has an
+expandable vocabulary of symbols. Bees and other insects do not seem to have the power
+to generalize-that is, to develop new class symbols from instances which we would
+perceive as nearly identical.
+
+A classic experiment with solitary wasps is reported in Dean Wooldridge's book,
+Mechanical Man, from which I quote:
+
+When the time comes for egg laying, the wasp Sphex builds a burrow for the
+purpose and seeks out a cricket which she stings in such a way as to paralyze but not
+kill it. She drags the cricket into the burrow, lays her eggs alongside, closes the
+burrow, then flies away, never to return. In due course, the eggs hatch and the wasp
+grubs feed off the paralyzed cricket, which has not decayed, having been kept in the
+wasp equivalent of a deepfreeze. To the human mind, such an elaborately organized
+and seemingly purposeful routine conveys a convincing flavor of logic and
+thoughtfulness-until more details are examined. For example, the wasp's routine is to
+bring the paralyzed cricket to the burrow, leave it on the threshold, go inside to see
+that all is well, emerge, and then drag the cricket in. If the cricket is moved a few
+inches away while the wasp is inside making her preliminary inspection, the wasp,
+on emerging from the burrow, will bring the cricket back to the threshold, but not
+inside, and will then repeat the preparatory procedure of entering the burrow to see
+that everything is all right. If again the cricket is removed a few inches while the
+wasp is inside, once again she will move the cricket up to the threshold and reenter
+the burrow for a final check. The wasp never thinks of pulling the cricket straight in.
+On one occasion this procedure was repeated forty times, always with the same
+result.'
+
+
+Brains and Thoughts
+
+
+360
+
+
+
+This seems to be completely hard-wired behavior. Now in the wasp brain, there may be
+rudimentary symbols, capable of triggering each other; but there is nothing like the
+human capacity to see several instances as instances of an as-yet-unformed class, and
+then to make the class symbol; nor is there anything like the human ability to wonder,
+"What if I did this-what would ensue in that hypothetical world%" This type of thought
+process requires an ability to manufacture instances and to manipulate them as if they
+were symbols standing for objects in a real situation, although that situation may not be
+the case, and may never be the case.
+
+Class Symbols and Imaginary Worlds
+
+Let us reconsider the April Fool's joke about the borrowed car, and the images conjured
+up in your mind during the telephone call. To begin with, you need to activate symbols
+which represent a road, a car, a person in a car. Now the concept "road" is a very general
+one, with perhaps several stock samples which you can unconsciously pull out of
+dormant memory when the occasion arises. "Road" is a class, rather than an instance. As
+you listen to the tale, you quickly activate symbols which are instances with gradually
+increasing-specificity. For instance, when you learn that the road' was wet, this conjures
+up a more specific image, though you realize that it is most likely quite different from the
+actual road where the incident took place. But that is not important; what matters is
+whether your symbol is sufficiently well suited for the story-that is, whether the symbols
+which it can trigger are the right kind.
+
+As the story progresses, you fill in more aspects of this road: there is a high bank against
+which a car could smash. Now does this mean that you are activating the symbol for
+"bank", or does it mean that you are setting some parameters in your symbol for "road
+Undoubtedly both. That is, the network of neurons which represents "road" has many
+different ways of firing, and you are selecting which subnetwork actually shall fire. At
+the same time, you are activating the symbol for "bank", and this is probably instrumental
+in the process of selecting the parameters for. "road", in that its neurons may send signals
+to some of those in "road"-and vice versa. (In case this seems a little confusing, it is
+because I am somewhat straddling levels of description-I am trying to set up an image of
+the symbols, as well as of their component neurons.)
+
+No less important than the nouns are the verbs, prepositions, etc: They, too, activate
+symbols, which send messages back and forth to each other. There are characteristic
+differences between the kinds of triggering patterns of symbols for verbs and symbols for
+nouns, of course, which means that they may be physically somewhat differently
+organized. For instance, nouns might have fairly localized symbols, while verbs and
+prepositions might have symbols with many "tentacles" reaching all around the cortex; or
+any number of other possibilities.
+
+After the story is all over, you learn it was all untrue. The power of
+
+
+Brains and Thoughts
+
+
+361
+
+
+
+"rubbing off instances from classes, in the way that one makes rubbings from brasses in
+churches, has enabled you to represent the situation, and has freed you from the need to
+remain faithful to the real world. The fact that symbols can act as templates for other
+symbols gives you some mental independence of reality: you can create artificial
+universes, in which there can happen nonreal events with any amount of detail that you
+care to imbue them with. But the class symbols themselves, from which all of this
+richness springs, are deeply grounded in reality.
+
+Usually symbols play isomorphic roles to events which seem like they could happen,
+although sometimes symbols are activated which represent situations which could not
+happen-for example, watches sizzling, tubas laying eggs, etc. The borderline between
+what could and what could not happen is an extremely fuzzy one. As we imagine a
+hypothetical event, we bring certain symbols into active states-and depending on how
+well they interact (which is presumably reflected in our comfort in continuing the train of
+thought), we say the event "could" or "could not" happen. Thus the terms "could" and
+"could not" are extremely subjective. Actually, there is a good deal of agreement among
+people about which events could or could not happen. This reflects the great amount of
+mental structure which we all share-but there is a borderline area where the subjective
+aspect of what kinds of hypothetical worlds we are willing to entertain is apparent. A
+careful study of the kinds of imaginary events that people consider could and could not
+happen would yield much insight into the triggering patterns of the symbols by which
+people think.
+
+
+Intuitive Laws of Physics
+
+When the story has been completely told, you have built up quite an elaborate mental
+model of a scene, and in this model all the objects obey physical law. This means that
+physical law itself must be implicitly present in the triggering patterns of the symbols. Of
+course, the phrase "physical law" here does not mean "the laws of physics as expounded
+by a physicist", but rather the intuitive, chunked laws which all of us have to have in our
+minds in order to survive.
+
+A curious sidelight is that one can voluntarily manufacture mental sequences of events
+which violate physical law, if one so desires. For instance, if I but suggest that you
+imagine a scene with two cars approaching each other and then passing right through
+each other, you won't have any trouble doing so. The intuitive physical laws can be
+overridden by imaginary laws of physics; but how this overriding is done, how such
+sequences of images are manufactured-indeed what any one visual image is-all of these
+are deeply cloaked mysteries-inaccessible pieces of knowledge.
+
+Needless to say, we have in our brains chunked laws not only of how inanimate objects
+act, but also of how plants, animals, people and societies act-in other words, chunked
+laws of biology, psychology, sociology, and so
+
+
+Brains and Thoughts
+
+
+362
+
+
+
+on. All of the internal representations of such entities involve the inevitable feature of
+chunked models: determinism is sacrificed for simplicity. Our representation of reality
+ends up being able only to predict probabilities of ending up in certain parts of abstract
+spaces of behavior-not to predict anything with the precision of physics.
+
+Procedural and Declarative Knowledge
+
+A distinction which is made in Artificial Intelligence is that between procedural and
+declarative types of knowledge. A piece of knowledge is said to be declarative if it is
+stored explicitly, so that not only the programmer but also the program can "read" it as if
+it were in an encyclopedia or an almanac. This usually means that it is encoded locally,
+not spread around. By contrast, procedural knowledge is not encoded as facts-only as
+programs. A programmer may be able to peer in and say, "I see that because of these
+procedures here, the program 'knows' how to write English sentences "-but the program
+itself may have no explicit awareness of how it writes those sentences. For instance, its
+vocabulary may include none of the words "English", "sentence", and "write" at all! Thus
+procedural knowledge is usually spread around in pieces, and you can't retrieve it, or
+"key" on it. It is a global consequence of how the program works, not a local detail. In
+other words, a piece of purely procedural knowledge is an epiphenomenon.
+
+In most people there coexists, along with a powerful procedural representation of
+the grammar of their native language, a weaker declarative representation of it. The two
+may easily be in conflict, so that a native speaker will often instruct a foreigner to say
+things he himself would never say, but which agree with the declarative "book learning"
+he acquired in school sometime. The intuitive or chunked laws of physics and other
+disciplines mentioned earlier fall mainly on the procedural side; the knowledge that an
+octopus has eight tentacles falls mainly on the declarative side.
+
+In between the declarative and procedural extremes, there are all possible shades.
+Consider the recall of a melody. Is the melody stored in your brain, note by note? Could a
+surgeon extract a winding neural filament from your brain, then stretch it straight, and
+finally proceed to pinpoint along it the successively stored notes, almost as if it were a
+piece of magnetic tape? If so, then melodies are stored declaratively. Or is the recall of a
+melody mediated by the interaction of a large number of symbols, some of which
+represent tonal relationships, others of which represent emotional qualities, others of
+which represent rhythmic devices, and so on? If so, then melodies are stored
+procedurally. In reality, there is probably a mixture of these extremes in the way a
+melody is stored and recalled.
+
+It is interesting that, in pulling a melody out of memory, most people do not
+discriminate as to key, so that they are as likely to sing "Happy Birthday" in the key of F-
+sharp as in the key of C. This indicates that tone relationships, rather than absolute tones,
+are stored. But there is no reason
+
+
+Brains and Thoughts
+
+
+363
+
+
+
+that tone relationships could not be stored quite declaratively. On the other hand, some
+melodies are very easy to memorize, whereas others are extremely elusive. If it were just
+a matter of storing successive notes, any melody could be stored as easily as any other.
+The fact that some melodies are catchy and others are not seems to indicate that the brain
+has a certain repertoire of familiar patterns which are activated as the melody is heard.
+So, to "play back" the melody, those patterns would have to be activated in the same
+order. This returns us to the concept of symbols triggering one another, rather than a
+simple linear sequence of declaratively stored notes or tone relationships.
+
+How does the brain know whether a piece of knowledge is stored declaratively?
+For instance, suppose you are asked, "What is the population of Chicago?" Somehow the
+number five million springs to mind, without your wondering, "Gee, how would I go
+about counting them all?" Now suppose I ask you, "How many chairs are there in your
+living room?" Here, the opposite happens-instead of trying to dredge the answer out of a
+mental almanac, you immediately either go to the room and count the chairs, or you
+manufacture the room in your head and count the chairs in the image of the room. The
+questions were of a single type-"how many?"-yet one of them caused a piece of
+declarative knowledge to be fetched, while the other one caused a procedural method of
+finding the answer to be invoked. This is one example where it is clear that you have
+knowledge about how you classify your own knowledge; and what is more, some of that
+metaknowledge may itself be stored procedurally, so that it is used without your even
+being aware of how it is done.
+
+
+Visual Imagery
+
+One of the most remarkable and difficult-to-describe qualities of consciousness is visual
+imagery. How do we create a visual image of our living room? Of a roaring mountain
+brook? Of an orange? Even more mysterious, how do we manufacture images
+unconsciously, images which guide our thoughts, giving them power and color and
+depth? From what store are they fetched? What magic allows us to mesh two or three
+images, hardly giving a thought as to how we should do it? Knowledge of how to do this
+is among the most procedural of all, for we have almost no insight into what mental
+imagery is.
+
+It may be that imagery is based on our ability to suppress motor activity. By this, I
+mean the following. If you imagine an orange, there may occur in your cortex a set of
+commands to pick it up, to smell it, to inspect it, and so on. Clearly these commands
+cannot be carried out, because the orange is not there. But they can be sent along the
+usual channels towards the cerebellum or other suborgans of the brain, until, at some
+critical point, a "mental faucet" is closed, preventing them from actually being carried
+out. Depending on how far down the line this "faucet" is situated, the images may be
+more or less vivid and real-seeming. Anger can cause us to
+
+
+Brains and Thoughts
+
+
+364
+
+
+
+imagine quite vividly picking up some object and throwing it, or kicking something; yet
+we don't actually do so. On the other hand, we feel so "near" to actually doing so.
+Probably the faucet catches the nerve impulses "at the last moment".
+
+Here is another way in which visualization points out the distinction between
+accessible and inaccessible knowledge. Consider how you visualized the scene of the car
+skidding on the mountain road. Undoubtedly you imagined the mountain as being much
+larger than the car. Now did this happen because sometime long ago you had occasion to
+note that "cars are not as big as mountains"; then you committed this statement to rote
+memory: and in imagining the story, you retrieved this fact, and made use of it in
+constructing your image? A most unlikely theory. Or did it happen instead as a
+consequence of some introspectively inaccessible interactions of the symbols which were
+activated in your brain? Obviously the latter seems far more likely. This knowledge that
+cars are smaller than mountains is not a piece of rote memorization, but a piece of
+knowledge which can be created by deduction. Therefore, most likely it is not stored in
+any single symbol in your brain, but rather it can be produced as a result of the activation,
+followed by the mutual interaction, of many symbols-for example, those for "compare",
+"size", "car", "mountain", and probably, others. This means that the knowledge is stored
+not explicitly, but implicitly, in a spread-about manner, rather than as a local "packet of
+information". Such simple facts as relative sizes of objects have to be assembled, rather
+than merely retrieved. Therefore, even in the case of a verbally accessible piece of
+knowledge, there are complex inaccessible processes which mediate its coming to the
+state of being ready to be said.
+
+We shall continue our exploration of the entities called "symbols" in different
+Chapters. In Chapters XVIII and XIX, on Artificial Intelligence, we shall discuss some
+possible ways of implementing active symbols in programs. And next Chapter, we shall
+discuss some of the insights that our symbol-based model of brain activity give into the
+comparison of brains.
+
+
+Brains and Thoughts
+
+
+365
+
+
+
+English French German Suite
+
+
+By Lewis Carroll...
+
+... et Frank L. Warrin.
+
+. and Robert Scott
+
+'Twas brillig, and the slithy toves
+Did gyre and gimble in the wabe:
+
+All mimsy were the borogoves.
+
+And the mome raths outgrabe.
+
+II brilgue: les toves lubricilleux
+Se gyrent en vrillant dans le guave.
+
+Enmimes sont les gougebosqueux
+Et le momerade horsgrave.
+
+Es brillig war. Die schlichten Toven
+Wirrten and wimmelten in Waben;
+Und aller-mumsige Burggoven
+Die mohmen Rath' ausgraben.
+
+"Beware the Jabberwock, my son!
+
+The jaws that bite, the claws that catch!
+
+Beware the Jubjub bird, and shun
+The framious Bandersnatch!"
+
+((Garde-toi du Jaseroque, mon fits!
+
+La gueule qui mord; la griffe qui prend!
+Garde-toi de I'oiseau Jube, evite
+Le frumieux Band-a-prend!))
+
+))Bewahre doch vor Jammerwoch!
+
+Die Zahne knirschen, Krallen kratzen!
+Bewahr' vor Jubjub-Vogel, vor
+Frumiosen Banderschnatzchen!)),
+
+He took his vorpal sword in hand:
+
+Long time the manxome foe he sought
+So rested he by the Tumtum tree,
+
+And stood awhile in thought.
+
+Son glaive vorpal en main, it va
+T-a la recherche du fauve manscant;
+
+Puis arrive a I'arbre Te-te,
+
+1 y reste, reflechissant
+
+
+English French German Suite
+
+
+366
+
+
+
+
+Er griff sein vorpals Schwertchen zu,
+
+Er suchte lang das manchsam' Ding;
+
+Dann, stehend unterm Tumtum Baum,
+
+Er an-zu-denken-fing.
+
+And, as in uffish thought he stood,
+
+The Jabberwock, with eyes of flame.
+
+Came whiffling through the tulgey wood,
+
+And burbled as it came!
+
+Pendant qu'il pense, tout uffuse,
+
+Le Jaseroque, a l'oeil flambant,
+
+Vient siblant par le bois tullegeais,
+
+Et burbule en venant.
+
+Als stand er tief in Andacht auf,
+
+Des Jammerwochen's Augen-feuer
+Durch turgen Wald mit Wiffek kam
+Fin burbelnd Ungeheuer!
+
+One, two! One, two! And through and through
+The vorpal blade went snicker-snack!
+
+He left it dead, and with its head
+He went galumphing back.
+
+Un deux, un deux, par le milieu,
+
+Le glaive vorpal fait pat-a-pan!
+
+La bete defaite, avec sa tete,
+
+II rentre gallomphant.
+
+Eins, Zwei! Eins, Zwei! Und durch and durch
+Sein vorpals Schwert zerschnifer-schnuck,
+
+Da blieb es todt! Er, Kopf in Hand,
+
+Gelaumfig zog zuriick.
+
+"And hast thou slain the Jabberwock?
+
+Come to my arms, my beamish boy!
+
+0 frabjous day! Callooh! Callay!"
+
+He chortled in his joy.
+
+((As-tu tue le Jaseroque?
+
+Viens amon coeur, fils rayonnais!
+
+O jour frabbejais! Calleau! Callai!))
+
+II cortule clans sa joie.
+
+))Und schlugst Du ja den Jammerwoch?
+Umarme mich, mein Bohm'sches Kind!
+
+O Freuden-Tag! 0 Halloo-Schlag!((
+
+Er schortelt froh-gesinnt.
+
+
+English French German Suite
+
+
+367
+
+
+
+'Twas brillig, and the slithy toves
+Did gyre and gimble in the wabe:
+
+All mimsy were the borogoves.
+
+And the mome raths outgrabe.
+
+II brilgue: les toves lubricilleux
+Se gyrent en vrillant dans le guave.
+Enmimes sont les gougebosqueux
+Et le momerade horsgrave.
+
+Es brillig war. Die schlichten Toven
+Wirrten and wimmelten in Waben:
+Und aller-mumsige Burggoven
+Die mohmen Rath' ausgraben.
+
+
+English French German Suite
+
+
+368
+
+
+
+CHAPTER XII
+
+
+Minds and Thoughts
+
+Can Minds Be Mapped onto Each Other?
+
+Now THAT WE have hypothesized the existence of very high-level active
+subsystems of the brain (symbols), we may return to the matter of a possible
+isomorphism, or partial isomorphism, between two brains. Instead of asking about an
+isomorphism on the neural level (which surely does not exist), or on the macroscopic
+suborgan level (which surely does exist but does not tell us very much), we ask about the
+possibility of an isomorphism between brains on the symbol level: a correspondence
+which not only maps symbols in one brain onto symbols in another brain, but also maps
+triggering patterns onto triggering patterns. This means that corresponding symbols in the
+two brains are linked in corresponding ways. This would be a true functional
+isomorphism-the same type of isomorphism as we spoke of when trying to characterize
+what it is that is invariant about all butterflies.
+
+It is clear from the outset that such an isomorphism does not exist between any
+pair of human beings. If it did, they would be completely indistinguishable in their
+thoughts; but in order for that to be true, they would have to have completely
+indistinguishable memories, which would mean they would have to have led one and the
+same life. Even identical twins do not approach, in the remotest degree, this ideal.
+
+How about a single individual When you look back over things which you
+yourself wrote a few years ago, you think "How awful!" and smile with amusement at the
+person you once were. What is worse is when you do the same thing with something you
+wrote or said five minutes ago. When this happens, it shows that you do not fully
+understand the person you were moments ago. The isomorphism from your brain now to
+your brain then is imperfect. What, then, of the isomorphisms to other people, other
+species ...
+
+The opposite side of the coin is shown by the power of the communication that
+arises between the unlikeliest partners. Think of the barriers spanned when you read lines
+of poetry penned in jail by Francois Villon, the French poet of the 1400's. Another human
+being, in another era, captive in jail, speaking another language ... How can you ever
+hope to have a sense of the connotations behind the facade of his words, translated into
+English% Yet a wealth of meaning comes through.
+
+Thus, on the one hand, we can drop all hopes of finding exactly isomorphic
+software in humans, but on the other, it is clear that some people think more alike than
+others do. It would seem an obvious conclu
+
+
+Minds and Thoughts
+
+
+369
+
+
+
+
+FIGURE 70. A tiny portion of the author's "semantic network".
+
+
+Minds and Thoughts
+
+
+370
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+
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+
+
+sion that there is some sort of partial software isomorphism connecting the brains of
+people whose style of thinking is similar-in particular, a correspondence of (1) the
+repertoire of symbols, and (2) the triggering patterns of symbols
+
+Comparing Different Semantic Networks
+
+But what is a partial isomorphism? This is a most difficult question to answer. It is made
+even more difficult by the fact that no one has found an adequate way to represent the
+network of symbols and their triggering patterns. Sometimes a picture of a small part of
+such a network of symbols is drawn, where each symbol is represented as a node into
+which, and out of which, lead some arcs. The lines represent triggering relationships-in
+some sense. Such figures attempt to capture something of the intuitively sensible notion
+of "conceptual nearness". However, there are many different kinds of nearness, and
+different ones are relevant in different contexts. A tiny portion of my own "semantic
+network" is shown in Figure 70. The problem is that representing a complex
+interdependency of many symbols cannot be carried out very easily with just a few lines
+joining vertices.
+
+Another problem with such a diagram is that it is not accurate to think of a symbol
+as simply "on" or "off. While this is true of neurons, it does not carry upwards, to
+collections of them. In this respect, symbols are quite a bit more complicated than
+neurons-as you might expect, since they are made up of many neurons. The messages that
+are exchanged between symbols are more complex than the mere fact, "I am now
+activated". That is more like the neuron-level messages. Each symbol can be activated in
+many different ways, and the type of activation will be influential in determining which
+other symbols it tries to activate. How these intertwining triggering relationships can be
+represented in a pictorial manner-indeed, whether they can be at all-is not clear.
+
+But for the moment, suppose that issue had been solved. Suppose we now agree
+that there are certain drawings of nodes, connected by links (let us say they come in
+various colors, so that various types of conceptual nearness can be distinguished from
+each other), which capture precisely the way in which symbols trigger other symbols.
+Then under what conditions would we feel that two such drawings were isomorphic, or
+nearly isomorphic? Since we are dealing with a visual representation of the network of
+symbols, let us consider an analogous visual problem. How would you try to determine
+whether two spiderwebs had been spun by spiders belonging to the same species? Would
+you try to identify individual vertices which correspond exactly, thereby setting up an
+exact map of one web onto the other, vertex by vertex, fiber by fiber, perhaps even angle
+by angle? This would be a futile effort. Two webs are never exactly the same: yet there is
+still some sort of "style", "form", what-have-you, that infallibly brands a given species'
+web.
+
+In any network-like structure, such as a spiderweb, one can look at local
+properties and global properties. Local properties require only a very
+
+
+Minds and Thoughts
+
+
+371
+
+
+
+nearsighted observer-for example an observer who can only see one vertex at a time; and
+global properties require only a sweeping vision, without attention to detail. Thus, the
+overall shape of a spiderweb is a global property, whereas the average number of lines
+meeting at a vertex is a local property. Suppose we agree that the most reasonable
+criterion for calling two spiderwebs "isomorphic" is that they should have been spun by
+spiders of the same species. Then it is interesting to ask which kind of observation-local
+or global-tends to be a more reliable guide in determining whether two spiderwebs are
+isomorphic. Without answering the question for spiderwebs, let us now return to the
+question of the closeness-or isomorphicness, if you will-of two symbol networks.
+
+Translations of "Jabberwocky"
+
+Imagine native speakers of English, French, and German, all of whom have excellent
+command of their respective native languages, and all of whom enjoy wordplay in their
+own language. Would their symbol networks be similar on a local level, or on a global
+level? Or is it meaningful to ask such a question? The question becomes concrete when
+you look at the preceding translations of Lewis Carroll's famous "Jabberwocky".
+
+I chose this example because it demonstrates, perhaps better than an example in
+ordinary prose, the problem of trying to find "the same node" in two different networks
+which are, on some level of analysis, extremely nonisomorphic. In ordinary language, the
+task of translation is more straightforward, since to each word or phrase in the original
+language, there can usually be found a corresponding word or phrase in the new
+language. By contrast, in a poem of this type, many "words" do not carry ordinary
+meaning, but act purely as exciters of nearby symbols. However, what is nearby in one
+language may be remote in another.
+
+Thus, in the brain of a native speaker of English, "slithy" probably activates such
+symbols as "slimy", "slither", "slippery", "lithe", and "sly", to varying extents. Does
+"lubricilleux" do the corresponding thing in the brain of a Frenchman? What indeed
+would be "the corresponding thing"? Would it be to activate symbols which are the
+ordinary translations of those words? What if there is no word, real or fabricated, which
+will accomplish that? Or what if a word does exist, but is very intellectual-sounding and
+Latinate ("lubricilleux"), rather than earthy and Anglo-Saxon ("slithy")? Perhaps
+"huilasse" would be better than "lubricilleux"? Or does the Latin origin of the word
+"lubricilleux" not make itself felt to a speaker of French in the way that it would if it were
+an English word ("lubricilious", perhaps)?
+
+An interesting feature of the translation into French is the transposition into the
+present tense. To keep it in the past would make some unnatural turns of phrase
+necessary, and the present tense has a much fresher flavor in French than the past. The
+translator sensed that this would be "more appropriate"-in some ill-defined yet
+compelling senseand made the switch. Who can say whether remaining faithful to the
+English tense would have been better?
+
+
+Minds and Thoughts
+
+
+372
+
+
+
+In the German version, the droll phrase "er an-zu-denken-fing" occurs; it does not
+correspond to any English original. It is a playful reversal of words, whose flavor vaguely
+resembles that of the English phrase "he out-to-ponder set", if I may hazard a reverse
+translation. Most likely this funny turnabout of words was inspired by the similar playful
+reversal in the English of one line earlier: "So rested he by the Tumtum tree". It
+corresponds, yet doesn't correspond.
+
+Incidentally, why did the Tumturn tree get changed into an "arbre T6-t6" in
+French? Figure it out for yourself.
+
+The word "manxome" in the original, whose "x" imbues it with many rich
+overtones, is weakly rendered in German by "manchsam", which hack-translates into
+English as "maniful". The French "manscant" also lacks the manifold overtones of
+"manxome". There is no end to the interest of this kind of translation task.
+
+When confronted with such an example, one realizes that it is utterly impossible
+to make an exact translation. Yet even in this pathologically difficult case of translation,
+there seems to be some rough equivalence obtainable. Why is this so, if there really is no
+isomorphism between the brains of people who will read the different versions? The
+answer is that there is a kind of rough isomorphism, partly global, partly local, between
+the brains of all the readers of these three poems.
+
+ASU's
+
+An amusing geographical fantasy will give some intuition for this kind of quasi¬
+isomorphism. (Incidentally, this fantasy is somewhat similar to a geographical analogy
+devised by M. Minsky in his article on "frames", which can be found in P. H. Winston's
+book The Psychology of Computer Vision.) Imagine that you are given a strange atlas of
+the USA, with all natural geological features premarked-such as rivers, mountains, lakes,
+and so on-but with nary a printed word. Rivers are shown as blue lines, mountains b
+color, and so on. Now you are told to convert it into a road atlas for a trip which you will
+soon make. You must neatly fill in the names of all states, their boundaries, time zones,
+then all counties, cities, towns, all freeways and highways and toll routes, all county
+roads, all state and national parks, campgrounds, scenic areas, dams, airports, and so on
+... All of this must be carried out down to the level that would appear in a detailed road
+atlas. And it must be manufactured out of your own head. You are not allowed access to
+any information which would help you for the duration of your task.
+
+You are told that it will pay off, in ways that will become clear at a later date, to
+make your map as true as you can. Of course, you will begin by filling in large cities and
+major roads, etc., which you know. And when you have exhausted your factual
+knowledge of an area, it will be to your advantage to use your imagination to help you
+reproduce at least the flavor of that area, if not its true geography, by making up fake
+town names, fake populations, fake roads, fake parks, and so on. This arduous task will
+take
+
+
+Minds and Thoughts
+
+
+373
+
+
+
+months. To make things a little easier, you have a cartographer on hand to print
+everything in neatly. The end product will be your personal map of the "Alternative
+Structure of the Union"-your own personal "ASU".
+
+Your personal ASU will be very much like the USA in the area where you grew
+up. Furthermore, wherever your travels have chanced to lead you, or wherever you have
+perused maps with interest, your ASU will have spots of striking agreement with the
+USA: a few small towns in North Dakota or Montana, perhaps, or the whole of
+metropolitan New York, might be quite faithfully reproduced in your ASU.
+
+A Surprise Reversal
+
+When your ASU is done, a surprise takes place. Magically, the country you have
+designed comes into being, and you are transported there. A friendly committee presents
+you with your favorite kind of-automobile, and explains that, "As a reward for your
+designing efforts, you may now enjoy an all-expense-paid trip, at a leisurely pace, around
+the good old A. S. of U. You may go wherever you want, do whatever you wish to do,
+taking as long as you wish-compliments of the Geographical Society of the ASU. And-to
+guide you around-here is a road atlas." To your surprise, you are given not the atlas
+which you designed, but a regular road atlas of the USA.
+
+When you embark on your trip, all sorts of curious incidents will take place. A
+road atlas is being used to guide you through a country which it only partially fits. As
+long as you stick to major freeways, you will probably be able to cross the country
+without gross confusions. But the moment you wander off into the byways of New
+Mexico or rural Arkansas, there will be adventure in store for you. The locals will not
+recognize any of the towns you're looking for, nor will they know the roads you're asking
+about. They will only know the large cities you name, and even then the routes to those
+cities will not be the same as are indicated on your map. It will happen occasionally that
+some of the cities which are considered huge by the locals are nonexistent on your map of
+the USA; or perhaps they exist, but their population according to the atlas is wrong by an
+order of magnitude.
+
+
+Centrality and Universality
+
+What makes an ASU and the USA, which are so different in some ways, nevertheless so
+similar? It is that their most important cities and routes of communication can be mapped
+onto each other. The differences between them are found in the less frequently traveled
+routes, the cities of smaller size, and so on. Notice that this cannot be characterized either
+as a local or a global isomorphism. Some correspondences do extend down to the very
+local level-for instance, in both New Yorks, the main street may be Fifth Avenue, and
+there may be a Times Square in both as well-yet there may not be a single town that is
+found in both Montanas. So the local-global
+
+
+Minds and Thoughts
+
+
+374
+
+
+
+distinction is not relevant here. What is relevant is the centrality of the city, in terms of
+economics, communication, transportation, etc. The more vital the city is, in one of these
+ways, the more certain it will be to occur in both the ASU and the USA.
+
+In this geographic analogy, one aspect is very crucial: that there are certain
+definite, absolute points of reference which will occur in nearly all ASU's: New York,
+San Francisco, Chicago, and so on. From these it is then possible to orient oneself. In
+other words, if we begin comparing my ASU with yours, I can use the known agreement
+on big cities to establish points of reference with which I can communicate the location
+of smaller cities in my ASU. And if I hypothesize a voyage from Kankakee to Fruto and
+you don't know where those towns are, I can refer to something we have in common, and
+thereby guide you. And if I talk about a voyage from Atlanta to Milwaukee, it may go
+along different freeways or smaller roads, but the voyage itself can still be carried out in
+both countries. And if you start describing a trip from Horsemilk to Janzo, I can plot out
+what seems to me to be an analogous trip in my ASU, despite not having towns by those
+names, as long as you constantly keep me oriented by describing your position with
+respect to nearby larger towns which are found in my ASU as well as in yours.
+
+My roads will not be exactly the same as yours, but, with our separate maps, we
+can each get from a particular part of the country to another. We can do this, thanks to the
+external, predetermined geological facts mountain chains, streams, etc.-facts which were
+available to us both as we worked on our maps. Without those external features, we
+would have no possibility of reference points in common. For instance, if you had been
+given only a map of France, and I had been given a map of Germany, and then we had
+both filled them in in great detail, there would he no way to try to find "the same place"
+in our fictitious lands. It is necessary to begin with identical external conditions-
+otherwise nothing will match.
+
+Now that we have carried our geographical analogy quite far, we return to the
+question of isomorphisms between brains. You might well wonder why this whole
+question of brain isomorphisms has been stressed so much. What does it matter if two
+brains are isomorphic, or quasi-isomorphic, or not isomorphic at all? The answer is that
+we have an intuitive sense that, although other people differ from us in important ways,
+they are still "the same" as we are in some deep and important ways. It would be
+instructive to be able to pinpoint what this invariant core of human intelligence is, and
+then to be able to describe the kinds of "embellishments" which can be added to it,
+making each one of us a unique embodiment of this abstract and mysterious quality
+called "intelligence".
+
+In our geographic analogy, cities and towns were the analogues of symbols, while
+roads and highways were analogous to potential triggering paths. The fact that all ASU's
+have some things in common, such as the East Coast, the West Coast, the Mississippi
+River, the Great Lakes, the Rockies, and many major cities and roads is analogous to the
+fact that we are all forced, by external realities, to construct certain class symbols and
+trigger
+
+
+Minds and Thoughts
+
+
+375
+
+
+
+ing paths in the same way. These core symbols are like the large cities, to which everyone
+can make reference without ambiguity. (Incidentally, the fact that cities are localized
+entities should in no way be taken as indicative that symbols in a brain are small, almost
+point-like entities. They are merely symbolized in that manner in a network.)
+
+The fact is that a large proportion of every human's network of symbols is
+universal. We simply take what is common to all of us so much for granted that it is hard
+to see how much we have in common with other people. It takes the conscious effort of
+imagining how much-or how little-we have in common with other types of entities, such
+as stones, cars, restaurants, ants, and so forth, to make evident the large amount of
+overlap that we have with randomly chosen people. What we notice about another person
+immediately is not the standard overlap, because that is taken for granted as soon as we
+recognize the humanity of the other person; rather, we look beyond the standard overlap
+and generally find some major differences, as well as some unexpected, additional
+overlap.
+
+Occasionally, you find that another person is missing some of what you thought
+was the standard, minimal core-as if Chicago were missing from their ASU, which is
+almost unimaginable. For instance, someone might not know what an elephant is, or who
+is President, or that the earth is round. In such cases, their symbolic network is likely to
+be so fundamentally different from your own that significant communication will be
+difficult. On the other hand, perhaps this same person will share some specialized kind of
+knowledge with you-such as expertise in the game of dominoes-so that you can
+communicate well in a limited domain. This would be like meeting someone who comes
+from the very same rural area of North Dakota as you do, so that your two ASU's
+coincide in great detail over a very small region, which allows you to describe how to get
+from one place to another very fluently.
+
+How Much Do Language and Culture Channel Thought?
+
+If we now go back to comparing our own symbol network with those of a Frenchman and
+a German, we can say that we expect them to have the standard core of class symbols,
+despite the fact of different native languages. We do not expect to share highly
+specialized networks with them, but we do not expect such sharing with a randomly
+chosen person who shares our native language, either. The triggering patterns of people
+with other languages will be somewhat different from our own, but still the major class
+symbols, and the major routes between them, will be universally available, so that more
+minor routes can be described with reference to them.
+
+Now each of our three people may in addition have some command of the
+languages of the other two. What is it that marks the difference between true fluency, and
+a mere ability to communicate? First of all, someone fluent in English uses most words at
+roughly their- regular frequencies. A non-native speaker will have picked up some words
+from
+
+
+Minds and Thoughts
+
+
+376
+
+
+
+dictionaries, novels, or classes-words which at some time may have been prevalent or
+preferable, but which are now far down in frequency-for example, "fetch" instead of
+"get", "quite" instead of "very", etc. Though the meaning usually comes through, there is
+an alien quality transmitted by the unusual choice of words.
+
+But suppose that a foreigner learns to use all words at roughly the normal
+frequencies. Will that make his speech truly fluent? Probably not. Higher than the word
+level, there is an association level, which is attached to the culture as a whole-its history,
+geography, religion, children's stories, literature, technological level, and so on. For
+instance, to be able to speak modern Hebrew absolutely fluently, you need to know the
+Bible quite well in Hebrew, because the language draws on a stock of biblical phrases
+and their connotations. Such an association level permeates each language very deeply.
+Yet there is room for all sorts of variety inside fluency-otherwise the only truly fluent
+speakers would be people whose thoughts were the most stereotyped possible!
+
+Although we should recognize the depth to which culture affects thought, we
+should not overstress the role of language in molding thoughts. For instance, what we
+might call two "chairs" might be perceived by a speaker of French as objects belonging to
+two distinct types: "chaise" and "fauteuil" ("chair" and "armchair"). People whose native
+language is French are more aware of that difference than we are-but then people who
+grow up in a rural area are more aware of, say, the difference between a pickup and a
+truck, than a city dweller is. A city dweller may call them both "trucks". It is not the
+difference in native language, but the difference in culture (or subculture), that gives rise
+to this perceptual difference.
+
+The relationships between the symbols of people with different native languages
+have every reason to be quite similar, as far as the core is concerned, because everyone
+lives in the same world. When you come down to more detailed aspects of the triggering
+patterns, you will find that there is less in common. It would he like comparing rural
+areas in Wisconsin in ASU's which had been made up by people who had never lived in
+Wisconsin. This will be quite irrelevant, however, as long as there is sufficient agreement
+on the major cities and major routes, so that there are common points of reference all
+over the map.
+
+
+Trips and Itineraries in ASU's
+
+Without making it explicit, I have been using an image of what a "thought" is in the
+ASU-analogy-namely, I have been implying that a thought corresponds to a trip. The
+towns which are passed through represent the symbols which are excited. This is not a
+perfect analogy, but it is quite strong. One problem with it is that when a thought recurs
+in someone's mind sufficiently often, it can get chunked into a single concept. This would
+correspond to quite a strange event in an ASU: a commonly taken trip would become, in
+some strange fashion, a new town or city! If one is to continue to use the ASU-metaphor,
+then, it is important to remember that
+
+
+Minds and Thoughts
+
+
+377
+
+
+
+the cities represent not only the elementary symbols, such as those for "grass", "house",
+and "car", but also symbols which get created as a result of the chunking ability of a
+brain-symbols for such sophisticated concepts as "crab canon", "palindrome", or "ASU".
+
+Now if it is granted that the notion of taking a trip is a fair counterpart to the
+notion of having a thought, then the following difficult issue comes up: virtually any
+route leading from one city to a second, then to a third, and so on, can be imagined, as
+long as one remembers that some intervening cities are also passed through. This would
+correspond to the activation of an arbitrary sequence of symbols, one after another,
+making allowance for some extra symbols-those which lie en route. Now if virtually any
+sequence of symbols can be activated in any desired order, it may seem that a brain is an
+indiscriminate system, which can absorb or produce any thought whatsoever. But we all
+know that that is not so. In fact, there are certain kinds of thoughts which we call
+knowledge, or beliefs, which play quite a different role from random fancies, or
+humorously entertained absurdities. How can we characterize the difference between
+dreams, passing thoughts, beliefs, and pieces of knowledge?
+
+Possible, Potential, and Preposterous Pathways
+
+There are some pathways-you can think of them as pathways either in an ASU or in a
+brain-which are taken routinely in going from one place to another. There are other
+pathways which can only be followed if one is led through them by the hand. These
+pathways are "potential pathways", which would be followed only if special external
+circumstances arose. The pathways which one relies on over and over again are pathways
+which incorporate knowledge-and here I mean not only knowledge of facts (declarative
+knowledge), but also knowledge of how-to's (procedural knowledge). These stable,
+reliable pathways are what constitute knowledge. Pieces of knowledge merge gradually
+with beliefs, which are also represented by reliable pathways, but perhaps ones which are
+more susceptible to replacement if, so to speak, a bridge goes out, or there is heavy fog.
+This leaves us with fancies, lies, falsities, absurdities, and other variants. These would
+correspond to peculiar routes such as: New York City to Newark via Bangor, Maine and
+Lubbock, Texas. They are indeed possible pathways, but ones which are not likely to be
+stock routes, used in everyday voyages.
+
+A curious, and amusing, implication of this model is that all of the "aberrant"
+kinds of thoughts listed above are composed, at rock bottom, completely out of beliefs or
+pieces of knowledge. That is, any weird and snaky indirect route breaks up into a number
+of non-weird, non-snaky direct stretches, and these short, straightforward symbol¬
+connecting routes represent simple thoughts that one can rely on-beliefs and pieces of
+knowledge. On reflection, this is hardly surprising, however, since it is quite reasonable
+that we should only be able to imagine fictitious things that are somehow grounded in the
+realities we have experienced, no matter how
+
+
+Minds and Thoughts
+
+
+378
+
+
+
+wildly they deviate from them. Dreams are perhaps just such random meanderings about
+the ASU's of our minds. Locally, they make sense-but globally ...
+
+
+Different Styles of Translating Novels
+
+A poem like 'Jabberwocky" is like an unreal journey around an ASU, hopping from one
+state to another very quickly, following very curious routes. The translations convey this
+aspect of the poem, rather than the precise sequence of symbols which are triggered,
+although they do their best in that respect. In ordinary prose, such leaps and bounds are
+not so common. However, similar problems of translation do occur. Suppose you are
+translating a novel from Russian to English, and come across a sentence whose literal
+translation is, "She had a bowl of borscht." Now perhaps many of your readers will have
+no idea what borscht is. You could attempt to replace it by the "corresponding" item in
+their culture-thus, your translation might run, "She had a bowl of Campbell's soup." Now
+if you think this is a silly exaggeration, take a look at the first sentence of Dostoevsky's
+novel Crime and Punishment in Russian and then in a few different English translations. I
+happened to look at three different English paperback translations, and found the
+following curious situation.
+
+The first sentence employs the street name "S. Pereulok" (as transliterated). What
+is the meaning of this? A careful reader of Dostoevsky's work who knows Leningrad
+(which used to be called "St. Petersburg "-or should I say "Petrograd"?) can discover by
+doing some careful checking of the rest of the geography in the book (which incidentally
+is also given only by its initials) that the street must be "Stoliarny Pereulok". Dostoevsky
+probably wished to tell his story in a realistic way, yet not so realistically that people
+would take literally the addresses at which crimes and other events were supposed to
+have occurred. In any case, we have a translation problem; or to be more precise, we have
+several translation problems, on several different levels.
+
+First of all, should we keep the initial so as to reproduce the aura of semi-mystery
+which appears already in this first sentence of the book? We would get "S. Lane" ("lane"
+being the standard translation of "pereulok"). None of the three translators took this tack.
+However, one chose to write "S. Place". The translation of Crime and Punishment which
+I read in high school took a similar option. I will never forget the disoriented feeling I
+experienced when I began reading the novel and encountered those streets with only
+letters for names. I had some sort of intangible malaise about the beginning of the book; I
+was sure that I was missing something essential, and yet I didn't know what it was ... I
+decided that all Russian novels were very weird.
+
+Now we could be frank with the reader (who, it may be assumed, probably won't
+have the slightest idea whether the street is real or fictitious anyway!) and give him the
+advantage of our modern scholarship, writing
+
+
+Minds and Thoughts
+
+
+379
+
+
+
+"Stoliarny Lane" (or "Place"). This was the choice of translator number 2, who gave the
+translation as "Stoliarny Place".
+
+What about number 3? This is the most interesting of all. This translation says
+"Carpenter's Lane". And why not, indeed? After all, "stoliar" means "carpenter" and "ny"
+is an adjectival ending. So now we might imagine ourselves in London, not Petrograd,
+and in the midst of a situation invented by Dickens, not Dostoevsky. Is that what we
+want-, Perhaps we should just read a novel by Dickens instead, with the justification that
+it is "the corresponding work in English". When viewed on a sufficiently high level, it is
+a "translation" of the Dostoevsky novel-in fact, the best possible one! Who needs
+Dostoevsky?
+
+We have come all the way from attempts at great literal fidelity to the author's
+style, to high-level translations of flavor. Now if this happens already in the first
+sentence, can you imagine how it must go on in the rest of the book? What about the
+point where a German landlady begins shouting in her German-style Russian% How do
+you translate broken Russian spoken with a German accent, into English?
+
+Then one may also consider the problems of how to translate slang and colloquial
+modes of expression. Should one search for an "analogous" phrase, or should one settle
+for a word-by-word translation? If you search for an analogous phrase, then you run the
+risk of committing a "Campbell's soup" type of blunder; but if you translate every
+idiomatic phrase word by word, then the English will sound alien. Perhaps this is
+desirable, since the Russian culture is an alien one to speakers of English. But a speaker
+of English who reads such a translation will constantly be experiencing, thanks to the
+unusual turns of phrase, a sense-an artificial sense-of strangeness, which was not intended
+by the author, and which is not experienced by readers of the Russian original.
+
+Problems such as these give one pause in considering such statements as this one,
+made by Warren Weaver, one of the first advocates of translation by computer, in the late
+1940's: "When I look at an article in Russian, I say, 'This is really written in English, but
+it has been coded in some strange symbols. I will now proceed to decode."" Weaver's
+remark simply cannot be taken literally; it must rather be considered a provocative way of
+saying that there is an objectively describable meaning hidden in the symbols, or at least
+something pretty close to objective; therefore, there would be no reason to suppose a
+computer could not ferret it out, if sufficiently well programmed.
+
+High-Level Comparisons between Programs
+
+Weaver's statement is about translations between different natural languages. Let's
+consider now the problem of translating between two computer languages. For instance,
+suppose two people have written programs which run on different computers, and we
+want to know if the two programs carry out the same task. How can we find out? We
+must compare the programs. But on what level should this be done? Perhaps one program
+
+
+Minds and Thoughts
+
+
+380
+
+
+
+mer wrote in a machine language, the other in a compiler language. Are two such
+programs comparable? Certainly. But how to compare them? One way might be to
+compile the compiler language program, producing a program in the machine language of
+its home computer.
+
+Now we have two machine language programs. But there is another problem:
+there are two computers, hence two different machine languages-and they may be
+extremely different. One machine may have sixteen-bit words; the other thirty-six-bit
+words. One machine may' have built-in stack-handling instructions (pushing and
+popping), while the other lacks them. The differences between the hardware of the two
+machines may make the two machine language programs seem incomparable-and yet we
+suspect they are performing the same task, and we would like to see that at a glance. We
+are obviously looking at the programs from much too close a distance.
+
+What we need to do is to step back, away from machine language, towards a
+higher, more chunked view. From this vantage point, we hope we will be able to perceive
+chunks of program which make each program seem rationally planned out on a global,
+rather than a local, scale-that is, chunks which fit together in a way that allows one to
+perceive the goals of the programmer. Let us assume that both programs were originally
+written in high-level languages. Then some chunking has already been done for us. But
+we will run into other troubles. There is a proliferation of such languages: Fortran, Algol,
+LISP, APL, and many others. How can you compare a program written in APL with one
+written in Algol: Certainly not by matching them up line by line. You will again chunk
+these programs in your mind, looking for conceptual, functional units which correspond.
+Thus, you are not comparing hardware, you are not comparing software-you are
+comparing "etherware"-the pure concepts which lie back of the software. There is some
+sort of abstract "conceptual skeleton" which must be lifted out of low levels before you
+can carry out a meaningful comparison of two programs in different computer languges,
+of two animals, or of two sentences in different natural languages.
+
+Now this brings us back to an earlier question which we asked about computers
+and brains: How can we make sense of a low-level description of a computer or a brain?
+Is there, in any reasonable sense, an objective way to pull a high-level description out of a
+low-level one, in such complicated systems? In the case of a computer, a full display of
+the contents of memory-a so-called memory dump-is easily available. Dumps were
+commonly printed out in the early days of computing, when something went wrong with
+a program. Then the programmer would have to go home and pore over the memory
+dump for hours, trying to understand what each minuscule piece of memory represented.
+In essence, the programmer would be doing the opposite of what a compiler does: he
+would be translating from machine language into a higher-level language, a conceptual
+language. In the end, the programmer would understand the goals of the program and
+could describe it in high-level terms-for example, "This program translates novels front
+Russian to English", or "This program composes an eight-voice fugue based on any
+theme which is fed in".
+
+
+Minds and Thoughts
+
+
+381
+
+
+
+High-Level Comparisons between Brains
+
+
+Now our question must be investigated in the case of brains. In this case, we are asking,
+"Are people's brains also capable of being 'read', on a high level? Is there some objective
+description of the content of a brain?" In the Ant Fugue, the Anteater claimed to be able
+to tell what Aunt Hillary was thinking about, by looking at the scurryings of her
+component ants. Could some superbeing-a Neuroneater, perhaps-conceivably look down
+on our neurons, chunk what it sees, and come up with an analysis of our thoughts?
+
+Certainly the answer must be yes, since we are all quite able to describe, in
+chunked (i.e., non-neural) terms, the activity of our minds at any given time. This means
+that we have a mechanism which allows us to chunk our own brain state to some rough
+degree, and to give a functional description of it. To be more precise, we do not chunk all
+of the brain state-we only chunk those portions of it which are active. However, if
+someone asks us about a subject which is coded in a currently inactive area of our brain,
+we can almost instantly gain access to the appropriate dormant area and come up with a
+chunked description of it-that is, some belief on that subject. Note that we come back
+with absolutely zero information on the neural level of that part of the brain: our
+description is so chunked that we don't even have any idea what part of our brain it is a
+description of. This can be contrasted with the programmer whose chunked description
+comes from conscious analysis of every part of the memory dump.
+
+Now if a person can provide a chunked description of any part of his own brain,
+why shouldn't an outsider too, given some nondestructive means of access to the same
+brain, not only be able to chunk limited portions of the brain, but actually to give a
+complete chunked description of it-in other words, a complete documentation of the
+beliefs of the person whose brain is accessible? It is obvious that such a description
+would have an astronomical size, but that is not of concern here. We are interested in the
+question of whether, in principle, there exists a well-defined, highlevel description of a
+brain, or whether, conversely, the neuron-level description-or something equally
+physiological and intuitively unenlightening-is the best description that in principle
+exists. Surely, to answer this question would be of the highest importance if we seek to
+know whether we can ever understand ourselves.
+
+Potential Beliefs, Potential Symbols
+
+It is my contention that a chunked description is possible, but when we get it, all will not
+suddenly be clear and light. The problem is that in order to pull a chunked description out
+of the brain state, we need a language to describe our findings. Now the most appropriate
+way to describe a brain, it would seem, would be to enumerate the kinds of thoughts it
+could entertain, and the kinds of thoughts it could not entertain-or, perhaps, to enumerate
+its beliefs and the things which it does not believe. If that is the
+
+
+Minds and Thoughts
+
+
+382
+
+
+
+kind of goal we will be striving for in a chunked description, then it is easy to see what
+kinds of troubles we will run up against.
+
+Suppose you wanted to enumerate all possible voyages that could be taken in an
+ASU; there are infinitely many. How do you determine which ones are plausible, though?
+Well, what does "plausible" mean? We will have precisely this kind of difficulty in trying
+to establish what a "possible pathway" from symbol to symbol in a brain is. We can
+imagine an upsidedown dog flying through the air with a cigar in its mouth-or a collision
+between two giant fried eggs on a freeway-or any number of other ridiculous images. The
+number of far-fetched pathways which can be followed in our brains is without bound,
+just as is the number of insane itineraries that could be planned on an ASU. But just what
+constitutes a "sane" itinerary, given an ASU? And just what constitutes a "reasonable"
+thought, given a brain state? The brain state itself does not forbid anv pathway, because
+for any pathway there are always circumstances which could force the following of that
+pathway. The physical status of a brain, if read correctly, gives information telling not
+which pathways could be followed, but rather how much resistance would be offered
+along the way.
+
+Now in an ASU, there are many trips which could be taken along two or more
+reasonable alternative routes. For example, the trip from San Francisco to New York
+could go along either a northern route or a southern route. Each of them is quite
+reasonable, but people tend to take them under different circumstances. Looking at a map
+at a given moment in time does not tell you anything about which route will be preferable
+at some remote time in the future-that depends on the external circumstances under which
+the trip is to be taken. Likewise, the "reading" of a brain state will reveal that several
+reasonable alternative pathways are often available, connecting a given set of symbols.
+However, the trip among these symbols need not be imminent; it may be simply one of
+billions of "potential" trips, all of which figure in the readout of the brain state. Lrom this
+follows an important conclusion: there is no information in the brain state itself which
+tells which route will be chosen. The external circumstances will play a large determining
+role in choosing the route.
+
+What does this imply? It implies that thoughts which clash totally may be
+produced by a single brain, depending on the circumstances. And any high-level readout
+of the brain state which is worth its salt must contain all such conflicting versions.
+Actually this is quite obvious-that we all are bundles of contradictions, and we manage to
+hang together by bringing out only one side of ourselves at a given time. The selection
+cannot be predicted in advance, because the conditions which will force the selection are
+not known in advance. What the brain state can provide, if properly read, is a conditional
+description of the selection of routes.
+
+Consider, for instance, the Crab's plight, described in the Prelude. He can react in
+various ways to the playing of a piece of music. Sometimes he will be nearly immune to
+it, because he knows it so well. Other times, he will be quite excited by it, but this
+reaction requires the right kind of triggering from the outside-for instance, the presence of
+an enthusiastic listener, to
+
+
+Minds and Thoughts
+
+
+383
+
+
+
+whom the work is new. Presumably, a high-level reading of the Crab's brain state would
+reveal the potential thrill (and conditions which would induce it), as well as the potential
+numbness (and conditions which would induce it). The brain state itself would not tell
+which one would occur on the next hearing of the piece, however: it could only say, "If
+such-&-such conditions obtain, then a thrill will result; otherwise ..."
+
+Thus a chunked description of a brain state would give a catalogue of beliefs
+which could be evoked conditionally, dependent on circumstances. Since not all possible
+circumstances can be enumerated, one would have to settle for those which one thinks are
+"reasonable". Furthermore, one would have to settle for a chunked description of the
+circumstances themselves, since they obviously cannot-and should not-be specified down
+to the atomic level! Therefore, one will not be able to make an exact, deterministic
+prediction saying which beliefs will be pulled out of the brain state by a given chunked
+circumstance. In summary, then, a chunked description of a brain state will consist of a
+probabilistic catalogue, in which are listed those beliefs which are most likely to be
+induced (and those symbols which are most likely to be activated) by various sets of
+"reasonably likely" circumstances, themselves described on a chunked level. Trying to
+chunk someone's beliefs without referring to context is precisely as silly as trying to
+describe the range of a single person's "potential progeny" without referring to the mate.
+
+The same sorts of problems arise in enumerating all the symbols in a given
+person's brain. There are potentially not only an infinite number of pathways in a brain,
+but also an infinite number of symbols. As was pointed out, new concepts can always be
+formed from old ones, and one could argue that the symbols which represent such new
+concepts are merely dormant symbols in each individual, waiting to be awakened. They
+may never get awakened in the person's lifetime, but it could be claimed that those
+symbols are nonetheless always there, just waiting for the right circumstances to trigger
+their synthesis. However, if the probability is very low, it would seem that "dormant"
+would be a very unrealistic term to apply in the situation. To make this clear, try to
+imagine all the "dormant dreams" which are sitting there inside your skull while you're
+awake. Is it conceivable that there exists a decision procedure which could tell
+"potentially dreamable themes" from "undreamable themes", given your brain State
+
+Where Is the Sense of Self?
+
+Looking back on what we have discussed, you might think to yourself, "These
+speculations about brain and mind are all well and good, but what about the feelings
+involved in consciousness. These symbols may trigger each other all they want, but
+unless someone perceives the whole thing, there's no consciousness."
+
+This makes sense to our intuition on some level, but it does not make much sense
+logically. For we would then be compelled to look for an
+
+
+Minds and Thoughts
+
+
+384
+
+
+
+explanation of the mechanism which does the perceiving of all the active symbols, if it is
+not covered by what we have described so far. Of course, a "soulist" would not have to
+look any further-he would merely assert that the perceiver of all this neural action is the
+soul, which cannot be described in physical terms, and that is that. However, we shall try
+to give a "nonsoulist" explanation of where consciousness arises.
+
+Our alternative to the soulist explanation-and a disconcerting one it is, too- is to
+stop at ohe symbol level and say, "This is it-this is what consciousness is. Consciousness
+is that property of a system that arises whenever there exist symbols in the system which
+obey triggering patterns somewhat like the ones described in the past several sections."
+Put so starkly, this may seem inadequate. How does it account for the sense of "I", the
+sense of self?
+
+
+Subsystems
+
+There is no reason to expect that "I", or "the self'", should not be represented by a
+symbol. In fact, the symbol for the self is probably the most complex of all the symbols
+in the brain. For this reason, I choose to put it on a new level of the hierarchy and call it a
+subsystem, rather than a symbol. To be precise, by "subsystem", I mean a constellation of
+symbols, each of which can be separately activated under the control of the subsystem
+itself. The image I wish to convey of a subsystem is that it functions almost as an
+independent "subbrain", equipped with its own repertoire of symbols which can trigger
+each other internally. Of course, there is also much communication between the
+subsystem and the "outside" world-that is, the rest of the brain. "Subsystem" is just
+another name for an overgrown symbol, one which has gotten so complicated that it has
+many subsymbols which interact among themselves. Thus, there is no strict level
+distinction between symbols and subsystems.
+
+Because of the extensive links between a subsystem and the rest of the brain
+(some of which will be described shortly), it would be very difficult to draw a sharp
+boundary between the subsystem and the outside; but even if the border is fuzzy, the
+subsystem is quite a real thing. The interesting thing about a subsystem is that, once
+activated and left to its own devices, it can work on its own. Thus, two or more
+subsystems of the brain of an individual may operate simultaneously. I have noticed this
+happening on occasion in my own brain: sometimes I become aware that two different
+melodies are running through my mind, competing for "my" attention. Somehow, each
+melody is being manufactured, or "played", in a separate compartment of my brain. Each
+of the systems responsible for drawing a melody out of my brain is presumably activating
+a number of symbols, one after another, completely oblivious to the other system doing
+the same thing. Then they both attempt to communicate with a third subsystem of my
+brain-mv self-symbol- and it is at that point that the "1" inside my brain gets wind of
+what’s going on: in other words, it starts picking up a chunked description of the activities
+of those two subsystems.
+
+
+Minds and Thoughts
+
+
+385
+
+
+
+Subsystems and Shared Code
+
+
+Typical subsystems might be those that represent the people we know intimately. They
+are represented in such a complex way in our brains that their symbols enlarge to the rank
+of subsystem, becoming able to act autonomously, making use of some resources in our
+brains for support. By this, I mean that a subsystem symbolizing a friend can activate
+many of the symbols in my brain just as I can. For instance, I can fire up my subsystem
+for a good friend and virtually feel myself in his shoes, running through thoughts which
+he might have, activating symbols in sequences which reflect his thinking patterns more
+accurately than my own. It could be said that my model of this friend, as embodied in a
+subsystem of my brain, constitutes my own chunked description of his brain.
+
+Does this subsystem include, then, a symbol for every symbol which I think is in
+his brain? That would be redundant. Probably the subsystem makes extensive use of
+symbols already present in my brain. For instance, the symbol for "mountain" in my brain
+can be borrowed by the subsystem, when it is activated. The way in which that symbol is
+then used by the subsystem will not necessarily be identical to the way it is used by my
+full brain. In particular, if I am talking with my friend about the Tien Shan mountain
+range in Central Asia (neither of us having been there), and I know that a number of years
+ago he had a wonderful hiking experience in the Alps, then my interpretation of his
+remarks will be colored in part by my imported images of his earlier Alpine experience,
+since I will be trying to imagine how he visualizes the area.
+
+In the vocabulary we have been building up in this Chapter, we could say that the
+activation of" the "mountain" symbol in me is under control of my subsystem
+representing him. The effect of this is to open up a different window onto to my
+memories from the one which I normally use-namely, my "default option" switches from
+the full range of my memories to the set of my memories of his memories. Needless to
+say, my representations of his memories are only approximations to his actual memories,
+which are complex modes of activation of the symbols in his brain, inaccessible to me.
+
+My representations of his memories are also complex modes of activation of my
+own symbols-those for "primordial" concepts, such as grass, trees, snow, sky, clouds, and
+so on. These are concepts which I must assume are represented in him "identically" to the
+way they are in me. I must also assume a similar representation in him of even more
+primordial notions: the experiences of gravity, breathing, fatigue, color, and so forth.
+Less primordial but perhaps a nearly universal human quality is the enjoyment of
+reaching a summit and seeing a view. Therefore, the intricate processes in my brain
+which are responsible for this enjoyment can be taken over directly by the friend-
+subsystem without much loss of fidelity.
+
+We could go on to attempt to describe how I understand an entire tale told by my
+friend, a tale filled with many complexities of human relationships and mental
+experiences. But our terminology would quickly become inadequate. There would be
+tricky recursions connected with representa
+
+
+Minds and Thoughts
+
+
+386
+
+
+
+tions in him of representations in me of representations in him of one thing and another.
+If mutual friends figured in the tale being told, I would unconsciously look for
+compromises between my image of his representations of them, and my own images of
+them. Pure recursion would simply be an inappropriate formalism for dealing with
+symbol amalgams of this type. And I have barely scratched the surface!
+
+We plainly lack the vocabulary today for describing the complex interactions that are
+possible between symbols. So let us stop before we get bogged down.
+
+We should note, however, that computer systems are beginning to run into some
+of the some kinds of complexity, and therefore some of these notions have been given
+names. For instance, my "mountain" symbol is analogous to what in computer jargon is
+called shared (or reentrant) codecode which can be used by two or more separate
+timesharing programs running on a single computer. The fact that activation of one
+symbol can have different results when it is part of different subsystems can be explained
+by saying that its code is being processed by different interpreters. Thus, the triggering
+patterns in the "mountain" symbol are not absolute; they are relative to the system within
+which the symbol is activated.
+
+The reality of such "subbrains" may seem doubtful to some. Perhaps the
+following quote from M. C. Escher, as he discusses how he creates his periodic plane¬
+filling drawings, will help to make clear what kind of phenomenon I am referring to:
+
+While drawing I sometimes feel as if I were a spiritualist medium, controlled
+by the creatures which I am conjuring up. It is as if they themselves decide on
+the shape in which they choose to appear. They take little account of my
+critical opinion during their birth and I cannot exert much influence on the
+measure of their development. They are usually very difficult and obstinate
+creatures
+
+Here is a perfect example of the near-autonomy of certain subsystems of the
+brain, once they are activated. Escher's subsystems seemed to him almost to be able to
+override his esthetic judgment. Of course, this opinion must be taken with a grain of salt,
+since those powerful subsystems came into being as a result of his many years of training
+and submission to precisely the forces that molded his esthetic sensitivities. In short, it is
+wrong to divorce the subsystems in Escher's brain from Escher himself or from his
+esthetic judgment. They constitute a vital part of his esthetic sense, where "he" is the
+complete being of the artist.
+
+The Self-Symbol and Consciousness
+
+A very important side effect of the self-subsystem is that it can play the role of "soul", in
+the following sense: in communicating constantly with the rest of the subsystems and
+symbols in the brain, it keeps track of what symbols are active, and in what way. This
+means that it has to have symbols for mental activity-in other words, symbols for
+symbols, and symbols for the
+actions of symbols.
+
+
+Minds and Thoughts
+
+
+387
+
+
+
+Of course, this does not elevate consciousness or awareness to any "magical",
+nonphysical level. Awareness here is a direct effect of the complex hardware and
+software we have described. Still, despite its earthly origin, this way of describing
+awareness-as the monitoring of brain activity by a subsystem of the brain itself-seems to
+resemble the nearly indescribable sensation which we all know and call "consciousness".
+Certainly one can see that the complexity here is enough that many unexpected effects
+could be created. For instance, it is quite plausible that a computer program with this kind
+of structure would make statements about itself which would have a great deal of
+resemblance to statements which people commonly make about themselves. This
+includes insisting that it has free will, that it is not explicable as a "sum of its parts", and
+so on. (On this subject, see the article "Matter, Mind, and Models" by M. Minsky in his
+book Semantic Information Processing.)
+
+What kind of guarantee is there that a subsystem, such as I have here postulated,
+which represents the self, actually exists in our brains? Could a whole complex network
+of symbols such as has been described above evolve without a self-symbol evolving,
+How could these symbols and their activities play out "isomorphic" mental events to real
+events in the surrounding universe, if there were no symbol for the host organism, All the
+stimuli coming into the system are centered on one small mass in space. It would be quite
+a glaring hole in a brain's symbolic structure not to have a symbol for the physical object
+in which it is housed, and which plays a larger role in the events it mirrors than any other
+object. In fact, upon reflection, it seems that the only way one could make sense of the
+world surrounding a localized animate object is to understand the role of that object in
+relation to the other objects around it. This necessitates the existence of a selfsymbol; and
+the step from symbol to subsystem is merely a reflection of the importance of the self¬
+symbol', and is not a qualitative change.
+
+Our First Encounter with Lucas
+
+The Oxford philosopher J. R. Lucas (not connected with the Lucas numbers described
+earlier) wrote a remarkable article in 1961, entitled "Minds, Machines, and Godel". His
+views are quite opposite to mine, and yet he manages to mix many of the same
+ingredients together in coming up with his opinions. The following excerpt is quite
+relevant to what we have just been discussing:
+
+At one's first and simplest attempts to philosophize, one becomes entangled in questions of
+whether when one knows something one knows that one knows it, and what, when one is
+thinking of oneself, is being thought about, and what is doing the thinking. After one has
+been puzzled and bruised by this problem for a long time, one learns not to press these
+questions: the concept of a conscious being is, implicitly, realized to be different from that
+of an unconscious object. In saying that a conscious being knows something, we are saying
+not onh that he knows it, but that he knows that he knows it, and that he knows that he
+knows that he knows it, and so on, as long as we care to pose the
+
+
+Minds and Thoughts
+
+
+388
+
+
+
+question: there is, we recognize, an infinity here, but it is not an infinite regress in the had
+sense, for it is the questions that peter out, as being pointless, rather than the answers. The
+questions are felt to be pointless because the concept contains within itself the idea of
+being able to go on answering such questions indefinitely. Although conscious beings have
+the power of going on, we do not wish to exhibit this simply as a succession of tasks they
+are able to perforin, nor do we see the mind as an infinite sequence of selves and super¬
+selves and super-super-selves. Rather, we insist that a conscious being is a unity, and
+though we talk about parts of the mind, we (to so only as a metaphor, and will not allow it
+to be taken literally.
+
+The paradoxes of consciousness arise because a conscious being can be aware of itself
+as well as of other things, and yet cannot really be construed as being divisible into parts. It
+means that a conscious being can deal with Godelian questions in a was in which a
+machine cannot, because a conscious being can both consider itself and its perform a rice
+and vet not be other than that which did the performance. A machine can be made in a
+manner of speaking to "consider" its performance, but it cannot take this "into account"
+without thereby becoming a different machine, namely the old machine with a "new part"
+added. Btu it is inherent in our idea of a conscious mind that it can reflect upon itself and
+criticize its own performances, and no extra part is required to (to this: it is already
+complete, and has no Achilles' heel.
+
+The thesis thus begins to become more of a matter of conceptual analysis than
+mathematical discovery. This is borne out by considering another argument put forward by
+Turing. So far, we have constructed only fairly simple and predictable artifacts. When we
+increase the complexity of our machines, there may, perhaps, be suiprises in store for us.
+He draws a parallel with a fission pile. Below a certain "critical" size, nothing much
+happens: but above the critical size, the sparks begin to fly. So too, perhaps, with brains
+and machines. Most brains and all machines are, at present, sub-critical"-they react to
+incoming stimuli in a stodgy and uninteresting way, have no ideas of their own, can
+produce only stock responses-but a few brains at present, and possibly some machines in
+the future, are super-critical, and scintillate on their own account. Turing is suggesting that
+it is only a matter of complexity, and that above a certain level of complexity a qualitative
+difference appears, so that "super-critical" machines will be quite unlike the simple ones
+hitherto envisaged.
+
+This may be so. Complexity often does introduce qualitative differences. Although it
+sounds implausible, it might turn out that above a certain level of complexity, a machine
+ceased to be predictable, even in principle, and started doing things on its own account, or,
+to use a very revealing phrase, it might begin to have a mind of its own. It might begin to
+have a mind of its own. It would begin to have a mind of its own when it was no longer
+entirely predictable and entirely docile, but was capable of doing things which we
+recognized as intelligent, and not just mistakes or random shots, but which we had not
+programmed into it. But then it would cease to be a machine, within the meaning of the act.
+What is at stake in the mechanist debate is not how minds are, or might be, brought into
+being, but how they operate. It is essential for the mechanist thesis that the mechanical
+model of the mind shall operate according to "mechanical principles," that is, that we can
+understand the operation of the whole in terms of the operations of its parts, and the
+operation of each part either shall be determined by its initial state and the construction of
+the machine, or shall be a random choice between a determinate number of determinate
+operations. If the mechanist produces a machine which is so complicated that this ceases to
+hold good of it, then it is no longer a
+
+
+Minds and Thoughts
+
+
+389
+
+
+
+machine for the purposes of our discussion, no matter how it was constructed. We should
+say, rather, that he had created a mind, in the same sort of sense as we procreate people at
+present. There would then be two ways of bringing new minds into the world, the
+traditional way, by begetting children born of women, and a new way by constructing very,
+very complicated systems of, say, valves and relays. When talking of the second way. we
+should take care to stress that although what was created looked like a machine, it was not
+one really, because it was not just the total of its parts. One could not tell what it was going
+to do merely by knowing the way in which it was built up and the initial state of its parts:
+one could not even tell the limits of what it could do, for even when presented with a
+Godel-type question, it got the answer right. In fact we should say briefly that any system
+which was not floored by the Godel question was eo ipso not a Turing machine, i.e. not a
+machine within the meaning of the act .3
+
+In reading this passage, my mind constantly boggles at the rapid succession of topics,
+allusions, connotations, confusions, and conclusions. We jump from a Carrollian paradox
+to Godel to Turing to Artificial Intelligence to holism and reductionism, all in the span of
+two brief pages. About Lucas one can say that he is nothing if not stimulating. In the
+following Chapters, we shall come back to many of the topics touched on so tantalizingly
+and fleetingly in this odd passage.
+
+
+Minds and Thoughts
+
+
+390
+
+
+
+Aria with Diverse Variations
+
+Achilles has been unable to 'sleep these past few nights. His friend the
+Tortoise has come over tonight, to keep him company during these annoying
+hours.
+
+Tortoise: I am so sorry to hear of the troubles that have been plaguing you, my dear
+Achilles. I hope my company will provide a welcome relief from all the unbearable
+stimulation which has kept you awake. Perhaps I will bore you sufficiently that you
+will at long last go to sleep. In that way, I will be of some service.
+
+Achilles: Oh, no, I am afraid that I have already had some of the world's finest bores try
+their hand at boring me to sleep-and all, sad to say, to no avail. So you will be no
+match for them. No, Mr. T, I invited you over hoping that perhaps you could
+entertain me with a little this or that, taken from number theory, so that I could at
+least while away these long hours in an agreeable fashion. You see, I have found that
+a little number theory does wonders for my troubled psyche.
+
+Tortoise: How quaint an idea! You know, it reminds me, just a wee bit, of the story of
+poor Count Kaiserling.
+
+'Achilles: Who was he?
+
+Tortoise: Oh, he was a Count in Saxony in the eighteenth century-a Count of no account,
+to tell the truth-but because of him-well, shall I tell you the story? It is quite
+entertaining.
+
+Achilles: In that case, by all means, do!
+
+Tortoise: There was a time when the good Count was suffering from sleeplessness, and it
+just so happened that a competent musician lived in the same town, and so Count
+Kaiserling commissioned this musician to compose a set of variations to be played
+by the Count's court harpsichordist for him during his sleepless nights, to make the
+hours pass by more pleasantly.
+
+Achilles: Was the local composer up to the challenge?
+
+Tortoise: I suppose so, for after they were done, the Count rewarded him most
+lucratively-he presented him with a gold goblet containing one hundred Louis d'or.
+
+Achilles: You don't say! I wonder where he came upon such a goblet and all those Louis
+d'or, in the first place.
+
+Tortoise. Perhaps he saw it in a museum, and took a fancy to it.
+
+Achilles: Are you suggesting he absconded with it?
+
+Tortoise: Now, now, I wouldn't put it exactly that way, but... Those days. Counts could
+get away with most anything. Anyway, it is clear that the Count was most pleased
+with the music, for he was constantly entreating his harpsichordist-a mere lad of a
+fellow, name of Goldberg-to
+
+
+Aria with Diverse Variations
+
+
+391
+
+
+
+play one or another of these thirty variations. Consequently (and somewhat
+ironically) the variations became attached to the name of young Goldberg, rather
+than to the distinguished Count's name.
+
+Achilles: You mean, the composer was Bach, and these were the so-called "Goldberg
+Variations"?
+
+Tortoise: Do I ever! Actually, the work was entitled Aria with Diverse Variations, of
+which there are thirty. Do you know how Bach structured these thirty magnificent
+variations?
+
+Achilles: Do tell.
+
+Tortoise: All the pieces-except the final one-are based on a single theme, which he called
+an "aria". Actually, what binds them all together is not a common melody, but a
+common harmonic ground. The melodies may vary, but underneath, there is a
+constant theme. Only in the last variation did Bach take liberties. It is a sort of "post¬
+ending ending". It contains extraneous musical ideas having little to do with the
+original Theme-in fact, two German folk tunes. That variation is called a
+"quodlibet".
+
+Achilles: What else is unusual about the Goldberg Variations?
+
+Tortoise: Well, every third variation is a canon. First a canon in two canonizing voices
+enter on the SAME note. Second, a canon in which one of the canonizing voices
+enters ONE NOTE HIGHER than the first. Third, one voice enters Two notes higher
+than the other. And so on, until the final canon has entries just exactly one ninth
+apart. Ten canons, all told. And
+
+Achilles: Wait a minute. Don't I recall reading somewhere or other about fourteen
+recently discovered Goldberg canons ...
+
+Tortoise: Didn't that appear in the same journal where they recently reported the
+discovery of fourteen previously unknown days in November?
+
+Achilles: No, it's true. A fellow named Wolff-a musicologist-heard about a special copy
+of the Goldberg Variations in Strasbourg. He went there to examine it, and to his
+surprise, on the back page, as a sort of "post-ending ending", he found these fourteen
+new canons, all based on the first eight notes of the theme of the Goldberg
+Variations. So now it is known that there are in reality forty-four Goldberg
+Variations, not thirty.
+
+Tortoise: That is, there are forty-four of them, unless some other musicologist discovers
+yet another batch of them in some unlikely spot. And although it seems improbable,
+it is still possible, even if unlikely, that still another batch will be discovered, and
+then another one, and on and on and on ... Why, it might never stop! We may never
+know if or when we have the full complement of Goldberg Variations.
+
+Achilles: That is a peculiar idea. Presumably, everybody thinks that this latest discovery
+was just a fluke, and that we now really do have all the Goldberg Variations. But just
+supposing that you are right, and some turn up sometime, we shall start to expect
+this kind of thing. At
+
+
+Aria with Diverse Variations
+
+
+392
+
+
+
+that point, the name "Goldberg Variations" will start to shift slightly in meaning, to
+include not only the known ones, but also any others which might eventually turn up.
+Their number-call it 'g'-is certain to be finite, wouldn't you agree?-but merely
+knowing that g is finite isn't the same as knowing how big g is. Consequently, this
+information won't tell us when the last Goldberg Variation has been located.
+
+Tortoise: That is certainly true.
+
+Achilles: Tell me-when was it that Bach wrote these celebrated variations?
+
+Tortoise: It all happened in the year 1742, when he was Cantor in Leipzig. Achilles:
+1742? Hmm ... That number rings a bell.
+
+Tortoise: It ought to, for it happens to be a rather interesting number, being a sum of two
+odd primes: 1729 and 13.
+
+Achilles: By thunder! What a curious fact' I wonder how often one runs across an even
+number with that property. Let's see
+
+6= 3+3
+8= 3+5
+
+10= 3+7= 5+5
+
+12= 5+7
+
+14=3+11= 7+7
+
+16=3+13= 5+11
+
+18=5+13= 7+11
+
+20=3+ 17= 7+ 13
+
+22 = 3 + 19=5 + 17= 11 +11
+
+24=5+19= 7+17=11+13
+
+26=3+23= 7+19=13+13
+
+28 = 5 +23 = 11 + 17
+30 = 7 +23 = 11 + 19= 13 + 17
+
+Now what do you know-according to my little table here, it seems to be quite a
+common occurrence. Yet I don't discern any simple regularity in the table so far.
+
+Tortoise: Perhaps there is no regularity to be discerned.
+
+Achilles: But of course there is! I am just not clever enough to spot it right off the bat.
+
+Tortoise: You seem quite convinced of it.
+
+Achilles: There's no doubt in my mind. I wonder ... Could it be that ALL even numbers
+(except 4) can be written as a sum of two odd primes?
+
+Tortoise: Hmm ... That question rings a bell ... Ah, I know why! You're not the first
+person to ask that question. Why, as a matter of fact, in the year 1742, a
+mathematical amateur put forth this very question in a
+
+Achilles: Did you say 1742? Excuse me for interrupting, but I just noticed that 1742
+happens to be a rather interesting number, being a difference of two odd primes:
+1747 and 5.
+
+Tortoise: By thunder! What a curious fact! I wonder how often one runs across an even
+number with that property.
+
+
+Aria with Diverse Variations
+
+
+393
+
+
+
+Achilles: But please don't let me distract you from your story.
+
+Tortoise: Oh, yes-as I was saying, in 1742. a certain mathematical amateur, whose name
+escapes me momentarily, sent a letter to Euler, who at the time was at the court of
+King Frederick the Great in Potsdam, and-well, shall I tell you the story? It is not
+without charm. Achilles: In that case, by all means, do!
+
+Tortoise: Very well. In his letter, this dabbler in number theory propounded an unproved
+conjecture to the great Euler: "Every even number can he represented as a sum of
+two odd primes." Now what was that fellow's name?
+
+Achilles: I vaguely recollect the story, from some number theory book or other. Wasn't
+the fellow named Iiupfergiidel
+
+Tortoise: Hmm ... No, that sounds too long.
+
+Achilles: Could it have been "Silberescher"?
+
+Tortoise: No, that's not it, either. There's a name on the tip of my tongue-ah-ah-oh yes! It
+was "Goldbach"! Goldbach was the fellow. Achilles: I knew it was something like
+that.
+
+Tortoise: Yes-your guesses helped jog my memory. It's quite odd, how one occasionally
+has to hunt around in one's memory as if for a book in a library without call numbers
+... But let us get back to 1742.
+
+Achilles: Indeed, let's. I wanted to ask you: did Euler ever prove that this guess by
+Goldbach was right?
+
+Tortoise: Curiously enough, he never even considered it worthwhile working on.
+However, his disdain was not shared by all mathematicians. In fact, it caught the
+fancy of many, and became known as the "Goldbach Conjecture".
+
+Achilles: Has it ever been proven correct?
+
+Tortoise: No, it hasn't. But there have been some remarkable near misses. For instance, in
+1931 the Russian number theorist Schnirelmann proved that any number-even or
+odd-can be represented as the sum of not more than 300,000 primes.
+
+Achilles: What a strange result. Of what good is it?
+
+Tortoise: It has brought the problem into the domain of the finite. Previous to
+Schnirelmann's proof, it was conceivable that as you took larger and larger even
+numbers, they would require more and more primes to represent them. Some even
+number might take a trillion primes to represent it! Now it is known that that is not
+so-a sum of 300,000 primes (or fewer) will always suffice.
+
+Achilles: I see.
+
+Tortoise: Then in 1937, a sly fellow named Vinogradov-a Russian too-managed to
+establish something far closer to the desired result: namely, every sufficiently large
+ODD number can be represented as a sum of no more than THREE odd primes. For
+example, 1937 = 641 + 643 + 653. We could say that an odd number which is
+representable as a sum of three odd primes has "the Vinogradov property. Thus, all
+sufficiently large odd numbers have the Vinigradov properties
+
+
+Aria with Diverse Variations
+
+
+394
+
+
+
+Achilles: Very well-but what does "sufficiently large" mean?
+
+Tortoise: It means that some finite number of odd numbers may fail to have the
+Vinogradov property, but there is a number-call it 'v'beyond which all odd numbers
+have the Vinogradov property. But Vinogradov was unable to say how big v is. So in
+a way, v is like g, the finite but unknown number of Goldberg Variations. Merely
+knowing that v is finite isn't the same as knowing how big v is. Consequently, this
+information won't tell us when the last odd number which needs more than three
+primes to represent it has been located.
+
+Achilles: I see. And so any sufficiently large even number 2N can be represented as a
+sum of FOUR primes, by first representing 2N - 3 as a sum of three primes, and then
+adding back the prime number 3.
+
+Tortoise: Precisely. Another close approach is contained in the Theorem which says, "All
+even numbers can be represented as a sum of one prime and one number which is a
+product of at most two primes."
+
+Achilles: This question about sums of two primes certainly leads you into strange
+territory. I wonder where you would be led if you looked at DIFFERENCES of two
+odd primes. I'll bet I could glean some insight into this teaser by making a little table
+of even numbers, and their representations as differences of two odd primes, just as I
+did for sums. Let's see ...
+
+
+2= 5-3,
+
+7-5,
+
+13-11,
+
+19-17, etc.
+
+4= 7-3,
+
+11-7,
+
+17 - 13,
+
+23 - 19,etc.
+
+6= 11 -5,
+
+13-7,
+
+17 - 11,
+
+19- 13, etc.
+
+8= 11-3,
+
+13-5,
+
+19-11,
+
+31 - 23,etc.
+
+10=13-3,
+
+17-7,
+
+23 - 13,
+
+29- 19, etc.
+
+
+My gracious! There seems to be no end to the number of different representations I can
+find for these even numbers. Yet I don't discern any simple regularity in the table so
+far.
+
+Tortoise: Perhaps there is no regularity to be discerned.
+
+Achilles: Oh, you and your constant rumblings about chaos! I'll hear none of that, thank
+you.
+
+Tortoise: Do you suppose that EVERY even number can be represented somehow as the
+difference of two odd primes?
+
+Achilles: The answer certainly would appear to be yes, from my table. But then again, I
+suppose it could also be no. That doesn't really get us very far, does it?
+
+Tortoise: With all due respect, I would say there are deeper insights to be had on the
+matter.
+
+Achilles: Curious how similar this problem is to Goldbach's original one. Perhaps it
+should be called a "Goldbach Variation".
+
+Tortoise: Indeed. But you know, there is a rather striking difference between the
+Goldbach Conjecture, and this Goldbach Variation, which I would like to tell you
+about. Let us say that an even number 2N has the “Goldbach property” if it is the
+SUM of two odd primes, and it has the “Tortoise property” if it is the DIFFERENCE
+of two odd primes
+
+
+Aria with Diverse Variations
+
+
+395
+
+
+
+Achilles: I think you should call it the "Achilles property". After all, I suggested the
+problem.
+
+Tortoise: I was just about to propose that we should say that a number which LACKS the
+Tortoise property has the "Achilles property". Achilles: Well, all right.. .
+
+Tortoise: Now consider, for instance, whether I trillion has the Goldbach property or the
+Tortoise property. Of course, it may have both.
+
+Achilles: I can consider it, but I doubt whether I can give you an answer to either
+question.
+
+Tortoise: Don't give up so soon. Suppose I asked you to answer one or the other question.
+Which one would you pick to work on?
+
+Achilles: I suppose I would flip a coin. I don't see much difference between them.
+
+Tortoise: Aha: But there's a world of difference' If you pick the Goldbach property,
+involving SUMS of primes, then you are limited to using primes which are bounded
+between 2 and 1 trillion, right?
+
+Achilles: Of course.
+
+Tortoise: So your search for a representation for 1 trillion as a sum of two primes is
+GUARANTEED TO TERMINATE.
+
+Achilles: Ahhh! I see your point. Whereas if I chose to work on representing 1 trillion as
+the DIFFERENCE of two primes, I would not have any bound on the size of the
+primes involved. They might be so big that it would take me a trillion years to find
+them.
+
+Tortoise: Or then again, they might not even EXIST! After all, that's what the question
+was asking-do such primes exist, It wasn't of much concern how big they might turn
+out to be.
+
+Achilles: You're right. If they didn't exist, then a search process would lead on forever,
+never answering yes, and never answering no. And nevertheless, the answer would
+be no.
+
+Tortoise: So if you have some number, and you wish to test whether it has the Goldbach
+property or the Tortoise property, the difference between the two tests will be this: in
+the former, the search involved is GUARANTEED TO TERMINATE; in the latter,
+it is POTENTIALLY ENDLESS-there are no guarantees of any type. It might just
+go merrily on forever, without yielding an answer. And yet, on the other hand, in
+some cases, it might stop on the first step.
+
+Achilles: I see there is a rather vast difference between the Goldbach and Tortoise
+properties.
+
+Tortoise: Yes, the two similar problems concern these vastly different properties. The
+Goldbach Conjecture is to the effect that all even numbers have the Goldbach
+property; the Goldbach Variation suggests that all even numbers have the Tortoise
+property. Both problems are unsolved, but what is interesting is that although they
+sound very much alike, they involve properties of whole numbers which are quite
+different.
+
+Achilles: I see what you mean. The Goldbach property is a detectable, or
+
+
+Aria with Diverse Variations
+
+
+396
+
+
+
+recognizable property of any even number, since I know how to test for its presence
+just embark on a search. It will automatically come to an end with a yes or no
+answer. The Tortoise property, however, is more elusive, since a brute force search
+just may never give an answer.
+
+Tortoise: Well, there may be cleverer ways of searching in the case of the Tortoise
+property, and maybe following one of them would always come to an end, and yield
+an answer.
+
+Achilles: Couldn't the search only end if the answer were yes%
+
+Tortoise: Not necessarily. There might be some way of proving that whenever the search
+lasts longer than a certain length of time, then the answer must be no. There might
+even be some OTHER way of searching for the primes, not such a brute force way,
+which is guaranteed to find them if they exist, and to tell if they don't. In either case,
+a finite search would be able to yield the answer no. But I don't know if such a thing
+can be proven or not. Searching through infinite spaces is always a tricky matter, you
+know.
+
+Achilles: So as things stand now, you know of no test for the Tortoise property which is
+guaranteed to terminate-and yet there MIGHT exist such a search.
+
+Tortoise: Right. I suppose one could embark on a search for such a search, but I can give
+no guarantee that that "meta-search" would terminate, either.
+
+Achilles: You know, it strikes me as quite peculiar that if some even number-for
+example, a trillion-failed to have the Tortoise property, it would be caused by an
+infinite number of separate pieces of information. It's funny to think of wrapping all
+that information up into one bundle, and calling it, as you so gallantly suggested,
+"the Achilles property" of 1 trillion. It is really a property of the number system as a
+"HOLE, not just of the number 1 trillion.
+
+Tortoise: That is an interesting observation, Achilles, but I maintain that it makes a good
+deal of sense to attach this fact to the number 1 trillion nevertheless. For purposes of
+illustration, let me suggest that you . consider the simpler statement "29 is prime".
+Now in fact, this statement really means that 2 times 2 is not 29, and 5 times 6 is not
+29, and so forth, doesn't it?
+
+Achilles: It must, I suppose.
+
+Tortoise: But you are perfectly happy to collect all such facts together, and attach them in
+a bundle to the number 29, saying merely, "29 is prime"
+
+Achilles: Yes ...
+
+Tortoise: And the number of facts involved is actually infinite, isn't it:, After all, such
+facts as "4444 times 3333 is not 29" are all part of it, aren't they%
+
+Achilles: Strictly speaking, I suppose so. But you and I both know that you can't produce
+29 by multiplying two numbers which are both bigger than 29. So in reality, saying
+"29 is prime" is only summarizing a FINITE number of facts about multiplication
+
+
+Aria with Diverse Variations
+
+
+397
+
+
+
+Tortoise: You can put it that way if you want, but think of this: the fact that two numbers
+which are bigger than 29 can't have a product equal to 29 involves the entire
+structure of the number system. In that sense, that fact in itself is a summary of an
+infinite number of facts. You can't get away from the fact, Achilles, that when you
+say "29 is prime'-', you are actually stating an infinite number of things.
+
+Achilles: Maybe so, but it feels like just one fact to me.
+
+Tortoise: That's because an infinitude of facts are contained in your prior knowledge-they
+are embedded implicitly in the way you visualize things. You don't see an explicit
+infinity because it is captured implicitly inside the images you manipulate.
+
+Achilles: I guess that you're right. It still seems odd to lump a property of the entire
+number system into a unit, and label the unit "primeness of 29"
+
+Tortoise: Perhaps it seems odd, but it is also quite a convenient way to look at things.
+Now let us come back to your hypothetical idea. If, as you suggested, the number 1
+trillion has the Achilles property, then no matter what prime you add to it, you do not
+get another prime. Such a state of affairs would be caused by an infinite number of
+separate mathematical "events". Now do all these "events" necessarily spring from
+the same source? Do they have to have a common cause? Because if they don't, then
+some sort of "infinite coincidence" has created the fact, rather than an underlying
+regularity.
+
+Achilles: An "infinite coincidence"? Among the natural numbers, NoTHING is
+coincidental-nothing happens without there being some underlying pattern. Take 7,
+instead of a trillion. I can deal with it more easily, because it is smaller. 7 has the
+Achilles property.
+
+Tortoise: You're sure?
+
+Achilles: Yes. Here's why. If you add 2 to it, you get 9, which isn't prime. And if you add
+any other prime to 7, you are adding two odd numbers, resulting in an even number-
+thus you again fail to get a prime. So here the "Achilleanity" of 7, to coin a term, is a
+consequence of just Two reasons: a far cry from any "infinite coincidence". Which
+just goes to support my assertion: that it never takes an infinite number of reasons to
+account for some arithmetical truth. If there WERE some arithmetical fact which
+were caused by an infinite collection of unrelated coincidences, then you could never
+give a finite proof for that truth. And that is ridiculous.
+
+Tortoise: That is a reasonable opinion, and you are in good company in making it.
+However
+
+Achilles: Are there actually those who disagree with this view? Such people would have
+to believe that there are "infinite coincidences", that there is chaos in the midst of the
+most perfect, harmonious, and beautiful of all creations: the system of natural
+numbers.
+
+Tortoise: Perhaps they do; but have you ever considered that such chaos might be an
+integral part of the beauty and harmony?
+
+
+Aria with Diverse Variations
+
+
+398
+
+
+
+FIGURE 71 Order and Chaos, by M. C. Escher (lithograph, 1950).
+
+
+Achilles: Chaos, part of perfection? Order and chaos make a pleasing unity? Heresy!
+
+Tortoise: Your favorite artist, M. C. Escher, has been known to suggest such a heretical
+point of view in one of his pictures ... And while we're on the subject of chaos, I
+believe that you might be interested in hearing about two different categories of
+search, both of which are guaranteed to terminate.
+
+Achilles: Certainly.
+
+
+Aria with Diverse Variations
+
+
+399
+
+
+
+Tortoise: The first type of search-the non-chaotic type-is exemplified by the test involved
+in checking for the Goldbach property. You just look at primes less than 2N, and if
+some pair adds up to 2N, then 2N has the Goldbach property; otherwise, it doesn't.
+This kind of test is not only sure to terminate, but you can predict BY '"HEN it will
+terminate, as well.
+
+Achilles: So it is a PREDICTABLY TERMINATING test. Are you going to tell me that
+checking for some number-theoretical properties involves tests which are guaranteed
+to terminate, but about which there is no way to know in advance how long they will
+take?
+
+Tortoise: How prophetic of you, Achilles. And the existence of such tests shows that
+there is intrinsic chaos, in a certain sense, in the natural number system.
+
+Achilles: Well, in that case, I would have to say that people just don't know enough about
+the test. If they did a little more research, they could figure out how long it will take,
+at most, before it terminates. After all, there must always be some rhyme or reason to
+the patterns among integers. There can't just be chaotic patterns which defy
+prediction'
+
+Tortoise: I can understand your intuitive faith, Achilles. However, it's not always
+justified. Of course, in many cases you are exactly right just because somebody
+doesn't know something, one can't conclude that it is unknowable' But there are
+certain properties of integers for which terminating tests can be proven to exist, and
+yet about which it can also be PROVEN that there is no way to predict in advance
+how long they will take.
+
+Achilles: I can hardly believe that. It sounds as if the devil himself managed to sneak in
+and throw a monkey wrench into God's beautiful realm of natural numbers'
+
+Tortoise: Perhaps it will comfort you to know that it is by no means easy, or natural, to
+define a property for which there is a terminating but not PREDICTABLY
+terminating test. Most "natural" properties of integers do admit of predictably
+terminating tests. For example, primeness, squareness, being a power of ten, and so
+on.
+
+Achilles: Yes, I can see that those properties are completely straightforward to test for.
+Will you tell me a property for which the only possible test is a terminating but
+nonpredictable one?
+
+Tortoise: That's too complicated for me in my sleepy state. Let me instead show you a
+property which is very easy to define, and yet for which no terminating test is
+known. I'm not saying there won't ever be one discovered, mind you just that none is
+known. You begin with a number-would you care to pick one?
+
+Achilles: How about 15?
+
+Tortoise: An excellent choice. We begin with your number, and if it is ODD, we triple it,
+and add 1. If it is EVEN, we take half of it. Then we repeat the process. Call a
+number which eventually reaches 1 this way a WONDROUS number, and a number
+which doesn't, an UNWONDROUS number
+
+
+Aria with Diverse Variations
+
+
+400
+
+
+
+Achilles: Is 15 wondrous, or unwondrous? Let's see:
+
+
+15 is ODD, so I make 3n + 1: 46
+
+46 is EVEN, so I take half: 23
+
+23 is ODD, so I make 3n + 1: 70
+
+70 is EVEN, so I take half: 35
+
+35 is ODD, so I make 3n + 1: 106
+
+106 is EVEN, so I take half: 53
+
+53 is ODD, so I make 3n + 1: 160
+
+160 is EVEN, so I take half: 80
+
+80 is EVEN, so I take half: 40
+
+40 is EVEN, so I take half: 20
+
+20 is EVEN, so I take half: 10
+
+10 is EVEN, so I take half: 5
+
+5 is ODD, so I make 3n + 1: 16
+
+16 is EVEN, so I take half: 8
+
+8 is EVEN, so I take half: 4
+
+4 is EVEN, so I take half: 2
+
+2 is EVEN, so I take half: 1
+
+
+Wow! That's quite a roundabout journey, from 15 to 1. But I finally reached it. That
+shows that 15 has the property of being wondrous. I wonder what numbers are
+UN wondrous ...
+
+Tortoise: Did you notice how the numbers swung up and down, in this simply defined
+process?
+
+Achilles: Yes. I was particularly surprised, after thirteen turns, to find myself at 16, only
+one greater than 15, the number I started with. In one sense, I was almost back where
+I started-yet in another sense, I' was nowhere near where I had started. Also, I found
+it quite curious that I had to go as high as 160 to resolve the question. I wonder how
+come.
+
+Tortoise: Yes, there is an infinite "sky" into which you can sail, and it is very hard to
+know in advance how high into the sky you will wind up sailing. Indeed, it is quite
+plausible that you might just sail up and up and up, and never come down.
+
+Achilles: Really? I guess that is conceivable-but what a weird coincidence it would
+require! You'd just have to hit odd number after odd number, with only a few evens
+mixed in. I doubt if that would ever happen-but I just don't know for sure.
+
+Tortoise: Why don't you try starting with 27? Mind you, I don't promise anything. But
+sometime, just try it, for your amusement. And I'd advise you to bring along a rather
+large sheet of paper.
+
+Achilles: Hmm ... Sounds interesting. You know, it still makes me feel funny to associate
+the wondrousness (or unwondrousness) with the starting number, when it is so
+obviously a property of the entire number system.
+
+Tortoise: I understand what you mean, but it's not that different from saying “29 is
+prime” or “gold is valuable” - both statements attribute to
+
+
+Aria with Diverse Variations
+
+
+401
+
+
+
+
+
+
+a single entity a property which it has only by virtue of being embedded in a
+particular context.
+
+Achilles: I suppose you're right. This "wondrousness" problem is wondrous tricky,
+because of the way in which the numbers oscillate-now increasing, now decreasing.
+The pattern OUGHT to be regular,-yet on the surface it appears to be quite chaotic.
+Therefore, I can well imagine why, as of yet, no one knows of a test for the property
+of wondrousness which is guaranteed to terminate.
+
+Tortoise: Speaking of terminating and nonterminating processes, and those which hover
+in between, I am reminded of a friend of mine, an author, who is at work on a book.
+
+Achilles: Oh, how exciting! What is it called?
+
+Tortoise: Copper, Silver, Gold: an Indestructible Metallic Alloy. Doesn't that sound
+interesting?
+
+Achilles: Frankly, I'm a little confused by the title. After all, what do Copper, Silver, and
+Gold have to do with each other? Tortoise: It seems clear to me.
+
+Achilles: Now if the title were, say, Giraffes, Silver, Gold, or Copper, Elephants, Gold,
+why, I could see it.. .
+
+Tortoise: Perhaps you would prefer Copper, Silver, Baboons?
+
+Achilles: Oh, absolutely! But that original title is a loser. No one would understand it.
+
+Tortoise: I'll tell my friend. He'll be delighted to have a catchier title (as will his
+publisher).
+
+Achilles: I'm glad. But how were you reminded of his book by our discussion?
+
+Tortoise: Ah, yes. You see, in his book there will be a Dialogue in which he wants to
+throw readers off by making them SEARCH for the ending.
+
+Achilles: A funny thing to want to do. How is it done?
+
+Tortoise: You've undoubtedly noticed how some authors go to so much trouble to build
+up great tension a few pages before the end of their stories-but a reader who is
+holding the book physically in his hands can FEEL that the story is about to end.
+Hence, he has some extra information which acts as an advance warning, in a way.
+The tension is a bit spoiled by the physicality of the book. It would be so much better
+if, for instance, there were a lot of padding at the end of novels.
+
+Achilles: Padding?
+
+Tortoise: Yes; what I mean is, a lot of extra printed pages which are not part of the story
+proper, but which serve to conceal the exact location of the end from a cursory
+glance, or from the feel of the book.
+
+Achilles: I see. So a story's true ending might occur, say, fifty or a hundred pages before
+the physical end of the book?
+
+Tortoise: Yes. This would provide an element of surprise, because the reader wouldn't
+know in advance how many pages are padding, and how many are story.
+
+Achilles: If this were standard practice, it might be quite effective. But
+
+
+Aria with Diverse Variations
+
+
+402
+
+
+
+there is a problem. Suppose your padding were very obvious-such as a lot of blanks,
+or pages covered with X's or random letters. Then, it would be as good as absent.
+
+Tortoise: Granted. You'd have to make it resemble normal printed pages.
+
+Achilles: But even a cursory glance at a normal page from one story will often suffice to
+distinguish it from another story. So you will have to make the padding resemble the
+genuine story rather closely.
+
+Tortoise: That's quite true. The way I've always envisioned it is this: you bring the story
+to an end; then without any break, you follow it with something which looks like a
+continuation but which is in reality just padding, and which is utterly unrelated to the
+true theme. The padding is, in a way, a "post-ending ending". It may contain
+extraneous literary ideas, having little to do with the original theme.
+
+Achilles: Sneaky! But then the problem is that you won't be able to tell when the real
+ending comes. It'll just blend right into the padding.
+
+Tortoise: That's the conclusion my author friend and I have reached as well. It's a shame,
+for I found the idea rather appealing.
+
+Achilles: Say, I have a suggestion. The transition between genuine story and padding
+material could be made in such a way that, by sufficiently assiduous inspection of
+the text, an intelligent reader will be able to detect where one leaves off and the other
+begins. Perhaps it will take him quite a while. Perhaps there will be no way to
+predict how long it will take ... But the publisher could give a guarantee that a
+sufficiently assiduous search for the true ending will always terminate, even if he
+can't say how long it will be before the test terminates.
+
+Tortoise: Very well-but what does "sufficiently assiduous" mean?
+
+Achilles: It means that the reader must be on the lookout for some small but telltale
+feature in the text which occurs at some point. That would signal the end. And he
+must be ingenious enough to think up, and hunt for, many such features until he
+finds the right one.
+
+Tortoise: Such as a sudden shift of letter frequencies or word lengths? Or a rash of
+grammatical mistakes?
+
+Achilles: Possibly. Or a hidden message of some sort might reveal the true end to a
+sufficiently assiduous reader. Who knows? One could even throw in some
+extraneous characters or events which are inconsistent with the spirit of the
+foregoing story. A naive reader would swallow the whole thing, whereas a
+sophisticated reader would be able to spot the dividing line exactly.
+
+Tortoise: That's a most original idea, Achilles. I'll relay it to my friend, and perhaps he
+can incorporate it in his Dialogue.
+
+Achilles: I would be highly honored.
+
+Tortoise: Well, I am afraid that I myself am growing a little groggy, Achilles. It would be
+well for me to take my leave, while I am still capable of navigating my way home.
+
+Achilles: I am most flattered' that you have stayed up for so long, and at such an odd hour
+of the night, just for my benefit. I assure you that
+
+
+Aria with Diverse Variations
+
+
+403
+
+
+
+your number-theoretical entertainment has been a perfect antidote to my usual
+tossing and turning. And who knows-perhaps I may even be able to go to sleep
+tonight. As a token of my gratitude, Mr. T, I would like to present you with a special
+gift.
+
+Tortoise: Oh, don't be silly, Achilles.
+
+Achilles: It is my pleasure, Mr. T. Go over to that dresser; on it, you will see an Asian
+box.
+
+(The Tortoise moseys over to Achilles' dresser.)
+
+Tortoise. You don't mean this very gold Asian box, do you?
+
+Achilles: That's the one. Please accept it, Mr. T, with my warmest compliments.
+
+Tortoise: Thank you very much indeed, Achilles. Hmm ... Why are all these
+mathematicians' names engraved on the top? What a curious list:
+
+De Morgan
+
+Abel
+
+Boole
+
+Brouwer
+S i e r p i n s k i
+Weierstrass
+
+Achilles: I believe it is supposed to be a Complete List of All Great Mathematicians.
+What I haven't been able to figure out is why the letters running down the diagonal
+are so much bolder.
+
+Tortoise: At the bottom it says, "Subtract 1 from the diagonal, to find Bach in Leipzig".
+Achilles: I saw that, but I couldn't make head or tail of it. Say, how about a shot of
+excellent whiskey? I happen to have some in that decanter on my shelf.
+
+Tortoise: No, thanks. I'm too tired. I'm just going to head home. (Casually, he opens the
+box.) Say, wait a moment, Achilles-there are one hundred Louis d'or in here!
+
+Achilles: I would be most pleased if you would accept them, Mr. T. Tortoise: But-but
+Achilles: No objections, now. The box, the gold-they're yours. And thank you for an
+evening without parallel.
+
+Tortoise: Now whatever has come over you, Achilles? Well, thank you for your
+outstanding generosity and I hope you have sweet dreams about the strange
+Goldbach Conjecture, and its Variation. Good night.
+
+(And he picks up the very gold Asian box filled with the one hundred Louis d'or, and
+walks towards the door. As he is about to leave, there is a loud knock.)
+
+Who could be knocking at this ungodly hour, Achilles?
+
+Achilles: I haven't the foggiest idea. It seems suspicious to me. Why don't you go hide
+behind the dresser, in case there's any funny business.
+
+
+Aria with Diverse Variations
+
+
+404
+
+
+
+Tortoise: Good idea. (Scrambles in behind the dresser.) Achilles: Who's there?
+
+Voice: Open up-it's the cops.
+
+Achilles: Come in, it's open.
+
+(Two burly policemen walk in, wearing shiny badges.)
+
+Cop: I'm Silva. This is Gould. (Points at his badge.) Is there an Achilles at this address?
+Achilles: That's me!
+
+Cop: Well, Achilles, we have reason to believe that there is a very gold Asian box here,
+filled with one hundred Louis d'or. Someone absconded with it from the museum
+this afternoon. Achilles: Heavens to Betsy!
+
+Cop: If it is here, Achilles, since you would be the only possible suspect, I' regret to say
+that I should have to take you into custody. Now I have here a search warrant
+Achilles: Oh, sirs, am I ever glad you arrived! All evening long, I have been being
+terrorized by Mr. Tortoise and his very Asian gold box. Now at last you have come
+to liberate me! Please, sirs, just take a look behind that dresser, and there you will
+find the culprit!
+
+(The cops look behind the dresser and spy the Tortoise huddled behind it, holding
+his very gold Asian box, and trembling.)
+
+Cop: So there it is! And so Mr. Tortoise is the varmint, eh? I never would have suspected
+HIM. But he's caught, red-handed.
+
+Achilles: Haul the villain away, kind sirs! Thank goodness, that's the last I'll have to hear
+of him, and the Very Asian Gold Box!
+
+
+Aria with Diverse Variations
+
+
+405
+
+
+
+CHAPTER XII1
+
+
+BlooP and FlooP and GlooP
+
+Self-Awareness and Chaos
+
+BLOOP, FLOOP, AND GLOOP are not trolls, talking ducks, or the sounds made by a
+sinking ship-they are three computer languages, each one with is own special purpose.
+These languages were invented specially for this chapter. They will be of use in
+explaining some new senses of the word 'recursive -in particular, the notions of primitive
+recursivity and general recursivity. They will prove very helpful in clarifying the
+machinery of self-reference in TNT.
+
+We seem to be making a rather abrupt transition from brains and hinds to
+technicalities of mathematics and computer science. Though the transition is abrupt in
+some ways, it makes some sense. We just saw how a certain kind of self-awareness
+seems to be at the crux of consciousness. Vow we are going to scrutinize "self-
+awareness" in more formal settings, such as TNT. The gulf between TNT and a mind is
+wide, but some of the ideas will be most illuminating, and perhaps metaphorically
+transportable back to our thoughts about consciousness.
+
+One of the amazing things about TNT's self-awareness is that it is intimately
+connected to questions about order versus chaos among the natural numbers. In
+particular, we shall see that an orderly system of sufficient complexity that it can mirror
+itself cannot be totally orderly-it must contain some strange, chaotic features. For readers
+who have some Achilles in them, this will be hard to take. However, there is a "magical"
+compensation: there is a kind of order to the disorder, which is now its own field of
+study, called "recursive function theory". Unfortunately, we will not be able to do much
+more than hint at the fascination of this subject.
+
+Representability and Refrigerators
+
+Phrases such as "sufficiently complex", "sufficiently powerful" and the like lave cropped
+up quite often earlier. Just what do they mean? Let us go back to the battle of the Crab
+and Tortoise, and ask, "What qualifies something as a record player?" The Crab might
+claim that his refrigerator s a "Perfect" record player. Then to prove it, he could set any
+record whatsoever atop it, and say, "You see-it's playing it!" The Tortoise, if he wanted to
+counter this Zen-like act, would have to reply, "No-your refrigerator is too low-fidelity to
+be counted as a phonograph: it cannot reproduce sounds-at all (let alone its self-breaking
+sound)." The Tortoise
+
+
+BlooP and FlooP and GlooP
+
+
+407
+
+
+
+can only make a record called "I Cannot Be Played on Record Player X" provided that
+Record Player X is really a record player! The Tortoise's method is quite insidious, as it
+plays on the strength, rather than on the weakness, of the system. And therefore he
+requires "sufficiently hi-fi" record players.
+
+Ditto for formal versions of number theory. The reason that TNT is a
+formalization of N is that its symbols act the right way: that is, its theorems are not silent
+like a refrigerator-they speak actual truths of N. Of course, so do the theorems of the pq-
+system. Does it, too, count as "a formalization of number theory", or is it more like a
+refrigerator? Well, it is a little better than a refrigerator, but it is still pretty weak. The pq-
+system does not include enough of the core truths of N to count as "a number theory".
+
+What, then, are these "core truths" of N? They are the primitive recursive truths',
+that means they involve only predictably terminating calculations. These core truths
+serve for N as Euclid's first four postulates served for geometry: they allow you to throw
+out certain candidates before the game begins, on the grounds of "insufficient power".
+From here on out, the representability of all primitive recursive truths will be the
+criterion for calling a system "sufficiently powerful".
+
+Ganto's Ax in Metamathematics
+
+The significance of the notion is shown by the following key fact: If you have a
+sufficiently powerful formalization of number theory, then Godel’s method is applicable,
+and consequently your system is incomplete. If, on the other hand, your system is not
+sufficiently powerful (i.e., not all primitive recursive truths are theorems), then your
+system is, precisely by virtue of that lack, incomplete. Here we have a reformulation of
+"Ganto's Ax" in metamathematics: whatever the system does, Godel’s Ax will chop its
+head off! Notice also how this completely parallels the high-fidelity-versus-low fidelity
+battle in the Contracrostipunctus.
+
+Actually, it turns out that much weaker systems are still vulnerable to the Godel method;
+the criterion that all primitive recursive truths need be represented as theorems is far too
+stringent. It is a little like a thief who will only rob "sufficiently rich" people, and whose
+criterion is that the potential victim should be carrying at least a million dollars in cash.
+In the case of TNT, luckily, we will be able to act in our capacity as thieves, for the
+million in cash is there-which is to say, TNT does indeed contain all primitive recursive
+truths as theorems.
+
+Now before we plunge into a detailed discussion of primitive recursive functions
+and predicates, I would like to tie thee themes of this Chapter to themes from earlier
+Chapters, so as to provide a bit better motivation.
+
+Finding Order by Choosing the Right Filter
+
+We saw at a very early stage that formal systems can be difficult and unruly beasts
+because they have lengthening and shortening rules, which can
+
+
+BlooP and FlooP and GlooP
+
+
+408
+
+
+
+possibly lead to never-ending searches among strings. The discovery of Godel-numbering
+showed that any search for a string having a special typographical property has an
+arithmetical cousin: an isomorphic search for an integer with a corresponding special
+arithmetical property. Consequently, the quest for decision procedures for formal systems
+involves solving the mystery of unpredictably long searches- chaos -among the integers.
+Now in the Aria with Diverse Variations, I gave perhaps too much weight to apparent
+manifestations of chaos in problems about integers. As a matter of fact, people have
+tamed wilder examples of apparent chaos than the "wondrousness" problem, finding them
+to be quite gentle beasts after all. Achilles' powerful faith in the regularity and
+predictability of numbers should therefore be accorded quite a bit of respect-especially as
+it reflects the beliefs of nearly all mathematicians up till the 1930's. To show why order
+versus chaos is such a subtle and significant issue, and to tie it in with questions about the
+location and revelation of meaning, I would like to quote a beautiful and memorable
+passage from Arc Quanta Reall-a Galilean Dialogue by the late J. M. Jauch:
+
+SALVIATI Suppose I give you two sequences of numbers, such as
+78539816339744830961566084...
+
+And
+
+
+1, -1/3, +1/5, -1/7, +1/9, -1/11, +1/13, -1/15, ...
+
+If I asked you, Simplicio, what the next number of the first sequence is, what
+would you say?
+
+SIMPLICIO I could not tell you. I think it is a random sequence and that there is
+no law in it.
+
+SALVIATI And for the second sequence?
+
+SIMPLICIO That would be easy. It must be +1/17.
+
+SALVIATI Right. But what would you say if I told you that the first
+
+sequence is also constructed by a law and this law is in fact identical with the
+
+one you have just discovered for the second sequence? SIMPLICIO This does not
+seem probable to me.
+
+SALVIATI But it is indeed so, since the first sequence is simply the beginning of
+the decimal fraction [expansion] of the sum of the second. Its value is Tr/4.
+
+SIMPLICIO You are full of such mathematical tricks, but I do not see what this
+has to do with abstraction and reality.
+
+SALVIATI The relationship with abstraction is easy to see. The first sequence
+looks random unless one has developed through a process of abstraction a kind of
+filter which sees a simple structure behind the apparent randomness.
+
+It is exactly in this manner that laws of nature are discovered. Nature
+
+presents us with a host of phenomena which appear mostly as chaotic randomness
+until we select some significant events, and abstract from their particular, irrelevant
+circumstances so that they become idealized. Only then can they exhibit their true
+structure in full splendor.
+
+SAGREDO This is a marvelous idea! It suggests that when we try to understand
+nature, we should look at the phenomena as if they were messages to be
+
+
+BlooP and FlooP and GlooP
+
+
+409
+
+
+
+understood. Except that each message appears to be random until we establish a code
+to read it. This code takes the form of an abstraction, that is, we choose to ignore
+certain things as irrelevant and we thus partially select the content of the message by
+a free choice. These irrelevant signals form the "background noise," which will limit
+the accuracy of our message.
+
+But since the code is not absolute there may be several messages in the same raw
+material of the data, so changing the code will result in a message of equally deep
+significance in something that was merely noise before, and conversely: In a new
+code a former message may be devoid of meaning.
+
+Thus a code presupposes a free choice among different, complementary aspects,
+each of which has equal claim to reality, if I may use this dubious word.
+
+Some of these aspects may be completely unknown to us now but they may
+reveal themselves to an observer with a different system of abstractions.
+
+But tell me, Salviati, how can we then still claim that we discover something out
+there in the objective real world? Does this not mean that we are merely creating
+things according to our own images and that reality is only within ourselves?
+
+SALVIATI I don't think that this is necessarily so, but it is a question which
+requires deeper reflection.'
+
+Jauch is here dealing with messages that come not from a "sentient being" but from
+nature itself. The questions that we raised in Chapter VI on the relation of meaning to
+messages can be raised equally well with messages from nature. Is nature chaotic, or is
+nature patterned? And what is the role of intelligence in determining the answer to this
+question?
+
+To back off from the philosophy, however, we can consider the point about the
+deep regularity of an apparently random sequence. Might the function Q(n) from Chapter
+V have a simple, nonrecursive explanation, too? Can every problem, like an orchard, be
+seen from such an angle that its secret is revealed? Or are there some problems in number
+theory which, no matter what angle they are seen from, remain mysteries?
+
+With this prologue, I feel it is time to move ahead to define the precise meaning
+of the term "predictably long search". This will be accomplished in terms of the language
+BlooP.
+
+
+Primordial Steps of the Language BlooP
+
+Our topic will be searches for natural numbers which have various properties. In order to
+talk about the length of any search, we shall have to define some primordial steps, out of
+which all searches are built, so that length can be measured in terms of number of steps.
+Some steps which we might consider primordial are:
+
+adding any two natural numbers;
+multiplying any two natural numbers;
+determining if two numbers are equal;
+determining the larger (smaller) of two numbers.
+
+
+BlooP and FlooP and GlooP
+
+
+410
+
+
+
+Loops and Upper Bounds
+
+
+If we try to formulate a test for, say, primality in terms of such steps, we shall soon see
+that we have to include a control structure -that is, descriptions of the order to do things
+in, when to branch back and try something again, when to skip over a set of steps, when
+to stop, and similar matters.
+
+It is typical of any algorithm-that is, a specific delineation of how to carry out a task-that
+it includes a mixture of (1) specific operations to be performed, and (2) control
+statements. Therefore, as we develop our language for expressing predictably long
+calculations, we shall have to incorporate primordial control structures also. In fact, the
+hallmark of BlooP is its limited set of control structures. It does not allow you to branch
+to arbitrary steps, or to repeat groups of steps without limit; in BlooP, essentially the only
+control structure is the bounded loop: a set of instructions which can be executed over
+and over again, up to a predefined maximum number of times, called the upper bound, or
+ceiling, of the loop. If the ceiling were 300, then the loop might be executed 0, 7, or 300
+times-but not 301.
+
+Now the exact values of all the upper bounds in a program need not be put in numerically
+by the programmer-indeed, they may not be known in advance. Instead, any upper bound
+may be determined by calculations carried out before its loop is entered. For instance, if
+you wanted to calculate the value of 2"', there would be two loops. First, you evaluate 3",
+which involves n multiplications. Then, you put 2 to that power, which involves 3"
+multiplications. Thus, the upper bound for the second loop is the result of the calculation
+of the first loop.
+
+Here is how you would express this in a BlooP program:
+
+DEFINE PROCEDURE "TWO-TO-THE-THREE-TO-THE" [N]:
+
+BLOCK 0: BEGIN
+
+CELL(O) <= 1;
+
+LOOP N TIMES:
+
+BLOCK 1: BEGIN
+
+CELL(0)' 3 x CELL(O);
+
+BLOCK 1: END;
+
+CELL(l) <= 1;
+
+LOOP CELL(O) TIMES:
+
+BLOCK 2: BEGIN
+
+CELL(l) # 2 X CELL(1);
+
+BLOCK 2: END;
+
+OUTPUT <= CELL( I);
+
+BLOCK 0: END.
+
+
+Conventions of BlooP
+
+Now it is an acquired skill to be able to look at an algorithm written in a computer
+language, and figure out what it is doing. However, I hope that this algorithm is simple
+enough that it makes sense without too much
+
+
+BlooP and FlooP and GlooP
+
+
+411
+
+
+
+scrutiny. A procedure is defined, having one input parameter, N; its output is the desired
+value.
+
+This procedure definition has what is called block structure , which means that
+certain portions of it are to be considered as units, or blocks. All the statements in a block
+get executed as a unit. Each block has a number (the outermost being BLOCK 0), and is
+delimited by a BEGIN and an END. In our example, BLOCK 1 and BLOCK 2 contain
+just one statement each but shortly you will see longer blocks. A LOOP statement always
+means to execute the block immediately under it repeatedly. As can be seen above,
+blocks can be nested.
+
+The strategy of the above algorithm is as described earlier. You begin by taking
+an auxiliary variable, called CELL(O); you set it initially to 1, and then, in a loop, you
+multiply it repeatedly by 3 until you've done so exactly N times. Next, you do the
+analogous thing for CELL(l)-set it to 1, multiply by 2 exactly CELL(O) times, then
+quit. Finally, you set OUTPUT to the value of CELL(l). This is the value returned to the
+outside world-the only externally visible behavior of the procedure.
+
+A number of points about the notation should be made here. First, the meaning of
+the left-arrow <= is this:
+
+Evaluate the expression to its right, then take the result and set the CELL (or OUTPUT)
+on its left to that value.
+
+So the meaning of a command such as CELL(l) <= 3 X CELL(l) is to triple the value
+stored in CELL(l). You may think of each CELL as being a separate word in the
+memory of some computer. The only difference between a CELL and a true word is that
+the latter can only hold integers up to some finite limit, whereas we allow a CELL to
+hold any natural number, no matter how big.
+
+Every procedure in BlooP, when called, yields a value-namely the value of the
+variable called OUTPUT. At the beginning of execution of any procedure, it is assumed
+as a default option that OUTPUT has the value 0. That way, even if the procedure never
+resets OUTPUT at all, OUTPUT has a well-defined value at all times.
+
+IF-Statements and Branching
+
+Now let us look at another procedure which will show us some other features of BlooP
+which give it more generality. How do you find out, knowing only how to add, what the
+value of M - N is? The trick is to add various numbers onto N until you find the one
+which yields M. However, what happens if M is smaller than N? What if we are trying to
+take 5 from 2? In the domain of natural numbers, there is no answer. But we would like
+our BlooP procedure to give an answer anyway-let's say 0. Here, then, is a BlooP
+procedure which does subtraction:
+
+
+BlooP and FlooP and GlooP
+
+
+412
+
+
+
+DEFINE PROCEDURE "MINUS" [M,N]:
+
+BLOCK 0: BEGIN
+
+IF M < N, THEN:
+
+QUIT BLOCK 0;
+
+LOOP AT MOST M + 1 TIMES:
+
+BLOCK 1: BEGIN
+
+IF OUTPUT + N = M, THEN:
+
+ABORT LOOP 1;
+
+OUTPUT, <= OUTPUT + 1;
+
+BLOCK 1: END;
+
+BLOCK 0: END.
+
+Here we are making use of the implicit feature that OUTPUT begins at 0. If M is
+less than N, then the subtraction is impossible, and we simply jump to the bottom of
+BLOCK 0 right away, and the answer is 0. That is what is meant by the line QUIT
+BLOCK 0. But if M is not less than N, then we skip over that QUIT-statement, and
+carry out the next command in sequence (here, a LOOP-statement). That is how IF-
+statements always work in BlooP.
+
+So we enter LOOP 1, so called because the block which it tells us to repeat is
+BLOCK 1. We try adding 0 to N, then 1, 2, etc., until we find a number that gives M. At
+that point, we ABORT the loop we are in, meaning we jump to the statement
+immediately following the END which marks the bottom of the loop's block. In this case,
+that jump brings us just below BLOCK 1: END, which is to say, to the last statement of
+the algorithm, and we are done. OUTPUT now contains the correct answer.
+
+Notice that there are two distinct instructions for jumping downwards: QUIT, and
+ABORT. The former pertains to blocks, the latter to loops. QUIT BLOCK n means to
+jump to the last line of BLOCK n, whereas ABORT LOOP n means to jump just below
+the last line of BLOCK n. This distinction only matters when you are inside a loop and
+want to continue looping but to quit the block this time around. Then you can say QUIT
+and the proper thing will happen.
+
+Also notice that the words AT MOST now precede the upper bound of the loop,
+which is a warning that the loop may be aborted before the upper bound is reached.
+
+Automatic Chunking
+
+Now there are two last features of BlooP to explain, both of them very important. The
+first is that, once a procedure has been defined , it may be called inside later procedure
+definitions. The effect of this is that once an operation has been defined in a procedure, it
+is considered as simple as a primordial step. Thus, BlooP features automatic chunking.
+You might compare it to the way a good ice skater acquires new motions: not by defining
+them as long sequences of primordial muscle-actions, but in terms of previously learned
+motions, which were themselves learned as compounds of earlier
+
+
+BlooP and FlooP and GlooP
+
+
+413
+
+
+
+learned motions, etc.-and the nestedness, or chunkedness, can go back many layers until
+you hit primordial muscle-actions And thus, the repertoire of BlooP programs, like the
+repertoire of a skater's tricks, grows, quite literally, by loops and bounds.
+
+BlooP Tests
+
+The other feature of BlooP is that certain procedures can have YES or NO as their
+output, instead of an integer value. Such procedures are tests, rather than functions. To
+indicate the difference, the name of a test must terminate in a question mark. Also, in a
+test, the default option for OUTPUT is not 0, of course, but NO.
+
+Let us see an example of these last two features of BlooP in an algorithm which tests its
+argument for primality:
+
+DEFINE PROCEDURE "PRIME?" [N]:
+
+BLOCK 0: BEGIN
+
+IF N = 0, THEN:
+
+QUIT BLOCK 0;
+
+CELL(O) <= 2;
+
+LOOP AT MOST MINUS [N,2] TIMES:
+
+BLOCK 1: BEGIN
+
+IF REMAINDER [N,CELL(0)] = 0, THEN:
+QUIT BLOCK 0;
+
+CELL(O) <= CELL(O) + 1;
+
+BLOCK 1: END;
+
+OUTPUT <= YES;
+
+BLOCK 0: END.
+
+Notice that I have called two procedures inside this algorithm: MINUS and
+REMAINDER. (The latter is presumed to have been previously defined, and you may
+work out its definition yourself.) Now this test for primality works by trying out potential
+factors of N one by one, starting at 2 and increasing to a maximum of N - 1. In case any
+of them divides N exactly (i.e., gives remainder 0), then we jump down to the bottom,
+and since OUTPUT still has its default value at this stage, the answer is NO. Only if N
+has no exact divisors will it survive the entirety of LOOP 1; then we will emerge
+smoothly at the statement OUTPUT <= YES, which will get executed, and then the
+procedure is over.
+
+BlooP Programs Contain Chains of Procedures
+
+We have seen how to define procedures in BlooP; however, a procedure definition is only
+a part of a program. A program consists of a chain of procedure definitions (each only
+calling previously defined procedures), optionally followed by one or more calls on the
+procedures defined. Thus, an
+
+
+BlooP and FlooP and GlooP
+
+
+414
+
+
+
+example of a full BlooP program would be the definition of the procedure TWO-TO-
+THE-THREE-TO-THE, followed by the call
+
+
+TWO-TO-THE-THREE-TO-THE [2]
+
+which would yield an answer of 512.
+
+If you have only a chain of procedure definitions, then nothing ever gets executed;
+they are all just waiting for some call, with specific numerical values, to set them in
+motion. It is like a meat grinder waiting for some meat to grind-or rather, a chain of meat
+grinders all linked together, each of which is fed from earlier ones ... In the case of meat
+grinders, the image is perhaps not so savory; however, in the case of BlooP programs,
+such a construct is quite important, and we will call it a "call-less program". This notion
+is illustrated in Figure 72.
+
+Now BlooP is our language for defining predictably terminating calculations. The
+standard name for functions which are BlooP-computable is primitive recursive
+functions; and the standard name for properties which can be detected by BlooP-tests is
+primitive recursive predicates. Thus, the function 2 3n is a primitive recursive function;
+and the statement "n is a prime number" is a primitive recursive predicate.
+
+It is clear intuitively that the Goldbach property is primitive recursive, and to
+make that quite explicit, here is a procedure definition in BlooP, showing how to test for
+its presence or absence:
+
+DEFINE PROCEDURE "GOLDBACH?" [N]:
+
+BLOCK 0: BEGIN
+
+CELL(O) 2;
+
+LOOP AT MOST N TIMES:
+
+BLOCK 1: BEGIN
+
+IF {PRIME? [CELL(O)]
+
+AND PRIME? [MINUS [N,CELL(0)]]},
+
+THEN:
+
+BLOCK 2: BEGIN
+
+OUTPUT,# YES;
+
+QUIT BLOCK 0-,
+
+BLOCK 2: END
+CELL(O) <= CELL(O) +
+
+BLOCK 1: END;
+
+BLOCK 0: END.
+
+
+As usual, we assume NO until proven YES, and we do a brute force search among pairs
+of numbers which sum up to N. If both are prime, we quit the outermost block; otherwise
+we just go back and try again, until all possibilities are exhausted.
+
+(Warning: The fact that the Goldbach property is primitive recursive does not make the
+question “Do all numbers have the Goldbach property?” a simple question—far from it!)
+
+
+BlooP and FlooP and GlooP
+
+
+415
+
+
+
+
+FIGURE 72. The structure of a call-less BlooP program. For this program to be self-
+contained, each procedure definition may only call procedures defined above it.
+
+Suggested Exercises
+
+Can you write a similar BlooP procedure which tests for the presence or absence of the
+Tortoise property (or the Achilles property)? If so, do it. If not, is it merely because you
+are ignorant about upper bounds, or could it be that there is a fundamental obstacle
+preventing the formulation of such an algorithm in BlooP? And what about the same
+questions, with respect to the property of wondrousness, defined in the Dialogue?
+
+Below, I list some functions and properties, and you ought to take the time to determine
+whether you believe they are primitive recursive (BlooP-programmable) or not. This
+means that you must carefully consider what kinds of operations will be involved in the
+calculations which they require, and whether ceilings can be given for all the loops
+involved.
+
+FACTORIAL [N] = NI (the factorial of N)
+
+(e.g., FACTORIAL [4] = 24)
+
+REMAINDER [M,N] = the remainder upon dividing M by N
+(e.g., REMAINDER [24,7] = 3)
+
+PI-DIGIT [N] = the Nth digit of pi, after the decimal point
+(e.g. PI-DIGIT [1] = 1,
+
+PI-DIGIT [2] = 4
+PI-DIGIT [1000000] = 1
+
+
+BlooP and FlooP and GlooP
+
+
+416
+
+
+
+
+
+
+
+
+
+
+FIBO [N] = the Nth Fibonacci number
+(e.g., FIBO [9] = 34)
+
+PRIME-BEYOND [N[ = the lowest prime beyond N
+(e g., PRIME-BEYOND [33] = 37)
+
+PERFECT [N] = the Nth "perfect" number (a number such as 28 whose divisors sum up
+to itself: 28 = 1 + 2 + 4 + 7 + 14)
+
+(e.g., PERFECT [2] = 28)
+
+PRIME? [N] = YES if N is prime, otherwise NO.
+
+PERFECT? [N] = YES if N is perfect, otherwise NO.
+
+TRIVIAL? [A,B,C,N] = YES if A"+B" = C n is correct; otherwise NO.
+
+(e.g., TRIVIAL? [3,4,5,2] = YES,
+
+TRIVIAL? [3,4,5,3] = NO)
+
+PIERRE? [A,B,C] = YES if A"+B" = C" is satisfiable for some value of N greater than
+1, otherwise NO.
+
+(e.g., PIERRE? [3,4,5] = YES,
+
+PIERRE? [1,2,3] = NO)
+
+FERMAT? [N] = YES if A"+B" = CN is satisfied by some positive
+values of A, B, C; otherwise NO.
+
+(e.g., FERMAT? [2] = YES)
+
+TORTOISE-PAIR? [M,N] = YES if both M and M + N are prime, otherwise NO.
+
+(e.g., ORTOISE-PAIR [5,1742] = YES,
+
+TORTOISE-PAIR [5,100] = NO)
+
+TORTOISE? [N] = YES if N is the difference of two primes, otherwise NO.
+
+(e.g., TORTOISE [1742] = YES,
+
+TORTOISE [7] = NO)
+
+MIU-WELL-FORMED? [N] = YES if N, when seen as a string of the MlU-System, is
+well-formed; otherwise NO.
+
+(e.g., MIU-WELL-FORMED? [310] = YES,
+
+MIU-WELL-FORMED? [415] = NO)
+
+MIU-PROOF-PAIR? [M,N] = YES If M, as seen as a sequence of strings of the MIU-
+system, is a derivation of N, as seen as a string of the MlU-system; otherwise NO.
+
+(e.g., MIU-PROOF-PAIR? [3131131111301,301] = YES,
+
+MIU-PROOF-PAIR? [311130,30] = NO)
+
+MIU-THEOREM? [N] = YES if N, seen as a MlU-system string, is a theorem;
+otherwise NO.
+
+(e.g., MIU-THEOREM? [311] = YES,
+
+MIU-THEOREM? [30] = NO,
+
+MIU-THEOREM? [701] = NO)
+
+TNT-THEOREM? [N] = YES if N, seen as a TNT-string, is a theorem.
+
+(e.g., TNT-THEOREM? [666111666] = YES,
+
+TNT-THEOREM? [123666111666] = NO,
+
+TNT-THEOREM? [7014] = NO)
+
+
+BlooP and FlooP and GlooP
+
+
+417
+
+
+
+FALSE? [N] = YES if N, seen as a TNT-string, is a false statement of number theory;
+otherwise NO.
+
+(e.g., FALSE? [6661 1 1666] = NO,
+
+FALSE? [2236661 1 1666] = YES,
+
+FALSE? [7014] = NO)
+
+The last seven examples are particularly relevant to our future metamathematical
+explorations, so they highly merit your scrutiny.
+
+Expressibility and Representability
+
+Now before we go on to some interesting questions about BlooP and are led to its
+relative, FlooP, let us return to the reason for introducing BlooP in the first place, and
+connect it to TNT. Earlier, I stated that the critical mass for Godel’s method to be
+applicable to a formal system is attained when all primitive recursive notions are
+representable in that system. Exactly what does this mean? First of all, we must
+distinguish between the notions of representability and expressibility. Expressing a
+predicate is a mere matter of translation from English into a strict formalism. It has
+nothing to do with theoremhood. For a predicate to be represented, on the other hand, is a
+much stronger notion. It means that
+
+(1) All true instances of the predicate are theorems;
+
+(2) All false instances are nontheorems.
+
+By "instance", I mean the string produced when you replace all free variables by
+numerals. For example, the predicate m + n = k is represented in the pq-system, because
+each true instance of the predicate is a theorem, each false instance is a nontheorem. Thus
+any specific addition, whether true or false, translates into a decidable string of the pq-
+system. However, the pq-system is unable to express-let alone represent-any other
+properties of natural numbers. Therefore it would be a weak candidate indeed in a
+competition of systems which can do number theory.
+
+Now TNT has the virtue of being able to express virtually any number-theoretical
+predicate; for example, it is easy to write a TNT-string which expresses the predicate "b
+has the Tortoise property". Thus, in terms of expressive power, TNT is all we want.
+
+However, the question "Which properties are represented in TNT?" is Precisely
+the question "How powerful an axiomatic system is TNT?" Are all Possible predicates
+represented in TNT? If so, then TNT can answer any question of number theory; it is
+complete.
+
+
+Primitive Recursive Predicates Are Represented in TNT
+
+Now although completeness will turn out to be a chimera. TNT is at least complete with
+respect to primitive recursive predicates. In other words, any statement of number theory
+whose truth or falsity can be decided by a
+
+
+BlooP and FlooP and GlooP
+
+
+418
+
+
+
+computer within a predictable length of time is also decidable inside TNT. Or, one final
+restatement of the same thing:
+
+If a BlooP test can be written for some property of natural numbers, then that
+property is represented in TNT.
+
+Are There Functions Which Are Not Primitive Recursive?
+
+Now the kinds of properties which can be detected by BlooP tests are widely varied,
+including whether a number is prime or perfect, has the Goldbach property, is a power of
+2, and so on and so forth. It would not be crazy to wonder whether every property of
+numbers can be detected by some suitable BlooP program. The fact that, as of the present
+moment, we have no way of testing whether a number is wondrous or not need not
+disturb us too much, for it might merely mean that we are ignorant about wondrousness,
+and that with more digging around, we could discover a universal formula for the upper
+bound to the loop involved. Then a BlooP test for wondrousness could be written on the
+spot. Similar remarks could be made about the Tortoise property.
+
+So the question really is, "Can upper bounds always be given for the length of
+calculations-or, is there an inherent kind of jumbliness to the natural number system,
+which sometimes prevents calculation lengths from being predictable in advance?" The
+striking thing is that the latter is the case, and we are about to see why. It is the sort of
+thing that would have driven Pythagoras, who first proved that the square root of 2 is
+irrational, out of his mind. In our demonstration, we will use the celebrated diagonal
+method, discovered by Georg Cantor, the founder of set theory.
+
+Pool B, Index Numbers, and Blue Programs
+
+We shall begin by imagining a curious notion: the pool of all possible BlooP programs.
+Needless to say, this pool-"Pool B"-is an infinite one. We want to consider a subpool of
+Pool B, obtained by three successive filtering operations. The first filter will retain for us
+only call-less programs. From this subpool we then eliminate all tests, leaving only
+functions. (By the way, in call-less programs, the last procedure in the chain determines
+whether the program as a whole is considered a test, or a function.) The third filter will
+retain only functions which have exactly one input parameter. (Again referring to the
+final procedure in the chain.) What is left?
+
+A complete pool of all call-less BlooP programs which calculate functions of
+exactly one input parameter.
+
+Let us call these special BlooP programs Blue Programs.
+
+What we would like to do now is to assign an unambiguous index
+number to each Blue Program. How can this be done? The easiest way—we shall use it—
+is to list them in order of length: the shortest possible. Blue
+
+
+BlooP and FlooP and GlooP
+
+
+419
+
+
+
+Program being # 1, the second shortest being #2, etc. Of course, there will be many
+programs tied for each length. To break such ties, we use alphabetical order. Here,
+"alphabetical order" is taken in an extended sense, where the alphabet includes all the
+special characters of BlooP, in some arbitrary order, such as the following:
+
+ABCDEFGHIJK LMN
+OPQRSTUVWXY Z + x
+0123456789 <==<>
+
+()[]{}-'?:; , •
+
+-and at the end comes the lowly blank! Altogether, fifty-six characters. For convenience's
+sake, we can put all Blue Programs of length 1 in Volume 1, programs of 2 characters in
+Volume 2, etc. Needless to say, the first few volumes will be totally empty, while later
+volumes will have many, many entries (though each volume will only have a finite
+number). The very first Blue Program would be this one:
+
+DEFINE PROCEDURE "A" [B]:
+
+BLOCK 0: BEGIN
+BLOCK 0: END.
+
+This rather silly meat grinder returns a value of 0 no matter what its input is. It occurs in
+Volume 56, since it has 56 characters (counting necessary blanks, including blanks
+separating successive lines).
+
+Soon after Volume 56, the volumes will get extremely fat, because there are just
+so many millions of ways of combining symbols to make Blue BlooP programs. But no
+matter-we are not going to try to print out this infinite catalogue. All that we care about is
+that, in the abstract, it is well-defined, and that each Blue BlooP program therefore has a
+unique and definite index number. This is the crucial idea.
+
+Let us designate the function calculated by the kth Blue Program this way:
+
+Blueprogramf# k} [N]
+
+Here, k is the index number of the program, and N is the single input parameter. For
+instance, Blue Program #12 might return a value twice the size of its input:
+
+Blueprogram{#12} [N] = 2 x N
+
+The meaning of the equation above is that the program named on the left-hand side
+returns the same value as a human would calculate from the ordinary algebraic
+expression on the right-hand side. As another example, perhaps the 5000th Blue Program
+calculates the cube of its input parameter:
+
+Blueprogram{#5000} [N] = N3
+
+
+BlooP and FlooP and GlooP
+
+
+420
+
+
+
+The Diagonal Method
+
+
+Very well-now we apply the "twist": Cantor's diagonal method. We shall take this
+catalogue of Blue Programs and use it to define a new function of one variabl e-Bluediag
+[N]-which will turn out not to be anywhere in the list (which is why its name is in italics).
+Yet Bluediag will clearly be a well-defined, calculable function of one variable, and so
+we will have to conclude that functions exist which simply are not programmable in
+BlooP.
+
+Here is the definition of Bluediag ~N]:
+
+Equation (1)... Bluediag [N] = 1 + Blueprogram{#N} [N]
+
+The strategy is: feed each meat grinder with its own index number, then add 1 to the
+output. To illustrate, let us find Bluediag [12], We saw that Blueprogram{# 12} is the
+function 2N; therefore, Bluediag [12] must have the value 1 + 2 x 12, or 25. Likewise,
+Bluediag [5000] would have the value 125,000,000,001, since that is 1 more than the
+cube of 5000. Similarly, you can find Bluediag of any particular argument you wish.
+
+The peculiar thing about Bluediag [N] is that it is not represented in the catalogue
+of Blue Programs. It cannot be. The reason is this. To be a Blue Program, it would have
+to have an index number-say it were Blue Program # X. This assumption is expressed by
+writing
+
+
+Equation (2)... Bluediag [N] = Blueprogram{# X] [N]
+
+But there is an inconsistency between the equations (1) and (2). It becomes apparent at
+the moment we try to calculate the value of Bluediag [ X], for we can do so by letting N
+take the value of X in either of the two equations. If we substitute into equation (1), we
+get:
+
+
+Bluediag [ X] = 1 + Blueprogram{# X] [ X]
+
+But if we substitute into equation (2) instead, we get:
+
+Bluediag [ X] = Blueprogram}# X] [ X]
+
+Now Bluediag [ X] cannot be equal to a number and also to the successor of that number.
+But that is what the two equations say. So we will have to go back and erase some
+assumption on which the inconsistency is based. The only possible candidate for erasure
+is the assumption expressed by Equation (2): that the function Bluediag [N] is able to be
+coded up as a Blue BlooP program. And that is the proof that Bluediag lies outside the
+realm of primitive recursive functions. Thus, we have achieved our aim of destroying
+Achilles' cherished but naive notion that every number-theoretical function must be
+calculable within a predictable number of steps.
+
+There are some subtle things going on here. You might ponder this, for instance:
+the number of steps involved in the calculation of Bluediag [N],for each specific value of
+N, is predictable—but the different methods of prediction cannot all be united into a
+general recipe for predict
+
+
+BlooP and FlooP and GlooP
+
+
+421
+
+
+
+ing the length of calculation of Bluediag [N]. This is an "infinite conspiracy", related to
+the Tortoise's notion of "infinite coincidences", and also to w-incompleteness. But we
+shall not trace out the relations in detail.
+
+
+Cantor's Original Diagonal Argument
+
+Why is this called a diagonal argument? The terminology comes from Cantor's original
+diagonal argument, upon which many other arguments (such as ours) have subsequently
+been based. To explain Cantor's original argument will take us a little off course, but it is
+worthwhile to do so. Cantor, too, was concerned with showing that some item is not in a
+certain list. Specifically, what Cantor wanted to show was that if a "directory" of real
+numbers were made, it would inevitably leave some real numbers out-so that actually, the
+notion of a complete directory of real numbers is a contradiction in terms.
+
+It must be understood that this pertains not just to directories of finite size, but
+also to directories of infinite size. It is a much deeper result than the statement "the
+number of reals is infinite, so of course they cannot be listed in a finite directory". The
+essence of Cantor's result is that there are (at least) two distinct types of infinity: one kind
+of infinity describes how many entries there can be in an infinite directory or table, and
+another describes how many real numbers there are (i.e., how many points there are on a
+line, or line segment)-and this latter is "bigger", in the sense that the real numbers cannot
+be squeezed into a table whose length is described by the former kind of infinity. So let
+us see how Cantor's argument involves the notion of diagonal, in a literal sense.
+
+Let us consider just real numbers between 0 and 1. Assume, for the sake of
+argument, that an infinite list could be given, in which each positive integer N is matched
+up with a real number r(N) between 0 and 1, and in which each real number between 0
+and 1 occurs somewhere down the line. Since real numbers are given by infinite
+decimals, we can imagine that the beginning of the table might look as follows:
+
+
+r(l): -1
+
+r(2): .3
+
+r(3): .7
+
+r(4): .4
+
+r(5): .5
+
+
+4 1
+3 3
+1 8
+1 4
+0 0
+
+
+5 9
+3 3
+2 8
+
+2 1
+0 0
+
+
+2 6
+
+3 3
+
+1 8
+
+3 5
+
+0 0
+
+
+5 3
+3 3
+2 8
+
+6 2
+0 0
+
+
+The digits that run down the diagonal are in boldface: 1, 3, 8, 2, 0.... Now those diagonal
+digits are going to be used in making a special real number d, which is between 0 and 1
+but which, we will see, is not in the list. To make d, you take the diagonal digits in order,
+and change each one of them to some other digit. When you prefix this sequence of digits
+by a decimal point you have d. There are of course many ways of changing a digit to
+some other digit, and correspondingly many different d's. Suppose for,
+
+
+BlooP and FlooP and GlooP
+
+
+422
+
+
+
+
+example, that we subtract 1 from the diagonal digits (with the convention that 1 taken
+from 0 is 9). Then our number d will be:
+
+.0 2 7 1 9.
+
+Now, because of the way we constructed it,
+
+d's 1st digit is not the same as the 1st digit of r(l);
+d's 2nd digit is not the same as the 2nd digit of r(2);
+d's 3rd digit is not the same as the 3rd digit of r(3);
+
+... and so on.
+
+
+Hence,
+
+
+d is different from r(l);
+d is different from r(2);
+d is different from r(3);
+
+... and soon.
+
+In other words, d is not in the list!
+
+What Does a Diagonal Argument Prove?
+
+Now comes the crucial difference between Cantor's proof and our proofit is in the matter
+of what assumption to go back and undo. In Cantor's argument, the shaky assumption
+was that such a table could be drawn up. Therefore, the conclusion warranted by the
+construction of d is that no exhaustive table of reals can be drawn up after all-which
+amounts to saying that the set of integers is just not big enough to index the set of reals.
+On the other hand, in our proof, we know that the directory of Blue BlooP programs can
+be drawn up-the set of integers is big enough to index the set of Blue BlooP programs.
+So, we have to go back and retract some shakier idea which we used. And that idea is that
+Bluediag [N] is calculable by some program in BlooP. This is a subtle difference in the
+application of the diagonal method.
+
+It may become clearer if we apply it to the alleged "List of All Great
+Mathematicians" in the Dialogue-a more concrete example. The diagonal itself is
+"Dboups". If we perform the desired diagonal-subtraction, we will get "Cantor". Now two
+conclusions are possible. If you have an unshakable belief that the list is complete, then
+you must conclude that Cantor is not a Great Mathematician, for his name differs from all
+those on the list. On the other hand, if you have an unshakable belief that Cantor is a
+Great Mathematician, then you must conclude that the List of All Great Mathematicians
+is incomplete, for Cantor's name is not on the list! (Woe to those who have unshakable
+beliefs on both sides!) The former case corresponds to our proof that Bluediag [N] is not
+primitive recursive; the latter case corresponds to Cantor's proof that the list of reals is
+incomplete;
+
+
+BlooP and FlooP and GlooP
+
+
+423
+
+
+
+FIGURE 73. Georg Cantor
+
+
+Cantor’s proof uses a diagonal in the literal sense of the word. Other "diagonal* proofs are
+based on a more general notion, which is abstracted from the geometric sense of the
+word. The essence of the diagonal method is the fact of using one integer in two different
+ways-or, one could say, using one integer on two different tevW.y-thanks to which one can
+construct an item which is outside of some predetermined list. One time, the integer
+serves as a vertical index, the other time as a horizontal index. In Cantor's construction
+this is very clear. As for the function Bluediag [N], it involves using one integer on two
+different levels-first, as a Blue Program index number; and second, as an input parameter.
+
+The Insidious Repeatability of the Diagonal Argument
+
+At first, the Cantor argument may seem less than fully convincing. Isn't there some way
+to get around it? Perhaps by throwing in the diagonally constructed number d, one might
+obtain an exhaustive list. If you consider this idea, you will see it helps not a bit to throw
+in the number d, for as soon as you assign it a specific place in the table, the diagonal
+method becomes applicable to the new table, and a new missing number d’ can be
+constructed, which is not in the new table. No matter how many times you repeat the
+operation of constructing a number by the diagonal method and then throwing it in to
+make a "more complete" table, you still are caught on the ineradicable hook of Cantor’s
+method. You might even try to build a table of reals which tries to outwit the Cantor
+diagonal method by taking
+
+
+BlooP and FlooP and GlooP
+
+
+424
+
+
+
+the whole trick, lock, stock, and barrel, including its insidious repeatability, into account
+somehow. It is an interesting exercise. But if you tackle it, you will see that no matter
+how you twist and turn trying to avoid the Cantor "hook", you are still caught on it. One
+might say that any self-proclaimed "table of all reals" is hoist by its own petard.
+
+The repeatability of Cantor's diagonal method is similar to the repeatability of the
+Tortoise's diabolic method for breaking the Crab's phonographs, one by one, as they got
+more and more "hi-fi" and-at least so the Crab hoped-more "Perfect". This method
+involves constructing, for each phonograph, a particular song which that phonograph
+cannot reproduce. It is not a coincidence that Cantor's trick and the Tortoise's trick share
+this curious repeatability; indeed, the Contracrostipunctus might well have been named
+" Cantorcrostipunctus" instead. Moreover, as the Tortoise subtly hinted to the innocent
+Achilles, the events in the Contracrostipunctus are a paraphrase of the construction which
+Godel used in proving his Incompleteness Theorem; it follows that the Godel
+construction is also very much like a diagonal construction. This will become quite
+apparent in the next two Chapters.
+
+From BIooP to FIooP
+
+We have now defined the class of primitive recursive functions and primitive recursive
+properties of natural numbers by means of programs written in the language BIooP. We
+have also shown that BIooP doesn't capture all the functions of natural numbers which we
+can define in words. We even constructed an "unBlooPable" function, Bluediag [N], by
+Cantor's diagonal method. What is it about BIooP that makes Bluediag unrepresentable in
+it? How could BIooP be improved so that Bluediag became representable?
+
+BlooP's defining feature was the boundedness of its loops. What if we drop that
+requirement on loops, and invent a second language, called "FIooP" ('F' for "free")?
+FIooP will be identical to BIooP except in one respect: we may have loops without
+ceilings, as well as loops with ceilings (although the only reason one would include a
+ceiling when writing a loop-statement in FIooP would be for the sake of elegance). These
+new loops will be called MU-LOOPS. This follows the convention of mathematical
+logic, in which "free" searches (searches without bounds) are usually indicated by a
+symbol called a "p-ope rat or" (mu-operator). Thus, loop statements in FIooP may look
+like this:
+
+
+MU-LOOP:
+BLOCK n: BEGIN
+
+
+BLOCK n: END
+
+
+BIooP and FIooP and GlooP
+
+
+425
+
+
+
+This feature will allow us to write tests in FlooP for such properties as wondrousness
+and the Tortoise property-tests which we did not know how to program in BlooP because
+of the potential open-endedness of the searches involved. I shall leave it to interested
+readers to write a FlooP test for wondrousness which does the following things:
+
+(1) If its input, N, is wondrous, the program halts and gives the answer YES.
+
+(2) If N is unwondrous, but causes a closed cycle other than 1-4-2-1-4-2-1- ... , the
+program halts and gives the answer NO.
+
+(3) If N is unwondrous, and causes an "endlessly rising progression", the program
+never halts. This is FlooP's way of answering by not answering. FlooP's
+nonanswer bears a strange resemblance to Joshu's nonanswer "MU".
+
+The irony of case 3 is that OUTPUT always has the value NO, but it is always
+inaccessible, since the program is still grinding away. That troublesome third alternative
+is the price that we must pay for the right to write free loops. In all FlooP programs
+incorporating the MU-LOOP option, nontermination will always be one theoretical
+alternative. Of course there will be many FlooP programs which actually terminate for all
+possible input values. For instance, as I mentioned earlier, it is suspected by most people
+who have studied wondrousness that a FlooP program such as suggested above will
+always terminate, and moreover with the answer YES each time.
+
+Terminating and Nonterminating FlooP Programs
+
+It would seem extremely desirable to be able to separate FlooP procedures into two
+classes: terminators and nonterminators. A terminator will eventually halt no matter what
+its input, despite the "MU-ness" of its loops. A nonterminator will go on and on forever,
+for at least one choice of input. If we could always tell, by some kind of complicated
+inspection of a FlooP program, to which class it belonged, there would be some
+remarkable repercussions (as we shall shortly see). Needless to say, the operation of
+class-checking would itself have to be a terminating operation-otherwise
+one would gain nothing!
+
+
+Turing's Trickery
+
+The idea springs to mind that we might let a BlooP procedure do the inspection. But
+BlooP procedures only accept numerical input, not programs! However, we can get
+around that ... by coding programs into numbers! This sly trick is just Godel-numbering
+in another of its many
+
+
+BlooP and FlooP and GlooP
+
+
+426
+
+
+
+a very long Godel number. For instance, the shortest BlooP function (which is also a
+terminating FlooP program)
+
+DEFINE PROCEDURE "A" [B]:
+
+BLOCK 0: BEGIN
+
+BLOCK 0: END.
+
+-would get the Godel number partially shown below:
+
+904, 905, 906, 909, 914, 905 905, 914.904, 955,
+
+DEFINE END.
+
+Now our scheme would be to write a BlooP test called TERMINATOR? which
+says YES if its input number codes for a terminating FlooP program, NO if not. This way
+we could hand the task over to a machine and with luck, distinguish terminators from
+non terminators. However, an ingenious argument given by Alan Turing shows that no
+BlooP program can make this distinction infallibly. The trick is actually much the same
+as Godel’s trick, and therefore closely related to the Cantor diagonal trick. We shall not
+give it here-suffice it to say that the idea is to feed the termination tester its own Godel
+number. This is not so simple, however, for it is like trying to quote an entire sentence
+inside itself. You have to quote the quote, and so forth; it seems to lead to an infinite
+regress. However, Turing figured out a trick for feeding a program its own Godel
+number. A solution to the same problem in a different context will be presented next
+Chapter. In the present Chapter, we shall take a different route to the same goal, which is
+namely to prove that a termination tester is impossible. For readers who wish to see an
+elegant and simple presentation of the Turing approach, I recommend the article by
+Hoare and Allison, mentioned in the Bibliography.
+
+A Termination Tester Would Be Magical
+
+Before we destroy the notion, let us delineate just why having a termination tester would
+be a remarkable thing. In a sense, it would be like having a magical dowsing rod which
+could solve all problems of number theory in one swell FlooP. Suppose, for instance, that
+we wished to know if the Goldbach Variation is a true conjecture or not. That is, do all
+numbers have the Tortoise property? We would begin by writing a FlooP test called
+TORTOISE? which checks whether its input has the Tortoise property. Now the defect
+of this procedure-namely that it doesn't terminate if the Tortoise property is absent-here
+turns into a virtue! For now we run the termination tester on the procedure TORTOISE?.
+If it says YES, that means that TORTOISE? terminates for all values of its input-in other
+words, all numbers have the Tortoise property. If it says NO, then we know there exists a
+number which has the Achilles property. The irony is that we never actually use the
+program TORTOISE at all—we just inspect it.
+
+This idea of solving any problem in number theory by coding it into a
+
+
+BlooP and FlooP and GlooP
+
+
+427
+
+
+
+program and then waving a termination tester over the program is not unlike the idea of
+testing a khan for genuineness by coding it into a folded string and then running a test for
+Buddha-nature on the string instead. As
+
+Achilles suggested, perhaps the desired information lies "closer to the surface" in one
+representation than in another.
+
+Pool F, Index Numbers, and Green Programs
+
+Well, enough daydreaming. How can we prove that the termination tester is impossible?
+Our argument for its impossibility will hinge on trying to apply the diagonal argument to
+FlooP, just as we did to BlooP. We shall see that there are some subtle and crucial
+differences between the two cases.
+
+As we did for BlooP, imagine the pool of all FlooP programs. We shall call it
+"Pool F". Then perform the same three filtering operations on Pool F, so that you get, in
+the end:
+
+A complete pool of all call-less FlooP programs which calculate functions of
+exactly one input parameter.
+
+Let us call these special FlooP-programs Green Programs (since they may go forever).
+
+Now just as we assigned index numbers to all Blue Programs, we can assign
+index numbers to Green Programs, by ordering them in a catalogue, each volume of
+which contains all Green Programs of a fixed length, arranged in alphabetical order.
+
+So far, the carry-over from BlooP to FlooP has been straightforward. Now let us see if we
+can also carry over the last part: the diagonal trick. What if we try to define a diagonal
+function?
+
+
+Greendiag [N] = 1 + Greenprogram{#N} [N]
+
+Suddenly, there is a snag: this function Greendiag [N] may not have a well-defined
+output value for all input values N. This is simply because we have not filtered out the
+non terminator programs from Pool F, and therefore we have no guarantee that we can
+calculate Greendiag [N] for all values of N. Sometimes we may enter calculations which
+never terminate. And the diagonal argument cannot be carried through in such a case, for
+it depends on the diagonal function having a value for all possible inputs.
+
+The Termination Tester Gives Us Red Programs
+
+To remedy this, we would have to make use of a termination tester, if one existed. So let
+us deliberately introduce the shaky assumption that one exists, and let us use it as our
+fourth filter. We run down the list of Green Programs, eliminating one by one all
+nonterminators, so that in the end we are left with:
+
+
+BlooP and FlooP and GlooP
+
+
+428
+
+
+
+A complete pool of all call-less FlooP programs which calculate functions of
+exactly one input parameter, and which terminate for all values of their input..
+
+
+Let us call these special FlooP programs Red Programs (since they all must stop). Now,
+the diagonal argument will go through. We define
+
+Reddiag [N] = 1 + Redprogram(#N} [N]
+
+and in an exact parallel to Bluediag, we are forced to conclude that Reddiag [N] is a well-
+defined, calculable function of one variable which is not in the catalogue of Red
+Programs, and is hence not even calculable in the powerful language FlooP. Perhaps it is
+time to move on to GlooP?
+
+
+GIooP ...
+
+Yes, but what is GlooP? If FlooP is BlooP unchained, then GlooP must be FlooP
+unchained. But how can you take the chains off twice% How do you make a language
+whose power transcends that of FlooP? In Reddiag , we have found a function whose
+values we humans know how to calculate-the method of doing so has been explicitly
+described in English-but which seemingly cannot be programmed in the language FlooP.
+This is a serious dilemma because no one has ever found any more powerful computer
+language than FlooP.
+
+Careful investigation into the power of computer languages has been carried out.
+We need not do it ourselves; let it just be reported that there is a vast class of computer
+languages all of which can be proven to have exactly the same expressive power as FlooP
+does, in this sense: any calculation which can be programmed in any one of the languages
+can be programmed in them all. The curious thing is that almost any sensible attempt at
+designing a computer language ends up by creating a member of this class-which is to
+say, a language of power equal to that of FlooP. It takes some doing to invent a
+reasonably interesting computer language which is weaker than those in this class. BlooP
+is, of course, an example of a weaker language, but it is the exception rather than the rule.
+The point is that there are some extremely natural ways to go about inventing algorithmic
+languages; and different people, following independent routes, usually wind up creating
+equivalent languages, with the only difference being style, rather than power.
+
+... Is a Myth
+
+In fact, it is widely believed that there cannot be any more powerful -language for
+describing calculations than languages that are equivalent to FlooP. This hypothesis was
+formulated in the 1930's by two people, independently of each other: Alan Turing—about
+whom we shall say more later—and Alonzo Church, one of the eminent logicians of this
+century. It
+
+
+BlooP and FlooP and GlooP
+
+
+429
+
+
+
+is called the Church-Turing Thesis. If we accept the CT-Thesis, we have to conclude that
+"GlooP" is a myth-there are no restrictions to remove in FlooP, no ways to increase its
+power by "unshackling" it, as we did BlooP.
+
+This puts us in the uncomfortable position of asserting that people can calculate
+Reddiag [N] for any value of N, but there is no way to program a computer to do so. For,
+if it could be done at all, it could be done in FlooP-and by construction, it can't be done in
+FlooP. This conclusion is so peculiar that it should cause us to investigate very carefully
+the pillars on which it rests. And one of them, you will recall, was our shaky assumption
+that there is a decision procedure which can tell terminating from nonterminating FlooP
+programs. The idea of such a decision procedure already seemed suspect, when we saw
+that its existence would allow all problems of number theory to be solved in a uniform
+way. Now we have double the reason for believing that any termination test is a myth-
+that there is no way to put FlooP programs in a centrifuge and separate out the
+terminators from the nonterminators.
+
+Skeptics might maintain that this is nothing like a rigorous proof that such a
+termination test doesn't exist. That is a valid objection; however, the Turing approach
+demonstrates more rigorously that no computer program can be written in a language of
+the FlooP class which can perform a termination test on all FlooP programs.
+
+The Church-Turing Thesis
+
+Let us come back briefly to the Church-Turing Thesis. We will talk about it-and
+variations on it-in considerable detail in Chapter XVII; for now it will suffice to state it in
+a couple of versions, and postpone discussion of its merits and meanings until then. Here,
+then, are three related ways to state the CT-Thesis:
+
+(1) What is human-computable is machine-computable.
+
+(2) What is machine-computable is FlooP-computable.
+
+(3) What is human-computable is FlooP-computable
+
+(i.e., general or partial recursive).
+
+Terminology: General and Partial Recursive
+
+We have made a rather broad survey, in this Chapter, of some notions from number
+theory and their relations to the theory of computable functions. It is a very wide and
+flourishing field, an intriguing blend of computer science and modern mathematics. We
+should not conclude this Chapter without introducing the standard terminology for the
+notions we have been dealing with.
+
+As has already been mentioned, “BlooP-computable” is synonymous with
+“primitive recursive”. Now FlooP computable functions can be di-
+
+
+BlooP and FlooP and GlooP
+
+
+430
+
+
+
+vided into two realms: (1) those which are computable by terminating FlooP programs:
+these are said to be general recursive', and (2) those which are computable only by
+nonterminating FlooP programs: these are said to be partial recursive. (Similarly for
+predicates.) People often just say "recursive" when they mean "general recursive".
+
+The Power of TNT
+
+It is interesting that TNT is so powerful that not only are all primitive recursive
+predicates represented, but moreover all general recursive predicates are represented. We
+shall not prove either of these facts, because such proofs would be superfluous to our
+aim, which is to show that TNT is incomplete. If TNT could not represent some
+primitive or general recursive predicates, then it would be incomplete in an uninteresting
+way-so we might as well assume that it can, and then show that it is incomplete in an
+interesting way.
+
+
+BlooP and FlooP and GlooP
+
+
+431
+
+
+
+Air on G's String
+
+The Tortoise and Achilles have just completed a tour of a porridge factory.
+
+
+Achilles: You don't mind if I change the subject, do you? Tortoise: Be my guest.
+
+Achilles: Very well, then. It concerns an obscene phone call I received a few days ago.
+Tortoise: Sounds interesting.
+
+Achilles: Yes. Well-the problem was that the caller was incoherent, at least as far as I
+could tell. He shouted something over the line and then hung up-or rather, now that I
+think of it, he shouted something, shouted it again, and then hung up.
+
+Tortoise: Did you catch what that thing was?
+
+Achilles: Well, the whole call went like this:
+
+Myself. Hello?
+
+Caller (shouting wildly ): Yields falsehood when preceded by its quotation! Yields
+falsehood when preceded by its quotation!
+
+(Click.)
+
+Tortoise: That is a most unusual thing to say to somebody on an obscene phone call.
+Achilles: Exactly how it struck me.
+
+Tortoise: Perhaps there was some meaning to that seeming madness.
+
+Achilles: Perhaps.
+
+(They enter a spacious courtyard framed by some charming three-story stone
+houses. At its center stands a palm tree, and to one side is a tower. Near the
+tower there is a staircase where a boy sits, talking to a young woman in a
+window.)
+
+Tortoise: Where are you taking me, Achilles?
+
+Achilles: I would like to show you the pretty view from the top of this tower.
+
+Tortoise: Oh, how nice.
+
+(They approach the boy, who watches them with curiosity, then says something to
+the young woman-they both chuckle. Achilles and Mr. T, instead of going up the
+boy's staircase, turn left and head down a short flight of stairs which leads to a
+small wooden door.)
+
+Achilles: We can just step inside right here. Follow me.
+
+
+Air on G's String
+
+
+431
+
+
+
+
+3URE 74. Above and Below, by M.C. Escher (lithograph J947).
+
+
+
+
+G’s String
+
+
+432
+
+
+
+
+(Achilles opens the door. They enter, and begin climbing the steep helical staircase
+inside the tower.)
+
+Tortoise (puffing slightly ): I'm a little out of shape for this sort of exercise,
+
+Achilles. How much further do we have to go?
+
+Achilles: Another few flights ... but I have an idea. Instead of walking on the top side of
+these stairs, why don't you walk on the underside?
+
+Tortoise: How do I do THAT?
+
+Achilles: Just hold on tightly, and climb around underneath-there's room enough for you.
+
+You'll find that the steps make just as much sense from below as from above ...
+Tortoise ( gingerly shifting himself about): Am I doing it right?
+
+Achilles: You've got it!
+
+Tortoise (his voice slightly muffled): Say-this little maneuver has got me confused.
+Should I head upstairs or downstairs, now?
+
+Achilles: Just continue heading in the same direction as you were before. On your side of
+the staircase, that means go DOWN, on mine it means UP.
+
+Tortoise: Now you're not going to tell me that I can get to the top of the tower by going
+down, are you?
+
+Achilles: I don't know, but it works ...
+
+(And so they begin spiraling in synchrony, with A always on one side, and T
+matching him on the other side. Soon they reach the end of the staircase.)
+
+Now just undo the maneuver, Mr. T. Here-let me help you up.
+
+(He lends an arm to the Tortoise, and hoists him back to the other side of the
+stairs.)
+
+Tortoise: Thanks. It was a little easier getting back up.
+
+(And they step out onto the roof, overlooking the town.)
+
+That's a lovely view, Achilles. I'm glad you brought me up here-or rather, DOWN
+here.
+
+Achilles: I figured you'd enjoy it.
+
+Tortoise: I've been thinking about that obscene phone call. I think I understand it a little
+better now.
+
+Achilles: You do? Would you tell me about it?
+
+Tortoise: Gladly. Do you perchance feel, as I do, that that phrase "preceded by its
+quotation" has a slightly haunting quality about it?
+
+Achilles: Slightly, yes-extremely slightly.
+
+Tortoise: Can you imagine something preceded by its quotation?
+
+Achilles: I guess I can conjure up an image of Chairman Mao walking into a banquet
+room in which there already hangs a large banner with some of his own writing on it.
+Here would be Chairman Mao, preceded by his quotation.
+
+Tortoise: A most imaginative example. But suppose we restrict the word
+
+
+Air on G's String
+
+
+433
+
+
+
+"preceded" to the idea of precedence on a printed sheet, rather than elaborate entries
+into a banquet room.
+
+Achilles: All right. But what exactly do you mean by "quotation" here? Tortoise: When
+you discuss a word or a phrase, you conventionally put it in quotes. For example, I can
+say.
+
+
+The word "philosopher" has five letters.
+
+Here, I put "philosopher" in quotes to show that I am speaking about the WORD
+"philosopher" rather than about a philosopher in the flesh. This is called the USE-
+MENTION distinction.
+
+Achilles: Oh?
+
+Tortoise: Let me explain. Suppose I were to say to you,
+
+Philosophers make lots of money.
+
+Here, I would be USING the word to manufacture an image in your mind of a twinkle¬
+eyed sage with bulging moneybags. But when I put this word-or any word-in quotes, I
+subtract out its meaning and connotations, and am left only with some marks on paper,
+or some sounds. That is called "MENTION". Nothing about the word matters, other
+than its typographical aspects-any meaning it might have is ignored.
+
+Achilles: It reminds me of using a violin as a fly swatter. Or should I say mentioning"?
+Nothing about the violin matters, other than its solidity-any meaning or function it
+might have is being ignored. Come to think of it, I guess the fly is being treated that
+way, too.
+
+Tortoise: Those are sensible, if slightly unorthodox, extensions of the use-mention
+distinction. But now, I want you to think about preceding something by its own
+quotation.
+
+Achilles: All right. Would this be correct?
+
+"HUBBA" HUBBA
+
+
+Tortoise: Good. Try another.
+
+Achilles: All right.
+
+"'PLOP' IS NOT THE TITLE OF ANY BOOK. SO FAR AS I KNOW"'
+
+'PLOP' IS NOT THE TITLE OF ANY BOOK, SO FAR AS I KNOW.
+
+Tortoise: Now this example can be modified into quite an interesting specimen, simply
+by dropping 'Plop 1 . Achilles: Really? Let me see what you mean. It becomes
+
+"IS NOT THE TITLE OF ANY BOOK, SO FAR AS I KNOW"
+
+IS NOT THE TITLE OF ANY BOOK, SO FAR AS I KNOW.
+
+Tortoise: You see, you have made a sentence.
+
+Achilles: So I have. It is a sentence about the pjrase “is not the toitle of any book, as far
+as I know”, and quite a silly one too.
+
+Tortoise: Why silly?
+
+
+Air on G's String
+
+
+434
+
+
+
+Achilles: Because it's so pointless. Here's another one for you:
+
+“WILL BE BOYS" WILL BE BOYS.
+
+Now what does that mean? Honestly, what a silly game.
+
+Tortoise: Not to my mind. It's very earnest stuff, in my opinion. In fact this operation of
+preceding some phrase by its quotation is so overwhelmingly important that I think I'll
+give it a name.
+
+Achilles: You will? What name will you dignify that silly operation by?
+
+Tortoise: I believe I'll call it "to quine a phrase", to quine a phrase.
+
+Achilles: "Quine"? What sort of word is that?
+
+Tortoise: A five-letter word, if I'm not in error.
+
+Achilles: What 1 was driving at is why you picked those exact five letters in that exact
+order.
+
+Tortoise: Oh, now I understand what you meant when you asked me "What sort of word
+is that?" The answer is that a philosopher by the name of "Willard Van Orman Quine"
+invented the operation, so I name it in his honor. However, I cannot go any further
+than this in my explanation. Why these particular five letters make up his name-not to
+mention why they occur in this particular order-is a question to which I have no ready
+answer. However, I'd be perfectly willing to go and
+Achilles: . Please don't bother! I didn't really want to know everything about Quine's
+name. Anyway, now I know how to quine a phrase. It's quite amusing. Here's a quined
+phrase:
+
+”IS A SENTENCE FRAGMENT" IS A SENTENCE FRAGMENT.
+
+It's silly but all the same I enjoy it. You take a sentence fragment, quine
+it, and lo and behold, you've made a sentence! A true sentence, in this case.
+
+Tortoise: How about quining the phrase "is a king with without no subject”?
+
+Achilles: A king without a subject would be—
+
+Tortoise: -an anomaly, of course. Don't wander from the point. Let's have quines first,
+and kings afterwards!
+
+Achilles: I'm to quine that phrase, am I? All right
+
+"IS A KING WITH NO SUBJECT" IS A KING WITH NO SUBJECT.
+
+It seems to me that it might make more sense if it said "sentence" instead of "king".
+Oh, well. Give me another!
+
+Tortoise: All right just one more. Try this one:
+
+"WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG"
+
+Achilles: That should be easy ... I'd say the quining gives this:
+
+"WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG"
+
+WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG
+
+Hmmm... There's something just a little peculiar here. Oh, I see what it is! The
+sentence is talking about itself! Do you see that?
+
+
+Air on G's String
+
+
+435
+
+
+
+Tortoise: What do you mean? Sentences can't talk.
+
+Achilles: No, but they REFER to things-and this one refers directly unambiguously-
+unmistakably-to the very sentence which it is! You just have to think back and
+remember what quining is all about.
+
+Tortoise: I don't see it saying anything about itself. Where does it say "me", or: "this
+sentence", or the like?
+
+Achilles: Oh, you are being deliberately thick-skulled. The beauty of it lies in just that:
+
+it talks about itself without having to come right out and say so!
+
+Tortoise: Well, as I'm such a simple fellow, could you just spell it all out for me,
+Achilles: Oh, he is such a Doubting Tortoise ... All right, let me see ... Suppose I make
+up a sentence-I'll call it "Sentence P"-with a blank in it.
+
+Tortoise: Such as?
+
+Achilles: Such as ...
+
+“_WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG".
+
+Now the subject matter of Sentence P depends on how you fill in the blank. But
+once you've chosen how to fill in the blank, then the subject matter is determined: it
+is the phrase which you get by QUINING the blank. Call that "Sentence Q", since it
+is produced by an act of quining.
+
+Tortoise: That makes sense. If the blank phrase were "is written on old jars of mustard
+to keep them fresh", then Sentence Q would have to be
+
+"IS WRITTEN ON OLD JARS OF MUSTARD TO KEEP THEM FRESH"
+
+IS WRITTEN ON OLD JARS OF MUSTARD TO KEEP THEM FRESH.
+
+Achilles: True, and Sentence P makes the claim (though whether it is valid or not, I do
+not know) that Sentence Q is a Tortoise's love song. In any case, Sentence P here is
+not talking about itself, but rather about Sentence Q. Can we agree on that much?
+Tortoise: By all means, let us agree-and what a beautiful song it is, too.
+
+Achilles: But now I want to make a different choice for the blank, namely
+
+: "WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG".
+
+Tortoise: Oh, heavens, you're getting a little involved here. I hope this all isn't going to
+be too highbrow for my modest mind.
+
+Achilles: Oh, don't worry-you'll surely catch on. With this choice, Sentence Q
+becomes .. .
+
+"WHEN QUINED, YIELDS A TORTOISE'S LOVE-SONG"
+
+WHEN QUINED, YIELDS A TORTOISE'S LOVE-SONG.
+
+Tortoise: Oh, you wily old warrior you, I catch on. Now Sentence Q is just the same as
+Sentence P.
+
+Achilles: And since Sentence Q is always the topic of Sentence P, there is a loop now,
+P points back to itself. But you see, the self-reference is a
+
+
+Air on G's String
+
+
+436
+
+
+
+sort of accident. Usually Sentences Q and P are entirely unlike each other; but with
+the right choice for the blank in Sentence-P, quining will do this magic trick for
+you.
+
+Tortoise: Oh, how clever. I wonder why I never thought of that myself. Now tell me:
+is the following sentence self-referential?
+
+"IS COMPOSED OF FIVE WORDS" IS COMPOSED OF FIVE WORDS.
+
+Achilles: Hmm ... I can't quite tell. The sentence which you just gave is not really
+about itself, but rather about the phrase "is composed of five words". Though, of
+course, that phrase is PART of the sentence ...
+
+Tortoise: So the sentence refers to some part of itself-so what? Achilles: Well,
+wouldn't that qualify as self-reference, too?
+
+Tortoise: In my opinion, that is still a far cry from true-self-reference. But don't worry
+too much about these tricky matters. You'll have ample time to think about them in
+the future. Achilles: I will?
+
+Tortoise: Indeed you will. But for now, why don't you try quining the phrase "yields
+falsehood when preceded by its quotation"?
+
+Achilles: I see what you're getting at-that old obscene phone call. Quining it produces
+the following:
+
+"YIEEDS FAFSEHOOD WHEN PRECEDED BY ITS QUOTATION"
+
+YIELDS FALSEHOOD WHEN PRECEDED BY ITS QUOTATION.
+
+So this is what that caller was saying! I just couldn't make out where the quotation
+marks were as he spoke. That certainly is an obscene remark! People ought to be
+jailed for saying things like that!
+
+Tortoise: Why in the world?
+
+Achilles: It just makes me very uneasy. Unlike the earlier examples, I can't quite make
+out if it is a truth or a falsehood. And the more I think about it, the more I can't
+unravel it. It makes my head spin. I wonder what kind of a lunatic mind would
+make something like that up, and torment innocent people in the night with it?
+
+Tortoise: I wonder ... Well, shall we go downstairs now?
+
+Achilles: We needn't go down-we're at ground level already. Let's go back inside -
+you'll see. ( They go into the tower, and come to a small wooden door.) We can just
+step outside right here. Follow me.
+
+Tortoise: Are you sure? I don't want to fall three floors and break my shell.
+
+Achilles: Would I fool you?
+
+(And he opens the door. In front of them sits, to all appearances, the same boy,
+talking to the same young woman. Achilles and Mr. T walk up what seem to be the
+same stairs they walked down to enter the tower, and find themselves in what looks
+like just the same courtyard they first came into.)
+Thank you, Mr. T, for your lucid clarification of that obscene telephone call.
+
+Tortoise: And thank you, Achilles, for the pleasant promenade. I hope we meet again
+soon.
+
+
+Air on G's String
+
+
+437
+
+
+
+CHAPTER XIV
+
+
+On Formally Undecidable
+Propositions of TNT
+and Related Systems'
+
+The Two Ideas of the "Oyster"
+
+THIS CHAPTER'S TITLE is an adaptation of the title of Godel’s famous 1931 paper-
+"TNT" having been substituted for "Principia Mathematica". Godel’s paper was a
+technical one, concentrating on making his proof watertight and rigorous; this Chapter
+will be more intuitive, and in it I will stress the two key ideas which are at the core of the
+proof. The first key idea is the deep discovery that there are strings of TNT which can be
+interpreted as speaking about other strings of TNT; in short, that TNT, as a language, is
+capable of "introspection", or self-scrutiny. This is what comes from Godel-numbering.
+The second key idea is that the property of self scrutiny can be entirely concentrated into
+a single string; thus that string's sole focus of attention is itself. This "focusing trick" is
+traceable, in essence, to the Cantor diagonal method.
+
+In my opinion, if one is interested in understanding Godel’s proof, in a deep way,
+then one must recognize that the proof, in its essence, consists of a fusion of these two
+main ideas. Each of them alone is a master stroke; to put them together took an act of
+genius. If I were to choose, however, which of the two key ideas is deeper, I would
+unhesitatingly pick the first one-the idea of Godel-numbering, for that idea is related to
+the whole notion of what meaning and reference are, in symbol-manipulating systems.
+This is an idea which goes far beyond the confines of mathematical logic, whereas the
+Cantor trick, rich though it is in mathematical consequences, has little if any relation to
+issues in real life.
+
+
+The First Idea: Proof-Pairs
+
+Without further ado, then, let us proceed to the elaboration of the proof itself. We have
+already given a fairly careful notion of what the Godel isomorphism is about, in Chapter
+IX. We now shall describe a mathematical notion which allows us to translate a statement
+such as "The string 0=0 is a theorem of TNT into a statement of number theory. This will
+involve the notion of proof-pairs. A proof-pair is a pair of natural numbers related in a
+particular way. Here is the idea:
+
+
+On Formally Undecidable Propositions
+
+
+438
+
+
+
+Two natural numbers, m and n respectively, form a TNT proof-pair if and only if m
+is the Godel number of a TNT derivation whose bottom line is the string with
+Godel number n.
+
+The analogous notion exists with respect to the MlU-system, and it is a little easier on the
+intuition to consider that case first. So, for a moment, let us back off from TNT -proof-
+pairs, and look at MlU-proof-pairs. Their definition is parallel:
+
+Two natural numbers, m and n respectively, form a MlU-proof pair if and only if m
+is the Godel number of a MlU-system derivation whose bottom line is the string
+with Godel number n.
+
+Let us see a couple of examples involving MlU-proof-pairs. First, let m =
+3131131111301, n = 301. These values of m and n do indeed form a MlU-proof-pair,
+because m is the Godel number of the MlU-derivation
+
+MI
+
+Mil
+
+MIIII
+
+MUI
+
+whose last line is MUI, having Godel number 301, which is n. By contrast, let m =
+31311311130, and n = 30. Why do these two values not form a MlU-proof-pair? To see
+the answer, let us write out the alleged derivation which m codes for:
+
+MI
+
+Mil
+
+Mill
+
+MU
+
+There is an invalid step in this alleged derivation! It is the step from the second to the
+third line: from Mil to Mill. There is no rule of inference in the MlU-system which
+permits such a typographical step. Correspondingly-and this is most crucial-there is no
+arithmetical rule of inference which carries you from 311 to 3111. This is perhaps a
+trivial observation, in light of our discussion in Chapter IX, yet it is at the heart of the
+Godel isomorphism. What we do in any formal system has its parallel in arithmetical
+manipulations.
+
+In any case, the values m = 31311311130, n = 30 certainly do not form a MIU-
+proof-pair. This in itself does not imply that 30 is not a MlU-number. There could be
+another value of m which forms a MIU proof-pair with 30. (Actually, we know by earlier
+reasoning that MU is not a MlU-theorem, and therefore no number at all can form a
+MlU-proof-pair with 30.)
+
+Now what about TNT proof pairs? Here are two parallel examples, one being
+merely an alleged TNT proof-pair, the other being a valid TNT proof-pair. Can you spot
+which is which? (Incidentally, here is where
+
+
+On Formally Undecidable Propositions
+
+
+439
+
+
+
+the '611' codon comes in. Its purpose is to separate the Godel numbers of successive lines
+in a TNT-derivation. In that sense, '611' serves as a punctuation mark. In the MIU-
+system, the initial '3' of all lines is sufficient-no extra punctuation is needed.)
+
+(1) m = 626.262,636,223,123,262,111,666,611,223,123.666.111,666
+
+n= 123,666.111,666
+
+(2) m=626,262.636,223.123,262,111,666,611223,333,262.636,123.262,111,666
+
+n = 223,333,262,636,123,262.111,666
+
+It is quite simple to tell which one is which, simply by translating back to the old
+notation, and making some routine examinations to see
+
+(1) whether the alleged derivation coded for by m is actually a legitimate derivation;
+
+(2) if so, whether the last line of the derivation coincides with the string which n codes
+
+for.
+
+Step 2 is trivial; and step 1 is also utterly straightforward, in this sense: there are no open-
+ended searches involved, no hidden endless loops. Think of the examples above
+involving the MlU-system, and now just mentally substitute the rules of TNT for the
+MlU-system's rules, and the axioms of TNT for the MlU-system's one axiom. The
+algorithm in both cases is the same. Let me make that algorithm explicit:
+
+Go down the lines in the derivation one by one. Mark those which are axioms.
+
+For each line which is not an axiom, check whether it follows by any of the
+rules of inference from earlier lines in the alleged derivation.
+
+If all nonaxioms follow by rules of inference from earlier lines, then you have a
+legitimate derivation; otherwise it is a phony derivation.
+
+At each stage, there is a clear set of tasks to perform, and the number of them is quite
+easily determinable in advance.
+
+Proof-Pair-ness Is Primitive Recursive...
+
+The reason I am stressing the boundedness of these loops is, as you may have
+sensed, that I am about to assert
+
+FUNDAMENTAL FACT 1: The property of being a proof-pair is a primitive
+recursive number-theoretical property, and can therefore be tested for by a BlooP
+program.
+
+There is a notable contrast to be made here with that other closely related number-
+theoretical property: that of being a theorem-number. To
+
+
+On Formally Undecidable Propositions
+
+
+440
+
+
+
+assert that n is a theorem-number is to assert that some value of in exists which forms a
+proof-pair with n. (Incidentally, these comments apply equally well to TNT and to the
+MlU-system; it may perhaps help to keep both in mind, the MlU-system serving as a
+prototype.) To check whether n is a theorem-number, you must embark on a search
+through all its potential proof-pair "partners" m-and here you may be getting into an
+endless chase. No one can say how far you will have to look to find a number which
+forms a proof-pair with n as its second element. That is the whole problem of having
+lengthening and shortening rules in the same system: they lead to a certain degree of
+unpredictability.
+
+The example of the Goldbach Variation may prove helpful at this point. It is
+trivial to test whether a pair of numbers ( m,n ) form a Tortoise pair, that is to say, both m
+and n + m should be prime. The test is easy because the property of primeness is
+primitive recursive: it admits of a predictably terminating test. But if we want to know
+whether n possesses the Tortoise property, then we are asking, "Does any number m form
+a Tortoise-pair with n as its second element?"-and this, once again, leads us out into the
+wild, MU-loopy unknown.
+
+... And Is Therefore Represented in TNT
+
+The key concept at this juncture, then, is Fundamental Fact 1 given above, for from it we
+can conclude
+
+FUNDAMENTAL FACT 2: The property of forming a proof-pair is testable in
+BlooP, and consequently, it is represented in TNT by some formula having two
+free variables.
+
+Once again, we are being casual about specifying which system these proof-pairs
+are relative to; it really doesn't matter, for both Fundamental Facts hold for any formal
+system. That is the nature of formal systems: it is always possible to tell, in a predictably
+terminating way, whether a given sequence of lines forms a proof, or not-and this carries
+over to the corresponding arithmetical notions.
+
+The Power of Proof-Pairs
+
+Suppose we assume we are dealing with the MlU-system, for the sake of concreteness.
+You probably recall the string we called "MUMON", whose interpretation on one level
+was the statement "MU is a theorem of the MlU-system". We can show how MUMON
+would be expressed in TNT, in terms of the formula which represents the notion of MIU-
+proof-pairs. Let us abbreviate that formula, whose existence we are assured of by
+Fundamental Fact 2, this way:
+
+
+MIU-PROOF-PAIR {a,a }
+
+
+On Formally Undecidable Propositions
+
+
+441
+
+
+
+Since it is a property of two numbers, it is represented by a formula with two free
+variables. (Note: In this Chapter we shall always use austere TNT-so be careful to
+distinguish between the variables a, a', a".) In order to assert "MU is a theorem of the
+MlU-system", we would have to make the isomorphic statement "30 is a theorem-
+number of the MlU-system", and then translate that into TNT-notation. With the aid of
+our abbreviation, this is easy (remember also from Chapter VIII that to indicate the
+replacement of every a' by a numeral, we write that numeral followed by "/a 1 1):
+
+3a:MIU-PROOF- PAIRja,SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSO/a'}
+
+Count the S's: there are 30. Note that this is a closed sentence of TNT, because one free
+variable was quantified, the other replaced by a numeral. A clever thing has been done
+here, by the way. Fundamental Fact 2 gave us a way to talk about proof-pairs; we have
+figured out how to talk about theorem-numbers, as well: you just add an existential
+quantifier in front! A more literal translation of the string above would be, "There exists
+some number a that forms a MllJ-proof-pair with 30 as its second element".
+
+Suppose that we wanted to do something parallel with respect to TNT-say, to
+express the statement "0=0 is a theorem of TNT". We may abbreviate the formula which
+Fundamental Fact 2 assures us exists, in an analogous way (with two free variables,
+again):
+
+
+TNT- PROOF- PAIR{a,a'}
+
+(The interpretation of this abbreviated TNT-formula is: "Natural numbers a and a' form a
+TNT-proof-pair.") The next step is to transform our statement into number theory,
+following the MUMON-model above. The statement becomes "There exists some
+number a which forms a TNT proof-pair with 666,111,666 as its second element". The
+TNT-formula which expresses this is:
+
+3a:TNT-PROOF-PAI R{a,SSSSS SSSSSO/a'}
+many, many 5's!
+
+(in fact, 666,111,666 of them)
+
+-a closed sentence of TNT. (Let us call it "JOSHtU", for reasons to appear
+momentarily.) So you see that there is a way to talk not only about the primitive recursive
+notion of TNT-proof-pairs, but also about the related but trickier notion of TNT-
+theorem-numbers.
+
+To check your comprehension of these ideas, figure out how to translate into TNT
+the following statements of meta-TNT:.
+
+(1) 0=0 is not a theorem o f TNT.
+
+(2) ~0=0 is a theorem of TNT.
+
+(3) ~0=0 is not a theorem of TNT.
+
+
+On Formally Undecidable Propositions
+
+
+442
+
+
+
+How do the solutions differ from the example done above, and from each other' Here are
+a few more translation exercises.
+
+
+(4) JOSHU is a theorem of TNT. (Call the TNT-string which expresses this
+":METAJOSH t"\)
+
+(5) META-JOSH[. is a theorem of TNT. (Call the TNT-string which expresses this
+"META-META-JOSHC".)
+
+(6) META-META-JOSHU is a theorem of TNT
+
+(7) META-META- ME IA -JOSHU is a theorem of TNT
+
+(etc., etc.)
+
+Example 5 shows that statements of meta-meta-TNT can be translated into TNT-notation;
+example 6 does the same for meta-meta-meta-TNT, etc.
+
+It is important to keep in mind the difference between expressing a property, and
+representing it, at this point. The property of being a TNT theorem-number, for instance,
+is expressed by the formula
+
+
+3a:TNT- PROOF- PAI R{a,a'}
+
+Translation: "a 1 is a TNT-theorem-number". However, we have no guarantee that this
+formula represents the notion, for we have no guarantee that this property is primitive
+recursive-in fact, we have more than a sneaking suspicion that it isn't. (This suspicion is
+well warranted. The property of being a TNT-theorem-number is not primitive recursive,
+and no TNT-formula can represent the property!) By contrast, the property of being a
+proof-pair, by virtue of its primitive recursivity, is both expressible and representable, by
+the formula already introduced.
+
+Substitution Leads to the Second Idea
+
+The preceding discussion got us to the point where we saw how TNT can "introspect" on
+the notion of TNT-theoremhood. This is the essence of the first part of the proof. We
+now wish to press on to the second major idea of the proof, by developing a notion which
+allows the concentration of this introspection into a single formula. To do this, we need to
+look at what happens to the Godel number of a formula when you modify the formula
+structurally in a simple way. In fact, we shall consider this specific modification:
+
+replacement of all free variables by a specific numeral
+
+. Below are shown a couple of examples of this operation in the left hand column, and in
+the right hand column are exhibited the parallel changes in Godel numbers.
+
+
+On Formally Undecidable Propositions
+
+
+443
+
+
+
+Godel number
+262,1 11,262
+
+
+123,123,666,111.123,123,666
+
+223,333,262,636,333,262,163,636,
+262,163,163,111,362,123,123,262,
+236,123,123,262,163,323
+
+
+223,333,262,636,333,262,163,636,
+
+—3a:3a':SSSSO=(SSa-SSa') 123,123,123,123,666,111,362,123,
+
+123,262,236,123,123,262,163,323
+
+An isomorphic arithmetical process is going on in the right-hand column, in
+which one huge number is turned into an even huger number. The function which makes
+the new number from the old one would not be too difficult to describe arithmetically, in
+terms of additions, multiplications, powers of 10 and so on-but we need not do so. The
+main point is this: that the relation among (1) the original Godel number, (2) the number
+whose numeral is inserted, and (3) the resulting Godel number, is a primitive recursive
+relation. That is to say, a BlooP test could be written which, when fed as input any three
+natural numbers, says YES if they are related in this way, and NO if they aren't. You may
+test yourself on your ability to perform such a test-and at the same time convince yourself
+that there are no hidden open-ended loops to the process-by checking the following two
+sets of three numbers:
+
+(1) 362,262,112,262,163,323,111,123,123,123,123,666;
+
+2 :
+
+362,123,123,666,112,123,123,666,323,111,123,123,123,123,666.
+
+(2) 223,362,123,666,236,123,666,323,111,262,163.
+
+1
+
+223,362,262,236,262,323,111,262,163;
+
+As usual, one of the examples checks, the other does not. Now this relationship between
+three numbers will be called the substitution relationship. Because it is primitive
+recursive, it is represented by some formula of TNT having three free variables. Lets us
+abbreviate that TNT - formula by the following notation
+
+SUB (a,a\a")
+
+
+Formula
+
+a=a
+
+We now replace all
+free variables by
+the numeral for 2:
+
+SSO=SSO
+
+3a:3a':a"=(SSa*SSa')
+
+We now replace all
+free variables by
+the numeral for 4:
+
+
+On Formally Undecidable Propositions
+
+
+444
+
+
+
+Because this formula represents the substitution relationship, the formula shown
+below must be a TNT-theorem:
+
+SU B{SSSSS SSSSSO/a,SSO/a\SSSSSS SSSSO/a"}
+
+262,111,262 S's 123,123,666,111,123,123,666 S's
+
+(This is based on the first example of the substitution relation shown in the parallel
+columns earlier in this section.) And again because the SUB formula represents the
+substitution relation, the formula shown below certainly is not a TNT-theorem:
+
+SU B{SSSO/a,SSO/a',SO/a"}
+
+Arithmoquining
+
+We now have reached the crucial point where we can combine all of our disassembled
+parts into one meaningful whole. We want to use the machinery of the TNT-PROOF-
+PAIR and SUB formulas in some way to construct a single sentence of TNT whose
+interpretation is: "This very string of TNT is not a TNT-theorem." How do we do it%
+Even at this point, with all the necessary machinery in front of us, the answer is not easy
+to find.
+
+A curious and perhaps frivolous-seeming notion is that of substituting a formula's
+own Godel number into itself. This is quite parallel to that other curious, and perhaps
+frivolous-seeming, notion of "quining" in the Air on G's String. Yet quining turned out to
+have a funny kind of importance, in that it showed a new way of making a self-referential
+sentence. Self reference of the Quine variety sneaks up on you from behind the first time
+you see it-but once you understand the principle, you appreciate that it is quite simple and
+lovely. The arithmetical version of quining-let's call it arithmoquining-'wiW allow us to
+make a TNT-sentence which is "about itself".
+
+Let us see an example of arithmoquining. We need a formula with at least one
+free variable. The following one will do:
+
+
+a=SO
+
+
+This formula's Godel number is 262,111,123,666, and we will stick this number into the
+formula itself-or rather, we will stick its numeral in. Here is the result:
+
+SSSSS SSSSSO=SO
+262,111,123,666 S's
+
+This new formula a asserts a silly falsity-that 262.111.123.666 equals 1: If we had begun
+with the string ~a=S0 and then arthmoquined, we would have cone up with a true
+statement—as you can see for yourself.
+
+When you arithmoquine, you are of course performing a special case
+
+
+On Formally Undecidable Propositions
+
+
+439
+
+
+
+of the substitution operation we defined earlier. If we wanted to speak about
+arithmoquining inside TNT, we would use the formula
+
+
+SUB{a" a” a’}
+
+
+where the first two variables are the same. This comes from the fact that we are using a
+single number in two different ways (shades of the Cantor diagonal method!). The
+number a" is both (1) the original Godel number, and (2) the insertion-number. Let us
+invent an abbreviation for the above formula:
+
+ARITHMOQUINE{a", a'}
+
+What the above formula says, in English, is:
+
+a' is the Godel number of the formula gotten by arithmoquining the formula with
+Godel number a".
+
+Now the preceding sentence is long and ugly. Let's introduce a concise and elegant term
+to summarize it. We'll say
+
+a' is the arithmoquinification of a"
+
+to mean the same thing. For instance, the arithmoquinification of 262,111,123,666 is this
+unutterably gigantic number:
+
+123,123,123 123,123,123,666,111,123,666
+
+262,111,123,666 copies of '1231
+
+(This is just the Godel number of the formula we got when we arithmoquined a=SO.) We
+can speak quite easily about arithmoquining inside TNT.
+
+The Last Straw
+
+Now if you look back in the Air on G's String, you will see that the ultimate trick
+necessary for achieving self-reference in Quine's way is to quine a sentence which itself
+talks about the concept of quining. It's not enough just to quine-you must quine a quine-
+mentioning sentence! All right, then the parallel trick in our case must be to arithmoquine
+some formula which itself is talking about the notion of arithmoquining!
+
+Without further ado, we'll now write that formula down, and call it G's uncle :
+
+-3a:3a':<TNT-PROOF-PAIR{a,a'}A.ARITHMOQUINE{a",a'}>
+
+You can see explicitly how arithmoquinification is thickly involved in the plot, Now this
+“uncle” has a Godel number, of course, which we’ll call 'u'
+
+
+On Formally Undecidable Propositions
+
+
+440
+
+
+
+The head and tail of u's decimal expansion, and even a teeny bit of its midsection, can be
+read off directly:
+
+u = 223,333,262,636,333,262,163,636,212,... ,161,... ,213
+
+For the rest, we'd have to know just how the formulas TNT-PROOF-PAIR and
+ARITHMOQUINE actually look when written out. That is too complex, and it is quite
+beside the point, in any case.
+
+Now all we need to do is-arithmoquine this very uncle! What this entails is
+"booting out" all free variables-of which there is only one, namely a"-and putting in the
+numeral for u everywhere. This gives us:
+
+-3a:3a’:<TNT-PROOF-PAIR{a,a'} A ,ARITHMOQUINE{SSS ... SSSO/a",a'}>
+
+u S's
+
+And this, believe it or not, is Godel’s string, which we can call 'G'. Now there are two
+questions we must answer without delay. They are
+
+(1) What Is G's Godel number?
+
+(2) What is the interpretation of G?
+
+Question 1 first. How did we make G? Well, we began with the uncle, and arithmoquined
+it. So, by the definition of arithmoquinification, G's Godel number is
+
+the arithmoquinification of u.
+
+Now question 2. We will translate G into English in stages, getting gradually more
+comprehensible as we go along. For our first rough try, we make a pretty literal
+translation:
+
+"There do not exist numbers a and a' such that both (1) they form a TNT-proof-
+pair. and (2) a' is the arithmoquinification of u."
+
+Now certainly there is a number a' which is the arithmoquinification of u-so the problem
+must lie with the other number, a. This observation allows us to rephrase the translation
+of G as follows:
+
+"There is no number a that forms a TNT-proof-pair with the arithmoquinification
+of u."
+
+(This step, which can be confusing, is explained below in more detail.) Do you see what
+is happening? G is saying this:
+
+"The formula whose Godel number is the arithmoquinification
+of u is not a theorem of TNT."
+
+But-and this should come as no surprise by now-that formula is none other than G itself;
+whence we can make the ultimate translation of G; as
+
+“G is not a theorem of TNT.”
+
+
+On Formally Undecidable Propositions
+
+
+441
+
+
+
+-or if you prefer,
+
+
+"I am not a theorem of TNT."
+
+We have gradually pulled a high-level interpretation-a sentence of meta-TNT-out of what
+was originally a low-level interpretation-a sentence of number theory.
+
+TNT Says "Uncle!"
+
+The main consequence of this amazing construction has already been delineated in
+Chapter IX: it is the incompleteness of TNT. To reiterate the argument:
+
+Is G a TNT-theorem? If so, then it must assert a truth. But what in fact does G
+assert? Its own nontheoremhood. Thus from its theoremhood would follow its
+nontheoremhood: a contradiction.
+
+Now what about G being a nontheorem? This is acceptable, in that it doesn't
+lead to a contradiction. But G's nontheoremhood is what G asserts-hence G asserts
+a truth. And since G is not a theorem, there exists (at least) one truth which is not a
+theorem of TNT.
+
+Now to explain that one tricky step again. I will use another similar example. Take this
+string:
+
+—3a:3a': <TORTOISE-PAIR{a,a' }.TENTH-POWER{SSO/a ", a' J >
+
+where the two abbreviations are for strings of TNT which you can write down yourself.
+TENTH-POWER{a",a'} represents the statement "a' is the tenth power of a"". The
+literal translation into English is then:
+
+"There do not exist numbers a and a' such that both (1) they form a Tortoise-pair,
+and (2) a' is the tenth power of 2."
+
+But clearly, there is a tenth power of 2-namely 1024. Therefore, what the string is really
+saying is that
+
+"There is no number a that forms a Tortoise-pair with 1024"
+which can be further boiled down to:
+
+"1024 does not have the Tortoise property."
+
+The point is that we have achieved a way of substituting a description of a number, rather
+than its numeral, into a predicate. It depends on using one “extra quantified variable (a').
+Here, it was the number 1024 that was described as the “tenth power of 2”; above it was
+the number described as the arithmoquinification of a”.
+
+
+On Formally Undecidable Propositions
+
+
+442
+
+
+
+"Yields Nontheoremhood When Arithmoquined"
+
+
+Let us pause for breath for a moment, and review what has been done. The best way I
+know to give some perspective is to set out explicitly how it compares with the version of
+the Epimenides paradox due to Quine. Here is a map:
+
+
+Falsehood
+
+<==>
+
+nontheoremhood
+
+quotation of a phrase
+
+<==>
+
+
+preceding a predicate
+by a subject
+
+<==>
+
+definite term) into an open formula
+
+preceding a predicate
+by a quoted phrase
+
+<==>
+
+substituting the Godel number of a
+string into an open formula
+
+preceding a predicate
+by itself, in quotes
+("quining")
+
+<==>
+
+substituting the Godel number of an
+open formula into the formula itself
+("arithmoquining")
+
+yields falsehood when quined
+(a predicate without a subject)
+
+<==>
+
+"uncle" of G”
+
+the(an open formula of TNT
+
+'yields falsehood when quined"
+(the above predicate, quoted)
+
+<==>
+
+the number a (the Godel number
+of the above open formula)
+
+'yields falsehood when quined"
+yields falsehood when quined
+(complete sentence formed by
+quining the above predicate)
+
+<==>
+
+G itself
+
+(sentence of TNT formed
+by substituting a into the uncle, i.e.
+arithmoquining the uncle)
+
+
+Godel’s Second Theorem
+
+
+Since G's interpretation is true, the interpretation of its negation —G is false. And we
+know that no false statements are derivable in TNT. Thus neither G nor its negation -G
+can be a theorem of TNT. We have found a "hole" in our system-an undecidable
+proposition. This has a number of ramifications. Here is one curious fact which follows
+from G's undecidability: although neither G nor -G is a theorem, the formula <GV -G> is
+a theorem, since the rules of the Propositional Calculus ensure that all well-formed
+formulas of the form <Pv—P> are theorems.
+
+This is one simple example where an assertion inside the system and an assertion about
+the system seem at odds with each other. It makes one wonder if the system really
+reflects itself accurately. Does the "reflected metamathematics" which exists inside TNT
+correspond well to the metamathematics which we do? This was one of the questions
+which intrigued Godel when he wrote his paper. In particular, he was interested in
+whether it was possible, in the “reflected metamathematics”, to prove TNT’s consistency.
+Recall that this was a great philosophical dilemma of
+
+
+On Formally Undecidable Propositions
+
+
+443
+
+
+
+the day: how to prove a system consistent. Godel found a simple way to express the
+statement "TNT is consistent" in a TNT formula; and then he showed that this formula
+(and all others which express the same idea) are only theorems of TNT under one
+condition: that TNT is inconsistent. This perverse result was a severe blow to optimists
+who expected that one could find a rigorous proof that mathematics is contradiction-free.
+
+How do you express the statement "TNT is consistent" inside TNT It hinges on
+this simple fact: that inconsistency means that two formulas, x and x, one the negation of
+the other, are both theorems. But if both x and — x are theorems, then according to the
+Propositional Calculus, all well-formed formulas are theorems. Thus, to show TNT's
+consistency, it would suffice to exhibit one single sentence of TNT which can be proven
+to be a nontheorem. Therefore, one way to express "TNT is consistent" is to say "The
+formula -0=0 is not a theorem of TNT". This was already proposed as an exercise a few
+pages back. The translation is:
+
+—3a:TNT-PROOF- PAIR{a,SSSSS SSSSSOIa'}
+
+223,666,111,666 S's
+
+It can be shown, by lengthy but fairly straightforward reasoning, that-as long as TNT is
+consistent-this oath-of-consistency by TNT is not a theorem of TNT. So TNT's powers
+of introspection are great when it comes to expressing things, but fairly weak when it
+comes to proving them. This is quite a provocative result, if one applies it metaphorically
+to the human problem of self-knowledge.
+
+TNT Is (o-incomplete
+
+Now what variety of incompleteness does TNT "enjoy? We shall see that TNT's
+incompleteness is of the "omega" variety-defined in Chapter VIII. This means that there
+is some infinite pyramidal family of strings all of which are theorems, but whose
+associated "summarizing string" is a nontheorem. It is easy to exhibit the summarizing
+string which is a non theorem:
+
+
+u S’s
+
+Va: 3a':<TNT-PROOF- PAIR{a,a’}nARITHMOQUINE{SSS ... SSSO/a",a'}>
+
+To understand why this string is a nontheorem, notice that it is extremely similar to G
+itself-in fact, G can be made from it in one step (viz., according to TNT's Rule of
+Interchange). Therefore, if it were a theorem, so would G be. But since G isn't a theorem,
+neither can this be.
+
+Now we want to show that all of the strings in the related pyramidal family are
+theorems. We can write them own easily enough:
+
+
+On Formally Undecidable Propositions
+
+
+444
+
+
+
+u S's
+
+-3a': <TNT-PROOF- PAIR,0/a,a’} ARITHMOQUINE;SSS ... SSSO/a" a'I>
+
+-3a': <TNT-PROOF- PAIR) SO/a,a'} ARITHMOQUINEISSS ... SSSO/a", a’}>
+
+-3a': <TNT-PROOF-PAIR{SSO/a,a'} ARITHMOQUINEI5SS ... SSSO/a", a'}>
+
+-3a': <TNT-PROOF- PAIR}SSSO/a,a’} ARITHMOQUINE{SSS ... SSSO/a", a'}>
+
+What does each one assert? Their translations, one by one, are:
+
+"0 and the arithmoquinification of u do not form a TNT-proof-pair."
+
+"1 and the arithmoquinification of u do not form a TNT-proof-pair."
+
+"2 and the arithmoquinification of u do not form a TNT-proof-pail."
+
+"3 and the arithmoquinification of u do not form a TNT-proof-pair."
+
+Now each of these assertions is about whether two specific integers form a proof-pair or
+not. (By contrast, G itself is about whether one specific integer is a theorem-number or
+not.) Now because G is a nontheorem, no integer forms a proof-pair with G's Godel
+number. Therefore, each of the statements of the family is true. Now the crux of the
+matter is that the property of being a proof-pair is primitive recursive, hence represented,
+so that each of the statements in the list above, being true, must translate into a theorem
+of TNT-which means that everything in our infinite pyramidal family is a theorem. And
+that shows why TNT is w-incomplete.
+
+Two Different Ways to Plug Up the Hole
+
+Since G's interpretation is true, the interpretation of its negation -G is false. And, using
+the assumption that TNT is consistent, we know that no false statements are derivable in
+TNT. Thus neither G nor its negation -G is a theorem of TNT. We have found a hole in
+our system-an undecidable proposition. Now this need be no source of alarm, if we are
+philosophically detached enough to recognize what this is a symptom of. It signifies that
+TNT can be extended, just as absolute geometry could be. In fact, TNT can be extended
+in two distinct directions, just as absolute geometry could be. It can be extended in a
+standard direction-which corresponds to extending absolute geometry in the Euclidean
+direction; or, it can be extended in a nonstandard direction-which corresponds, of course,
+to extending absolute geometry in the non-Euclidean direction. Now the standard type of
+extension would involve
+
+
+Adding G as a new axiom.
+
+
+On Formally Undecidable Propositions
+
+
+445
+
+
+
+This suggestion seems rather innocuous and perhaps even desirable, since, after all, G
+asserts something true about the natural number system. But what about the nonstandard
+type of extensions If it is at all parallel to the case of the parallel postulate, it must
+involve
+
+
+adding the negation of G as a new axiom.
+
+But how can we even contemplate doing such a repugnant, hideous thing? After all, to
+paraphrase the memorable words of Girolamo Saccheri, isn't what --G says "repugnant to
+the nature of the natural numbers'?
+
+Supernatural Numbers
+
+I hope the irony of this quotation strikes you. The exact problem with Saccheri's approach
+to geometry was that he began with a fixed notion of what was true and what was not
+true, and he set out only to prove what he'd assessed as true to start with. Despite the
+cleverness of his approach-which involved denying the fifth postulate, and then proving
+many "repugnant" propositions of the ensuing geometry-Saccheri never entertained the
+possibility of other ways of thinking about points and lines. Now we should be wary of
+repeating this famous mistake. We must consider impartially, to the extent that we can,
+what it would mean to add -G as an axiom to TNT. Just think what mathematics would
+be like today if people had never considered adding new axioms of the following sorts:
+
+3a:(a+a)=S0
+3a:Sa=0
+3a:(a»a)=SSO
+3a:S(a»a) =0
+
+While each of them is "repugnant to the nature of previously known number systems",
+each of them also provides a deep and wonderful extension of the notion of whole
+numbers: rational numbers, negative numbers, irrational numbers, imaginary numbers.
+Such a possibility is what -G is trying to get us to open our eyes to. Now in the past, each
+new extension of the notion of number was greeted with hoots and catcalls. You can hear
+this particularly loudly in the names attached to the unwelcome arrivals, such as
+"irrational numbers", "imaginary numbers". True to this tradition, we shall name the
+numbers which -'-G is announcing to us the supernatural numbers, showing how we feel
+they violate all reasonable and commonsensical notions.
+
+If we are going to throw -G in as the sixth axiom of TNT, we had better
+understand how in the world it could coexist, in one system, with the infinite pyramidal
+family we just finished discussing. To put it bluntly, -G says:
+
+“There exists some number which forms a TNT-proof-pair with the
+arithmoquinification of u”
+
+
+On Formally Undecidable Propositions
+
+
+446
+
+
+
+-but the various members of the pyramidal family successively assert:
+
+
+"0 is not that number"
+
+" 1 is not that number"
+
+"2 is not that number"
+
+This is rather confusing, because it seems to be a complete contradiction (which is why it
+is called "co-inconsistency"). At the root of our confusion-much as in the case of the
+splitting of geometry-is our stubborn resistance to adopt a modified interpretation for the
+symbols, despite the fact that we are quite aware that the system is a modified system.
+We want to get away without reinterpreting any symbols-and of course that will prove
+impossible.
+
+The reconciliation comes when we reinterpret 3 as "There exists a generalized
+natural number", rather than as "There exists a natural number". As we do this, we shall
+also reinterpret V in the corresponding way. This means that we are opening the door to
+some extra numbers besides the natural numbers. These are the supernatural numbers.
+The naturals and supematurals together make up the totality of generalized naturals.
+
+The apparent contradiction vanishes into thin air, now, for the pyramidal family
+still says what it said before: "No natural number forms a TNT-proof-pair with the
+arithmoquinification of u." The family doesn't say anything about supernatural numbers,
+because there are no numerals for them. But now, -G says, "There exists a generalized
+natural number which forms a TNT-proof-pair with the arithmoquinification of u." It is
+clear that taken together, the family and -G tell us something: that there is a supernatural
+number which forms a TNT-proof-pair with the arithmoquinification of u. That is all-
+there is no contradiction any more. TNT+-G is a consistent system, under an
+interpretation which includes supernatural numbers.
+
+Since we have now agreed to extend the interpretations of the two quantifiers, this
+means that any theorem which involves either of them has an extended meaning. For
+example, the commutativity theorem
+
+Va:da':(a+a')=(a'+a)
+
+now tells us that addition is commutative for all generalized natural numbers-in other
+words, not only for natural numbers, but also for supernatural numbers. Likewise, the
+TNT-theorem which says "2 is not the square of a natural number"—
+
+-3a:(a • a)=SSO
+
+—now tells us that 2 is not the square of a supernatural number, either. In fact,
+supernatural numbers share all the properties of natural numbers, as
+
+
+On Formally Undecidable Propositions
+
+
+447
+
+
+
+long as those properties are given to us in theorems of TNT. In other words, everything
+that can be formally proven about natural numbers is thereby established also for
+supernatural numbers. This means, in particular, that supernatural numbers are not
+anything already familiar to you, such as fractions, or negative numbers, or complex
+numbers, or whatever. The supernatural numbers are, instead, best visualized as integers
+which are greater than all natural numbers-as infinitely large integers. Here is the point:
+although theorems of TNT can rule out negative numbers, fractions, irrational numbers,
+and complex numbers, still there is no way to rule out infinitely large integers. The
+problem is, there is no way even to express the statement "There are no infinite
+quantities".
+
+This sounds quite strange, at first. Just exactly how big is the number which
+makes a TNT-proof-pair with G's Godel number= (Let's call it 7 . for no particular
+reason.) Unfortunately, we have not got any good vocabulary for describing the sizes of
+infinitely large integers, so I am afraid I cannot convey a sense of I's magnitude. But then
+just how big is i (the square root of -1)? Its size cannot be imagined in terms of the sizes
+of familiar natural numbers. You can't say, "Well, i is about half as big as 14, and 9/10 as
+big as 24." You have to say, "i squared is -1", and more or less leave it at that. A quote
+from Abraham Lincoln seems a propos here. When he was asked, "How long should a
+man's legs be?" he drawled, "Long enough to reach the ground." That is more or less how
+to answer the question about the size of I-it should be just the size of a number which
+specifies the structure of a proof of G-no bigger, no smaller.
+
+Of course, any theorem of TNT has many different derivations, so you might
+complain that my characterization of I is nonunique. That is so. But the parallel with 1-
+the square root of -1-still holds. Namely, recall that there is another number whose square
+is also minus one: -i. Now i and -i are not the same number. They just have a property in
+common. The only trouble is that it is the property which defines them! We have to
+choose one of them-it doesn't matter which one-and call it "i". In fact there's no way of
+telling them apart. So for all we know we could have been calling the wrong one "i" for
+all these centuries and it would have made no difference. Now, like i, I is also
+nonuniquely defined. So you just have to think of I as being some specific one of the
+many possible supernatural numbers which form TNT-proof-pairs with the
+arithmoquinification of u.
+
+Supernatural Theorems Have Infinitely Long Derivations.
+
+We haven't yet faced head on what it means to throw -G in as an axiom. We have said it
+but not stressed it. The point is that -G asserts that G has a proof. How can a system
+survive, when one of its axioms asserts that its own negation has a proof? We must be in
+hot water now! Well, it is not so bad as you might think. As long as we only construct
+finite proofs, we will never prove G Therefore, no calamitous collision between G and its
+negative ~G will ever take place. The supernatural number -I won’t cause any disaster.
+
+
+On Formally Undecidable Propositions
+
+
+448
+
+
+
+However, we will have to get used to the idea that —G is now the one which asserts a
+truth ("G has a proof"), while G asserts a falsity ("G has no proof'). In standard number
+theory it is the other way around-but then, in standard number theory there aren't any
+supernatural numbers. Notice that a supernatural theorem of TNT -namely G-may assert a
+falsity, but all natural theorems still assert truths.
+
+Supernatural Addition and Multiplication
+
+There is one extremely curious and unexpected fact about supematurals which I would
+like to tell you, without proof. (I don't know the proof either.) This fact is reminiscent of
+the Heisenberg uncertainty principle in quantum mechanics. It turns out that you can
+"index" the supematurals in a simple and natural way by associating with each
+supernatural number a trio of ordinary integers (including negative ones). Thus, our
+original supernatural number, 7, might have the index set (9,-8,3), and its successor, 7+1,
+might have the index set (9,-8,4). Now there is no unique way to index the supematurals;
+different methods offer different advantages and disadvantages. Under some indexing
+schemes, it is very easy to calculate the index triplet for the sum of two supernaturals,
+given the indices of the two numbers to be added. Under other indexing schemes, it is
+very easy to calculate the index triplet for the product of two supernaturals, given the
+indices of the two numbers to be multiplied. But under no indexing scheme is it possible
+to calculate both. More precisely, if the sum's index can be calculated by a recursive
+function, then the product's index will not be a recursive function; and conversely, if the
+product's index is a recursive function, then the sum's index will not be. Therefore,
+supernatural schoolchildren who learn their supernatural plus-tables will have to be
+excused if they do not know their supernatural times-tables-and vice versa! You cannot
+know both at the same time.
+
+
+Supernaturals Are Useful...
+
+One can go beyond the number theory of supematurals, and consider supernatural
+fractions (ratios of two supernaturals), supernatural real numbers, and so on. In fact, the
+calculus can be put on a new footing, using the notion of supernatural real numbers.
+Infinitesimals such as dx and dy, those old bugaboos of mathematicians, can be
+completely justified, by considering them to be reciprocals of infinitely large real
+numbers! Some theorems in advanced analysis can be proven more intuitively with the
+aid of "nonstandard analysis".
+
+
+But Are They Real?
+
+Nonstandard number theory is a disorienting thing when you first meet up with it. But,
+then, non-Euclidean geometry is also a disorienting subject. In
+
+
+On Formally Undecidable Propositions
+
+
+449
+
+
+
+both instances, one is powerfully driven to ask, "But which one of these two rival theories
+is correct? Which is the truthT In a certain sense, there is no answer to such a question.
+(And vet, in another sense-to be discussed later-there is an answer.) The reason that there
+is no answer to the question is that the two rival theories, although they employ the same
+terms, do not talk about the same concepts. Therefore, they are only superficially rivals,
+just like Euclidean and non-Euclidean geometries. In geometry, the words "point", "line",
+and so on are undefined terms, and their meanings are determined by the axiomatic
+system within which they are used.
+
+Likewise for number theory. When we decided to formalize TNT. we preselected
+the terms we would use as interpretation words-for instance, words such as "number",
+"plus", "times", and so on. By taking the step of formalization, we were committing
+ourselves to accepting whatever passive meanings these terms might take on. But just like
+Saccheri-we didn't anticipate any surprises. We thought we knew what the true, the real,
+the only theory of natural numbers was. We didn't know that there would be some
+questions about numbers which TNT would leave open, and which could therefore be
+answered ad libitum by extensions of TNT heading off in different directions. Thus, there
+is no basis on which to say that number theory "really" is this way or that, just as one
+would be loath to say that the square root of -1 "really" exists, or "really" does not.
+
+Bifurcations in Geometry, and Physicists
+
+There is one argument which can be, and perhaps ought to be, raised against the
+preceding. Suppose experiments in the real, physical world can be explained more
+economically in terms of one particular version of geometry than in terms of any other.
+Then it might make sense to say that that geometry is "true". From the point of view of a
+physicist who wants to use the "correct" geometry, then it makes some sense to
+distinguish between the "true" geometry, and other geometries. But this cannot be taken
+too simplistically. Physicists are always dealing with approximations and idealizations of
+situations. For instance, my own Ph.D. work, mentioned in Chapter V, was based on an
+extreme idealization of the problem of a crystal in a magnetic field. The mathematics
+which emerged was of a high degree of beauty and symmetry. Despite-or rather, because
+of-the artificiality of the model, some fundamental features emerged conspicuously in the
+graph. These features then suggest some guesses about the kinds of things that might
+happen in more realistic situations. But without the simplifying assumptions which
+produced my graph, there could never be such insights. One can see this kind of thing
+over and over again in physics, where a physicist uses a "nonreal" situation to learn about
+deeply hidden features of reality. Therefore, one should be extremely cautious in saying
+that the brand of geometry which physicists might wish to use would represent “the
+
+
+On Formally Undecidable Propositions
+
+
+450
+
+
+
+true geometry", for in fact, physicists will always use a variety of different geometries,
+choosing in any given situation the one that seems simplest and most convenient.
+
+Furthermore-and perhaps this is even more to the point-physicists do not study
+just the 3-D space we live in. There are whole families of "abstract spaces" within which
+physical calculations take place, spaces which have totally different geometrical
+properties from the physical space within which we live. Who is to say, then, that "the
+true geometry" is defined by the space in which Uranus and Neptune orbit around the
+sun? There is "Hilbert space", where quantum-mechanical wave functions undulate; there
+is "momentum space", where Fourier components dwell; there is "reciprocal space",
+where wave-vectors cavort; there is "phase space", where many-particle configurations
+swish; and so on. There is absolutely no reason that the geometries of all these spaces
+should be the same; in fact, they couldn't possibly be the same! So it is essential and vital
+for physicists that different and "rival" geometries should exist.
+
+Bifurcations in Number Theory, and Bankers
+
+So much for geometry. What about number theory? Is it also essential and vital that
+different number theories should coexist with each other? If you asked a bank officer, my
+guess is that you would get an expression of horror and disbelief. How could 2 and 2 add
+up to anything but 4? And moreover, if 2 and 2 did not make 4, wouldn't world
+economies collapse immediately under the unbearable uncertainty opened up by that
+fact? Not really. First of all, nonstandard number theory doesn't threaten the age-old idea
+that 2 plus 2 equals 4. It differs from ordinary number theory only in the way it deals with
+the concept of the infinite. After all, every theorem of TNT remains a theorem in any
+extension of TNT\ So bankers need not despair of the chaos that will arrive when
+nonstandard number theory takes over.
+
+And anyway, entertaining fears about old facts being changed betrays a
+misunderstanding of the relationship between mathematics and the real world.
+Mathematics only tells you answers to questions in the real world after you have taken
+the one vital step of choosing which kind of mathematics to apply. Even if there were a
+rival number theory which used the symbols '2', '3', and '+', and in which a theorem said
+"2 + 2 = 3", there would be little reason for bankers to choose to use that theory! For that
+theory does not fit the way money works. You fit your mathematics to the world, and not
+the other way around. For instance, we don't apply number theory to cloud systems,
+because the very concept of whole numbers hardly fits. There can be one cloud and
+another cloud, and they will come together and instead of there being two clouds, there
+will still only be one. This doesn't prove that 1 plus 1 equals 1; it just proves that our
+number theoretical concept of “one” is not applicable in its full power to cloud counting.
+
+
+On Formally Undecidable Propositions
+
+
+451
+
+
+
+Bifurcations in Number Theory, and Metamathematicians
+
+
+So bankers, cloud-counters, and most of the rest of us need not worry ,about the advent of
+supernatural numbers: they won't affect our everyday perception of the world in the
+slightest. The only people who might actually be a little worried are people whose
+endeavors depend in some crucial way on the nature of infinite entities. There aren't too
+many such people around-but mathematical logicians are members of this category. How
+can the existence of a bifurcation in number theory affect them Well, number theory
+plays two roles in logic: (1) when axiomatized, it is an object of study; and (2) when used
+informally, it is an indispensable tool with which formal systems can be investigated.
+This is the use-mention distinction once again, in fact: in role (1), number theory is
+mentioned, in role (2) it is used.
+
+Now mathematicians have judged that number theory is applicable to the study of
+formal systems even if not to cloud-counting, just as bankers have judged that the
+arithmetic of real numbers is applicable to their transactions. This is an
+extramathematical judgement, and shows that the thought processes involved in doing
+mathematics, just like those in other areas, involve "tangled hierarchies" in which
+thoughts on one level can affect thoughts on any other level. Levels are not cleanly
+separated, as the formalist version of what mathematics is would have one believe.
+
+The formalist philosophy claims that mathematicians only deal with abstract
+symbols, and that they couldn't care less whether those symbols have any applications to
+or connections with reality. But that is quite a distorted picture. Nowhere is this clearer
+than in metamathematics. If the theory of numbers is itself used as an aid in gaining
+factual knowledge about formal systems, then mathematicians are tacitly showing that
+they believe these ethereal things called "natural numbers" are actually part of reality not
+just figments of the imagination. This is why I parenthetically remarked earlier that, in a
+certain sense, there is an answer to the question of which version of number theory is
+"true". Here is the nub of the matter: mathematical logicians must choose which version
+of number theory to put their faith in. In particular, they cannot remain neutral on the
+question of the existence or nonexistence of supernatural numbers, for the two different
+theories may give different answers to questions in metamathematics.
+
+For instance, take this question: "Is -G finitely derivable in TNT?" No one
+actually knows the answer. Nevertheless, most mathematical logicians would answer no
+without hesitation. The intuition which motivates that answer is based on the fact that if -
+G were a theorem, TNT would be w-inconsistent, and this would force supematurals
+down your throat if you wanted to interpret TNT meaningfully-a most unpalatable
+thought for most people. After all, we didn't intend or expect supematurals to be part of
+TNT when we invented it. That is, we-or most of us-believe that it is possible to make a
+formalization of number theory which does not force you into believing that supernatural
+numbers are every bit as real as naturals. It is that intuition about reality which
+determines which “fork” of number theory mathematicians will put their faith in, when
+the chips are
+
+
+On Formally Undecidable Propositions
+
+
+452
+
+
+
+down. But this faith may be wrong. Perhaps every consistent formalization of number
+theory which humans invent will imply the existence of supernaturals, by being co-
+inconsistent. This is a queer thought, but it is conceivable.
+
+If this were the case-which I doubt, but there is no disproof available-then G
+would not have to be undecidable. In fact, there might be no undecidable formulas of
+TNT at all. There could simply be one unbifurcated theory of numbers-which necessarily
+includes supernaturals. This is not the kind of thing mathematical logicians expect, but it
+is something which ought not to be rejected outright. Generally, mathematical logicians
+believe that TNT-and systems similar to it-are co-consistent, and that the Godel string
+which can be constructed in any such system is undecidable within that system. That
+means that they can choose to add either it or its negation as an axiom.
+
+Hilbert's Tenth Problem and the Tortoise
+
+I would like to conclude this Chapter by mentioning one extension of Godel’s
+Theorem. (This material is more fully covered in the article "Hilbert's Tenth Problem" by
+Davis and Hersh, for which see the Bibliography.) For this, I must define what a
+Diophantine equation is. This is an equation in which a polynomial with fixed integral
+coefficients and exponents is set to 0. For instance,
+
+
+a —0
+
+
+and
+
+5jc+13v-1=0
+
+And
+
+5p5 + \lq\l - 177-0
+and
+
+123,666,111,666 + ^123,-666,111,666 c 123,666,111,666 _ q
+
+are Diophantine equations. It is in general a difficult matter to know whether a given
+Diophantine equation has any integer solutions or not. In fact, in a famous lecture at the
+beginning of the century, Hilbert asked mathematicians to look for a general algorithm by
+which one could determine in a finite number of steps if a given Diophantine equation
+has integer solutions or not. Little did he suspect that no such algorithm exists!
+
+
+On Formally Undecidable Propositions
+
+
+453
+
+
+
+Now for the simplification of G. It has been shown that whenever you have a sufficiently
+powerful formal number theory and a Godel-numbering for it, there is a Diophantine
+equation which is equivalent to G. The equivalence lies in the fact that this equation,
+when interpreted on a metamathematical level, asserts of itself that it has no solutions.
+Turn it around: if you found a solution to it, you could construct from it the Godel
+number of a proof in the system that the equation has no solutions! This is what the
+Tortoise did in the Prelude, using Fermat's equation as his Diophantine equation. It is
+nice to know that when you do this, you can retrieve the sound of Old Bach from the
+molecules in the air!
+
+
+On Formally Undecidable Propositions
+
+
+454
+
+
+
+Birthday Cantatatata
+
+
+One (tine May day, the Tortoise and Achilles meet, wandering in the woods.
+
+The latter, all decked out handsomely, is doing a jiggish sort of thing to a
+tune which he himself is humming. On his vest he is wearing a great big
+button with the words "Today is my Birthday!"
+
+Tortoise: Hello there, .Achilles. What makes you so joyful today? Is it your birthday, by
+any chance?
+
+Achilles: Yes, yes! Yes it is, today is my birthday!
+
+Tortoise: That is what I had suspected, on account of that button which you are wearing,
+and also because unless I am mistaken, you are singing a tune from a Birthday
+Cantata by Bach, one written in 1727 for the fifty-seventh birthday of Augustus, King
+of Saxony.
+
+Achilles: You're right. And Augustus' birthday coincides with mine, so THIS Birthday
+Cantata has double meaning. However, I shan't tell you my age.
+
+Tortoise: Oh, that's perfectly all right. However, I would like to know one other thing.
+From what you have told me so far, would it be correct to conclude that today is your
+birthday?
+
+Achilles: Yes, yes, it would be. Today is my birthday.
+
+Tortoise: Excellent. That's just as I suspected. So now, I WILL conclude it is your
+birthday, unless
+
+Achilles: Yes-unless what?
+
+Tortoise: Unless that would be a premature or hasty conclusion to draw, you know.
+Tortoises don't like to jump to conclusions, after all. (We don't like to jump at all, but
+especially not to conclusions.) So let me just ask you, knowing full well of your
+fondness for logical thought, whether it would be reasonable to deduce logically from
+the foregoing sentences, that today is in fact your birthday.
+
+Achilles: I do believe I detect a pattern to your questions, Mr. T. But rather than jump to
+conclusions myself, I shall take your question at face value, and answer it
+straightforwardly. The answer is: YES.
+
+Tortoise: Fine! Fine! Then there is only one more thing I need to know, to be quite
+certain that today is
+
+Achilles: Yes, yes, yes, yes ... I can already see the line of your questioning, Mr. T. I'll
+have you know that I am not so gullible as I was when we discussed Euclid's proof, a
+while back.
+
+Tortoise: Why, who would ever have thought you to be gullible? Quite to the contrary, I
+regard you as an expert in the forms of logical thought, an authority in the science of
+valid deductions, a fount of knowledge about certain correct methods of reasoning. . .
+To tell the truth, Achilles, you are, in my opinion, a veritable titan in the art of rational
+cogitation.
+
+
+Birthday Cantatatata . .
+
+
+461
+
+
+
+And it is only for that reason that I would ask you, "Do the foregoing sentences present
+enough evidence that I should conclude without further puzzlement that today is your
+birthday
+
+Achilles: You flatten me with your weighty praise, Mr. T-FLATTER, I mean. But I am
+struck by the repetitive nature of your questioning and in my estimation, you, just as
+well as I, could have answered 'yes' each time.
+
+Tortoise: Of course I could have, Achilles. But you see, to do so would have been to
+make a Wild Guess-and Tortoises abhor Wild Guesses. Tortoises formulate only
+Educated Guesses. Ah, yes-the power of the Educated Guess. You have no idea how
+many people fail to take into account all the Relevant Factors when they're guessing.
+
+Achilles: It seems to me that there was only one relevant factor in this rigmarole, and that
+was my first statement.
+
+Tortoise: Oh, to be sure, it's at least ONE of the factors to take into account, I'd say-but
+would you have me neglect Logic, that venerated science of the ancients? Logic is
+always a Relevant Factor in making Educated Guesses, and since I have with me a
+renowned expert in Logic, I thought it only Logical to take advantage of that fact, and
+confirm my hunches, by directly asking him whether my intuitions were correct. So
+let me finally come out and ask you point blank: "Do the preceding sentences allow
+me to conclude, with no room for doubt, that Today is your Birthday?"
+
+Achilles: For one more time, YES. But frankly speaking, I have the distinct impression
+that you could have supplied that answer-as well as all the previous ones-yourself.
+
+Tortoise: How your words sting! Would I were so wise as your insinuation suggests! But
+as merely a mortal Tortoise, profoundly ignorant and longing to take into account all
+the Relevant Factors, I needed to know the answers to all those questions.
+
+Achilles: Well then, let me clear the matter up for once and for all. The answer to all the
+previous questions, and to all the succeeding ones which you will ask along the same
+line, is just this: YES.
+
+Tortoise: Wonderful! In one fell swoop, you have circumvented the whole mess, in your
+characteristically inventive manner. I hope you won't mind if I call this ingenious
+trick an ANSWER SCHEMA. It rolls up yes-answers numbers 1, 2, 3, etc., into one
+single ball. In fact, coming as it does at the end of the line, it deserves the title
+"Answer Schema Omega", 'w' being the last letter of the Greek alphabet-as if YOU
+needed to be told THAT!
+
+Achilles: I don't care what you call it. I am just very relieved that you finally agree that it
+is my birthday, and we can go on to some other topic-such as what you are going to
+give me as a present.
+
+Tortoise: Hold on—not so fast. I WILL agree it is your birthday, provided on thing
+
+Achilles: What? That I Ask for no present?
+
+
+Birthday Cantatatata . .
+
+
+462
+
+
+
+Tortoise: Not at all. In fact, Achilles, I am looking forward to treating you to a fine
+
+birthday dinner, provided merely that I am convinced that knowledge of all those yes-
+answers at once (as supplied by Answer Schema w) allows me to proceed directly and
+without any further detours to the conclusion that today is your birthday. That's the
+case, isn't it?
+
+Achilles: Yes, of course it is.
+
+Tortoise: Good. And now I have yes-answer co + 1. Armed with it, I can proceed to
+accept the hypothesis that today is your birthday, if it is valid to do so. Would you be
+so kind as to counsel me on that matter, Achilles?
+
+Achilles: What is this? I thought I had seen through your infinite plot. Now doesn't yes-
+answer co + 1 satisfy you? All right. I'll give you not only yes-answer co + 2, but also
+yes-answers co + 3, co + 4, and so on.
+
+Tortoise: How generous of you, Achilles. And here it is your birthday, when I should be
+giving YOU presents instead of the reverse. Or rather, I SUSPECT it is your birthday.
+I guess I can conclude that it IS your birthday, now, armed with the new Answer
+Schema, which I will call "Answer Schema 2co ". But tell me, Achilles: Does Answer
+Schema 2co REALLY allow me to make that enormous leap, or am I missing
+something?
+
+Achilles: You won't trick me any more, Mr. T. I've seen the way to end this silly game. I
+hereby shall present you with an Answer Schema to end all Answer Schemas! That is,
+I present you simultaneously with Answer Schemas co, 2 co, 3 co, 4 co, 5 co, etc. With
+this Meta-Answer-Schema, I have JUMPED OUT of the whole system, kit and
+caboodle, transcended this silly game you thought you had me trapped in-and now we
+are DONE!
+
+Tortoise: Good grief! I feel honored, Achilles, to be the recipient of such a powerful
+Answer Schema. I feel that seldom has anything so gigantic been devised by the mind
+of man, and I am awestruck by its power. Would you mind if I give a name to your
+gift?
+
+Achilles: Not at all.
+
+Tortoise: Then I shall call it "Answer Schema co"". And we can shortly proceed to other
+matters-as soon as you tell me whether the possession of Answer Schema co 2 allows
+me to deduce that today is your birthday.
+
+Achilles: Oh, woe is me! Can't I ever reach the end of this tantalizing trail? What comes
+next?
+
+Tortoise: Well, after Answer Schema co 2 there's answer co 2 + 1. And then answer co 2 + 2.
+And so forth. But you can wrap those all together into a packet, being Answer
+Schema co 2 + co. And then there are quite a few other answer-packets, such as co 2 + 2co,
+and co 2 + 3co.... Eventually you come to Answer Schema 2co 2, and after a while.
+Answer Schemas 3co 2 and 4co 2 . Beyond them there
+
+
+Birthday Cantatatata. .
+
+
+463
+
+
+
+are yet further Answer Schemas, such as co 3 ;, co 4 , or, and so on. It goes on quite a
+ways, you know.
+
+Achilles: I can imagine, I suppose it comes to Answer Schema are yet further Answer
+Schemas, such as w;, w4, w5, and so on. It goes on quite a ways, you know.
+
+Achilles: I can imagine, I suppose it comes to Answer Schema co“ after a while.
+
+Tortoise: Of course.
+
+Achilles: And then co““, and co“““',
+
+Tortoise: You're catching on mighty fast, Achilles. I have a suggestion for you, if you
+don't mind. Why don't you throw all of those together into a single Answer Schema?
+
+Achilles: All right, though I'm beginning to doubt whether it will do any good.
+
+Tortoise: It seems to me that within our naming conventions as so far set up, there is no
+obvious name for this one. So perhaps we should just arbitrarily name it Answer
+Schema (Eo.
+
+Achilles: Confound it all! Every time you give one of my answers a NAME, it seems to
+signal the imminent shattering of my hopes that that answer will satisfy you. Why
+don't we just leave this Answer Schema nameless?
+
+Tortoise: We can hardly do that, Achilles. We wouldn't have any way to refer to it
+without a name. And besides, there is something inevitable and rather beautiful about
+this particular Answer Schema. It would be quite ungraceful to leave it nameless! And
+you wouldn't want to do something lacking in grace on your birthday, would you? Or
+is it your birthday? Say, speaking of birthdays, today is MY' birthday!
+
+Achilles: It is?
+
+Tortoise: Yes, it is. Well, actually, it's my uncle's birthday, but that's almost the same.
+How would you like to treat me to a delicious birthday dinner this evening?
+
+Achilles: Now just a cotton-picking minute, Mr. T. Today is MY birthday. You should do
+the treating!
+
+Tortoise: Ah, but you never did succeed in convincing me of the veracity of that remark.
+You kept on beating around the bush with answers, Answer Schemas, and whatnot.
+All I wanted to know was if it was your birthday or not, but you managed to befuddle
+me entirely. Oh, well, too bad. In any case. I'll be happy to let you treat me to a
+birthday dinner this evening.
+
+Achilles: Very well. I know just the place. They have a variety of delicious soups. And I
+know exactly what kind we should have ...
+
+
+Birthday Cantatatata .
+
+
+464
+
+
+
+CHAPTER XV
+
+
+Jumping out of the System
+
+A More Powerful Formal System
+
+ONE OF THE things which a thoughtful critic of GodeFs proof might do would be to
+examine its generality. Such a critic might, for example, suspect that Godel has just
+cleverly taken advantage of a hidden defect in one particular formal system, TNT. If this
+were the case, then perhaps a formal system superior to TNT could be developed which
+would not be subject to the Godelian trick, and GodeFs Theorem would lose much of its
+sting. In this Chapter we will carefully scrutinize the properties of TNT which made it
+vulnerable to the arguments of last Chapter.
+
+A natural thought is this: If the basic trouble with TNT is that it contains a "hole"-
+in other words, a sentence which is undecidable, namely G-then why not simply plug up
+the hole? Why not just tack G onto TNT as a sixth axiom? Of course, by comparison to
+the other axioms, G is a ridiculously huge giant, and the resulting system-TNT+G-would
+have a rather comical aspect due to the disproportionateness of its axioms. Be that as it
+may, adding G is a reasonable suggestion. Let us consider it done. Now, it is to be hoped,
+the new system, TNT+G, is a superior formal system-one which is not only supernatural-
+free, but also complete. It is certain that TNT+G is superior to TNT in at least one
+respect: the string G is no longer undecidable in this new system, since it is a theorem.
+
+What was the vulnerability of TNT due to? The essence of its vulnerability was
+that it was capable of expressing statements about itself-in particular, the statement
+
+"I Cannot Be Proven in Formal System TNT"
+
+or, expanded a bit,
+
+"There does not exist a natural number which forms a TNT-proof-pair with
+the Godel number of this string."
+
+Is there any reason to expect or hope that TNT+G would be invulnerable to GodeFs
+proof? Not really. Our new system is just as expressive as TNT. Since GodeFs proof
+relies primarily on the expressive power of a formal system, we should not be surprised
+to see our new system succumb,
+
+
+Jumping out of the System
+
+
+465
+
+
+
+too. The trick will be to find a string which expresses the statement
+
+"I Cannot Be Proven in Formal System TNT+G."
+
+Actually, it is not much of a trick, once you have seen it done for TNT. All the same
+principles are employed: only, the context shifts slightly. (Figuratively speaking, we take
+a tune we know and simply sing it again, only in a higher key.) As before, the string
+which we are looking for-let us call it "G"'-is constructed by the intermediary of an
+"uncle", But instead of being based on the formula which represents TNT-proof-pairs, it
+is based on the similar but slightly more complicated notion of TNT+G-proofpairs. This
+notion of TNT+G-proof-pairs is only a slight extension of the original notion of TNT-
+proof-pairs.
+
+A similar extension could be envisaged for the MlU-system. We have seen the
+unadulterated form of MlU-proof-pairs. Were we now to add MU as a second axiom, we
+would be dealing with a new system-the MIU+MU system. A derivation in this extended
+system is presented:
+
+
+MU axiom
+
+MUU rule 2
+
+There is a MIU+MU-proof-pair which corresponds-namely, m = 30300, n = 300. Of
+course, this pair of numbers does not form a MlU-proof-pair-only a MIU+MU-proof-
+pair. The addition of an extra axiom does not substantially complicate the arithmetical
+properties of proof-pairs. The significant fact about them-that being a proof-pair is
+primitive recursive-is preserved.
+
+The Godel Method Reapplied
+
+Vow, returning to TNT+G, we will find a similar situation. TNT+G proof-pairs, like
+their predecessors, are primitive recursive, so they are represented inside TNT+G by a
+formula which we abbreviate in an obvious manner.
+
+(TNT+G)-PROOF-PAIR{a,a'}
+
+Vow we just do everything all over again. We make the counterpart of G by beginning
+with an "uncle", just as before:
+
+3a:3a':<(TNT+G)-PROOF-PAIR{a,a'}ARITHMOQUINE{a",a'}>
+
+.et us say its Godel-number is u'. Now we arithmoquine this very uncle. That will give us
+G':
+
+
+3a:3a’: < (TNT+G)-PROOF-PAIR{a,a’}
+ARITHMOQUINE {SSS....SSSo/a ",a'}>
+
+U' S's
+
+
+Jumping out of the System
+
+
+466
+
+
+
+Its interpretation is
+
+
+More concisely.
+
+
+"There is no number a that forms a TNT +G-proof-pair
+with the arithmoquinification of w'."
+
+
+"I Cannot Be Proven in Formal System TNT+G."
+
+Multifurcation
+
+Well (yawn), the details are quite boring from here on out. G’ is precisely to TNT+G as
+G was to TNT Itself. One finds that either G' or -G’ can be added to TNT+G, to yield a
+further splitting of number theory. And, lest you think this only happens to the "good
+guys", this very same dastardly trick can be played upon TNT+—G-that is, upon the
+nonstandard extension of TNT gotten by adding G’s negation. So now we see (Fig. 75)
+that there are all sorts of bifurcations in number theory:
+
+
+
+FIGURE 75. "Multif urccitioif of TNT. Each extension of TNT has its very own Gtdel
+sentence; that sentence, or its negation, can be added on, so that from each extension
+there sprouts a pair of further extensions, a process which goes on ad infinitum.
+
+Of course, this is just the beginning. Let us imagine moving down the leftmost branch of
+this downwards-pointing tree, where we always toss in the Godel sentences (rather than
+their negations). This is the best we can do by way of eliminating supernaturals. After
+adding G, we add G'. Then we add G", and G m , and so on. Each time we make a new
+extension of TNT, its vulnerability to the Tortoise's method-pardon me, I mean Godel’s
+method., allows a new string to be devised, having the interpretation.
+
+“I cannot be proven in formal system X”
+
+
+Jumping out of the System
+
+
+467
+
+
+Naturally, after a while, the whole process begins to seem utterly predictable and routine.
+Why, all the "holes" are made by one single technique! This means that, viewed as
+typographical objects, they are all cast from one single mold, which in turn means that
+one single axiom schema suffices to represent all of them! So if this is so, why not plug
+up all :he holes at once and be done with this nasty business of incompleteness 3nce and
+for all? This would be accomplished by adding an axiom schema to TNT, instead of just
+one axiom at a time. Specifically, this axiom schema would be the mold in which all of
+G, G', G", G'", etc., are cast. By adding :his axiom schema (let's call it "G~"), we would
+be outsmarting the "Godelization" method. Indeed, it seems quite clear that adding G. to
+TNT would :)e the last step necessary for the complete axiomatization of all of number-
+theoretical truth.
+
+It was at about this point in the Contracrostipunctus that the Tortoise related the
+Crab's invention of "Record Player Omega". However, readers were left dangling as to
+the fate of that device, since before completing his tale, the tuckered-out Tortoise decided
+that he had best go home to sleep; but not before tossing off a sly reference to Godel’s
+Incompleteness Theorem). Now, at last, we can get around to clearing up that dangling
+detail... Perhaps you already have an inkling, after reading the Birthday Cantatatata.
+
+Essential Incompleteness
+
+As you probably suspected, even this fantastic advance over TNT suffers the same fate.
+And what makes it quite weird is that it is still for, in essence, the same reason. The
+axiom schema is not powerful enough, and the Godel construction can again be effected.
+Let me spell this out a little. (One can do it much more rigorously than I shall here.) If
+there is a way of capturing the various strings G, G', G", G'" . . in a single typographical
+mold, then there is a way of describing their Godel numbers in a single arithmetical mold.
+And this arithmetical portrayal of an infinite class of numbers can then be represented
+inside TNT+G. by some formula OMEGA-AXIOM {a} whose interpretation is: "a is the
+Godel number of one of the axioms coming from G.". When a is replaced by any specific
+numeral, the formula which results will be a theorem of TNT+G. if and only if the
+numeral stands for the Godel number of an axiom coming from the schema.
+
+With the aid of this new formula, it becomes possible to represent even such a
+complicated notion as TNT+G. -proof-pairs inside TNT+Gto:
+
+(TNT+G.)- PROOF- PAIR{a, a')
+
+sing this formula, we can construct a new uncle, which we proceed to Arithmoquine in
+the by now thoroughly familiar way, making yet another undecidable string, which will
+be called "TNT+Gto+i". At this point, you might well wonder, "Why isn't Gco+i among
+the axioms created by the axiom schema Gto?” The answer is that G was not clever
+enough to foresee its own embeddability inside number theory.
+
+
+Jumping out of the System
+
+
+468
+
+
+
+In the Contracrostipunctus, one of the essential steps in the Tortoise's making an
+"unplayable record" was to get a hold of a manufacturer's blueprint of the record player
+which he was out to destroy. This was necessary so that he could figure out to what kinds
+of vibrations it was vulnerable, and then incorporate into his record such grooves as
+would code for sounds which would induce those vibrations. It is a close analogue to the
+Godel trick, in which the system's own properties are reflected inside the notion of proof-
+pairs, and then used against it. Any system, no matter how complex or tricky it is, can be
+Godel-numbered, and then the notion of its proof-pairs can be defined-and this is the
+petard by which it is hoist. Once a system is well-defined, or "boxed", it becomes
+vulnerable.
+
+This principle is excellently illustrated by the Cantor diagonal trick, which finds
+an omitted real number for each well-defined list of reals between 0 and 1. It is the act of
+giving an explicit list-a "box" of reals which causes the downfall. Let us see how the
+Cantor trick can be repeated over and over again. Consider what happens if, starting with
+some list L, you do the following:
+
+(la) Take list L, and construct its diagonal number d.
+
+(lb) Throw d somewhere into list L, making a new list L+d.
+
+(2a) Take list L +d, and construct its diagonal number d'.
+
+(2b) Throw d' somewhere into list L+d, making a new list L+d+d'.
+
+Now this step-by-step process may seem a doltish way to patch up L, for we could have
+made the entire list d, d', d", d”\ ... at once, given L originally. But if you think that
+making such a list will enable you to complete your list of reals, you are very wrong. The
+problem comes at the moment you ask, "Where to incorporate the list of diagonal
+numbers inside L?" No matter how diabolically clever a scheme you devise for
+ensconcing the d-numbers inside L, once you have done it, then the new list is still
+vulnerable. As was said above, it is the act of giving an explicit list-a "box" of reals-that
+causes the downfall.
+
+Now in the case of formal systems, it is the act of giving an explicit recipe for
+what supposedly characterizes number-theoretical truth that causes the incompleteness.
+This is the crux of the problem with TNT+Gco,. Once you insert all the G's in a well-
+defined way into TNT, there is seen to be some other G-some unforeseen G-which you
+didn't capture in your axiom schema. And in the case of the TC-battle inside the
+ContracrostiPunctus, the instant a record player's "architecture" is determined, the record
+player becomes capable of being shaken to pieces.
+
+So what is to be done? There is no end in sight. It appears that TNT, even when
+extended ad infinitum, cannot be made complete. TNT is therefore said to suffer from
+essential incompleteness because the income-
+
+
+jumping out of the System
+
+
+469
+
+
+
+pleteness here is part and parcel of TNT ; it is an essential part of the nature of TNT and
+cannot be eradicated in any way, whether simpleminded or ingenious. What's more, this
+problem will haunt any formal version of number theory, whether it is an extension of
+TNT, a modification of TNT, or an alternative to TNT. The fact of the matter is this: the
+possibility of constructing, in a given system, an undecidable string via Godel’s self¬
+reference method, depends on three basic conditions:
+
+(1) That the system should be rich enough so that all desired statements about
+numbers, whether true or false, can be expressed in it. (Failure on this count
+means that the system is from the very start too weak to be counted as a rival to
+TNT, because it can't even express number-theoretical notions that TNT can.
+
+In the metaphor of the Contracrosttpunctus, it is as if one did not have a
+phonograph but a refrigerator or some other kind of object.)
+
+(2) That all general recursive relations should be represented by formulas in the
+system. (Failure on this count means the system fails to capture in a theorem
+some general recursive truth, which can only be considered a pathetic bellyflop
+if it is attempting to produce all of number theory's truths. In the
+Contracrostipunctus metaphor, this is like having a record player, but one of
+low fidelity.)
+
+(3) That the axioms and typographical patterns defined by its rules be recognizable
+by some terminating decision procedure. (Failure on this count means that there
+is no method to distinguish valid derivations in the system from invalid ones-
+thus that the "formal system" is not formal after all, and in fact is not even well-
+defined. In the Contracrostipunctus metaphor, it is a phonograph which is still
+on the drawing board, only partially designed.)
+
+Satisfaction of these three conditions guarantees that any consistent system will be
+incomplete, because Godel’s construction is applicable.
+
+The fascinating thing is that any such system digs its own hole; the system's own
+richness brings about its own downfall. The downfall occurs essentially because the
+system is powerful enough to have self-referential sentences. In physics, the notion exists
+of a "critical mass" of a fissionable substance, such as uranium. A solid lump of the
+substance will just sit there, if its mass is less than critical. But beyond the critical mass,
+such a lump will undergo a chain reaction, and blow up. It seems that with formal
+systems there is an analogous critical point. Below that point, a system is "harmless" and
+does not even approach defining arithmetical truth formally; but beyond the critical point,
+the system suddenly attains the capacity for self-reference, and thereby dooms itself to
+incompleteness. The threshold seems to be roughly when a system attains the three
+properties listed above.
+
+
+Jumping out of the System
+
+
+470
+
+
+
+Once this ability for self-reference is attained, the system has a hole which is tailor-made
+for itself; the hole takes the features of the system into account and uses them against the
+system.
+
+
+The Passion According to Lucas
+
+The baffling repeatability of the Godel argument has been used by various people-notably
+J. R. Lucas-as ammunition in the battle to show that there is some elusive and ineffable
+quality to human intelligence, which makes it unattainable by "mechanical automata"-that
+is, computers. Lucas begins his article "Minds, Machines, and Godel" with these words:
+
+Godel’s theorem seems to me to prove that Mechanism is false, that is, that minds
+
+cannot be explained as machines.'
+
+Then he proceeds to give an argument which, paraphrased, runs like this. For a
+computer to be considered as intelligent as a person is, it must be able to do every
+intellectual task which a person can do. Now Lucas claims that no computer can do
+"Godelization" (one of his amusingly irreverent terms) in the manner that people can.
+Why not? Well, think of any particular formal system, such as TNT, or TNT+G, or even
+TNT+G.. One can write a computer program rather easily which will systematically
+generate theorems of that system, and in such a manner that eventually, any preselected
+theorem will be printed out. That is, the theorem-generating program won't skip any
+portion of the "space" of all theorems. Such a program would be composed of two major
+parts: (1) a subroutine which stamps out axioms, given the "molds" of the axiom schemas
+(if there are any), and (2) a subroutine which takes known theorems (including axioms, of
+course) and applies rules of inference to produce new theorems. The program would
+alternate between running first one of these subroutines, and then the other.
+
+We can anthropomorphically say that this program "knows" some facts of number
+theory-namely, it knows those facts which it prints out. If it fails to print out some true
+fact of number theory, then of course it doesn't "know" that fact. Therefore, a computer
+program will be inferior to human beings if it can be shown that humans know something
+which the program cannot know. Now here is where Lucas starts rolling. He says that we
+humans can always do the Godel trick on any formal system as powerful as TNT-and
+hence no matter what the formal system, we know more than it does. Now this may only
+sound like an argument about formal systems, but it can also be slightly modified so that
+it becomes, seemingly, an invincible argument against the possibility of Artificial
+Intelligence ever reproducing the human level of intelligence. Here is the gist of it:
+
+Rigid internal codes entirely rule computers and robots; ergo ...
+
+Computers are isomorphic to formal systems. Now .. .
+
+Any computer which wants to be as smart as we are has got to be able to do
+
+number theory as well as we can, so....
+
+
+Jumping out of the System
+
+
+471
+
+
+
+Among other things, it has to be able to do primitive recursive arithmetic. But for
+this very reason .. .
+
+It is vulnerable to the Godelian "hook", which implies that...
+
+We, with our human intelligence, can concoct a certain statement of number
+theory which is true, but the computer is blind to that statement's truth (i.e., will
+never print it out), precisely because of Godel’s boomeranging argument.
+
+This implies that there is one thing which computers just cannot be programmed
+to do, but which we can do. So we are smarter.
+
+Let us enjoy, with Lucas, a transient moment of anthropocentric glory:
+
+However complicated a machine we construct, it will, if it is a machine, correspond
+to a formal system, which in turn will be liable to the Godel procedure for finding a
+formula unprovable-in-that-system. This formula the machine will be unable to
+produce as being true, although a mind can see it is true. And so the machine will
+still not be an adequate model of the mind. We are trying to produce a model of the
+mind which is mechanical-which is essentially "dead"-but the mind, being in fact
+"alive," can always go one better than any formal, ossified, dead system can. Thanks
+to Godel’s theorem, the mind always has the last word. 2
+
+On first sight, and perhaps even on careful analysis, Lucas' argument appears
+compelling. It usually evokes rather polarized reactions. Some ;eize onto it as a nearly
+religious proof of the existence of souls, while others laugh it off as being unworthy of
+comment. I feel it is wrong, but Fascinatingly so-and therefore quite worthwhile taking
+the time to rebut. In fact, it was one of the major early forces driving me to think over the
+matters in this book. I shall try to rebut it in one way in this Chapter, and in ether ways in
+Chapter XVII.
+
+We must try to understand more deeply why Lucas says the computer cannot be
+programmed to "know" as much as we do. Basically the idea is :hat we are always
+outside the system, and from out there we can always perform the "Godelizing"
+operation, which yields something which the program, from within, can't see is true. But
+why can't the "Godelizing operator", as Lucas calls it, be programmed and added to the
+program as a third major component, Lucas explains:
+
+The procedure whereby the Godelian formula is constructed is a standard
+procedure-only so could we be sure that a Godelian formula can be constructed for
+every formal system. But if it is a standard procedure, then a machine should be
+able to be programmed to carry it out too.... This would correspond to having a
+system with an additional rule of inference which allowed one to add, as a theorem,
+the Godelian formula of the rest of the formal system, and then the Godelian
+formula of this new, strengthened, formal system, and so on. It would be
+tantamount to adding to the original formal system an infinite sequence of axioms,
+each the Godelian formula of the system hitherto obtained... We might expect a
+mind, faced with a machine that possessed a Godelizing operator, to take this into
+account, and
+
+
+Jumping out of the System
+
+
+472
+
+
+
+out-Godel the new machine, Godelizing operator and all. This has, in fact, proved
+to be the case. Even if we adjoin to a formal system the infinite set of axioms
+consisting of the successive Godelian formulae, the resulting system is still
+incomplete, and contains a formula which cannot be proved-in-the system, although
+a rational being can, standing outside the system, see that it is true. We had
+expected this, for even if an infinite set of axioms were added, they would have to
+be specified by some finite rule or specification, and this further rule or
+specification could then be taken into account by a mind considering the enlarged
+formal system. In a sense, just because the mind has the last word, it can always
+pick a hole in any formal system presented to it as a model of its own workings.
+The mechanical model must be, in some sense, finite and definite: and then the
+mind can always go one better.'
+
+Jumping Up a Dimension
+
+A visual image provided by M. C. Escher is extremely useful in aiding the intuition here:
+his drawing Dragon (Fig. 76). Its most salient feature is, of course, its subject matter-a
+dragon biting its tail, with all the Godelian connotations which that carries. But there is a
+deeper theme to this picture. Escher himself wrote the following most interesting
+comments. The first comment is about a set of his drawings all of which are concerned
+with "the conflict between the flat and the spatial"; the second comment is about Dragon
+in particular.
+
+I Our three-dimensional space is the only true reality we know. The two-
+dimensional is every bit as fictitious as the four-dimensional, for nothing is flat, not
+even the most finely polished mirror. And yet we stick to the convention that a wall
+or a piece of paper is flat, and curiously enough, we still go on, as we have done
+since time immemorial, producing illusions of space on just such plane surfaces as
+these. Surely it is a bit absurd to draw a few lines and then claim: "This is a house".
+This odd situation is the theme of the next five pictures (Including Dragon)
+
+II. However much this dragon tries to be spatial, he remains completely flat. Two
+incisions are made in the paper on which he is printed. Then it is folded in such a
+way as to leave two square openings. But this dragon is an obstinate beast, and in'
+spite of his two dimensions he persists in assuming that he has three; so he sticks his
+head through one of the holes and his tail through the others 5
+
+This second remark especially is a very telling remark. The message is that no matter
+how cleverly you try to simulate three dimensions in two, you are always missing some
+"essence of three-dimensionality". The dragon tries very hard to fight his two-
+dimensionality. He defies the two-dimensionality of the paper on which he thinks he is
+drawn, by sticking his head through it; and yet all the while, we outside the drawing can
+see the pathetic futility of it all, for the dragon and the holes and the folds are all merely
+two-dimensional simulations of those concepts, and not a one of them is real. But the
+dragon cannot step out of his two-dimensional space, and cannot
+
+
+Jumping out of the System
+
+
+473
+
+
+
+
+FIGURE 76. Dragon, by M. C. Escher (wood-engraving, 1952).
+
+know it as we do. We could, in fact, carry the Escher picture any number of steps further.
+For instance, we could tear it out of the book, fold it, cut holes in it, pass it through itself,
+and photograph the whole mess, so that it again becomes two-dimensional. And to that
+photograph, we could once again do the same trick. Each time, at the instant that it
+becomes two- Matter how 'cleverly we seem to have simulated three dimensions inside
+two—it becomes vulnerable to being cut and folded again.
+
+
+Jumping out of the System
+
+
+474
+
+
+
+Now with this wonderful Escherian metaphor, let us return to the program versus
+the human. We were talking about trying to encapsulate the "Godelizing operator" inside
+the program itself. Well, even if we had written a program which carried the operation
+out, that program would not capture the essence of Godel’s method. For once again, we,
+outside the system, could still "zap" it in a way which it couldn't do. But then are we
+arguing with, or against, Lucas
+
+The Limits of Intelligent Systems
+
+Against. For the very fact that we cannot write a program to do "Godelizing" must make
+us somewhat suspicious that we ourselves could do it in every case. It is one thing to
+make the argument in the abstract that Godelizing "can be done"; it is another thing to
+know how to do it in every particular case. In fact, as the formal systems (or programs)
+escalate in complexity, our own ability to "Godelize" will eventually begin to waver. It
+must, since, as we have said above, we do not have any algorithmic way of describing
+how to perform it. If we can't tell explicitly what is involved in applying the Godel
+method in all cases, then for each of us there will eventually come some case so
+complicated that we simply can't figure out how to apply it.
+
+Of course, this borderline of one's abilities will be somewhat ill-defined, just as is
+the borderline of weights which one can pick up off the ground. While on some days you
+may not be able to pick up a 250-pound object, on other days maybe you can.
+Nevertheless, there are no days whatsoever on which you can pick up a 250-ton object.
+And in this sense, though everyone's Godelization threshold is vague, for each person,
+there are systems which lie far beyond his ability to Godelize.
+
+This notion is illustrated in the Birthday Cantatatata. At first, it seems obvious
+that the Tortoise can proceed as far as he wishes in pestering Achilles. But then Achilles
+tries to sum up all the answers in a single swoop. This is a move of a different character
+than any that has gone before, and is given the new name 'co'. The newness of the name
+is quite important. It is the first example where the old naming scheme-which only
+included names for all the natural numbers-had to be transcended. Then come some more
+extensions, some of whose names seem quite obvious, others of which are rather tricky.
+But eventually, we run out of names once again-at the point where the answer-schemas
+
+O , , .0000
+
+14, (0 , CO .
+
+are all subsumed into one outrageously complex answer schema. The altogether new
+name 'e,,' is supplied for this one. And the reason a new name is needed is that some
+fundamentally new kind of step has been taken—a sort of irregularity has been
+encountered. Thus a new name must be applied ad hoc.
+
+
+Jumping out of the System
+
+
+475
+
+
+
+
+There Is No Recursive Rule for Naming Ordinals.
+
+
+Now offhand you might think that these irregularities in the progression >m ordinal to
+ordinal (as these names of infinity are called) could be handled by a computer program.
+That is, there would be a program to produce new names in a regular way, and when it
+ran out of gas, it would invoke the "irregularity handler", which would supply a new
+name, and pass control back to the simple one. But this will not work. It turns out that
+irregularities themselves happen in irregular ways, and one would need o a second-order
+program-that is, a program which makes new programs which make new names. And
+even this is not enough. Eventually, a third-order program becomes necessary. And so on,
+and so on.
+
+All of this perhaps ridiculous-seeming complexity stems from a deep “theorem,
+due to Alonzo Church and Stephen C. Kleene, about the structure of these "infinite
+ordinals", which says:
+
+There is no recursively related notation-system which gives a
+
+name to every constructive ordinal.
+
+hat "recursively related notation-systems" are, and what "constructive ordinals" are, we
+must leave to the more technical sources, such as Hartley )gets' book, to explain. But the
+intuitive idea has been presented. As the ordinals get bigger and bigger, there are
+irregularities, and irregularities in e irregularities, and irregularities in the irregularities in
+the irregularities, etc. No single scheme, no matter how complex, can name all e ordinals.
+And from this, it follows that no algorithmic method can tell w to apply the method of
+Godel to all possible kinds of formal systems, ad unless one is rather mystically inclined,
+therefore one must conclude at any human being simply will reach the limits of his own
+ability to 5delize at some point. From there on out, formal systems of that complex,
+though admittedly incomplete for the Godel reason, will have as much power as that
+human being.
+
+
+Other Refutations of Lucas
+
+Now this is only one way to argue against Lucas' position. There are others, possibly
+more powerful, which we shall present later. But this counterargument has special
+interest because it brings up the fascinating concept trying to create a computer program
+which can get outside of itself, see itself completely from the outside, and apply the
+Godel zapping-trick to itself. Of course this is just as impossible as for a record player to
+be able to ay records which would cause it to break.
+
+But-one should not consider TNT defective for that reason. If there a defect anywhere, it
+is not in TNT, but in our expectations of what it should he able to do. Furthermore, it is
+helpful to realize that we are equally vulnerable to the word trick which Godel
+transplanted into mathematical formalisms: the Epimenides paradox. This was quite
+cleverly pointed out
+
+
+Jumping out of the System
+
+
+476
+
+
+
+by C. H. Whitely, when he proposed the sentence "Lucas cannot consistently assert this
+sentence." If you think about it, you will see that (1) it is true, and yet (2) Lucas cannot
+consistently assert it. So Lucas is also "incomplete" with respect to truths about the
+world. The way in which he mirrors the world in his brain structures prevents him from
+simultaneously being "consistent" and asserting that true sentence. But Lucas is no more
+vulnerable than any of us. He is just on a par with a sophisticated formal system.
+
+An amusing way to see the incorrectness of Lucas' argument is to translate it into
+a battle between men and women ... In his wanderings, Loocus the Thinker one day
+comes across an unknown object-a woman. Such a thing he has never seen before, and at
+first he is wondrous thrilled at her likeness to himself: but then, slightly scared of her as
+well, he cries to all the men about him, "Behold! I can look upon her face, which is
+something she cannot do-therefore women can never be like me!" And thus he proves
+man's superiority over women, much to his relief, and that of his male companions.
+Incidentally, the same argument proves that Loocus is superior to all other males, as well-
+but he doesn't point that out to them. The woman argues back: "Yes, you can see my face,
+which is something I can't do-but I can see your face, which is something you can't do!
+We're even." However, Loocus comes up with an unexpected counter: "I'm sorry, you're
+deluded if you think you can see my face. What you women do is not the same as what
+we men do-it is, as I have already pointed out, of an inferior caliber, and does not deserve
+to be called by the same name. You may call it 'womanseeing'. Now the fact that you can
+'womansee' my face is of no import, because the situation is not symmetric. You see?" "I
+womansee," womanreplies the woman, and womanwalks away .. .
+
+Well, this is the kind of "heads-in-the-sand" argument which you have to be
+willing to stomach if you are bent on seeing men and women running ahead of computers
+in these intellectual battles.
+
+Self-Transcendence-A Modern Myth
+
+It is still of great interest to ponder whether we humans ever can jump out of ourselves-or
+whether computer programs can jump out of themselves. Certainly it is possible for a
+program to modify itself-but such modifiability has to be inherent in the program to start
+with, so that cannot be counted as an example of "jumping out of the system". No matter
+how a program twists and turns to get out of itself, it is still following the rules inherent
+in itself. It is no more possible for it to escape than it is for a human being to decide
+voluntarily not to obey the laws of physics. Physics is an overriding system, from which
+there can be no escape. However, there is a lesser ambition which it is possible to
+achieve: that is, one can certainly Jump from a subsystem of one's brain into a wider
+subsystem. One can step out of ruts on occasion. This is still due to the interaction of
+various subsystems of one’s brain, but it can feel very much like stepping entirely out of
+oneself. Similarly, it is entirely conceivable that a partial ability to “step outside of itself’
+could be embodied in a computer program.
+
+
+Jumping out of the System
+
+
+477
+
+
+
+However, it is important to see the distinction between perceiving oneself, and
+transcending oneself. You can gain visions of yourself in all sorts of rays-in a mirror, in
+photos or movies, on tape, through the descriptions if others, by getting psychoanalyzed,
+and so on. But you cannot quite break out of your own skin and be on the outside of
+yourself (modern occult movements, pop psychology fads, etc. notwithstanding). TNT
+can talk about itself, but it cannot jump out of itself. A computer program can modify
+itself but it cannot violate its own instructions-it can at best change some parts of itself by
+obeying its own instructions. This is reminiscent of the numerous paradoxical question,
+"Can God make a stone so heavy that he can’t lift it?"
+
+Advertisement and Framing Devices
+
+[his drive to jump out of the system is a pervasive one, and lies behind all progress in art,
+music, and other human endeavors. It also lies behind such trivial undertakings as the
+making of radio and television commercials, [his insidious trend has been beautifully
+perceived and described by Irving Goffman in his book Frame Analysis:
+
+For example, an obviously professional actor completes a commercial pitch and,
+with the camera still on him, turns in obvious relief from his task, now to take real
+pleasure in consuming the product he had been advertising.
+
+This is, of course, but one example of the way in which TV and radio
+commercials are coming to exploit framing devices to give an appearance of
+naturalness that (it is hoped) will override the reserve auditors have developed.
+Thus, use is currently being made of children's voices, presumably because these
+seem unschooled; street noises, and other effects to give the impression of
+interviews with unpaid respondents; false starts, filled pauses, byplays, and
+overlapping speech to simulate actual conversation; and, following Welles, the
+interception of a firm's jingle commercials to give news of its new product,
+alternating occasionally with interception by a public interest spot, this presumably
+keeping the faith of the auditor alive.
+
+The more that auditors withdraw to minor expressive details as a test of
+genuineness, the more that advertisers chase after them. What results is a sort of
+interaction pollution, a disorder that is also spread by the public relations
+consultants of political figures, and, more modestly, by micro-sociology.'
+
+
+Here we have yet another example of an escalating "TC-battle"-the antagonists this time
+being Truth and Commercials.
+
+
+Simplicio, Salviati, Sagredo: Why Three?
+
+There is a fascinating connection between the problem of jumping out of ie system and
+the quest for complete objectivity. When I read Jauch's four dialogues in Are Quanta
+Real? based on Galileo's four Dialogues Concerning Two New Sciences, I found myself
+
+
+Jumping out of the System
+
+
+478
+
+
+
+wondering why there were three characters participating. Simplico, Salviati and Sagredo.
+Why wouldn’t two have
+
+
+Jumping out of the System
+
+
+479
+
+
+
+sufficed: Simplicio, the educated simpleton, and Salviati, the knowledgeable thinker?
+What function does Sagredo have? Well, he is supposed to be a sort of neutral third party,
+dispassionately weighing the two sides and coming out with a "fair" and "impartial"
+judgment. It sounds very balanced, and yet there is a problem: Sagredo is always
+agreeing with Salviati, not with Simplicio. How come Objectivity Personified is playing
+favorites? One answer, of course, is that Salviati is enunciating correct views, so Sagredo
+has no choice. But what, then, of fairness or "equal time"?
+
+By adding Sagredo, Galileo (and Jauch) stacked the deck more against Simplicio,
+rather than less. Perhaps there should be added a yet higher level Sagredo-someone who
+will be objective about this whole situation ... You can see where it is going. We are
+getting into a never-ending series of "escalations in objectivity", which have the curious
+property of never getting any more objective than at the first level: where Salviati is
+simply right, and Simplicio wrong. So the puzzle remains: why add Sagredo at all? And
+the answer is, it gives the illusion of stepping out of the system, in some intuitively
+appealing sense.
+
+
+Zen and "Stepping Out"
+
+In Zen, too, we can see this preoccupation with the concept of transcending the system.
+For instance, the koan in which Tozan tells his monks that "the higher Buddhism is not
+Buddha". Perhaps, self-transcendence is even the central theme of Zen. A Zen person is
+always trying to understand more deeply what he is, by stepping more and more out of
+what he sees himself to be, by breaking every rule and convention which he perceives
+himself to be chained by-needless to say, including those of Zen itself. Somewhere along
+this elusive path may come enlightenment. In any case (as I see it), the hope is that by
+gradually deepening one's self-awareness, by gradually widening the scope of "the
+system", one will in the end come to a feeling of being at one with the entire universe.
+
+
+Jumping out of the System
+
+
+480
+
+
+
+Edifying Thoughts
+of a Tobacco Smoker
+
+Achilles has been invited to the Crab's home.
+
+Achilles: I see you have made a few additions since I was last here, Mr. Crab. Your new
+paintings are especially striking.
+
+Crab: Thank you. I am quite fond of certain painters-especially Rene Magritte. Most of
+the paintings I have are by him. He's my favorite artist.
+
+Achilles: They are very intriguing images, I must say. In some ways, these paintings by
+Magritte remind me of works by MY favorite artist, M. C. Escher.
+
+Crab: I can see that. Both Magritte and Escher use great realism in exploring the worlds
+of paradox and illusion; both have a sure sense for the evocative power of certain
+visual symbols, and-something which even their admirers often fail to point out-both
+of them have a sense of the graceful line.
+
+Achilles: Nevertheless, there is something quite different about them. I wonder how one
+could characterize that difference.
+
+Crab: It would be fascinating to compare the two in detail.
+
+Achilles: I must say, Magritte's command of realism is astonishing. For instance, I was
+quite taken in by that painting over there of a tree with a giant pipe behind it.
+
+
+
+FIGURE 77. The Shadows, by Rene Magritte (1966).
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+480
+
+
+
+Crab: You mean a normal pipe with a tiny tree in front of it!
+
+Achilles: Oh, is that what it is? Well, in any case, when I first spotted it, I was convinced
+I was smelling pipe smoke! Can you imagine how silly I felt?
+
+Crab: I quite understand. My guests are often taken in by that one.
+
+(So saying, he reaches up, removes the pipe from behind the tree in the painting,
+turns over and taps it against the table, and the room begins to reek of pipe tobacco.
+
+He begins packing in a new wad of tobacco.)
+
+This is a fine old pipe, Achilles. Believe it or not, the bowl has a copper lining,
+which makes it age wonderfully.
+
+Achilles: A copper lining! You don't say!
+
+Crab (pulls out a box of matches, and lights his pipe): Would you care for a smoke,
+Achilles?
+
+Achilles: No, thank you. I only smoke cigars now and then.
+
+Crab: No problem! I have one right here! (Reaches out towards another Magritte
+painting, featuring a bicycle mounted upon a lit cigar.)
+
+Achilles: Uhh-no thank you, not now.
+
+Crab: As you will. I myself am an incurable tobacco smoker. Which reminds me-you
+undoubtedly know of Old Bach's predilection for pipe smoking?
+
+Achilles: I don't recall exactly.
+
+Crab: Old Bach was fond of versifying, philosophizing, pipe smoking, and
+FIGURE 78. State of Grace, by Rene Magritte (1959).
+
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+481
+
+
+music making (not necessarily in that order). He combined all four into a droll poem
+which he set to music. It can be found in the famous musical notebook he kept for his
+wife, Anna Magdalena, and it is called
+
+
+Edifying Thoughts of a Tobacco Smoker'
+
+Whene'er I take my pipe and stuff it
+And smoke to pass the time away.
+
+My thoughts, as I sit there and puff it,
+
+Dwell on a picture sad and gray:
+
+It teaches me that very like
+Am I myself unto my pipe.
+
+Like me, this pipe so fragrant burning
+Is made of naught but earth and clay;
+
+To earth I too shall be returning.
+
+It falls and, ere I'd think to say.
+
+It breaks in two before my eyes;
+
+In store for me a like fate lies.
+
+No stain the pipe's hue yet cloth darken;
+
+It remains white. Thus do I know
+That when to death's call I must harken
+My body, too, all pale will grow.
+
+To black beneath the sod 'twill turn,
+
+Likewise the pipe, if oft it burn.
+
+Or when the pipe is fairly glowing,
+
+Behold then, instantaneously.
+
+The smoke off into thin air going.
+
+Till naught but ash is left to see.
+
+Man's fame likewise away will burn
+And unto dust his body turn.
+
+How oft it happens when one's smoking:
+
+The stopper's missing from its shelf,
+
+And one goes with one's finger poking
+Into the bowl and burns oneself.
+
+If in the pipe such pain cloth dwell,
+
+How hot must be the pains of hell.
+
+Thus o'er my pipe, in contemplation
+Of such things, I can constantly
+Indulge in fmitful meditation,
+
+And so, puffing contentedly.
+
+On land, on sea, at home, abroad
+I smoke my pipe and worship God.
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+482
+
+
+
+A charming philosophy, is it not?
+
+Achilles: Indeed. Old Bach was a turner of phrases quite pleasin'.
+
+Crab: You took the very words from my mouth. You know, in my time I have tried to
+write clever verses. But I fear mine don't measure up to much. I don't have such a way
+with words.
+
+Achilles: Oh, come now, Mr. Crab. You have-how to put it?-quite a penchant for trick'ry
+and teasin'. I'd be honored if you'd sing me one of your songs, Mr. C.
+
+Crab: I'm most flattered. How about if I play you a record of myself singing one of my
+efforts? I don't remember when it dates from. Its title is "A Song Without Time or
+Season".
+
+Achilles: How poetic!
+
+(The Crab pulls a record from his shelves, and walks over to a huge, complex piece
+of apparatus. He opens it up, and inserts the record into an ominous-looking
+mechanical mouth. Suddenly a bright flash of greenish light sweeps over the
+surface of the record, and after a moment, the record is silently whisked into some
+hidden belly of the fantastic machine. A moment passes, and then the strains of the
+Crab's voice ring out.)
+
+
+A turner of phrases quite pleasin',
+
+Had a penchant for trick'ry and teasin'.
+
+In his songs, the last line
+Might seem sans design;
+
+What I mean is, without why or wherefore.
+
+Achilles: Lovely! Only, I'm puzzled by one thing. It seems to me your song, the last line
+is
+
+Crab: Sans design?
+
+Achilles: No ... What I mean is, without rhyme or reason. Crab: You could be right.
+
+Achilles: Other than that, it's a very nice song, but I must say I am even more intrigued
+by this monstrously complex contraption. Is it merely an oversized record player?
+
+Crab: Oh, no, it's much more than that. This is my Tortoise-chomping record player.
+Achilles: Good grief!
+
+Crab: Well, I don't mean that it chomps up Tortoises. But it chomps up records produced
+by Mr. Tortoise.
+
+Achilles: Whew! That's a little milder. Is this part of that weird musical battle that
+evolved between you and Mr. T some time ago?
+
+Crab: In a way. Let me explain a little more fully. You see, Mr. Tortoise's sophistication
+had reached the point where he seemed to be able to destroy almost any record player
+I would obtain.
+
+Achilles: But when I heard about your rivalry, it seemed to me you had at last come into
+possession of an invincible phonograph—one with a
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+483
+
+
+
+built-in TV camera, minicomputer and so on, which could take itself apart and rebuild
+itself in such a way that it would not be destroyed.
+
+Crab: Alack and alas! My plan was foiled. For Mr. Tortoise took advantage of one small
+detail which I had overlooked: the subunit which directed the disassembly and
+reassembly processes was itself stable during the entire process. That is, for obvious
+reasons, it could not take itself apart and rebuild itself, so it stayed intact.
+
+Achilles: Yes, but what consequences did that have.
+
+Crab: Oh, the direst ones! For you see, Mr. T focused his method down onto that subunit
+entirely.
+
+Achilles: How is that=
+
+Crab: He simply made a record which would induce fatal vibrations in the one structure
+he knew would never change-the disassembly reassembly subunit.
+
+Achilles: Oh, I see ... Very sneaky.
+
+Crab: Yes, so I thought, too. And his strategy worked. Not the first time, mind you. I
+thought I had outwitted him when my phonograph survived his first onslaught. I
+laughed gleefully. But the next time, he returned with a steely glint in his eye, and I
+knew he meant business. I placed his new record on my turntable. Then, both of us
+eagerly watched the computer-directed subunit carefully scan the grooves, then
+dismount the record, disassemble the record player, reassemble it in an astonishingly
+different way, remount the record-and then slowly lower the needle into the outermost
+groove.
+
+Achilles: Golly!
+
+Crab: No sooner had the first strains of sound issued forth than a loud SMASH! filled the
+room. The whole thing fell apart, but particularly badly destroyed was the assembler-
+disassembler. In that painful instant I finally realized, to my chagrin, that the Tortoise
+would ALWAYS be able to focus down upon-if you'll pardon the phrase-the Achilles'
+heel of the system.
+
+Achilles: Upon my soul! You must have felt devastated.
+
+Crab: Yes, I felt rather forlorn for a while. But, happily, that was not the end of the story.
+There is a sequel to the tale, which taught me a valuable lesson, which I may pass on
+to you. On the Tortoise's recommendation, I was browsing through a curious book
+filled with strange Dialogues about many subjects, including molecular biology,
+fugues, Zen Buddhism, and heaven knows what else.
+
+Achilles: Probably some crackpot wrote it. What is the book called:'
+
+Crab: If I recall correctly, it was called Copper, Silver, Gold: an Indestructible Metallic
+Alloy.
+
+Achilles: Oh, Mr. Tortoise told me about it, too. It's by a friend of his, who, it appears, is
+quite taken with metal-logic.
+
+Crab- I wonder which friend it is ... Anyway_ in one of the Dialogues, I encountered
+some Edifying Thoughts on the Tobacco Mosaic Virus, ribosomes, and other strange
+things I had never heard of.
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+484
+
+
+
+FIGURE 79. Tobacco Mosaic Virus.
+
+From A. Lehninger, Biochemistry (New York: Worth Publishers, 1976).
+
+
+Achilles: What is the Tobacco Mosaic Virus? What are ribosomes?
+
+Crab: I can't quite say, for I'm a total dunce when it comes to biology. All I know is what
+I gathered from that Dialogue. There, it said that Tobacco Mosaic Viruses are tiny
+cigarette-like objects that cause a disease in tobacco plants.
+
+Achilles: Cancer?
+
+Crab: No, not exactly, but
+
+Achilles: What next? A tobacco plant smoking and getting cancer! Serves it right!
+
+Crab: I believe you've jumped to a hasty conclusion, Achilles. Tobacco plants don't
+SMOKE these "cigarettes". The nasty little "cigarettes" just come and attack them,
+uninvited.
+
+Achilles: I see. Well, now that I know all about Tobacco Mosaic Viruses, tell me what a
+ribosome is.
+
+Crab: Ribosomes are apparently some sort of sub cellular entities which take a message
+in one form and convert it into a message in another form.
+
+Achilles: Something like a teeny tape recorder or phonograph?
+
+Crab: Metaphorically, I suppose so. Now the thing which caught my eye was a line where
+this one exceedingly droll character mentions the fact that ribosomes-as well as
+Tobacco Mosaic Viruses and certain other bizarre biological structures-possess "the
+baffling ability to spontaneously self-assemble. Those were his exact words.
+
+Achilles: That was one of his droller lines, I take it.
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+485
+
+
+
+Crab: That's just what the other character in the Dialogue thought. But that's a
+preposterous interpretation of the statement. ( The Crab draws deeply from his pipe,
+and puffs several billows of smoke into the air.)
+
+Achilles: Well, what does "spontaneous self-assembly" mean, then?
+
+Crab: The idea is that when some biological units inside a cell are taken apart, they can
+spontaneously reassemble themselves-without being directed by any other unit. The
+pieces just come together, and presto!-they stick.
+
+Achilles: That sounds like magic. Wouldn't it be wonderful if a full-sized record player
+could have that property? I mean, if a miniature "record player" such as a ribosome
+can do it, why not a big one? That would allow you to create an indestructible
+phonograph, right? Any time it was broken, it would just put itself together again.
+
+Crab: Exactly my thought. I breathlessly rushed a letter off to my manufacturer
+explaining the concept of self-assembly, and asked him if he could build me a record
+player which could take itself apart and spontaneously self-assemble in another form.
+
+Achilles: A hefty bill to fill.
+
+Crab: True; but after several months, he wrote to me that he had succeeded, at long last-
+and indeed he sent me quite a hefty bill. One fine day, ho! My Grand Self-assembling
+Record Player arrived in the mail, and it was with great confidence that I telephoned
+Mr. Tortoise, and invited him over for the purpose of testing my ultimate record
+player.
+
+Achilles: So this magnificent object before us must be the very machine of which you
+speak.
+
+Crab: I'm afraid not, Achilles.
+
+Achilles: Don't tell me that once again ...
+
+Crab: What you suspect, my dear friend is unfortunately the case. I don't pretend to
+understand the reasons why. The whole thing is too painful to recount. To see all those
+springs and wires chaotically strewn about on the floor, and puffs of smoke here and
+there-oh, me ...
+
+Achilles: There, there, Mr. Crab, don't take it too badly.
+
+Crab: I'm quite all right; I just have these spells every so often. Well, to go on, after Mr.
+Tortoise's initial gloating, he at last realized how sorrowful I was feeling, and took
+pity. He tried to comfort me by explaining that it couldn't be helped-it all had to do
+with somebody-or-other's "Theorem", but I couldn't follow a word of it. It sounded
+like "Turtle's Theorem".
+
+Achilles: I wonder if it was that "Godel’s Theorem" which he spoke of once before to me
+... It has a rather sinister ring to it. Crab: It could be. I don't recall.
+
+Achilles: I can assure you, Mr. Crab, that I have followed this tale with the utmost
+empathy for your position. It is truly sad. But, you mentioned that there was a silver
+lining. Pray tell, what was that?
+
+Crab: Oh, yes—the silver lining. Well eventually, I abandoned my quest after
+
+“Perfection” in phonographs, and decided that I might do better
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+486
+
+
+
+to tighten up my defenses against the Tortoise's records. I concluded that a more
+modest aim than a record player which can play anything is simply a record player that
+can SURVIVE: one that will avoid getting destroyed-even if that means that it can
+only play a few particular records.
+
+Achilles: So you decided you would develop sophisticated anti-Tortoise mechanisms at
+the sacrifice of being able to reproduce every possible sound, eh?
+
+Crab: Well ... I wouldn't exactly say I "decided" it. More accurate would be to say that I
+was FORCED into that position.
+
+Achilles: Yes, I can see what you mean.
+
+Crab: My new idea was to prevent all "alien" records from being played on my
+phonograph. I knew my own records are harmless, and so if I prevented anyone else
+from infiltrating THEIR records, that would protect my record player, and still allow
+me to enjoy my recorded music.
+
+Achilles: An excellent strategy for your new goal. Now does this giant thing before us
+represent your accomplishments to date along those lines?
+
+Crab: That it does. Mr. Tortoise, of course, has realized that he must change HIS strategy,
+as well. His main goal is now to devise a record which can slip past my censors-a new
+type of challenge.
+
+Achilles: For your part, how are you planning to keep his and other "alien" records out?
+
+Crab: You promise you won't reveal my strategy to Mr. T, now?
+
+Achilles: Tortoise's honor.
+
+Crab: What!?
+
+Achilles: Oh-it's just a phrase I've picked up from Mr. T. Don't worry-I swear your secret
+will remain secret with me.
+
+Crab: All right, then. My basic plan is to use a LABELING technique. To each and every
+one of my records will be attached a secret label. Now the phonograph before you
+contains, as did its predecessors, a television camera for scanning the records, and a
+computer for processing the data obtained in the scan and controlling subsequent
+operations. My idea is simply to chomp all records which do not bear the proper label!
+
+Achilles: Ah, sweet revenge! But it seems to me that your plan will be easy to foil. All
+Mr. T needs to do is to get a hold of one of your records, and copy its label!
+
+Crab: Not so simple, Achilles. What makes you think he will be able to tell the label from
+the rest of the record? It may be better integrated than you suspect.
+
+Achilles: Do you mean that it could be mixed up somehow with the actual music?
+
+Crab: Precisely. But there is a way to disentangle the two. It requires sucking the data off
+the record visually and then-
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+487
+
+
+
+Achilles: Is that what that bright green Hash was for?
+
+Crab: That's right. That was the TV camera scanning the grooves. The groove-patterns
+were sent to the minicomputer, which analyzed the musical style of the piece I had put
+on-all in silence. Nothing had been played yet.
+
+Achilles: Then is there a screening process, which eliminates pieces which aren't in the
+proper styles?
+
+Crab: You've got it, Achilles. The only records which can pass this second test are
+records of pieces in my own style-and it will be hopelessly difficult for Mr. T to
+imitate that. So you see, I am convinced I will win this new musical battle. However, I
+should mention that Mr. T is equally convinced that somehow, he will manage to slip
+a record past my censors.
+
+Achilles: And smash your marvelous machine to smithereens?
+
+Crab: Oh, no-he has proved his point on that. Now he just wants to prove to me that he
+can slip a record-an innocuous one-by me, no matter what measures I take to prevent
+it. He keeps on muttering things about songs with strange titles, such as "I Can Be
+Played on Record Player X". But he can't scare MtE! The only thing that worries me a
+little is that, as before, he seems to have some murky arguments which ... which ... (He
+trails off into silence. Then, looking quite pensive, he takes a few puffs on his pipe.)
+
+Achilles: Hmm ... I'd say Mr. Tortoise has an impossible task on his hands. He's met his
+match, at long last!
+
+Crab: Curious that you should think so ... I don't suppose that you know Henkin's
+Theorem forwards and backwards, do you?
+
+Achilles: Know WHOSE Theorem forwards and backwards? I've never heard of anything
+that sounds like that. I'm sure it's fascinating, but I'd rather hear more about "music to
+infiltrate phonographs by". It's an amusing little story. Actually, I guess I can fill in the
+end. Obviously, Mr. T will find out that there is no point in going on, and so he will
+sheepishly admit defeat, and that will be that. Isn't that exactly it?
+
+Crab: That's what I'm hoping, at least. Would you like to see a little bit of the inner
+workings of my defensive phonograph?
+
+Achilles: Gladly. I've always wanted to see a working television camera.
+
+Crab: No sooner said than done, my friend. (Reaches into the gaping"mouth" of the large
+phonograph, undoes a couple of snaps, and pulls out a neatly packaged instrument.)
+You see, the whole thing is built of independent modules, which can be detached and
+used independently. This TV camera, for instance, works very well by itself. Watch
+the screen over there, beneath the painting with the flaming tuba. (He points the
+camera at Achilles, whose face instantly appears on the large screen.)
+
+Achilles: Terrific! May I try it out?
+
+Crab: Certainly.
+
+Achilles: (pointing the camera at the Crab. There YOU are, Mt Crab, on the screen.
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+488
+
+
+
+
+FIGURE 80. The Fair Captive, by Rene Magritte (1947).
+
+
+Crab: So I am.
+
+Achilles: Suppose I point the camera at the painting with the burning tuba. Now it is on
+the screen, too!
+
+Crab: The camera can zoom in and out, Achilles. You ought to try it. Achilles: Fabulous!
+Let me just focus down onto the tip of those flames, where they meet the picture frame
+... It's such a funny feeling to be able to instantaneously "copy" anything in the room-
+anything I want-onto that screen. I merely need to point the camera at it, and it pops
+like magic onto the screen.
+
+Crab: ANYTHING in the room, Achilles? Achilles: Anything in sight, yes. That's
+obvious.
+
+Crab: What happens, then, if you point the camera at the flames on the TV screen?
+
+(Achilles shifts the camera so that it points directly at that part of the television
+screen on which the flames are-or were-displayed.)
+
+Achilles: Hey, that’s funny! That very act makes the flames DISAPPEAR from the
+screen! Where did they go?
+
+Crab: You can't keep an image still on the screen and move the camera at the same time.
+
+Achilles: So I see... But I don’t understand what’s on the screen now—not at all! It
+seems to be a strange long corridor. Yet I’m certainly not
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+489
+
+
+
+
+
+
+
+
+
+(h) A "’galaxy" u born.
+
+
+(hi The galaxy has burned tseij out. and became —
+a black hole!
+
+
+
+t'l The galaxy rvo/iv> . . .
+
+
+
+(!) A "pulsating petal pattern", caught in the nuddle
+of one of Us pulMtians,
+
+
+FIGURE 81. Twelve self-engulfing TV screens. I would have included one more, had 13
+not been prime
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+490
+
+
+
+
+
+
+
+
+
+
+
+'> Ar hit lei ''corridur"
+
+
+(*) H 'hat hapftrn i U'hrn you uxim in.
+
+
+
+) W'hal happen* u>hen you rnlale ihr camera (J) Comhinrri rffrtl of ivlaliun and ioomWi.
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+491
+
+
+
+
+
+
+
+pointing the camera down any corridor. I'm merely pointing it at an ordinary TV
+screen.
+
+Crab: Look more carefully, Achilles. Do you really see a corridor?
+
+Achilles: Ahhh, now I see. It's a set of nested copies of the TV screen itself, getting
+smaller and smaller and smaller ... Of course! The image of the flames HAD to go
+away, because it came from my- pointing the camera at the PAINTING. When I point
+the camera at the SCREEN, then the screen itself appears, with whatever is on the
+screen at the time which is the screen itself, with whatever is on the screen at the time
+which is the screen itself, with
+
+Crab: I believe I can fill in the rest, Achilles. Why- don't you try rotating the camera?
+
+Achilles: Oh! I get a beautiful spiraling corridor! Each screen is rotated inside its framing
+screen, so that the littler they get, the more rotated they are, with respect, to the
+outermost screen. This idea of having a TV screen "engulf itself" is weird.
+
+Crab: What do you mean by "self-engulfing", Achilles?
+
+Achilles: I mean, when I point the camera at the screen-or at part of the screen. THAT'S
+self-engulfing.
+
+Crab: Do you mind if I pursue that a little further? I'm intrigued by this new notion.
+
+Achilles: So am I.
+
+Crab: Very well, then. If you point the camera at a CORNER of the screen, is that still
+what you mean by "self-engulfing"?
+
+Achilles: Let me try it. Hmm-the "corridor" of screens seems to go off the edge, so there
+isn't an infinite nesting any more. It's pretty, but it doesn't seem to me to have the spirit
+of self-engulfing. It's a "failed self-engulfing".
+
+Crab: If you were to swing the TV camera back towards the center of the screen, maybe
+you could fix it up again ...
+
+Achilles (slowly and cautiously turning the camera): Yes! The corridor is getting longer
+and longer ... There it is! Now it's all back. I can look down it so far that it vanishes in
+the distance. The corridor became infinite again precisely at the moment when the
+camera took in the WHOLE screen. Hmm-that reminds me of something Mr. Tortoise
+was saying a while back, about self-reference only occurring when a sentence talks
+about ALL of itself ...
+
+Crab: Pardon me?
+
+Achilles: Oh, nothing just muttering to myself.
+
+(As Achilles plays with the lens and other controls on the camera, a profusion of new
+kinds of self-engulfing images appear: swirling spirals that resemble galaxies,
+kaleidoscopic flower-like shapes, and other assorted patterns ...)
+
+Crab: You seem to be having a grand time.
+
+Achilles: (turns away from the camera); I’ll say! What a wealth of images this simple
+idea can produce! (He glances back at the screen, and a look of
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+492
+
+
+
+astonishment crosses his face.) Good grief, Mr. Crab! There's a pulsating petal-pattern
+on the screen! Where do the pulsations come from? The TV is still, and so is the
+camera.
+
+Crab: You can occasionally set up patterns which change in time. This is because there is
+a slight delay in the circuitry between the moment the camera "sees" something, and
+the moment it appears on the screen around a hundredth of a second. So if you have a
+nesting of depth fifty or so, roughly a half-second delay will result. If somehow a
+moving image gets onto the screen-for example, by you putting your finger in front of
+the camera-then it takes a while for the more deeply nested screens to "find out" about
+it. This delay then reverberates through the whole system, like a visual echo. And if
+things are set up so the echo doesn't die away, then you can get pulsating patterns.
+
+Achilles: Amazing! Say-what if we tried to make a TOTAL self-engulfing?
+
+Crab: What precisely do you mean by that?
+
+Achilles: Well, it seems to me that this stuff with screens within screens is interesting, but
+I'd like to get a picture of the TV camera AND the screen, ON the screen. Only then
+would I really have made the system engulf itself. For the screen is only PART of the
+total system.
+
+Crab: I see what you mean. Perhaps with this mirror, you can achieve the effect you
+want.
+
+(The Crab hands him a mirror, and Achilles maneuvers the mirror and camera in such
+a way that the camera and the screen are both pictured on the screen.)
+
+Achilles: There! I've created a TOTAL self-engulfing!
+
+Crab: It seems to me you only have the front of the mirror-what about its back? If it
+weren't for the back of the mirror, it wouldn't be reflective-and you wouldn't have the
+camera in the picture.
+
+Achilles: You're right. But to show both the front and back of this mirror, I need a second
+mirror.
+
+Crab: But then you'll need to show the back of that mirror, too. And what about including
+the back of the television, as well as its front? And then there's the electric cord, and
+the inside of the television, and
+
+Achilles: Whoa, whoa! My head's beginning to spin! I can see that this "total self¬
+engulfing project" is going to pose a wee bit of a problem. I'm feeling a little dizzy.
+
+Crab: I know exactly how you feel. Why don't you sit down here and take your mind off
+all this self-engulfing? Relax! Look at my paintings, and you'll calm down.
+
+(Achilles lies down, and sighs.)
+
+Oh-perhaps my pipe smoke is bothering you? Here, I'll put my pipe away. ( Takes the pipe
+from his mouth, and carefully places it above some written words in another .Magritte
+painting.) There! Feeling any better?
+
+Achilles: I’m still a little woozy, (Points at the Magritte.) That’s an interesting painting. I
+like the way it’s framed, especially the shiny inlay inside the wooden frame.
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+493
+
+
+
+
+FIGURE 82. The Air and the Song, by Rene Magritte (1964).
+
+Crab: Thank you. I had it specially done-it's a gold lining.
+
+Achilles: A gold lining? What next? What are those words below the pipe? They aren't in
+English, are they?
+
+Crab: No, they are in French. They say, "Ceci n’est pas une pipe." That means, "This is
+not a pipe". Which is perfectly true. 4chilles: But it is a pipe! You were just smoking
+it!
+
+Crab: Oh, you misunderstand the phrase, I believe. The word "ceci" refers to the painting,
+not to the pipe. Of course the pipe is a pipe. But a painting is not a pipe.
+
+Achilles: I wonder if that "ceci" inside the painting refers to the WHOLE painting, or just
+to the pipe inside the painting. Oh, my gracious! That would be ANOTHER self¬
+engulfing! I'm not feeling at all well, Mr. Crab. I think I'm going to be sick ...
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+494
+
+
+
+
+CHAPTER XVI
+
+
+Self-Ref and Self-Rep
+
+IN THIS CHAPTER, we will look at some of the mechanisms which create self-reference in
+various contexts, and compare them to the mechanisms which allow some kinds of systems
+to reproduce themselves. Some remarkable and beautiful parallels between these mechanisms
+will come to light.
+
+
+Implicitly and Explicitly Self-Referential Sentences
+
+To begin with, let us look at sentences which, at first glance, may seem to provide the
+simplest examples of self-reference. Some such sentences are these:
+
+(1) This sentence contains five words.
+
+(2) This sentence is meaningless because it is self-referential.
+
+(3) This sentence no verb.
+
+(4) This sentence is false. (Epimenides paradox)
+
+(5) The sentence I am now writing is the sentence you are now reading.
+
+All but the last one (which is an anomaly) involve the simple-seeming mechanism contained
+in the phrase "this sentence". But that mechanism is in reality far from simple. All of these
+sentences are "floating" in the context of the English language. They can be compared to
+icebergs, whose tips only are visible. The word sequences are the tips of the icebergs, and the
+processing which must be done to understand them is the hidden part. In this sense their
+meaning is implicit, not explicit. Of course, no sentence's meaning is completely explicit, but
+the more explicit the self-reference is, the more exposed will be the mechanisms underlying
+it. In this case, for the self-reference of the sentences above to be recognized, not only has
+one to be comfortable with a language such as English which can deal with linguistic subject
+matter, but also one has to be able to figure out the referent of the phrase "this sentence". It
+seems simple, but it depends on our very complex yet totally assimilated ability to handle
+English. What is especially important here is the ability to figure out the referent of a noun
+phrase with a demonstrative adjective in it. This ability is built up slowly, and should by no
+means be considered trivial. The difficulty is perhaps underlined when a sentence such as
+number 4 is presented to someone naive about paradoxes and linguistic tricks, such as a
+child. They may say, "What sentence is false and it may take a bit of persistence to get across
+the idea that the sentence is talking about itself. The whole idea is a little mind
+
+
+Self-Rep and Self-Rep
+
+
+495
+
+
+
+boggling at first. A couple of pictures may help (Figs. 83, 84). Figure 83 is a picture which
+can be interpreted on two levels. #n one level, it is a sentence pointing at itself; on the other
+level, it is a picture of Epimenides executing his own death sentence.
+
+
+
+Figure 84, showing visible and invisible portions of the iceberg, suggests the relative
+proportion of sentence to processing required for the recognition of self-reference:
+
+
+
+It is amusing to try to create a self-referring sentence without using the trick of saving
+this sentence". #ne could try to quote a sentence inside itself. Here is an attempt:
+
+The sentence "The sentence contains five words" contains five words.
+
+But such an attempt must fail, for any sentence that could be quoted entirely inside itself
+would have to be shorter than itself. This is actually possible, but only if you are willing to
+entertain infinitely long sentences, such as:
+
+
+Self-Rep and Self-Rep
+
+
+496
+
+
+
+The sentence
+
+"The sentence
+
+"The sentence
+
+"The sentence
+
+
+etc,.,etc.
+
+
+is infinitely long"
+is infinitely long"
+is infinitely long'
+is infinitely long.
+
+But this cannot work for finite sentences. For the same reason, Godel's string G could not
+contain the explicit numeral for its Godel number: it would not fit. No string of TNT can
+contain the TNT-numeral for its own Godel number, for that numeral always contains more
+symbols than the string itself does. But you can get around this by having G contain a
+description of its own Godel number, by means of the notions of "sub" and
+"arithmoquinification".
+
+•ne way of achieving self-reference in an English sentence by means of description
+instead of by self-quoting or using the phrase "this sentence" is the Quine method, illustrated
+in the dialogue Air on G's String. The understanding of the Quine sentence requires less
+subtle mental processing than the four examples cited earlier. Although it may appear at first
+to be trickier, it is in some ways more explicit. The Quine construction is quite like the Godel
+construction, in the way that it creates self-reference by describing another typographical
+entity which, as it turns out, is isomorphic to the Quine sentence itself. The description of the
+new typographical entity is carried out by two parts of the Quine sentence. #ne part is a set
+of instructions telling how to build a certain phrase, while the other part contains the
+construction materials to be used; that is, the other part is a template. This resembles a
+floating cake of soap more than it resembles an iceberg (See Fig. S5).
+
+FIGURE 85.
+
+
+
+Self-Rep and Self-Rep
+
+
+497
+
+
+
+
+The self-reference of this sentence is achieved in a more direct way than in the Epimenides
+paradox; less hidden processing is needed. By the way, it is interesting to point out that the
+phrase "this sentence" appears in the previous sentence; yet it is not there to cause self¬
+reference: you probably understood that its referent was the Quine sentence, rather than the
+sentence in which it occurs. This just goes to show how pointer phrases such as "this
+sentence" are interpreted according to context, and helps to show that the processing of such
+phrases is indeed quite involved.
+
+A Self-Reproducing Program
+
+The notion of quining, and its usage in creating self-reference, have already been explained
+inside the Dialogue itself, so we need not dwell on such matters here. Let us instead show
+how a computer program can use precisely the same technique to reproduce itself. The
+following selfreproducing program is written in a BlooP-like language and is based on
+following a phrase by its own quotation (the opposite order from quining, so I reverse the
+name "quine" to make "eniuq"):
+
+DEFINE PROCEDURE "ENIUQ" [TEMPLATE]: PRINT [TEMPLATE, LEFT-
+BRACKET, QUOTE-MARK, TEMPLATE, QUOTE-MARK, RIGHT-BRACKET,
+PERIOD].
+
+ENIUQ
+
+[ DEFINE PROCEDURE "ENIUQ" [TEMPLATE]: PRINT [TEMPLATE, LEFT-
+BRACKET, QUOTE-MARK, TEMPLATE, QUOTE-MARK, RIGHT-BRACKET,
+PERIOD]. ENIUQ'].
+
+ENIUQ is a procedure defined in the first two lines, and its input is called "TEMPLATE". It
+is understood that when the procedure is called, TEMPLATE'S value will be some string of
+typographical characters. The effect of ENIUQ is to carry out a printing operation, in which
+TEMPLATE gets printed twice: the first time just plain; the second time wrapped in (single)
+quotes and brackets, and garnished with a final period. Thus, if TEMPLATE'S value were
+the string DOUBLE-BUBBLE, then performing ENIUQ on it would yield:
+
+DOUBLE-BUBBLE ['DOUBLE-BUBBLE'].
+
+Now in the last four lines of the program above, the procedure ENIUQ is called with a
+specific value of TEMPLATE-namely the long string inside the single quotes: DEFINE ...
+ENIUQ. That value has been carefully chosen; it consists of the definition of ENIUQ,
+followed by the word ENIUQ. This makes the program itself-or, if you prefer, a perfect copy
+of it-get printed out. It is very similar to Quine's version of the Epimenides sentence:
+
+"yields falsehood when preceded by its quotation"
+yields falsehood when preceded by its quotation.
+
+It is very important to realize that the character string which appears n quotes in the last three
+lines of the program above-that is, the value of
+
+
+Self-Rep and Self-Rep
+
+
+498
+
+
+
+TEMPLATE -is never interpreted as a sequence of instructions. That it happens to be one is,
+in a sense, just an accident. As was pointed out above, it could just as well have been
+DOUBLE-BUBBLE or any other string of characters. The beauty of the scheme is that when
+the same string appears in the top two lines of this program, it is treated as a program
+(because it is not in quotes). Thus in this program, one string functions in two ways: first as
+program, and second as data. This is the secret of self-reproducing programs, and, as we shall
+see, of self-reproducing molecules. It is useful, incidentally, to call any kind of self-
+•reproducing object or entity a self-rep; and likewise to call any self-referring object or entity
+a self-ref. I will use those terms occasionally from here on.
+
+The preceding program is an elegant example of a self-reproducing program written
+in a language which was not designed to make the writing of self-reps particularly easy.
+Thus, the task had to be carried out using those notions and operations which were assumed
+to be part of the language-such as the word QUOTE-MARK, and the command PRINT. But
+suppose a language were designed expressly for making self-reps easy to write. Then one
+could write much shorter self-reps. For example, suppose that the operation of eniuq-ing
+were a built-in feature of the language, needing no explicit definition (as we assumed PRINT
+was). Then a teeny self-rep would be this:
+
+ENIUQ ['ENIUQ'].
+
+It is very similar to the Tortoise's version of Quine's version of the Epimenides self-ref,
+where the verb "to quine" is assumed to be known:
+
+"yields falsehood when quined" yields falsehood when quined
+
+But self-reps can be even shorter. For instance, in some computer language it might
+be a convention that any program whose first symbol is an asterisk is to be copied before
+being executed normally. Then the program consisting of merely one asterisk is a self-rep!
+You may complain that this is silly and depends on a totally arbitrary convention. In doing
+so, you are echoing my earlier point that it is almost cheating to use the phrase "this
+sentence" to achieve self-reference-it relies too much on the processor, and not enough on
+explicit directions for self-reference. Using an asterisk as an example of a self-rep is like
+using the word "I" as an example of a self-ref: both conceal all the interesting aspects of their
+respective problems.
+
+This is reminiscent of another curious type of self-reproduction: via photocopy
+machine. It might be claimed that any written document is a self-rep because it can cause a
+copy of itself to be printed when it is placed in a photocopy machine and the appropriate
+button is pushed. But somehow this violates our notion of self-reproduction; the piece of
+paper is not consulted at all, and is therefore not directing its own reproduction. Again,
+everything is in the processor. Before we call something a self-rep, we want to have the
+feeling that, to the maximum extent possible, it explicitly contains the directions for copying
+itself.
+
+
+Self-Rep and Self-Rep
+
+
+499
+
+
+
+To be sure, explicitness is a matter of degree; nonetheless there is an intuitive borderline on
+one side of which we perceive true self-directed self-reproduction, and on the other side of
+which we merely see copying being carried out by an inflexible and autonomous copying
+machine.
+
+
+What Is a Copy?
+
+Now in any discussion of self-refs and self-reps, one must sooner or later come to grips with
+the essential issue: what is a copy? We already dealt with that question quite seriously in
+Chapters V and VI; and now we come back to it. To give the flavor of the issue, let us
+describe some highly fanciful, yet plausible, examples of self-reps.
+
+A Self-Reproducing Song
+
+Imagine that there is a nickelodeon in the local bar which, if you press buttons I l-U, will
+play a song whose lyrics go this way:
+
+Put another nickel in, in the nickelodeon,
+
+All 1 want is II-U, and music, music, music.
+
+We could make a little diagram of what happens one evening (Fig. 86).
+
+
+person
+
+
+
+FIGURE 86. A self-reproducing song.
+
+Although the effect is that the song reproduces itself, it would feel strange to call the song a
+self-rep, because of the fact that when it passes through the I I-U stage, not all of the
+information is there. The infonnation only gets put back by virtue of the fact that it is fully
+stored in the nickelodeon that is, in one of the arrows in the diagram, not in one of the ovals.
+It is questionable whether this song contains a complete description of how to get itself
+played again, because the symbol pair "I I-U" is only a trigger, not a copy.
+
+A "Crab" Program
+
+Consider next a computer program which prints itself out backwards. (Some readers might
+enjoy thinking about how to write such a program in
+
+
+511
+
+
+Self-Rep and Self-Rep
+
+
+the blooP-like language above, using the given sell-rep as a inouel.) vvouiu this funny
+program count as a self-rep. Yes, in a way, because a trivial transformation performed on its
+output will restore the original program. It seems fair to say that the output contains the same
+information as the program itself, just recast in a simple way. Yet it is clear that someone
+might look at the output and not recognize it as a program printed backwards. To recall
+terminology from Chapter VI, we could say that the "inner messages" of the output and the
+program itself are the same, but they have different "outer messages"-that -is, they must be
+read by using different decoding mechanisms. Now if one counts the outer message as part of
+the information-which seems quite reasonable-then the total information is not the same after
+all, so the program can't be counted as a self-rep.
+
+However, this is a disquieting conclusion, because we are accustomed to considering
+something and its mirror image as containing the same information. But recall that in Chapter
+VI, we made the concept of "intrinsic meaning" dependent on a hypothesized universal
+notion of intelligence. The idea was that, in determining the intrinsic meaning of an object,
+we could disregard some types of outer message-those which would be universally
+understood. That is, if the decoding mechanism seems fundamental enough, in some still ill-
+defined sense, then the inner message which it lets be revealed is the only meaning that
+counts. In this example, it seems reasonably safe to guess that a "standard intelligence" would
+consider two mirror images to contain the same information as each other; that is, it would
+consider the isomorphism between the two to be so trivial as to be ignorable. And thus our
+intuition that the program is in some sense a fair self-rep, is allowed to stand.
+
+Epimenides Straddles the Channel
+
+Now another far-fetched example of a self-rep would be a program which prints itself our,
+but translated into a different computer language. One might liken this to the following
+curious version of the Quine version of the Epimenides self-ref:
+
+'lest une expression qui, quand elle est precedee de sa traduction, mise entre
+guillemets, clans la langue provenant de l'autre tote de la Manche. tree une faussete"
+is an expression which, when it is preceded by its translation, placed in quotation
+marks, into the language originating on the other side of the Channel, yields a
+falsehood.
+
+You might try to write down the sentence which is described by this weird concoction. (Hint:
+It is not itself-or at least it is not if "itself' is taken in a naive sense.) If the notion of "self-rep
+by retrograde motion" (i.e., a program which writes itself out backwards) is reminiscent of a
+crab canon, the notion of "self-rep by translation" is no less reminiscent of "a canon which
+involves a transposition of the theme into another key.
+
+
+Self-Rep and Self-Rep
+
+
+501
+
+
+
+A Program That Prints Out Its Own Godel Number
+
+
+The idea of printing out a translation instead of an exact copy of the original program may
+seem pointless. However, if you wanted to write a self-rep program in BlooP or FlooP, you
+would have to resort to some such device, for in those languages, OUTPUT is always a
+number, rather than a typographical string. Therefore, you would have to make the program
+print out its own Godel number: a very huge integer whose decimal expansion codes for the
+program, character by character, by using three digit codons. The program is coming as close
+as it can to printing itself, within the means available to it: it prints out a copy of itself in
+another "space", and it is easy to switch back and forth between the space of integers and the
+space of strings. Thus, the value of OUTPUT is not a mere trigger, like "11-12". Instead, all
+the information of the original program lies "close to the surface" of the output.
+
+Godelian Self-Reference
+
+This comes very close to describing the mechanism of Godel's self-ref G. After all, that string
+of TNT contains a description not of itself, but of an integer (the arithmoquinification of u). It
+just so happens that that integer is an exact "image" of the string G, in the space of natural
+numbers. Thus, G refers to a translation of itself into another space. We still feel comfortable
+in calling G a self-referential string, because the isomorphism between the two spaces is so
+tight that we can consider them to be identical.
+
+This isomorphism that mirrors TNT inside the abstract realm of natural numbers can
+be likened to the quasi-isomorphism that mirrors the real world inside our brains, by means
+of symbols. The symbols play quasi-isomorphic roles to the objects, and it is thanks to them
+that we can think. Likewise, the Godel numbers play isomorphic roles to strings, and it is
+thanks to them that we can find metamathematical meanings in statements about natural
+numbers. The amazing, nearly magical, thing about G is that it manages to achieve self¬
+reference despite the fact that the language in which it is written, TNT, seems to offer no
+hope of referring to its own structures, unlike English, in which it is the easiest thing in the
+world to discuss the English language.
+
+So G is an outstanding example of a self-ref via translation-hardly the most
+straightforward case. One might also think back to some of the Dialogues, for some of them,
+too, are self-refs via translation. For instance, take the Sonata for Unaccompanied Achilles.
+In that Dialogue, several references are made to the Bach Sonatas for unaccompanied violin,
+and the Tortoise's suggestion of imagining harpsichord accompaniments is particularly
+interesting. After all, if one applies this idea to the Dialogue itself, one invents lines which
+the Tortoise is saying; but if one assumes that Achilles' part stands alone (as does the violin),
+then it is quite wrong to attribute any lines at all to the Tortoise. In any case, here again is a
+self-ref by means of a mapping which maps Dialogues onto pieces by Bach. And this
+mapping is
+
+
+Self-Rep and Self-Rep
+
+
+502
+
+
+
+left, of course, for the reader to notice. Yet even if the reader does not notice it, the mapping
+is still there, and the Dialogue is still a self-ref.
+
+A Self-Rep by Augmentation
+
+We have been likening self-reps to canons. What, then, would be a fair analogue to a canon
+by augmentation? Here is a possibility: consider a program which contains a dummy loop
+whose only purpose is to slow up the program. A parameter might tell how often to repeat the
+loop. A self-rep could be made which prints out a copy of itself, but with the parameter
+changed, so that when that copy is run, it will run at half the speed of its parent program; and
+its "daughter" will in turn run at half again the speed, and so on . . . None of these programs
+prints itself out precisely; yet all clearly belong to a single "family".
+
+This is reminiscent of the self-reproduction of living organisms. Clearly, an
+individual is never identical to either of its parents; why, then, is the act of making young
+called "self-reproduction'? The answer is that there is a coarse-grained isomorphism between
+parent and child; it is an isomorphism which preserves the information about species. Thus,
+what is reproduced is the class, rather than the instance. This is also the case in the recursive
+picture Gplot, in Chapter V: that is, the mapping between "magnetic butterflies" of various
+sizes and shapes is coarse-grained; no two are identical, but they all belong to a single
+"species", and the mapping preserves precisely that fact. In terms of self-replicating
+programs, this would correspond to a family of programs, all written in "dialects" of a single
+computer language; each one can write itself out, but slightly modified, so that it comes out
+in a dialect of its original language.
+
+
+A Kimian Self-Rep
+
+Perhaps the sneakiest example of a self-rep is the following: instead of writing a legal
+expression in the compiler language, you type one of the compiler's own error messages.
+When the compiler looks at your "program", the first thing it does is get confused, because
+your "program" is ungrammatical; hence the compiler prints out an error message. All you
+need to do is arrange that the one it prints out will be the one you typed in. This kind of self¬
+rep, suggested to me by Scott Kim, exploits a different level of the system from the one you
+would normally approach. Although it may seem frivolous, it may have counterparts in
+complex systems where self-reps vie against each other for survival, as we shall soon discuss.
+
+What Is the Original?
+
+Besides the question "What constitutes a copy?", there is another fundamental philosophical
+question concerning self-reps. That is the obverse
+
+
+Self-Rep and Self-Rep
+
+
+503
+
+
+
+side of the coin: "What is the original?" This can best be explained by referring to some
+examples:
+
+(1) a program which, when interpreted by some interpreter running on some
+computer, prints itself out;
+
+(2) a program which, when interpreted by some interpreter running on some
+computer, prints itself out along with a complete copy of the interpreter (which,
+after all, is also a program);
+
+(3) a program which, when interpreted by some interpreter running on some
+computer, not only prints itself out along with a complete copy of the interpreter,
+but also directs a mechanical assembly process in which a second computer,
+identical to the one on which the interpreter and program are running, is put
+together.
+
+It is clear that in (1), the program is the self-rep. But in (3), is it the program which is the self¬
+rep, or the compound system of program plus interpreter, or the union of program,
+interpreter, and processor?
+
+Clearly, a self-rep can involve more than just printing itself out. In fact, most of the
+rest of this Chapter is a discussion of self-reps in which data, program, interpreter, and
+processor are all extremely intertwined, and in which self-replication involves replicating all
+of them at once.
+
+
+Typogenetics
+
+We are now about to broach one of the most fascinating and profound topics of the twentieth
+century: the study of "the molecular logic of the living state", to borrow Albert Lehninger's
+richly evocative phrase. And logic it is, too but of 'a sort more complex and beautiful than
+any a human mind ever imagined. We will come at it from a slightly novel angle: via an
+artificial solitaire game which I call Typogenetics-short for "Typographical Genetics". In
+Typogenetics I have tried to capture some ideas of molecular genetics in a typographical
+system which, on first sight, resembles very much the formal systems exemplified by the
+MlU-system. Of course, Typogenetics involves many simplifications, and therefore is useful
+primarily for didactic purposes.
+
+T should explain immediately that the field of molecular biology is a field in which
+phenomena on several levels interact, and that Typogenetics is only trying to illustrate
+phenomena from one or two levels. In particular, purely chemical aspects have been
+completely avoided-they belong to a level lower than is here dealt with; similarly, all aspects
+of classical genetics (viz., nonmolecular genetics) have also been avoided-they belong to a
+level higher than is here dealt with. I have intended in Typogenetics only to give an intuition
+for those processes centered on the celebrated Central Dogma of
+
+
+Self-Rep and Self-Rep
+
+
+504
+
+
+
+Molecular Biology, enunciated by Francis Crick (one of the co-discoverers of the double¬
+helix structure of DNA):
+
+
+DNA => RNA => proteins.
+
+It is my hope that with this very skeletal model I have constructed the reader will perceive
+some simple unifying principles of the field principles which might otherwise be obscured by
+the enormously intricate interplay of phenomena at many different levels. What is sacrificed
+is, of course, strict accuracy; what is gained is, I hope, a little insight.
+
+Strands, Bases, Enzymes
+
+The game of Typogenetics involves typographical manipulation on sequences of letters.
+There are four letters involved:
+
+
+A CG T.
+
+Arbitrary sequences of them are called strands. Thus, some strands are:
+
+GGGG
+
+ATTACCA
+
+CATCATCATCAT
+
+Incidentally, "STRAND" spelled backwards begins with "DNA". This is appropriate since
+strands, in Typogenetics, play the role of pieces of DNA (which, in real genetics, are often
+called "strands"). Not only this, but "STRAND" fully spelled out backwards is "DNA RTS",
+which may be taken as an acronym for "DNA Rapid Transit Service". This, too, is
+appropriate, for the function of "messenger RNA"-which in Typogenetics is represented by
+strands as well-is quite well characterized by the phrase "Rapid Transit Service" for DNA, as
+we shall see later.
+
+I will sometimes refer to the letters A, C, G, T as bases, and to the positions which
+they occupy as units. Thus, in the middle strand, there are seven units, in the fourth of which
+is found the base A.
+
+If you have a strand, you can operate on it and change it in various ways. You can
+also produce additional strands, either by copying, or by cutting a strand in two. Some
+operations lengthen strands, some shorten them, and some leave their length alone.
+
+Operations come in packets-that is, several to be performed together, in order. Such a
+packet of operations is a little like a programmed machine which moves up and down the
+strand doing things to it. These mobile machines are called "typographical enzymes"-
+enzymes for short. Enzymes operate on strands one unit at a time, and are said to be "bound"
+to the unit they are operating on at any given moment.
+
+I will show how some sample enzymes act on particular strings. The first thing to
+know is that each enzyme likes to start out bound to a particular letter. Thus, there are four
+kinds of enzyme-those which prefer
+
+
+Self-Rep and Self-Rep
+
+
+505
+
+
+
+A, those which prefer C, etc. Given the sequence of operations which an enzyme performs,
+you can figure out which letter it prefers, but for now I'll just give them without explanation.
+Here's a sample enzyme, consisting of three operations:
+
+(1) Delete the unit to which the enzyme is bound (and then bind to the next unit to the
+right).
+
+(2) Move one unit to the right.
+
+(3) Insert a T (to the immediate right of this unit).
+
+This enzyme happens to like to bind to A initially. And here's a sample strand:
+
+ACA
+
+What happens if our enzyme binds to the left A and begins acting? Step I deletes the A, so we
+are left with CA-and the enzyme is now bound to the C. Step 2 slides the enzyme rightwards,
+to the A, and Step 3 appends a T onto the end to form the strand CAT. And the enzyme has
+done its complete duty: it has transformed ACA into CAT.
+
+What if it had bound itself to the right A of ACA? It would have deleted that A and
+moved off the end of the strand. Whenever this happens, the enzyme quits (this is a general
+principle). So the entire effect would just be to lop off one symbol.
+
+Let's see some more examples. Here is another enzyme:
+
+(1) Search for the nearest pyrimidine to the right of this unit.
+
+(2) Go into Copy mode.
+
+(3) Search for the nearest purine to the right of this unit.
+
+(4) Cut the strand here (viz., to the right of the present unit).
+
+Now this contains the terms "pyrimidine" and "purine". They are easy terms. A and G are
+called purines, and C and T are called pyrimidines. So searching for a pyrimidine merely
+means searching for the nearest C or T.
+
+Copy Mode and Double Strands
+
+The other new term is Copy mode. Any strand can be "copied" onto another strand, but in a
+funny way. Instead of copying A onto A, you copy it onto T, and vice versa. And instead of
+copying C onto C, you copy it onto G, and vice versa. Note that a purine copies onto a
+pyrimidine, and vice versa. This is called complementary base pairing. The complements are
+shown below
+
+
+Purinas
+
+
+Complement
+
+A <====> T
+G <====> C
+
+
+pyrimidines
+
+
+Self-Rep and Self-Rep
+
+
+506
+
+
+
+You can perhaps remember this molecular pairing scheme by recalling that Achilles is paired
+with the Tortoise, and the Crab with his Genes.
+
+When "copying" a strand, therefore, you don't actually copy it, but you manufacture
+its complementary strand. And this one will be written upside down above the original strand.
+Let's see this in concrete terms. Let the previous enzyme act on the following strand (and that
+enzyme also happens to like to start at A):
+
+CAAAGAGAATCCTCTTTGAT
+
+There are many places it could start. Let's take the second A, for example. The enzyme binds
+to it, then executes step 1: Search for the nearest pyrimidine to the right. Well, this means a C
+or a T. The first one is a T somewhere near the middle of the strand, so that's where we go.
+Now step 2: Copy mode. Well, we just put an upside-down A above our T. But that's not all,
+for Copy mode remains in effect until it is shut off-or until the enzyme is done, whichever
+comes first. This means that every base which is passed through by the enzyme while Copy
+mode is on will get a complementary base put above it. Step 3 says to look for a purine to the
+right of our T. That is the G two symbols in from the right-hand end. Now as we move up to
+that G, we must "copy"-that is, create a complementary strand. Here's what that gives;
+
+(editor’s note, I can’t print upside down ie it is too much hard work so V = A and D = G
+upside down)
+
+
+ovwovoov
+
+CAAAGAGAATCCTCTTTGAT
+
+
+The last step is to cut the strand. This will yield two pieces;
+
+VDDVDVVVJ
+
+CAAAGAGAATCCTCTTTG
+
+and AT.
+
+And the instruction packet is done. We are left with a double strand, however. Whenever this
+happens, we separate the two complementary strands from each other (general principle); so
+in fact our end product is a set of three strands:
+
+AT, CAAAGAGGA, an 4 CAAAGAGAATCCTCTTTG
+
+Notice that the upside-down strand has been turned right side up, and thereby right and left
+have been reversed.
+
+Now you have seen most of the typographical operations which can be carried out on
+strands. There are two other instructions which should be mentioned. #ne shuts off Copy
+mode; the other switches the enzyme from a strand to the upside-down strand above it. When
+this happens, if you keep the paper right side up, then you must switch "left" and "right" in all
+the instructions. #r better, you can keep the wording and just turn the paper around so the top
+strand becomes legible. If the "switch" command is
+
+
+Self-Rep and Self-Rep
+
+
+5#7
+
+
+
+given, but there is no complementary base where the enzyme is bound at that instant, then the
+enzyme just detaches itself from the strand, and its job is done.
+
+It should be mentioned that when a "cut" instruction is encountered, this pertains to
+both strands (if there are two): however, "delete" pertains only to the strand on which the
+enzyme is working. If Copy mode is on, then the "insert" command pertains to both strands-
+the base itself into the strand the enzyme is working on, and its complement into the other
+strand.
+
+If Copy mode is off', then the "insert" command pertains only to the one strand, so a blank
+space must he inserted into the complementary strand.
+
+And, whenever Copy mode is on, "move" and "search" commands require that one
+manufacture complementary bases to all bases which the sliding enzyme touches.
+Incidentally, Copy mode is always off when an enzyme starts to work. If Copy mode is off,
+and the command "Shut off copy mode" is encountered, nothing happens. Likewise, If Copy
+mode is already on, and the command "Turn copy mode on" is encountered, then nothing
+happens.
+
+
+Amino Acids
+
+There are fifteen types of command, listed below:
+
+Cut cut strand(s)
+
+del delete a base from strand
+
+swi switch enzyme to other strand
+
+mvr move one unit to the right
+
+mvl move one unit to the left
+
+cop turn on Copy mode
+
+off turn off Copy mode
+
+ina insert A to the right of this unit
+
+inc insert C to the right of this unit
+
+ing insert G to the right of this unit
+
+int insert T to the right of this unit
+
+rpy search for the nearest pyrimidine to the right
+
+rpu search for the nearest purine to the right
+
+Ipy search for the nearest pyrimidine to the left
+
+lpu search for the nearest purine to the left
+
+Each one has a three-letter abbreviation. We shall refer to the three-letter abbreviations of
+commands as amino acids. Thus, every enzyme is made up of a sequence of amino acids. Let
+us write down an arbitrary enzyme:
+
+rpu - inc - cop - myr - tnyl - swi - Tpu - int
+and an arbitrary strand:
+
+
+TAGATCCAGTCCATCGA
+
+
+Self-Rep and Self-Rep
+
+
+508
+
+
+
+and see how the enzyme acts on the strand. It so happens that the enzyme binds to G only.
+Let us bind to the middle G and begin. Search rightwards for a purine (viz., A or G). We (the
+enzyme) skip over TCC and land on A.
+
+Insert a C. Now we have
+
+
+TAGATCCAGTCCACTCGA
+
+where the arrow points to the unit to which the enzyme is bound. Set Copy mode. This puts
+an upside-down G above the C. Move right, move left, then switch to the other strand. Here's
+what we have so far:
+
+
+7V
+
+TAGATCCAGTCCACTCGA
+
+Let's turn it upside down, 'so that the enzyme is attached to the lower strand:
+
+VDJIDV» ID VDD1VDV1
+AG
+
+Now we search leftwards for a purine and find A. Copy mode is on, but the complementary
+bases are already there, so nothing is added. Finally, we insert a T (in Copy mode), and quit:
+
+VD)IVJVJJ17V)DIVDV 1
+ATG
+
+
+Our final product is thus two strands:
+
+ATG, and TAGATCCAGTCCACATCGA
+The old one is of course gone.
+
+Translation and the Typogenetic Code
+
+Now you might be wondering where the enzymes and strands come from, and how to tell the
+initial binding-preference of a given enzyme. One way might be just to throw some random
+strands and some random enzymes together, and see what happens when those enzymes act
+on those strands and their progeny. This has a similar flavor to the MU-puzzle, where there
+were some given rules of inference and an axiom, and you just began. The only difference is
+that here, every time a strand is acted on, its original form is gone forever. In the MU-puzzle,
+acting on MI to make MIU didn't destroy MI
+
+But in Typogenetics, as in real genetics, the scheme is quite a bit trickier. We do
+begin with some arbitrary strand, somewhat like an axiom in a formal system. But we have,
+initially, no "rules of inference"-that is, no enzymes. However, we can translate each strand
+into one or more enzymes! Thus, the strands themselves will dictate the operations which
+will be performed upon them, and those operations will in turn produce
+
+
+Self-Rep and Self-Rep
+
+
+509
+
+
+
+new strands which will dictate further enzymes, etc. etc.! This is mixing levels with a
+vengeance! Think, for the sake of comparison, how different the MU-puzzle would have
+been if each new theorem produced could have been turned into a new rule of inference by
+means of some code.
+
+How is this "translation" done? It involves a Typogenetic Code by which adjacent pairs of
+bases-called "duplets"-in a single strand represent different amino acids. There are sixteen
+possible duplets: AA, AC, AG, AT, CA, CC, etc. And there are fifteen amino acids. The
+Typogenetic Code is shown in Figure S7.
+
+
+Second Base
+
+
+
+A
+
+c
+
+G
+
+T
+
+A
+
+
+cut
+
+5
+
+del
+
+s
+
+swi
+
+r
+
+U
+
+| c
+
+mvr
+
+$
+
+mvl
+
+s
+
+cop
+
+r
+
+off
+
+/
+
+G
+
+ina
+
+s
+
+inc
+
+r|
+
+ing
+
+r
+
+int
+
+/
+
+T
+
+r PV
+
+r \
+
+rpu
+
+/I
+
+ip>
+
+/
+
+Ipu
+
+/
+
+
+EJCL RE H/. The Typogenetic Code, by
+whirl, each duplet in a strand codes for 'one
+of fif<em “ammo acids " (or a punctuation
+mark).
+
+
+According to the table, the translation of the duplet GC is "inc" ("insert a C"); that of AT is
+"swi" ("switch strands"); and so on. Therefore it becomes clear that a strand can dictate an
+enzyme very straightforwardly. For example, the strand
+
+TAGATCCAGTCCACATCGA
+
+breaks up into duplets as follows:
+
+TA GA TC CA GT CC AC AT CG A
+
+with the A left over at the end. Its translation into an enzyme is:
+
+rpy - ina - rpu - mvr - int - mvl - cut - swi - cop.
+
+(Note that the leftover A contributes nothing.)
+
+Tertiary Structure of Enzymes
+
+What about the little letters ’s', T, and 'r' in the lower righthand comer of each box% They are
+crucial in detennining the enzyme's binding-preference, and in a peculiar way. In order to
+figure out what letter an enzyme likes to bind to, you have to figure out the enzyme’s "tertiary
+structure", which is itself determined by the enzyme's "primary structure". By its
+
+
+Self-Rep and Self-Rep
+
+
+51#
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+primary structure is meant its amino acid sequence. By its tertiary structure is meant the way
+it likes to "fold up". The point is that enzymes don't like being in straight lines, as we have so
+far exhibited them. At each internal amino acid (all but the two ends), there is a possibility of
+a "kink", which is dictated by the letters in the corners. In particular, T and V stand for "left"
+and "right", and's' stands for "straight". So let us take our most recent sample enzyme, and let
+it fold itself up to show its tertiary structure. We will start with the enzyme's primary
+structure, and move along it from left to right. At each amino acid whose comer-letter is T
+we'll put a left turn, for those with 'r', we'll put a right turn, and at 's' we'll put no turn. In
+Figure 88 is shown the two-dimensional conformation for our enzyme.
+
+Cop
+
+
+swi <= cut <= mvl <= int
+
+mvr
+
+ipy -=> ima => rpu
+
+FIGURE 88. The tertiary structure of a typoenzyme.
+
+Note the left-kink at "rpu", the right-kink at "swi", and so on. Notice also that the first
+segment ("rpy z> ina") and the last segment ("swi => cop") are perpendicular. This is the key
+to the binding-preference. In fact, the relative orientation of the first and last segments of an
+enzyme's tertiaiy structure determines the binding-preference of the enzyme. We can always
+orient the enzyme so that its first segment points to the right. If we do so, then the last
+segment determines the binding-preference, as shown in Figure 89.
+
+
+FIGURE 89.
+
+Tahir of binding-preferences j
+
+o> hpocniymes.
+
+First Segment
+
+Last Segment
+
+Binding-letter
+
+
+=>
+
+A
+
+
+
+C
+
+
+
+G
+
+
+
+T
+
+
+Self-Rep and Self-Rep
+
+
+511
+
+
+So in our case, we have an enzyme which likes the letter C. If, in folding up, an enzyme
+happens to cross itself, that's okay-just think of it as going under or over itself. Notice that all
+its amino acids play a role in the determination of an enzyme's tertiary structure.
+
+Punctuation, Genes, and Ribosomes
+
+Now one thing remains to he explained. Why is there a blank in box AA of the typogenetic
+Code' The answer is that the duplet AA acts as a punctuation mark inside a strand, and it
+signals the end of the code for an enzyme.
+
+That is to say, one strand may code for two or more enzymes if it has one or more duplets
+AA in it. For example, the strand
+
+CG GA TA CT AA AC CG A
+
+
+Codes for two enzymes
+
+cop - ina - rpy - off
+and
+
+cut - cop
+
+with the AA serving to divide the strand up into two "genes". The definition of gene is: that
+portion of a strand which codes for a single enzyme. Note that the mere presence of AA
+inside a strand does not mean that the strand codes for two enzymes. For instance, CAAG
+codes for "mvr - del". The AA begins on an even-numbered unit and therefore is not read as a
+duplet!
+
+The mechanism which reads strands and produces the enzymes which are coded
+inside them is called a ribosome. (In Typogenetics, the player of the game does the work of
+the ribosomes.) Ribosomes are not in any way responsible for the tertiary structure of
+enzymes, for that is entirely determined once the primary structure is created. Incidentally,
+the process of translation always goes from strands to enzymes, and never in the reverse
+direction.
+
+
+Puzzle: A Typogenetical Self-Rep
+
+Now that the rules of Typogenetics have been fully set out, you may find it interesting to
+experiment with the game. In particular, it would he most interesting to devise a self-
+replicating strand. This would mean something along the following lines. A single strand is
+written down. A ribosome acts on it, to produce any or all of the enzymes which are coded
+for in the strand. Then those enzymes are brought into contact with the original strand, and
+allowed to work on it. This yields a set of "daughter strands". The daughter strands
+themselves pass through the rihosomes, to yield a second generation of enzymes, which act
+on the daughter strands; and the
+
+
+Self-Rep and Self-Rep
+
+
+512
+
+
+
+cycle goes on and on. This can go on for any number of stages; the hope is that eventually,
+among the strands which are present at some point, there
+
+will be found two copies of the original strand (one of the copies may be, in fact, the original
+strand).
+
+
+The Central Dogma of Typogenetics
+
+
+Typogenetical processes can be represented in skeletal fortn in a diagram (Fig. 90).
+enzymes
+
+
+translation
+ina nbosomes
+
+
+\ . FIGURE 90. The ‘'Central Dogma of
+
+1 typographical Typogenetics ‘; an example of a 'Tangled
+
+1 manipulation Hierarchy’.
+
+
+strands
+
+
+This diagram illustrates the Central Dogma of Typogenetics. It shows how strands define
+enzymes (via the Typogenetic Code); and how in turn, enzymes act back on the strands
+which gave rise to them, yielding new strands. Therefore, the line on the left portrays how
+old information flows upwards, in the sense that an enzyme is a translation of a strand, and
+contains therefore the same information as the strand, only in a different fonn-in particular;
+in an active form. The line on the right, however, does not show information flowing
+downwards; instead, it shows how new information gets created: by the shunting of symbols
+in strands.
+
+An enzyme in Typogenetics, like a rule of inference in a fonnal system, blindly
+shunts symbols in strands without regard to any "meaning" which may lurk in those symbols.
+So there is a curious mixture of levels here. On the one hand, strands are acted upon, and
+therefore play the role of data (as is indicated by the arrow on the right); on the other hand,
+they also dictate the actions which are to be performed on the data, and therefore they play
+the role of programs (as is indicated by the arrow on the left). It is the player of Typogenetics
+who acts as interpreter and processor, of course. The two-way street which links "upper" and
+"lower" levels of Typogenetics shows that, in fact, neither strands nor enzymes can be
+thought of as being on a higher level than the other. By contrast, a picture of the Central
+Dogma of the M/F/-system looks this way:
+
+rules of inference
+
+(typographical manipulation)
+
+■ \ i
+
+strings
+
+
+In the MlU-system, there is a clear distinction of levels: rules of inference simply belong to a
+higher level than strings. Similarly for TNT, and all fonnal systems.
+
+
+Self-Rep and Self -Rep
+
+
+513
+
+
+
+Strange Loops, TNT, and Real Genetics
+
+
+However, we have seen that in TNT, levels are mixed, in another sense. In fact, the
+distinction between language and metalanguage breaks down: statements about the system
+get mirrored inside the system. It turns out that if we make a diagram showing the
+relationship between TNT and its metalanguage, we will produce something which resembles
+in a remarkable way the diagram which represents the Central Dogma of Molecular Biology.
+In fact, it is our goal to make this comparison in detail; but to do so, we need to indicate the
+places where Typogenetics and true genetics coincide, and where they differ. Of course, real
+genetics is far more complex than Typogenetics-but the "conceptual skeleton" which the
+reader has acquired in understanding Typogenetics will be very useful as a guide in the
+labyrinth of true genetics.
+
+
+DNA and Nucleotides
+
+We begin by discussing the relationship between "strands", and DNA. The initials "DNA"
+stand for "deoxyribonucleic acid". The DNA of most cells resides in the cell's nucleus, which
+is a small area protected by a membrane. Gunther Stent has characterized the nucleus as the
+"throne room" of the cell, with DNA acting as the ruler. DNA consists of long chains of
+relatively simple molecules called nucleotides. Each nucleotide is made up of three parts: (1)
+a phosphate group stripped of one special oxygen atom, whence the prefix "deoxy"; (2) a
+sugar called "ribose", and (3) a base. It is the base alone which distinguishes one nucleotide
+from another; thus it suffices to specify its base to identify a nucleotide. The four types of
+bases which occur in DNA nucleotides are:
+
+
+A:
+
+adenine
+
+C:
+
+guanine
+
+C:
+
+cytosine
+
+X:
+
+thymine
+
+
+purines
+^ pyrimidines
+
+
+(Also see Fig. 91.) It is easy to remember which ones are pyrimidines because the first vowel
+in "cytosine", "thymine", and "pyrimidine" is 'y\ Later, when we talk about RNA, "uracil"-
+also a pyrimidine-will come in and wreck the pattern, unfortunately. (Note: Letters
+representing nucleotides in real genetics will not be in the Quadrata font, as they were in
+Typogenetics.)
+
+A single strand of DNA thus consists of many nucleotides strung together like a chain
+of beads. The chemical bond which links a nucleotide to its two neighbors is very strong;
+such bonds are called covalent bonds, and the "chain of beads" is often called the covalent
+backbone of DNA.
+
+Now DNA usually comes in double strands-that is, two single strands which are paired up,
+nucleotide by nucleotide (see Fig. 92). It is the bases
+
+
+Self-Rep and Self-Rep
+
+
+514
+
+
+
+THVMINE H CYTOSINE H
+
+FIGURE 91. The four constituent bases of DNA: Adenine, Guanine, Thymine, Cytosine.
+[From Hanawalt and Haynes, The Chemical Basis of Life (San Francisco: W. H. Freeman,
+1973), p. 142 J
+
+
+FIGURE 92. DNA structure resembles a ladder in which the side pieces consist of alternating
+units of deoxyrihose and phosphate. The rungs are formed by the bases paired in a special
+way, A with T and G with C, and held together respectively by two and three hydrogen
+bonds. [From Hanawalt and Haynes, The Chemical Basis of Life, p. 142.
+
+
+
+Self-Rep and Self-Rep
+
+
+515
+
+
+which are responsible for the peculiar kind of pairing which takes place between strands.
+Each base in one strand faces a complementary base in the other strand, and binds to it. The
+complements are as in Typogenetics: A pairs up with T, and C with G. Always one purine
+pairs up with a pyrimidine.
+
+Compared to the strong covalent bonds along the backbone, the interstrand bonds are
+quite weak. They are not covalent bonds, but hydrogen bonds. A hydrogen bond arises when
+two molecular complexes are aligned in such a way that a hydrogen atom which originally
+belonged to one of them becomes "confused" about which one it belongs to, and it hovers
+between the two complexes, vacillating as to which one to join. Because the two halves of
+double-stranded DNA are held together only by hydrogen bonds, they may come apart or be
+put together relatively easily; and this fact is of great import for the workings of the cell.
+
+When DNA forms double strands, the two strands curl around each other like
+twisting vines (Fig. 93). There are exactly ten nucleotide pairs per revolution; in other words,
+at each nucleotide, the "twist" is 36 degrees. Single-stranded DNA does not exhibit this kind
+of coiling, for it is a consequence of the base-pairing.
+
+
+
+FIGURE 93. Molecular model of the DNA double helix. [From Vernon M. Ingram,
+Biosynthesis (Menlo Park, Calif.: W. A. Benjamin, 1972)
+
+
+Self-Rep and Self-Rep
+
+
+516
+
+
+
+
+
+
+
+Messenger KNA and Ribosomes
+
+
+As was mentioned above, in many cells, DNA, the ruler of the cell, dwells in its private
+"throne room": the nucleus of the cell. But most of the "living" in a cell goes on outside of
+the nucleus, namely in the cytoplasm -the "ground" to the nucleus' "figure". In particular,
+enzymes, which make practically every life process go, are manufactured by ribosomes in the
+cytoplasm, and they do most of their work in the cytoplasm. And just as in Typogenetics, the
+blueprints for all enzymes are stored inside the strands-that is, inside the DNA, which
+remains protected in its little nuclear home. So how does the information about enzyme
+structure get from the nucleus to the ribosomes'
+
+Here is where messenger RNA-mRNA-comes in. Earlier, mRNA strands were
+humorously said to constitute a kind of DNA Rapid Transit Service; by this is meant not that
+mRNA physically carries DNA anywhere, but rather that it serves to carry the information, or
+message, stored in the DNA in its nuclear chambers, out to the ribosomes in the cytoplasm.
+How is this done? The idea is easy: a special kind of enzyme inside the nucleus faithfully
+copies long stretches of the DNA's base sequence onto a new strand-a strand of messenger
+RNA. This mRNA then departs from the nucleus and wanders out into the cytoplasm, where
+it runs into many ribosomes which begin doing their enzyme-creating work on it.
+
+The process by which DNA gets copied onto mRNA inside the nucleus is called
+transcription', in it, the double-stranded DNA must be temporarily separated into two single
+strands, one of which serves as a template for the mRNA. Incidentally, "RNA" stands for
+"ribonucleic acid", and it is very much like DNA except that all of its nucleotides possess that
+special oxygen atom in the phosphate group which DNA's nucleotides lack. Therefore the
+"deoxy" prefix is dropped. Also, instead of thymine, RNA uses the' base uracil, so the
+information in strands of RNA can be represented by arbitrary sequences of the four letters
+'A', 'C', 'G', 'U'. Now when mRNA is transcribed off of DNA, the transcription process
+operates via the usual base-pairing (except with U instead of T), so that a DNA-template and
+its mRNA-mate might look something like this:
+
+DNA: GGTAAATCAAGTCA (template)
+
+mRNA: GGCAUUUAGUCAGU (copy")
+
+RNA does not generally form long double strands with itself, although it can. Therefore it is
+prevalently found not in the helical form which so characterizes DNA, but rather in long,
+somewhat randomly curving strands.
+
+Once a strand of mRNA has escaped the nucleus, it encounters those strange
+subcellular creatures called "ribosomes"-but before we go on to explain how a ribosome uses
+mRNA, I want to make some comments about enzymes and proteins. Enzymes belong to the
+general category of biomolecules called proteins, and the job of ribosomes is to make all pro
+
+
+Self-Rep and Self-Rep
+
+
+517
+
+
+
+teins, not just enzymes. Proteins which are not enzymes are much more passive kinds of
+beings: many of them, for instance, are structural molecules, which means that they are like
+girders and beams and so forth in buildings: they hold the cell's parts together. There are
+other kinds of proteins, but for our purposes, the principal proteins are enzymes, and I will
+henceforth not make a sharp distinction.
+
+
+Amino Acids
+
+Proteins are composed of sequences of amino acids, which come in twenty primary varieties,
+each with a three-letter abbreviation:
+
+
+ala alanine
+arg arginine
+asn asparagines
+asp aspartic acid
+cys cysteine
+gin glutamine
+glu glutamic acid
+gly glycine
+his histidine
+He isoleucine
+leu leucine
+lys lysine
+met methionine
+phe phenylalanine
+pro praline
+ser serine
+thr threonine
+trp tryptophan
+tyr tyrosine
+val valine
+
+
+Notice the slight numerical discrepancy with Typogenetics, where we had only fifteen
+"amino acids" composing enzymes. An amino acid is a small molecule of roughly the same
+complexity as a nucleotide; hence the building blocks of proteins and of nucleic acids (DNA,
+RNA) are roughly of the same size. However, proteins are composed of much shorter
+sequences of components: typically, about three hundred amino acids make a complete
+protein, whereas a strand of DNA can consist of hundreds of thousands or millions of
+nucleotides.
+
+
+Ribosomes and Tape Recorders
+
+Now when a strand of mRNA, after its escape into the cytoplasm, encounters a ribosome, a
+very intricate and beautiful process called translation takes place. It could be said that this
+process of translation is at the very heart of
+
+
+Self-Rep and Self-Rep
+
+
+518
+
+
+
+all of life, and there are many mysteries connected with it. But in essence it is easy to
+describe. Let us first give a picturesque image, and then render it more precise. Imagine the
+mRNA to be like a long piece of magnetic recording tape, and the ribosome to be like a tape
+recorder. As the tape passes through the playing head of the recorder, it is "read" and
+converted into music, or other sounds. Thus magnetic markings are "translated" into notes.
+Similarly, when a "tape" of mRNA passes through the "playing head" of a ribosome, the
+"notes" which are produced are amine acids, and the "pieces of music" which they make up
+are proteins. This is what translation is all about: it is shown in Figure 96.
+
+
+The Genetic Code
+
+But how can a ribosome produce a chain of amino acids when it is reading a chain of
+nucleotides This mystery was solved in the early 1960's by the efforts of a large number
+of people, and at the core of the answer lies the Genetic Code-a mapping from triplets of
+nucleotides into amino acids (see Fig. 94). This is in spirit extremely similar to the
+Typogenetic Code, except that here, three consecutive bases (or
+
+
+nucleotides)
+whereas there,
+needed. Thus
+4x4x4 (equals
+entries in the
+sixteen. A
+down a strand
+nucleotides at
+to say, one.
+-and each
+
+
+CUA GAU
+, C u A g A u
+
+A typical segment of mRNA
+read first as two triplets
+(above), and second as three
+duplets (below): an example
+of hemiolia in biochemistry
+
+
+form a codon,
+only two were
+there must be
+64) different
+table, instead of
+ribosome clicks
+of RNA three
+a time-which is
+codon at a time
+time it does so,
+
+
+it appends a single new amino acid to the protein it is presently manufacturing. Thus, a
+protein comes out of the ribosome amino acid by amino acid.
+
+
+Tertiary Structure
+
+
+However, as a protein emerges from a ribosome, it is not only getting longer and longer, but
+it is also continually folding itself up into an extraordinary three-dimensional shape, very
+much in the way that those funny little Fourth-of-July fireworks called "snakes"
+simultaneously grow longer and curl up, when they are lit. This fancy shape is called the
+protein's tertiary structure (Fig. 95), while the amino acid sequence per se is called the
+primary structure of the protein. The tertiary structure is implicit in the primary structure, just
+as in Typogenetics. However, the recipe for deriving the tertiary structure, if you know only
+the primary structure, is by far more complex than that given in Typogenetics. In fact, it is
+one of the outstanding problems of contemporary molecular biology to figure out some rules
+by which the tertiary structure of a protein can be predicted if only its primary structure is
+known.
+
+
+Self-Rep and Self-Rep
+
+
+519
+
+
+
+The Genetic Code.
+
+
+
+U
+
+c
+
+A
+
+G
+
+
+
+phe
+
+ser
+
+tyr
+
+cys
+
+u
+
+u
+
+phe
+
+ser
+
+tyr
+
+cys
+
+c
+
+leu
+
+ser
+
+punt.
+
+punt.
+
+A
+
+
+leu
+
+ser
+
+punc.
+
+trp
+
+G
+
+
+leu
+
+pro
+
+his
+
+arg
+
+u
+
+
+leu
+
+pro
+
+his
+
+arg
+
+c
+
+c
+
+leu
+
+pro
+
+gin
+
+arg
+
+A
+
+
+leu
+
+pro
+
+gin
+
+arg
+
+G
+
+
+ile
+
+thr
+
+asn
+
+ser
+
+U
+
+
+ile
+
+thr
+
+asn
+
+ser
+
+c
+
+A
+
+ile
+
+thr
+
+lys
+
+arg
+
+A
+
+
+met
+
+thr
+
+lys
+
+arg
+
+G
+
+G
+
+val
+
+ala
+
+asp
+
+giy
+
+U
+
+
+val
+
+ala
+
+asp
+
+giy
+
+c
+
+
+val
+
+ala
+
+glu
+
+giy
+
+A
+
+
+val
+
+ala
+
+glu
+
+giy
+
+G
+
+
+FIGURE 94. The Genetic Code, by which each triplet in a strand of messenger RNA codes
+for one of twenty amino acids (or a punctuation mark).
+
+Reductionistic Explanation of Protein Function
+
+Another discrepancy between Typogenetics and true genetics-and this is probably the most
+serious one of all-is this: whereas in Typogenetics, each component amino acid of an enzyme
+is responsible for some specific "piece of the action", in real enzymes, individual amino acids
+cannot be assigned such clear roles. It is the tertiary structure as a whole which determines
+the mode in which an enzyme will function; there is no way one can say, "This
+
+
+Self-Rep and Self-Rep
+
+
+520
+
+
+
+amino acid’s presence means that such-and-such an operation will get performed". In other
+words, in real genetics, an individual amino acid's contribution to the enzyme's overall
+function is not "context-free". However, this fact should not be construed in any way as
+ammunition for an anti reductionist argument to the effect that "the whole [enzyme] cannot
+be explained as the sum of its parts". That would he wholly unjustified. What is justified is
+rejection of the simpler claim that "each amino acid contributes to the sum in a manner which
+is independent of the other amino acids present". In other words, the function of a protein
+cannot be considered to be built up from context-free functions of its parts; rather, one must
+consider how the parts interact. It is still possible in principle to write a computer program
+which takes as input the primary structure of a protein,
+
+FIGURE 95. The structure of myoglobin, deduced from high-resolution X-ray data. The
+large-scale "twisted pipe" appearance is the tertiary structure; the finer helix inside-the
+"alpha helix"-is the secondary structure. [From A. Lehninger, Biochemistry]
+
+
+
+Self-Rep and Self-Rep
+
+
+521
+
+
+
+
+
+
+
+and firstly determines its tertiary structure, and secondly determines the function of the
+enzyme. This would be a completely reductionistic explanation of the workings of proteins,
+but the determination of the "sum" of the parts would require a highly complex algorithm.
+The elucidation of the function of an enzyme, given its primary, or even its tertiary, structure,
+is another great problem of contemporary molecular biology.
+
+Perhaps, in the last analysis, the function of the whole enzyme can be considered to
+be built up from functions of parts in a context-free manner, but where the parts are now
+considered to be individual particles, such as electrons and protons, rather than "chunks",
+such as amino acids. This exemplifies the "Reductionist's Dilemma": In order to explain
+everything in terms of context free sums, one has to go down to the level of physics; but then
+the number of particles is so huge as to make it only a theoretical "in-principle" kind of thing.
+So, one has to settle for a context-dependent sum, which has two disadvantages. The first is
+that the parts are much larger units, whose behavior is describable only on a high level, and
+therefore indeterminately. The second is that the word "sum" carries the connotation that
+each part can be assigned a simple function and that the function of the whole is just a
+context-free sum of those individual functions. This just cannot be done when one tries to
+explain a whole enzyme's function, given its amino acids as parts. But for better or for worse,
+this is a general phenomenon which arises in the explanations of complex systems. In order
+to acquire an intuitive and manageable understanding of how parts interact-in short, in order
+to proceed-one often has to sacrifice the exactness yielded by a microscopic, context-free
+picture, simply because of its unmanageability. But one does not sacrifice at that time the
+faith that such an explanation exists in principle.
+
+
+Transfer RNA and Ribosomes
+
+Returning, then, to ribosomes and RNA and proteins, we have stated that a protein is
+manufactured by a ribosome according to the blueprint carried from the DNA's "royal
+chambers" by its messenger, RNA. This seems to imply that the ribosome can translate from
+the language of codons into the language of amino acids, which amounts to saying that the
+ribosome "knows" the Genetic Code. However, that amount of information is simply not
+present in a ribosome. So how does it do it? Where is the Genetic Code stored? The curious
+fact is that the Genetic Code is stored-where else?-in the DNA itself. This certainly calls for
+some explanation.
+
+Let us back off from a total explanation for a moment, and give a partial explanation.
+There are, floating about in the cytoplasm at any given moment, large numbers of four-leaf-
+clover-shaped molecules; loosely fastened (i.e., hydrogen-bonded) to one leaf is an amino
+acid, and on the opposite leaf there is a triplet of nucleotides called an anticodon. For our
+purposes, the other two leaves are irrelevant. Here is how these "clovers" are used by the
+ribosomes in their production of proteins. When a new
+
+
+Self-Rep and Self-Rep
+
+
+522
+
+
+
+
+FIGURE 96. A section of mRNA passing through a ribosome. Floating nearby are tRNA
+molecules, carrying amino acids which are stripped off by the ribosome and appended to the
+growing protein. The Genetic Code is contained in the tRNA molecules, collectively. Note
+how the base-pairing (A-U, C-G) is represented by interlocking letter-forms in the diagram.
+[Drawing by Scott E. Kim]
+
+codon of mRNA clicks into position in the ribosome’s "playing head", the ribosome reaches
+out into the cytoplasm and latches onto a clover whose anticodon is complementary to the
+mRNA codon. Then it pulls the clover into such a position that it can rip off the clover's
+amino acid, and stick it covalently onto the growing protein. (Incidentally, the bond between
+an amino acid and its neighbor in a protein is a very strong covalent bond, called a "peptide
+bond". For this reason, proteins are sometimes called "polypeptides".) Of course it is no
+accident that the "clovers" carry the proper amino acids, for they have all been manufactured
+according to precise instructions emanating from the "throne room".
+
+
+Self-Rep and Self-Rep
+
+
+523
+
+
+
+
+The real name for such a clover is transfer RNA. A molecule of tRNA is quite small-about
+the size of a very small protein-and consists of a chain of about eighty nucleotides. Like
+mRNA, tRNA molecules are made by transcription off of the grand cellular template, DNA.
+However, tRNA's are tiny by comparison with the huge mRNA molecules, which may
+contain thousands of nucleotides in long, long chains. Also, tRNA's resemble proteins (and
+are unlike strands of mRNA) in this respect: they have fixed, well-defined tertiary structures-
+determined by their primary structure. A tRNA molecule's tertiary structure allows precisely
+one amino acid to bind to its amino-acid site: to be sure, it is that one dictated according to
+the Genetic Code by the anticodon on the opposite arm. A vivid image of the function of
+tRNA molecules is as flashcards floating in a cloud around a simultaneous interpreter, who
+snaps one out of the air-invariably the right one!-whenever he needs to translate a word. In
+this case, the interpreter is the ribosome, the words are codons, and their translations are
+amino acids.
+
+In order for the inner message of DNA to get decoded by the ribosomes, the tRNA
+flashcards must be floating about in the cytoplasm. In some sense, the tRNAs contain the
+essence of the outer message of the DNA, since they are the keys to the process of
+translation. But they themselves came from the DNA. Thus, the outer message is trying to be
+part of the inner message, in a way reminiscent of the message-in-a-bottle which tells what
+language it is written in. Naturally, no such attempt can be totally successful: there is no way
+for the DNA to hoist itself by its own bootstraps. Some amount of knowledge of the Genetic
+Code must already be present in the cell beforehand, to allow the manufacture of those
+enzymes which transcribe tRNA's themselves off of the master copy of DNA. And this
+knowledge resides in previously manufactured tRNA molecules. This attempt to obviate the
+need for any outer message at all is like the Escher dragon, who tries as hard as he can,
+within the context of the two-dimensional world to which he is constrained, to be
+threedimensional. He seems to go a long way-but of course he never makes it, despite the
+fine imitation he gives of three-dimensionality.
+
+Punctuation and the Reading Frame
+
+How does a ribosome know when a protein is done? Just as in Typogenetics, there is a signal
+inside the mRNA which indicates the termination or initiation of a protein. In fact, three
+special codons-UAA, CAG, UGA act as punctuation marks instead of coding for amino
+acids. Whenever such a triplet clicks its way into the "reading head" of a ribosome, the
+ribosome releases the protein under construction and begins a new one.
+
+Recently, the entire genome of the tiniest known virus, cp)X174, has been laid bare.
+One most unexpected discovery was made en route: some of its nine genes overlap-that is,
+two distinct proteins are coded for by the same stretch of DNA\ There is even one gene
+contained entirely inside another!
+
+
+Self-Rep and Self-Rep
+
+
+524
+
+
+
+This is accomplished by having the reading frames of the two genes shifted relative to each
+other, by exactly one unit. The density of information packing in such a scheme is incredible.
+This is, of course, the inspiration behind the strange "5/17 haiku" in Achilles' fortune cookie,
+in the Canon bylntervallic Augmentation,
+
+
+Recap
+
+In brief, then, this picture emerges: from its central throne, DNA sends off long strands of
+messenger RNA to the ribosomes in the cytoplasm; and the ribosomes, making use of the
+"flashcards" of tRNA hovering about them, efficiently construct proteins, amino acid by
+amino acid, according to the blueprint contained in the mRNA. Only the primary structure of
+the proteins is dictated by the DNA; but this is enough, for as they emerge from the
+ribosomes, the proteins "magically" fold up into complex conformations which then have the
+ability to act as powerful chemical machines.
+
+Levels of Structure and Meaning in Proteins and Music
+
+We have been using this image of ribosome as tape recorder, rnRNA as tape, and protein as
+music. It may seem arbitrary, and yet there are some beautiful parallels. Music is not a mere
+linear sequence of notes. Our minds perceive pieces of music on a level far higher than that.
+We chunk notes into phrases, phrases into melodies, melodies into movements, and
+movements into full pieces. Similarly, proteins only make sense when they act as chunked
+units. Although a primary structure carries all the information for the tertiary structure to be
+created, it still "feels" like less, for its potential is only realized when the tertiary structure is
+actually physically created.
+
+Incidentally, we have been referring only to primary and tertiary structures, and you may
+well wonder whatever happened to the secondary structure. Indeed, it exists, as does a
+quaternary structure, as well. The folding-up of a protein occurs at more than one level.
+Specifically, at some points along the chain of amino acids, there may be a tendency to form
+a kind of helix, called the alpha helix (not to be confused with the DNA double helix). This
+helical twisting of a protein is on a lower level than its tertiary structure. This level of
+structure is visible in Figure 95. Quaternary structure can be directly compared with the
+building of 'a musical piece out of independent movements, for it involves the assembly of
+several distinct polypeptides, already in their full-blown tertiary beauty, into a larger
+structure. The binding of these independent chains is usually accomplished by hydrogen
+bonds, rather than covalent bonds; this is of course just as with pieces of music composed of
+several movements, which are far less tightly bound to each other than they are internally, but
+which nevertheless form a tight "organic" whole.
+
+The four levels of primary, secondary, tertiary, and quaternary structure can also be
+compared to the four levels of the MU-picture (Fig. 60) in
+
+
+Self-Rep and Self-Rep
+
+
+525
+
+
+
+50S SOS
+
+Incoming
+ribosomal
+subunits
+
+
+
+\ /
+
+
+
+70S
+
+
+
+4
+
+Complete
+
+peptide
+
+chain
+
+
+
+FIGURE 97. A polyribosome. A single
+strand of mRNA passes through one ribo¬
+some after another, like one tape passmg
+through several tape recorders in a rote.
+The result is a set of growing proteins tn
+various stages of completion: the analogue
+to a musical canon produced by the
+staggered tape recorders. [From A.
+Lehnmget, Biochemistry.]
+
+
+Self-Rep and Self-Rep
+
+
+526
+
+
+
+
+the Prelude, Ant Fugue. The global structure-consisting of the letters 'M' and ’U'-is its
+quaternary structure; then each of those two parts has a tertiary structure, consisting of
+"HOLISM" or "REDUCTIONISM"; and then the opposite word exists on the secondary
+level, and at bottom, the primary structure is once again the word "MU", over and over again.
+
+Polyribosomes and Two-Tiered Canons
+
+Now we come to another lovely parallel between tape recorders translating tape into music
+and ribosomes translating mRNA into proteins. Imagine a collection of many tape recorders,
+arranged in a row, evenly spaced. We might call this array a "polyrecorder". Now imagine a
+single tape passing serially through the playing heads of all the component recorders. If the
+tape contains a single long melody, then the output will be a many-voiced canon, of course,
+with the delay determined by the time it takes the tape to get from one tape recorder to the
+next. In cells, such "molecular canons" do indeed exist, where many ribosomes, spaced out in
+long lines-forming what is called a polyribosome-all "play" the same strand of mRNA,
+producing identical proteins, staggered in time (see Fig. 97).
+
+Not only this, but nature goes one better. Recall that mRNA is made by transcription
+off of DNA; the enzymes which are responsible for this process are called RNA polymerases
+("-ase" is a general suffix for enzymes). It happens often that a series of RNA polymerases
+will be at work in parallel on a single strand of DNA, with the result that many separate (but
+identical) strands of mRNA are being produced, each delayed with respect to the other by the
+time required for the DNA to slide from one RNA polymerase to the next. At the same time,
+there can be several different ribosomes working on each of the parallel emerging mRNA's.
+Thus one arrives at a double-decker, or two-tiered, "molecular canon" (Fig. 98). The
+corresponding image in music is a rather fanciful but amusing scenario: several
+
+FIGURE 98. Here, an even more complex scheme. Not just one but several strands of
+mRNA, all emerging by transcription from a single strand of DNA, are acted upon by
+polyribosomes. The result is a two-tiered molecular canon. [From Hanawalt and Haynes, The
+Chemical Basis of Life, p. 271]
+
+
+
+Self-Rep and Self-Rep
+
+
+527
+
+
+
+
+
+
+
+
+
+
+
+different copyists are all at work simultaneously, each one of them copying the same original
+manuscript from a clef which flutists cannot read into a clef which they can read. As each
+copyist finishes a page of the original manuscript, he passes it on to the next copyist, and
+starts transcribing a new page himself. Meanwhile, from each score emerging from the pens
+of the copyists, a set of flutists are reading and tooting the melody, each flutist delayed with
+respect to the others who are reading from the same sheet. This rather wild image gives,
+perhaps, an idea of some of the complexity of the processes which are going on in each and
+every cell of your body during every second of every day ...
+
+Which Came First-The Ribosome or the Protein?
+
+We have been talking about these wonderful beasts called ribosomes; but what are they
+themselves composed of? How are they made? Ribosomes are composed of two types of
+things: (1) various kinds of proteins, and (2) another kind of RNA, called ribosomal RNA
+(rRNA). Thus, in order for a ribosome to be made, certain kinds of proteins must be present,
+and rRNA must be present. Of course, for proteins to be present, ribosomes must be there to
+make them. So how do you get around the vicious circle? Which comes first-the ribosome or
+the protein? Which makes which? Of course there is no answer because one always traces
+things back to previous members of the same class just as with the chicken-and-the-egg
+question-until everything vanishes over the horizon of time. In any case, ribosomes are made
+of two pieces, a large and a small one, each of which contains some rRNA and some proteins.
+Ribosomes are about the size of large proteins; they are much much smaller than the strands
+of mRNA which they take as input, and along which they move.
+
+Protein Function
+
+We have spoken somewhat of the structure of proteins-specifically enzymes; but we have not
+really mentioned the kinds of tasks which they perform in the cell, nor how they do them. All
+enzymes are catalysts, which means that in a certain sense, they do no more than selectively
+accelerate various chemical processes in the cell, rather than make things happen which
+without them never could happen. An enzyme realizes certain pathways out of the myriad
+myriad potentialities. Therefore, in choosing which enzymes shall be present, you choose
+what shall happen and what shall not happen-despite the fact that, theoretically speaking,
+there is a nonzero probability for any cellular process to happen spontaneously, without the
+aid of catalysts.
+
+Now how do enzymes act upon the molecules of the cell? As has been mentioned,
+enzymes are folded-up polypeptide chains. In every enzyme, there is a cleft or pocket or
+some other clearly-defined surface feature where the enzyme hinds to some other kind of
+molecule. This location is
+
+
+Self-Rep and Self-Rep
+
+
+528
+
+
+
+called its active site, and any molecule which gets bound there is called a substrate. Enzymes
+may have more than one active site, and more than one substrate. Just as in Typogenetics,
+enzymes are indeed very choosy about what they will operate upon. The active site usually is
+quite specific, and allows just one kind of molecule to bind to it, although there are
+sometimes "decoys"-other molecules which can fit in the active site and clog it up, fooling
+the enzyme and in fact rendering it inactive.
+
+Once an enzyme and its substrate are bound together, there is some disequilibrium of
+electric charge, and consequently charge-in the form of electrons and protons-flows around
+the bound molecules and readjusts itself. By the time equilibrium has been reached, some
+rather profound chemical changes may have occurred to the substrate. Some examples are
+these: there may have been a "welding", in which some standard small molecule got tacked
+onto a nucleotide, amino acid, or other common cellular molecule; a DNA strand may have
+been "nicked" at a particular location; some piece of a molecule may have gotten lopped off;
+and so forth. In fact, bio-enzymes do operations on molecules which are quite similar to the
+typographical operations which Typo-enzymes perform. However, most enzymes perform
+essentially only a single task, rather than a sequence of tasks. There is one other striking
+difference between Typoenzymes and bio-enzymes, which is this: whereas Typo-enzymes
+operate only on strands, bio-enzymes can act on DNA, RNA, other proteins, ribosomes, cell
+membranes-in short, on anything and everything in the cell. In other words, enzymes are the
+universal mechanisms for getting things done in the cell. There are enzymes which stick
+things together and take them apart and modify them and activate them and deactivate them
+and copy them and repair them and destroy them .. .
+
+Some of the most complex processes in the cell involve "cascades" in which a single
+molecule of some type triggers the production of a certain kind of enzyme; the manufacturing
+process begins and the enzymes which come off the "assembly line" open up a new chemical
+pathway which allows a second kind of enzyme to be produced. This kind of thing can go on
+for three or four levels, each newly produced type of enzyme triggering the production of
+another type. In the end a "shower" of copies of the final type of enzyme is produced, and all
+of the copies go off and do their specialized thing, which may be to chop up some "foreign"
+DNA, or to help make some amino acid for which the cell is very "thirsty", or whatever.
+
+
+Need for a Sufficiently Strong Support System
+
+Let us describe nature's solution to the puzzle posed for Typogenetics: "What kind of strand
+of DNA can direct its own replication?" Certainly not every strand of DNA is inherently a
+self-rep. The key point is this: any strand which wishes to direct its own copying must
+contain directions for assembling precisely those enzymes which can carry out the task. Now
+it is futile to hope that a strand of DNA in isolation could be a self-rep; for in
+
+
+Self-Rep and Self-Rep
+
+
+529
+
+
+
+order for those potential proteins to be pulled out of the DNA, there must not only be
+ribosomes, but also RNA polymerase, which makes the mRNA that gets transported to the
+ribosomes. And so we have to begin by assuming a kind of "minimal support system" just
+sufficiently strong that it allows transcription and translation to be carried out. This minimal
+support system will thus consist in (1) some proteins, such as RNA polymerase, which allow
+mRNA to be made from DNA, and (2) some ribosomes.
+
+How DNA Self-Replicates
+
+It is not by any means coincidental that the phrases "sufficiently strong support system" and
+"sufficiently powerful formal system" sound alike. One is the precondition for a self-rep to
+arise, the other for a self-ref to arise. In fact there is in essence only one phenomenon going
+on in two very different guises, and we shall explicitly map this out shortly. But before we do
+so, let us finish the description of how a strand of DNA can be a self-rep.
+
+The DNA must contain the codes for a set of proteins which will copy it. Now there
+is a very efficient and elegant way to copy a double-stranded piece of DNA, whose two
+strands are complementary. This involves two steps:
+
+(1) unravel the two strands from each other;
+
+(2) mate" a new strand to each of the two new single strands.
+
+This process will create two new double strands of DNA, each identical to the original one.
+Now if our solution is to be based on this idea, it must involve a set of proteins, coded for in
+the DNA itself, which will carry out these two steps.
+
+It is believed that in cells, these two steps are performed together in a coordinated
+way, and that they require three principal enzymes: DNA endonuclease, DNA polymerase,
+and DNA ligase. The first is an "unzipping enzyme": it peels the two original strands apart
+for a short distance, and then stops. Then the other two enzymes come into the picture. The
+DNA polymerase is basically a copy-and-move enzyme: it chugs down the short single
+strands of DNA, copying them complementarity in a fashion reminiscent of the Copy mode
+in Typogenetics. In order to copy, it draws on raw materials-specifically nucleotides-which
+are floating about in the cytoplasm. Because the action proceeds in fits and starts, with some
+unzipping and some copying each time, some short gaps are created, and the DNA ligase is
+what plugs them up. The process is repeated over and over again. This precision three-
+enzyme machine proceeds in careful fashion all the way down the length of the DNA
+molecule, until the whole thing has been peeled apart and simultaneously replicated, so that
+there are now two copies of it.
+
+
+Self-Rep and Self-Rep
+
+
+530
+
+
+
+Comparison of DNA's Self-Rep Method with Quining
+
+
+Note that in the enzymatic action on the DNA strands, the fact that information is stored in
+the DNA is just plain irrelevant; the enzymes are merely carrying out their symbol-shunting
+functions, just like rules of inference in the MMIU-system. It is of no interest to the three
+enzymes that at some point they are actually copying the very genes which coded for them.
+The DNA, to them, is just a template without meaning or interest.
+
+It is quite interesting to compare this with the Quine sentence's method of describing
+how to construct a copy of itself. There, too, one has a sort of "double strand"-two copies of
+the same information, where one copy acts as instructions, the other as template. In DNA, the
+process is vaguely parallel, since the three enzymes (DNA endonuclease, DNA polymerase,
+DNA ligase) are coded for in just one of the two strands, which therefore acts as program,
+while the other strand is merely a template. The parallel is not perfect, for when the copying
+is carried out, both strands are used as template, not just one. Nevertheless, the analogy is
+highly suggestive. There is a biochemical analogue to the use-mention dichotomy: when
+DNA is treated as a mere sequence of chemicals to be copied, it is like mention of
+typographical symbols; when DNA is dictating what operations shall he carried out, it is like
+use of typographical symbols.
+
+
+Levels of Meaning of DNA
+
+There are several levels of meaning which can be read from a strand of DNA, depending on
+how big the chunks are which you look at, and how powerful a decoder you use. On the
+lowest level, each DNA strand codes for an equivalent RNA strand-the process of decoding
+being transcription. If one chunks the DNA into triplets, then by using a "genetic decoder",
+one can read the DNA as a sequence of amino acids. This is translation (on top of
+transcription). On the next natural level of the hierarchy, DNA is readable as a code for a set
+of proteins. The physical pulling-out of proteins from genes is called gene expression.
+Currently, this is the highest level at which we understand what DNA means.
+
+However, there are certain to be higher levels of DNA meaning which are harder to
+discern. For instance, there is every reason to believe that the DNA of, say, a human being
+codes for such features as nose shape, music talent, quickness of reflexes, and so on. Could
+one, in principle, learn to read off such pieces of information directly from a strand of DNA,
+without going through the actual physical process of epigenesis-the physical pulling-out of
+phenotype from genotype Presumably, yes, since-in theory-one could have an incredibly
+powerful computer program simulating the entire process, including every cell, every protein,
+every tiny feature involved in the replication of DNA, of cells, to the bitter end. The output of
+such a pseudo-epigenesis program would be a high-level description of the phenotype.
+
+
+Self-Rep and Self-Rep
+
+
+531
+
+
+
+There is another (extremely faint) possibility: that we could learn to read the phenotype off
+of the genotype without doing an isomorphic simulation of the physical process of
+epigenesis, but by finding some simpler sort of decoding mechanism. This could be called
+"shortcut pseudoepigenesis". Whether shortcut or not, pseudo-epigenesis is, of course, totally
+beyond reach at the present time-with one notable exception: in the species Felis catus, deep
+probing has revealed that it is indeed possible to read the phenotype directly off of the
+genotype. The reader will perhaps better appreciate this remarkable fact after directly
+examining the following typical section of the DNA of Felis catus:
+
+CATCATCATCATCATCATCATCATCATCAT...
+
+
+Below is shown a summary of the levels of DNA-readability, together with the names of the
+different levels of decoding. DNA can be read as a sequence of:
+
+
+bases (nucleotides)
+amino acids
+
+proteins (primary structure)
+proteins (tertiary structure)
+protein clusters
+
+unknown levels of DNA meaning
+
+(N-l) ????
+
+(N) physical, mental, and
+
+psychological traits pseudo-epigenesis
+
+
+transcription
+translation
+. gene expression
+
+higher levels of gene expression
+
+
+The Central Dogmap
+
+With this background, now we are in a position to draw an elaborate comparison between F.
+Crick's "Central Dogma of Molecular Biology" (.DOGMA I) upon which all cellular
+processes are based; and what I, with poetic license, call the "Central Dogma of
+Mathematical Logic" (.DOGMA II), upon which G6del's Theorem is based. The mapping
+from one onto the other is laid out in Figure 99 and the following chart, which together
+constitute the Central Dogmap.
+
+FIGURE 99. The Central Dogmap. An analogy is established between two fundamental
+Tangled Hierarchies: that of molecular biology and that of mathematical logic.
+
+
+Self-Rep and Self-Rep
+
+
+532
+
+
+
+LKJGNH 1
+(Molcculai Biologv)
+
+
+.DOOM A 11
+(Mathematical logic)
+
+strand* of DNA
+
+<J=4>
+
+strings of TNT
+
+sttands of mRNA
+
+
+statements of N
+
+proteins
+
+
+statements of meu-TNT
+
+pitiicin* which att
+on proteins
+
+
+statements about stateineiiis
+of meta-TNT
+
+proteins which act on
+proteins which att
+on proteins
+
+
+sialeinents about statements
+about statements
+of mela-1 NT
+
+ti ansenption
+(DNA => RNA)
+
+
+interpretation
+(TNT * N)
+
+T ranslation
+(RNA ^ proteins)
+
+<£=>
+
+Arithmeti/atioil
+(N ^ meta-TNT)
+
+Crick
+
+<£4>
+
+G«>del
+
+Cenctic (axJe
+(aibitian contention)
+
+
+Godel C<xle
+(arbitrary convention)
+
+todon (triplet of bases)
+
+<=^>
+
+codon (triplet of digits)
+
+mniiio acid
+
+
+quoted m mbol of TNT
+used in meta-TNT
+
+self-reproduction
+
+
+setf-relerence
+
+sufficiently strong
+cellular support system
+to permit self-rep
+
+
+sufficient]! powerful
+arithmetical formal system
+to permit self-ref
+
+
+Central Doginap
+
+
+
+Self-Rep and Self-Rep
+
+
+533
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Note the base-pairi ng of A and T (Arithmetization and Translation), as well as of G
+and C (Godel and Crick). Mathematical logic gets the purine side, and molecular biology gets
+the pyrimidine side.
+
+To complete the esthetic side of this mapping, I chose to model my Godel-numbering
+scheme on the Genetic Code absolutely faithfully. In fact, under the following
+correspondence, the table of the Genetic Code becomes the table of the Godel Code:
+
+
+(odd)
+
+i
+
+
+(even)
+
+2
+
+
+(odd)
+
+3
+
+
+(even)
+
+6
+
+
+
+A (purine)
+
+C (pyrimidine)
+G (purine)
+l (pvrimidine)
+
+
+Each amino acid-of which there are twenty-corresponds to exactly one symbol of TNT-of
+which there are twenty. Thus, at last, my motive for concocting "austere TNT" comes out-so
+that there would be exactly twenty symbols! The Godel Code is shown in Figure 100.
+Compare it with the Genetic Code (Fig. 94).
+
+There is something almost mystical in seeing the deep sharing of such an abstract
+structure by these two esoteric, yet fundamental, advances in knowledge achieved in our
+century. This Central Dogmap is by no means a rigorous proof of identity of the two theories;
+but it clearly shows a profound kinship, which is worth deeper exploration.
+
+Strange Loops in the Central Dogmap
+
+One of the more interesting similarities between the two sides of the map is the way in which
+"loops" of arbitrary complexity arise on the top level of both: on the left, proteins which act
+on proteins which act on proteins and so on, ad infinitum; and on the right, statements about
+statements about statements of meta-TNT and so on, ad infinitum. These are like
+heterarchies, which we discussed in Chapter V, where a sufficiently complex substratum
+allows high-level Strange Loops to occur and to cycle around, totally sealed off from lower
+levels. We will explore this idea in greater detail in Chapter XX.
+
+Incidentally, you may be wondering about this question: "What, according to the
+Central Dogmap, is Godel's Incompleteness Theorem itself mapped onto?" This is a good
+question to think about before reading ahead.
+
+The Central Dogmap and the Contracrostipuntus
+
+It turns out that the central dogmap is quite similar to the mapping that was laid out in
+Chapter IV between the Contracrostipunctus and Godel’s Theorem. One can therefore drew
+parallels between all three systems.
+
+
+Self-Rep and Self-Rep
+
+
+534
+
+
+The God el Code.
+
+
+
+FIGURE 100. The Godel Code. Under this Godel-numbering scheme, each T V7- s'mbol gets
+one or more codons. The small ovals show how this table subsumes the earlier
+Godelnumhering table of Chapter IX.
+
+(1) formal systems and strings
+
+(2) cells and strands of DNA
+
+(3) record players and records
+
+
+In the following chart, the mapping between systems 2 and 3 is explained carefully
+
+
+Self-Rep and Self-Rep
+
+
+535
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Contracrostipuncius
+
+
+Molecular Biology
+
+phonograph
+
+
+cell
+
+“Perfect" phonograph
+
+
+“Perfect” cell
+
+record
+
+
+strand of DNA
+
+record playable
+by a given phonograph
+
+
+strand of DNA
+reproducible by a given cell
+
+record unplayable
+by that phonograph
+
+
+strand of DNA
+unreproducible by that cell
+
+process of converting
+record grooves into sounds
+
+
+process of transcription
+of DNA onto mRNA
+
+sounds produced by
+record player
+
+
+strands of messenger RNA
+
+translation of sounds
+into vibrations of phonograph
+
+
+translation of mRNA
+into proteins
+
+mapping from external
+sounds ont* vibrations
+of phonograph
+
+
+Genetic Code
+
+(mapping from mRNA triplets
+onto amino acids)
+
+breaking of phonograph
+
+
+destruction of the cell
+
+Title of song specially
+tailored for Record Player X:
+
+“I Cannot Be Played on
+Record Player X’’
+
+
+High-level interpretation of
+DNA strand specially tailored
+for Cell X: “I Cannot Be
+Replicated by Cell X"
+
+"Imperfect” Record Player
+
+
+Cell for which there exists at
+least one DNA strand which it
+cannot reproduce
+
+“Todel's Theorem":
+“There always exists an
+unplayable record, given
+a particular phonograph."
+
+<£=>
+
+Immunity Theorem:
+“There always exists an
+un reproducible DNA strand,
+given a particular cell."
+
+
+The analogue of Godel's Theorem is seen to be a peculiar fact, probably little useful to
+molecular biologists (to whom it is likely quite obvious):
+
+It is always possible to design a strand ofDNA which, if injected into a cell, would,
+upon being transcribed, cause such proteins to be manufactured as would destroy the cell
+(or the DNA), and thus result in the non-reproduction of that DNA
+
+This conjures tip a somewhat droll scenario, at least if taken in light of evolution: an invading
+species of virus enters a cell by some surreptitious
+
+
+Self-Rep and Self-Rep
+
+
+536
+
+
+
+
+COLLAR
+
+
+HEAD
+
+
+FIGURE 111. The T4 bacterial virus is an assembly of protein components (a). The "head" is
+a protein membrane, shaped like a kind of prolate irosahedron with thirty facets and filled
+with DNA. It is attached by a neck to a tail consisting of a hollow core surrounded by a
+contractile sheathh and based on a spiked end plate to which six fibers are attached. The
+spikes and fibers affix the virus to a bacterial cell wall (h). The sheath contracts, driving the
+core through the wall, and viral DNA enters the cell. [From Hanawalt and Haynes, The
+Chemical Basis of Life, p. 230.1
+
+
+means, and then carefully ensures the manufacture of proteins which will have the effect of
+destroying the virus itself! It is a sort of suicide-or Epimenides sentence, if you will-on the
+molecular level. Obviously it would not prove advantageous from the point of view of
+survival of the species. However, it demonstrates the spirit, if not the letter, of the
+mechanisms of protection and subversion which cells and their invaders have developed.
+
+E. Coli vs. T4
+
+Let us consider the biologists' favorite cell, that of the bacterium Escherichia coli (no relation
+to M. C. Escher), and one of their favorite invaders of that cell: the sinister and eerie T4
+phage, pictures of which you can see in Figure 101. (Incidentally, the words "phage" and
+"virus" are synonymous and mean "attacker of bacterial cells".) The weird tidbit looks like a
+little like a cross between a LEM (Lunar Excursion Module) and a mosquito-and it is much
+more sinister than the latter. It has a "head" wherein is stored all its "knowledge"-namely its
+DNA; and it has six "legs" wherewith to fasten itself to the cell it has chosen to invade; and it
+has a "stinging tube" (more properly called its "tail") like a mosquito. The major difference is
+that unlike a mosquito, which uses its stinger for sucking blood, the T4 phage uses its stinger
+for injecting its hereditary substance into the cell against the will of its victim. Thus the
+phage commits "rape" on a tiny scale.
+
+
+Self-Rep and Self-Rep
+
+
+537
+
+
+
+
+
+
+r\
+
+
+
+
+< • »!* ,■» ,
+* 2 * '•*■
+
+• . . . .•*
+
+
+/
+
+
+FIGURE 102. Viral infection begins when viral DNA enters a bacterium. Bacterial DNA is
+disrupted and viral DNA replicated. Synthesis of viral structural proteins and their assembly
+into virus continues until the cell bursts, releasing particles. [From Hanawalt and Haynes,
+The Chemical Basis of Life, p. 230.]
+
+A Molecular Trojan Horse
+
+What actually happens when the viral DNA enters a cell? The virus "hopes", to speak
+anthropomorphically, that its DNA will get exactly the same treatment as the DNA of the
+host cell. This would mean getting transcribed and translated, thus allowing it to direct the
+synthesis of its own special proteins, alien to the host cell, which will then begin to do their
+thing. This amounts to secretly transporting alien proteins "in code" (viz., the Genetic Code)
+into the cell, and then "decoding" (viz., producing) them. In a way this resembles the story of
+the Trojan horse, according to which hundreds of soldiers were sneaked into Troy inside a
+harmless seeming giant wooden horse; but once inside the city, they broke loose and captured
+it. The alien proteins, once they have been "decoded" (synthesized) from their carrier DNA,
+now jump into action. The sequence of actions directed by the T4 phage has been carefully
+studied, and is more or less as follows (see also Figs. 102 and 103):
+
+Time elapsed Action taking place
+
+0 min. Injection of viral DNA.
+
+1 min. Breakdown of host DNA. Cessation of production of native proteins and
+
+initiation of production of alien (T4) proteins. Among the earliest produced
+proteins are those which direct the replication of the alien (T4) DNA.
+
+5 min. Replication of viral DNA begins.
+
+8 min. Initiation of production of structural proteins which will form the "bodies" of
+
+new phages.
+
+
+Self-Rep and Self-Rep
+
+
+538
+
+
+
+
+
+
+Head
+
+
+Tail
+
+
+
+Tail fiber
+
+
+FlGl FtF 103. Thf morphogmftu path-
+
+of the T4 trims has three principal
+branches leading tndependentiy to the for¬
+int on of heads, tails, and tailfibers, which
+len combine to farm complete virus parti -
+[from Hanowalt and Haynes. The
+Basis of Life, p. 237.]
+
+
+
+Self-Rep and Self-Rep
+
+
+539
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+13 min. First complete replica of T4 invader is produced.
+
+25 min. Lysozyme (a protein) attacks host cell wall, breaking open the bacterium, and
+
+the "bicentuplets" emerge.
+
+Thus, when a T4 phage invades an E. coli cell, after the brief span of about twenty-four or
+twenty-five minutes, the cell has been completely subverted, and breaks open. Out pop about
+two hundred exact copies of the original virus-"bicentuplets"-ready to go attack more
+bacterial cells, the original cell having been largely consumed in the process.
+
+Although from a bacterium's point of view this kind of thing is a deadly serious
+menace, from our large-scale vantage point it can be looked upon as an amusing game
+between two players: the invader, or "T" player (named after the T-even class of phages,
+including the T2, T4, and others), and the "C" player (standing for "Cell"). The objective of
+the T player is to invade and take over the cell of the C player from within, for the purpose of
+reproducing itself. The objective of the C player is to protect itself and destroy the invader.
+When described this way, the molecular TC-game can be seen to be quite parallel to the
+macroscopic TC-game described in the preceding Dialogue. (The reader can doubtless figure
+out which player-T or C-corresponds to the Tortoise, and which to the Crab.)
+
+Recognition, Disguises, Labeling
+
+This "game" emphasizes the fact that recognition is one of the central themes of cellular and
+subcellular biology. How do molecules (or higher-level structures) recognize each other? It is
+essential for the functioning of enzymes that they should be able to latch onto special
+"binding sites" on their substrates; it is essential that a bacterium should be able to distinguish
+its own DNA from that of phages; it is essential that two cells should be able to recognize
+each other and interact in a controlled way. Such recognition problems may remind you of
+the original, key problem about formal systems: How can you tell if a string has, or does not
+have, some property such as theoremhood? Is there a decision procedure? This kind of
+question is not restricted to mathematical logic: it permeates computer science and, as we are
+seeing, molecular biology.
+
+The labeling technique described in the Dialogue is in fact one of E. colt's tricks for
+outwitting its phage invaders. The idea is that strands of DNA can be chemically labeled by
+tacking on a small molecule-methyl-to various nucleotides. Now this labeling operation does
+not change the usual biological properties of the DNA; in other words, methylated (labeled)
+DNA can be transcribed just as well as unmethylated (unlabeled) DNA, and so it can direct
+the synthesis of proteins. But if the host cell has some special
+
+
+Self-Rep and Self-Rep
+
+
+540
+
+
+
+mechanisms for examining whether DNA is labeled or not, then the label may make all the
+difference in the world. In particular, the host cell may have an enzyme system which looks
+for unlabeled DNA, and destroys any that it finds by unmercifully chopping it to pieces. In
+that case, woe to all unlabeled invaders!
+
+The methyl labels on the nucleotides have been compared to serifs on letters. Thus,
+using this metaphor, we could say that the E. colt cell is looking for DNA written in its
+"home script", with its own particular typeface-and will chop up any strand of DNA written
+in an "alien" typeface. One counterstrategy, of course, is for phages to learn to label
+themselves, and thereby become able to fool the cells which they are invading into
+reproducing them.
+
+This TC-battle can continue to arbitrary levels of complexity, but we shall not pursue
+it further. The essential fact is that it is a battle between a host which is trying to reject all
+invading DNA, and a phage which is trying to infiltrate its DNA into some host which will
+transcribe it into mRNA (after which its reproduction is guaranteed). Any phage DNA which
+succeeds in getting itself reproduced this way can be thought of as having this high-level
+interpretation: "I Can Be Reproduced in'Cells of Type X". This is to be distinguished from
+the evolutionarily pointless kind of phage mentioned earlier, which codes for proteins that
+destroy it, and whose high-level interpretation is the self-defeating sentence: "I Cannot Be
+Reproduced in Cells of Type X".
+
+Henkin Sentences and Viruses
+
+Now both of these contrasting types of self-reference in molecular biology have their
+counterparts in mathematical logic. We have already discussed the analogue of the self-
+defeating phages-namely, strings of the G6del type, which assert their own unproducibility
+within specific formal sstems. But one can also make a counterpart sentence to a real phage:
+the' phage asserts its own producibility in a specific cell, and the sentence asserts its own
+producibility in a specific formal system. Sentences of this type are called Henkin sentences,
+after the mathematical logician Leon Henkin. They can be constructed exactly along the lines
+of Godel sentences, the only difference being the omission of a negation. One begins with an
+"uncle", of course:
+
+3a:3a':<TNT-PROOF-PAIR{a,a'}-°ARITHMOQUINE{a",a'}>
+
+and then proceeds by the standard trick. Say the Godel number of the above "uncle" is h.
+Now by arithmoquining this very uncle, you get a Henkin sentence:
+
+3a:3a':<TNT-PROOF-PAIR{a,a'} A ARITHMOQUTNE{SSS ... SSSO/a",a'}> M
+
+
+Self-Rep and Self-Rep
+
+
+541
+
+
+
+(By the way, can you spot how this sentence differs from -G?) The reason I show it
+explicitly is to point out that a Henkin sentence does not give a full recipe for its own
+derivation; it just asserts that there exists one. You might well wonder whether its claim is
+justified. Do Henkin sentences indeed possess derivations? Are they, as they claim,
+theorems? It is useful to recall that one need not believe a politician who says. "I am honest"-
+he may be honest, and yet he may not be. Are Henkin sentences any more trustworthy than
+politicians? Or do Henkin sentences, like politicians, lie in cast-iron sinks?
+
+It turns out that these Henkin sentences are invariably truth tellers. Why this is so is
+not obvious; but we will accept this curious fact without proof.
+
+Implicit vs. Explicit Henkin Sentences
+
+I mentioned that a Henkin sentence tells nothing about its own derivation; it just asserts that
+one exists. Now it is possible to invent a variation on the theme of Henkin sentences-namely
+sentences which explicitly describe their own derivations. Such a sentence's high-level
+interpretation would not be "Some Sequence of Strings Exists Which is a Derivation of Me",
+but rather, "The Herein-described Sequence of Strings Is a Derivation of Me". Let us
+
+call the first type of sentence an implicit Henkin sentence. The new sentences will be called
+explicit Henkin sentences, since they explicitly describe their own derivations. Note that,
+unlike their implicit brethren, explicit Henkin sentences need not be theorems. In fact, it is
+quite easy to write a string which asserts that its own derivation consists of the single string
+0=0-a false statement, since 0=0 is not a derivation of anything. However, it is also possible
+to write an explicit Henkin sentence which is a theorem-that is, a sentence which in fact gives
+a recipe for its own derivation.
+
+Henkin Sentences and Self-Assembly
+
+The reason I bring up this distinction between explicit and implicit Henkin sentences is that it
+corresponds very nicely to a significant distinction between types of virus. There are certain
+viruses, such as the so-called "tobacco mosaic virus", which are called self-assembling
+viruses; and then there are others, such as our favorite T-evens, which are non-self-
+assembling. Now what is this distinction? It is a direct analogue to the distinction between
+implicit and explicit Henkin sentences.
+
+The DNA of a self-assembling virus codes only for the parts of a new virus, but not
+for any enzymes. Once the parts are produced, the sneaky virus relies upon them to link up to
+each other without help from any enzymes. Such a process depends on chemical affinities
+which the parts have for each other, when swimming in the rich chemical brew of a cell. Not
+only viruses, but also some organelles-such as ribosomes-assemble
+
+
+Self-Rep and Self-Rep
+
+
+542
+
+
+
+themselves. Somtiems enzymes may be needed - but in such cases, they are recruited from
+the host cell, and enslaved. This is what is meant by self-assembly.
+
+By contrast, the DNA of more complex viruses, such as the T-evens, codes not only
+for the parts, but in addition for various enzymes which play special roles in the assembly of
+the parts into wholes. Since the assembly process is not spontaneous but requires "machines",
+such viruses are not considered to be self-assembling. The essence of the distinction, then,
+between self-assembling units and non-self-assembling units is that the former get away with
+self-reproduction without telling the cell anything about their construction, while the latter
+need to give instructions as to how to assemble themselves.
+
+Now the parallel to Henkin sentences, implicit and explicit, ought to be quite clear.
+Implicit Henkin sentences are self-proving but do not tell anything at all about their proofs-
+they are analogous to self-assembling viruses; explicit Henkin sentences direct the
+construction of their own proofs-they are analogous to more complex viruses which direct
+their host cells in putting copies of themselves together.
+
+The concept of self-assembling biological structures as complex as viruses raises the
+possibility of complex self-assembling machines as well. Imagine a set of parts which, when
+placed in the proper supporting environment, spontaneously group themselves in such a way
+as to form a complex machine. It seems unlikely, yet this is quite an accurate way to describe
+the process of the tobacco mosaic virus' method of selfreproduction via self-assembly. The
+information for the total conformation of the organism (or machine) is spread about in its
+parts; it is not concentrated in some single place.
+
+Now this concept can lead in some strange directions, as was shown in the Edifying
+Thoughts of a Tobacco Smoker. There, we saw how the Crab used the idea that information
+for self-assembly can be distributed around, instead of being concentrated in a single place.
+His hope was that this would prevent his new phonographs from succumbing to the Tortoise's
+phonograph-crashing method. Unfortunately, just as with the most sophisticated axiom
+schemata, once the system is all built and packaged into a box, its well-defmedness renders it
+vulnerable to a sufficiently clever "Godelizer"; and that was the sad tale related by the Crab.
+Despite its apparent absurdity, the fantastic scenario of that Dialogue is not so far from
+reality, in the strange, surreal world of the cell.
+
+
+Two Outstanding Problems:
+
+Differentiation and Morphogenesis
+
+Now self-assembly may be the trick whereby certain subunits of cells are constructed, and
+certain viruses-but what of the most complex macroscopic structures, such as the body of an
+elephant or a spider, or the shape of a Venus's-Hyt-ap? How are homing instincts built into
+the brain of' a
+
+
+Self-Rep and Self-Rep
+
+
+543
+
+
+
+bird, or hunting instincts into the brain of a dog% In short, how is it that merely by dictating
+which proteins are to be produced in cells, DNA exercises such spectacularly precise control
+over the exact structure and function of macroscopic living objects? There are two major
+distinct problems here. One is that of cellular differentiation: how do different cells, sharing
+exactly the same DNA, perform different roles-such as a kidney cell, a bone marrow cell, and
+a brain cell? The other is that of morphogenesis ("birth of form"): how does intercellular
+communication on a local level give rise to large-scale, global structures and organizations-
+such as the various organs of the body, the shape of the face, the suborgans of the brain, and
+so on? Although both cellular differentiation and morphogenesis are poorly understood at
+present, the trick appears to reside in exquisitely fine-tuned feedback and "feedforward"
+mechanisms within cells and between cells, which tell a cell when to "turn on" and when to
+"turn off' production of various proteins.
+
+Feedback and Feedforward
+
+Feedback takes place when there is too much or too little of some desired substance in the
+cell: then the cell must somehow regulate the production line which is assembling that
+substance. Feedforward also involves the regulation of' an assembly line, but not according
+to the amount of end product present: rather, according to the amount of some precursor of
+the end product of that assembly line. There are two major devices for achieving negative
+feedforward or feedback. One way is to prevent the relevant enzymes from being able to
+perform-that is, to "clog up" their active sites. This is called inhibition. The other way is to
+prevent the relevant enzymes from ever being manufactured! This is called repression.
+Conceptually, inhibition is simple: you just block up the active site of the first enzyme in the
+assembly line, and the whole process of synthesis gets stopped dead.
+
+Repressors and Inducers
+
+Repression is trickier. How does a cell stop a gene from being expressed? The answer is, it
+prevents it from ever getting transcribed. This means that it has to prevent RNA polymerase
+from doing its job. This can be accomplished by placing a huge obstacle in its path, along the
+DNA. precisely in front of that gene which the cell wants not to get transcribed. Such
+obstacles do exist, and are called repressors. They are themselves proteins, and they bind to
+special obstacle-holding sites on the DNA, called (I am not sure why) operators. An operator
+therefore is a site of control for the gene (or genes) which immediately follow it: those genes
+are called its operon. Because a series of enzymes often act in concert in carrying out a long
+chemical transformation, they are often coded for in sequence; and this is why operons often
+contain several genes, rather than just one. The effect of the successful repression of an
+operon is that a whole series of genes is
+
+
+Self-Rep and Self-Rep
+
+
+544
+
+
+
+prevented from being transcribed, which means that a whole set of related enzymes remains
+unsynthesized.
+
+What about positive feedback and feedforward? Here again, there are two options: (1)
+unclog the clogged enzymes, or (2) stop the repression of the relevant operon. (Notice how
+nature seems to love double-negations! Probably there is some very deep reason for this.)
+The mechanism by which repression is repressed involves a class of molecules called
+inducers. The role of an inducer is simple: it combines with a repressor protein before the
+latter has had a chance to bind to an operator on a DNA molecule; the resulting "repressor-
+inducer complex" is incapable of binding to an operator, and this leaves the door open for the
+associated operon to be transcribed into mRNA and subsequently translated into protein.
+Often the end product or some precursor of the end product can act as an
+inducer.
+
+
+Feedback and Strange Loops Compared
+
+Incidentally, this is a good time to distinguish between simple kinds of feedback, as in the
+processes of inhibition and repression, and the looping-hack between different informational
+levels, shown in the Central Dogmap. Both are "feedback" in some sense; but the latter is
+much deeper than the former. When an amino acid, such as tryptophan or isoleucine, acts as
+feedback (in the form of an inducer) by binding to its repressor so that more of it gets made,
+it is not telling how to construct itself; it is just telling enzymes to make more of it. This
+could be compared to a radio's volume, which, when fed through a listener's ears, may cause
+itself to be turned down or up. This is another thing entirely from the case in which the
+broadcast itself tells you explicitly to turn your radio on or off, or to tune to another
+wavelength-or even how to build another radio! The latter is much more like the looping-
+back between informational levels, for here, information inside the radio signal gets
+"decoded" and translated into mental structures. The radio signal is composed of symbolic
+constituents whose symbolic meaning matters-a case of use, rather than mention. On the
+other hand, when the sound is just too loud, the symbols are not conveying meaning: they are
+merely being perceived as loud sounds, and might as well be devoid of meaning-a case of
+mention, rather than use. This case more resembles the feedback loops by which proteins
+regulate their own rates of synthesis.
+
+It has been theorized that the difference between two neighboring cells which share
+the exact same genotype and yet have different functions is that different segments of their
+genome have been repressed, and therefore they have different working sets of proteins. A
+hypopothesis like this could account for the phenomenal differences between cells in
+different organs of the body of a human being.
+
+
+Self-Rep and Self-Rep
+
+
+545
+
+
+
+Two Simple Examples of Differentiation
+
+
+The process by which one initial cell replicates over and over, giving rise to a myriad of
+differentiated cells with specialized functions, can be likened to the spread of a chain letter
+from person to person, in which each new participant is asked to propagate the message
+faithfully, but also to add some extra personal touch. Eventually, there will be letters which
+are tremendously different from each other.
+
+Another illustration of the ideas of differentiation is provided by this extremely
+simple computer analogue of a differentiating self-rep. Consider a very short program which
+is controlled by an up-down switch, and which has an internal parameter N-a natural number.
+This program can run in two modes-the up-mode, and the down-mode. When it runs in the
+upmode, it self-replicates into an adjacent part of the computer's memoryexcept it makes the
+internal parameter N of its "daughter" one greater than in itself. When it runs in the down¬
+mode, it does not self-rep, but instead calculates the number
+
+(-1 )'/(2N + 1)
+
+
+and adds it to a running total.
+
+Well, suppose that at the beginning, there is one copy of the program in memory, N =
+0, and the mode is up. Then the program will copy itself next door in memory, with N = 1.
+Repeating the process, the new program will self-rep next door to itself, with a copy having
+N = 2. And over and over again ... What happens is that a very large program is growing
+inside memory. When memory is full, the process quits. Now all of memory can be looked
+upon as being fdled with one big program, composed of many similar, but differentiated,
+modules-or "cells". Now suppose we switch the mode to down, and run this big program.
+What happens? The first "cell" runs, and calculates 1/1. The second "cell" runs, calculating -
+1/3, and adding it to the previous result. The third "cell" runs, calculating + 1/5 and adding it
+on. .. The end result is that the whole "organism"-the big program-calculates the sum
+
+1-1/3 +1/5 -117+1/9 -1/11 +1/13 -1/15 + .. .
+
+to a large number of terms (as many terms as "cells" can fit inside memory). And since this
+series converges (albeit slowlv) to 7r/4, we have a "phenotype" whose function is to calculate
+the value of a famous mathematical constant.
+
+Level Mixing in the Cell
+
+I hope that the descriptions of processes such as labeling, self-assembly, differentiation,
+morphogenesis, as well as transcription and translation, have helped to convey some notion
+of the immensely complex system which is a cell-an information-processing system with
+some strikingly
+
+
+Self-Rep and Self-Rep
+
+
+546
+
+
+
+novel features. We have seen, in the Central Dogmap, that although we can try to draw a
+clear line between program and data, the distinction is somewhat arbitrary. Carrying this line
+of thought further, we find that not only are program and data intricately woven together, but
+also the interpreter of programs, the physical processor, and even the language are included
+in this intimate fusion. Therefore, although it is possible (to some extent) to draw boundaries
+and separate out the levels, it is just as important-and ust as fascinating-to recognize the
+level-crossings and mixings. Illustrative of this is the amazing fact that in biological systems,
+all the various features necessary for self-rep (viz., language, program, data, interpreter, and
+processor) cooperate to such a degree that all of them are replicated simultaneously-which
+shows how much deeper is biological self-rep'ing than anything yet devised along those lines
+by humans. For instance, the self-rep program exhibited at the beginning of this Chapter
+takes for granted the pre-existence of three external aspects: a language, an interpreter, and a
+processor, and does not replicate those.
+
+Let us try to summarize various ways in which the subunits of a cell can be classified
+in computer science terms. First, let us take DNA. Since DNA contains all the information
+for construction of proteins., which are the active agents of the cell, DNA can be viewed as a
+program written in a higher-level language, which is subsequently translated (or interpreted)
+into the "machine language" of the cell (proteins). On the other hand, DNA is itself a passive
+molecule which undergoes manipulation at the hands of various kinds of enzymes; in this
+sense, a DNA molecule is exactly like a long piece of data, as well. Thirdly, DNA contains
+the templates off of which the tRNA "flashcards" are rubbed, which means that DNA also
+contains the definition of its own higher-level language.
+
+Let us move on to proteins. Proteins are active molecules, and carry out all the
+functions of the cell; therefore it is quite appropriate to think of them as programs in the
+"machine language" of the cell (the cell itself being the processor). On the other hand, since
+proteins are hardware and most programs are software, perhaps it is better to think of the
+proteins as processors. Thirdly, proteins are often acted upon by other proteins, which means
+that proteins are often data. Finally, one can view proteins as interpreters; this involves
+viewing DNA as a collection of high-level language programs, in which case enzymes are
+merely carrying out the programs written in the DNA code, which is to say, the proteins are
+acting as interpreters.
+
+Then there are ribosomes and tRNA molecules. They mediate the translation from
+DNA to proteins, which can be compared to the translation of a program from a high-level
+language to a machine language; in other words, the ribosomes are functioning as interpreters
+and the tRNA molecules provide the definition of the higher-level language. But an
+alternative view of translation has it that the ribosomes are processors, while the tRNA's are
+interpreters.
+
+We have barely scratched the surface in this analysis of interrelations between all
+these biomolecules. What we have seen is that nature feels quite
+
+
+Self-Rep and Self-Rep
+
+
+547
+
+
+
+comfortable in mixing levels which we tend to see as quite distinct. Actually, in computer
+science there is already a visible tendency to nix all these seemingly distinct aspects of an
+information-processing system. This is particularly so in Artificial Intelligence research,
+which is usually at the forefront of computer language design.
+
+The Origin of Life
+
+A natural and fundamental question to ask, on learning of these incredibly intricately
+interlocking pieces of software and hardware is: "How did they ever get started in the first
+place?" It is truly a baffling thing. One has to imagine some sort of a bootstrap process
+occurring, somewhat like that which is used in the development of new computer languages-
+but a bootstrap from simple molecules to entire cells is almost beyond one's power to
+imagine. There are various theories on the origin of life. They all run aground on this most
+central of all central questions: "How did the Genetic Code, along with the mechanisms for
+its translation (ribosomes and tRNA molecules), originate" For the moment, we will have to
+content ourselves with a sense of wonder and awe, rather than with an answer. And perhaps
+experiencing that sense of wonder and awe is more satisfying than having an answer-at least
+for a while.
+
+
+Self-Rep and Self-Rep
+
+
+548
+
+
+
+The Magnificrab, Indeed,
+
+
+It is spring, and the Tortoise and Achilles are taking a Sunday promenade in
+the woods together. They have decided to climb a hill at the top of which, it is
+said, there is a wonderful teahouse, with all sorts of delicious pastries.
+
+Achilles: Man oh man! If a crab
+
+Tortoise: If a crab??
+
+Achilles: I was about to say, if a crab ever were intelligent, then surely it would be our
+mutual friend the Crab. Why, he must be at least two times as smart as any crab alive.
+Or maybe even three times as smart as any crab alive. Or perhaps
+
+Tortoise: My soul! How you magnify the Crab!
+
+Achilles: Well, I just happen to be an admirer of his ...
+
+Tortoise: No need to apologize. I admire him, too. Speaking of Crab admirers, did I tell
+you about the curious fan letter which the Crab received not too long ago?
+
+Achilles: I don't believe so. Who sent it?
+
+Tortoise: It bore a postmark from India, and was from someone neither of us had ever
+heard of before-a Mr. Najunamar, I believe.
+
+Achilles: I wonder why someone who never knew Mr. Crab would send him a letter-or
+for that matter, how they would get his address. Tortoise: Apparently whoever it was
+was under the illusion that the Crab is a mathematician. It contained numerous results,
+all of which were But, ho! Speak of the devil! Here comes Mr. Crab now, down the
+hill. Crab: Good-bye! It was nice to talk with you again. Well, I guess I had best be
+off. But I'm utterly stuffed-couldn't eat one more bite if I had to! I've just been up there
+myself-recommend it highly. Have you ever been to the teahouse at the crest of the
+hill? How are you, Achilles? Oh, there's Achilles. Hello, hello. Well, well, if it isn't
+Mr. T!
+
+Tortoise: Hello, Mr. C. Are you headed up to the hilltop teahouse? Crab: Why, yes
+indeed, I am; how did you guess it? I'm quite looking forward to some of their special
+napoleons-scrumptious little morsels. I'm so hungry I could eat a frog. Oh, there's
+Achilles. How are you, Achilles?
+
+Achilles: Could be worse, I suppose.
+
+Crab: Wonderful! Well, don't let me interrupt your discussion. I'll just tag along.
+
+Tortoise: Curiously enough, I was just about to describe your mysterious letter from that
+Indian fellow a few weeks back-but now that you're here. I'll let Achilles get the story
+from the Crab’s mouth.
+
+
+The Magnificrab, Indeed,
+
+
+549
+
+
+
+
+FIGURE 104. Castrovalva, by’M. C. Escher (lithograph, 1930).
+
+
+The Magnificrab, Indeed,
+
+
+550
+
+
+Crab: Well, it was this way. This fellow Najunamar had apparently never had any formal
+training in mathematics, but had instead worked out some of his own methods for
+deriving new truths of mathematics. Some of his discoveries defeated me completely;
+I had never seen anything in the least like them before. For instance, he exhibited a
+map of India that he had managed to color using no fewer than 1729 distinct colors.
+
+Achilles: 1729! Did you say 1729? Crab: Yes-why do you ask?
+
+Achilles: Well, 1729 is a very interesting number, you know. Crab: Indeed. I wasn't
+aware of it.
+
+Achilles: In particular, it so happens that 1729 is the number of the taxicab which I took
+to Mr. Tortoise's this morning!
+
+Crab: How fascinating! Could you possibly tell me the number of the trolley car which
+you'll take to Mr. Tortoise's tomorrow morning?
+
+Achilles (after a moment's thought): It's not obvious to me; however, I should think it
+would be very large.
+
+Tortoise: Achilles has a wonderful intuition for these things.
+
+Crab: Yes. Well, as I was saying, Najunamar in his letter also proved that every even
+prime is the sum of two odd numbers, and that there are no solutions in positive
+integers to the equation
+
+
+a n + b n =c n for n = 0.
+
+Achilles: What? All these old classics of mathematics resolved in one fell swoop? He
+must be a genius of the first rank! Tortoise: But Achilles-aren't you even in the
+slightest skeptical?
+
+Achilles: What? Oh, yes-skeptical. Well, of course I am. You don't think I believe that
+Mr. Crab got such a letter, do you? I don't fall for just anything, you know. So it must
+have been 5'ou, Mr. T, who received the letter!
+
+Tortoise: Oh, no, Achilles, the part about Mr. C receiving the letter is quite true. What I
+meant was, aren't you skeptical about the content of the letter-its extravagant claims?
+
+Achilles: Why should I be? Hmm ... Well, of course I am. I'm a very skeptical person, as
+both of you should well know by now. It's very hard to convince me of anything, no
+matter how true or false it is.
+
+Tortoise: Very well put, Achilles. You certainly have a first-class awareness of your own
+mental workings.
+
+Achilles: Did it ever occur to you, my friends, that these claims of Najunamar might be
+incorrect?
+
+Crab: Frankly, Achilles, being rather conservative and orthodox myself, I was a bit
+concerned about that very point on first receiving the letter. In fact, I suspected at first
+that here was an out-and-out fraud. But on second thought, it occurred to me that not
+many types of people could manufacture such strange-sounding and complex results
+purely from their imagination. In fact, what it boiled down to was this question:
+
+
+The Magnificrab, Indeed,
+
+
+551
+
+
+
+"Which is the more likely: a charlatan of such extraordinary ingenuity, or a
+mathematician of great genius?" And before long, I realized that the probabilities clearly
+favored the former.
+
+Achilles: Didn't you directly checkout any of his amazing claims, however?
+
+Crab: Why should I? The probability argument was the most convincing thing I had ever
+thought of; no mathematical proof would have equaled it. But Mr. T here insisted on
+rigor. I finally gave in to his insistence, and checked all of Najunamar's results. To my
+great surprise, each one of them was right. How he discovered them, I'll never know,
+however. He must have some amazing and inscrutable Oriental type of insight which we
+here in the Occident can have no inkling of. At present, that's the only theory which
+makes an sense to me.
+
+Tortoise: Mr. Crab has always been a little more susceptible to mystical or fanciful
+explanations than I am. I have full confidence that whatever Najunamar did in his way
+has a complete parallel inside orthodox mathematics. There is no way of doing
+mathematics which is fundamentally different from what we now know, in my opinion.
+
+Achilles: That is an interesting opinion. I suppose it has something to do with the Church-
+Turing Thesis and related topics.
+
+Crab: Oh, well, let us leave these technical matters aside on such a fine day, and enjoy the
+quiet of the forest, the chirping of the birds, and the play of sunlight on the new leaves
+and buds. Ho!
+
+Tortoise: I second the motion. After all, all generations of Tortoises have reveled in such
+delights of nature.
+
+Crab: As have all generations of Crabs.
+
+Achilles: You don't happen to have brought your flute along, by any chance, Mr. C?
+
+Crab: Why, certainly! I take it with me everywhere. Would you like to hear a tune or two?
+
+Achilles: It would be delightful, in this pastoral setting. Do you play from memory?
+
+Crab: Sad to say, that is beyond my capability. I have to read my music
+
+from a sheet. But that is no problem. I have several very pleasant pieces here in this case.
+
+(He opens up a thin case and draws out a few pieces of paper. The topmost one has the
+following symbols on it:
+
+
+Va:-Sa=0
+
+
+He sticks the top sheet into a little holder attached to his flute, and plays. The tune is very
+short.)
+
+Achilles: That was charming. (Peers over at the sheet on the flute, and a quizzical expression
+beclouds his face.) What is that statement of number theory doing, attached to your flute
+like that?
+
+(The Crab looks at his flute, then his music, turns his head all around, and appears slightly
+confused.)
+
+
+The Magnificrab, Indeed,
+
+
+552
+
+
+
+Crab: I don't understand. What statement of number theory?
+
+Achilles: "Zero is not the successor of any natural number." Right there, in the holder on
+your flute!
+
+Crab: That's the third Piano Postulate. There are five of them, and I've arranged them all for
+flute. They're obvious, but catchy.
+
+Achilles: What's not obvious to me is how a number-theoretical statement can be played as
+music.
+
+Crab: But I insist, it's 'NOT a number-theoretical statement-it's a Piano Postulate! Would
+you like to hear another?
+
+Achilles: I'd be enchanted.
+
+(The Crab places another piece of paper on his flute, and this time Achilles watches more
+carefully.)
+
+Well, I watched your eyes, and they were looking at that FORMULA on the sheet. Are
+you sure that that is musical notation? I swear, it most amazingly resembles the notation
+which one might use in a formalized version of number theory.
+
+Crab: How odd! But certainly that is music, not any kind of statement of mathematics, as
+far as I can tell! Of course, I am not a mathematician in any sense of the word. Would
+you like to hear any other tunes?
+
+Achilles: By all means. Have you some others?
+
+Crab: Scads.
+
+(He takes a new sheet, and attaches it to his flute. It contains the following symbols:
+
+3a:3b:(SSa. SSb)=SSSSSSSSSSSSSO
+Achilles peers at it, while the Crab plays it.)
+
+Isn't it lovely?
+
+Achilles: Yes, it certainly is a tuneful little piece. But I have to say, it's looking more and
+more like number theory to me.
+
+Crab: Heavens! It is just my usual music notation, nothing more. I simply don't know how
+you read all these extramusical connotations into a straightforward representation for
+sounds.
+
+Achilles: Would you be averse to playing a piece of my own composition?
+
+Crab: Not in the least. Have you got it with you?
+
+Achilles: Not yet, but I have a hunch I might be able to compose some tunes all by myself.
+Tortoise: I must, tell you, Achilles, that Mr. C is a harsh judge of music composed by others,
+so do not be disappointed if, by some chance, he is not an enthusiast for your efforts.
+Achilles: That is very kind of you to forewarn me. Still, I'm willing to give it a try .
+
+(He writes:
+
+
+((SSSO . SSSO) +(SSSSO. SSSSO))=(SSSSSO. SSSSSO)
+
+
+The Magnificrab, Indeed,
+
+
+553
+
+
+
+The Crab takes it, looks it over for a moment, then sets it in his music holder, and pipes.)
+
+Crab: Why, that's quite nice, Achilles. I enjoy strange rhythms.
+
+Achilles: What's strange about the rhythms in that piece?
+
+Crab: Oh, naturally, to you as the composer it must seem quite bland, but to my ears,
+shifting from a 3/3 rhythm to 4/4 and then to 5/5 is quite exotic. If you have any other
+songs. I'd be glad to play them. Achilles: Thank you very much. I've never composed
+anything before, and I must say composing is quite different from how I had imagined it
+to be. Let me try my hand at another one. ( jots down a line.)
+
+3a:3b:(SSa - SSb) =SSSSSSSSSSSSSSO
+
+Crab: Hmmm ... Isn't that just a copy of my earlier piece?
+
+Achilles: Oh, no! I've added one more S. Where you had thirteen in a row, I have fourteen.
+
+Crab: Oh, yes. Of course. {He plays it, and looks very stern.)
+
+Achilles: I do hope you didn't dislike my piece!
+
+Crab: I am afraid, Achilles, that you completely failed to grasp the subtleties of my piece,
+upon which yours is modeled. But how could I expect you to understand it on first
+hearing? One does not always understand what is at the root of beauty. It is so easy to
+mistake the superficial aspects of a piece for its beauty, and to imitate them, when the
+beauty itself is locked deep inside the music, in a way which seems always to elude
+analysis.
+
+Achilles: I am afraid that you have lost me a little in your erudite commentary. I understand
+that my piece does not measure up to your high standards, but I do not know exactly
+where I went astray. Could you perhaps tell me some specific way in which you find fault
+with my composition?
+
+Crab: One possible way to save your composition, Achilles, would be to insert another three
+S's-five would do as well-into that long group of S's near the end. That would create a
+subtle and unusual effect.
+
+Achilles: I see.
+
+Crab: But there are other ways you might choose to change your piece. Personally, I would
+find it most appealing to put another tilde in the front. Then there would be a nice balance
+between the beginning and the end. Having two tildes in a row never fails to give a gay
+little twist to .a piece, you know.
+
+Achilles: How about if I take both of your suggestions, and make the following piece?
+
+-3a:3b:(SSa.SSb)=SSSSSSSSSSSSSSSSSO
+
+Crab (a painful grimace crossing his face): Now, Achilles, it is important to learn the
+following lesson: never try to put too much into any single piece. There is always a point
+beyond which it cannot be improved,
+
+
+The Magnificrab, Indeed,
+
+
+554
+
+
+
+and further attempts to improve it will in fact destroy it. Such is the case in this example.
+Your idea of incorporating both of my suggestions together does not yield the desired
+extra amount of beauty, but on the contrary creates an imbalance which quite takes away
+all the charm.
+
+Achilles: How is it that two very similar pieces, such as yours with thirteen 5's, and mine
+with fourteen S's, seem to you to be so different in their musical worth? Other than in that
+minor respect, the two are identical.
+
+Crab: Gracious! There is a world of difference between your piece and mine. Perhaps this is
+a place where words fail to convey what the spirit can feel. Indeed, I would venture to say
+that there exists no set of rules which delineate what it is that makes a piece beautiful, nor
+could there ever exist such a set of rules. The sense of Beauty is the exclusive domain of
+Conscious Minds, minds which through the experience of living have gained a depth that
+transcends explanation by any mere set of rules.
+
+Achilles: I will always remember this vivid clarification of the nature of Beauty. I suppose
+that something similar applies to the concept of Truth, as well?
+
+Crab: Without doubt. Truth and Beauty are as interrelated as-as
+
+Achilles: As interrelated as, say, mathematics and music?
+
+Crab: Oh! You took the words right out of my mouth! How did you know that that is what I
+was thinking?
+
+Tortoise: Achilles is very clever, Mr. C. Never underestimate the potency of his insight.
+
+Achilles: Would you say that there could conceivably be any relationship between the truth
+or falsity of a particular statement of mathematics, and the beauty, or lack of beauty, of
+an associated piece of music? Or is that just a far-fetched fancy of mine, with no basis in
+reality?
+
+Crab: If you are asking me, that is carrying things much too far. When I spoke of the
+interrelatedness of music and mathematics, I was speaking very figuratively, you know.
+As for a direct connection between specific pieces of music and specific statements of
+mathematics, however, I harbor extremely grave doubts about its possibility. I would
+humbly counsel you not to give too' much time to such idle speculations.
+
+Achilles: You are no doubt right. It would be most unprofitable. Perhaps I ought to
+concentrate on sharpening my musical sensitivity by composing some new pieces. Would
+you be willing to serve as my mentor,
+
+Mr. C?
+
+Crab: I would be very happy to aid you in your steps towards musical understanding.
+
+(So Achilles takes pen in hand, and, with what appears to be a great deal of
+concentration, writes:
+
+AOOaV'V--nn:b+cS(33=OAD((-d)<v(VS-+(>v
+
+
+The Magnificrab, Indeed,
+
+
+555
+
+
+
+The Crab looks very startled.)
+
+Y ou really want me to play that-that-that whatever-it-is?
+
+Achilles: Oh, please do!
+
+(So the Crab plays it, with evident difficulty.)
+
+Tortoise: Bravo! Bravo! Is John Cage your favorite composer, Achilles? Achilles: Actually,
+he's my favorite anti-composer. Anyway, I'm glad you liked MY music.
+
+Crab: The two of you may find it amusing to listen to such totally meaningless cacophony,
+but I assure you it is not at all pleasant for a sensitive composer to be subjected to such
+excruciating, empty dissonances and meaningless rhythms. Achilles, I thought you had a
+good feeling for music. Could it be that your previous pieces had merit merely by
+coincidence?
+
+Achilles: Oh, please forgive me, Mr. Crab. I was trying to explore the limits of your musical
+notation. I wanted to learn directly what kinds of sound result when I write certain types
+of note sequences, and also how you evaluate pieces written in various styles.
+
+Crab: Harrumph! I am not just an automatic music-machine, you know. Nor am I a garbage
+disposal for musical trash.
+
+Achilles: I am very sorry. But I feel that I have learned a great deal by writing that small
+piece, and I am convinced that I can now write much better music than I ever could have
+if I hadn't tried that idea. And if you'll just play one more piece of mine, I have high
+hopes that you will feel better about my musical sensitivities.
+
+Crab: Well, all right. Write it down and I'll give it a chance.
+
+(Achilles writes:
+
+
+Ya:Vb:<(a -a) =(SSO -(b > b))Da=0>
+
+and the Crab plays.)
+
+You were right, Achilles. You seem to have completely regained your musical acuity.
+This is a little gem! How did you come to compose it? I have never heard anything like it.
+It obeys all the rules of harmony, and yet has a certain-what shall I say?-irrational appeal
+to it. I can't put my finger on it, but I like it for that very reason.
+
+Achilles: I kind of thought you might like it.
+
+Tortoise: Have you got a name for it, Achilles? Perhaps you might call it "The Song of
+Pythagoras". You remember that Pythagoras and his followers were among the first to
+study musical sound.
+
+Achilles: Yes, that's true. That would be a fine title.
+
+Crab: Wasn't Pythagoras also the first to discover that the ratio of two squares can never be
+equal to 2? Tortoise: I believe you're right. It was considered a truly sinister discovery at
+the time, for never before had anyone realized that there are
+
+
+The Magnificrab, Indeed,
+
+
+556
+
+
+
+numbers-such as the square root of 2-which are not ratios of integers. And thus the
+discovery was deeply disturbing to the Pythagoreans, who felt that it revealed an
+unsuspected and grotesque defect in the abstract world of numbers. But I don't know
+what this has to do with the price of tea in China.
+
+Achilles: Speaking of tea, isn't that the teahouse just up there ahead of us?
+
+Tortoise: Yes, that's it, all right. We ought to be there in a couple of minutes.
+
+Achilles: Hmm ... That's just enough time for me to whistle for you the tune which the taxi
+driver this morning had on his radio. It went like this.
+
+Crab: Hold on for a moment; I'll get some paper from my case, and jot
+
+down your tune. ( Scrounges around inside his case, and finds a blank sheet.)
+
+Go ahead; I'm ready.
+
+(Achilles whistles a rather long tune, and the Crab scrambles to keep up with him.)
+
+Could you whistle the last few bars again?
+
+Achilles: Why, certainly.
+
+(After a couple of such repeats, the session is complete, and the Crab proudly displays
+his transcription'.
+
+<((SSSSSO.SSSSSO)+(SSSSSO.SSSSSO))=((SSSSSSSO.SSSSSSSO)+(SO.SO))n—
+3b:<3c:(Sc+b)=((SSSSSSSO . SSSSSSSO)+(SO • SO))n3d:3d':3e:3e':
+
+<—<d=evd=e' >n<b=((Sd • Sd)+(Sd' • Sd'))nb=((Se • Se)+(Se' • Se’))» »
+
+The Crab then plays it himself.)
+
+Tortoise: It's peculiar music, isn't it? It sounds a wee bit like music from India, to me.
+
+Crab. Oh, I think it's too simple to be from India. But of course I know precious little about
+such things.
+
+Tortoise: Well, here we are at the teahouse. Shall we sit outside here, on the verandah?
+
+Crab: If you don't mind, I'd prefer to go inside. I've gotten perhaps enough sun for the day.
+
+(They go inside the teahouse and are seated at a nice wooden table, and order cakes and
+tea. Soon a cart of scrumptious-looking pastries is wheeled up, and each of them chooses
+his favorite.)
+
+Achilles: You know, Mr. C, I would love to know what you think of another piece which I
+have just composed in my head. Crab: Can you show it to me? Here, write it down on this
+napkin.
+
+(Achilles writes :
+
+da:3b:3c:<- 3d:3e:<(SSd*SSe)=bv(SSd*SSe)=c>n(a+a)=(b+c)>
+
+The Crab and Tortoise study it with interest.)
+
+
+The Magnificrab, Indeed,
+
+
+557
+
+
+
+Tortoise: Is it another beautiful piece, Mr. C, in your opinion?
+
+Crab: Well, uh ... (Shifts in his chair, and looks somewhat uncomfortable.)
+
+Achilles: What's the matter? Is it harder to decide whether this piece is beautiful than it is for
+other pieces?
+
+Crab: Ahm ... No, it's not that-not at all. It's just that, well ... I really have to HEAR a piece
+before I can tell how much I like it.
+
+Achilles: So go ahead and play it! I'm dying to know whether you find it beautiful or not.
+
+Crab: Of course, I'd be extremely glad to play it for you. The only thing is
+
+Achilles: Can't you play it for me? What's the matter? Why are you balking?
+
+Tortoise: Don't you realize, Achilles, that for Mr. Crab to fulfill your request would be most
+impolite and disturbing to the clientele and employees of this fine establishment?
+
+Crab (suddenly looking relieved): That's right. We have no right to impose our music on
+others.
+
+Achilles (dejectedly): Oh, PHOOEY! And I so much wanted to know what he thinks of this
+piece!
+
+Crab: Whew! That was a close call!
+
+Achilles: What was that remark?
+
+Crab: Oh-nothing. It's just that that waiter over there, he got knocked into by another waiter,
+and almost dropped a whole pot of tea into a lady's lap. A narrow escape, I must say.
+What do you say, Mr. Tortoise? Tortoise: Very good teas, I'd say. Wouldn't you agree,
+Achilles? Achilles: Oh, yes. Prime teas, in fact.
+
+Crab: Definitely. Well, I don't know about you two, but I should perhaps be going, for I've a
+long steep trail back to my house, on the other side of this hill.
+
+Achilles: You mean this is a big bluff?
+
+Crab: You said it, Achilles.
+
+Achilles: I see. Well, I'll have to remember that.
+
+Crab: It has been such a jolly afternoon, Achilles, and I sincerely hope we will exchange
+more musical compositions another day.
+
+Achilles: I'm looking forward to that very much, Mr. C. Well, good-bye. Tortoise: Good¬
+bye, Mr. C.
+
+(And the Crab heads off down his side of the hill.)
+
+Achilles: Now there goes a brilliant fellow ... In my estimation, he's at least four times as
+smart as any crab alive. Or he might even be five
+
+Tortoise: As you said in the beginning, and probably shall be saying forevermore, words
+without end.
+
+
+The Magnificrab, Indeed,
+
+
+558
+
+
+
+CHAPTER XVII
+
+
+Church, Turing, Tarski, and Others
+
+Formal and Informal Systems
+
+WE HAVE COME to the point where we can develop one of the main theses of this
+book: that every aspect of thinking can be viewed as a high-level description of a system
+which, on a low level, is governed by simple, even formal, rules. The "system", of course,
+is a brain-unless one is' speaking of thought processes flowing in another medium, such
+as a computer's circuits. The image is that of a formal system underlying an "informal
+system"-a system which can, for instance, make puns, discover number patterns, forget
+names, make awful blunders in chess, and so forth. This is what one sees from the
+outside: its informal, overt, software level. By contrast, it has a formal, hidden, hardware
+level (or "substrate") which is a formidably complex mechanism that makes transitions
+from state to state according to definite rules physically embodied in it, and according to
+the input of signals which impinge on it.
+
+A vision of the brain such as this has many philosophical and other consequences,
+needless to say. I shall try to spell some of them out in this Chapter. Among other things,
+this vision seems to imply that, at bottom, the brain is some sort of a "mathematical"
+object. Actually, that is at best a very awkward way to look at the brain. The reason is
+that, even if a brain is, in a technical and abstract sense, some sort of formal system, it
+remains true that mathematicians only work with simple and elegant systems, systems in
+which everything is extremely clearly defined-and the brain is a far cry from that, with its
+ten billion or more semi-independent neurons, quasi-randomly connected up to each
+other. So mathematicians would never study a real brain's networks. And if you define
+"mathematics" as what mathematicians enjoy doing, then the properties of brains are not
+mathematical.
+
+The only way to understand such a complex system as a brain is by chunking it on
+higher and higher levels, and thereby losing some precision at each step. What emerges at
+the top level is the "informal system" which obeys so many rules of such complexity that
+we do not yet have the vocabulary to think about it. And that is what Artificial
+Intelligence research is hoping to find. It has quite a different 'flavor from mathematics
+research. Nevertheless, there is a loose connection to mathematics: A1 people often come
+from a strong mathematics background, and
+
+
+Church, Turing, Tarski, and Others
+
+
+559
+
+
+
+mathematicians sometimes are intrigued by the workings of their own brains. The
+following passage, quoted from Stanislaw Ulam's autobiographical Adventures of a
+Mathematician , illustrates this point:
+
+It seems to me that more could be done to elicit ... the nature of associations, with
+computers providing the means for experimentation. Such a study would have to
+involve a gradation of notions, of symbols, of classes of symbols, of classes of
+classes, and so on, in the same way that the complexity of mathematical or
+physical structures is investigated.
+
+There must be a trick to the train of thought, a recursive formula. A group of
+neurons starts working automatically, sometimes without external impulse. It is a
+kind of iterative process with a growing pattern. It wanders about in the brain, and
+the way it happens must depend on the memory of similar patterns.'
+
+Intuition and the Magnificent Crab
+
+Artificial Intelligence is often referred to as "Al". Often, when I try to explain what is
+meant by the term, I say that the letters "AI" could just as well stand for "Artificial
+Intuition", or even "Artificial Imagery". The aim of Al is to get at what is happening
+when one's mind silently and invisibly chooses, from a myriad alternatives, which one
+makes most sense in a very complex situation. In many real-life situations, deductive
+reasoning is inappropriate, not because it would give wrong answers, but because there
+are too many correct but irrelevant statements which can be made; there are just too many
+things to take into account simultaneously for reasoning alone to be sufficient. Consider
+this mini-dialogue:
+
+"The other day I read in the paper that the—
+
+"Oh-you were reading? It follows that you have eyes. Or at least one eye. Or
+rather, that you had at least one eye then."
+
+A sense of judgment-"What is important here, and what is not?"-is called for. Tied up
+with this is a sense of simplicity, a sense of beauty. Where do these intuitions come from?
+How can they emerge from an underlying formal system?
+
+In the Magnificrab , some unusual powers of the Crab's mind are revealed. His
+own version of his powers is merely that he listens to music and distinguishes the
+beautiful from the non-beautiful. (Apparently for him there is a sharp dividing line.) Now
+Achilles finds another way to describe the Crab's abilities: the Crab divides statements of
+number theory into the categories true and false. But the Crab maintains that, if he
+chances to do so, it is only by the purest accident, for he is, by his own admission,
+incompetent in mathematics. What makes the Crab's performance all the more mystifying
+to Achilles, however, is that it seems to be in direct violation of a celebrated result of
+metamathematics with which Achilles is familiar:
+
+CHURCH'S THEOREM: There is no infallible method for telling theorems of TNT from
+nontheorems.
+
+
+Church, Turing, Tarski, and Others
+
+
+560
+
+
+
+It was proven in 1936 by the American logician Alonzo Church. Closely related is what I
+call the
+
+TARSKI-CHURCH-TURING THEOREM: There is no infallible method for telling true
+from false statements of number theory.
+
+The Church-Turing Thesis
+
+To understand Church's Theorem and the Tarski-Church-Turing Theorem better, we
+should first describe one of the ideas on which they are based; and that is the Church-
+Turing Thesis (often called "Church's Thesis"). For the Church-Turing Thesis is certainly
+one of the most important concepts in the philosophy of mathematics, brains, and
+thinking.
+
+Actually, like tea, the Church-Turing Thesis can be given in a variety of different
+strengths. So I will present it in various versions, and we will consider what they imply.
+The first version sounds very innocent-in fact almost pointless:
+
+CHURCH-TURING THESIS, TAUTOLOGICAL VERSION: Mathematics
+problems can be solved only by doing mathematics.
+
+Of course, its meaning resides in the meaning of its constituent terms. By "mathematics
+problem" I mean the problem of deciding whether some number possesses or does not
+possess a given arithmetical property. It turns out that by means of Godel-numbering and
+related coding tricks, almost any problem in any branch of mathematics can be put into
+this form, so that "mathematics problem" retains its ordinary meaning. What about "doing
+mathematics"? When one tries to ascertain whether a number has a property, there seem
+to be only a small number of operations which one uses in combination over and over
+again-addition, multiplication, checking for equality or inequality. That is, loops
+composed of such operations seem to be the only tool we have that allows us to probe the
+world of numbers. Note the word "seem". This is the critical word which the Church-
+Turing Thesis is about. We can give a revision:
+
+CHURCH-TURING THESIS, STANDARD VERSION: Suppose there is a
+method which a sentient being follows in order to sort numbers into two classes.
+Suppose further that this method always yields an answer within a finite amount
+of time, and that it always gives the same answer for a given number. Then:
+Some terminating FlooP program (i.e., some general recursive function) exists
+which gives exactly the same answers as the sentient being's method does.
+
+The central hypothesis, to make it very clear, is that any mental process which divides
+numbers into two sorts can be described in the form of a FlooP program. The intuitive
+belief is that there are no other tools than those in FlooP, and that there are no ways to use
+those tools other than by
+
+
+Church, Turing, Tarski, and Others
+
+
+561
+
+
+
+unlimited iterations (which FlooP allows). The Church-Turing Thesis is not a provable
+fact in the sense of a Theorem of mathematics-it is a hypothesis about the processes
+which human brains use.
+
+
+The Public-Processes Version
+
+Now some people might feel that this version asserts too much. These people might put
+their objections as follows: "Someone such as the Crab might exist-someone with an
+almost mystical insight into mathematics, but who is just as much in the dark about his
+own peculiar abilities as anyone else-and perhaps that person's mental mechanisms carry
+out operations which have no counterpart in FlooP." The idea is that perhaps we have a
+subconscious potential for doing things which transcend the conscious processes-things
+which are somehow inexpressible in terms of the elementary FlooP operations. For these
+objectors, we shall give a weaker version of the Thesis, one which distinguishes between
+public and private mental processes:
+
+CHURCH-TURING THESIS, PUBLIC-PROCESSES VERSION: Suppose there
+is a method which a sentient being follows in order to sort numbers into two
+classes. Suppose further that this method always yields an answer within a finite
+amount of time, and that it always gives the same answer for a given number.
+Proviso. Suppose also that this method can be communicated reliably from one
+sentient being to another by means of language. Then: Some terminating FlooP
+program (i.e., general recursive function) exists which gives exactly the same
+answers as the sentient beings' method does.
+
+This says that public methods are subject to "FlooPification", but asserts nothing about
+private methods. It does not say that they are un-FlooP-able, but it at least leaves the door
+open.
+
+
+Srinivasa Ramanujan
+
+As evidence against any stronger version of the Church-Turing Thesis, let us consider the
+case of the famous Indian mathematician of the first quarter of the twentieth century,
+Srinivasa Ramanujan (1887-1920). Ramanujan (Fig. 105) came from Tamilnadu, the
+southernmost part of India, and studied mathematics a little in high school. One day,
+someone who recognized Ramanujan's talent for math presented him with a copy of a
+slightly out-of-date textbook on analysis, which Ramanujan devoured (figuratively
+speaking). He then began making his own forays into the world of analysis, and by the
+time he was twenty-three, he had made a number of discoveries which he considered
+worthwhile. He did not know to whom to turn, but somehow was told about a professor
+of mathematics in faraway England, named G. H. Hardy. Ramanujan compiled his best
+
+
+Church, Turing, Tarski, and Others
+
+
+562
+
+
+
+
+FIGURE 105. Srinivasa Ramanujan and
+one of his strange Indian melodies.
+
+
+1 + e
+
+
+Tt7T
+
+
+V5
+
+
+1 ♦ e'
+
+
+
+
+£i}\
+2 ’/
+
+
+
+
+1 + e
+
+
+
+
+1 +
+
+
+results together in a packet of papers, and sent them allto the, unforewamed Hardy with a
+covering letter which friends helped him express
+
+in English. Below are some excerpts taken from Hardy's description of his reaction upon
+receiving the bundle:
+
+... It soon became obvious that Ramanujan must possess much more general
+theorems and was keeping a great deal up his sleeve.... [Some formulae]defeated
+me completely; I had never seen anything in the least like them before. A single
+look at them is enough to show that they could only be written down by a
+mathematician of the highest class. They must be true because, if they were not
+true, no one would have had the imagination to invent them. Finally ... the writer
+must be completely honest, because great mathematicians are commoner than
+thieves or humbugs of such incredible skill 2
+
+What resulted from this correspondence was that Ramanujan came to England in 1913,
+sponsored by Hardy; and then followed an intense collaboration which terminated in
+Ramanujan's early demise at age thin thirty-three from tuberculosis.
+
+Ramanujan had several extraordinary characteristics which set him apart from the
+majority of mathematicians. One was his lack of rigor. Very often he would simply state
+a result which, he would insist, had just come to
+
+
+Church, Turing, Tarski, and Others
+
+
+563
+
+
+
+him from a vague intuitive source, far out of the realm of conscious probing. In fact, he
+often said that the goddess Namagiri inspired him in his dreams. This happened time and
+again, and what made it all the more mystifying-perhaps even imbuing it with a certain
+mystical quality-was the fact that many of his "intuition-theorems" were wrong. Now
+there is a curious paradoxical effect where sometimes an event which you think could not
+help but make credulous people become a little more skeptical, actually has the reverse
+effect, hitting the credulous ones in some vulnerable spot of their minds, tantalizing them
+with the hint of some baffling irrational side of human nature. Such was the case with
+Ramanujan's blunders: many educated people with a yearning to believe in something of
+the sort considered Ramanujan's intuitive powers to be evidence of a mystical insight into
+Truth, and the fact of his fallibility seemed, if anything, to strengthen, rather than
+weaken, such beliefs.
+
+Of course it didn't hurt that he was from one of the most backward parts of India,
+where fakirism and other eerie Indian rites had been practiced for millennia, and were
+still practiced with a frequency probably exceeding that of the teaching of higher
+mathematics. And his occasional wrong flashes of insight, instead of suggesting to people
+that he was merely human, paradoxically inspired the idea that Ramanujan's wrongness
+always had some sort of "deeper rightness" to it-an "Oriental" rightness, perhaps touching
+upon truths inaccessible to Western minds. What a delicious, almost irresistible thought!
+Even Hardy-who would have been the first to deny that Ramanujan had any mystical
+powers-once wrote about one of Ramanujan's failures, "And yet I am not sure that, in
+some ways, his failure was not more wonderful than any of his triumphs."
+
+The other outstanding feature of Ramanujan's mathematical personality was his
+"friendship with the integers", as his colleague Littlewood put it. This is a characteristic
+that a fair number of mathematicians share to some degree or other, but which
+Ramanujan possessed to an extreme. There are a couple of anecdotes which illustrate this
+special power. The first one is related by Hardy:
+
+I remember once going to see him when he was lying ill at Putney. I had ridden in
+taxi-cab No. 1729, and remarked that the number seemed to me rather a dull one,
+and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very
+interesting number; it is the smallest number expressible as a sum of two cubes in
+two different ways." .1 asked him, naturally, whether he knew the answer to the
+corresponding problem for fourth powers; and he replied, after a moment's
+thought, that he could see no obvious example, and thought that the first such
+number must be very large.'
+
+It turns out that the answer for fourth powers is:
+
+635318657 = 134 + 1334 = 1584 + 594
+
+The reader may find it interesting to tackle the analogous problem for squares, which is
+much easier.
+
+It is actually quite interesting to ponder why it is that Hardy im-
+
+
+Church, Turing, Tarski, and Others
+
+
+560
+
+
+
+mediately jumped to fourth powers. After all, there are several other reasonably natural
+generalizations of the equation
+
+
+3,3 3,3
+
+u + v = x + y
+
+
+along different dimensions. For instance, there is the question about representing a
+number in three distinct ways as a sum of two cubes:
+
+
+3,3 3 , 3 3 , 3
+
+r + =u+v=x+y
+
+
+Or, one can use three different cubes:
+
+u3 + v3 + w 3 = x 3 + y 3 + z 3 .
+
+Or one can even make a Grand Generalization in all dimensions at once:
+
+
+44444 4444
+
+r + s + t = u + v + w = x + v + z
+
+
+There is a sense, however, in which Hardy's generalization is "the most mathematician¬
+like". Could this sense of mathematical esthetics ever be programmed?
+
+The other anecdote is taken from a biography of Ramanujan by his
+countryman S. R. Ranganathan, where it is called "Ramanujan's Flash". It is related by a
+Indian friend of Ramanujan's from his Cambridge days. Dr. P. C. Mahalanobis.
+
+On another occasion, I went to his room to have lunch with him. The First World
+War had started some time earlier. I had in my hand a copy of the monthly "Strand
+Magazine" which at that time used to publish a number of puzzles to be solved by
+readers. Ramanujan was stirring something in a pan over the fire for our lunch. I
+was sitting near the table, turning over the pages of the Magazine. I got interested in
+a problem involving a relation between two numbers. I have forgotten the details;
+but I remember the type of the problem. Two British officers had been billeted in
+Paris in two different houses in a long street; the door numbers of these houses
+were related in a special way; the problem was to find out the two numbers. It was
+not at all difficult. I got the solution in a few minutes by trial and error.
+
+MAHALANOBIS (in a joking way): Now here is a problem for you.
+
+RAMANUJAN: What problem, tell me.'(He went on stirring the pan.) I read out the
+question from the "Strand Magazine".
+
+RAMANUJAN: Please take down the solution. (He dictated a continued fraction.)
+
+The first term was the solution which I had obtained. Each successive term
+represented successive solutions for the same type of relation between two numbers,
+as the number of houses in the street would increase indefinitely. I was amazed.
+MAHALANOBIS: Did you get the solution in a flash?
+
+RAMANUJAN: Immediately I heard the problem, it was clear that the Solution was
+obviously a continued fraction; I then thought, "Which continued fraction?" and the
+answer came to my mind. It was just as simple as this.'
+
+
+Hardy, as Ramanujan's closest co-worker, was often asked after
+
+
+Church, Turing, Tarski, and Others
+
+
+561
+
+
+
+Ramanujan's death if there had been any occult or otherwise exotically flavored elements
+to Ramanujan's thinking style. Here is one comment which he gave:
+
+I have often been asked whether Ramanujan had any special secret; whether his
+methods differed in kind from those of other mathematicians; whether there was
+anything really abnormal in his mode of thought. I cannot answer these questions
+with any confidence or conviction; but I do not believe it. My belief is that all
+mathematicians think, at bottom, in the same kind of way, and that Ramanujan was
+no exception , 5
+
+Here Hardy states in essence his own version of the Church-Turing Thesis. I paraphrase:
+CHURCH-TURING THESIS, HARDY'S VERSION: At bottom, all mathematicians are
+isomorphic.
+
+This does not equate the mathematical potential of mathematicians with that of general
+recursive functions; for that, however, all you need is to show that some mathematician's
+mental capacity is no more general than recursive functions. Then, if you believe Hardy's
+Version, you know it for all mathematicians.
+
+Then Hardy compares Ramanujan with calculating prodigies:
+
+His memory, and his powers of calculation, were very unusual, but they could not
+reasonably be called "abnormal". If he had to multiply two large numbers, he
+multiplied them in the ordinary way; he could do it with unusual rapidity and
+accuracy, but not more rapidly and accurately than any mathematician who is
+naturally quick and has the habit of computations
+
+Hardy describes what he perceived as Ramanujan's outstanding intellectual attributes:
+
+With his memory, his patience, and his power of calculation, he combined a power
+of generalisation, a feeling for form, and a capacity for rapid modification of his
+hypotheses, that were often really startling, and made him, in his own field, without a
+rival in his day.'
+
+The part of this passage which I have italicized seems to me to be an excellent
+characterization of some of the subtlest features of intelligence in general. Finally, Hardy
+concludes somewhat nostalgically:
+
+[His work has not the simplicity and inevitableness of the very greatest work; it
+would be greater if it were less strange. One gift it has which no one can deny-
+profound and invincible originality. He would probably have been a greater
+mathematician if he had been caught and tamed a little in his youth; he would have
+discovered more that was new, and that, no doubt, of greater importance. On the
+other hand he would have been less of a Ramanujan, and more of a European
+
+o
+
+professor and the loss might have been greater than the gain.
+
+The esteem in which Hardy held Ramanujan is revealed by the romantic way in which he
+speaks of him.
+
+
+Church, Turing, Tarski, and Others
+
+
+562
+
+
+
+"Idiots 'Savants"
+
+
+There is another class of people whose mathematical abilities seem to defy rational
+explanation-the so-called "idiots savants", who can perform complex calculations at
+lightning speeds in their heads (or wherever they do it). Johann Martin Zacharias Dase,
+who lived from 1824 to 1861 and was employed by various European governments to
+perform computations, is an outstanding example. He not only could multiply two
+numbers each of 100 digits in his head; he also had an uncanny sense of quantity. That is,
+he could just "tell", without counting, how many sheep were in a field, or words in a
+sentence, and so forth, up to about 30-this in contrast to most of us, who have such a
+sense up to about 6, with reliability. Incidentally, Dase was not an idiot.
+
+I shall not describe the many fascinating documented cases of "lightning
+calculators", for that is not my purpose here. But I do feel it is important to dispel the idea
+that they do it by some mysterious, unanalyzable method. Although it is often the case
+that such wizards' calculational abilities far exceed their abilities to explain their results,
+every once in a while, a person with other intellectual gifts comes along who also has this
+spectacular ability with numbers. From such people's introspection, as well as from
+extensive research by psychologists, it has been ascertained that nothing occult takes
+place during the performances of lightning calculators, but simply that their minds race
+through intermediate steps with the kind of self-confidence that a natural athlete has in
+executing a complicated motion quickly and gracefully. They do not reach their answers
+by some sort of instantaneous flash of enlightenment (though subjectively it may feel that
+way to some of them), but-like the rest of us-by sequential calculation, which is to say, by
+FlooP-ing (or BlooP-ing) along. -
+
+Incidentally, one of the most obvious clues that no "hot line to God" is involved is
+the mere fact that when the numbers involved get bigger, the answers are slower in
+coming. Presumably, if God or an "oracle" were supplying the answers, he wouldn't have
+to slow up when the numbers got bigger. One could probably make a nice plot showing
+how the time taken by a lightning calculator varies with the sizes of the numbers
+involved, and the operations involved, and from it deduce some features of the algorithms
+employed.
+
+
+The Isomorphism Version of the Church-Turing Thesis
+
+This finally brings us to a strengthened standard version of the Church-Turing Thesis:
+
+CHURCH-TURING THESIS, ISOMORPHISM VERSION: Suppose there is a
+method which a sentient being follows in order to sort numbers into two classes.
+Suppose further that this method always yields an answer within a finite amount
+of time, and that it always gives the same answer for a given number. Then:
+Some terminating FlooP program (i.e..
+
+
+Church, Turing, Tarski, and Others
+
+
+563
+
+
+
+general recursive function) exists which gives exactly the same answers as the
+sentient being's method does. Moreover: The mental process and the FlooP
+program are isomorphic in the sense that on some level there is a correspondence
+between the steps being carried out in both computer and brain.
+
+Notice that not only has the conclusion been strengthened, but also the proviso of
+communicability of the faint-hearted Public-Processes Version has been dropped. This
+bold version is the one which we now shall discuss.
+
+In brief, this version asserts that when one computes something, one's mental
+activity can be mirrored isomorphically in some FlooP program. And let it be very clear
+that this does not mean that the brain is actually running a FlooP program, written in the
+FlooP language complete with BEGIN's, END'S, ABORT'S, and the rest-not at all. It is
+just that the steps are taken in the same order as they could be in a FlooP program, and
+the logical structure of the calculation can be mirrored in a FlooP program.
+
+Now in order to make sense of this idea, we shall have to make some level
+distinctions in both computer and brain, for otherwise it could be misinterpreted as utter
+nonsense. Presumably the steps of the calculation going on inside a person's head are on
+the highest level, and are supported by lower levels, and eventually by hardware. So if we
+speak of an isomorphism, it means we've tacitly made the assumption that the highest
+level can be isolated, allowing us to discuss what goes on there independently of other
+levels, and then to map that top level into FlooP. To be more precise, the assumption is
+that there exist software entities which play the roles of various mathematical constructs,
+and which are activated in ways which can be mirrored exactly inside FlooP (see Fig.
+106). What enables these software entities to come into existence is the entire
+infrastructure discussed in Chapters XI and XI I, as well as in the Prelude, Ant Fugue.
+There is no assertion of isomorphic activity on the lower levels of brain and computer
+(e.g., neurons and bits).
+
+The spirit of the Isomorphism Version, if not the letter, is gotten across by saying
+that what an idiot savant does in calculating, say, the logarithm of 7r, is isomorphic to
+what a pocket calculator does in calculating it-where the isomorphism holds on the
+arithmetic-step level, not on the lower levels of, in the one case, neurons, and in the other,
+integrated circuits. (Of course different routes can be followed in calculating anything-but
+presumably the pocket calculator, if not the human, could be instructed to calculate the
+answer in any specific manner.)
+
+FIGURE 106. The behavior of natural numbers can be mirrored in a human brain or in
+the programs of a computer. These two different representations can then be mapped
+onto each other on an appropriately abstract level.
+
+
+
+Church, Turing, Tarski, and Others
+
+
+564
+
+
+
+
+
+
+Representation of Knowledge about the Real World
+
+
+Now this seems quite plausible when the domain referred to is number theory, for there
+the total universe in which things happen is very small and clean. Its boundaries and
+residents and rules are well-defined, as in a hard-edged maze. Such a world is far less
+complicated than the open-ended and ill-defined world which we inhabit. A number
+theory problem, once stated, is complete in and of itself. A real-world problem, on the
+other hand, never is sealed off from any part of the world with absolute certainty. For
+instance, the task of replacing a burnt-out light bulb may turn out to require moving a
+garbage bag; this may unexpectedly cause the spilling of a box of pills, which then forces
+the floor to be swept so that the pet dog won't eat any of the spilled pills, etc., etc. The
+pills and the garbage and the dog and the light bulb are all quite distantly related parts of
+the world-yet an intimate connection is created by some everyday happenings. And there
+is no telling what else could be brought in by some other small variations on the
+expected. By contrast, if you are given a number theory problem, you never wind up
+having to consider extraneous things such as pills or dogs or bags of garbage or brooms
+in order to solve your problem. (Of course, your intuitive knowledge of such objects may
+serve you in good stead as you go about unconsciously trying to manufacture mental
+images to help you in visualizing the problem in geometrical terms-but that is another
+matter.)
+
+Because of the complexity of the world, it is hard to imagine a little pocket
+calculator that can answer questions put to it when you press a few buttons bearing labels
+such as "dog", "garbage", "light bulb", and so forth. In fact, so far it has proven to be
+extremely complicated to have a full-size high-speed computer answer questions about
+what appear to us to be rather simple subdomains of the real world. It seems that a large
+amount of knowledge has to be taken into account in a highly integrated way for
+"understanding" to take place. We can liken real-world thought processes to a tree whose
+visible part stands sturdily above ground but depends vitally on its invisible roots which
+extend way below ground, giving it stability and nourishment. In this case the roots
+symbolize complex processes which take place below the conscious level of the mind-
+processes whose effects permeate the way we think but of which we are unaware. These
+are the "triggering patterns of symbols" which were discussed in Chapters XI and XII.
+
+Real-world thinking is quite different from what happens when we do a
+multiplication of two numbers, where everything is "above ground", so to speak, open to
+inspection. In arithmetic, the top level can be "skimmed off " and implemented equally
+well in many different sorts of hardware: mechanical adding machines, pocket
+calculators, large computers, people's brains, and so forth. This is what the Church-
+Turing Thesis is all about. But when it comes to real-world understanding, it seems that
+there is no simple way to skim off the top level, and program it. alone. The triggering
+patterns of symbols are just too complex. There must he several levels through which
+thoughts may "percolate" and "bubble".
+
+
+Church, Turing, Tarski, and Others
+
+
+565
+
+
+
+In particular-and this comes back to a major theme of Chapters XI ' and XH-the
+representation of the real world in the brain, although rooted in isomorphism to some
+extent, involves some elements which have no counterparts at all in the outer world. That
+is, there is much more to it than simple mental structures representing "dog", "broom",
+etc. All of these symbols exist, to be sure-but their internal structures are extremely
+complex and to a large degree are unavailable for conscious inspection. Moreover, one
+would hunt in vain to map each aspect of a symbol’s internal structure onto some specific
+feature of the real world.
+
+Processes That Are Not So Skimmable
+
+For this reason, the brain begins to look like a very peculiar formal system, for on its
+bottom level-the neural level-where the "rules" operate and change the state, there may be
+no interpretation of the primitive elements (neural firings, or perhaps even lower-level
+events). Yet on the top level, there emerges a meaningful interpretation-a mapping from
+the large "clouds" of neural activity which we have been calling "symbols", onto the real
+world. There is some resemblance to the Godel construction, in that a high-level
+isomorphism allows a high level of meaning to be read into strings; but in the Godel
+construction, the higher-level meaning "rides" on the lower level-that is, it is derived
+from the lower level, once the notion of Godel-numbering has been introduced. But in the
+brain, the events on the neural level are not subject to real-world interpretation; they are
+simply not imitating anything. They are there purely as the substrate to support the higher
+level, much as transistors in a pocket calculator are there purely to support its number¬
+mirroring activity. And the implication is that there is no way to skim off just the highest
+level and make an isomorphic copy in a program; if one is to mirror the brain processes
+which allow real-world understanding, then one must mirror some of the lower-level
+things which are taking place: the "languages of the brain". This doesn't necessarily mean
+that one must go all the way down to the level of the hardware, though that may turn out
+to be the case.
+
+In the course of developing a program with the aim of achieving an "intelligent"
+(viz., human-like) internal representation of what is "out there", at some point one will
+probably be forced into using structures and processes which do not admit of any
+straightforward interpretations-that is, which cannot be directly mapped onto elements of
+reality. These lower layers of the program will be able to be understood only by virtue of
+their catalytic relation to layers above them, rather than because of some direct
+connection they have to the outer world. (A concrete image of this idea was suggested by
+the Anteater in the Ant Fugue: the "indescribably boring nightmare" of trying to
+understand a book on the letter level.)
+
+Personally, I would guess that such multilevel architecture of concept-handling
+systems becomes necessary just when processes involving images and analogies become
+significant elements of the program-in
+
+
+Church, Turing, Tarski, and Others
+
+
+566
+
+
+
+contrast to processes which are supposed to carry out strictly deductive reasoning.
+Processes which carry out deductive reasoning can be programmed in essentially one
+single level, and are therefore skimmable, by definition. According to my hypothesis,
+then, imagery and analogical thought processes intrinsically require several layers of
+substrate and are therefore intrinsically non-skim mable. I believe furthermore that it is
+precisely at this same point that creativity starts to emerge-which would imply that
+creativity intrinsically depends upon certain kinds of "uninterpretable" lower-level
+events. The layers of underpinning of analogical thinking are, of course, of extreme
+interest, and. some speculations on their nature will be offered in the next two Chapters.
+
+Articles of Reductionistic Faith
+
+
+One way to think about the relation between higher and lower levels in the brain is this.
+One could assemble a neural net which, on a local (neuron-to-neuron) level, performed in
+a manner indistinguishable from a neural net in a brain, but which had no higher-level
+meaning at all. The fact that the lower level is composed of interacting neurons does not
+necessarily force any higher level of meaning to appear-no more than the fact that
+alphabet soup contains letters forces meaningful sentences to be found, swimming about
+in the bowl. High-level meaning is an optional feature of a neural network-one which
+may emerge as a consequence of evolutionary environmental pressures.
+
+Figure 107 is a diagram illustrating the fact that emergence of a higher level of
+meaning is optional. The upwards-pointing arrow indicates that a substrate can occur
+without a higher level of meaning, but not vice versa: the higher level must be derived
+from properties of a lower one.
+
+FIGURE 107. Floating on neural activity, the symbol level of the brain mirrors the
+world. But neural activity per se, which can be simulated on a computer, does not create
+thought; that calls for higher levels of organization.
+
+
+higher level
+of brain
+(symbol level)
+
+
+$cm-
+isomorphism
+(meaning)
+
+
+world
+
+
+A optional link
+
+
+I computer
+I model of
+yneural network/
+
+
+isomorphism
+
+
+(substrate: brain)
+as collection
+of neurons
+
+
+Church, Turing, Tarski, and Others
+
+
+567
+
+
+
+The diagram includes an indication of a computer simulation of a neural network. This is
+in principle feasible, no matter how complicated the network, provided that the behavior
+of individual neurons can be described in terms of computations which a computer can
+carry out. This is a subtle postulate which few people even think of questioning.
+Nevertheless it is a piece of "reductionistic faith"; it could be considered a "microscopic
+version" of the Church-Turing Thesis. Below we state it explicitly:
+
+CHURCH-TURING THESIS, MICROSCOPIC VERSION: The behavior of the
+components of a living being can be simulated on a computer. That is, the
+behavior of any component (typically assumed to be a cell) can be calculated by
+a FlooP program (i.e., general recursive function) to any desired degree of
+accuracy, given a sufficiently precise description of the component's internal
+state and local environment.
+
+This version of the Church-Turing Thesis says that brain processes do not possess any
+more mystique-even though they possess more levels of organization-than, say, stomach
+processes. It would be unthinkable in this day and age to suggest that people digest their
+food, not by ordinary chemical processes, but by a sort of mysterious and magic
+"assimilation". This version of the CT-Thesis simply extends this kind of commonsense
+reasoning to brain processes. In short, it amounts to faith that the brain operates in a way
+which is, in principle, understandable. It is a piece of reductionist faith.
+
+A corollary to the Microscopic CT-Thesis is this rather terse new macroscopic
+version:
+
+CHURCH-TURING THESIS, REDUCTIONIST'S VERSION: All brain processes are
+derived from a computable substrate.
+
+This statement is about the strongest theoretical underpinning one could give in support
+of the eventual possibility of realizing Artificial Intelligence.
+
+Of course, Artificial Intelligence research is not aimed at simulating neural
+networks, for it is based on another kind of faith: that probably there are significant
+features of intelligence which can be floated on top of entirely different sorts of substrates
+than those of organic brains. Figure 108 shows the presumed relations among Artificial
+Intelligence, natural intelligence, and the real world.
+
+Parallel Progress in A1 and Brain Simulation?
+
+The idea that, if A1 is to be achieved, the actual hardware of the brain might one day have
+to be simulated or duplicated, is, for the present at least, quite an abhorrent thought to
+many A1 workers. Still one wonders, "How finely will we need to copy the brain to
+achieve Al?" The real answer is probably that it all depends on how many of the features
+of human consciousness you want to simulate.
+
+
+Church, Turing, Tarski, and Others
+
+
+568
+
+
+
+A , program f 0 „ 0r/}M '
+
+
+lOK'i'l
+
+sofrwaic levels
+
+
+electronic
+
+substrare
+
+
+12
+
+
+yiniml" level
+brain (mind)/
+
+
+scm-
+
+
+macrascopic
+
+world
+
+
+brain
+(inieniiediaLtr
+lewis)
+
+
+Lwmorptusm
+(representation
+of knowledge)
+
+
+microscopic
+
+world
+
+
+neural
+
+substrate
+
+
+“ultimate -
+substrate”,
+((laws of physics)/
+
+
+FIGURE 108. Crucial to the endeavor of Artificial Intelligence research is the notion that
+the symbolic levels of the mind can be "skimmed off " of their neural substrate and
+implemented in other media, such as the electronic substrate of computers. To what depth
+the copying of brain must go is at present completely unclear.
+
+Is an ability to play checkers well a sufficient indicator of intelligence? If so, then
+A1 already exists, since checker-playing programs are of world class. Or is intelligence an
+ability to integrate functions symbolically, as in a freshman calculus class? If so, then AI
+already exists, since symbolic integration routines outdo the best people in most cases. Or
+is intelligence the ability to play chess well? If so, then AI is well on its way, since chess¬
+playing programs can defeat most good amateurs; and the level of artificial chess will
+probably continue to improve slowly.
+
+Historically, people have been naive about what qualities, if mechanized, would
+undeniably constitute intelligence. Sometimes it seems as though each new step towards
+AI, rather than producing something which everyone agrees is real intelligence, merely
+reveals what real intelligence is not. If intelligence involves learning, creativity,
+emotional responses, a sense of beauty, a sense of self, then there is a long road ahead,
+and it may be that these will only be realized when we have totally duplicated a living
+brain.
+
+
+Beauty, the Crab, and the Soul
+
+Now what, if anything, does all this have to say about the Crab's virtuoso performance in
+front of Achilles? There are two issues clouded together here. They are:
+
+
+Church, Turing, Tarski, and Others
+
+
+569
+
+
+
+
+
+
+
+(1) Could any brain process, under any circumstances, distinguish completely
+reliably between true and false statements of TNT without being in violation of
+the Church-Turing Thesis-or is such an act in principle impossible?
+
+(2) Is perception of beauty a brain process?
+
+First of all, in response to (1), if violations of the Church-Turing Thesis are allowed, then
+there seems to be no fundamental obstacle to the strange events in the Dialogue. So what
+we are interested in is whether a believer in the Church-Turing Thesis would have to
+disbelieve in the Crab's ability. Well, it all depends on which version of the CT-Thesis
+you believe. For example, if you only subscribe to the Public-Processes Version, then you
+could reconcile the Crab's behavior with it very easily by positing that the Crab's ability is
+not communicable. Contrariwise, if you believe the Reductionist's Version, you will have
+a very hard time believing in the Crab's ostensible ability (because of Church's Theorem-
+soon to be demonstrated). Believing in intermediate versions allows you a certain amount
+of wishy-washiness on the issue. Of course, switching your stand according to
+convenience allows you to waffle even more.
+
+It seems appropriate to present a new version of the CT-Thesis, one which is
+tacitly held by vast numbers of people, and which has been publicly put forth by several
+authors, in various manners. Some of the more famous ones are: philosophers Hubert
+Dreyfus, S. Jaki, Mortimer Taube, and J. R. Lucas; the biologist and philosopher Michael
+Polanyi (a holist par excellence); the distinguished Australian neurophysiologist John
+Eccles. I am sure there are many other authors who have expressed similar ideas, and
+countless readers who are sympathetic. I have attempted below to summarize their joint
+position. I have probably not done full justice to it, but I have tried to convey the flavor as
+accurately as I can:
+
+CHURCH-TURING THESIS, SOULISTS' VERSION: Some kinds of things which
+a brain can do can be vaguely approximated on a computer but not most, and
+certainly not the interesting ones. But anyway, even if they all could, that would
+still leave the soul to explain, and there is no way that computers have any
+bearing on that.
+
+This version relates to the tale of the Magnificrab in two ways. In the first place, its
+adherents would probably consider the tale to be silly and implausible, but-not forbidden
+in principle. In the second place, they would probably claim that appreciation of qualities
+such as beauty is one of those properties associated with the elusive soul, and is therefore
+inherently possible only for humans, not for mere machines.
+
+We will come back to this second point in a moment; but first, while we are on
+the subject of "soulists", we ought to exhibit this latest version in an even more extreme
+form, since that is the form to which large numbers of well-educated people subscribe
+these days:
+
+CHURCH-TURING THESIS, THEODORE ROSZAK VERSION: Computers are
+ridiculous. So is science in general.
+
+
+Church, Turing, Tarski, and Others
+
+
+570
+
+
+
+This view is prevalent among certain people who see in anything smacking of numbers or
+exactitude a threat to human values. It is too bad that they do not appreciate the depth and
+complexity and beauty involved in exploring abstract structures such as the human mind,
+where, indeed, one comes in intimate contact with the ultimate questions of what to be
+human is.
+
+Getting back to beauty, we were about to consider whether the appreciation of
+beauty is a brain process, and if so, whether it is imitable by a computer. Those who
+believe that it is not accounted for by the brain are very unlikely to believe that a
+computer could possess it. Those who believe it is a brain process again divide up
+according to which version of the CT-Thesis they believe. A total reductionist would
+believe that any brain process can in principle be transformed into a computer program;
+others, however, might feel that beauty is too ill-defined a notion for a computer program
+ever to assimilate. Perhaps they feel that the appreciation of beauty requires an element of
+irrationality, and therefore is incompatible with the very fiber of computers.
+
+Irrational and Rational Can Coexist on Different Levels
+
+However, this notion that "irrationality is incompatible with computers" rests on a
+severe confusion of levels. The mistaken notion stems from the idea that since
+computers are faultlessly functioning machines, they are therefore bound to be
+"logical" on all levels. Yet it is perfectly obvious that a computer can be
+instructed to print out a sequence of illogical statements-or, for variety's sake, a
+batch of statements having random truth values. Yet in following such
+instructions, a computer would not be making any mistakes! On the contrary, it
+would only be a mistake if the computer printed out something other than the
+statements it had been instructed to print. This illustrates how faultless
+functioning on one level may underlie symbol manipulation on a higher level-and
+the goals of the higher level may be completely unrelated to the propagation of
+Truth.
+
+Another way to gain perspective on this is to remember that a brain, too, is a collection of
+faultlessly functioning elements-neurons. Whenever a neuron's threshold is surpassed by
+the sum of the incoming signals, BANG!-it fires. It never happens that a neuron forgets
+its arithmetical knowledge-carelessly adding its inputs and getting a wrong answer. Even
+when a neuron dies, it continues to function correctly, in the sense that its components
+continue to obey the laws of mathematics and physics. Yet as we all know, neurons are
+perfectly capable of supporting high-level behavior that is wrong, on its own level, in the
+most amazing ways. Figure 109 is meant to illustrate such a clash of levels: an incorrect
+belief held in the software of a mind, supported by the hardware of a faultlessly
+functioning brain.
+
+The point-a point which has been made several times earlier in various contexts-is
+simply that meaning can exist on two or more different levels of a symbol-handling
+system, and along with meaning, rightness and wrongness can exist on all those levels.
+The presence of meaning on a given
+
+
+Church, Turing, Tarski, and Others
+
+
+571
+
+
+
+v> <*VT - .
+
+
+v> «£'v r T •
+
+'.-V' v *i>
+
+
+# . »•»•» >
+
++*?• iV.,t
+
+^ 7 / if* IT
+
+
+r &»•<!
+
+- %.^-T
+i*‘w i •‘ , l
+
+
+AvV*
+
+
+* •*-^\
+
+if,T 3 *»»■«
+
+;•*■« C-7
+
+
+4$P
+
+• * •■.,*
+
+
+
+# * Hi
+
+
+Sgi*** \
+
+>M*m **«|
+
+W ! * ♦./ 'J »
+
+iVT.r
+
+f; 'v'
+
+**.*.».:* % **
+
+\'.t' -. '
+
+"t «»
+
+: <;
+
+’►Vs’
+
+
+FIGURE 109. The brain is rational; the mind may not be. I Dr awing by the author.]
+
+
+Church, Turing, Tarski, and Others
+
+
+
+
+
+
+
+
+
+
+
+
+
+level is determined by whether or- not reality is mirrored in an isomorphic (or looser)
+fashion on that level. So the fact that neurons always perform correct additions (in fact,
+much more complex calculations) has no bearing whatsoever on the correctness of the
+top-level conclusions supported by their machinery. Whether one's top level is engaged in
+proving koans of Boolean Buddhism or in meditating on theorems of Zerl Algebra, one's
+neurons are functioning rationally. By the same token, the high-level symbolic processes
+which in a brain create the experience of appreciating beauty are perfectly rational on the
+bottom level, where the faultless functioning is taking place; any irrationality, if there is
+such, is on the higher level, and is an epiphenomenon-a consequence-of the events on the
+lower level.
+
+To make the same point in a different way, let us say you are having a hard time
+making up your mind whether to order a cheeseburger or a pineappleburger. Does this
+imply that your neurons are also balking, having difficulty deciding whether or not to
+fire? Of course not. Your hamburger-confusion is a high-level state which fully depends
+on the efficient firing of thousands of neurons in very organized ways. This is a little
+ironic, yet it is perfectly obvious when you think about it. Nevertheless, it is probably fair
+to say that nearly all confusions about minds and computers have their origin in just such
+elementary level-confusions.
+
+There is no reason to believe that a. computer's faultlessly functioning hardware
+could not support high-level symbolic behavior which would represent such complex
+states as confusion, forgetting, or appreciation of beauty. It would require that there exist
+massive subsystems interacting with each other according to a complex "logic". The
+overt behavior could appear either rational or irrational; but underneath it would be the
+performance of reliable, logical hardware.
+
+More Against Lucas
+
+Incidentally, this kind of level distinction provides us with some new fuel in arguing
+against Lucas. The Lucas argument is based on the idea that Godel’s Theorem is
+applicable, by definition, to machines. In fact, Lucas makes a most emphatic
+pronunciation:
+
+Godel’s theorem must apply to cybemetical machines, because it is of the essence
+
+of being a machine, that it should be a concrete instantiation of a formal system. 0
+
+This is, as we have seen, true on the hardware level-but since there may be higher levels,
+it is not the last word on the subject. Now Lucas gives the impression that in the mind-
+imitating machines he discusses, there is only one level on which manipulation of
+symbols takes place. For instance, the Rule of Detachment (called "Modus Ponens" in his
+article) would be wired into the hardware and would be an unchangeable feature of such
+a machine. He goes further and intimates that if Modus Ponens were not an
+
+
+Church, Turing, Tarski, and Others
+
+
+573
+
+
+
+immutable pillar of the machine's system, but could be overridden on occasion, then:
+
+The system will have ceased to be a formal logical system, and the machine will
+barely qualify for the title of a model for the mind. 10
+
+Now many programs which are being developed in At research have very little in
+common with programs for generating truths of number theory—programs with inflexible
+rules of inference and fixed sets of axioms. Yet they are certainly intended as "models for
+the mind". On their top level the "informal" level-there may be manipulation of images,
+formulation of analogies, forgetting of ideas, confusing of concepts, blurring of
+distinctions, and so forth. But this does not contradict the fact that they rely on the correct
+functioning of their underlying hardware as much as brains rely on the correct
+functioning of their neurons. So At programs are still "concrete instantiations of formal
+systems"-but they are not machines to which Lucas' transmogrification of Godel’s proof
+can be applied. Lucas' argument applies merely to their bottom level, on which their
+intelligence-however great or small it may be-does not lie.
+
+There is one other way in which Lucas betrays his oversimplified vision of how
+mental processes would have to be represented inside computer programs. In discussing
+the matter of consistency, he writes
+
+If we really were inconsistent machines, we should remain content with our
+inconsistencies, and would happily affirm both halves of a contradiction. Moreover,
+we would be prepared to say absolutely anything-which we are not. It is easily
+shown that in an inconsistent formal system everything is provable."
+
+This last sentence shows that Lucas assumes that the Propositional Calculus must of
+necessity be built into any formal system which carries out reasoning. In particular, he is
+thinking of the theorem «PA-P>DQ> of the Propositional Calculus; evidently he has
+the erroneous belief that it is an inevitable feature of mechanized reasoning. However, it
+is perfectly plausible that logical thought processes, such as propositional reasoning, will
+emerge as consequences of the general intelligence of an At program, rather than being
+preprogrammed. This is what happens in humans! And there is no particular reason to
+assume that the strict Propositional Calculus, with its rigid rules and the rather silly
+definition of consistency that they entail, would emerge from such a program.
+
+An Underpinning of A1
+
+We can summarize this excursion into level distinctions and come away with one final,
+strongest version of the Church-Turing Thesis:
+
+CHURCH-TURING THESIS, At VERSION: Mental processes of any sort can be
+simulated by a computer program whose underlying language is of
+
+
+Church, Turing, Tarski, and Others
+
+
+574
+
+
+
+power equal to that of FlooP-that is, in which all partial recursive functions can be
+programmed.
+
+It should also be pointed out that in practice, many AI researchers rely on another
+article of faith which is closely related to the CT-Thesis, and which I call the AI Thesis. It
+runs something like this:
+
+AI THESIS: As the intelligence of machines evolves, its underlying
+
+mechanisms will gradually converge to the mechanisms underlying human
+intelligence.
+
+In other words, all intelligences are just variations on a single theme; to create true
+intelligence, At workers will just have to keep pushing to ever lower levels, closer and
+closer to brain mechanisms, if they wish their machines to attain the capabilities which
+we have.
+
+
+Church's Theorem
+
+Now let us come back to the Crab and to the question of whether his decision procedure
+for theoremhood (which is presented in the guise of a filter for musical beauty) is
+compatible with reality. Actually, from the events which occur in the Dialogue, we have
+no way of deducing whether the Crab's gift is an ability to tell theorems from
+nontheorems, or alternatively, an ability to tell true statements from false ones. Of course
+in many cases this amounts to the same thing but Godel’s Theorem shows that it doesn't
+always. But no matter: both of these alternatives are impossible, if you believe the At
+Version of the Church-Turing Thesis. The proposition that it is impossible to have a
+decision procedure for theoremhood in any formal system with the power of TNT is
+known as Church's Theorem. The proposition that it is impossible to have a decision
+procedure for number theoretical truth-if such truth exists, which one can well doubt after
+meeting up with all the bifurcations of TNT-follows quickly from Tarski's Theorem
+(published in 1933, although the ideas were known to Tarski considerably earlier).
+
+The proofs of these two highly important results of metamathematics are very
+similar. Both of them follow quite quickly from self-referential constructions. Let us first
+consider the question of a decision procedure for TNT-theoremhood. If there were a
+uniform way by which people could decide which of the classes "theorem" and
+"nontheorem" any given formula X fell into, then, by the CT-Thesis (Standard Version),
+there would exist a terminating FlooP program (a general recursive function) which could
+make the same decision, when given as input the Godel number of formula X. The
+crucial step is to recall that any property that can be tested for by a terminating FlooP
+program is represented in TNT. This means that the property of TNT-theoremhood would
+be represented (as distinguished from merely expressed) inside TNT. But as we shall see
+in a moment, this.
+
+
+Church, Turing, Tarski, and Others
+
+
+575
+
+
+
+would put us in hot water, for if theoremhood is a representable attribute, then Godel’s
+formula G becomes as vicious as the Epimenides paradox.
+
+It all hinges on what G says: "G is not a theorem of TNT". Assume that G were a
+theorem. Then, since theoremhood is supposedly represented, the TNT-formula which
+asserts "G is a theorem" would be a theorem of TNT. But this formula is -G, the negation
+of G, so that TNT is inconsistent. On the other hand, assume G were not a theorem. Then
+once again by the supposed representability of theoremhood, the formula which asserts
+"G is not a theorem" would be a theorem of TNT. But this formula is G, and once again
+we get into paradox. Unlike the situation before, there is no resolution of the paradox.
+The problem is created by the assumption that theoremhood is represented by some
+formula of TNT, and therefore we must backtrack and erase that assumption. This forces
+us also to conclude that no FlooP program can tell the Godel numbers of theorems from
+those of nontheorems. Finally, if we accept the A1 Version of the CT-Thesis, then we
+must backtrack further, and conclude that no method whatsoever could exist by which
+humans could reliably tell theorems from nontheorems-and this includes determinations
+based on beauty. Those who subscribe only to the Public-Processes Version might still
+think the Crab's performance is possible; but of all the versions, that one is perhaps the
+hardest one to find any justification for.
+
+Tarski's Theorem
+
+Now let us proceed to Tarski's result. Tarski asked whether there could be a way of
+expressing in TNT the concept of number-theoretical truth. That theoremhood is
+expressible (though not representable) we have seen; Tarski was interested in the
+analogous question regarding the notion of truth. More specifically, he wished to
+determine whether there is any TNT-formula with a single free variable a which can be
+translated thus:
+
+
+"The formula whose Godel number is a expresses a truth."
+
+Fet us suppose, with Tarski, that there is one-which we'll abbreviate as TRUEja}. Now
+what we'll do is use the diagonalization method to produce a sentence which asserts about
+itself that it is untme. We copy the Godel method exactly, beginning with an "uncle":
+
+3 a: <-TRUE {a} n ARITHMOQUINE {a", a} >
+
+Let us say the Godel number of the uncle is t. We arithmoquine this very uncle, and
+produce the Tarski formula T:
+
+3a:<—TRUE {a} A ARITHMOQUINE {SSS ... SSSO/a",a}>
+
+tS's
+
+
+Church, Turing, Tarski, and Others
+
+
+576
+
+
+
+When interpreted, it says:
+
+
+"The arithmoquinification of t is the
+Godel number of a false statement."
+
+
+But since the arithmoquinification of t is T's own Godel number, Tarski's formula T
+reproduces the Epimenides paradox to a tee inside TNT, saying of itself, "I am a falsity".
+Of course, this leads to the conclusion that it must be simultaneously true and false (or
+simultaneously neither). There arises now an interesting matter: What is so bad about
+reproducing the Epimenides paradox? Is it of any consequence? After all, we already
+have it in English, and the English language has not gone up in smoke.
+
+The Impossibility of the Magnificrab
+
+The answer lies in remembering that there are two levels of meaning involved here. One
+level is the level we have just been using; the other is as a statement of number theory. If
+the Tarski formula T actually existed, then it would be a statement about natural numbers
+that is both true and false at once! There is the rub. While we can always just sweep the
+English-language Epimenides paradox under the rug, saying that its subect matter (its
+own truth) is abstract, this is' not so when it becomes a concrete statement about
+numbers! If we believe this is a ridiculous state of affairs, then we have to undo our
+assumption that the formula TRUE {a} exists. Thus, there is no way of expressing the
+notion of truth inside TNT. Notice that this makes truth a far more elusive property than
+theoremhood, for the latter is expressible. The same backtracking reasons as before
+(involving the Church-Turing Thesis, A1 Version) lead us to the conclusion that
+
+The Crab's mind cannot be a truth-recognizer any more than it is a TNT-
+
+theorem-recognizer.
+
+The former would violate the Tarski-Church-Turing Theorem ("There is no decision
+procedure for arithmetical truth"), while the latter would violate Church's Theorem.
+
+Two Types of Form
+
+It is extremely interesting, then, to think about the meaning of the word "form" as it
+applies to constructions of arbitrarily complex shapes. For instance, what is it that we
+respond to when we look at a painting and feel its beauty? Is it the "form" of the lines and
+dots on our retina? Evidently it must be, for that is how it gets passed along to the
+analyzing mechanisms in our heads-but the complexity of the processing makes us feel
+that we are not merely looking at a two-dimensional surface; we are responding to
+
+
+Church, Turing, Tarski, and Others
+
+
+577
+
+
+
+some sort of inner meaning inside the picture, a multidimensional aspect trapped
+somehow inside those two dimensions. It is the word "meaning" which is important here.
+Our minds contain interpreters which accept
+
+two-dimensional patterns and then "pull" from them high-dimensional notions which are
+so complex that we cannot consciously describe them. The same can be said about how
+we respond to music, incidentally.
+
+It feels subjectively that the pulling-out mechanism of inner meaning is not at all
+akin to a decision procedure which checks for the presence or absence of some particular
+quality such as well-formedness in a string. Probably this is because inner meaning is
+something which reveals more of itself over a period of time. One can never be sure, as
+one can about well-formedness, that one has finished with the issue.
+
+This suggests a distinction that could be drawn between two senses of "form" in
+patterns which we analyze. First, there are qualities such as well-formedness, which can
+be detected by predictably terminating tests, as in BlooP programs. These I propose to
+call syntactic qualities of form. One intuitively feels about the syntactic aspects of form
+that they lie close to the surface, and therefore they do not provoke the creation of
+multidimensional cognitive structures.
+
+By contrast, the semantic aspects of form are those which cannot be tested for in
+predictable lengths of time: they require open-ended tests. Such an aspect is theoremhood
+of TNT-strings, as we have seen. You cannot just apply some standard test to a string and
+find out if it is a theorem. Somehow, the fact that its meaning is involved is crucially
+related to the difficulty of telling whether or not a string is a TNT-theorem. The act of
+pulling out a string's meaning involves, in essence, establishing all the implications of its
+connections to all other strings, and this leads, to be sure, down an open-ended trail. So
+"semantic" properties are connected to open-ended searches because, in an important
+sense, an object's meaning is not localized within the object itself. This is not to say that
+no understanding of any object's meaning is possible until the end of time, for as time
+passes, more and more of the meaning unfolds. However, there are always aspects of its
+meaning which will remain hidden arbitrarily long.
+
+Meaning Derives from Connections to Cognitive Structures
+
+Let us switch from strings to pieces of music, just for variety. You may still
+substitute the term "string" for every reference to a piece of music, if you prefer.
+The discussion is meant to be general, but its flavor is better gotten across, I feel,
+by referring to music. There is a strange duality about the meaning of a piece of
+music: on the one hand, it seems to be spread around, by virtue of its relation to
+many other things in the world-and yet, on the other hand, the meaning of a piece
+of music is obviously derived from the music itself, so it must be localized
+somewhere inside the music.
+
+The resolution of this dilemma comes from thinking about the interpreter-the
+mechanism which does the pulling-out of meaning. (By "inter
+
+
+Church, Turing, Tarski, and Others
+
+
+578
+
+
+
+preter in this context, I mean not -the performer of the piece, but the mental mechanism
+in the listener which derives meaning when the piece is played.) The interpreter may
+discover many important aspects of a piece's meaning while hearing it for the first time;
+this seems to confirm the notion that the meaning is housed in the piece itself, and is
+simply being read off. But that is only part of the story. The music interpreter works by
+setting up a multidimensional cognitive structure-a mental representation of *_he piece-
+which it tries to integrate with pre-existent information by finding links to other
+multidimensional mental structures which encode previous experiences. As this process
+takes place, the full meaning gradually unfolds. In fact, years may pass before someone
+comes to feel that he has penetrated to the core meaning of a piece. This seems to support
+the opposite view: that musical meaning is spread around, the interpreter's role being to
+assemble it gradually.
+
+The truth undoubtedly lies somewhere in between: meanings-both musical and
+linguistic-are to some extent localizable, to some extent spread around. In the
+terminology of Chapter VI, we can say that musical pieces and pieces of text are partly
+triggers, and partly carriers of explicit meaning. A vivid illustration of this dualism of
+meaning is provided by the example of a tablet with an ancient inscription: the meaning
+is partially stored in the libraries and the brains of scholars around the world, and yet it is
+also obviously implicit in the tablet itself.
+
+Thus, another way of characterizing the difference between "syntactic" and
+"semantic" properties (in the just-proposed sense) is that the syntactic ones reside
+unambiguously inside the object under consideration, whereas semantic properties
+depend on its relations with a potentially infinite class of other objects, and therefore are
+not completely localizable. There is nothing cryptic or hidden, in principle, in syntactic
+properties, whereas hiddenness is of the essence in semantic properties. That is the reason
+for my suggested distinction between "syntactic" and "semantic" aspects of visual form.
+
+Beauty, Truth, and Form
+
+What about beauty? It is certainly not a syntactic property, according to the ideas above.
+Is it even a semantic property? Is beauty a property which, for instance, a particular
+painting has? Let us immediately restrict our consideration to a single viewer. Everyone
+has had the experience of finding something beautiful at one time, dull another time-and
+probably intermediate at other times. So is beauty an attribute which varies in time? One
+could turn things around and say that it is the beholder who has varied in time. Given a
+particular beholder of a particular painting at a particular time, is it reasonable to assert
+that beauty is a quality that is definitely present or absent? Or is there still something ill-
+defined and intangible about it?
+
+Different levels of interpreter probably could be invoked in every
+
+
+Church, Turing, Tarski, and Others
+
+
+579
+
+
+
+person, depending on the circumstances. These various interpreters pull out different
+meanings, establish different connections, and generally evaluate all deep aspects
+differently. So it seems that this notion of beauty is extremely hard to pin down. It is for
+this reason that I chose to link beauty, in the Magnificrab, with truth, which we have seen
+is also one of the most intangible notions in all of metamathematics.
+
+The Neural Substrate of the Epimenides Paradox
+
+I would like to conclude this Chapter with some ideas about that central problem of truth,
+the Epimenides paradox. I think the Tarski reproduction of the Epimenides paradox
+inside TNT points the way to a deeper understanding of the nature of the Epimenides
+paradox in English. What Tarski found was that his version of the paradox has two
+distinct levels to it. On one level, it is a sentence about itself which would be true if it
+were false, and false if it were true. On the other level-which I like to call the arithmetical
+substrate-it is a sentence about integers which is true if and only if false.
+
+Now for some reason this latter bothers people a lot more than the former. Some
+people simply shrug off the former as "meaningless", because of its self-referentiality.
+But you can't shrug off paradoxical statements about integers. Statements about integers
+simply cannot be both true and false.
+
+Now my feeling is that the Tarski transformation of the Epimenides paradox
+teaches us to look for a substrate in the English-language version. In the arithmetical
+version, the upper level of meaning is supported by the lower arithmetical level. Perhaps
+analogously, the self-referential sentence which we perceive ("This sentence is false") is
+only the top level of a dual-level entity. What would be the lower level, then? Well, what
+is the mechanism that language rides on? The brain. Therefore one ought to look for a
+neural substrate to the Epimenides paradox-a lower level of physical events which clash
+with each other. That is, two events which by their nature cannot occur simultaneously. If
+this physical substrate exists, then the reason we cannot make heads or tails of the
+Epimenides sentence is that our brains are trying to do an impossible task.
+
+Now what would be the nature of the conflicting physical events? Presumably
+when you hear the Epimenides sentence, your brain sets up some "coding" of the
+sentence-an internal configuration of interacting symbols. Then it tries to classify the
+sentence as "true" or "false". This classifying act must involve an attempt to force several
+symbols to interact in a particular way. (Presumably this happens when any sentence is
+processed.) Now if it happens that the act of classification would physically disrupt the
+coding of the sentence-something which would ordinarily never happen-then one is in
+trouble, for it is tantamount to trying to force a record player to play its self-breaking
+record. We have described the conflict in physical terms, but not in neural terms. If this
+analysis is right so
+
+
+Church, Turing, Tarski, and Others
+
+
+580
+
+
+
+far, then presumably the rest of the discussion could be carried on when we know •
+something about the constitution of the "symbols" in the brain out of neurons and their
+firings, as well as about the way that sentences become converted into "codings".
+
+This sketch of the neural substrate of the Epimenides paradox suggests (to me, at
+least) that the resolution of the English version of the Epimenides paradox might be
+similar to that for the Tarski version. The resolution involves abandoning the notion that
+a brain could ever provide a fully accurate representation for the notion of truth. The
+novelty of this resolution lies in its suggestion that a total modeling of truth is impossible
+for quite physical reasons: namely, such a modeling would require physically
+incompatible events to occur in a brain.
+
+
+Church, Turing, Tarski, and Others
+
+
+581
+
+
+
+SHRDLU ,
+
+Toy of Man’s Designing
+
+
+One day, Eta Oin wanders into the MIT Artificial Intelligence Laboratory,
+where she runs into the brilliant young computer program, SHRDLU. It
+happens that SHRDLU is just dying for someone t* try out the newly
+developed human being "Dr. Tony Earrwig”. SHRDLU explains that this
+Earrwig is fairly intelligent in the limited domain of analyzing a conversation
+about a "toy world" containing blocks of various shapes, sizes, and colors-
+blocks which sit on a table, and which can be picked up and moved around.
+
+Eta Oin, fascinated, begins enthusiastically typing to SHRDLU. Earrwig
+dangles over her shoulder and provides a sort of running commentary, as
+promised.
+
+1. Eta Oin: pick up a big red block. [Refer to Fig. 110.
+
+SHRDLU: OK.
+
+Dr. Tony Earrwig: SHRDLU answers "OK" when it carries out a command. In order
+to pick up the red block, it had to clear it off by finding a space for the green one
+and moving the green one away.
+
+2. Eta Oin: grasp the pyramid.
+
+SHRDLU: I DON'T UNDERSTAND WHICH PYRAMID YOU MEAN.
+
+Dr. Tony Earrwig: There are three pyramids in the scene, and SHRDLU knows that
+phrases beginning with "the" are intended to refer to a specific object the speaker has in
+mind.
+
+FIGURE 110. "Pick up a big red block.” [From Terry Winograd, Understanding Natural
+Language, (New York: Academic Press, 1972), p. 8.1
+
+
+
+SHRDLU, Toy of Man’s Designing
+
+
+586
+
+
+
+
+
+
+
+
+
+FIGURE III. "Find a block which is bigger than the one you are holding and put it into the
+box.” Adapted from Terry Winograd, Understanding Natural Language, p.9.]
+
+3. Eta Oin: find a block which is taller than the one you are holding and put it into the box.
+[Refer to Fig. 111.]
+
+Dr. Tony Earrwig: SHRDLU has to figure out what is meant by "one" and "it", as well as
+using deduction to decide which block it is holding.
+
+SHRDLU: BY "IT", I ASSUME YOU MEAN THE BLOCK WHICH IS TALLER
+THAN THE ONE I AM HOLDING.
+
+Dr. Tony Earrwig: There were two possible referents for "it": the one selected, and "the
+one you are holding". SHRDLU has heuristics for cfwosing between them.
+
+SHRDLU: OK.
+
+Dr. Tony Earrwig: SHRDLU’s hand has to set down the red block pick up the taller blue
+one.
+
+Eta Oin: what does the box contain?
+
+SHRDLU: THE BLUE PYRAMID AND THE BLUE BLOCK.
+
+Eta Oin: what is the pyramid supported by?
+
+Dr. Tony Earrwig: Earlier SHRDLU responded to "the pyramid" by saying "I DON'T
+UNDERSTAND WHICH PYRAMID YOU MEAN." Now, it assumes that you mean the
+one just mentioned.
+
+SHRDLU: THE BOX.
+
+Eta Oin: how many blocks are not in the box?
+
+
+SHRDLU, Toy of Man's Designing
+
+
+587
+
+
+
+
+
+
+
+
+
+
+Dr. Tony Earrwig: Negatives such as "not" and "no" are handled whether they appear in
+clauses or noun groups (such as "no boxes").
+
+SHRDLU: FOUR OF THEM.
+
+7. Eta Oin: is at least one of them narrower than the one which i told you to pick up?
+
+Dr. Tony Earrwig: The phrase "at least one of them" must be analyzed to refer
+back to the previous question, and "the one which ..." must also refer to blocks.
+Noun groups can have complicated numbers such as "at least one", "more than
+three", "exactly seven", etc.. The memory of past events is used to deduce which
+block is "the one which / told you to pick up". Notice the complicated syntactic
+structure in which "the one" is actually the object of a clause embedded two levels
+deeper. SHRDLU's parser incorporates a grammar that handles this kind of
+complexity in a general way.
+
+SHRDLU: YES, THE RED CUBE.
+
+8. Eta Oin: is it supported?
+
+SHRDLU: YES, BY THE TABLE.
+
+Dr. Tony Earrwig: SHRDLU has heuristics about what information is relevant to
+a person who asks a question, even if the person does not request that information
+specifically.
+
+9. Eta Oin: can the table pick up blocks?
+
+SHRDLU: NO,
+
+Dr. Tony Earrwig: SHRDLU's semantic rules include the knowledge that "picking
+up" must be done by an animate object, and that the table is not animate.
+
+10. Eta Oin: can a pyramid be supported by a block?
+
+SHRDLU: YES.
+
+Dr. Tony Earrwig:• SHRDLU's deductive system finds an actual example, so it
+knows this is possible.
+
+11. Eta Oin: can a pyramid support a pyramid?
+
+SHRDLU: I DON'T KNOW.
+
+Dr. Tony Earrwig: The question makes sense, but there is no example in the
+scene. SHRDLU has no way of looking into the details of its programs, even
+though these ultimately define its capabilities.
+
+12. Eta Oin: stack up two pyramids.
+
+SHRDLU: I CANT.
+
+Dr. Tony Earrwig: SHRDLU tries and fails.
+
+13. Eta Oin: the blue pyramid is mine.
+
+SHRDLU: I UNDERSTAND.
+
+Dr. Tony Earrwig: All of the information about colors, shapes, sizes and locations
+of the objects is known to SHRDLU initially. Possession is left as an unknown
+property of objects about which it can be told. It stores this information in its data
+base.
+
+14. Eta Gin: i own all blocks which are not red, but i don't own anything which supports a
+
+pyramid.
+
+SHRDLU: I UNDERSTAND.
+
+
+SHRDLU, Toy of Man's Designing
+
+
+588
+
+
+
+FIGURE 112. "Will you please stack up both of the red blocks and either a green cube or
+
+a pyramid?'’ [Adapted from Terry Winograd, Understanding Natural Language, p. 12.
+
+Dr. Tony Earrwig: The result of understanding these sentences is the creation of
+
+new procedures within SHRDLUfor deducing whether you own something.
+
+15. Eta Oin: do i own the box?
+
+SHRDLU: NO.
+
+Dr. Tony Earrwig: The box supports whatever it contains, including a pyramid.
+
+16. Eta Oin: do i own anything in the box?
+
+SHRDLU: YES, TWO THINGS: THE BLUE BLOCK AND PYRAMID.
+
+17. Eta Oin: will you please stack up both of the red blocks and either a green cube or a
+
+pyramid? [Refer to Fig. 112.
+
+SHRDLU: OK.
+
+Dr. Tony Earrwig: Logical connectives, such as "and", "or", "either”, etc. are
+handled in both the grammar and semantics. A command is executed even if it
+looks like a question. The. command is carried out by the following series of
+steps: SHRDLU puts a green cube on the large red block (it chooses the green
+cube with nothing on it), then removes the small pyramid from the little red
+cube, so that it can use that cube to complete the stack as specified.
+
+
+SHRDLU, Toy of Man's Designing
+
+
+589
+
+
+
+
+
+
+18. Eta Oin: which cube is sitting on the table?
+
+SHRDLU: THE LARGE GREEN ONE WHICH SUPPORTS THE RED
+PYRAMID.
+
+Dr. Tony Earrwig: Ordinarily SHRDLU would be satisfied with calling this "A
+BIG GREEN CUBE", but since you asked "which cube" it has to be more spec/lc
+in naming it. It has procedures for using substitute nouns (such as "one”) and
+pronouns in answers to make the discourse more natural.
+
+19. Eta Oin: is there a large block behind a pyramid?
+
+SHRDLU: YES, THREE OF THEM: A LARGE RED ONE, A LARGE GREEN
+CUBE AND THE BLUE ONE.
+
+20. Eta Oin: put a small one onto the green cube which supports a pyramid.
+
+Dr. Tony Earrwig: "A small one" must contrast with your earlier phrase "a large
+block", so this means "a small block". SHRDLU: OK,
+
+21. Eta Oin: put the littlest pyramid on top of it.
+
+Dr. Tony Earrwig: Words like "littlest" are not in SHRDEU's dictionary but are
+interpreted from the root farms like "littleSHRDLU compares the sizes of all
+the pyramids to carry out the action.
+
+SHRDLU* OK.
+
+22. Eta Oin: how many things are on top of green cubes?
+
+SHRDLU: I'M NOT SURE WHAT YOU MEAN BY "ON TOP OF" IN THE
+PHRASE "ON TOP OF GREEN CUBES"
+
+DO YOU MEAN
+
+1- DIRECTLY ON THE SURFACE
+
+2- ANYWHERE ON TOP OF IT?
+
+Dr. Tony Earrwig: In the previous sentence, the phrase "on top of " was
+interpreted without trouble because the context made it clear what was meant. In
+this one, the heuristics for disambiguation do not come up with a clearcut
+definition, so SHRDLU had to ask you for more information. The paraphrases
+were included with the dictionary definitions of "on". A request like this can
+also occur when the meaning of a pronoun such as "it" or "they" is unclear.
+
+23. Eta Din: had you touched any pyramid before you put the green one on the little
+
+cube?
+
+SHRDLU: YES, THE GREEN ONE.
+
+Dr. Tony Earrwig: SHRDLU deduces from its memory what event is being
+referred to, and can relate other events to it in time. The analysis includes
+complex tenses of verb groups, and there are special facilities for keeping
+track of the various forms of irregular verbs like "have”.
+
+24. Eta Oin: when did you pick it up?
+
+SHRDLU: WHILE I WAS STACKING UP THE RED CUBE, A LARGE RED
+BLOCK AND A LARGE GREEN ONE.
+
+Dr. Tony Earrwig: A time is described by naming the main goal which was being
+worked on.
+
+25. Eta Oin: why?
+
+
+SHRDLU, Toy of Man's Designing
+
+
+590
+
+
+
+Dr. Tony Earrwig: SHRDLU remembers immediately previous sentences to
+understand questions like this and the following ones.
+
+SHRDLU: TO GET RID OF IT.
+
+Dr. Tony Earrwig: By keeping track of selected parts of the original subgoal tree,
+SHRDLU has some understanding of its own motives.
+
+26. Eta Oin: why did you do that?
+
+SHRDLU: TO CLEAN OFF THE RED CUBE.
+
+Dr. Tony Earrwig: "That" refers to the action mentioned in SHRDLU's last
+answer.
+
+27. Eta Oin: why did you clear off that cube?
+
+Dr. Tony Earrwig: Here, "that" means the one just referred to. SHRDLU: TO
+P UT IT ON A LARGE GREEN CUBE.
+
+28. Eta Oin: why did you do that?
+
+SHRDLU: TO STACK UP THE RED CUBE, A LARGE RED BLOCK, AND A
+LARGE GREEN CUBE.
+
+29. Eta Oin: why did you do that?
+
+SHRDLU.' BECAUSE YOU ASKED ME TO.
+
+Dr. Tony Earrwig: SHRDLU remembers what it did, not how the request was
+worded.
+
+30. Eta Oin: how did you do it?
+
+Dr. Tony Earrwig: "It" refers to the action mentioned in the last question,
+rather than the answer.
+
+SHRDLU: BY PUTTING A LARGE RED BLOCK ON THE TABLE; THEN
+LETTING GO OF IT; THEN PUTTING A LARGE GREEN CUBE ON IT;
+THEN LETTING GO OF THAT CUBE; THEN PUTTING THE RED CUBE
+‘ ON THAT CUBE; THEN LETTING GO OF THAT CUBE.
+
+Dr. Tony Earrwig: This is a description of the top-level sub goals used to
+achieve the goal.
+
+31. Eta Oin: how many objects did you touch while you were doing it?
+
+Dr. Tony Earrwig: Time references can refer to a time span, using words like
+"while ".
+
+SHRDLU: FOUR OF THEM.
+
+32. Eta Oin: what did the red cube support before you started to clean it off?
+
+SHRDLU: THE GREEN PYRAMID.
+
+Dr. Tony Earrwig: By remembering what has been moved, it is possible to
+deduce things about earlier states of the scene. You can use complex nestings
+of clauses like "before you started to clean it off ".
+
+33. Eta Oin: there were five blocks to the left of the box then.
+
+Dr. Tony Earrwig: "Then" must refer to the time mentioned in the last sentence.
+SHRDLU: NO, ONLY FOUR OF THEM: THE RED CUBE, TWO LARGE
+GREEN CUBES, AND A LARGE RED BLOCK.
+
+Dr. Tony Earrwig: Earlier in the dialogue, information about "owning" was
+accepted at face value. But SHRDLU knows that it has complete infornia-
+
+
+SHRDLU, Toy of Man's Designing
+
+
+591
+
+
+
+tion about the basic properties of objects, including location. It checks any
+statements about these to see whether it agrees or disagrees.
+
+34. Eta Oin: put the blue pyramid on the block in the box.
+
+Dr. Tony Earrwig: This is syntactically ambiguous, but is understood without
+ambiguity since the parsing corresponding to "put the blue pyramid which is
+on the block into the box" does not make sense in the current scene, whereas
+"put the blue pyramid onto the block which is in the box" does. SHRDLU: OK.
+
+35. Eta Oin: is there anything which is bigger than every pyramid but is not as wide as
+
+the thing that supports it?
+
+Dr. Tony Earrwig: SHRDLU understands quantifiers such as "every", "any",
+"some", etc., and structures such as "bigger than", "as wide as", etc.. Here "it"
+refers to a noun group within the same sentence but two levels up in the parsing
+tree.
+
+SHRDLU: YES, THE BLUE BLOCK.
+
+36. Eta Oin: does a steeple
+
+SHRDLU: SORRY, I DON'T KNOW THE WORD "STEEPLE".
+
+Dr. Tony Earrwig: SHRDLU interrupts immediately when it sees a word it doesn't
+know.
+
+37. Eta Oin: a "steeple" is a stack which contains two green cubes and a pyramid.
+
+Dr. Tony Earrwig: The quote marks indicate that you are defining a new word.
+The definition is added to SHRDLU's dictionary in a form compatible with all the
+words it already knows.
+
+SHRDLU: I UNDERSTAND.
+
+38. Eta Oin: are there any steeples now?
+
+SHRDLU: NO.
+
+Dr. Tony Earrwig: The new definition is used as a description to recognize
+steeples.
+
+39. Eta Oin: build one.
+
+SHRDLU: OK.
+
+Dr. Tony Earrwig: The description given in the definition is enough to deduce a
+plan for building new objects as well as recognizing them. The word "one" means
+"a steeple", and the plan is carried out. First, one of the large green cubes is
+cleared off by removing the red cube and two pyramids. Then the two green cubes
+and the tall red pyramid are stacked up to fit the description.
+
+40. Eta Oin: call the biggest block "superblock".
+
+SHRDLU: OK.
+
+Dr. Tony Earrwig: You can also give objects names.
+
+41. Eta Oin: have you picked up superblock since we began?
+
+SHRDLU: YES.
+
+42. Eta Oin: why did you drop it?
+
+SHRDLU: BY "IT", I ASSUME YOU MEAN THE SUPERBLOCK. I CAN'T
+EXPLAIN A NON-EXISTENT EVENT.
+
+
+SHRDLU, Toy of Man's Designing
+
+
+592
+
+
+
+Dr. Tony Earrwig: There is no record in SHRDLU's memory of any block being
+dropped. The answer routines have special checks for this type of "loaded
+question" which presupposes things which aren't true.
+
+43. Eta Oin: is there anything to the right of the red pyramid?
+
+SHRDLU: YES, FOUR THINGS: THE BOX, THE RED CUBE, THE BLUE
+PYRAMID AND THE BLOCK NAMED SUPERBLOCK.
+
+Dr. Tony Earrwig: Once an object has a name, the name is used to help identify it
+in responses.
+
+44. Eta Oin: thank you.
+
+SHRDLU: YOU'RE WELCOME!
+
+Dr. Tony Earrwig: SHRDLU's grammar is written as a procedure, which gives
+
+SHRDLU the capability to carry out special sorts of parsing for
+
+idioms.
+
+
+SHRDLU, Toy of Man's Designing
+
+
+593
+
+
+
+CHAPTER XVI11
+
+
+Artificial Intelligence:
+Retrospects
+
+Turing
+
+
+
+Artificial Intelligence: Retrospects
+
+
+IN 1951, ALAN TURING wrote a most
+prophetic and provocative article on
+Artificial Intelligence. It was entitled
+"Computing Machinery and
+Intelligence" and appeared in the journal
+Mind! I will say some things about that
+article, but I would like to precede them
+with some remarks about Turing the
+man.
+
+Alan Mathison Turing was born in
+London in 1912. He was a child full of
+curiosity and humor. Gifted in
+mathematics, he went to Cambridge
+where his interests in machinery and
+mathematical logic cross-fertilized and
+resulted in his famous paper on
+"computable numbers", in which he
+invented the theory of Turing machines
+and demonstrated the unsolvability of
+the halting problem; it was published in
+1937. In the 194t’s, his interests turned
+from the theory of computing machines
+to the actual building of real computers.
+He was a major figure in the
+development of computers in Britain,
+and a staunch defender of Artificial In-
+
+FIGURE 113. Alan Turing, after a
+successful race (May, 1950). [From
+Sara Turing, Alan M. Turing
+(Cambridge, U. K.:W. Hefjer & Sons,
+1959).
+
+
+594
+
+
+telligence when it first came under attack. One of his best friends was David
+Champemowne (who later worked on computer composition of music). Champemowne
+and Turing were both avid chess players and invented "round-the-house" chess: after
+your move, run around the house-if you get back before your opponent has moved, you're
+entitled to another move. More seriously, Turing and Champemowne invented the first
+chess playing program, called "Turochamp . Turing died young, at 41-apparently of an
+accident with chemicals. Or some say suicide. His mother, Sara Turing, wrote his
+biography. From the people she quotes, one gets the sense that Turing was highly
+unconventional, even gauche in some ways, but so honest and decent that he was
+vulnerable to the world. He loved games, chess, children, and bike riding; he was a strong
+long-distance runner. As a student at Cambridge, he bought himself a second-hand violin
+and taught himself to play. Though not very musical, he derived a great deal of
+enjoyment from it. He was somewhat eccentric, given to great bursts of energy in the
+oddest directions. One area he explored was the problem of morphogenesis in biology.
+According to his mother, Turing "had a particular fondness for the Pickwick Papers ", but
+"poetry, with the exception of Shakespeare's, meant nothing to him." Alan Turing was
+one of the true pioneers in the field of computer science.
+
+The Turing Test
+
+Turing's article begins with the sentence: "I propose to consider the question 'Can
+machines think?"' Since, as he points out, these are loaded terms, it is obvious that we
+should search for an operational way to approach the question. This, he suggests, is
+contained in what he calls the "imitation game"; it is nowadays known as the Turing test.
+Turing introduces it as follows:
+
+It is played with three people: a man (A), a woman (B), and an interrogator (C)
+who may be of either sex. The interrogator stays in a room apart from the other
+two. The object of the game for the interrogator is to determine which of the other
+two is the man and which is the woman. He knows them by labels X and Y, and at
+the end of the game he says either "X is A and Y is B" or "X is B and Y is A". The
+interrogator is allowed to put questions to A and B thus:
+
+C: Will X please tell me the length of his or her hair?
+
+Now suppose X is actually A, then A must answer. It is A's object in the game to
+try to cause C to make the wrong identification. His answer might therefore be
+
+"My hair is shingled, and the longest strands are about nine inches long."
+
+In order that tones of voice may not help the interrogator the answers should be
+written, or better still, typewritten. The ideal arrangement is to have a teleprinter
+communicating between the two rooms. Alternatively the questions and answers
+can be repeated by an intermediary. The object of the game for the third player (B)
+is to help the interrogator. The best strategy for her is probably to give truthful
+answers. She can add such things as "I am the woman, don't listen to him!" to her
+answers, but it will avail nothing as the man can make similar remarks.
+
+
+Artificial Intelligence: Retrospects
+
+
+595
+
+
+
+We now ask the question, "What will happen when a machine takes the part of A
+in this game Will the interrogator decide wrongly as often when the game is
+played like this as he does when the game is played between a man and a woman?
+These questions replace our original, "Can machines think?"'
+
+After having spelled out the nature of his test, Turing goes on to make some
+commentaries on it, which, given the year he was writing in, are quite sophisticated. To
+
+3
+
+begin with, he gives a short hypothetical dialogue between interrogator and interrogatee:
+
+Q. Please write me a sonnet on the subject of the Forth Bridge [a bridge over the
+Firth of Forth, in Scotland].
+
+A. Count me out on this one. I never could write poetry.
+
+Q. Add 34957 to 70764.
+
+A. (Pause about 30 seconds and then give as answer) 105621.
+
+Q. Do you play chess?
+
+A. Yes.
+
+Q. I have K at my Kl, and no other pieces. You have only K at K6 and R at Rl. It
+is your move. What do you play?
+
+A. (After a pause of 15 seconds) R-R8 mate.
+
+Few readers notice that in the arithmetic problem, not only is there an inordinately long
+delay, but moreover, the answer given is wrong! This would be easy to account for if the
+respondent were a human: a mere calculational error. But if the respondent were a
+machine, a variety of explanations are possible. Here are some:
+
+(1) a run-time error on the hardware level (i.e., an irreproducible fluke);
+
+(2) an unintentional hardware (or programming) (reproducibly) causes
+arithmetical mistakes;
+
+(3) a ploy deliberately inserted by the machine's programmer (or builder) to
+introduce occasional arithmetical mistakes, so as to trick interrogators;
+
+(4) an unanticipated epiphenomenon: the program has a hard time thinking
+abstractly, and simply made "an honest mistake", which it might not make the
+next time around;
+
+(5) a joke on the part of the machine itself, deliberately teasing its interrogator.
+
+Reflection on what Turing might have meant by this subtle touch opens up just about all
+the major philosophical issues connected with Artificial Intelligence.
+
+Turing goes on to point out that
+
+The new problem has the advantage of drawing a fairly sharp line between the
+physical and the intellectual capacities of a man. . . . We do not wish to
+penalize the machine for its inability to shine in beauty competitions, nor to
+penalize a man for losing in a race against an airplane.'
+
+One of the pleasures of the article is to see how far Turing traced out each
+
+
+Artificial Intelligence: Retrospects
+
+
+596
+
+
+
+line of thought, usually turning up a seeming contradiction at some stage and, by refining
+his concepts, resolving it at a deeper level of analysis. Because of this depth of
+penetration into the issues, the article still shines after nearly thirty years of tremendous
+progress in computer development and intensive work in Al. In the following short
+excerpt you can see some of this rich back-and-forth working of ideas:
+
+The game may perhaps be criticized on the ground that the odds are weighted too
+heavily against the machine. If the man were to try to pretend to be the machine he
+would clearly make a very poor showing. He would be given away at once by
+slowness and inaccuracy in arithmetic. May not machines carry out something
+which ought to be described as thinking but which is very different from what a
+man does: This objection is a very strong one, but at least we can say that if,
+nevertheless, a machine can be constructed to play the imitation game satisfactorily,
+we need not be troubled by this objection.
+
+It might be urged that when playing the "imitation game" the best strategy
+for the machine may possibly be something other than imitation of the behaviour of
+a man. This may be, but I think it is unlikely that there is any greet effect of this
+kind. In any case there is ,no intention to investigate here the theory of the game,
+and it will be assumed that the best strategy is to try to provide answers that would
+naturally be given by a mans
+
+Once the test has been proposed and discussed, Turing remarks:
+
+The original question "Can machines think 1 believe to be too meaningless to
+deserve discussion. Nevertheless, I believe that at the end of the century the use of
+words and general educated opinion will have altered so much that one will be able
+to speak of machines thinking without expecting to be contradicted. 6
+
+Turing Anticipates Objections
+
+Aware of the storm of opposition that would undoubtedly greet this opinion, he then
+proceeds to pick apart, concisely and with wry humor, a series of objections to the notion
+that machines could think. Below I list the nine types of objections he counters, using his
+own descriptions of them .7 Unfortunately there is not space to reproduce the humorous
+and ingenious responses he formulated. You may enjoy pondering the objections
+yourself, and figuring out your own responses.
+
+(1) The Theological Objection. Thinking is a function of man's immortal soul. God
+has given an immortal soul to every man and woman, but not to any other animal
+or to machines. Hence no animal or machine can think.
+
+(2) The "Heads in the Sand" Objection. The consequences of machines thinking
+would be too dreadful. Let us hope and believe that they cannot do so.
+
+(3) The Mathematical Objection. [This is essentially the Lucas argument.
+
+(4) The Argument from Consciousness. "Not until a machine can write a sonnet or
+compose a concerto because of thoughts and emotions felt, and not by the chance
+fall of symbols, could we agree that machine equals brainy that is, not only write
+it but know that it had written it. No mechanism
+
+
+Artificial Intelligence: Retrospects
+
+
+597
+
+
+
+could feel (and not merely artificially signal, an easy contrivance) pleasure at its
+successes, grief when its valves fuse, be warmed by flattery, be made miserable
+by its mistakes, be charmed by sex, be angry or depressed when it cannot get
+what it wants." [A quote from a certain Professor Jefferson.]
+
+Turing is quite concerned that he should answer this serious objection in full detail.
+Accordingly, he devotes quite a bit of space to his answer, and in it he offers another
+short hypothetical dialogue:'
+
+Interrogator: In the first line of your sonnet which reads "Shall I compare thee to a
+summer's day", would not "a spring day" do as well or better, Witness: It
+wouldn't scan.
+
+Interrogator: How about' a winter's day"? That would scan all right. Witness: Yes,
+but nobody wants to be compared to a winter's day. Interrogator: Would you say
+Mr. Pickwick reminded you of Christmas? Witness: In a way.
+
+Interrogator: Yet Christmas is a winter's day, and I do not think Mr. Pickwick
+would mind the comparison.
+
+Witness: I don't think you're serious. By a winter's day one means a typical winter's
+day, rather than a special one like Christmas.
+
+After this dialogue, Turing asks, "What would Professor Jefferson say if the sonnet¬
+writing machine was able to answer like this in the viva voce?"
+
+Further objections:
+
+(5) Arguments from various Disabilities. These arguments take the form, "I grant you that
+you can make machines do all the things that you have mentioned but you will never be
+able to make one to do X." Numerous features X are suggested in this connection. I offer
+a selection:
+
+Be kind, resourceful, beautiful, friendly, have initiative, have a sense of humor, tell right
+from wrong, make mistakes, fall in love, enjoy strawberries and cream, make someone
+fall in love with it, learn from experience, use words properly, be the subject of its own
+thought, have as much diversity of behaviour as a man, do something really new.
+
+(6) Lady Lovelace's Objection. Our most detailed information of Babbage's Analytical
+Engine comes from a memoir by Lady Lovelace. In it she states, "The Analytical Engine
+has no pretensions to originate anything. It can do whatever we know how to order it to
+perform" (her italics).
+
+(7) Argument from Continuity in the Nervous System. The nervous system is certainly not
+a discrete state machine. A small error in the information about the size of a nervous
+impulse impinging on a neuron may make a large difference to the size of the outgoing
+impulse. It may be argued that, this being so, one cannot expect to be able to mimic the
+behaviour of the nervous system with a discrete state system.
+
+(8) The Argument from Informality of Behaviour. It seems to run something like this. "If
+each man had a definite set of rules of conduct by which he regulated his life he would
+be no better than a machine. But there are no such rules, so men cannot be machines."
+
+(9) The Argument from Extra-Sensory Perception. Let us play the imitation game, using as
+witnesses a man who is good as a telepathic receiver, and a digital computer. The
+interrogator can ask such questions as "What suit
+
+
+Artificial Intelligence: Retrospects
+
+
+598
+
+
+
+does the card in my right hand belong to?" The man by telepathy or clairvoyance gives
+the right answer 130 times out of 400 cards. The machine can only guess at random, and
+perhaps get 104 right, so the interrogator makes the right identification.
+
+As you can see, many of the objections overlap and are related to each other. I have
+attempted in this book to answer most of them in some way or other, usually far less
+concisely than Turing did.
+
+Objection (9) I find remarkable. I have seen the Turing paper reprinted in a book-
+but with objection (9) omitted-which I find equally remarkable. My own point of
+view-contrary to Turing's-is that ESP does not exist. Turing was reluctant to accept
+the idea that ESP is real, but did so nonetheless, being compelled by his outstanding
+scientific integrity to accept the consequences of what he viewed as powerful
+statistical evidence in favor of ESP. I disagree, though I consider it an exceedingly
+complex and fascinating question.
+
+"Parry Encounters the Doctor"
+
+In the Dialogue preceding this Chapter, you have seen an authentic exchange between
+a computer program and a human. Now there have been many programs with less
+ability to "converse" than SHRDLU. Two rather famous ones are "Doctor", created by
+Joseph Weizenbaum, and "Parry", created by Kenneth Colby. The former is supposed
+to simulate a psychiatrist using "nondirective therapy", the latter to simulate the belief
+structure of a paranoid. An amusing head-on collision between the two programs was
+arranged over a computer network, and the following is a short excerpt from the full
+transcript 9
+
+Parry: I don't understand your motives.
+
+Doctor: Why don't you understand my motives?
+
+Parry: Shouldn't I be?
+
+Doctor: Why do you say that just now?
+
+Parry: That's just the way it is.
+
+Doctor: Suppose that's just the way it were not.
+
+Parry: So?
+
+Doctor: I am not sure I understand you fully.
+
+Parry: You harp on the same questions too much.
+
+Doctor: What does that similarity suggest to you?
+
+Parry: I don't get you.
+
+Doctor: Do you wish that you did get I?
+
+Parry: It's about time you showed some interest in my feelings.
+
+Doctor: Oh? I showed some interest in your feelings?
+
+These programs, of course, were made to talk to humans, so they are shown up rather
+poorly here, pitted against each other. Mostly they rely on shrewd guesses as to the
+nature of the input (which they analyze quite shallowly) and spit back canned answers
+that have been carefully selected from a large repertoire. The answer may be only
+partially canned: for example, a template with blanks that can be filled in. It is
+assumed that their
+
+
+Artificial Intelligence: Retrospects
+
+
+599
+
+
+
+human partners will read much more into what they say than is actually underlying it.
+And in fact, according to Weizenbaum, in his book Computer Power and Human
+Reason, just that happens. He writes:
+
+ELIZA [the program from which Doctor was made created the most remarkable
+illusion of having understood in the minds of the many people who conversed with
+it.... They would often demand to be permitted to converse with the system in
+private, and would, after conversing with it for a time, insist, in spite of my
+explanations, that the machine really understood them. 10
+
+Given the above excerpt, you may find this incredible. Incredible, but true.
+Weizenbaum has an explanation:
+
+Most men don't understand computers to even the slightest degree. So, unless they
+
+are capable of very great skepticism (the kind we bring to bear while
+
+watching a stage magician), they can explain the computer's intellectual feats only
+
+by bringing to hear the single analogy available to them, that is, their
+
+model of their own capacity to think. No wonder, then, that they overshoot the
+
+mark: it is truly impossible to imagine a human who could imitate ELIZA,
+
+for example, but for whom ELIZA's language abilities were his limit."
+
+Which amounts to an admission that this kind of program is based on a shrewd
+mixture of bravado and bluffing, taking advantage of people's gullibility.
+
+In light of this weird "ELIZA-effect", some people have suggested that the Turing
+test needs revision, since people can apparently be fooled by simplistic gimmickry. It
+has been suggested that the interrogator should be a Nobel Prize-winning scientist. It
+might be more advisable to turn the Turing test on its head, and insist that the
+interrogator should be another computer. Or perhaps there should be two
+interrogators-a human and a computer-and one witness, and the two interrogators
+should try to figure out whether the witness is a human or a computer.
+
+In a more serious vein, I personally feel that the Turing test, as originally proposed, is
+quite reasonable. As for the people who Weizenbaum claims were sucked in by
+ELIZA, they were not urged to be skeptical, or to use all their wits in trying to
+determine if the "person" typing to them were human or not. I think that Turing's
+insight into this issue was sound, and that the Turing test, essentially unmodified, will
+survive.
+
+
+A Brief History of AI
+
+I would like in the next few pages to present the story, perhaps from an unorthodox point
+of view, of some of the efforts at unraveling the algorithms behind intelligence: there
+have been failures and setbacks and there will continue to be. Nonetheless, we are
+learning a great deal, and it is an exciting period.
+
+Ever since Pascal and Leibniz, people have dreamt of machines that could perform
+intellectual tasks. In the nineteenth century, Boole and De Morgan devised "laws of
+thought"-essentially the Propositional
+
+
+Artificial Intelligence: Retrospects
+
+
+600
+
+
+
+Calculus-and thus took the first step towards At software; also Charles Babbage designed
+the first "calculating engine"-the precursor to the hardware of computers and hence of AI.
+One could define AI as coming into existence at the moment when mechanical devices
+took over any tasks previously performable only by human minds. It is hard to look back
+and imagine the feelings of those who first saw toothed wheels performing additions and
+multiplications of large numbers. Perhaps they experienced a sense of awe at seeing
+"thoughts" flow in their very physical hardware. In any case, we do know that nearly a
+century later, when the first electronic computers were constructed, their inventors did
+experience an awesome and mystical sense of being in the presence of another kind of
+"thinking being". To what extent real thought was taking place was a source of much
+puzzlement; and even now, several decades later, the question remains a great source of
+stimulation and vitriolics.
+
+It is interesting that nowadays, practically no one feels that sense of awe any longer-
+even when computers perform operations that are incredibly more sophisticated than
+those which sent thrills down spines in the early days. The once-exciting phrase "Giant
+Electronic Brain" remains only as a sort of "camp" cliche, a ridiculous vestige of the era
+of Flash Gordon and Buck Rogers. It is a bit sad that we become blase so quickly.
+
+There is a related "Theorem" about progress in AI: once some mental function is
+programmed, people soon cease to consider it as an essential ingredient of "real
+thinking". The ineluctable core of intelligence is always in that next thing which hasn't
+yet been programmed. This "Theorem" was first proposed to me by Larry Tesler, so I call
+it Tesler's Theorem. "AI is whatever hasn't been done vet."
+
+A selective overview of AI is furnished below. It shows several domains in which
+workers have concentrated their efforts, each one seeming in its own way to require the
+quintessence of intelligence. With some of the domains I have included a breakdown
+according to methods employed, or more specific areas of concentration.
+
+mechanical translation
+
+direct (dictionary look-up with some word rearrangement)
+indirect (via some intermediary internal language)
+
+game playing
+
+chess
+
+with brute force look-ahead
+with heuristically pruned look-ahead
+with no look-ahead checkers
+go
+
+kalah
+
+bridge (bidding; playing)
+poker
+
+variations on tic-tac-toe
+etc.
+
+
+Artificial Intelligence: Retrospects
+
+
+601
+
+
+
+proving theorems in various parts, of mathematics
+symbolic logic
+
+"resolution" theorem-proving
+elementary geometry
+
+symbolic manipulation of mathematical expressions
+symbolic integration
+algebraic simplification
+summation of infinite series
+
+vision
+
+printed matter:
+
+recognition of individual hand-printed characters drawn
+from a small class (e.g., numerals)
+reading text in variable fonts reading passages in handwriting
+reading Chinese or Japanese printed characters
+reading Chinese or Japanese handwritten characters
+pictorial:
+
+locating prespecified objects in photographs
+decomposition of a scene into separate objects
+identification of separate objects in a scene
+recognition of objects portrayed in sketches by people
+recognition of human faces
+
+hearing
+
+understanding spoken words drawn from a limited vocabulary (e.g., names of
+the ten digits)
+
+understanding continuous speech in fixed domains finding boundaries between
+phonemes
+
+identifying phonemes
+
+finding boundaries between morphemes
+
+identifying morphemes
+
+putting together whole words and sentences
+
+understanding natural languages
+
+answering questions in specific domains
+parsing complex sentences
+making paraphrases of longer pieces of text
+using knowledge of the real world in order to understand passages
+resolving ambiguous references
+
+producing natural language
+
+abstract poetry (e.g., haiku)
+
+random sentences, paragraphs, or longer pieces of text producing output from
+internal representation of knowledge
+
+
+Artificial Intelligence: Retrospects
+
+
+602
+
+
+
+creating original thoughts or works of art
+poetry writing (haiku) story writing
+computer art
+musical composition
+atonal
+tonal
+
+analogical thinking
+
+geometrical shapes ("intelligence tests")
+constructing proofs in one domain of mathematics based on
+those in a related domain
+
+
+learning
+
+adjustment of parameters
+concept formation
+
+
+Mechanical Translation
+
+Many of the preceding topics will not be touched upon in my selective discussion
+below, but the list would not be accurate without them. The first few topics are
+listed in historical order. In each of them, early efforts fell short of expectations.
+For example, the pitfalls in mechanical translation came as a great surprise to many
+who had thought it was a nearly straightforward task, whose perfection, to be sure,
+would be arduous, but whose basic implementation should be easy. As it turns out,
+translation is far more complex than mere dictionary look-up and word rearranging.
+Nor is the difficulty caused by a lack of knowledge of idiomatic phrases. The fact is
+that translation involves having a mental model of the world being discussed, and
+manipulating symbols in that model. A program which makes no use of a model of
+the world as it reads the passage will soon get hopelessly bogged down in
+ambiguities and multiple meanings. Even people-who have a huge advantage over
+computers, for they come fully equipped with an understanding of the world-when
+given a piece of text and a dictionary of a language they do not know, find it next to
+impossible to translate the text into their own language. Thus-and it is not
+surprising in retrospect-the first problem of AI led immediately to the issues at the
+heart of AI.
+
+
+Computer Chess
+
+Computer chess, too, proved to be much more difficult than the early intuitive
+estimates had suggested. Here again it turns out that the way humans represent a
+chess situation in their minds is far more complex than just knowing which piece is
+on which square, coupled with knowledge of the rules of chess. It involves
+perceiving configurations of several related pieces, as well as knowledge of
+heuristics, or rules of thumb, which pertain to
+
+
+Artificial Intelligence: Retrospects
+
+
+603
+
+
+
+such higher-level chunks. Even though heuristic rules are not rigorous in the way
+that the official rules are, they provide shortcut insights into what is going on on the
+board, which knowledge of the official rules does not. This much was recognized
+from the start; it was simply underestimated how large a role the intuitive, chunked
+understanding of the chess world plays in human chess skill. It was predicted that a
+program having some basic heuristics, coupled with the blinding speed and
+accuracy of a computer to look ahead in the game and analyze each possible move,
+would easily beat top-flight human players-a prediction which, even after twenty-
+five years of intense work by various people, still is far from being realized.
+
+People are nowadays tackling the chess problem from various angles. One of
+the most novel involves the hypothesis that looking ahead is a silly thing to do. One
+should instead merely look at what is on the board at present, and, using some
+heuristics, generate a plan, and then find a move which advances that particular
+plan. Of course, rules for the formulation of chess plans will necessarily involve
+heuristics which are, in some sense, "flattened" versions of looking ahead. That is,
+the equivalent of many games' experience of looking ahead is "squeezed" into
+another form which ostensibly doesn't involve looking ahead. In some sense this is
+a game of words. But if the "flattened" knowledge gives answers more efficiently
+than the actual look-ahead-even if it occasionally misleads- then something has
+been gained. Now this kind of distillation of knowledge into more highly usable
+forms is just what intelligence excels at-so look-ahead-less chess is probably a
+fruitful line of research to push. Particularly intriguing would be to devise a
+program which itself could convert knowledge gained from looking ahead into
+"flattened" rules-but that is an immense task.
+
+Samuel's Checker Program
+
+As a matter of fact, such a method was developed by Arthur Samuel in his
+admirable checker-playing program. Samuel's trick was to use both dynamic (look¬
+ahead) and static (no-look-ahead) ways of evaluating any given board position. The
+static method involved a simple mathematical function of several quantities
+characterizing any board position, and thus could be calculated practically
+instantaneously, whereas the dynamic evaluation method involved creating a "tree"
+of possible future moves, responses to them, responses to the responses, and so
+forth (as was shown in Fig. 38). In the static evaluation function there were some
+parameters which could vary; the effect of varying them was to provide a set of
+different possible versions of the static evaluation function. Samuel's strategy was
+to select, in an evolutionary way, better and better values of those parameters.
+
+Here's how this was done: each time the program evaluated a board position, it
+did so both statically and dynamically. The answer gotten by looking ahead-let us
+call it D-was used in determining the move to be made. The purpose of S, the static
+evaluation, was trickier: on each move, the variable parameters were readjusted
+slightly so that S approximated D
+
+
+Artificial Intelligence: Retrospects
+
+
+604
+
+
+
+as accurately as possible. The effect was to partially encode in the values of the
+static evaluation’s parameters the knowledge gained by dynamically searching the
+tree. In short, the idea was to "flatten" the complex dynamic evaluation method into
+the much simpler and more efficient static evaluation function.
+
+There is a rather nice recursive effect here. The point is that the dynamic
+evaluation of any single board position involves looking ahead a finite number of
+moves-say seven. Now each of the scads of board positions which might turn up
+seven turns down the road has to be itself evaluated somehow as well. But when the
+program evaluates these positions, it certainly cannot look another seven moves
+ahead, lest it have to look fourteen positions ahead, then twenty-one, etc., etc.-an
+infinite regress. Instead, it relies on static evaluations of positions seven moves
+ahead. Therefore, in Samuel's scheme, an intricate sort of feedback takes place,
+wherein the program is constantly trying to "flatten" look-ahead evaluation into a
+simpler static recipe; and this recipe in turn plays a key role in the dynamic look¬
+ahead evaluation. Thus the two are intimately linked together, and each benefits
+from improvements in the other in a recursive way.
+
+The level of play of the Samuel checkers program is extremely high: of the
+order of the top human players in the world. If this is so, why not apply the same
+techniques to chess? An international committee, convened in 1961 to study the
+feasibility of computer chess, including the Dutch International Grandmaster and
+mathematician Max Euwe, came to the bleak conclusion that the Samuel technique
+would be approximately one million times as difficult to implement in chess as in
+checkers, and that seems to close the book on that.
+
+The extraordinarily great skill of the checkers program cannot be taken as
+saying "intelligence has been achieved"; yet it should not be minimized, either. It is
+a combination of insights into what checkers is, how to think about checkers, and
+how to program. Some people might feel that all it shows is Samuel's own checkers
+ability. But this is not true, for at least two reasons. One is that skillful game players
+choose their moves according to mental processes which they do not fully
+understand-they use their intuitions. Now there is no known way that anyone can
+bring to light all of his own intuitions; the best one can do via introspection is to
+use "feeling" or "meta-intuition"-an intuition about one's intuitions-as a guide, and
+try to describe what one thinks one's intuitions are all about. But this will only give
+a rough approximation to the true complexity of intuitive methods. Hence it is
+virtually certain that Samuel has not mirrored his own personal methods of play in
+his program. The other reason that Samuel's program's play should not be confused
+with Samuel's own play is that Samuel does not play checkers as well as his
+program-it beats him. This is not a paradox at all-no more than is the fact that a
+computer which has been programmed to calculate 7T can outrace its programmer
+in spewing forth digits of ji.
+
+
+Artificial Intelligence: Retrospects
+
+
+605
+
+
+
+When Is a Program Original?
+
+
+This issue of a program outdoing its programmer is connected with the question of
+"originality" in AI. What if an AI program comes up with an idea, or a line of play in a
+game, which its programmer has never entertained-who should get the credit? There are
+various interesting instances of this having happened, some on a fairly trivial level, some
+on a rather deep level. One of the more famous involved a program to find proofs of
+theorems in elementary Euclidean geometry, written by E. Gelemter. One day the
+program came up with a sparklingly ingenious proof of one of the basic theorems of
+geometry-the so-called "pons asinorum", or "bridge of asses".
+
+This theorem states that the base angles of an isosceles triangle are equal. Its
+standard proof requires constructing an altitude which divides the triangle into
+symmetrical halves. The elegant method found by the program (see Fig. 114) used no
+construction lines. Instead, it considered
+
+
+
+FIGURE 114. Pons Asinorum Proof (found
+by Pappus [- 39 $ A.D. / and Gelemter's
+program [—1969 A.D. ]). Problem: To show
+that the base angles of an isosceles triangle
+are equal. Solution: As the triangle is
+isosceles, AP and AP' are of equal length.
+Therefore triangles PAP' and PAP are
+congruent (side-side-side). This implies that
+corresponding angles are equal. In
+particular, the two base angles are equal.
+
+
+the triangle and its mirror image as two different triangles. Then, having proved
+them congruent, it pointed out that the two base angles matched each other in this
+congruence-QED.
+
+This gem of a proof delighted the program's creator and others; some saw evidence
+of genius in its performance. Not to take anything away from this feat, it happens that in
+A.D. 300 the geometer Pappus had actually found this proof, too. In any case, the
+question remains: "Who gets the credit?" Is this intelligent behavior? Or was the proof
+lying deeply hidden within the human (Gelemter), and did the computer merely bring it
+to the surface? This last question comes close to hitting the mark. We can turn it around:
+Was the proof lying deeply hidden in the program? Or was it close to the surface? That is,
+how easy is it to see why the program did what it did? Can the discovery be attributed to
+some simple mechanism, or simple combination of mechanisms, in the program? Or was
+there a complex interaction which, if one heard it explained, would not diminish one’s
+awe at its having happened?
+
+It seems reasonable to say that if one can ascribe the performance to certain
+operations which are easily traced in the program, then in some sense the program was
+just revealing ideas which were in essence hiddenthough not too deeply-inside the
+programmer's own mind. Conversely, if
+
+
+Artificial Intelligence: Retrospects
+
+
+606
+
+
+
+following the program does not serve to enlighten one as to why this particular discovery
+popped out, then perhaps one should begin to separate the program's "mind" from that of
+its programmer. The human gets credit for having invented the program, but not for
+having had inside his own head the ideas produced by the program. In such cases, the
+human can be referred to as the "meta-author"-the author of the author of the result, and
+the program as the (just plain) author.
+
+In the particular case of Gelemter and his geometry machine, while Gelemter
+probably would not have rediscovered Pappus', proof, still the mechanisms which
+generated that proof were sufficiently close to the surface of the program that one
+hesitates to call the program a geometer in its own right. If it had kept on astonishing
+people by coming up with ingenious new proofs over and over again, each of which
+seemed to be based on a fresh spark of genius rather than on some standard method, then
+one would have no qualms about calling the program a geometer-but this did not happen.
+
+Who Composes Computer Music?
+
+The distinction between author and meta-author is sharply pointed up in the case of
+computer composition of music. There are various levels of autonomy which a program
+may seem to have in the act of composition. One level is illustrated by a piece whose
+"meta-author" was Max Mathews of Bell Laboratories. He fed in the scores of the two
+marches "When Johnny Comes Marching Home" and "The British Grenadiers", and
+instructed the computer to make a new score-one which starts out as "Johnny", but slowly
+merges into "Grenadiers". Halfway through the piece, "Johnny" is totally gone, and one
+hears "Grenadiers" by itself... Then the process is reversed, and the piece finishes with
+"Johnny", as it began. In Mathews' own words, this is
+
+... a nauseating musical experience but one not without interest, particularly in
+the rhythmic conversions. "The Grenadiers" is written in 2/4 time in the key of F
+major. "Johnny" is written in 6/8 time in the key of E minor. The change from 2/4
+to 6/8 time can be clearly appreciated, yet would be quite difficult for a human
+musician to play. The modulation from the key of F major to E minor, which
+involves a change of two notes in the scale, is jarring, and a smaller transition
+would undoubtedly have been a better choice."
+
+The resulting piece has a somewhat droll quality to it, though in spots it is turgid and
+confused.
+
+Is the computer composing? The question is best unasked, but it cannot be
+completely ignored. An answer is difficult to provide. The algorithms are
+deterministic, simple, and understandable. No complicated or hard-to understand
+computations are involved; no "learning" programs are used; no random processes
+occur; the machine functions in a perfectly mechanical and straightforward manner.
+However, the result is sequences of sound that are unplanned in fine detail by the
+composer, even though the over-all structure
+
+
+Artificial Intelligence: Retrospects
+
+
+607
+
+
+
+of the section is completely and precisely specified. Thus the composer is often
+surprised, and pleasantly surprised, ar the details of the realization of his ideas. To
+this extent only is the computer composing. We call the process algorithmic
+composition, but we immediately re-emphasize that the algorithms are
+transparently simple."
+
+This is Mathews' answer to a question which he would rather "unask". Despite his
+disclaimer, however, many people find it easier to say simply that the piece was
+"composed by a computer". I believe this phrase misrepresents the situation totally. The
+program contained no structures analogous to the brain's "symbols", and could not be said
+in any sense to be "thinking" about what it was doing. To attribute the composition of
+such a piece of music to the computer would be like attributing the authorship of this
+book to the computerized automatically (often incorrectly) hyphenating phototypesetting
+machine with which it was set.
+
+This brings up a question which is a slight digression from Al, but actually not a
+huge one. It is this: When you see the word "I" or "me" in a text, what do you take it to be
+referring to? For instance, think of the phrase "WASH ME" which appears occasionally
+on the back of dirty trucks. Who is this "me"? Is this an outcry of some forlorn child who,
+in desperation to have a bath, scribbled the words on the nearest surface? Or is the truck
+requesting a wash? Or, perhaps, does the sentence itself wish to be given a shower? Or, is
+it that the filthy English language is asking to be cleansed? One could go on and on in
+this game. In this case, the phrase is a joke, and one is supposed to pretend, on some
+level, that the truck itself wrote the phrase and is requesting a wash. On another level, one
+clearly recognizes the writing as that of a child, and enjoys the humor of the misdirection.
+Here, in fact, is a game based on reading the "me" at the wrong level.
+
+Precisely this kind of ambiguity has arisen in this book, first in the
+Contracrostipunctus, and later in the discussions of Godel’s string G (and its relatives).
+The interpretation given for unplayable records was "I Cannot Be Played on Record
+Player X", and that for unprovable statements was, "I Cannot Be Proven in Formal
+System X". Let us take the latter sentence. On what other occasions, if any, have you
+encountered a sentence containing the pronoun "I" where you automatically understood
+that the reference was not to the speaker of the sentence, but rather to the sentence itself?
+Very few, I would guess. The word "I", when it appears in a Shakespeare sonnet, is
+referring not to a fourteen-line form of poetry printed on a page, but to a flesh-and-blood
+creature behind the scenes, somewhere off stage.
+
+How far back do we ordinarily trace the "I" in a sentence? The answer, it seems to
+me, is that we look for a sentient being to attach the authorship to. But what is a sentient
+being? Something onto which we can map ourselves comfortably. In Weizenbaum's
+"Doctor" program, is there a personality? If so, whose is it? A small debate over this very
+question recently raged in the pages of Science magazine.
+
+This brings us back to the issue of the "who" who composes computer music. In
+most circumstances, the driving force behind such pieces is a
+
+
+Artificial Intelligence: Retrospects
+
+
+608
+
+
+
+human intellect, and the computer has been employed, with more or less ingenuity, as a
+tool for realizing an idea devised by the human. The program which carries this out is not
+anything which we can identify with. It is a simple and single-minded piece of software
+with no flexibility, no perspective on what it is doing, and no sense of self. If and when,
+however, people develop programs which have those attributes, and pieces of music start
+issuing forth from them, then I suggest that will be the appropriate, time to start splitting
+up one's admiration: some to the programmer for creating such an amazing program, and
+some to the program itself for its sense of music. And it seems to me that that will only
+take place when the internal structure of such a program is based on something similar to
+the "symbols" in our brains and their triggering patterns, which are responsible for the
+complex notion of meaning. The fact of having this kind of internal structure would
+endow the program with properties which would make us feel comfortable in identifying
+with it, to some extent. But until then, I will not feel comfortable in saying "this piece
+was composed by a computer".
+
+Theorem Proving and Problem Reduction
+
+Let us now return to the history of AI. One of the early things which people attempted to
+program was the intellectual activity of theorem proving. Conceptually, this is no
+different from programming a computer to look for a derivation of MU in the MIU-
+system, except that the formal systems involved were often more complicated than the
+MlU-system. They were versions of the Predicate Calculus, which is an extension of the
+Propositional Calculus involving quantifiers. Most of the rules of the Predicate Calculus
+are included in TNT, as a matter of fact. The trick in writing such a program is to instill a
+sense of direction, so that the program does not wander all over the map, but works only
+on "relevant" pathways-those which, by some reasonable criterion, seem to be leading
+towards the desired string.
+
+In this book we have not dealt much with such issues. How indeed can you know
+when you are proceeding towards a theorem, and how can you tell if what you are doing
+is just empty fiddling? This was one thing which I hoped to illustrate with the MU-
+puzzle. Of course, there can be no definitive answer: that is the content of the limitative
+Theorems, since if you could always know which way to go, you could construct an
+algorithm for proving any desired theorem, and that would violate Church's Theorem.
+There is no such algorithm. (I will leave it to the reader to see exactly why this follows
+from Church's Theorem.) However, this doesn't mean that it is impossible to develop any
+intuition at all concerning what is and what is not a promising route; in fact, the best
+programs have very sophisticated heuristics, which enable them to make deductions in
+the Predicate Calculus at speeds which are comparable to those of capable humans.
+
+The trick in theorem proving is to use the fact that you have an overall goal-namely
+the string you want to produce-in guiding you locally. One technique which was
+developed for converting global goals
+
+
+Artificial Intelligence: Retrospects
+
+
+609
+
+
+
+into local strategies for derivations is called problem reduction. It is based on the idea that
+whenever one has a long-range goal, there are usually subgoals whose attainment will aid
+in the attainment of the main goal. Therefore if one breaks up a given problem into a
+series of new subproblems, then breaks those in turn into subsubproblems, and so on, in a
+recursive fashion, one eventually comes down to very modest goals which can
+presumably be attained in a couple of steps. Or at least so it would seem ...
+
+Problem reduction got Zeno into hot water. Zeno’s method, you recall, for getting
+from A to B (think of B as the goal), is to "reduce” the problem into two subproblems:
+first go halfway, then go the rest of the way. So now you have "pushed"-in the sense of
+Chapter V-two subgoals onto your "goal stack". Each of these, in turn, will be replaced
+by two subsubgoals and so on ad infinitum. You wind up with an infinite goal-stack,
+instead of a single goal (Fig. 115). Popping an infinite number of goals off your stack will
+prove to be tricky-which is just Zeno’s point, of course.
+
+Another example of an infinite recursion in problem reduction occurred in the
+Dialogue Little Harmonic Labyrinth, when Achilles wanted to have a Typeless Wish
+granted. Its granting had to be deferred until permission was gotten from the Meta-Genie;
+but in order to get permission to give permission, she had to summon the Meta-Meta-
+Genie-and so on. Despite
+
+FIGURE 1 15. Zeno’s endless goal tree, for getting from A to B.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+the infmiteness of the goal stack, Achilles got his wish. Problem reduction wins the day!
+
+Despite my mockery, problem reduction is a powerful technique for converting
+global problems into local problems. It shines in certain situations, such as in the
+endgame of chess, where the look-ahead technique often performs miserably, even when
+it is carried to ridiculous lengths, such as fifteen or more plies. This is because the look¬
+ahead technique is not based on planning; it simply has no goals and explores a huge
+number of pointless alternatives. Having a goal enables you to develop a strategy for the
+achievement of that goal, and this is a completely different philosophy from looking
+ahead mechanically. Of course, in the look-ahead technique, desirability or its absence is
+measured by the evaluation function for positions, and that incorporates indirectly a
+number of goals, principally that of not getting checkmated. But that is too indirect. Good
+chess players who play against look-ahead chess programs usually come away with the
+impression that their opponents are very weak in formulating plans or strategies.
+
+Shandy and the Bone
+
+There is no guarantee that the method of problem reduction will work. There are
+many situations where it flops. Consider this simple problem, for instance. You are a dog,
+and a human friend has just thrown your favorite bone over a wire fence into another
+yard. You can see your bone through the fence, just lying there in the grass-how luscious!
+There is an open gate in the fence about fifty feet away from the bone. What do you do?
+Some dogs will just run up to the_ fence, stand next to it, and bark; others will dash up to
+the open gate and double back to the lovely bone. Both dogs can be said to be exercising
+the problem reduction technique; however, they represent the problem in their minds in
+different ways, and this makes all the difference. The barking dog sees the subproblems
+as (1) running to the fence, (2) getting through it, and (3) running to the bone-but that
+second subproblem is a "toughie", whence the barking. The other dog sees the
+subproblems as (1) getting to the gate; (2) going through the gate; (3) running to the
+bone. Notice how everything depends on the way you represent the "problem space"-that
+is, on what you perceive as reducing the problem (forward motion towards the overall
+goal) and what you perceive as magnifying the problem (backward motion away from the
+goal).
+
+
+Changing the Problem Space
+
+Some dogs first try running directly towards the bone, and when they encounter the
+fence, something clicks inside their brain; soon they change course, and run over to the
+gate. These dogs realize that what on first
+
+
+Artificial Intelligence: Retrospects
+
+
+611
+
+
+
+glance seemed as if it would increase the distance between the initial situation and the
+desired situation-namely, running away from the bone but towards the open gate-actually
+would decrease it. At first, they confuse physical distance with problem distance. Any
+motion away from the bone seems, by definition, a Bad Thing. But then-somehow-they
+realize that they can shift their perception of what will bring them "closer" to the bone. In
+a properly chosen abstract space, moving towards the gate is a trajectory bringing the dog
+closer to the bone! At every moment, the dog is getting "closer"-in the new sense-to the
+bone. Thus, the usefulness of problem reduction depends on how you represent your
+problem mentally. What in one space looks like a retreat can in another space look like a
+revolutionary step forward.
+
+In ordinary life, we constantly face and solve variations on the dog and-bone
+problem. For instance, if one afternoon 1 decide to drive one hundred miles south, but am
+at my office and have ridden my bike to work, 1 have to make an extremely large number
+of moves in what are ostensibly "wrong" directions before 1 am actually on my way in car
+headed south. 1 have to leave my office, which means, say, heading east a few feet; then
+follow the hall in the building which heads north, then west. Then 1 ride my bike home,
+which involves excursions in all the directions of the compass; and 1 reach my home. A
+succession of short moves there eventually gets me into my car, and 1 am off. Not that 1
+immediately drive due south, of course-1 choose a route which may involve some
+excursions north, west, or east, with the aim of getting to the freeway as quickly as
+possible.
+
+All of this doesn't feel paradoxical in the slightest; it is done without even any
+sense of amusement. The space in which physical backtracking is perceived as direct
+motion towards the goal is built so deeply into my mind that 1 don't even see any irony
+when 1 head north. The roads and hallways and so forth act as channels which 1 accept
+without much fight, so that part of the act of choosing how to perceive the situation
+involves just accepting what is imposed. But dogs in front of fences sometimes have a
+hard time doing that, especially when that bone is sitting there so close, staring them in
+the face, and looking so good. And when the problem space is just a shade more abstract
+than physical space, people are often just as lacking in insight about what to do as the
+barking dogs.
+
+In some sense all problems are abstract versions of the dog-and-bone problem.
+Many problems are not in physical space but in some sort of conceptual space. When you
+realize that direct motion towards the goal in that space runs you into some sort of
+abstract "fence", you can do one of two things: (1) try moving away from the goal in
+some sort of random way, hoping that you may come upon a hidden "gate" through
+which you can pass and then reach your bone; or (2) try to find a new "space" in which
+you can represent the problem, and in which there is no abstract fence separating you
+from your goal-then you can proceed straight towards the goal in this new space. The first
+method may seem like the lazy way to go, and the second method may seem like a
+difficult and complicated way to go. And yet, solutions which involve restructuring the
+problem space more often
+
+
+Artificial Intelligence: Retrospects
+
+
+612
+
+
+
+than not come as sudden flashes of insight rather than as products of a series of slow,
+deliberate thought processes. Probably these intuitive flashes come from the extreme core
+of intelligence-and, needless to say, their source is a closely protected secret of our
+jealous brains.
+
+In any case, the trouble is not that problem reduction per se leads to failures; it is
+quite a sound technique. The problem is a deeper one: how do you choose a good internal
+representation for a problem? What kind of "space" do you see it in? What kinds of
+action reduce the "distance" between you and your goal in the space you have chosen?
+This can be expressed in mathematical language as the problem of hunting for an
+approprate metric (distance function) between states. You want to find a metric in which
+the distance between you and your goal is very small.
+
+Now since this matter of choosing an internal representation is itself a type of
+problem-and a most tricky one, too-you might think of turning the technique of problem
+reduction back on it! To do so, you would have to have a way of representing a huge
+variety of abstract spaces, which is an exceedingly complex project. I am not aware of
+anyone's having tried anything along these lines. It may be just a theoretically appealing,
+amusing suggestion which is in fact wholly unrealistic. In any case, what A1 sorely lacks
+is programs which can "step back" and take a look at what is going on, and with this
+perspective, reorient themselves to the task at hand. It is one thing to write a program
+which excels at a single task which, when done by a human being, seems to require
+intelligence-and it is another thing altogether to write an intelligent program! It is the
+difference between the Sphex wasp (see Chapter XI), whose wired-in routine gives the
+deceptive appearance of great intelligence, and a human being observing a Sphex wasp.
+
+The I-Mode and the M-Mode Again
+
+An intelligent program would presumably be one which is versatile enough to solve
+problems of many different sorts. It would learn to do each different one and would
+accumulate experience in doing so. It would be able to work within a set of rules and yet
+also, at appropriate moments, to step back and make a judgment about whether working
+within that set of rules is likely to be profitable in terms of some overall set of goals
+which it has. It would be able to choose to stop working within a given framework, if
+need be, and to create a new framework of rules within which to work for a while.
+
+Much of this discussion may remind you of aspects of the MU-puzzle. For
+instance, moving away from the goal of a problem is reminiscent of moving away from
+MU by making longer and longer strings which you hope may in some indirect way
+enable you to make MU. If you are a naive "dog", you may feel you are moving away
+from your "MU-bone" whenever your string increases beyond two characters; if you are a
+more sophisticated dog, the use of such lengthening rules has an indirect justification,
+something like heading for the gate to get your MU-bone.
+
+
+Artificial Intelligence: Retrospects
+
+
+613
+
+
+
+Another connection between the previous discussion and the MU puzzle is the
+two modes of operation which led to insight about the nature of the MU-puzzle: the
+Mechanical mode, and the Intelligent mode. In the former, you are embedded within
+some fixed framework; in the latter, you can always step back and gain an overview of
+things. Having an overview is tantamount to choosing a representation within which to
+work; and working within the rules of the system is tantamount to trying the technique of
+problem reduction within that selected framework. Hardy's comment on Ramanujan's
+style-particularly his willingness to modify his own hypotheses-illustrates this interplay
+between the M-mode and the I-mode in creative thought.
+
+The Sphex wasp operates excellently in the M-mode, but it has absolutely no
+ability to choose its framework or even to alter its M-mode in the slightest. It has no
+ability to notice when the same thing occurs over and over and over again in its system,
+for to notice such a thing would be to jump out of the system, even if only ever so
+slightly. It simply does not notice the sameness of the repetitions. This idea (of not
+noticing the identity of certain repetitive events) is interesting when we apply it to
+ourselves. Are there highly repetitious situations which occur in our lives time and time
+again, and which we handle in the identical stupid way each time, because we don't have
+enough of an overview to perceive their sameness? This leads back to that recurrent
+issue, "What is sameness?" It will soon come up as an A1 theme, when we discuss pattern
+recognition.
+
+
+Applying A1 to Mathematics
+
+Mathematics is in some ways an extremely interesting domain to study from the A1 point
+of view. Every mathematician has the sense that there is a kind of metric between ideas in
+mathematics-that all of mathematics is a network of results between which there are
+enormously many connections. In that network, some ideas are very closely linked;
+others require more elaborate pathways to be joined. Sometimes two theorems in
+mathematics are close because one can be proven easily, given the other. Other times two
+ideas are close because they are analogous, or even isomorphic. These are two different
+senses of the word "close" in the domain of mathematics. There are probably a number of
+others. Whether there is an objectivity or a universality to our sense of mathematical
+closeness, or whether it is largely an accident of historical development is hard to say.
+Some theorems of different branches of mathematics appear to us hard to link, and we
+might say that they are unrelated-but something might turn up later which forces us to
+change our minds. If we could instill our highly developed sense of mathematical
+closeness-a "mathematician's mental metric", so to speak-into a program, we could
+perhaps produce a primitive "artificial mathematician". But that depends on being able to
+convey a sense of simplicity or "naturalness" as well, which is another major stumbling
+block.
+
+These issues have been confronted in a number of AI projects. There
+
+
+Artificial Intelligence: Retrospects
+
+
+614
+
+
+
+is a collection of programs developed at MIT which go under the name «MACSYMA",
+whose purpose it is to aid mathematicians in symbolic manipulation of complex
+mathematical, expressions. This program has in it some sense of "where to go"-a sort of
+"complexity gradient" which guides it from what we would generally consider complex
+expressions to simpler ones. Part of MACSYMA's repertoire is a program called "SIN",
+which does symbolic integration of functions; it is generally acknowledged to be superior
+to humans in some categories. It relies upon a number of different skills, as intelligence
+in general must: a vast body of knowledge, the technique of problem reduction, a large
+number of heuristics, and also some special tricks.
+
+Another program, written by Douglas Lenat at Stanford, had as its aim to invent
+concepts and discover facts in very elementary mathematics. Beginning with the notion
+of sets, and a collection of notions of what is "interesting" which had been spoon-fed into
+it, it "invented" the idea of counting, then the idea of addition, then multiplication, then-
+among other things-the notion of prime numbers, and it went so far as to rediscover
+Goldbach's conjecture! Of course these "discoveries" were all hundreds-even thousands-
+of years old. Perhaps this may be explained in part by saying that the sense of
+"interesting" was conveyed by Lenat in a large number of rules which may have been
+influenced by his twentieth century training; nonetheless it is impressive. The program
+seemed to run out of steam after this very respectable performance. An interesting thing
+about it was that it was unable to develop or improve upon its own sense of what is
+interesting. That seemed another level of difficulty up-or perhaps several levels up.
+
+The Crux of Al: Representation of Knowledge
+
+Many of the examples above have been cited in order to stress that the way a domain is
+represented has a huge bearing on how that domain is "understood". A program which
+merely printed out theorems of TNT in a preordained order would have no understanding
+of number theory; a program such as Lenat's with its extra layers of knowledge could be
+said to have a rudimentary sense of number theory; and one which embeds mathematical
+knowledge in a wide context of real-world experience would probably be the most able to
+"understand" in the sense that we think we do. It is this' representation of knowledge that
+is at the crux of Al.
+
+In the early days it was assumed that knowledge came in sentence-like packets",
+and that the best way to implant knowledge into a program was to develop a simple way
+of translating facts into small passive packets of data. Then every fact would simply be a
+piece of data, accessible to the programs using it. This is exemplified by chess programs,
+where board Positions are coded into matrices or lists of some sort and stored efficiently
+in memory where they can be retrieved and acted upon by subroutines.
+
+The fact that human beings store facts in a more complicated way was
+
+
+Artificial Intelligence: Retrospects
+
+
+615
+
+
+
+Known to psychologists for quite a while and has only recently been rediscovered by AI
+workers, who are now confronting the problems of "chunked" knowledge, and the
+difference between procedural and declarative types of knowledge, which is related, as
+we saw in Chapter XI, to the difference between knowledge which is accessible to
+introspection and knowledge which is inaccessible to introspection.
+
+The naive assumption that all knowledge should be coded into passive pieces of
+data is actually contradicted by the most fundamental fact about computer design: that is,
+how to add, subtract, multiply, and so on is not coded into pieces of data and stored in
+memory; it is, in fact, represented nowhere in memory, but rather in the wiring patterns of
+the hardware. A pocket calculator does not store in its memory knowledge of how to add;
+that knowledge is encoded into its "guts". There is no memory location to point to if
+somebody demands, "Show me where the knowledge of how to add resides in this
+machine!"
+
+A large amount of work in Al has nevertheless gone into systems in which the
+bulk of the knowledge is stored in specific places-that is, declaratively. It goes without
+saying that some knowledge has to be embodied in programs; otherwise one would not
+have a program at all, but merely an encyclopedia. The question is how to split up
+knowledge between program and data. Not that it is always easy to distinguish between
+program and data, by any means. I hope that was made clear enough in Chapter XVI. But
+in the development of a system, if the programmer intuitively conceives of some
+particular item as data (or as program), that may have significant repercussions on the
+system's structure, because as one programs one does tend to distinguish between data¬
+like objects and program-like objects.
+
+It is important to point out that in principle, any manner of coding information
+into data structures or procedures is as good as any other, in the sense that if you are not
+too concerned about efficiency, what you can do in one scheme, you can do in the other.
+However, reasons can be given which seem to indicate that one method is definitely
+superior to the other. For instance, consider the following argument in favor of using
+procedural representations only: "As soon as you try to encode features of sufficient
+complexity into data, you are forced into developing what amounts to a new language, or
+formalism. So in effect your data structures become program-like, with some piece of
+your program serving as their interpreter; you might as well represent the same
+information directly in procedural form to begin with, and obviate the extra level of
+interpretation."
+
+
+DNA and Proteins Help Give Some Perspective
+
+This argument sounds quite convincing, and yet, if interpreted a little loosely, it can be
+read as an argument for the abolishment of DNA and RNA. Why encode genetic
+information in DNA, when by representing it directly in proteins, you could eliminate not
+just one, but two levels of interpretation? The answer is: it turns out that it is extremely
+useful to have
+
+
+Artificial Intelligence: Retrospects
+
+
+616
+
+
+
+the same information in several different forms for different purposes. One advantage of
+storing genetic information in the modular and data-like form of DNA is that two
+individuals' genes can be easily recombined to form a new genotype. This would be very
+difficult if the information were only in proteins. A second reason for storing information
+in DNA is that it is easy to transcribe and translate it into proteins. When it is not needed,
+it does not take up much room; when it is needed, it serves as a template. There is no
+mechanism for copying one protein off of another; their folded tertiary structures would
+make copying highly unwieldy. Complementarily, it is almost imperative to be able to get
+genetic information into three-dimensional structures such as enzymes, because the
+recognition and manipulation of molecules is by its nature a three-dimensional operation.
+Thus the argument for purely procedural representations is seen to be quite fallacious in
+the context of cells. It suggests that there are advantages to being able to switch back and
+forth between procedural and declarative representations. This is probably true also in AI.
+
+This issue was raised by Francis Crick in a conference on communication with
+extraterrestrial intelligence:
+
+We see on Earth that there are two molecules, one of which is good for
+replication [DNA] and one of which is good for action [proteins]. Is it possible to
+devise a system in which one molecule does both jobs, or are there perhaps strong
+arguments, from systems analysis, which might suggest (if they exist) that to divide
+the job into two gives a great advantage. This is a question to which I do not know
+the answer. 14
+
+
+Modularity of Knowledge
+
+Another question which comes up in the representation of knowledge is modularity. How
+easy is it to insert new knowledge? How easy is it to revise old knowledge? How modular
+are books? It all depends. If from a tightly structured book with many cross-references a
+single chapter is removed, the rest of the book may become virtually incomprehensible. It
+is like trying to pull a single strand out of a spider web-you ruin the whole in doing so.
+On the other hand, some books are quite modular, having independent chapters.
+
+Consider a straightforward theorem-generating program which uses TNT's
+axioms and rules of inference. The "knowledge" of such a program has two aspects. It
+resides implicitly in the axioms and rules, and explicitly in the body of theorems which
+have so far been produced. Depending on which way you look at the knowledge, you will
+see it either as modular or as spread all around and completely nonmodular. For instance,
+suppose you had written such a program but had forgotten to include TNT's Axiom I in
+the list of axioms. After the program had done many thousands of derivations, you
+realized your oversight, and inserted the new axiom. The fact that you can do so in a trice
+shows that the system's implicit knowledge is modular; but the new axiom's contribution
+to the explicit knowledge of the system will only be reflected after a long time-after its
+effects have "dif-
+
+
+Artificial Intelligence: Retrospects
+
+
+617
+
+
+
+fused" outwards, as the odor of perfume slowly diffuses in a room when the bottle is
+broken. In that sense the new knowledge takes a long time to be incorporated.
+Furthermore, if you wanted to go back and replace Axiom I by its negation, you could not
+just do that by itself; you would have to delete all theorems which had involved Axiom 1
+in their derivations. Clearly this system's explicit knowledge is not nearly so modular as
+its implicit knowledge.
+
+It would be useful if we learned how to transplant knowledge modularly. Then to
+teach everyone French, we would just open up their heads and operate in a fixed way on
+their neural structures-then they would know how to speak French. Of course, this is only
+a hilarious pipe dream.
+
+Another aspect of knowledge representation has to do with the way in which one
+wishes to use the knowledge. Are inferences supposed to be drawn as pieces of
+information arrive? Should analogies and comparisons constantly be being made between
+new information and old information? In a chess program, for instance, if you want to
+generate look-ahead trees, then a representation which encodes board positions with a
+minimum of redundancy will be preferable to one which repeats the information in
+several different ways. But if you want your program to "understand" a board position by
+looking for patterns and comparing them to known patterns, then representing the same
+information several times over in different forms will be more useful.
+
+Representing Knowledge in a Logical Formalism
+
+There are various schools of thought concerning the best way to represent and manipulate
+knowledge. One which has had great influence advocates representations using formal
+notations similar to those for TNT-using propositional connectives and quantifiers. The
+basic operations in such representations are, not surprisingly, formalizations of deductive
+reasoning. Logical deductions can be made using rules of inference analogous to some of
+those in TNT. Querying the system about some particular idea sets up a goal in the form
+of a string to be derived. For example: "Is MUMON a theorem?" Then the automatic
+reasoning mechanisms take over in a goal oriented way, using various methods of
+problem reduction.
+
+For example, suppose that the proposition "All formal arithmetics are incomplete"
+were known, and the program were queried, "Is Principia Mathematica incomplete?" In
+scanning the list of known facts-often called the data base-the system might notice that if
+it could establish that Principia Mathematica is a formal arithmetic, then it could answer
+the question. Therefore the proposition " Principia Mathematica is a formal arithmetic"
+would be set up as a subgoal, and then problem reduction would take over. If it could find
+further things which would help in establishing (or refuting) the goal or the subgoal, it
+would work on them-and so on, recursively. This process is given the name of backwards
+chaining, since it begins with the goal and works its way backwards, presumably towards
+things which may already be known. If one makes a graphic representation of the main
+goal,
+
+
+Artificial Intelligence: Retrospects
+
+
+618
+
+
+
+subsidiary goals, subsubgoals, etc., a tree-like structure will arise, since the main goal
+may involve several different subgoals, each of which in turn involves several
+subsubgoals, etc.
+
+Notice that this method is not guaranteed to resolve the question, for there may be
+no way of establishing within the system that Principia Mathematica is a formal
+arithmetic. This does not imply, however, that either the goal or the subgoal is a false
+statement-merely that they cannot be derived with the knowledge currently available to
+the system. The system may print out, in such a circumstance, "I do not know" or words
+to that effect. The fact that some questions are left open is of course similar to the
+incompleteness from which certain well-known formal systems suffer.
+
+Deductive vs. Analogical Awareness
+
+This method affords a deductive awareness of the domain that is represented, in that
+correct logical conclusions can be drawn from known facts. However, it misses
+something of the human ability to spot similarities and to compare situations-it misses
+what might be called analogical awareness-a crucial side of human intelligence. This is
+not to say that analogical thought processes cannot be forced into such a mold, but they
+do not lend themselves naturally to being captured in that kind of formalism. These days,
+logic-oriented systems are not so much in vogue as other kinds, which allow complex
+forms of comparisons to be carried out rather naturally.
+
+When you realize that knowledge representation is an altogether different ball
+game than mere storage of numbers, then the idea that "a computer has the memory of an
+elephant" is an easy myth to explode. What is stored in memory is not necessarily
+synonymous with what a program knows; for even if a given piece of knowledge is
+encoded somewhere inside a complex system, there may be no procedure, or rule, or
+other type of handler of data, which can get at it-it may be inaccessible. In such a case,
+you can say that the piece of knowledge has been "forgotten" because access to it has
+been temporarily or permanently lost. Thus a computer program may "forget" something
+on a high level which it "remembers" on a low level. This is another one of those ever-
+recurring level distinctions, from which we can probably learn much about our own
+selves. When a human forgets, it most likely means that a high-level pointer has been
+lost-not that any information has been deleted or destroyed. This highlights the extreme
+importance of keeping track of the ways in which you store incoming experiences, for
+you never know in advance under what circumstances, or from what angle, you will want
+to pull something out of storage.
+
+From Computer Haiku to an RTN-Grammar
+
+The complexity of the knowledge representation in human heads first hit home with me
+when I was working on a program to generate English sentences "out of the blue". I had
+come to this project in a rather interest-
+
+
+Artificial Intelligence: Retrospects
+
+
+619
+
+
+
+ing way. I had heard on the radio a few examples of so-called "Computer Haiku".
+Something about them struck me deeply. There was a large element of humor and
+simultaneously mystery to making a computer generate something which ordinarily
+would be considered an artistic creation. I was highly amused by the humorous aspect,
+and I was very motivated by the mystery-even contradiction-of programming creative
+acts. So I set out to write a program even more mysteriously contradictory and humorous
+than the haiku program.
+
+At first I was concerned with making the grammar flexible and recursive, so that
+one would not have the sense that the program was merely filling in the blanks in some
+template. At about that time 1 ran across a Scientific American article by Victor Yngve in
+which he described a simple but flexible grammar which could produce a wide variety of
+sentences of the type found in some children's books. 1 modified some of the ideas I'd
+gleaned from that article and came up with a set of procedures which formed a Recursive
+Transition Network grammar, as described in Chapter V. In this grammar, the selection
+of words in a sentence was determined by a process which began by selecting-at random-
+the overall structure of the sentence; gradually the decision-making process trickled down
+through lower levels of structure until the word level and the letter level were reached. A
+lot had to be done below the word level, such as inflecting verbs and making plurals of
+nouns; also irregular verb and noun forms were first formed regularly, and then if they
+matched entries in a table, substitutions of the proper (irregular) forms were made. As
+each word reached its final form, it was printed out. The program was like the proverbial
+monkey at a typewriter, but operating on several levels of linguistic structure
+simultaneously-not just the letter level.
+
+In the early stages of developing the program, I used a totally silly vocabulary-
+deliberately, since I was aiming at humor. It produced a lot of nonsense sentences, some
+of which had very complicated structures, others of which were rather short. Some
+excerpts are shown below:
+
+A male pencil who must laugh clumsily would quack. Must program not
+always crunch girl at memory? The decimal bug which spits clumsily might
+tumble. Cake who does sure take an unexpected man within relationship might
+always dump card.
+
+Program ought run cheerfully.
+
+The worthy machine ought not always paste the astronomer.
+
+Oh, program who ought really run off of the girl writes musician for theater.
+The businesslike relationship quacks.
+
+The lucky girl which can always quack will never sure quack.
+
+The game quacks. Professor will write pickle. A bug tumbles. Man takes the box
+who slips.
+
+The effect is strongly surrealistic and at times a little reminiscent of
+
+
+Artificial Intelligence: Retrospects
+
+
+620
+
+
+
+haiku-for example, the final sample of four consecutive short sentences. At first it seemed
+very funny and had a certain charm, but soon it became rather stale. After reading a few
+pages of output one could sense the limits of the space in which the program was
+operating; and after that, seeing random points inside that space-even though each one
+was "new"-was nothing new. This is, it seems to me, a general principle: you get bored
+with something not when you have exhausted its repertoire of behavior, but when you
+have mapped out the limits of the space that contains its behavior. The behavior space of
+a person is just about complex enough that it can continually surprise other people; but
+that wasn't true of my program. I realized that my goal of producing truly humorous
+output would require that far more subtlety be programmed in. But what, in this case, was
+meant by "subtlety It was clear that absurd juxtapositions of words were just too
+unsubtle; I needed a way to ensure that words would be used in accordance with the
+realities of the world. This was where thoughts about representation of knowledge began
+to enter the picture.
+
+
+From RTN's to ATN's
+
+The idea I adopted was to classify each word-noun, verb, preposition, etc.-in several
+different "semantic dimensions". Thus, each word was a member of classes of various
+sorts; then there were also superclasses-classes of classes (reminiscent of the remark by
+Ulam). In principle, such aggregation could continue to any number of levels, but I
+stopped at two. At any given moment, the choice of words was now semantically
+restricted, because it was required that there should be agreement between the various
+parts of the phrase being constructed. The idea was, for instance, that certain kinds of acts
+could be performed only by animate objects; that only certain kinds of abstractions could
+influence events, and so on. The decisions about what categories were reasonable, and
+whether each category was better thought of as a class or a superclass, were quite
+complicated. All words were branded in several different dimensions. Common
+prepositions-"of', "in", etc.-had several distinct entries, corresponding to their distinct
+usages. Now, the output began to be much more comprehensible-and for that reason it
+was funny in a new way.
+
+
+A Little Turing Test
+
+Below, I have reproduced nine selections, carefully culled from many pages of output
+from later versions of my program. Along with them are three (seriously intended)
+human-written sentences. Which?
+
+(1) Blurting may be considered as the reciprocal substitution of semiotic material
+(dubbing) for a semiotic dialogical product in a dynamic reflexion.
+
+
+Artificial Intelligence: Retrospects
+
+
+621
+
+
+
+(2) Rather think of a pathway of a 'sequence' of gedankenexperiment simpletons
+where heir-lines are a prima facie case of a paradiachronic transitivity.
+
+(3) Think of that as a chain strength possibility of what, eventually, comes out as a
+product (epistemic conditions:,) and the product is not a Frankfurt-ish packing-
+it-all-in.
+
+(4) Despite the efforts, the reply, if you will, had been supported by the Orient;
+hence a fallacy will thereafter be suspended by the attitude which will be being
+held by the ambassador.
+
+(5) Of course, until the upheavals, the ambassador was slightly gradually
+mollycoddling the rabble.
+
+(6) Supposedly, refined liberty caused the attitudes insofar as peace is distilled by
+the consequences which will not eventually be caused by the command
+irrevocably insofar as peace of it is sometimes causing the intransigency
+infinitesimally surprisingly.
+
+(7) According to the sophists, the campaigns in the city-states, in other words, have
+been accepted by the Orient cunningly. Of course, the Orient has been separated
+by the states particularly violently.
+
+The Orient supports the efforts which had been supported by mankind.
+
+(8) Admittedly, the hierarchical origin of the fallacy, nevertheless, will be
+prophesied by the enemies of it. By the same token, the individualists will have
+testified that intransigency will not have suspended the campaigns.
+
+(9) Needless to say, during the upheaval which will have warranted the secrecy, the
+replies do not separate the Orient. Of course, the countries, ipso facto, are
+always probing liberty.
+
+(10) Although a Nobel Prize was being achieved by the humanists, yet in addition,
+it was being achieved by the serf.
+
+(11) An attitude will often be held by the serfs of a strife-tom nation.
+
+(12) Moreover, the Nobel Prizes will be achieved. By the same token, despite the
+consequence, the Nobel Prizes which will be achieved will sometimes be
+achieved by a woman.
+
+The human-written sentences are numbers 1 to 3; they were drawn from the
+contemporary journal Art-Language 15 and are-as far as I can tellcompletely serious
+efforts among literate and sane people to communicate something to each other. That
+they appear here out of context is not too misleading, since their proper context sounds
+just the same as they do.
+
+
+Artificial Intelligence: Retrospects
+
+
+622
+
+
+
+My program produced the rest. Numbers 10 to 12 were chosen to show that there
+were occasional bursts of total lucidity; numbers 7 to 9 are more typical of the output,
+floating, in that curious and provocative netherworld between meaning and no-meaning;
+and then numbers 4 to 6 pretty much transcend meaning. In a generous mood, one could
+say that they stand on their own as pure "language objects", something like pieces of
+abstract sculpture carved out of words instead of stone; alternatively, one could say that
+they are pure pseudointellectual drivel.
+
+My choice of vocabulary was still aimed at producing humorous effects. The
+flavor of the output is hard to characterize. Although much of it "makes sense", at least
+on a single-sentence level, one definitely gets the feeling that the output is coming from a
+source with no understanding of what it is saying and no reason to say it. In particular,
+one senses an utter lack of visual imagery behind the words. When I saw such sentences
+come pouring out of the line printer, I experienced complex emotions. I was very amused
+by the silliness of the output. I was also very proud of my achievement and tried to
+describe it to friends as similar to giving rules for building up meaningful stories in
+Arabic out of single strokes of the pen-an exaggeration, but it pleased me to think of it
+that way. And lastly I was deeply thrilled by the knowledge that this enormously
+complicated machine was shunting around long trains of symbols inside it according to
+rules, and that these long trains of symbols were something like thoughts in my own head
+... something like them.
+
+
+Images of What Thought Is
+
+Of course I didn't fool myself into thinking that there was a conscious being behind those
+sentences-far from it. Of all people, I was the most aware of the reasons that this program
+was terribly remote from real thought. Tester's Theorem is quite apt here: as soon as this
+level of language handling ability had been mechanized, it was clear that it did not
+constitute intelligence. But this strong experience left me with an image: a glimmering
+sense that real thought was composed of much longer, much more complicated trains of
+symbols in the brain-many trains moving simultaneously down many parallel and
+crisscrossing tracks, their cars being pushed and pulled, attached and detached, switched
+from track to track by a myriad neural shunting-engines ...
+
+It was an intangible image which I cannot convey in words, and it was only an
+image. But images and intuitions and motivations lie mingled close in the mind, and my
+utter fascination with this image was a constant spur to think more deeply about what
+thought really could be. I have tried in other parts of this book to communicate some of
+the daughter images of this original image-particularly in the Prelude, Ant Fugue.
+
+What stands out in my mind now, as I look back at this program from the
+perspective of a dozen years, is how there is no sense of imagery behind what is being
+said. The program had no idea what a serf is, what a person is, or what anything at all is.
+The words were empty formal symbols, as empty
+
+
+Artificial Intelligence: Retrospects
+
+
+623
+
+
+
+'/•ft
+
+^1
+
+l/’J'
+
+
+
+
+
+
+
+
+
+•;>
+
+
+
+csu-
+
+
+'Tr->
+
+iL_i»>oklf>l) oUvJjJ/ti *— 1
+
+>7*0^ ^ jLZ^jy V^jl^J/WjJy
+
+FIGURE 116. A meaningful story in Arabic. I From A. Khatibi and M. S~elmassi, The
+Splendour of Islamic Calligraphy (New York: Rizzoli, 1976).
+
+
+Artificial Intelligence: Retrospects
+
+
+624
+
+
+
+as-perhaps emptier than-the p and q of the pq-system. My program took advantage of the
+fact that when people read text, they quite naturally tend to imbue each word with its full
+flavor-as if that were necessarily attached to the group of letters which form the word.
+
+My program could be looked at as a formal system, whose "theorems"-the output
+sentences-had ready-made interpretations (at least to speakers of English). But unlike the
+pq-system, these "theorems" were not all true statements when interpreted that way.
+
+Many were false, many were nonsense.
+
+In its humble way, the pq-system mirrored a tiny corner of the world. But when
+my program ran, there was no mirror inside it of how the world works, except for the
+small semantic constraints which it had to follow. To create such a mirror of
+understanding, I would have had to wrap each concept in layers and layers of knowledge
+about the world. To do this would have been another kind of effort from what I had
+intended to do. Not that I didn't often think of trying to do it-but I never got around to
+trying it out.
+
+
+Higher-Level Grammars ...
+
+In fact, I often pondered whether I could write an ATN-grammar (or some other kind of
+sentence-producing program) which would only produce true sentences about the world.
+Such a grammar would imbue the words with genuine meanings, in the way it happened
+in the pq-system and in TNT. This idea of a language in which false statements are
+ungrammatical is an old one, going back to Johann Amos Comenius, in 1633. It is very
+appealing because you have a crystal ball embodied in your grammar: just write down the
+statement you want to know about, and check to see if it is grammatical... . Actually,
+Comenius went even further, for in his language, false statements were not only
+ungrammatical-they were inexpressible!
+
+Carrying this thought in another direction, you might imagine a highlevel
+grammar which would produce random koans. Why not? Such a grammar would be
+equivalent to a formal system whose theorems are koans. And if you had such a program,
+could you not arrange it to produce only genuine koans? My friend Marsha Meredith was
+enthusastic about this idea of "Artificial Ism", so she tackled the project of writing a
+koan-writing program. One of her early efforts produced this curious quasi-koan:
+
+A SMALL YOUNG MASTER WANTED A SMALL WHITE GNARLED
+BOWL. "HOW CAN WE LEARN AND UNDERSTAND WITHOUT STUDY"
+THE YOUNG MASTER ASKED A LARGE CONFUSED MASTER. THE
+CONFUSED MASTER WALKED FROM A BROWN HARD MOUNTAIN TO A
+WHITE SOFT MOUNTAIN WITH A SMALL RED STONY BOWL. THE
+CONFUSED MASTER SAW A RED SOFT HUT.THE CONFUSED MASTER
+WANTED THE HUT. "WHY DID BODHIDHARMA COME INTO CHINA?"
+THE CONFUSED MASTER
+
+
+Artificial Intelligence: Retrospects
+
+
+625
+
+
+
+ASKED A LARGE ENLIGHTENED STUDENT. "THE PEACHES ARE
+LARGE", THE STUDENT ANSWERED THE CONFUSED MASTER.w "HOW
+CAN WE LEARN AND UNDERSTAND WITHOUT STUDY?" THE
+CONFUSED MASTER ASKED A LARGE OLD MASTER. THE OLD MASTER
+WALKED FROM A WHITE STONY G0025. THE OLD MASTER GOT LOST.
+
+Your personal decision procedure for koan genuineness probably reached a
+verdict without need of the Geometric Code or the Art of Zen Strings. If the lack of
+pronouns or the unsophisticated syntax didn't arouse your suspicions, that strange
+"00025" towards the end must have. What is it? It is a strange fluke-a manifestation of a
+bug which caused the program to print out, in place of the English word for an object, the
+program's internal name for the "node" (a LISP atom, in fact) where all information
+concerning that particular object was stored. So here we have a "window" onto a lower
+level of the underlying Zen mind-a level that should have remained invisible.
+Unfortunately, we don't have such clear windows onto the lower levels of human Zen
+minds.
+
+The sequence of actions, though a little arbitrary, comes from a recursive LISP
+procedure called "CASCADE", which creates chains of actions linked in a vaguely
+causal way to each other. Although the degree of comprehension of the world possessed
+by this koan generator is clearly not stupendous, work is in progress to make its output a
+little more genuine seeming.
+
+
+Grammars for Music?
+
+Then there is music. This is a domain which you might suppose, on first thought, would
+lend itself admirably to being codified in an ATN grammar, or some such program.
+Whereas (to continue this naive line of thought) language relies on connections with the
+outside world for meaning, music is purely formal. There is no reference to things "out
+there" in the sounds of music; there is just pure syntax-note following note, chord
+following chord, measure following measure, phrase following phrase...
+
+But wait. Something is wrong in this analysis. Why is some music so much
+deeper and more beautiful than other music? It is because form, in music, is expressive-
+expressive to some strange subconscious regions of our minds. The sounds of music do
+not refer to serfs or city-states, but they do trigger clouds of emotion in our innermost
+selves; in that sense musical meaning is dependent on intangible links from the symbols
+to things in the world-those "things", in this case, being secret software structures in our
+minds. No, great music will not come out of such an easy formalism as an ATN-
+grammar. Pseudomusic, like pseudo-fairy tales, may well come out-and that will be a
+valuable exploration for people to make-but the secrets of meaning in music lie far, far
+deeper than pure syntax.
+
+I should clarify one point here: in principle, ATN-grammars have all the power of
+any programming formalism, so if musical meaning is captur-
+
+
+Artificial Intelligence: Retrospects
+
+
+626
+
+
+
+able in any way at all (which I believe it is), it is capturable in an A I N - grammar. True.
+But in that case, I maintain, the grammar will be defining not just musical structures, but
+the entire structures of the mind of a beholder. The "grammar" will be a full grammar of
+thought-not just a grammar of music.
+
+Winograd's Program SHRDLU
+
+What kind of program would it take to make human beings admit that it had some
+"understanding”, even if begrudgingly? What would it take before you wouldn't feel
+intuitively that there is "nothing there"?
+
+In the years 1968-70, Terry Winograd (alias Dr. Tony Earrwig) was a doctoral
+student at MIT, working on the joint problems of language and understanding. At that
+time at MIT, much AI research involved the so-called blocks world-a. relatively simple
+domain in which problems concerning both vision and language-handling by computer
+could fit easily. The blocks world consists of a table with various kinds of toy-like blocks
+on it-square ones, oblong ones, triangular ones, etc., in various colors. (For a "blocks
+world" of another kind, see Figure 117: the painting Mental Arithmetic by Magritte I find
+its title singularly appropriate in this context.) The vision problems in the MIT blocks
+world are very tricky: how can a computer figure out, from a TV-scan of a scene with
+many blocks in it, just what kinds of blocks are present, and what their relationships are?
+Some blocks may be perched on top of others, some may be in front of others, there may
+be shadows, and so on.
+
+
+FIGURE 117. Mental Arithmetic, by Rene Magritte ( 1931).
+
+
+
+Artificial Intelligence: Retrospects
+
+
+627
+
+
+
+Winograd s work was separate from the issues of vision, however. Beginning with
+the assumption that the blocks world was well represented h inside the computer's
+memory, he confronted the many-faceted problem of how to get the computer to:
+
+(1) understand questions in English about the situation;
+
+(2) give answers in English to questions about the situation;
+
+(3) understand requests in English to manipulate the blocks;
+
+(4) break down each request into a sequence of operations it could do;
+
+(5) understand what it had done, and for what reasons;
+
+(6) describe its actions and their reasons, in English.
+
+It might seem reasonable to break up the overall program into modular
+subprograms, with one module for each different part of the problem; then, after the
+modules have been developed separately, to integrate them smoothly. Winograd found
+that this strategy of developing independent modules posed fundamental difficulties. He
+developed a radical approach, which challenged the theory that intelligence can be
+compartmentalized into independent or semi-independent pieces. His program SHRDLU
+named after the old code "ETAOIN SHRDLU", used by linotype operators to mark
+typos in a newspaper column-did not separate the problem into clean conceptual parts.
+The operations of parsing sentences, producing internal representations, reasoning about
+the world represented inside itself, answering questions, and so on, were all deeply and
+intricately meshed together in a procedural representation of knowledge. Some critics
+have charged that his program is so tangled that it does not represent any "theory" at all
+about language, nor does it contribute in any way to our insights about thought processes.
+Nothing could be more wrong than such claims, in my opinion. A tour de force such as
+SHRDLU may not be isomorphic to what we do-in fact, in no way should you think that
+in SHRDLU, the "symbol level" has been attained-but the act of creating it and thinking
+about it offers tremendous insight into the way intelligence works.
+
+The Structure of SHRDLU
+
+In fact, SHRDLU does consist of separate procedures, each of which contains some
+knowledge about the world; but the procedures have such a strong interdependency that
+they cannot be cleanly teased apart. The program is like a very tangled knot which resists
+untangling; but the fact that you cannot untangle it does not mean that you cannot
+understand it. There may be an elegant geometrical description of the entire knot even if
+it is physically messy. We could go back to a metaphor from the Mu Offering, and
+compare it to looking at an orchard from a "natural" angle.
+
+Winograd has written lucidly about SHRDLU. I quote here from his article in
+Schank and Colby's book:
+
+
+Artificial Intelligence: Retrospects
+
+
+628
+
+
+
+One of the basic viewpoints underlying the model is that all language use can be
+thought of as a way of activating procedures within the hearer. We can think of
+any utterance as a program-one that indirectly causes a set of operations to he
+carried out within the hearer's cognitive system. This "program writing" is indirect
+in the sense that we are dealing with an intelligent interpreter, who may take a set
+of actions which are quite different from those the speaker intended. The exact
+form is determined by his knowledge of the world, his expectations about the
+person talking to him, etc. in this program we have a simple version of this
+process of interpretation as it takes place in the robot. Each sentence interpreted by
+the robot is converted to a set of instructions in PLANNER. The program that is
+created is then executed to achieve the desired effect."
+
+
+PLANNER Facilitates Problem Reduction
+
+The language PLANNER, referred to here, is an At language whose principal feature is
+that some of the operations necessary for problem reduction are built in-namely, the
+recursive process of creating a tree of jsubgoals, subsubgoals, etc. What this means is that
+such processes, instead of having to be spelled out time and time again by the
+programmer, are automatically implied by so-called GOAL-statements. Someone who
+reads a PLANNER program will see no explicit reference to such operations; in argon,
+they are user-transparent. If one path-in the tree fails to achieve the desired goal, then the
+PLANNER program will "backtrack" and try another route. "Backtracking" is the magic
+word as far as PLANNER is concerned.
+
+Winograd's program made excellent use of these features of PLANNER-more
+exactly, of MICROPLANNER, a partial implementation of the plans for PLANNER. In
+the past few years, however, people with the goal of developing At have concluded that
+automatic backtrack ing, as in PLANNER, has definite disadvantages, and that it will
+probably not lead to their goal; therefore they have backed off from it, preferring to try
+other routes to AI.
+
+Let us listen to further comments from Winograd on SHRDLU:
+
+The definition of every word is a program which is called at an appropriate point in
+the analysis, and which can do arbitrary computations involving the sentence and
+the present physical situation."
+
+Among the examples which Winograd cites is the following:
+
+The different possibilities for the meaning of "the" are procedures which check
+various facts about the context, then prescribe actions such as "Look for a unique
+object in the data base which fits this description", or "Assert that the object being
+described is unique as far as the speaker is concerned." The program incorporates a
+variety of heuristics for deciding what part of the context is relevant. 18
+
+It is amazing how deep this problem with the word "the" is. It is probably safe to say that
+writing a program which can fully handle the top five words
+
+
+Artificial Intelligence: Retrospects
+
+
+629
+
+
+
+of English-"the", "of 1 , "and", "a", and "to"-would be equivalent to solving the entire
+problem of AI, and hence tantamount to knowing what intelligence and consciousness
+are. A small digression: the five most common nouns in English are-according to the
+Word Frequency Book compiled by John B. Carroll et al-"time", "people", "way",
+"water", and "words" (in that order). The amazing thing about this is that most people
+have no idea that we think in such abstract terms. Ask your friends, and 10 to 1 they'll
+guess such words as "man", "house", "car", "dog", and "money". And while we're on the
+subject of frequencies-the top twelve letters in English, in order, according to
+Mergenthaler, are: "ETAOIN SHRDLU".
+
+One amusing feature of SHRDLU which runs totally against the stereotype of
+computers as "number crunchers" is this fact, pointed out by Winograd: "Our system does
+not accept numbers in numeric form, and has only been taught to count to ten." 19 With
+all its mathematical underpinning, SHRDLU is a mathematical ignoramus! Just like Aunt
+Hillary, SHRDLU doesn't know anything about the lower levels which make it up. Its
+knowledge is largely procedural (see particularly the remark by "Dr, Tony Earrwig" in
+section 11 of the previous Dialogue).
+
+It is interesting to contrast the procedural embedding of knowledge in SHRDLU
+with the knowledge in my sentence-generation program. All of the syntactical knowledge
+in my program was procedurally embedded in Augmented Transition Networks, written
+in the language Algol; but the semantic knowledge-the information about semantic class
+membership-was static: it was contained in a short list of numbers after each word. There
+were a few words, such as the auxiliary verbs "to be", "to have", and others, which were
+represented totally in procedures in Algol, but they were the exceptions. By contrast, in
+SHRDLU, all words were represented as programs. Here is a case which demonstrates
+that, despite the theoretical equivalence of data and programs, in practice the choice of
+one over the other has major consequences.
+
+Syntax and Semantics
+
+And now, a few more words from Winograd:
+
+Our program does not operate by first parsing a sentence, then doing semantic
+analysis, and finally by using deduction to produce a response. These three
+activities go on concurrently throughout the understanding of a sentence. As soon
+as a piece of syntactic structure begins to take shape, a semantic program
+is called to see whether it might make sense, and the resultant answer can direct the
+parsing. In deciding whether it makes sense, the semantic routine may call
+deductive processes and ask questions about the real world. As an
+example, in sentence 34 of the Dialogue ("Put the blue pyramid on the block in the
+box"), the parser first comes up with "the blue pyramid on the block" as a candidate
+for a noun group. At this point, semantic analysis is done, and since "the" is
+definite, a check is made in the data base for the object being referred to. When no
+such object is found, the parsing is redirected to find the noun group "the blue
+pyramid". It will then go on to find "on the block
+
+
+Artificial Intelligence: Retrospects
+
+
+630
+
+
+
+in the box" as a single phrase indicating a location Thus there is a continuing
+interplay between the different sorts of analysis, with the results of one affecting
+the others r %
+
+It is extremely interesting that in natural language, syntax and semantics are so
+deeply intertwined. Last Chapter, in discussing the elusive concept of "form", we had
+broken the notion into two categories: syntactic form, which is detectable by a
+predictably terminating decision procedure, and semantic form, which is not. But here,
+Winograd is telling us that-at least when the usual senses of "syntax" and "semantics" are
+taken-they merge right into each other, in natural language. The external form of a
+sentence-that is, its composition in terms of elementary signs-does not divide up so neatly
+into syntactic and semantic aspects. This significant point for linguistics.
+
+Here are some final comments on SHRDLU by Winograd.
+
+Let us look at what the system would do with a simple description like "a red cube which
+supports a pyramid". The description will use concepts like BLOCK, RED, PYRAMID,
+and EQUIDIMENSIONAL-all parts of the sys
+
+FIGURE 118. Procedural representation of "a red cube which supports a pyramid."
+Adapted from Roger Scliank and Kenneth Colby, Computer Models of Thought and
+Language (San Francisco: W. H. Freeman, 1973), p. 172.
+
+
+
+Artificial Intelligence: Retrospects
+
+
+631
+
+
+
+
+
+tern's underlying categorization of the world. The result can be represented in a
+flow chart like that in Figure 118. Note that this is a program for finding an object
+fitting the description. It would_ then be incorporated into a command for doing
+something with the object, a question asking something about it, or, if it appeared in
+a statement, it would become part of the program which was generated to represent
+the meaning for later use. Note that this bit of program could also be used as a test
+to see whether an object fit the description, if the first FIND instruction were told
+in advance to look only at that particular object.
+
+At first glance, it seems that there is too much structure in this program, as
+we don't like to think of the meaning of a simple phrase as explicitly containing
+loops, conditional tests, and other programming details. The solution is to provide
+an internal language that contains the appropriate looping and checking as its
+primitives, and in which the representation of the process is as simple as the
+description. The program described in Figure 11S would be written in PLANNER
+looking something like what is below:
+
+(GOAL (IS ?X 1 BLOCK))
+
+(GOAL (COLOR-OF ?X1 RED))
+
+(GOAL (EQUIDIMENSIONAL ?X1))
+
+(GOAL (IS ?X2 PYRAMID))
+
+(GOAL (SUPPORT ?X1 ?X2))
+
+The loops of the flowchart are implicit in PLANNER'S backtrack control structure.
+The description is evaluated by proceeding down the list until some goal fails, at
+which time the system backs up automatically to the last point where a decision
+was made, trying a different possibility. A decision can be made whenever a new
+object name or VARIABLE (indicated by the prefix
+
+") such as "?X 1" or "?X2" appears. The variables are used by the pattern matcher.
+
+If they have already been assigned to a particular item, it checks to see whether the
+GOAL is true for that item. If not, it checks for all possible items which satisfy the
+GOAL, by choosing one, and then taking successive ones whenever backtracking
+occurs to that point. Thus, even the distinction between testing and choosing is
+implicit. 21
+
+One significant strategy decision in devising this program was to not translate all the way
+from English into LISP, but only partway-into PLANNER. Thus (since the PLANNER
+interpreter is itself written in LISP), a new intermediate level-PLANNER-was inserted
+between the top-level language (English) and the bottom-level language (machine
+language). Once a PLANNER program had been made from an English sentence
+fragment, then it could be sent off to the PLANNER interpreter, and the higher levels of
+SHRDLU would be freed up, to work on new tasks.
+
+This kind of decision constantly crops up: How many levels should a system
+have? How much and what kind of "intelligence" should be placed on which level? These
+are some of the hardest problems facing AI today. Since we know so little about natural
+intelligence, it is hard for us to figure out which level of an artificially intelligent system
+should carry out what part of a task.
+
+
+Artificial Intelligence: Retrospects
+
+
+632
+
+
+
+This gives you a glimpse behind the scenes of the Dialogue preceding this
+Chapter. Next Chapter, we shall meet new and speculative ideas for AI.
+
+
+Artificial Intelligence: Retrospects
+
+
+633
+
+
+
+Contrafactus
+
+
+The Crab has invited a small group of friends over to watch the Saturday
+afternoon football game on television. Achilles has already arrived, but the
+Tortoise and his friend the Sloth are still awaited.
+
+
+Achilles: Could that be our friends, a-riding up on that unusual one-wheeled vehicle?
+
+(The Sloth and Tortoise dismount and come in.)
+
+Crab: Ah, my friends, I'm so glad you could make it. May I present my old and beloved
+acquaintance, Mr. Sloth-and this is Achilles. I believe you know the Tortoise.
+
+Sloth: This is the first time I can recall making the acquaintance of a Bicyclops. Pleased
+to meet you, Achilles. I've heard many fine things said about the bicyclopean species.
+
+Achilles: Likewise, I'm sure. May I ask about your elegant vehicle? Tortoise: Our tandem
+unicycle, you mean? Hardly elegant. It's just a way for two to get from A to B, at the
+same speed.
+
+Sloth: It's built by a company that also makes teeter-teeters.
+
+Achilles: I see, I see. What is that knob on it?
+
+Sloth: That's the gearshift.
+
+Achilles: Aha! And how many speeds does it have?
+
+Tortoise: One, including reverse. Most models have fewer, but this is a special model.
+
+Achilles: It looks like a very nice tandem unicycle. Oh, Mr. Crab, I wanted to tell you
+how much I enjoyed hearing your orchestra perform last night.
+
+Crab: Thank you, Achilles. Were you there by any chance, Mr. Sloth? Sloth: No, I
+couldn't make it, I'm sad to say. I was participating in a mixed singles ping-ping
+tournament. It was quite exciting because my team was involved in a one-way tie for
+first place.
+
+Achilles: Did you win anything?
+
+Sloth: Certainly did-a two-sided Mobius strip made out of copper; it is silver-plated on
+one side, and gold-plated on the other. Crab: Congratulations, Mr. Sloth.
+
+Sloth: Thank you. Well, do tell me about the concert.
+
+Crab: It was a most enjoyable performance. We played some pieces by the Bach twins
+
+Sloth: The famous Job and Sebastian?
+
+Crab: One and the same. And there was one work that made me think of you, Mr. Sloth-a
+marvelous piano concerto for two left hands. The
+
+
+Contrafactus
+
+
+633
+
+
+
+next-to-last (and only) movement was a one-voice fugue. You can't imagine its
+intricacies. For our finale, we played Beethoven's Ninth Zenfunny. At the end,
+everyone in the audience rose and clapped with one hand. It was overwhelming.
+
+Sloth: Oh, I'm sorry I missed it. But do you suppose it's been recorded: At home I have a
+fine hi-fi to play it on-the best two-channel monaural system money can buy.
+
+Crab: I'm sure you can find it somewhere. Well, my friends, the game is about to begin.
+
+Achilles: Who is playing today, Mr. Crab?
+
+Crab: I believe it's Home Team versus Visitors. Oh, no-that was last week. I think this
+week it's Out-of-Towners.
+
+Achilles: I'm rooting for Home Team. I always do.
+
+Sloth: Oh, how conventional. I never root for Home Team. The closer a team lives to the
+antipodes, the more I root for it.
+
+Achilles: Oh, so you live in the Antipodes? I've heard it's charming to live there, but I
+wouldn't want to visit them. They're so far away.
+
+Sloth: And the strange thing about them is that they don't get any closer no matter which
+way you travel.
+
+Tortoise: That's my kind of place.
+
+Crab: It's game time. I think I'll turn on the TV.
+
+(He walks over to an enormous cabinet with a screen, underneath which is an
+instrument panel as complicated as that of a jet airplane. He flicks a knob, and the
+football stadium a ears in bright vivid color on the screen.)
+
+Announcer: Good afternoon, fans. Well, it looks like that time of year has rolled around
+again when Home Team and Out-of-Town face each other on the gridiron and play out
+their classic pigskin rivalry. It's been drizzling on and off this afternoon, and the field's
+a little wet, but despite the weather it promises to be a fine game, especially with that
+GREAT pair of eighth-backs playing for Home Team, Tedzilliger and Palindromi.
+
+And now, here's Pilipik, kicking off for Home Team. It's in the air! Flampson takes it
+for Out-of-Towners, and runs it back he's to the 20, the 25, the 30, and down at the 32.
+That was Mool in on the tackle for Home Team.
+
+Crab: A superb runback! Did you see how he was ALMOST tackled by Quilker-but
+somehow broke away?
+
+Sloth: Oh, don't be silly, Crab. Nothing of the kind happened. Quilker did NOT tackle
+Flampson. There's no need to confuse poor Achilles (or the rest of us) with hocus-
+pocus about what "almost" happened. It's a fact-with no "almost" 's, "if "'s, "and" 's, or
+"but" 's.
+
+Announcer: Here's the instant replay. Just watch number 79, Quilker, come in from the
+side, surprising Flampson, and just about tackle him!
+
+Sloth: "Just about"! Bah!
+
+'Achilles: Such a graceful maneuver! What would we do without instant replays?
+
+
+Contraf actus
+
+
+634
+
+
+
+Announcer: It's first down and 10 for Out-of-Town. Noddle takes the ball, hands off to
+Orwix-it's a reverse-Orwix runs around to the right, handing off to Flampson-a double
+reverse, folks!-and now
+
+Flampson hands it to Treefig, who's downed twelve yards behind scrimmage. A twelve-
+yard loss on a triple reverse!
+
+Sloth: I love it! A sensational play!
+
+Achilles: But, Mr. S, I thought you were rooting for Out-of-Town. They lost twelve yards
+on the play.
+
+Moth: They did? Oh, well-who cares, as long as it was a beautiful play?
+
+Let's see it again.
+
+(... and so the first half of the game passes. Towards the end of the third quarter, a
+particularly crucial play comes up for Home Team. They are behind by eight points.
+It's third down and 10, and they badly need a first down.)
+
+Announcer: The ball is hiked to Tedzilliger, who fades back, looking-for a receiver, and
+fakes to Quilker. There's Palindromi, playing wide right, with nobody near him.
+Tedzilliger spots him and fires a low pass to him. Palindromi snatches it out of the air,
+and- ( There is an audible groan from the crowd.)— oh, he steps out of bounds! What a
+crushing blow for Home Team, folks! If Palindromi hadn't stepped out of bounds, he
+could've run all the way to the end zone for a touchdown!
+
+Let's watch the subjunctive instant replay.
+
+(And on the screen the same lineup appears as before.)
+
+The ball is hiked to Tedzilliger, who fades back, looking for a receiver, and fakes to
+Quilker. There's Palindromi, playing wide right, with nobody near him. Tedzilliger
+spots him, and fires a low pass to him. Palindromi snatches it out of the air, and-
+(There is an audible gasp from the crowd.)- he almost steps out of bounds! But he's
+still in bounds, and it's clear all the way to the end zone! Palindromi streaks in, for a
+touchdown for Home Team! (The stadium breaks into a giant roar of approval.) Well,
+folks, that's what would've happened if Palindromi hadn't stepped out of bounds.
+
+Achilles: Wait a minute ... WAS there a touchdown, or WASN'T there?
+
+Crab: Oh, no. That was just the subjunctive instant replay. They simply followed a
+hypothetical a little way out, you know.
+
+Sloth: That is the most ridiculous thing I ever heard of! Next thing you know, they'll be
+inventing concrete earmuffs.
+
+Tortoise: Subjunctive instant replays are a little unusual, aren't they?
+
+Crab: Not particularly, if you have a Subjunc-TV.
+
+Achilles: Is that one grade below a junk TV?
+
+Crab: Not at all! It's a new kind of TV, which can go into the subjunctive mode. They're
+particularly good for football games and such. I just got mine.
+
+Achilles: Why does it have so many knobs and fancy dials?
+
+
+Contraf actus
+
+
+635
+
+
+
+Crab: So that you can tune it to the proper channel. There are many channels
+
+broadcasting in the subjunctive mode, and you want to be able to select from them
+easily.
+
+Achilles: Could you show us what you mean? I'm afraid I don't quite understand what all
+this talk of "broadcasting in the subjunctive mode" is about.
+
+Crab: Oh, it's quite simple, really. You can figure it out yourself. I'm going into the
+kitchen to fix some French fries, which I know are Mr. Sloth's weakness.
+
+Sloth: Mmmmm! Go to it, Crab! French fries are my favorite food. Crab: What about the
+rest of you?
+
+Tortoise: I could devour a few.
+
+Achilles: Likewise. But wait-before you go into the kitchen, is there some trick to using
+your Subjunc-TV?
+
+Crab: Not particularly. Just continue watching the game, and whenever there's a near
+miss of some sort, or whenever you wish things had gone differently in some way, just
+fiddle with the dials, and see what happens. You can't do it any harm, though you may
+pick up some exotic channels. (And he disappears into the kitchen.)
+
+Achilles: I wonder what he means by that. Oh well, let's get back to this game. I was
+quite wrapped up in it.
+
+Announcer: It's fourth down for Out-of-Town, with Home Team receiving. Out-of-Town
+is in punt formation, with Tedzilliger playing deep. Orwix is back to kick-and he gets
+a long high one away. It's coming down near Tedzilliger
+
+Achilles: Grab it, Tedzilliger! Give those Out-of-Towners a run for their money!
+
+Announcer: -and lands in a puddle-KERSPLOSH! It takes a weird bounce! Now Sprunk
+is madly scrambling for the ball! It looks like it just barely grazed Tedzilliger on the
+bounce, and then slipped away from himit's ruled a fumble. The referee is signaling
+that the formidable Sprunk has recovered for Out-of-Town on the Home Team 7! It's a
+bad break for Home Team. Oh, well, that's the way the cookie crumbles.
+
+Achilles: Oh, no! If only it hadn't been raining ... ( Wrings his hands in despair .)
+
+Sloth: ANOTHER of those confounded hypotheticals! Why are the rest of you always
+running off into your absurd worlds of fantasy? If I were you, I would stay firmly
+grounded in reality. "No subjunctive nonsense" is my motto. And I wouldn't abandon
+it even if someone offered me a hundred-nay, a hundred and twelve-French fries.
+
+Achilles: Say, that gives me an idea. Maybe by suitably fiddling with these knobs, I can
+conjure up a subjunctive instant replay in which it isn't raining, there's no puddle, no
+weird bounce, and Tedzilliger doesn't
+
+fumble. I wonder ... ( Walks up to the Subjunc-TV and stares at it.) But I haven't any idea
+what these different knobs do. ( Spins a few at random .)
+
+Announcer: It's fourth down for Out-of-Town, with Home Team receiv-
+
+
+Contraf actus
+
+
+636
+
+
+
+ing. Out-of-Town is in punt formation, with Tedzilliger playing deep. Orwix is back to
+kick-and he gets a long high one away. It's coming down near Tedzilliger
+
+Achilles: Grab it, Tedzilliger! Give those Out-of-Towners a run for their money!
+
+Announcer: -and lands in a puddle-KERSPLOSH! Oh-it bounces right into his arms!
+
+Now Sprunk is madly scrambling after him, but he's got - good blocking, and he steers
+his way clear of the formidable Sprunk, and now he's got an open field ahead of him.
+Look at that, folks! He's to the 50, the 40, the 30, the 20, the 10-touchdown, Home
+Team! ( Huge cheers from the Home Team side.) Well, fans, that's how it would have
+gone, if footballs were spheres instead of oblate spheroids! But in reality. Home Team
+loses the ball, and Out-of-Towners take over on the Home Team 7-yard line. Oh, well,
+that's the way the ball bounces.
+
+Achilles: What do you think of THAT, Mr. Sloth?
+
+(And Achilles gives a smirk in the direction of the Sloth, but the latter is completely
+oblivious to its devastating effect, as he is busy watching ,the Crab arrive with, a large
+platter with a hundred and twelve-nay, a hundred-large and delicious French fries,
+and napkins for all.)
+
+Crab: So how do you three find my Subjunc-TV?
+
+Sloth: Most disappointing, Crab, to be quite frank. It seems to be badly out of order. It
+makes pointless excursions into nonsense at least half the time. If it belonged to me, I
+would give it away immediately to someone like you, Crab. But of course it doesn't
+belong to me.
+
+Achilles: It's quite a strange device. I tried to rerun a play to see how it would have gone
+under different weather conditions, but the thing seems to have a will of its own!
+Instead of changing the weather, it changed the football shape to ROUND instead of
+FOOTBALL-SHAPED! Now tell me-how can a football not be shaped like a football?
+That's a contradiction in terms. How preposterous!
+
+Crab: Such tame games! I thought you'd surely find more interesting subjunctives. How
+would you like to see how the last play would have looked if the game had been
+baseball instead of football?
+
+Tortoise: Oh! An outstanding idea!
+
+(The Crab twiddles two knobs, and steps back.)
+
+Announcer: There are four away, and—
+
+Achilles: FOUR away!?
+
+Announcer: That's right, fans-four away. When you turn football into baseball,
+SOMETHING'S got to give! Now as I was about to say, there are four away, with
+Out-of-Town in the field, and Home Team up. Tedzilliger is at bat. Out-of-Town is in
+bunt formation. Orwix raises his arm to pitch-and he gets a long high ball away. It's
+heading straight for Tedzilliger
+
+Achilles: Smash it, Tedzilliger! Give those Out-of-Towners a home run for their money!
+
+
+Contraf actus
+
+
+637
+
+
+
+Announcer: -but it seems to be a spitball, as it takes a strange curve. Now Sprunk is
+madly scrambling for the ball! It looks like it just barely grazed Tedzilliger's bat, then
+bounced off it-it's ruled a fly ball. The umpire is signaling that the formidable Sprunk
+has caught it for Out-of-Town, to end the seventh inning. It's a bad break for Home
+Team. That's how the last play would have looked, football fans, if this had been a
+game of baseball.
+
+Sloth: Bah! You might as well transport this game to the Moon.
+
+Crab: No sooner said than done! Just a twiddle here, a twiddle there ...
+
+(On the screen there appears a desolate crater-pitted field, with two teams in space
+suits facing each other, immobile. All at once, the two teams fly into motion, and the
+players are making great bounds into the air, sometimes over the heads of other
+players. The ball is thrown into the air, and sails so high that it almost disappears,
+and then slowly comes floating down into the arms of one space-suited player, roughly
+a quarter-mile from where it was released.)
+
+Announcer: And there, friends, you have the subjunctive instant replay as it would have
+happened on the Moon. We'll be right back after this important commercial message
+from the friendly folks who brew Glumpf Beer-my favorite kind of beer!
+
+Sloth: If I weren't so lazy, I would take that broken TV back to the dealer myself! But
+alas, it's my fate to be a lazy Sloth ... ( Helps himself to a large gob of French fries.)
+
+Tortoise: That's a marvelous invention, Mr. Crab. May I suggest a hypothetical?
+
+Crab: Of course!
+
+Tortoise: What would that last play have looked like if space were four-dimensional?
+
+Crab: Oh, that's a complicated one, Mr. T, but I believe I can code it into the dials. Just a
+moment.
+
+(He steps up, and, for the first time, appears to be using the full power of the control
+panel of his Subjunc-TV, turning almost every knob two or three times, and carefully
+checking various meters. Then he steps back with a satisfied expression on his face.)
+
+I think this should do it.
+
+Announcer: And now let's watch the subjunctive instant replay.
+
+(A confusing array of twisted pipes appears on the screen. It grows larger, then
+smaller, and for a moment seems to do something akin to rotation. Then it turns into a
+strange mushroom-shaped object, and back to a bunch of pipes. As it metamorphoses
+from this into other bizarre shapes, the announcer gives his commentary.)
+
+Tedzilliger's fading back to pass. He spots Palindromi ten yards outfield, and passes it to
+the right and outwards-it looks good! Palindromi's at the 35-yard plane, the 40, and
+he's tackled on his own
+
+
+Contraf actus
+
+
+638
+
+
+
+43-yard plane. And there you nave it, 3-L tans, as it would have looked if football
+were played in four spatial dimensions.
+
+Achilles: What is it you are doing, Mr. Crab, when you twirl these various dials on the
+control panel?
+
+Crab: I am selecting the proper subjunctive channel. You see, there are all sorts of
+
+subjunctive channels broadcasting simultaneously, and I want to tune in precisely that
+one which represents the kind of hypothetical which has been suggested.
+
+Achilles: Can you do this on any TV?
+
+Crab: No, most TV's can't receive subjunctive channels. They require a special kind of
+circuit which is quite difficult to make.
+
+Sloth: How do you know which channel is broadcasting what? Do you look it up in the
+newspaper?
+
+Crab: I don't need to know the channel's call letters. Instead, I tune it in by coding, in
+these dials, the hypothetical situation which I want to be represented. Technically, this
+is called "addressing a channel by its counterfactual parameters". There are always a
+large number of channels broadcasting every conceivable world. All the channels
+which carry worlds that are "near" to each other have call letters that are near
+
+to each other, too.
+
+Tortoise: Why did you not have to turn the dials at all, the first time we saw a subjunctive
+instant replay?
+
+Crab: Oh, that was because I was tuned in to a channel which is very near to the Reality
+Channel, but ever so slightly off. So every once in a while, it deviates from reality. It's
+nearly impossible to tune EXACTLY into the Reality Channel. But that's all right,
+because it's so dull. All their instant replays are straight! Can you imagine? What a
+bore!
+
+Sloth: I find the whole idea of Subjunc-TV's one giant bore. But perhaps I could change
+my mind, if I had some evidence that your machine here could handle an
+INTERESTING counterf actual. For example, how would that last play have looked if
+addition were not commutative?
+
+Crab: Oh me, oh my! That change is a little too radical, I'm afraid, for this model. I
+unfortunately don't have a Superjunc-TV, which is the top of the line. Superjunc-TV's
+can handle ANYTHING you throw at them.
+
+Sloth: Bah!
+
+Crab: But look-I can do ALMOST as well. Wouldn't you like to see how the last play
+would have happened if 13 were not a prime number? Sloth: No thanks! THAT
+doesn't make any sense! Anyway, if I were the last play, I'd be getting pretty tired of
+being trotted out time and again in new garb for the likes of you fuzzy-headed
+concept-slippers. Let's get on with the game!
+
+Achilles: Where did you get this Subjunc-TV, Mr. Crab?
+
+Crab: Believe it or not, Mr. Sloth and I went to a country fair the other evening, and it
+was offered as the first prize in a lottery. Normally I don't indulge in such frivolity, but
+some crazy impulse grabbed me, and I bought one ticket.
+
+
+Contraf actus
+
+
+639
+
+
+
+Achilles: What about you Mr. Sloth?
+
+Sloth: I admit, I bought one, just to humor old Crab.
+
+Crab: And when the winning number was announced, I found, to my amazement, that I'd
+won the lottery!
+
+Achilles: Fantastic! I've never known anyone who won anything in a lottery before!
+
+Crab: I was flabbergasted at my good fortune.
+
+Sloth: Don't you have something else to tell us about that lottery, Crab?
+
+Crab: Oh, nothing much. It's just that my ticket number was 129. Now when they
+
+announced the winning number, it was 128 just one off. Sloth: So you see, he actually
+didn't win it at all. Achilles: He ALMOST won, though ...
+
+Crab: I prefer to say that I won it, you see. For I came so terribly close . . If my number
+had been only one smaller, I would have won. Sloth: But unfortunately. Crab, a miss is
+as good as a mile.
+
+Tortoise: Or as bad. What about you, Mr. Sloth? What was your number:
+
+Sloth: Mine was 256-the next power of 2 above 128. Surely, that counts as a hit, if
+anything does! I can't understand why, however, those fair officials-those UNfair
+officials-were so thickheaded about it. They refused to award me my fully deserved
+prize. Some other joker claimed HE deserved it, because his number was 128. 1 think
+my number was far closer than His, but you can't fight City Hall.
+
+Achilles: I'm all confused. If you didn't win the Subjunc-TV after all, Mr. Crab, then how
+can we have been sitting here all afternoon watching it? It seems as if we ourselves
+have been living in some sort of hypothetical world that would have been, had
+circumstances just been ever so slightly different...
+
+Announcer: And that, folks, was how the afternoon at Mr. Crab's would have been spent,
+had he won the Subjunc-TV. But since he didn't, the four friends simply spent a
+pleasant afternoon watching Home Team get creamed, 128-0. Or was it 256-0? Oh
+well, it hardly matters, in five-dimensional Plutonian steam hockey.
+
+
+Contraf actus
+
+
+640
+
+
+
+CHAPTER XIX
+
+
+Artificial Intelligence:
+
+Prospects
+
+"Almost" Situations and Subjunctives
+
+AFTER READING Contrafactus, a friend said to me, "My uncle was almost President of
+the U.S.!" "Really?" I said. "Sure," he replied, "he was skipper of the PT 108." (John F.
+Kennedy was skipper of the PT 109.)
+
+That is what Contrafactus is all about. In everyday thought, we are constantly
+manufacturing mental variants on situations we face, ideas we have, or events that
+happen, and we let some features stay exactly the same while others "slip". What features
+do we let slip? What ones do we not even consider letting slip? What events are
+perceived on some deep intuitive level as being close relatives of ones which really
+happened? What do we think "almost" happened or "could have" happened, even though
+it unambiguously did not? What alternative versions of events pop without any conscious
+thought into our minds when we hear a story? Why do some counterfactuals strike us as
+"less counterf actual" than other counterf actuals? After all, it is obvious that anything that
+didn't happen didn't happen. There aren't degrees of "didn't-happen-ness". And the same
+goes for "almost" situations. There are times when one plaintively says, "It almost
+happened", and other times when one says the same thing, full of relief. But the "almost"
+lies in the mind, not in the external facts.
+
+Driving down a country road, you run into a swarm of bees. You don't just duly
+take note of it; the whole situation is immediately placed in perspective by a swarm of
+"replays" that crowd into your mind. Typically, you think, "Sure am lucky my window
+wasn't open!"-or worse, the reverse: "Too bad my window wasn't closed!" "Lucky I
+wasn't on my bike!" "Too bad I didn't come along five seconds earlier." Strange but-
+possible replays: "If that had been a deer, I could have been killed!" "I bet those bees
+would have rather had a collision with a rosebush." Even stranger replays: "Too bad
+those bees weren't dollar bills!" "Lucky those bees weren't made of cement!" "Too bad it
+wasn't just one bee instead of a swarm." "Lucky I wasn't the swarm instead of being me."
+What slips naturally and what doesn't-and why?
+
+In a recent issue of The New Yorker magazine, the following excerpt from the
+"Philadelphia Welcomat" was reprinted:'
+
+If Leonardo da Vinci had been born a female the ceiling of the Sistine Chapel
+might never have been painted. 1
+
+
+Artificial Intelligence:Prospects
+
+
+641
+
+
+
+The New The New Yorker commented:
+
+
+And if Michelangelo had been Siamese twins, the work would have been
+completed in half the time.
+
+The point of The New Yorker's comment is not that such counterfactuals are false; it is
+more that anyone who would entertain such an idea-anyone who would "slip" the sex or
+number of a given human being-would have to be a little loony. Ironically, though, in the
+same issue, the following sentence, concluding a book review, was printed without
+blushing:
+
+I think he [Professor Philipp Frank would have enjoyed both of these books
+enormously. 2
+
+Now poor Professor Frank is dead; and clearly it is nonsense to suggest that someone
+could read books written after his death. So why wasn't this serious sentence also scoffed
+at? Somehow, in some difficult-to-pin-down sense, the parameters slipped in this
+sentence do not violate our sense of "possibility" as much as in the earlier examples.
+Something allows us to imagine "all other things being equal" better in this one than in
+the others. But why? What is it about the way we classify events and people that makes
+us know deep down what is "sensible" to slip, and what is "silly":
+
+Consider how natural it feels to slip from the valueless declarative "I don't know
+Russian" to the more charged conditional "I would like to know Russian" to the
+emotional subjunctive "I wish I knew Russian" and finally to the rich counterf actual "If I
+knew Russian, I would read Chekhov and Lermontov in the original". How flat and dead
+would be a mind that saw nothing in a negation but an opaque barrier! A live mind can
+see a window onto a world of possibilities.
+
+I believe that "almost" situations and unconsciously manufactured subjunctives
+represent some of the richest potential sources of insight into how human beings organize
+and categorize their perceptions of the world.
+
+An eloquent co-proponent of this view is the linguist and translator George
+Steiner, who, in his book After Babel , has written:
+
+Hypothetical, 'imaginaries', conditionals, the syntax of counter-factuality and
+contingency may well be the generative centres of human speech.... [They] do more
+than occasion philosophical and grammatical perplexity. No less than future tenses
+to which they are, one feels, related, and with which they ought probably to be
+classed in the larger set of 'suppositionals' or 'alternates', these 'if propositions are
+fundamental to the dynamics of human feeling... .
+
+Ours is the ability, the need, to gainsay or 'un-say' the world, to image and speak it
+otherwise.... We need a word which will designate the power, the compulsion of
+language to posit 'otherness'. . . . Perhaps 'alternity' will do: to define the 'other than
+the case 1 , the counter-factual propositions, images, shapes of will and evasion with
+which we charge our mental being and by means of which we build the changing,
+largely fictive milieu of our somatic and our social existence... .
+
+Finally, Steiner sings a counterfactual hymn to counterf actuality:
+
+
+Artificial Intelligence:Prospects
+
+
+642
+
+
+
+It is unlikely that man, as we know him, would have survived without the fictive,
+counter-factual, anti-determinist means of language, without the semantic capacity,
+generated and stored in the 'superfluous, zones of the cortex, to conceive of, to articulate
+possibilities beyond the treadmill of organic decay and death .
+
+The manufacture of "subjunctive worlds" happens so casually, -so naturally, that
+we hardly notice what we are doing. We select from our fantasy a world which is close,
+in some internal mental sense, to the real world. We compare what is real with what we
+perceive as almost real. In so doing, what we gain is some intangible kind of perspective
+on reality. The Sloth is a droll example of a variation on reality-a thinking being without
+the ability to slip into subjunctives (or at least, who claims to be without the ability-but
+you may have noticed that what he says is full of counterfactuals'.). Think how
+immeasurably poorer our mental lives would be if we didn't have this creative capacity
+for slipping out of the midst of reality into soft "what if-s! And from the point of view of
+studying human thought processes, this slippage is very interesting, for most of the time it
+happens completely without conscious direction, which means that observation of what
+kinds of things slip, versus what kinds don't, affords a good window on the unconscious
+mind.
+
+One way to gain some perspective on the nature of this mental metric is to "fight
+fire with fire". This is done in the Dialogue, where our "subjunctive ability" is asked to
+imagine a world in which the very notion of
+
+subjunctive ability is slipped, compared to what we expect. In the Dialogue, the first
+subjunctive instant replay-that where Palindromi stays in bounds-is quite a normal thing
+to imagine. In fact, it was inspired by a completely ordinary, casual remark made to me
+by a person sitting next to me at a football game. For some reason it struck me and I
+wondered what made it seem so natural to slip that particular thing, but not, say, the
+number of the down, or the present score. From those thoughts, I went on to consider
+other, probably less slippable features, such as the weather (that's in the Dialogue), the
+kind of game (also in the Dialogue), and then even loonier variations (also in the
+Dialogue). I noticed, though, that what was completely ludicrous to slip in one situation
+could be quite slippable in another. For instance, sometimes you might spontaneously
+wonder how things would be if the ball had a different shape (e.g., if you are playing
+basketball with a half-inflated ball); other times that would never enter your mind (e.g.,
+when watching a football game on TV).
+
+Layers of Stability
+
+It seemed to me then, and still does now, that the slippability of a feature of some event
+(or circumstance) depends on a set of nested contexts in which the event (or
+circumstance) is perceived to occur. The terms constant, parameter, and variable,
+borrowed from mathematics, seem useful here. Often mathematicians, physicists, and
+others will carry out a calculation, saying "c is a constant, p is a parameter, and v is a
+variable". What they
+
+
+Artificial Intelligence:Prospects
+
+
+643
+
+
+
+mean.is that any of them can vary (including the "constant"); however, there is a kind of
+hierarchy of variability. In the situation which is being represented b~ the symbols, c
+establishes a global condition; p establishes some less global condition which can vary
+while c is held fixed; and finally, v can run around while c and p are held fixed. It makes
+little sense to think of holding v fixed while c and p vary, for c and p establish the context
+in which v has meaning. For instance, think of a dentist who has a list of patients, and for
+each patient, a list of teeth. It makes perfect sense (and plenty of money) to hold the
+patient fixed and vary his teeth-but it makes no sense at all to hold one tooth fixed and
+vary the patient. (Although sometimes it makes good sense to vary the dentist...)
+
+We build up our mental representation of a situation layer by layer. The lowest
+layer establishes the deepest aspect of the context-sometimes being so low that it cannot
+vary at all. For instance, the three-dimensionality of our world is so ingrained that most
+of us never would imagine letting it slip mentally. It is a constant constant. Then there are
+layers which establish temporarily, though not permanently, fixed aspects of situations,
+which could be called background assumptions-things which, in the back of your mind,
+you know can vary, but which most of the time you unquestioningly accept as
+unchanging aspects. These could still be called "constants". For instance, when you go to
+a football game, the rules of the game are constants of that sort. Then there are
+"parameters": you think of them as more variable, but you temporarily hold them
+constant. At a football game, parameters might include the weather, the opposing team,
+and so forth. There could be-and probably are-several layers of parameters. Finally, we
+reach the "shakiest" aspects of your mental representation of the situation-the variables.
+These are things such as Palindromi's stepping out of bounds, which are mentally "loose"
+and which you don't mind letting slip away from their real values, for a short moment.
+
+Frames and Nested Contexts
+
+The word frame is in vogue in A1 currently, and it could be defined as a computational
+instantiation of a context. The term is due to Marvin Minsky, as are many ideas about
+frames, though the general concept has been floating around for a good number of years.
+In frame language, one could say that mental representations of situations involve frames
+nested within each other. Each of the various ingredients of a situation has its own frame.
+It is interesting to verbalize explicitly one of my mental images concerning nested
+frames. Imagine a large collection of chests of drawers. When you choose a chest, you
+have a frame, and the drawer holes are places where "subframes" can be attached. But
+subframes are themselves chests of drawers. How can you stick a whole chest of drawers
+into the slot for a single drawer in another chest of drawers? Easy: you shrink and distort
+the second chest, since, after all, this is all mental, not physical. Now in the outer frame,
+there may be several different drawer slots that need to be
+
+
+Artificial Intelligence:Prospects
+
+
+644
+
+
+
+filled; then you may need to fill slots in some of the inner chests of drawers (or
+subframes). This can go on, recursively.
+
+The vivid surrealistic image of squishing and bending a chest of drawers so that it
+can fit into a slot of arbitrary shape is probably quite important, because it hints that your
+concepts are squished and bent by the contexts you force them into. Thus, what does your
+concept of "person" become when the people you are thinking about are football
+players? It certainly is a distorted concept, one which is forced on you by the overall
+context. You have stuck the "person" frame into a slot in the "football game" frame. The
+theory of representing knowledge in frames relies on the idea that the world consists of
+quasi-closed subsystems, each of which can serve as a context for others without being
+too disrupted, or creating too much disruption, in the process.
+
+One of the main ideas about frames is that each frame comes with its own set of
+expectations. The corresponding image is that each chest, of drawers comes with a built-
+in, but loosely bound, drawer in each of its
+
+drawer slots, called a default. If I tell you, "Picture a river bank", you will invoke a visual
+image which has various features, most of which you could override if I added extra
+phrases such as "in a drought" or "in Brazil" or "without a merry-go-round". The
+existence of default values for slots allows the recursive process of filling slots to come to
+an end. In effect, you say, "I will fill in the slots myself as far as three layers down;
+beyond that I will take the default options." Together with its default expectations, a
+frame contains knowledge of its limits of applicability, and heuristics for switching to
+other frames in case it has been stretched beyond its limits of tolerance.
+
+The nested structure of a frame gives you a way of "zooming in" and looking at
+small details from as close up as you wish: you just zoom in on the proper subframe, and
+then on one of its subframes, etc., until you have the desired amount of detail. It is like
+having a road atlas of the USA which has a map of the whole country in the front, with
+individual state maps inside, and even maps of cities and some of the larger towns if you
+want still more detail. One can imagine an atlas with arbitrary amounts of detail, going
+down to single blocks, houses, rooms, etc. It is like looking through a telescope with
+lenses of different power; each lens has its own uses. It is important that one can make
+use of all the different scales; often detail is irrelevant and even distracting.
+
+Because arbitrarily different frames can be stuck inside other frames' slots, there
+is great potential for conflict or "collision". The nice neat scheme of a single, global set of
+layers of "constants", "parameters", and "variables" is an oversimplification. In fact, each
+frame will have its own hierarchy of variability, and this is what makes analyzing how we
+perceive such a complex event as a football game, with its many subframes,
+subsubframes, etc., an incredibly messy operation. How do all these many frames interact
+with each other? If there is a conflict where one frame says, "This item is a constant" but
+another frame says, "No, it is a variable!", how does it get resolved? These are deep and
+difficult problems of frame theory to
+
+
+Artificial Intelligence:Prospects
+
+
+645
+
+
+
+which I can give no answers. There has as yet been no complete agreement on what a
+frame really is, or on how to implement frames in A1 programs. I make my own stab at
+discussing some of these questions in the following section, where I talk about some
+puzzles in visual pattern recognition, which I call "Bongard problems".
+
+Bongard Problems
+
+Bongard problems (BP’s) are problems of the general type given by the Russian scientist
+M. Bongard in his book Pattern Recognition. A typical BP-number 51 in his collection of
+one hundred-is shown in Figure 119.
+
+
+
+FIGURE 119. Bongard problem 57. From RJ. Bongard, Pattern Recognition (Rochelle
+Park, N.,J.j Hayden Book Co., Spartan Books, 1970).]
+
+These fascinating problems are intended for pattern-recognizers, whether human or
+machine. (One might also throw in ETI's-extraterrestrial intelligences.) Each problem
+consists of twelve boxed figures (henceforth called boxes): six on the left, forming Class
+/, and six on the right, forming Class II. The boxes may be indexed this way:
+
+
+I-A
+
+I-B
+
+II-A
+
+II-B
+
+I-C
+
+1-D
+
+II-C
+
+ll-D
+
+IE
+
+IF
+
+II-E
+
+II-F
+
+
+The problem is "How do Class I boxes differ from Class II boxes?"
+
+A Bongard problem-solving program would have several stages, in which raw
+data gradually get converted into descriptions. The early stages are relatively inflexible,
+and higher stages become gradually more flexible. The final stages have a property which
+I call tentativity, which means simply that the way a picture is represented is always
+tentative. Upon the drop of a hat, a high-level description can be restructured, using all
+the devices of the
+
+
+Artificial lntelligence:Prospects
+
+
+646
+
+
+
+
+
+
+later stages. The ideas presented below also have a tentative quality to them. I will try to
+convey overall ideas first, glossing over significant difficulties. Then I will go back and
+try to explain subtleties and tricks and so forth. So your notion of how it all works may
+also undergo some revisions as you read. But that is in the spirit of the discussion.
+
+Preprocessing Selects a Mini-vocabulary
+
+Suppose, then, that we have some Bongard problem which we want to solve. The
+problem is presented to a TV camera and the raw data are read in. Then the raw data are
+preprocessed. This means that some salient features are detected. The names of these
+features constitute a "mini-vocabulary" for the problem; they are drawn from a general
+"salient-feature vocabulary". Some typical terms of the salient-feature vocabulary are:
+
+line segment, curve, horizontal, vertical, black, white, big, small, pointy, round ...
+
+In a second stage of preprocessing, some knowledge about elementary shapes is used;
+and if any are found, their names are also made available. Thus, terms such as
+
+triangle, circle, square, indentation, protrusion, right angle, vertex, cusp, arrow ...
+
+may be selected. This is roughly the point at which the conscious and the unconscious
+meet, in humans. This discussion is primarily concerned with describing what happens
+from here on out.
+
+
+High-Level Descriptions
+
+Now that the picture is "understood", to some extent, in terms of familiar concepts, some
+looking around is done. Tentative descriptions are made for one or a few of the twelve
+boxes. They will typically use simple descriptors such as
+
+above, below, to the right of, to the left of, inside, outside of, close to, far from,
+parallel to, perpendicular to, in a row, scattered, evenly spaced, irregularly spaced, etc.
+
+Also, definite and indefinite numerical descriptors can be used:
+
+1, 2, 3, 4, 5,... many, few, etc.
+
+More complicated descriptors may be built up, such as
+
+further to the right of, less close to, almost parallel to, etc.
+
+
+Artificial Intelligence:Prospects
+
+
+647
+
+
+
+®
+
+
+® A
+
+O
+
+
+
+A
+
+
+\7 A
+v o
+
+v®
+
+
+°(gf
+
+
+
+
+
+V^A
+
+,y v
+
+[>®
+
+
+A®
+
+4
+
+7
+
+o a
+
+Oo
+
+
+TJa!
+
+o o
+
+
+FIGURE 120. Bongard problem 47. [From M. Bongard, Pattern Recognition.]
+Thus, a typical box-say I-F of BP 47 (Fig. 120)-could be variously described as having:
+
+three shapes
+or
+
+three white shapes
+or
+
+a circle on the right
+or
+
+two triangles and a circle
+or
+
+two upwards-pointing triangles
+or
+
+one large shape and two small shapes
+or
+
+one curved shape and two straight-edged shapes
+
+or
+
+a circle with the same kind of shape on the inside and outside.
+
+Each of these descriptions sees the box through a "filter". Out of context, any of them
+might be a useful description. As it turns out, though, all of them are "wrong", in the
+context of the particular Bongard problem they are part of. In other words, if you knew
+the distinction between Classes I and II in BP 47, and were given one of the preceding
+lines as a description of an unseen drawing, that information would not allow you to tell
+to which Class the drawing belonged. The essential feature of this box, in context, is that
+it includes
+
+
+a circle containing a triangle.
+
+Note that someone who heard such a description would not be able to reconstruct the
+original drawing, but would be able to recognize drawings
+
+
+Artificial lntelligence:Prospects
+
+
+648
+
+
+
+
+
+
+A
+
+
+
+
+□
+
+o
+
+
+□
+
+©
+
+
+
+j
+
+°oS
+
+
+_l_Li
+
+s.
+
+
+1
+
+%
+
+0000
+
+
+FIGURE 121. Bongarcl problem 91. I From M. Bongard, Pattern Recognition ./
+
+which have this property. It is a little like musical style: you may be an infallible
+recognizer of Mozart, but at the same time unable to write anything which would fool
+anybody into thinking it was by Mozart.
+
+Now consider box I-D of BP 91 (Fig. 121). An overloaded but ’'right 1 '' description
+in the context of BP 91 is
+
+a circle with three rectangular intrusions.
+
+Notice the sophistication of such a description, in which the word "with" functions as a
+disclaimer, implying that the "circle” is not really a circle: it is almost a circle, except that
+. . . Furthermore, the intrusions are not full rectangles. There is a lot of "play" in the way
+we use language to describe
+
+things. Clearly, a lot of information has been thrown away, and even more could
+be thrown away. A priori, it is very hard to know what it would be smart to throw away
+and what to keep So some sort of method for an intelligent compromise has to be
+encoded, via heuristics. Of course, there is always recourse to lower levels of description
+(i.e., less chunked descriptions) if discarded information has to be retrieved, just as
+people can constantly look at the puzzle for help in restructuring their ideas about it. The
+trick, then, is to devise explicit rules that say how to
+
+make tentative descriptions for each box;
+
+compare them with tentative descriptions for other boxes of either Class;
+restructure the descriptions, by
+
+(i) adding information,
+
+(ii) discarding information,
+
+or (iii) viewing the same information from another angle; iterate this process until
+finding out what makes the two Classes differ.
+
+
+Artificial lntelligence:Prospects
+
+
+649
+
+
+
+Templates and Sameness-Detectors
+
+One good strategy would be to try to make descriptions structurally similar to
+each other, to the extent this is possible. Any structure they have in common will
+make comparing them that much easier. Two important elements of this theory
+deal with this strategy. One is the idea of "description-schemas” or templates', the
+other is the idea of Sam-a "sameness detector".
+
+First Sam. Sam is a special agent present on all levels of the program.
+(Actually there may be different kinds of Sams on different levels.) Sam
+constantly runs around within individual descriptions and within different
+descriptions, looking for descriptors or other things which are repeated. When
+some sameness is found, various restructuring operations can be triggered, either
+on the single-description level or on the level of several descriptions at once.
+
+Now templates. The first thing that happens after preprocessing is an attempt to
+manufacture a template, or description-schema-a un form format for the
+descriptions of all the boxes in a problem. The idea is that a description can often
+be broken up in a natural way into subdescriptions, and those in turn into subs
+ubdescriptions, if need be. The bottom is hit when you come to primitive concepts
+which belong to the level of the preprocessor. Now it is important to choose the
+way of breaking descriptions into parts so as to reflect commonality among all the
+boxes; otherwise you are introducing a superfluous and meaningless kind of
+"pseudo-order" into the world.
+
+On the basis of what information is a template built? It is best to look at an
+example. Take BP 49 (Fig. 122). Preprocessing yields the information that each
+box consists of several little o's, and one large closed curve. This is a valuable
+observation, and deserves to be incorporated in the template. Thus a first stab at a
+template would be:
+
+large closed curve:-
+
+small o’s:-
+
+
+FIGURE 122. Bongarcl problem 49. [From M. Bongard, Pattern Recognition.]
+
+
+
+
+
+
+
+
+
+49
+
+
+
+
+o
+
+
+
+
+
+Artificial lntelligence:Prospects
+
+
+650
+
+
+
+
+
+
+
+
+It is very simple: the description-template has two explicit slots where
+subdescriptions are to be attached.
+
+
+A Heterarchical Program
+
+Sow an interesting thing happens, triggered by the term "closed curve", one of the
+most important modules in the program is a kind of semantic net-the concept
+network-in which all the known nouns, adjectives, etc., are linked in ways which
+indicate their interrelations. For instance, "closed curve" is strongly linked with
+the terms "interior" and "exterior". The concept net is just brimming with
+information about relations between terms, such as what is the opposite of what,
+what is similar to what, what often occurs with what, and so on. A little portion of
+a concept network, to be explained shortly, is shown in Figure 123. But let us
+follow what happens now, in the solution of problem 49. The concepts "interior"
+and "exterior" are activated by their proximity in the net to "closed curve". This
+suggests to the template-builder that it might be a good idea to make distinct slots
+for the interior and exterior of the curve. Thus, in the spirit of tentativity, the
+template is tentatively restructured to be this:
+
+large closed curve: —
+little o's in interior: —
+little o's in exterior:—
+
+Now when subdescriptions are sought, the terms "interior" and "exterior" will
+cause procedures to inspect those specific regions of the box. What is found in BP
+49, box I-A is this:
+
+large closed curve: circle
+little o's in interior: three
+little o's in exterior: three
+
+And a description of box II-A of the same BP might be
+
+large closed curve: cigar
+little o's in interior: three
+little o's in exterior: three
+
+Now Sam, constantly active in parallel with other operations, spots the
+recurrence of the concept "three" in all the slots dealing with o's, and this is strong
+reason to undertake a second template-restructuring operation. Notice that the first
+was suggested by the concept net, the second by Sam. Now our template for
+problem 49 becomes:
+
+large closed curve:—
+
+three little o's in interior:-
+
+three little o's in exterior:-
+
+
+Artificial Intelligence:Prospects
+
+
+651
+
+
+
+
+
+
+FIGURE 123. A small portion of a concept network for a program to solve
+Bongard Problems. "Nodes" are joined by "links", which in turn can be linked. By
+considering a link as a verb and the nodes it joins as subject and object, you can
+pull out some English sentences from this diagram.
+
+
+Artificial lntelligence:Prospects
+
+
+652
+
+
+
+
+
+
+
+
+
+
+
+
+Now that "three" has risen" one level of generality-namely, into the template-it
+becomes worthwhile to explore its neighbors in the concept network. One of them
+is "triangle", which suggests that triangles of o's may be important. As it happens,
+this leads down a blind alley-but how could you know in advances It is a typical
+blind alley that a human would explore, so it is good if our program finds it too!
+For box II-E, a description such as the following might get generated:
+
+large closed curve: circle
+
+three little o's in interior: equilateral triangle
+
+three little o's in exterior: equilateral triangle
+
+Of course an enormous amount of information has been thrown away concerning
+the sizes, positions, and orientations of these triangles, and many other things as
+well. But that is the whole point of making descriptions instead of just using the
+raw data! It is the same idea as funneling, which we discussed in Chapter XI.
+
+The Concept Network
+
+We need not run through the entire solution of problem 49; this suffices to show
+the constant back-and-forth interaction of individual descriptions, templates, the
+sameness-detector Sam, and the concept network. We should now look a little
+more in detail at the concept network and its function. A simplified portion shown
+in the figure codes the following ideas:
+
+"High" and "low" are opposites.
+
+"Up" and "down" are opposites.
+
+"High" and "up" are similar.
+
+"Low" and "down" are similar.
+
+"Right" and "left" are opposites.
+
+The "right-left" distinction is similar to the "high-low" distinction.
+
+"Opposite" and "similar" are opposites.
+
+Note how everything in the net-both nodes and links-can be talked about. In that
+sense nothing in the net is on a higher level than anything else. Another portion of
+the net is shown; it codes for the ideas that
+
+A square is a polygon.
+
+A triangle is a polygon.
+
+A polygon is a closed curve.
+
+
+Artificial Intelligence:Prospects
+
+
+653
+
+
+
+The difference between a triangle and a square is that one has 3 sides and the
+other has 4.
+
+4 is similar to 3.
+
+A circle is a closed curve.
+
+A closed curve has an interior and an exterior.
+
+"Interior" and "exterior" are opposites.
+
+The network of concepts is necessarily very vast. It seems to store knowledge only
+statically, or declaratively, but that is only half the story. Actually, its knowledge
+borders on being procedural as well, by the fact that the proximities in the net act
+as guides, or "programs", telling the main program how to develop its
+understanding of the drawings in the boxes.
+
+For instance, some early hunch may turn out to be wrong and yet have the germ of
+the right answer in it. In BP 33 (Fig. 124), one might at first
+
+
+F
+
+
+
+
+0
+
+Q
+
+☆
+
+co
+
+
+
+0
+
+0
+
+m
+
+A
+
+3
+
+3
+
+□
+
+o
+
+
+FIGURE 124. Bongard problem 33. [From M. Bongard, Pattern Recognition.]
+
+
+jump to the idea that Class I boxes contain "pointy" shapes. Class II boxes contain
+"smooth" ones. But on closer inspection, this is wrong. Nevertheless, there is a
+worthwhile insight here, and one can try to push it further, by sliding around in the
+network of concepts beginning at "pointy". It is close to the concept "acute",
+which is precisely the distinguishing feature of Class I. Thus one of the main
+functions of the concept network is to allow early wrong ideas to be modified
+slightly, to slip into variations which may be correct.
+
+Slippage and Tentativity
+
+Related to this notion of slipping between closely related terms is the notion of
+seeing a given object as a variation on another object. An excellent example has
+been mentioned already-that of the "circle with three indentations", where in fact
+there is no circle at all. One has to be able to bend concepts, when it is appropriate.
+Nothing should be absolutely rigid. On
+
+
+Artificial lntelligence:Prospects
+
+
+654
+
+
+
+
+
+the other hand, things shouldn't be so wishy-washy that nothing has any meaning
+at all, either. The trick is to know when and how to slip one concept into another.
+
+An extremely interesting set of examples where slipping from one description
+to another is the crux of the matter is given in Bongard problems 85-87 (Fig. 125).
+BP 85 is rather trivial. Let us assume that our program identifies "line segment" in
+its preprocessing stage. It is relatively simple for it then to count line segments and
+arrive at the difference
+
+FIGURE 125 .Bongard problems 85-87. I From M. Bongard, Pattern Recognition .]
+
+
+K
+
+ZL
+
+
+
+O
+
+
+Y
+
+
+Y
+
+*
+
+Y
+
+A
+
+85
+
+
+Y
+
+Y
+
+hv
+
+
+Y
+
+
+
+Y
+
+
+45
+
+&
+
+YJ
+
+y
+
+86 .
+
+\Y
+
+
+K
+
+X
+
+
+Y
+
+Y
+
+T
+
+H/
+
+
+-F
+
+0
+
+□
+
+L
+
+87 _
+
+H
+
+Y
+
+
+Artificial lntelligence:Prospects
+
+
+655
+
+
+
+
+
+
+
+
+
+
+between Class I and Class II in BP 85. Now it goes on to BP 86. A general
+heuristic which it uses is to try out recent ideas which have worked. Successful
+repetition of recent methods is very common in the real world, and Bongard does
+not try to outwit this kind of heuristic in his collection-in fact, he reinforces it,
+fortunately. So we plunge right into problem 86 with two ideas ("count" and "line
+segment") fused into one: "count line segments". But as it happens, the trick of BP
+86 is to count line trains rather than line segments, where "line train" means an
+end-to-end concatenation of (one or more) line segments. One way the program
+might figure this out is if the concepts "line train" and "line segment" are both
+known, and are close in the concept network. Another way is if it can invent the
+concept of "line train"-a tricky proposition, to say the least.
+
+Then comes BP 87, in which the notion of "line segment" is further played
+with. When is a line segment three line segments? (See box II-A.) The program
+must be sufficiently flexible that it can go back and forth between such different
+representations for a given part of a drawing. It is wise to store old representations,
+rather than forgetting them and perhaps having to reconstruct them, for there is no
+guarantee that a newer representation is better than an old one. Thus, along with
+each old representation should be stored some of the reasons for liking it and
+disliking it. (This begins to sound rather complex, doesn't it?)
+
+Meta- Descriptions
+
+Now we come to another vital part of the recognition process, and that has to do
+with levels of abstraction and meta-descriptions. For this let us consider BP 91
+(Fig. 121) again. What kind of template could be constructed here? There is such
+an amount of variety that it is hard to know where to begin. But this is in itself a
+clue! The clue says, namely, that the class distinction very likely exists on a higher
+level of abstraction than that of geometrical description. This observation clues the
+program that it should construct descriptions of descriptions -that is, meta¬
+descriptions. Perhaps on this second level some common feature will emerge; and
+if we are lucky, we will discover enough commonality to guide us towards the
+formulation of a template for the meta-descriptions! So we plunge ahead without a
+template, and manufacture descriptions for various boxes; then, once these
+descriptions have been made, we describe them. What kinds of slot will our
+template for meta-descriptions have? Perhaps these, among others:
+
+concepts used: —
+
+recurring concepts-:
+
+names of slots:-
+
+filters used:—
+
+There are many other kinds of slots which might be needed in metadescriptions,
+but this is a sample. Now suppose we have described box I-E of BP 91. Its
+(template-less) description might look like this:
+
+
+Artificial Intelligence:Prospects
+
+
+656
+
+
+
+
+
+horizontal line segment
+
+vertical line segment mounted on the horizontal line segment
+vertical line segment mounted on the horizontal line segment
+vertical line segment mounted on the horizontal line segment
+
+Of course much information has been thrown out: the fact that the three vertical
+lines are of the same length, are spaced equidistantly, etc. But it is plausible that
+the above description would be made. So the meta description might look like this:
+
+concepts used: vertical-horizontal, line segment, mounted on
+
+repetitions in description: 3 copies of "vertical line segment mounted on the
+
+horizontal line segment"
+
+names of slots-
+
+filters used:-
+
+Not all slots of the meta-description need be filled in; information can be thrown
+away on this level as well as on the Just- plain-description" level.
+
+‘Now if we were to make a description for any of the other boxes of Class I, and
+then a meta-description of it, we would wind up filling the slot "repetitions in
+description" each time with the phrase "3 copies of ..." The sameness-detector
+would notice this, and pick up three-ness as a salient feature, on quite a high level
+of abstraction, of the boxes of Class I. Similarly, four-ness would be recognized,
+via the method of metadescriptions, as the mark of Class II.
+
+Flexibility Is Important
+
+Now you might object that in this case, resorting to the method of
+metadescriptions is like shooting a fly with an elephant gun, for the three-ness
+versus four-ness might as easily have shown up on the lower level if we had
+constructed our descriptions slightly differently. Yes, true-but it is important to
+have the possibility of solving these problems by different routes. There should be
+a large amount of flexibility in the program; it should not be doomed if,
+malaphorically speaking, it "barks up the wrong alley" for a while. (The amusing
+term "malaphor" was coined by the newspaper columnist Lawrence Harrison; it
+means a cross between a malapropism and a metaphor. It is a good example of
+"recombinant ideas".) In any case, I wanted to illustrate the general principle that
+says: When it is hard to build a template because the preprocessor finds too much
+diversity, that should serve as a clue that concepts on a higher level of abstraction
+are involved than the preprocessor knows about.
+
+Focusing and Filtering
+
+Now let us deal with another question: ways to throw information out. This
+involves two related notions, which I call "focusing" and "filtering". Focus-
+
+
+Artificial Intelligence:Prospects
+
+
+657
+
+
+
+
+
+df
+
+CD
+
+
+
+
+A
+
+a
+
+
+
+
+
+5/
+
+o
+
+D
+
+in
+
+in
+
+<0
+
+O
+
+
+FIGURE 126. Bongarci problem 55. /From M. Bongcird, Pattern Recognition.]
+
+
+□ o
+
+>
+
+w
+
+
+°A
+
+D o
+
+
+o
+
+o
+
+
+u 0 v
+
+D°
+
+%
+
+
+22
+
+96
+
+93
+
+
+FIGURE 127. Bongarci problem 22. /From M. Bongarci, Pattern Recognition.]
+
+ing involves making a description whose focus is some part of the drawing in the
+box, to the exclusion of everything else. Filtering involves making a description
+which concentrates on some particular way of viewing the contents of the box, and
+deliberately ignores all other aspects. Thus they are complementary: focusing has
+to do with objects (roughly, nouns), and filtering has to do with concepts (roughly,
+adjectives). For an example of focusing, let's look at BP 55 (Fig. 126). Here, we
+focus on the indentation and the little circle next to it, to the exclusion of the
+everything else in the box. BP 22 (Fig. 127) presents an example of filtering. Here,
+we must filter out every concept but that of size. A combination of focusing and
+filtering is required to solve problem BP 58 (Fig. 128).
+
+One of the most important ways to get ideas for focusing and filtering is by
+another sort of "focusing": namely, by inspection of a single particularly simple
+box-say one with as few objects in it as possible. It can be
+
+
+Artificial lntelligence:Prospects
+
+
+658
+
+
+
+
+
+
+
+
+
+
+
+
+A m
+
+■
+
+
+■ □
+
+
+
+A ■
+
+■
+
+□
+
+■ □
+
+■ A
+
+
+■ □
+
+
+
+FIGURE 128. Bongard problem 58. [From M. Bongard, Pattern Recognition.]
+
+
+
+FIGURE 129. Bongard problem 61. /From M. Bongard, Pattern Recognition.J
+
+extremely helpful to compare the starkest boxes from the two Classes. But how
+can you tell which boxes are stark until you have descriptions for them? Well, one
+way of detecting starkness is to look for a box with a minimum of the features
+provided by the preprocessor. This can be done very early, for it does not require a
+pre-existing template; in fact, this can be one useful way of discovering features to
+build into a template. BP 61 (Fig. 129) is an example where that technique might
+quickly lead to a solution.
+
+Science and the World of Bongard Problems
+
+One can think of the Bongard-problem world as a tiny place where "science" is
+done-that is, where the purpose is to discern patterns in the world. As patterns are
+sought, templates are made, unmade, and remade;
+
+
+Artificial lntelligence:Prospects
+
+
+659
+
+
+
+
+
+
+
+FIGURE 130. Bongard problems 70-71. [From M. Bongord, Pattern
+Recognition .]
+
+slots are shifted from one level of generality to another: filtering and focusing are
+done; and so on. There are discoveries on all levels of complexity. The Kuhnian
+theory that certain rare events called "paradigm shifts" mark the distinction
+between "normal" science and "conceptual revolutions" does not seem to work,
+for we can see paradigm shifts happening all throughout the system, all the time.
+The fluidity of descriptions ensures that paradigm shifts will take place on all
+scales.
+
+Of course, some discoveries are more "revolutionary" than others, because
+they have wider effects. For instance, one can make the discovery that problems
+70 and 71 (Fig. 130) are ’’the same problem", when looked at on a sufficiently
+abstract level. The key observation is that both involve depth-2 versus depth-l
+nesting. This is a new level of discovery that can he made about Bongard
+problems. There is an even higher level, concerning the collection as a whole. If
+someone has never seen the collection, it can be a good puzzle just to figure out
+what it is. To figure it out is a revolutionary insight, but it must be pointed out that
+the mechanisms of thought which allow such a discovery to be made are no
+different from those which operate in the solution of a single Bongard problem.
+
+
+Artificial I ntell igence:Prospects
+
+
+660
+
+
+
+
+
+
+
+
+
+
+
+By the same token, real science does not divide up into "normal" periods versus
+"conceptual, revolutions"; rather, paradigm shifts pervade-there are just bigger and
+smaller ones, paradigm shifts on different levels. The recursive plots of INT and
+Gplot (Figs. 32 and 34) provide a geometric model for this idea: they have the same
+structure full of discontinuous jumps on every level, not just the top level-only the
+lower the level, the smaller the jumps
+
+Connections to Other Types of Thought
+
+To set this entire program somewhat in context, let me suggest two ways in which it is
+related to other aspects of cognition. Not only does it depend on other aspects of
+cognition, but also they in turn depend on it. First let me comment on how it depends on
+other aspects of cognition. The intuition which is required for knowing when it makes
+sense to blur distinctions, to try redescriptions, to backtrack, to shift levels, and so forth,
+is something which probably comes only with much experience in thought in general.
+Thus it would be very hard to define heuristics for these crucial aspects of the program.
+Sometimes one's experience with real objects in the world has a subtle effect on how one
+describes or redescribes boxes. For instance, who can say how much one's familiarity
+with living trees helps one to solve BP 70% It is very doubtful that in humans, the
+subnetwork of concepts relevant to these puzzles can be easily separated out from the
+whole network. Rather, it is much more likely that one's intuitions gained from seeing
+and handling real objects-combs, trains, strings, blocks, letters, rubber bands, etc., etc.-
+play an invisible but significant guiding role in the solution of these puzzles.
+
+Conversely, it is certain that understanding real-world situations heavily depends
+on visual imagery and spatial intuition, so that having a powerful and flexible way of
+representing patterns such as these Bongard patterns can only contribute to the general
+efficiency of thought processes.
+
+It seems to me that Bongard's problems were worked out with great care, and that they
+have a quality of universality to them, in the sense that each one has a unique correct
+answer. Of course one could argue with this and say that what we consider "correct"
+depends in some deep way on our being human, and some creatures from some other star
+system might disagree entirely. Not having any concrete evidence either way, I still have
+a certain faith that Bongard problems depend on a sense of simplicity which is not just
+limited to earthbound human beings. My earlier comments about the probable importance
+of being acquainted with such surely earth-limited objects as combs, trains, rubber bands,
+and so on, are not in conflict with the idea that our notion of simplicity is universal, for
+what matters is not any of these individual objects, but the fact that taken together they
+span a wide space. And it seems likely that any other civilization would have as vast a
+repertoire of artifacts and natural objects and varieties of experience on which to draw as
+we do. So I believe that the skill of solving Bongard
+
+
+Artificial Intelligence:Prospects
+
+
+661
+
+
+
+problems lies very close to the core of "pure" intelligence, if there is such a thing.
+Therefore it is a good place to begin if one wants to investigate the ability to discover
+"intrinsic meaning" in patterns or messages. Unfortunately we have reproduced only a
+small selection of his stimulating collection. I hope that many readers will acquaint
+themselves with the entire collection, to be found in his book (see Bibliography).
+
+Some of the problems of visual pattern recognition which we human beings seem
+to have completely "flattened" into our unconscious are quite amazing. They include:
+
+recognition of faces (invariance of faces under age change, expression change,
+lighting change, distance change, angle change, etc.)
+
+recognition of hiking trails in forests and mountains-somehow this has always
+impressed me as one of our most subtle acts of pattern recognition-and yet
+animals can do it, too
+
+reading text without hesitation in hundreds if not thousands of different typefaces
+
+Message-Passing Languages, Frames, and Symbols
+
+One way that has been suggested for handling the complexities of pattern recognition and
+other challenges to A1 programs is the so-called "actor" formalism of Carl Hewitt (similar
+to the language "Smalltalk", developed by Alan Kay and others), in which a program is
+written as a collection of interacting actors, which can pass elaborate messages back and
+forth among themselves. In a way, this resembles a heterarchical collection of procedures
+which can call each other. The major difference is that where procedures usually only
+pass a rather small number of arguments back and forth, the messages exchanged by
+actors can be arbitrarily long and complex.
+
+Actors with the ability to exchange messages become somewhat autonomous
+agents-in fact, even like autonomous computers, with messages being somewhat like
+programs. Each actor can have its own idiosyncratic way of interpreting any given
+message; thus a message's meaning will depend on the actor it is intercepted by. This
+comes about by the actor having within it a piece of program which interprets messages;
+so there may be as many interpreters as there are actors. Of course, there may be many
+actors with identical interpreters; in fact, this could be a great advantage, just as it is
+extremely important in the cell to have a multitude of identical ribosomes floating
+throughout the cytoplasm, all of which will interpret a message-in this case, messenger
+RNA-in one and the same way.
+
+It is interesting to think how one might merge the frame-notion with the actor-notion. Let
+us call a frame with the capability of generating and interpreting complex messages a
+symbol :
+
+
+frame + actor = symbol
+
+
+Artificial Intelligence:Prospects
+
+
+662
+
+
+
+We now have reached the point where we are talking about ways or implementing those
+elusive active symbols of Chapters XI and XII; henceforth in this Chapter, "symbol" will
+have that meaning. By the way, don't feel dumb if you don't immediately see just how
+this synthesis is to be made. It is not clear, though it is certainly one of the most
+fascinating directions to go in AI. Furthermore, it is quite certain that even the best
+synthesis of these notions will turn out to have much less power than the actual symbols
+of human minds. In that sense, calling these frame-actor syntheses "symbols" is
+premature, but it is an optimistic way of looking at things.
+
+Let us return to some issues connected with message passing. Should each
+message be directed specifically at a target symbol, or should it be thrown out into the
+grand void, much as mRNA is thrown out into the cytoplasm, to seek its ribosome? If
+messages have destinations, then each symbol must have an address, and messages for it
+should always be sent to that address. On the other hand, there could be one central
+receiving dock for messages, where a message would simply sit until it got picked up by
+some symbol that wanted it. This is a counterpart to General Delivery. Probably the best
+solution is to allow both types of message to exist; also to have provisions for different
+classes of urgency-special delivery, first class, second class, and so on. The whole postal
+system provides a rich source of ideas for message-passing languages, including such
+curios as self addressed stamped envelopes (messages whose senders want answers
+quickly), parcel post (extremely long messages which can be sent some very slow way),
+and more. The telephone system will give you more inspiration when you run out of
+postal-system ideas.
+
+
+Enzymes and AI
+
+Another rich source of ideas for message passing-indeed, for information processing in
+general-is, of course, the cell. Some objects in the cell are quite comparable to actors-in
+particular, enzymes. Each enzyme's active site acts as a filter which only recognizes
+certain kinds of substrates (messages). Thus an enzyme has an "address", in effect. The
+enzyme is "programmed" (by virtue of its tertiary structure) to carry out certain
+operations upon that "message", and then to release it to the world again. Now in this
+way, when a message is passed from enzyme to enzyme along a chemical pathway, a lot
+can be accomplished. We have already described the elaborate kinds of feedback
+mechanisms which can take place in cells (either by inhibition or repression). These kinds
+of mechanisms show that complicated control of processes can arise through the kind of
+message passing that exists in the cell.
+
+One of the most striking things about enzymes is how they sit around idly,
+waiting to be triggered by an incoming substrate. Then, when the substrate arrives,
+suddenly the enzyme springs into action, like a Venus's flytrap. This kind of "hair-
+trigger" program has been used in AI, and goes by the name of demon. The important
+thing here is the idea of having many different "species" of triggerable subroutines just
+lying around waiting to
+
+
+Artificial Intelligence:Prospects
+
+
+663
+
+
+
+be triggered. In cells, all the complex molecules and organelles are built up, simple step
+by simple step. Some of these new structures are often enzymes themselves, and they
+participate in the building of new enzymes, which in turn participate in the building of yet
+other types of enzyme, etc. Such recursive cascades of enzymes can have drastic effects
+on what a cell is doing. One would like to see the same kind of simple step-by-step
+assembly process imported into AI, in the construction of useful subprograms. For
+instance, repetition has a way of burning new circuits into our mental hardware, so that
+oft-repeated pieces of behavior become encoded below the conscious level. It would be
+extremely useful if there were an analogous way of synthesizing efficient pieces of code
+which can carry out the same sequence of operations as something which has been
+learned on a higher level of "consciousness". Enzyme cascades may suggest a model for
+how this could be done. (The program called "HACKER", written by Gerald Sussman,
+synthesizes and debugs small subroutines in a way not too much unlike that of enzyme
+cascades.)
+
+The sameness-detectors in the Bongard problem-solver (Sams) could be
+implemented as enzyme-like subprograms. Like an enzyme, a Sam would meander about
+somewhat at random, bumping into small data structures here and there. Upon filling its
+two "active sites" with identical data structures, the Sam would emit a message to other
+parts (actors) of the program. As long as programs are serial, it would not make much
+sense to have several copies of a Sam, but in a truly parallel computer, regulating the
+number of copies of a subprogram would be a way of regulating the expected waiting¬
+time before an operation gets done, just as regulating the number of copies of an enzyme
+in a cell regulates how fast that function gets performed. And if new Sams could be
+synthesized, that would be comparable to the seepage of pattern detection into lower
+levels of our minds.
+
+
+Fission and Fusion
+
+Two interesting and complementary ideas concerning the interaction of symbols are
+"fission" and "fusion". Fission is the gradual divergence of a new symbol from its parent
+symbol (that is, from the symbol which served as a template off of which it was copied).
+Fusion is what happens when two (or more) originally unrelated symbols participate in a
+"joint activation", passing messages so tightly back and forth that they get bound together
+and the combination can thereafter be addressed as if it were a single symbol. Fission is a
+more or less inevitable process, since once a new symbol has been "rubbed off of an old
+one, it becomes autonomous, and its interactions with the outside world get reflected in
+its private internal structure; so what started out as a perfect copy will soon become
+imperfect, and then slowly will become less and less like the symbol off of which it was
+"rubbed". Fusion is a subtler thing. When do two concepts really become 'one? Is there
+some precise instant when a fusion takes place?
+
+
+Artificial Intelligence:Prospects
+
+
+664
+
+
+
+This notion of joint activations opens up a Pandora's box of questions. For
+instance, how much coo we hear "dough" and "nut" when we say "doughnut"? Does a
+German who thinks of gloves ("Handschuhe") hear "hand-shoes" or not? How about
+Chinese people, whose word "dong-xi" ("East-West") means "thing"? It is a matter of
+some political concern, too, since some people claim that words like "chairman" are
+heavily charged with undertones of the male gender. The degree to which the parts
+resonate inside the whole probably varies from person to person and according to
+circumstances.
+
+The real problem with this notion of "fusion" of symbols is that it is very hard to
+imagine general algorithms which will create meaningful new symbols from colliding
+symbols. It is like two strands of DNA which come together. How do you take parts from
+each and recombine them into a meaningful and viable new strand of DNA which codes
+for an individual of the same species? Or a new kind of species? The chance is
+infinitesimal that a random combination of pieces of DNA will code for anything that
+will survive-something like the chance that a random combination of words from two
+books will make another book. The chance that recombinant DNA will make sense on
+any level but the lowest is tiny, precisely because there are so many levels of meaning in
+DNA. And the same goes for "recombinant symbols".
+
+Epigenesis of the Crab Canon
+
+I think of my Dialogue Crab Canon as a prototype example where two ideas collided in
+my mind, connected in a new way, and suddenly a new kind of verbal structure came
+alive in my mind. Of course I can still think about musical crab canons and verbal
+dialogues separately-they can still be activated independently of each other; but the fused
+symbol for crab canonical dialogues has its own characteristic modes of activation, too.
+To illustrate this notion of fusion or "symbolic recombination" in some detail, then, I
+would like to use the development of my Crab Canon as a case study, because, of course,
+it is very familiar to me, and also because it is interesting, yet typical of how far a single
+idea can be pushed. I will recount it in stages named after those of meiosis, which is the
+name for cell division in which "crossing-over", or genetic recombination, takes place-the
+source of diversity in evolution.
+
+PROPHASE: I began with a rather simple idea-that a piece of music, say a canon,
+could be imitated verbally. This came from the observation that, through a shared abstract
+form, a piece of text and a piece of music may be connected. The next step involved
+trying to realize some of the potential of this vague hunch; here, I hit upon the idea that
+"voices" in canons can be mapped onto "characters" in dialogues-still a rather obvious
+idea.
+
+Then I focused down onto specific kinds of canons, and remembered that there
+was a crab canon in the Musical Offering. At that time, I had just
+
+
+Artificial Intelligence:Prospects
+
+
+665
+
+
+
+begun writing Dialogues, and there were only two characters: Achilles and the Tortoise.
+Since the Bach crab canon has two voices, this mapped perfectly: Achilles should be one
+voice, the Tortoise the other, with the one doing forwards what the other does backwards.
+But here I was faced with a problem: on what level should the reversal take place? The
+letter level? The word level? The sentence level? After some thought, I concluded that
+the "dramatic line" level would be most appropriate.
+
+Now that the "skeleton" of the Bach crab canon had been transplanted, at least in
+plan, into a verbal form, there was just one problem. When the two voices crossed in the
+middle, there would be a short period of extreme repetition: an ugly blemish What to do
+about it? Here, a strange thing happened, a kind of level-crossing typical of creative acts:
+the word "crab" in "crab canon" flashed into my mind, undoubtedly because of some
+abstract shared quality with the notion of "tortoise"-and immediately I realized that at the
+dead center, I could block the repetitive effect, by inserting one special line, said by a
+new character: a Crab! This is how, in the "prophase" of the Crab Canon, the Crab was
+conceived: at the crossing over of Achilles and the Tortoise. (See Fig. 131.)
+
+
+
+A—c—li—i—I—1—c—Ss
+
+
+I -o-r-l-o-i-5-e
+
+
+FIGURE 131. A schematic diagram of the Dialogue Crab Canon.
+
+METAPHASE: This was the skeleton of my Crab Canon. I then entered the
+second stage-the "metaphase"-in which I had to fill in the flesh, which was of course an
+arduous task. I made a lot of stabs at it, getting used to the way in which pairs of
+successive lines had to make sense when read from either direction, and experimenting
+around to see what kinds of dual meanings would help me in writing such a form (e.g.,
+"Not at all"). There were two early versions both of which were interesting, but weak. I
+abandoned work on the book for over a year, and when I returned to the Crab Canon, I
+had a few new ideas. One of them was to mention a Bach canon inside it. At first my plan
+was to mention the "Canon per augmentationem, contrario motu", from the Musical
+Offering (Sloth Canon, as I call it). But that started to seem a little silly, so reluctantly I
+decided that inside my Crab Canon, I could talk about Bach’s own Crab Canon instead.
+Actually, this was a crucial turning point, but I didn't know it then.
+
+Now if one character was going to mention a Bach piece, wouldn't it be awkward
+for the other to say exactly the same thing in the corresponding place? Well, Escher was
+playing a similar role to Bach in my thoughts and my book, so wasn't there some way of
+just slightly modifying the line so that it would refer to Escher? After all, in the strict art
+of canons, note-perfect imitation is occasionally foregone for the sake of elegance or
+beauty. And
+
+
+Artificial lntelligence:Prospects
+
+
+666
+
+
+
+no sooner did that idea occur to me than the picture Day and Night (Fig. 49) popped into
+my mind. "Of course!" I thought, "It is a sort of pictorial crab canon, with essentially two
+complementary voices carrying the same theme both leftwards and rightwards, and
+harmonizing with each other!" Here again was the notion of a single "conceptual
+skeleton" being instantiated in two different media-in this case, music and art. So I let the
+Tortoise talk about Bach, and Achilles talk about Escher, in parallel language; certainly
+this slight departure from strict imitation retained the spirit of crab cano.is.
+
+At this point, I began realizing that something marvelous was happening namely,
+the Dialogue was becoming self-referential, without my even having intended it! What's
+more, it was an indirect self-reference, in that the characters did not talk directly about
+the Dialogue they were in, but rather about structures which were isomorphic to it (on a
+certain plane of abstraction). To put it in the terms I have been using, my Dialogue now
+shared a "conceptual skeleton" with Godel’s G, and could therefore be mapped onto G in
+somewhat the way that the Central Dogma was, to create in this case a "Central
+Crabmap". This was most exciting to me, since out of nowhere had come an esthetically
+pleasing unity of Godel, Escher, and Bach.
+
+ANAPHASE: The next step was quite startling. I had had Caroline MacGillavry's
+monograph on Escher's tessellations for years, but one day, as I flipped through it, my
+eye was riveted to Plate 23 (Fig. 42), for I saw it in a way I had never seen it before: here
+was a genuine crab canon-crab-like in both form and content! Escher himself had given
+the picture no title, and since he had drawn similar tessellations using many other animal
+forms, it is probable that this coincidence of form and content was just something which I
+had noticed. But fortuitous or not, this untitled plate was a miniature version of one main
+idea of my book: to unite form and content. So with delight I christened it Crab Canon,
+substituted it for Day and Night, and modified Achilles' and the Tortoise's remarks
+accordingly.
+
+Yet this was not all. Having become infatuated with molecular biology, one day I
+was perusing Watson's book in the bookstore, and in the index saw the word
+"palindrome". When I looked it up, I found a magical thing: crab-canonical structures in
+DNA. Soon the Crab's comments had been suitably modified to include a short remark to
+the effect that he owed his predilection for confusing retrograde and forward motion to
+his genes.
+
+TELOPHASE: The last step came months later, when, as I was talking about the
+picture of the crab-canonical section of DNA (Fig. 43), 1 saw that the 'A', 'T\ 'C' of
+Adenine, Thymine, Cytosine coincided- mirabile dictu-wdh the 'A', 'T', 'C' of Achilles,
+Tortoise, Crab; moreover, just as Adenine and Thymine are paired in DNA, so are
+Achilles and the Tortoise paired in the Dialogue. I thought for a moment and, in another
+of those level-crossings, saw that 'G', the letter paired with 'C' in DNA, could stand for
+"Gene". Once again, I jumped back to the Dialogue, did a little surgery on the Crab's
+speech to reflect this new discovery, and now I had a mapping between the DNA's
+structure, and the Dialogue's structure. In that sense, the DNA could be said to be a
+genotype coding for a phenotype: the
+
+
+Artificial Intelligence:Prospects
+
+
+667
+
+
+
+Structure of the Dialogue. This final touch dramatically heightened the self-reference,
+and gave the Dialogue a density of meaning which I had never anticipated.
+
+Conceptual Skeletons and Conceptual Mapping
+
+That more or less summarizes the epigenesis of the Crab Canon. The whole process can
+be seen as a succession of mappings of ideas onto each other, at varying levels of
+abstraction. This is what I call conceptual mapping, and the abstract structures which
+connect up two different ideas are conceptual skeletons. Thus, one conceptual skeleton is
+that of the abstract notion of a crab canon:
+
+a structure having two parts which do the same thing,
+only moving in opposite directions.
+
+This is a concrete geometrical image which can be manipulated by the mind
+almost as a Bongard pattern. In fact, when I think of the Crab Canon today, I visualize it
+as two strands which cross in the middle, where they are joined by a "knot” (the Crab's
+speech). This is such a vividly pictorial image that it instantaneously maps, in my mind,
+onto a picture of two homologous chromosomes joined by a centromere in their middle,
+which is an image drawn directly from meiosis, as shown in Figure 132.
+
+
+
+FIGURE 132.
+
+In fact, this very image is what inspired me to cast the description of the Crab Canon's
+evolution in terms of meiosis-which is itself, of course, vet another example of
+conceptual mapping.
+
+
+Recombinant Ideas
+
+There are a variety of techniques of fusion of two symbols. One involves lining the two
+ideas up next to each other (as if ideas were linear!), then judiciously choosing pieces
+from each one, and recombining them in a new symbol. This strongly recalls genetic
+recombination. Well, what do chromosomes exchange, and how do they do it? They
+exchange genes. What in a symbol is comparable to a gene? If symbols have frame-like
+slots, then slots, perhaps. But which slots to exchange, and why? Here is where the
+crabcanonical fusion may offer some ideas. Mapping the notion of "musical crab canon"
+onto that of "dialogue" involved several auxiliary mappings; in
+
+
+Artificial lntelligence:Prospects
+
+
+668
+
+
+
+fact it induced them. That is, once it had been decided that these two notions ,ere to be
+fused, it became a matter of looking at them on a level where analogous parts emerged
+into view, then going ahead and mapping the parts onto each other, and so on,
+recursively, to any level that was found desirable. Here, for instance, "voice" and
+"character" emerged as corresponding slots when "crab canon" and "dialogue" were
+viewed abstractly. Where did these abstract views come from, though? This is at the crux
+of the mapping-problem-where do abstract views come from? How do you make abstract
+views of specific notions?
+
+Abstractions, Skeletons, Analogies
+
+A view which has been abstracted from a concept along some dimension is what I call a
+conceptual skeleton. In effect, we have dealt with conceptual skeletons all along, without
+often using that name. For instance, many of the ideas concerning Bongard problems
+could be rephrased using this terminology. It is always of interest, and possibly of
+importance, when two or more ideas are discovered to share a conceptual skeleton. An
+example is the bizarre set of concepts mentioned at the beginning of the Contrafactus: a
+Bicyclops, a tandem unicycle, a teeter-teeter, the game of ping-ping, a one-way tie, a
+two-sided Mobius strip, the "Bach twins", a piano concerto for two left hands, a one-
+voice fugue, the act of clapping with one hand, a two-channel monaural phonograph, a
+pair of eighth-backs. All of these ideas are "isomorphic" because they share this
+conceptual skeleton:
+
+a plural thing made singular and re-pluralized wrongly.
+
+Two other ideas in this book which share that conceptual skeleton are (1) the Tortoise's
+solution to Achilles' puzzle, asking for a word beginning and ending in "HE" (the
+Tortoise's solution being the pronoun "HE", which collapses two occurrences into one),
+and (2) the Pappus-Gelernter proof of the Pons As' norum Theorem, in which one triangle
+is reperceived as two. Incidentally, these droll concoctions might be dubbed "demi-
+doublets".
+
+A conceptual skeleton is like a set of constant features (as distinguished from
+parameters or variables)-features which should not be slipped in a subjunctive instant
+replay or mapping-operation. Having no parameters or variables of its own to vary, it can
+be the invariant core of several different ideas. Each instance of it, such as "tandem
+unicycle", does have layers of variability and so can be "slipped" in various ways.
+
+Although the name "conceptual skeleton" sounds absolute and rigid, actually
+there is a lot of play in it. There can be conceptual skeletons on several different levels of
+abstraction. For instance, the "isomorphism" between Bongard problems 70 and 71,
+already pointed out, involves a higher-level conceptual skeleton than that needed to solve
+either problem in isolation.
+
+
+Artificial Intelligence:Prospects
+
+
+669
+
+
+
+Multiple Representations
+
+
+Not only must conceptual skeletons exist on different levels of abstraction; also, they
+must exist along different conceptual dimensions. Let us take the following sentence as
+an example:
+
+"The Vice President is the spare tire on the automobile of government."
+
+How do we understand what it means (leaving aside its humor, which is of course a vital
+aspect)? If you were told, "See our government as an automobile" without any prior
+motivation, you might come up with any number of correspondences: steering wheel =
+president, etc.. What are checks and balances? What are seat belts? Because the two
+things being mapped are so different, it is almost inevitable that the mapping will involve
+functional aspects. Therefore, you retrieve from your store of conceptual skeletons
+representing parts of automobiles, only those having to do with function, rather than, say,
+shape. Furthermore, it makes sense to work at a pretty high level of abstraction, where
+"function" isn't taken in too narrow a context. Thus, of the two following definitions of
+the function of a spare tire: (1) "replacement for a flat tire", and (2) "replacement for a
+certain disabled part of a car", certainly the latter would be preferable, in this case. This
+comes simply from the fact that an auto and a government are so different that they have
+to be mapped at a high level of abstraction.
+
+Now when the particular sentence is examined, the mapping gets forced in one
+respect-but it is not an awkward way, by any means. In fact, you already have a
+conceptual skeleton for the Vice President, among many others, which says,
+"replacement for a certain disabled part of government". Therefore the forced mapping
+works comfortably. But suppose, for the sake of contrast, that you had retrieved another
+conceptual skeleton for "spare tire"-say, one describing its physical aspects. Among other
+things, it might say that a spare tire is "round and inflated". Clearly, this is not the right
+way to go. (Or is it? As a friend of mine pointed out, some Vice Presidents are rather
+portly, and most are quite inflated!)
+
+
+Ports of Access
+
+One of the major characteristics of each idiosyncratic style of thought is how new
+experiences get classified and stuffed into memory, for that defines the "handles" by
+which they will later be retrievable. And for events, objects, ideas, and so on-for
+everything that can be thought about-there is a wide variety of "handles". I am struck by
+this each time I reach down to turn on my car radio, and find, to my dismay, that it is
+already on! What has happened is that two independent representations are being used for
+the radio. One is "music producer", the other is "boredom reliever". I am aware that the
+music is on, but I am bored anyway, and before the two realizations have a chance to
+interact, my reflex to reach
+
+
+Artificial Intelligence:Prospects
+
+
+670
+
+
+
+down has been triggered. The same reaching-down reflex one day occurred just after I'd
+left the radio at a repair shop and was driving away, wanting to hear some music. Odd.
+Many other representations for the same object exist, such as
+
+shiny silver-knob haver
+overheating-problems haver
+lying-on-my-back-over-hump-to-fix thing
+buzz-maker
+slipping-dials object
+
+multidimensional representation example
+
+All of them can act as ports of access. Though they all are attached to my symbol for my
+car radio, accessing that symbol through one does not open up all the others. Thus it is
+unlikely that I will be inspired to remember lying on my back to fix the radio when I
+reach down and turn it on. And conversely, when I'm lying on my back, unscrewing
+screws, I probably won't think about the time I heard the Art of the Fugue on it. There are
+"partitions" between these aspects of one symbol, partitions that prevent my thoughts
+from spilling over sloppily, in the manner of free associations. My mental partitions are
+important because they contain and channel the flow of my thoughts.
+
+One place where these partitions are quite rigid is in sealing off words for the
+same thing in different languages. If the partitions were not strong, a bilingual person
+would constantly slip back and forth between languages, which would be very
+uncomfortable. Of course, adults learning two new languages at once often confuse
+words in them. The partitions between these languages are flimsier, and can break down.
+Interpreters are particularly interesting, since they can speak any of their languages as if
+their partitions were inviolable and yet, on command, they can negate those partitions to
+allow access to one language from the other, so they can translate. Steiner, who grew up
+trilingual, devotes several pages in After Babel to the intermingling of French, English,
+and German in the layers of his mind, and how his different languages afford different
+ports of access onto concepts.
+
+
+Forced Matching
+
+When two ideas are seen to share conceptual skeletons on some level of abstraction,
+different things can happen. Usually the first stage is that you zoom in on both ideas, and,
+using the higher-level match as a guide, you try to identify corresponding subideas.
+Sometimes the match can be extended recursively downwards several levels, revealing a
+profound isomorphism. Sometimes it stops earlier, revealing an analogy or similarity.
+And then there are times when the high-level similarity is so compelling that, even if
+there is no apparent lower-level continuation of the map, you just go ahead and make
+one: this is the forced match.
+
+
+Artificial Intelligence:Prospects
+
+
+671
+
+
+
+Forced matches occur every day in the political cartoons of newspapers: a
+political figure is portrayed as an airplane, a boat, a fish, the Mona Lisa; a government is
+a human, a bird, an oil rig; a treaty is a briefcase, a sword, a can of worms; on and on and
+on. What is fascinating is how easily we can perform the suggested mapping, and to the
+exact depth intended. We don't carry the mapping out too deeply or too shallowly.
+
+Another example of forcing one thing into the mold of another occurred when I
+chose to describe the development of my Crab Canon in terms of meiosis. This happened
+in stages. First, I noticed the common conceptual skeleton shared by the Crab Canon and
+the image of chromosomes joined by a centromere; this provided the inspiration for the
+forced match. Then I saw a high-level resemblance involving "growth", "stages", and
+"recombination". Then I simply pushed the analogy as hard as I could. Tentativity-as in
+the Bongard problem-solver-played a large role: I went forwards and backwards before
+finding a match which I found appealing.
+
+A third example of conceptual mapping is provided by the Central Dogmap. I
+initially noticed a high-level similarity between the discoveries of mathematical logicians
+and those of molecular biologists, then pursued it on lower levels until I found a strong
+analogy. To strengthen it further, I chose a Godel-numbering which imitated the Genetic
+Code. This was the lone element of forced matching in the Central Dogmap.
+
+Forced matches, analogies, and metaphors cannot easily be separated out.
+Sportscasters often use vivid imagery which is hard to pigeonhole. For instance, in a
+metaphor such as "The Rams [football team are spinning their wheels", it is hard to say
+just what image you are supposed to conjure up. Do you attach wheels to the team as a
+whole% Or to each player? Probably neither one. More likely, the image of wheels
+spinning in mud or snow simply flashes before you for a brief instant, and then in some
+mysterious way, just the relevant parts get lifted out and transferred to the team's
+performance. How deeply are the football team and the car mapped onto each other in the
+split second that you do this?
+
+
+Recap
+
+Let me try to tie things together a little. I have presented a number of related ideas
+connected with the creation, manipulation, and comparison of symbols. Most of them
+have to do with slippage in some fashion, the idea being that concepts are composed of
+some tight and some loose elements, coming from different levels of nested contexts
+(frames). The loose ones can be dislodged and replaced rather easily, which, depending
+on the circumstances, can create a "subjunctive instant replay", a forced match, or an
+analogy. A fusion of two symbols may result from a process in which parts of each
+symbol are dislodged and other parts remain.
+
+
+Artificial Intelligence:Prospects
+
+
+672
+
+
+
+Creativity and Randomness
+
+
+It is obvious that we are talking about mechanization of creativity. But the this not a
+contradiction in terms? Almost, but not really. Creativity s essence of that which is not
+mechanical. Yet every creative act is mechanical-it has its explanation no less than a case
+of the hiccups does. The mechanical substrate of creativity may be hidden from view, but
+it exists. Conversely, there is something unmechanical in flexible programs, even today.
+It may not constitute creativity, but when programs cease to be transparent to their
+creators, then the approach to creativity has begun.
+
+It is a common notion that randomness is an indispensable ingredient of creative
+acts. This may be true, but it does not have any bearing on the mechanizability-or rather,
+programmability!-of creativity. The world is a giant heap of randomness; when you
+mirror some of it inside your head, your head's interior absorbs a little of that
+randomness. The triggering patterns of symbols, therefore, can lead you down the most
+randomseeming paths, simply because they came from your interactions with a crazy,
+random world. So it can be with a computer program, too. Randomness is an intrinsic
+feature of thought, not something which has to be "artificially inseminated", whether
+through dice, decaying nuclei, random number tables, or what-have-you. It is an insult to
+human creativity to imply that it relies on such arbitrary sources.
+
+What we see as randomness is often simply an effect of looking at something
+symmetric through a "skew" filter. An elegant example was provided by Salviati's two
+ways of looking at the number it/4. Although the decimal expansion of 7r/4 is not literally
+random, it is as random as one would need for most purposes: it is "pseudorandom".
+Mathematics is full of pseudorandomness-plenty enough to supply all would-be creators
+for all time.
+
+Just as science is permeated with "conceptual revolutions" on all levels at all
+times, so the thinking of individuals is shot through and through with creative acts. They
+are not just on the highest plane; they are everywhere. Most of them are small and have
+been made a million times before-but they are close cousins to the most highly creative
+and new acts. Computer programs today do not yet seem to produce many small
+creations. Most of what they do is quite "mechanical" still. That just testifies to the fact
+that they are not close to simulating the way we think-but they are getting closer.
+
+Perhaps what differentiates highly creative ideas from ordinary ones is some
+combined sense of beauty, simplicity, and harmony. In fact, I have a favorite "meta¬
+analogy", in which I liken analogies to chords. The idea is simple: superficially similar
+ideas are often not deeply related; and deeply related ideas are often superficially
+disparate. The analogy to chords is natural: physically close notes are harmonically
+distant (e.g., E-F-G); and harmonically close notes are physically distant (e.g., G-E-B).
+Ideas that share a conceptual skeleton resonate in a sort of conceptual analogue to
+harmony; these harmonious "idea-chords" are often widely separated, as
+
+
+Artificial Intelligence:Prospects
+
+
+673
+
+
+
+measured on an imaginary "keyboard of concepts". Of course, it doesn't suffice to reach
+wide and plunk down any old way-you may hit a seventh or a ninth! Perhaps the present
+analogy is like a ninth-chord-wide but dissonant.
+
+Picking up Patterns on All Levels
+
+Bongard problems were chosen as a focus in this Chapter because when you study them,
+you realize that the elusive sense for patterns which we humans inherit from our genes
+involves all the mechanisms of representation of knowledge, including nested contexts,
+conceptual skeletons and conceptual mapping, slippability, descriptions and meta¬
+descriptions and their interactions, fission and fusion of symbols, multiple representations
+(along different dimensions and different levels of abstraction), default expectations, and
+more.
+
+These days, it is a safe bet that if some program can pick up patterns in one area,
+it will miss patterns in another area which, to us, are equally obvious. You may remember
+that I mentioned this back in Chapter 1, saying that machines can be oblivious to
+repetition, whereas people cannot. For instance, consider SHRDLU. If Eta Oin typed the
+sentence "Pick up a big red block and put it down" over and over again, SHRDLU would
+cheerfully react in the same way over and over again, exactly as an adding machine will
+print out "4" over and over again, if a human being has the patience to type "2+2" over
+and over again. Humans aren't like that; if some pattern occurs over and over again, they
+will pick it up. SHRDLU wasn't built with the potential for forming new concepts or
+recognizing patterns: it had no sense of over and overview.
+
+The Flexibility of Language
+
+SHRDLU's language-handling capability is immensely flexible-within limits. SHRDLU
+can figure out sentences of great syntactical complexity, or sentences with semantic
+ambiguities as long as-they can- be resolved by inspecting the data base-but it cannot
+handle "hazy" language. For instance, consider the sentence "How many blocks go on top
+of each other to make a steeple?" We understand it immediately, yet it does not make
+sense if interpreted literally. Nor is it that some idiomatic phrase has been used. "To go
+on top of each other" is an imprecise phrase which nonetheless gets the desired image
+across quite well to a human. Few people would be misled into visualizing a paradoxical
+setup with two blocks each of which is on top of the other-or blocks which are "going"
+somewhere or other.
+
+The amazing thing about language is how imprecisely we use it and still manage
+to get away with it. SHRDLU uses words in a "metallic" way, while people use them in a
+"spongy" or "rubbery" or even "Nutty-Puttyish" way. If words were nuts and bolts,
+people could make any bolt fit into any nut: they'd just squish the one into the other, as in
+some surrealistic
+
+
+Artificial Intelligence:Prospects
+
+
+674
+
+
+
+painting where everything goes soft. Language, in human hands, becomes almost like a
+fluid, despite, the coarse grain of its components.
+
+Recently, A1 research in natural language understanding has turned away
+somewhat from the understanding of single sentences in isolation, and more towards
+areas such as understanding simple children's stories. Here is a well-known children's
+joke which illustrates the open-endedness of real-life situations:
+
+A man took a ride in an airplane.
+
+Unfortunately, he fell out.
+
+Fortunately, he had a parachute on.
+
+Unfortunately, it didn't work.
+
+Fortunately, there was a haystack below him.
+
+Unfortunately, there was a pitchfork sticking out of it.
+
+Fortunately, he missed the pitchfork.
+
+Unfortunately, he missed the haystack.
+
+It can be extended indefinitely. To represent this silly story in a frame-based system
+would be extremely complex, involving jointly activating frames for the concepts of man,
+airplane, exit, parachute, falling, etc., etc.
+
+Intelligence and Emotions
+
+Or consider this tiny yet poignant story:
+
+Margie was holding tightly to the string of her beautiful new balloon. Suddenly, a
+gust of wind caught it. The wind carried it into a tree. The balloon hit a branch
+and burst. Margie cried and cried.'
+
+To understand this story, one needs to read many things between the lines. For instance:
+Margie is a little girl. This is a toy balloon with a string for a child to hold. It may not be
+beautiful to an adult, but in a child's eye, it is. She is outside. The "it" that the wind
+caught was the balloon. The wind did not pull Margie along with the balloon; Margie let
+go. Balloons can break on contact with any sharp point. Once they are broken, they are
+gone forever. Little children love balloons and can be bitterly disappointed when they
+break. Margie saw that her balloon was broken. Children cry when they are sad. "To cry
+and cry" is to cry very long and hard. Margie cried and cried because of her sadness at
+her balloon's breaking.
+
+This is probably only a small fraction of what is lacking at the surface level. A
+program must have all this knowledge in order to get at what is going on. And you might
+object that, even if it "understands" in some intellectual sense what has been said, it will
+never really understand, until it, too, has cried and cried. And when will a computer do
+that? This is the kind of humanistic point which Joseph Weizenbaum is concerned with
+making in his book Computer Power and Human Reason , and I think it is an important
+issue; in fact, a very, very deep issue. Unfortunately, many A1 workers at this time are
+unwilling, for various reasons, to take this sort of point
+
+
+Artificial Intelligence:Prospects
+
+
+675
+
+
+
+seriously, taut in some ways, those A1 workers are right: it is a little premature to think
+about computers crying; we must first think about rules for computers to deal with
+language and other things; in time, we'll find ourselves face to face with the deeper
+issues.
+
+
+AI Has Far to Go
+
+Sometimes it seems that there is such a complete absence of rule-governed behavior that
+human beings just aren't rule-governed. But this is an illusion-a little like thinking that
+crystals and metals emerge from rigid underlying laws, but that fluids or flowers don't.
+We'll come back to this question in the next Chapter.
+
+The process of logic itself working internally in the brain may be more analogous
+to a succession of operations with symbolic pictures, a sort of abstract analogue of
+the Chinese alphabet or some Mayan description of events-except that the
+elements are not merely words but more like sentences or whole stories with
+linkages between them forming a sort of meta- or super-logic with its own rules.'
+
+It is hard for most specialists to express vividly-perhaps even to remember-what
+originally sparked them to enter their field. Conversely, someone on the outside may
+understand a field's special romance and may be able to articulate it precisely. I think that
+is why this quote from Ulam has appeal for me, because it poetically conveys the
+strangeness of the enterprise of Al, and yet shows faith in it. And one must run on faith at
+this point, for there is so far to go!
+
+Ten Questions and Speculations
+
+To conclude this Chapter, I would like to present ten "Questions and Speculations" about
+Al. I would not make so bold as to call them "Answers"-these are my personal opinions.
+They may well change in some ways, as I learn more and as Al develops more. (In what
+follows, the term "Al program" means a program which is far ahead of today's programs;
+it means an "Actually Intelligent" program. Also, the words "program" and "computer"
+probably carry overly mechanistic connotations, but let us stick with them anyway.)
+
+Question: Will a computer program ever write beautiful music?
+
+Speculation: Yes, but not soon. Music is a language of emotions, and until programs
+have emotions as complex as ours, there is no way a program will write anything
+beautiful. There can be "forgeries” shallow imitations of the syntax of earlier music-
+but despite what one might think at first, there is much more to musical expression
+than can be captured in syntactical rules. There will be no new kinds of beauty
+turned up for a long time by computer music-composing programs. Let me carry this
+thought a little further. To think-and I have heard this suggested-that we might soon
+be able to command a preprogrammed mass-produced mail-order twenty-dollar
+desk-model "music box" to bring forth from its sterile circuitry pieces which Chopin
+or Bach might have written had they lived longer is a grotesque and shameful
+misestimation of the depth of the human spirit. A "program" which could produce
+
+
+Artificial Intelligence:Prospects
+
+
+676
+
+
+
+music as they did would have to wander around the world on its own, fighting its
+way through the maze of life and feeling every moment of it. It would have to
+understand the joy and loneliness of a chilly night wind, the longing for a cherished
+hand, the inaccessibility of a distant town, the heartbreak and regeneration after a
+human death. It would have to have known resignation and worldweariness, grief
+and despair, determination and victory, piety and awe. In it would have had to
+commingle such opposites as hope and fear, anguish and jubilation, serenity and
+suspense. Part and parcel of it would have to be a sense of grace, humor, rhythm, a
+sense of the unexpected-and of course an exquisite awareness of the magic of fresh
+creation. Therein, and therein only, lie the sources of meaning in music.
+
+Question: Will emotions be explicitly programmed into a machine?
+
+Speculation: No. That is ridiculous. Any direct simulation of emotions-PARRY, for
+example-cannot approach the complexity of human emotions, which arise
+indirectly from the organization of our minds. Programs or machines will acquire
+emotions in the same way: as by-products of their structure, of the way in which
+they are organized-not by direct programming. Thus, for example, nobody will
+write a "falling-in-love" subroutine, any more than they would write a "mistake¬
+making" subroutine. "Falling in love" is a description which we attach to a complex
+process of a complex system; there need be no single module inside the system
+which is solely responsible for it, however!
+
+Question: Will a thinking computer be able to add fast?
+
+Speculation: Perhaps not. We ourselves are composed of hardware which does fancy
+calculations but that doesn't mean that our symbol level, where "we" are, knows
+how to carry out the same fancy calculations. Let me put it this way: there's no way
+that you can load numbers into your own neurons to add up your grocery bill.
+Luckily for you, your symbol level (i.e., you) can't gain access to the neurons which
+are doing your thinking-otherwise you'd get addle-brained. To paraphrase Descartes
+again:
+
+
+"I think; therefore I have no access
+to the level where I sum."
+
+Why should it not be the same for an intelligent program? It mustn't be allowed to gain
+access to the circuits which are doing its thinking otherwise it'll get addle-CPU'd.
+Quite seriously, a machine that can pass the Turing test may well add as slowly as
+you or I do, and for
+
+
+Artificial Intelligence:Prospects
+
+
+677
+
+
+
+similar reasons. It will represent the number 2 not just by the two bits "10", but as a
+full-fledged concept the way we do, replete with associations such as its homonyms
+"too" and "to", the words "couple" and "deuce", a host of mental images such as
+dots on dominos, the shape of the numeral '2', the notions of alternation, evenness,
+oddness, and on and on ... With all this "extra baggage" to carry around, an
+intelligent program will become quite slothful in its adding. Of course, we could
+give it a ' pocket calculator , so to speak (or build one in). Then it could answer
+very fast, but its performance would be just like that of a person with a pocket
+calculator. There would be two separate parts to the machine: a reliable but
+mindless part and an intelligent but fallible part. You couldn't rely on the composite
+system to be reliable, any more than a composite of person and machine is
+necessarily reliable. So if it's right answers you're after, better stick to the pocket
+calculator alone-don't throw in the intelligence!
+
+Question: Will there be chess programs that can beat anyone?
+
+Speculation: No. There may be programs which can beat anyone at chess, but they
+will not be exclusively chess players. They will be programs of general
+intelligence, and they will be just as temperamental as people. "Do you want to play
+chess?" "No, I'm bored with chess. Let's talk about poetry." That may be the kind of
+dialogue you could have with a program that could beat everyone. That is because
+real intelligence inevitably depends on a total overview capacity-that is, a
+programmed ability to "jump out of the system", so to speak-at least roughly to the
+extent that we have that ability. Once that is present, you can't contain the program;
+it's gone beyond that certain critical point, and you just have to face the facts of
+what you've wrought.
+
+Question: Will there be special locations in memory which store parameters governing
+the behavior of the program, such that if you reached in and changed them, you would
+be able to make the program smarter or stupider or more creative or more interested in
+baseball? In short, would you be able to "tune" the program by fiddling with it on a
+relatively low level?
+
+Speculation: No. It would be quite oblivious to changes of any particular elements in
+memory, just as we stay almost exactly the same though thousands of our neurons
+die every day(!). If you fuss around too heavily, though, you'll damage it, just as if
+you irresponsibly did neurosurgery on a human being. There will be no "magic"
+location in memory where, for instance, the "IQ" of the program sits. Again, that
+will be a feature which emerges as a consequence of lower-level behavior, and
+nowhere will it sit explicitly. The same goes for such things as "the number of
+items it can hold in short-term memory", "the amount it likes physics", etc., etc.
+
+Question: Could you "tune" an A1 program to act like me, or like you-or halfway between
+
+us?
+
+
+Artificial Intelligence:Prospects
+
+
+678
+
+
+
+Speculation: No. An intelligent program will not be chameleon-like, any more than
+people are. ,It will rely on the constancy of its memories, and will not be able to flit
+between personalities. The idea of changing internal parameters to "tune to a new
+personality" reveals a ridiculous underestimation of the complexity of personality.
+
+Question: Will there be a "heart" to an A1 program, or will it simply consist of "senseless
+loops and sequences of trivial operations" (in the words of Marvin Minskys)?
+Speculation: If we could see all the way to the bottom, as we can a shallow pond, we
+would surely see only "senseless loops and sequences of trivial operations"-and we
+would surely not see any "heart". Now there are two kinds of extremist views on
+AI: one says that the human mind is, for fundamental and mysterious reasons,
+unprogrammable. The other says that you merely need to assemble the appropriate
+"heuristic devices-multiple optimizers, pattern-recognition tricks, planning
+algebras, recursive administration procedures, and the like",' and you will have
+intelligence. I find myself somewhere in between, believing that the "pond" of an
+Al program will turn out to be so deep and murky that we won't be able to peer all
+the way to the bottom. If we look from the top, the loops will be invisible, just as
+nowadays the current-carrying electrons are invisible to most programmers. When
+we create a program that passes the Turing test, we will see a "heart" even though
+we know it's not there.
+
+Question: Will Al programs ever become "superintelligent"?
+
+Speculation: I don't know. It is not clear that we would be able to understand or relate
+to a "superintelligence", or that the concept even makes sense. For instance, our
+own intelligence is tied in with our speed of thought. If our reflexes had been ten
+times faster or slower, we might have developed an entirely different set of
+concepts with which to describe the world. A creature with a radically different
+view of the world may simply not have many points of contact with us. I have often
+wondered if there could be, for instance, pieces of music which are to Bach as Bach
+is to folk tunes: "Bach squared", so to speak. And would I be able to understand
+them? Maybe there is such music around me already, and I just don't recognize it,
+just as dogs don't understand language. The idea of superintelligence is very
+strange. In any case, I don't think of it as the aim of Al research, although if we ever
+do reach the level of human intelligence, superintelligence will undoubtedly be the
+next goal-not only for us, but for our Al-program colleagues, too, who will be
+equally curious about Al and superintelligence. It seems quite likely that Al
+programs will be extremely curious about Al in general-understandably.
+
+Question: You seem to be saying that AI programs will be virtually identical to people,
+then. Won't there be any differences?
+
+
+Artificial Intelligence:Prospects
+
+
+679
+
+
+
+Speculation: Probably the differences between A1 programs and people will be larger
+than the differences between most people. It is almost impossible to imagine that
+the "body" in which an A1 program is housed would not affect it deeply. So unless
+it had an amazingly faithful replica of a human body-and why should it?-it would
+probably have enormously different perspectives on what is important, what is
+interesting, etc. Wittgenstein once made the amusing comment, "If a lion could
+speak, we would not understand him." It makes me think of Rousseau's painting of
+the gentle lion and the sleeping gypsy on the moonlit desert. But how does
+Wittgenstein know? My guess is that any A1 program would, if comprehensible to
+us, seem pretty alien. For that reason, we will have a very hard time deciding when
+and if we really are dealing with an A1 program, or just a "weird" program.
+
+Question: Will we understand what intelligence and consciousness and free will and "I"
+are when we have made an intelligent program?
+
+Speculation: Sort of-it all depends on what you mean by "understand". On a gut level,
+each of us probably has about as good an understanding as is possible of those
+things, to start with. It is like listening to music. Do you really understand Bach
+because you have taken him apart? Or did you understand it that time you felt the
+exhilaration in every nerve in your body? Do we understand how the speed of light
+is constant in every inertial reference frame? We can do the math, but no one in the
+world has a truly relativistic intuition. And probably no one will ever understand
+the mysteries of intelligence and consciousness in an intuitive way. Each of us can
+understand people , and that is probably about as close as you can come.
+
+
+Artificial Intelligence:Prospects
+
+
+680
+
+
+
+Sloth Canon
+
+
+This time, we find Achilles and the Tortoise visiting
+the dwelling of their new friend, the Sloth.
+
+Achilles: Shall I tell you of my droll footrace with Mr. T?
+
+Sloth: Please do.
+
+Achilles: It has become quite celebrated in these parts. I believe it's even been written up,
+by Zeno.
+
+Sloth: It sounds very exciting.
+
+Achilles: It was. You see, Mr. T began way ahead of me. He had such a huge head start,
+and yet
+
+Sloth: You caught up, didn't you?
+
+Achilles: Yes-being so fleet of foot, I diminished the distance between us at a constant
+rate, and soon overtook him.
+
+Sloth: The gap kept getting shorter and shorter, so you could.
+
+Achilles: Exactly. Oh, look-Mr. T has brought his violin. May I try playing on it, Mr. T?
+
+Tortoise: Please don't. It sounds very flat.
+
+Achilles: Oh, all right. But I'm in a mood for music. I don't know why. Sloth: You can
+play the piano, Achilles.
+
+Achilles: Thank you. I'll try it In a moment. I just wanted to add that I also had another
+kind of "race" with Mr. T at a later date. Unfortunately, in that race
+
+Tortoise: You didn't catch up, did you? The gap kept getting longer and longer, so you
+couldn't.
+
+Achilles: That's true. I believe THAT race has been written up, too, by Lewis Carroll.
+Now, Mr. Sloth, I'll take up your offer of trying out the piano. But I'm so bad at the
+piano. I'm not sure I dare. Sloth: You should try.
+
+(Achilles sits down and starts playing a simple tune.)
+
+Achilles: Oh-it sounds very strange. That's not how it's supposed to sound at all!
+Something is very wrong.
+
+Tortoise: You can't play the piano, Achilles. You shouldn't try.
+
+Achilles: It's like a piano in a mirror. The high notes are on the left, and the low notes are
+on the right. Every melody comes out inverted, as if upside down. Who would have
+ever thought up something so cockeyed as that?
+
+Tortoise: That's so characteristic of sloths. They hang from
+
+Achilles: Yes, I know-from tree branches-upside down, of course. That sloth-piano would
+he appropriate for playing inverted melodies such
+
+
+Sloth Canon
+
+
+681
+
+
+
+cVYQOV)
+
+
+
+SLDTit C/M
+
+J.5. J5ACH
+
+
+FIGURE 133. "Sloth Canon”,from the Musical Offering, by J. S. Bach. /Music printed
+by Donald Byrd's program ",SMUT
+
+
+Sloth Canon
+
+
+682
+
+
+
+
+
+
+
+
+as occur in some canons and fugues. But to learn to play a piano while hanging from a
+tree must he very difficult. You must have to devote a great deal of energy to it.
+
+Sloth: That's not so characteristic of sloths.
+
+Achilles: No, I gather sloths like to take life very easy. They do everything about half as
+fast as normal. And upside down, to boot. What a peculiar way to go through life!
+Speaking of things that are both upside- and slowed-down, there's a "Canon per
+augmentationem, contrario motu" in the Musical Offering. In my edition, the letters
+'S', 'A', 'T' are in front of the three staves. I don't know why. Anyway, I think Bach
+carried it off very skillfully. What's your opinion, Mr. T?
+
+Tortoise: He outdid himself. As for those letters "SAT", you could guess what they stand
+for.
+
+Achilles: "Soprano", "Alto", and "Tenor", I suppose. Three-part pieces are often written
+for that combination of voices. Wouldn't you agree, Mr. Sloth?
+
+Sloth: They stand for-
+
+Achilles: Oh, just a moment, Mr. Sloth. Mr. Tortoise-why are you putting on your coat?
+You're not leaving, are you? We were just going to fix a snack to eat. You look very
+tired. How do you feel?
+
+Tortoise: Out of gas. So long! (Trudges wearily out the door)
+
+Achilles: The poor fellow-he certainly looked exhausted. He was jogging all morning.
+He's in training for another race with me. Sloth: He did himself in.
+
+Achilles: Yes, but in vain. Maybe he could beat a Sloth ... but me? Never! Now-weren't
+you about to tell me what those letters "SAT" stand for? Sloth: As for those letters
+"SAT", you could never guess what they stand for.
+
+Achilles: Well, if they don't stand for what I thought, then my curiosity is piqued.
+
+Perhaps I'll think a little more about it. Say, how do you cook French fries? Sloth: In
+oil.
+
+Achilles: Oh, yes-I remember. I'll cut up this potato into strips an inch or two in length.
+
+Sloth: So short?
+
+Achilles: All right, already, I'll cut four-inch strips. Oh, boy, are these going to be good
+French fries! Too bad Mr. T won't be here to share them.
+
+
+Sloth Canon
+
+
+683
+
+
+
+CHAPTER XX
+
+Strange Loops,
+
+Or Tangled Hierarchies
+
+Can Machines Possess
+Originality?
+
+IN THE CHAPTER before last, I
+described Arthur Samuel's very
+successful checkers program-the one
+which can beat its designer. In light of
+that, it is interesting to hear how Samuel
+himself feels about the issue of
+computers and originality. The following
+extracts are taken from a rebuttal by
+Samuel, written in 1960, to an article by
+Norbert Wiener.
+
+It is my conviction that machines
+cannot possess originality in the
+sense implied by Wiener in his
+thesis that "machines can and do
+transcend some of the limitations
+of their designers, and that in
+doing so they may be both
+effective and dangerous." .. .
+
+A machine is not a genie, it does
+not work by magic, it does not
+possess a will, and, Wiener to the
+contrary, nothing comes out which
+has not been put in, barring, of
+course, an infrequent case of
+malfunctioning... .
+
+The "intentions" which the
+machine seems to manifest are the
+intentions of the human
+programmer, as specified in
+advance, or they are subsidiary
+intentions derived from these,
+following rules specified by the
+programmer. We can even
+anticipate higher levels of
+abstraction, just as Wiener does, in
+which the program will not only
+
+Strange Loops, Or Tangled Hierarchies
+
+
+modify the subsidiary intentions
+but will also modify the rules
+which are used in their derivation,
+or in which it will modify the ways
+in which it modifies the rules, and
+so on, or even in which one
+machine will design and construct
+a second machine with enhanced
+capabilities. However, and this is
+important, the machine will not
+and cannot [italics are his do any
+of these things until it has been
+instructed as to how to proceed.
+There is and logically there must
+always remain a complete hiatus
+between (i) any ultimate extension
+and elaboration in this process of
+carrying out man's wishes and (ii)
+the development within the
+machine of a will of its own. To
+believe otherwise is either to
+believe in magic or to believe that
+the existence of man's will is an
+illusion and that man's actions are
+as mechanical as the machine's.
+Perhaps Wiener's article and my
+rebuttal have both been
+mechanically determined, but this
+I refuse to believe.'
+
+This reminds me of the Lewis Carroll
+Dialogue (the Two-Part Invention)', I'll
+try to explain why. Samuel bases his
+argument against machine consciousness
+(or will) on the notion that any
+mechanical instantiation of will would
+require an infinite regress. Similarly,
+Carroll's Tortoise argues that no step of
+reasoning, no matter how simple, can be
+done without invoking some rule on a
+higher level to justify the step in
+question. But that being
+
+
+684
+
+
+
+also a step of reasoning, one must resort
+to a yet higher-level rule, and so on.
+Conclusion: Reasoning involves an
+infinite regress.
+
+Of course something is wrong
+with the Tortoise's argument, and I
+believe something analogous is wrong
+with Samuel's argument. To show how
+the fallacies are analogous, I now shall
+"help the Devil", by arguing
+momentarily as Devil's advocate. (Since,
+as is well known, God helps those who
+help themselves, presumably the Devil
+helps all those, and only those, who don't
+help themselves. Does the Devil help
+himself?) Here are my devilish
+conclusions drawn from the Carroll
+Dialogue:
+
+The conclusion "reasoning is
+impossible" does not apply to
+people, because as is plain to
+anyone, we do manage to carry out
+many steps of reasoning, all the
+higher levels notwithstanding.
+That shows that we humans
+operate without need of rules: we
+are "informal systems". On the
+other hand, as an argument against
+the possibility of any mechanical
+instantiation of reasoning, it is
+valid, for any mechanical
+reasoning-system would have to
+depend on rules explicitly, and so
+it couldn't get off the ground
+unless it had metarules telling it
+when to apply its rules,
+metametarules telling it when to
+apply its metarules, and so on. We
+may conclude that the ability to
+reason can never be mechanized. It
+is a uniquely human capability.
+
+What is wrong with this Devil's advocate
+point of view? It is obviously the
+assumption that a machine cannot do
+
+Strange Loops, Or Tangled Hierarchies
+
+
+anything without having a rule telling it
+to do so. In fact, machines get around the
+Tortoise's silly objections as easily as
+people do, and moreover for exactly the
+same reason: both machines and people
+are made of hardware which runs all by
+itself, according to the laws of physics.
+There is no need to rely on "rules that
+permit you to apply the rules", because
+the lowest- level rules-those without any
+"meta'"s in front-are embedded in the
+hardware, and they run without
+permission. Moral: The Carroll Dialogue
+doesn't say anything about the
+differences between people and
+machines, after all. (And indeed,
+reasoning is mechanizable.)
+
+So much for the Carroll
+Dialogue. On to Samuel's argument.
+Samuel's point, if I may caricature it, is
+this:
+
+No computer ever "wants" to do
+anything, because it was
+programmed by someone else.
+Only if it could program itself
+from zero on up-an absurdity-
+would it have its own sense of
+desire.
+
+In his argument, Samuel reconstructs the
+Tortoise's position, replacing "to reason"
+by "to want". He implies that behind any
+mechanization of desire, there has to be
+either an infinite regress or worse, a
+closed loop. If this is why computers
+have no will of their own, what about
+people? The same criterion would imply
+that
+
+
+685
+
+
+
+Unless a person designed himself
+and chose his own wants (as well
+as choosing to choose his own
+wants, etc.), he cannot be said to
+have a will of his own.
+
+It makes you pause to think where your
+sense of having a will comes from.
+Unless you are a soulist, you'll probably
+say that it comes from your brain-a piece
+of hardware which you did not design or
+choose. And yet that doesn't diminish
+your sense that you want certain things,
+and not others. You aren't a "self-
+programmed object" (whatever that
+would be), but you still do have a sense
+of desires, and it springs from the
+physical substrate of your mentality.
+Likewise, machines may someday have
+wills despite the fact that no magic
+program spontaneously appears in
+memory from out of nowhere (a "self-
+programmed program"). They will have
+wills for much the same reason as you
+do-by reason of organization and
+structure on many levels of hardware
+and software. Moral: The Samuel
+argument doesn't say anything about the
+differences between people and
+machines, after all. (And indeed, will
+will be mechanized.)
+
+Below Every Tangled Hierarchy
+Lies An Inviolate Level
+
+Right after the Two-Part Invention , I
+wrote that a central issue of this book
+would be: "Do words and thoughts
+follow formal rules?" One major thrust
+of the book has been to point out the
+many-leveledness of the mind/brain, and
+I have tried to show why the ultimate
+answer to the question is, "Yes-provided
+that you go down to the lowest level-the
+hardware-to find the rules."
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+Now Samuel's statement brought
+up a concept which I want to pursue. It is
+this: When we humans think, we
+certainly do change our own mental
+rules, and we change the rules that
+change the rules, and on and on-but
+these are, so to speak, "software rules".
+However, the rules at bottom do not
+change. Neurons run in the same simple
+way the whole time. You can't "think"
+your neurons into running some
+nonneural way, although you can make
+your mind change style or subject of
+thought. Like Achilles in the Prelude ,
+Ant Fugue, you have access to your
+thoughts but not to your neurons.
+Software rules on various levels can
+change; hardware rules cannot-in fact, to
+their rigidity is due the software's
+flexibility! Not a paradox at all, but a
+fundamental, simple fact about the
+mechanisms of intelligence.
+
+This distinction between self-
+modifiable software and inviolate
+hardware is what I wish to pursue in this
+final Chapter, developing it into a set of
+variations on a theme. Some of the
+variations may seem to be quite far¬
+fetched, but I hope that by the time I
+close the loop by returning to brains,
+minds, and the sensation of
+consciousness, you will have found an
+invariant core in all the variations.
+
+My main aim in this Chapter is to
+communicate some of the images which
+help me to visualize how consciousness
+rises out of the jungle of neurons; to
+communicate a set of intangible
+intuitions, in the hope that
+
+
+686
+
+
+
+these intuitions are valuable and may
+perhaps help others a 14tle to come to
+clearer formulations of their own images
+of what makes minds run. I could not
+hope for more than that my own mind's
+blurry images of minds and images
+should catalyze the formation of sharper
+images of minds and images in other
+minds.
+
+A Self-Modifying Game
+
+A first variation, then, concerns games in
+which on your turn, you may modify the
+rules. Think of chess. Clearly the rules
+stay the same, just the board position
+changes on each move. But let's invent a
+variation in which, on your turn, you can
+either make a move or change the rules.
+But how? At liberty? Can you turn it into
+checkers? Clearly such anarchy would
+be pointless. There must be some
+constraints. For instance, one version
+might allow you to redefine the knight's
+move. Instead of being l-and-then-2, it
+could be m-and-then-n where m and n
+are arbitrary natural numbers; and on
+your turn you could change either m or n
+by plus or minus l.-So it could go from
+1-2 to 1-3 to 0-3 to 0-4 to 0-5 to 1-5 to 2-
+5 ... Then there could be rules about
+redefining the bishop's moves, and the
+other pieces' moves as well. There could
+be rules about adding new squares, or
+deleting old squares .. .
+
+Now we have two layers of rules:
+those which tell how to move pieces, and
+those which tell how to change the rules.
+So we have rules and metarules. The
+next step is obvious: introduce
+metametarules by which we can change
+the metarules. It is not so obvious how to
+do this. The reason it is easy to
+formulate rules for moving pieces is that
+pieces move in a formalized space: the
+
+Strange Loops, Or Tangled Hierarchies
+
+
+checkerboard. If you can devise a simple
+formal notation for expressing rules and
+metarules, then to manipulate them will
+be like manipulating strings formally, or
+even like manipulating chess pieces. To
+carry things to their logical extreme, you
+could even express rules and metarules
+as positions on auxiliary chess boards.
+Then an arbitrary chess position could be
+read as a game, or as a set of rules, or as
+a set of metarules, etc., depending on
+which interpretation you place on it. Of
+course, both players would have to agree
+on conventions for interpreting the
+notation.
+
+Now we can have any number of
+adjacent chess boards: one for the game,
+one for rules, one for metarules, one for
+metametarules, and so on, as far as you
+care to carry it. On your turn, you may
+make a move on any one of the chess
+boards except the top-level one, using
+the rules which apply (they come from
+the next chess board up in the hierarchy).
+Undoubtedly both players would get
+quite disoriented by the fact that almost
+anything-though not everything!-can
+change. By definition, the top-level
+chess board can't be changed, because
+you don't have rules telling how to
+change it. It is inviolate. There is more
+that is inviolate: the conventions by
+which the different boards are
+interpreted, the agreement to take turns,
+the agreement that each person may
+change one chess board each turn-and
+you will find more if you examine the
+idea carefully.
+
+
+687
+
+
+
+Now it is possible to go considerably
+further in removing the pillars by which
+orientation is achieved. One step at a
+time. .. We begin by collapsing the
+whole array of boards into a single
+board. What is meant by this? There will
+be two ways of interpreting the board:
+(1) as pieces to be moved; (2) as rules
+for moving the pieces. On your turn, you
+move pieces-and perforce, you change
+rules! Thus, the rules constantly change
+themselves. Shades of Typogenetics-or
+for that matter, of real genetics. The
+distinction between game, rules,
+metarules, metametarules, has been lost.
+What was once a nice clean hierarchical
+setup has become a Strange Loop, Or
+Tangled Hierarchy. The moves change
+the rules, the rules determine the moves,
+round and round the mulberry bush ...
+There are still different levels, but the
+distinction between "lower" and "higher"
+has been wiped out.
+
+Now, part of what was inviolate
+has been made changeable. But there is
+still plenty that is inviolate. Just as
+before, there are conventions between
+you and your opponent by which you
+interpret the board as a collection of
+rules. There is the agreement to take
+turns-and probably other implicit
+conventions, as well. Notice, therefore,
+that the notion of different levels has
+survived, in an unexpected way. There is
+an Inviolate level-let's call it the I-level-
+on which the interpretation conventions
+reside; there is also a Tangled level-the
+T-level-on which the Tangled Hierarchy
+resides. So these two levels are still
+hierarchical: the I-level governs what
+happens on the T-level, but the T-level
+does not and cannot affect the I-level.
+No matter that the T-level itself is a
+Tangled Hierarchy-it is still governed by
+a set of conventions outside of itself.
+And that is the important point.
+
+Strange Loops, Or Tangled Hierarchies
+
+
+As you have no doubt imagined,
+there is nothing to stop us from doing the
+"impossible"-namely, tangling the I-
+level and the T-level by making the
+interpretation conventions themselves
+subject to revision, according to the
+position on the chess board. But in order
+to carry out such a "supertangling",
+you'd have to agree with your opponent
+on some further conventions connecting
+the two levels-and the act of doing so
+would create a new level, a new sort of
+inviolate level on top of the
+"supertangled" level (or underneath it, if
+you prefer). And this could continue
+going on and on. In fact, the 'jumps"
+which are being made are very similar to
+those charted in the Birthday
+Cantatatata, and in the repeated
+Godelization applied to various
+improvements on TNT. Each time you
+think you have reached the end, there is
+some new variation on the theme of
+jumping out of the system which
+requires a kind of creativity to spot.
+
+The Authorship Triangle Again
+
+But I am not interested in pursuing the
+strange topic of the ever more abstruse
+tanglings which can arise in self¬
+modifying chess. The point of this has
+been to show, in a somewhat graphic
+way, how in any system there is always
+some "protected" level which is
+unassailable by the rules on other levels,
+no matter how tangled their interaction
+may be among themselves. An amusing
+riddle from Chapter IV illustrates this
+same idea in a slightly different context.
+Perhaps it will catch you off guard:
+
+
+688
+
+
+
+
+There are three authors-Z, T, and E. Now it happens that Z exists only in a novel by
+T. Likewise, T exists only in a novel by E. And strangely, E, too, exists only in a
+novel-by Z, of course. Now, is such an ’’authorship triangle" really possible? (See
+Fig. 134.)
+
+Of course it's possible. But there's a trick ... All three authors Z, T, E, are themselves
+characters in another novel-by H! You can think of the Z-T-E triangle as a Strange Loop,
+Or Tangled Hierarchy; but author H is outside of the space in which that tangle takes
+place-author H is in an inviolate space. Although Z, T, and E all have access-direct or
+indirect-to each other, and can do dastardly things to each other in their various novels,
+none of them can touch H's life! They can't even imagine him-no more than you can
+imagine the author of the book you're a character in. If I were to draw author H, I would
+represent him somewhere off the page. Of course that would present a problem, since
+drawing a thing necessarily puts it onto the page ... Anyway, H is really outside of the
+world of Z, T, and E, and should be represented as being so.
+
+Escher's Drawing Hands
+
+Another classic variation on our theme is the Escher picture of Drawing Hands (Fig.
+135). Here, a left hand (LH) draws a right hand (RH), while at the same time, RH draws
+LH. Once again, levels which ordinarily are seen as hierarchical-that which draws, and
+that which is drawn-turn back on each other, creating a Tangled Hierarchy. But the theme
+of the Chapter is borne out, of course, since behind it all lurks the undrawn but drawing
+hand of M. C. Escher, creator of both LH and RH. Escher is outside of the two-hand
+space, and in my schematic version of his picture (Fig. 136), you can see that explicitly.
+In this schematized representation of the Escher picture, you see the Strange Loop, Or
+Tangled Hierarchy at the top; also, you see the Inviolate Level below it, enabling it to
+come into being. One could further Escherize the Escher picture, by taking a photograph
+of a hand drawing it. And so on.
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+689
+
+
+
+Wr
+
+
+
+FIGURE 135 Drawing Hands , by M. C. Escher (lithograph, 1948).
+
+
+
+FIGURE 136. Abstract diagram ofM. C. Escher’s Drawing Hands. On top, a seeming
+
+paradox. Below, its resolution.
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+685
+
+
+
+Brain and Mind:
+
+A Neural Tangle Supporting a Symbol Tangle
+
+
+Now we can relate this to the brain, as well as to A1 programs. In our thoughts, symbols
+activate other symbols, and all interact heterarchically. Furthermore, the symbols may
+cause each other to change internally, in the fashion of programs acting on other
+programs. The illusion is created, because of the Tangled Hierarchy of symbols, that
+there is no inviolate level, one thinks there is no such level because that level is shielded
+from our view.
+
+If it were possible to schematize this whole image, there would be a gigantic
+forest of symbols linked to each other by tangly lines like vines in a tropical jungle-this
+would be the top level, the Tangled Hierarchy where thoughts really flow back and forth.
+This is the elusive level of mind: the analogue to LH and RH. Far below in the schematic
+picture, analogous to the invisible "prime mover" Escher, there would be a representation
+of the myriad neurons-the "inviolate substrate" which lets the tangle above it come into
+being. Interestingly, this other level is itself a tangle in a literal sense-billions of cells and
+hundreds of billions of axons, joining them all together.
+
+This is an interesting case where a software tangle, that of the symbols, is
+supported by a hardware tangle, that of the neurons. But only the symbol tangle is a
+Tangled Hierarchy. The neural tangle is* ust a "simple" tangle. This distinction is pretty
+much the same as that between Strange Loops and feedback, which I mentioned in
+Chapter XVI. A Tangled Hierarchy occurs when what you presume are clean hierarchical
+levels take you by surprise and fold back in a hierarchy-violating way. The surprise
+element is important; it is the reason I call Strange Loops "strange". A simple tangle, like
+feedback, doesn't involve violations of presumed level distinctions. An example is when
+you're in the shower and you wash your left arm with your right, and then vice versa.
+There is no strangeness to the image. Escher didn't choose to draw hands drawing hands
+for nothing!
+
+Events such as two arms washing each other happen all the time in the world, and
+we don't notice them particularly. I say something to you, then you say something back to
+me. Paradox % No; our perceptions of each other didn't involve a hierarchy to begin with,
+so there is no sense of strangeness.
+
+On the other hand, where language does create strange loops is when it talks
+about itself, whether directly or indirectly. Here, something in the system jumps out and
+acts on the system, as if it were outside the system. What bothers us is perhaps an ill-
+defined sense of topological wrongness: the inside-outside distinction is being blurred, as
+in the famous shape called a "Klein bottle". Even though the system is an abstraction, our
+minds use spatial imagery with a sort of mental topology.
+
+Getting back to the symbol tangle, if we look only at it, and forget the neural
+tangle, then we seem to see a self-programmed object-in just the same way as we seem to
+see a self-drawn picture if we look at Drawing Hands and somehow fall for the illusion,
+by forgetting the existence of Escher. For
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+686
+
+
+
+the picture, this is unlikely-but for humans and the way they look at their minds, this is
+usually what happens. W t feel self-programmed. Indeed, we couldn't feel any other way,
+for we are shielded from the lower levels, the neural tangle. Our thoughts seem to run
+about in their own space, creating new thoughts and modifying old ones, and we never
+notice any neurons helping us out! But that is to be expected. We can't.
+
+An analogous double-entendre can happen with LISP programs that are designed
+to reach in and change their own structure. If you look at them on the LISP level, you
+will say that they change themselves; but if you shift levels, and think of LISP programs
+as data to the LISP interpreter (see Chapter X), then in fact the sole program that is
+running is the interpreter, and the changes being made are merely changes in pieces of
+data. The LISP interpreter itself is shielded from changes.
+
+How you describe a tangled situation of this sort depends how far back you step
+before describing. If you step far enough back, you can often see the clue that allows you
+to untangle things.
+
+
+Strange Loops in Government
+
+A fascinating area where hierarchies tangle is government-particularly in the courts.
+Ordinarily, you think of two disputants arguing their cases in court, and the court
+adjudicating the matter. The court is on a different level from the disputants. But strange
+things can start to happen when the courts themselves get entangled in legal cases.
+Usually there is a higher court which is outside the dispute. Even if two lower courts get
+involved in some sort of strange fight, with each one claiming jurisdiction over the other,
+some higher court is outside, and in some sense it is analogous to the inviolate
+interpretation conventions which we discussed in the warped version of chess.
+
+But what happens when there is no higher court, and the Supreme Court itself gets
+all tangled up in legal troubles? This sort of snarl nearly happened in the Watergate era.
+The President threatened to obey only a "definitive ruling" of the Supreme Court-then
+claimed he had the right to decide what is "definitive". Now that threat never was made
+good; but if it had been, it would have touched off a monumental confrontation between
+two levels of government, each of which, in some ways, can validly claim to be "above"
+the other-and to whom is there recourse to decide which one is right? To say "Congress"
+is not to settle the matter, for Congress might command the President to obey the
+Supreme Court, yet the President might still refuse, claiming that he has the legal right to
+disobey the Supreme Court (and Congress!) under certain circumstances. This would
+create a new court case, and would throw the whole system into disarray, because it
+would be so unexpected, so Tangled-so Strange!
+
+The irony is that once you hit your head against the ceiling like this, where you
+are prevented from jumping out of the system to a yet higher authority, the only recourse
+is to forces which seem less well defined by
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+687
+
+
+
+
+rules, but which are the only source of higher-level rules anyway: the lower-level rules,
+which in this case means the general reaction of society. It is well to remember that in a
+society like ours, the legal system is, in a sense, a polite gesture granted collectively by
+millions of people-and it can be overridden just as easily as a river can overflow its
+banks. Then a seeming anarchy takes over; but anarchy has its own kinds of rules, no less
+than does civilized society: it is just that they operate from the bottom up, not from the
+top down. A student of anarchy could try to discover rules according to which anarchic
+situations develop in time, and very likely there are some such rules.
+
+An analogy from physics is useful here. As was mentioned earlier in the book,
+gases in equilibrium obey simple laws connecting their temperature, pressure, and
+volume. However, a gas can violate those laws (as a President can violate laws)-provided
+it is not in a state of equilibrium. In nonequilibrium situations, to describe what happens,
+a physicist has recourse only to statistical mechanics-that is, to a level of description
+which is not macroscopic, for the ultimate explanation of a gas's behavior always lies on
+the molecular level, just as the ultimate explanation of a society's political behavior
+always lies at the "grass roots level". The field of nonequilibrium thermodynamics
+attempts to find macroscopic laws to describe the behavior of gases (and other systems)
+which are out of equilibrium. It is the analogue to the branch of political science which
+would search for laws governing anarchical societies.
+
+Other curious tangles which arise in government include the FBI investigating its
+own wrongdoings, a sheriff going to jail while in office, the self-application of the
+parliamentary rules of procedure, and so on. One of the most curious legal cases I ever
+heard of involved a person who claimed to have psychic powers. In fact, he claimed to be
+able to use his psychic powers to detect personality traits, and thereby to aid lawyers in
+picking juries. Now what if this "psychic" has to stand trial himself one day? What effect
+might this have on a jury member who believes staunchly in ESP? How much will he feel
+affected by the psychic (whether or not the psychic is genuine)? The territory is ripe for
+exploitation-a great area for selffulfilling prophecies.
+
+Tangles Involving Science and the Occult
+
+Speaking of psychics and ESP, another sphere of life where strange loops abound is
+fringe science. What fringe science does is to call into question many of the standard
+procedures or beliefs of orthodox science, and thereby challenge the objectivity of
+science. New ways of interpreting evidence that rival the established ones are presented.
+But how do you evaluate a way of interpreting evidence? Isn't this precisely the problem
+of objectivity all over again, just on a higher plane? Of course. Lewis Carroll's infinite-
+regress paradox appears in a new guise. The Tortoise would argue that if you want to
+show that A is a fact, you need evidence: B. But what makes you sure that B is evidence
+of A?' To show that, you need meta-
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+688
+
+
+
+evidence: C. And for the validity of that meta-evidence, you need metameta-evidence-
+and so on, ad nauseam. Despite this argument, people have an intuitive sense of evidence.
+This is because-to repeat an old refrain-people have built-in hardware in their brains that
+includes some rudimentary ways of interpreting evidence. We can build on this, and
+accumulate new ways of interpreting evidence; we even learn how and when to override
+our most basic mechanisms of evidence interpretation, as one must, for example, in trying
+to figure out magic tricks.
+
+Concrete examples of evidence dilemmas crop up in regard to many phenomena
+of fringe science. For instance, ESP often seems to manifest itself outside of the
+laboratory, but when brought into the laboratory, it vanishes mysteriously. The standard
+scientific explanation for this is that ESP is a nonreal phenomenon which cannot stand up
+to rigorous scrutiny. Some (by no means all) believers in ESP have a peculiar way of
+fighting back, however. They say, "No, ESP is real; it simply goes away when one tries
+to observe it scientifically-it is contrary to the nature of a scientific worldview." This is
+an amazingly brazen technique, which we might call "kicking the problem upstairs".
+What that means is, instead of questioning the matter at hand, you call into doubt theories
+belonging to a higher level of credibility. The believers in ESP insinuate that what is
+wrong is not their ideas, but the belief system of science. This is a pretty grandiose claim,
+and unless there is overwhelming evidence for it, one should be skeptical of it. But then
+here we are again, talking about "overwhelming evidence" as if everyone agreed on what
+that means!
+
+
+The Nature of Evidence
+
+The Sagredo-Simplicio-Salviati tangle, mentioned in Chapters XIII and XV, gives
+another example of the complexities of evaluation of evidence. Sagredo tries to find some
+objective compromise, if possible, between the opposing views of Simplicio and Salviati.
+But compromise may not always be possible. How can one compromise "fairly" between
+right and wrong? Between fair and unfair? Between compromise and no compromise?
+These questions come up over and over again in disguised form in arguments about
+ordinary things.
+
+Is it possible to define what evidence is? Is it possible to lay down laws as to how
+to make sense out of situations? Probably not, for any rigid rules would undoubtedly have
+exceptions, and nonrigid rules are not rules. Having an intelligent AI program would not
+solve the problem either, for as an evidence processor, it would not be any less fallible
+than humans are. So, if evidence is such an intangible thing after all, why am I warning
+against new ways of interpreting evidence? Am I being inconsistent? In this case, I don't
+think so. My feeling is that there are guidelines which one can give, and out of them an
+organic synthesis can be made. But inevitably some amount of judgment and intuition
+must enter the picture-things which are different in different people. They will also be
+different in
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+689
+
+
+
+different AI programs. Ultimately, there are complicated criteria for deciding if a method
+of evaluation of evidence is good. One involves the "usefulness" of ideas which are
+arrived at by that kind of reasoning. Modes of thought which lead to useful new things in
+life are deemed "valid" in some sense. But this word "useful" is extremely subjective.
+
+My feeling is that the process by which we decide what is valid or what is true is
+an art; and that it relies as deeply on a sense of beauty and simplicity as it does on rock-
+solid principles of logic or reasoning or anything else which can be objectively
+formalized. I am not saying either (1) truth is a chimera, or (2) human intelligence is in
+principle not programmable. I am saying (1) truth is too elusive for any human or any
+collection of humans ever to attain fully; and (2) Artificial Intelligence, when it reaches
+the level of human intelligence-or even if it surpasses it-will still be plagued by the
+problems of art, beauty, and simplicity, and will run up against these things constantly in
+its own search for knowledge and understanding.
+
+"What is evidence?" is not just a philosophical question, for it intrudes into life in
+all sorts of places. You are faced with an extraordinary number of choices as to how to
+interpret evidence at every moment. You can hardly go into a bookstore (or these days,
+even a grocery store!) without seeing books on clairvoyance, ESP, UFO's, the Bermuda
+triangle, astrology, dowsing, evolution versus creation, black holes, psi fields,
+biofeedback, transcendental meditation, new theories of psychology ... In science, there
+are fierce debates about catastrophe theory, elementary particle theory, black holes, truth
+and existence in mathematics, free will, Artificial Intelligence, reductionism versus
+holism ... On the more pragmatic side of life, there are debates over the efficacy of
+vitamin C or of laetrile, over the real size of oil reserves (either underground or stored),
+over what causes inflation and unemployment-and on and on. There is Buckminster
+Fullerism, Zen Buddhism, Zeno's paradoxes, psychoanalysis, etc., etc. From issues as
+trivial as where books ought to be shelved in a store, to issues as vital as what ideas are to
+be taught to children in schools, ways of interpreting evidence play an inestimable role.
+
+Seeing Oneself
+
+One of the most severe of all problems of evidence interpretation is that of trying to
+interpret all the confusing signals from the outside as to who one is. In this case, the
+potential for intralevel and interlevel conflict is tremendous. The psychic mechanisms
+have to deal simultaneously with the individual's internal need for self-esteem and the
+constant flow of evidence from the outside affecting the self-image. The result is that
+information flows in a complex swirl between different levels of the personality; as it
+goes round and round, parts of it get magnified, reduced, negated, or otherwise distorted,
+and then those parts in turn get further subjected to the same sort of swirl, over and over
+again-all of this in an attempt to reconcile what is, with what we wish were (see Fig. 81).
+
+
+Strange Foops, Or Tangled Hierarchies
+
+
+690
+
+
+
+The upshot is that the total picture of "who I am" is integrated in some
+enormously complex way inside the entire mental structure, and contains in each one of
+us a large number of unresolved, possibly unresolvable, inconsistencies. These
+undoubtedly provide much of the dynamic tension which is so much a part of being
+human. Out of this tension between the inside and outside notions of who we are come
+the drives towards various goals that make each of us unique. Thus, ironically, something
+which we all have in common-the fact of being self-reflecting conscious beings-leads to
+the rich diversity in the ways we have of internalizing evidence about all sorts of things,
+and in the end winds up being one of the major forces in creating distinct individuals.
+
+Godel’s Theorem and Other Disciplines
+
+It is natural to try to draw parallels between people and sufficiently complicated formal
+systems which, like people, have "self-images" of a sort. Godel’s Theorem shows that
+there are fundamental limitations to consistent formal systems with self-images. But is it
+more general? Is there a "Godel’s Theorem of psychology", for instance?
+
+If one uses Godel’s Theorem as a metaphor, as a source of inspiration, rather than
+trying to translate it literally into the language of psychology or of any other discipline,
+then perhaps it can suggest new truths in psychology or other areas. But it is quite
+unjustifiable to translate it directly into a statement of another discipline and take that as
+equally valid. It would be a large mistake to think that what has been worked out with the
+utmost delicacy in mathematical logic should hold without modification in a completely
+different area.
+
+
+Introspection and Insanity: A Godelian Problem
+
+I think it can have suggestive value to translate Godel’s Theorem into other domains,
+provided one specifies in advance that the translations are metaphorical and are not
+intended to be taken literally. That having been said, I see two major ways of using
+analogies to connect Godel’s Theorem and human thoughts. One involves the problem of
+wondering about one's sanity. How can you figure out if you are sane? This is a Strange
+Loop indeed. Once you begin to question your own sanity, you can get trapped in an
+ever-tighter vortex of self-fulfilling prophecies, though the process is by no means
+inevitable. Everyone knows that the insane interpret the world via their own peculiarly
+consistent logic; how can you tell if your own logic is "peculiar" or not, given that you
+have only your own logic to judge itself? I don't see any answer. I am just reminded of
+Godel’s second Theorem, which implies that the only versions of formal number theory
+which assert their own consistency are inconsistent...
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+691
+
+
+
+Can We Understand Our Own" Minds or Brains?
+
+
+The other metaphorical analogue to Godel’s Theorem which I find provocative suggests
+that ultimately, we cannot understand our own minds/ brains. This is such a loaded,
+many-leveled idea that one must be extremely cautious in proposing it. What does
+"understanding our own minds/brains" mean? It could mean having a general sense of
+how they work, as mechanics have a sense of how cars work. It could mean having a
+complete explanation for why people do any and all things they do. It could mean having
+a complete understanding of the physical structure of one's own brain on all levels. It
+could mean having a complete wiring diagram of a brain in a book (or library or
+computer). It could mean knowing, at every instant, precisely what is happening in one's
+own brain on the neural level-each firing, each synaptic alteration, and so on. It could
+mean having written a program which passes the Turing test. It could mean knowing
+oneself so perfectly that such notions as the subconscious and the intuition make no
+sense, because everything is out in the open. It could mean any number of other things.
+
+Which of these types of self-mirroring, if any, does the self-mirroring in Godel’s
+Theorem most resemble? I would hesitate to say. Some of them are quite silly. For
+instance, the idea of being able to monitor your own brain state in all its detail is a pipe
+dream, an absurd and uninteresting proposition to start with; and if Godel’s Theorem
+suggests that it is impossible, that is hardly a revelation. On the other hand, the age-old
+goal of knowing yourself in some profound way-let us call it "understanding your own
+psychic structure "-has a ring of plausibility to it. But might there not be some vaguely
+Godelian loop which limits the depth to which any individual can penetrate into his own
+psyche? Just as we cannot see our faces with our own eyes, is it not reasonable to expect
+that we cannot mirror our complete mental structures in the symbols which carry them
+out?
+
+All the limitative Theorems of metamathematics and the theory of computation
+suggest that once the ability to represent your own structure has reached a certain critical
+point, that is the kiss of death: it guarantees that you can never represent yourself totally.
+Godel’s Incompleteness Theorem, Church's Undecidability Theorem, Turing's Halting
+Theorem, Tarski's Truth Theorem-all have the flavor of some ancient fairy tale which
+warns you that "To seek self-knowledge is to embark on a journey which ... will always
+be incomplete, cannot be charted on any map, will never halt, cannot be described."
+
+But do the limitative Theorems have any bearing on people? Here is one way of arguing
+the case. Either I am consistent or I am inconsistent. (The latter is much more likely, but
+for completeness' sake, I consider both possibilities.) If I am consistent, then there are
+two cases. (1) The "low-fidelity" case: my self-understanding is below a certain critical
+point. In this case, I am incomplete by hypothesis. (2) The "high-fidelity" case: My self¬
+understanding has reached the critical point where a metaphorical analogue of the
+limitative Theorems does apply, so my self-understanding
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+692
+
+
+
+undermines itself in a Godelian way, and I am incomplete for that reason. Cases (1) and
+(2) are predicated on my being 100 per cent consistent-a very unlikely state of affairs.
+More likely is that I am inconsistent-but that's worse, for then inside me there are
+contradictions, and how can I ever understand that?
+
+Consistent or inconsistent, no one is exempt from the mystery of the self.
+Probably we are all inconsistent. The world is just too complicated for a person to be able
+to afford the luxury of reconciling all of his beliefs with each other. Tension and
+confusion are important in a world where many decisions must be made quickly, Miguel
+de Unamuno once said, "If a person never contradicts himself, it must be that he says
+nothing." I would say that we all are in the same boat as the Zen master who, after
+contradicting himself several times in a row, said to the confused Doko, "I cannot
+understand myself."
+
+Godel’s Theorem and Personal Nonexistence
+
+Perhaps the greatest contradiction in our lives, the hardest to handle, is the knowledge
+"There was a time when I was not alive, and there will come a time when I am not alive."
+On one level, when you "step out of yourself" and see yourself as "just another human
+being", it makes complete sense. But on another level, perhaps a deeper level, personal
+nonexistence makes no sense at all. All that we know is embedded inside our minds, and
+for all that to be absent from the universe is not comprehensible. This is a basic
+undeniable problem of life; perhaps it is the best metaphorical analogue of Godel’s
+Theorem. When you try to imagine your own nonexistence, you have to try to jump out
+of yourself, by mapping yourself onto someone else. You fool yourself into believing that
+you can import an outsider's view of yourself into you, much as TNT "believes" it
+mirrors its own metatheory inside itself. But TNT only contains its own metatheory up to
+a certain extent-not fully. And as for you, though you may imagine that you have jumped
+out of yourself, you never can actually do so-no more than Escher's dragon can jump out
+of its native two-dimensional plane into three dimensions. In any case, this contradiction
+is so great that most of our lives we just sweep the whole mess under the rug, because
+trying to deal with it just leads nowhere.
+
+Zen minds, on the other hand, revel in this irreconcilability. Over and over again,
+they face the conflict between the Eastern belief: "The world and I are one, so the notion
+of my ceasing to exist is a contradiction in terms" (my verbalization is undoubtedly too
+Westernized-apologies to Zenists), and the Western belief: "I am just part of the world,
+and I will die, but the world will go on without me."
+
+Science and Dualism
+
+Science is often criticized as being too "Western" or "dualistic"-that is, being permeated
+by the dichotomy between subject and object, or observer
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+693
+
+
+
+and observed. While it is true that up until this century, science was exclusively
+concerned with things which can be readily distinguished from their human observers-
+such as oxygen and carbon, light and heat, stars and planets, accelerations and orbits, and
+so on-this phase of science was a necessary prelude to the more modern phase, in which
+life itself has come under investigation. Step by step, inexorably, "Western" science has
+moved towards investigation of the human mind-which is to say, of the observer.
+Artificial Intelligence research is the furthest step so far along that route. Before AI came
+along, there were two major previews of the strange consequences of the mixing of
+subject and object in science. One was the revolution of quantum mechanics, with its
+epistemological problems involving the interference of the observer with the observed.
+The other was the mixing of subject and object in metamathematics, beginning with
+Godel's Theorem and moving through all the other limitative'Theorems we have
+discussed. Perhaps the next step after Al will be the self-application of science: science
+studying itself as an object. This is a different manner of mixing subject and object-
+perhaps an even more tangled one than that of humans studying their own minds.
+
+By the way, in passing, it is interesting to note that all results essentially
+dependent on the fusion of subject and object have been limitative results. In addition to
+the limitative Theorems, there is Heisenberg's uncertainty principle, which says that
+measuring one quantity renders impossible the simultaneous measurement of a related
+quantity. I don't know why all these results are limitative. Make of it what you will.
+
+Symbol vs. Object in Modern Music and Art
+
+Closely linked with the subject-object dichotomy is the symbol-object dichotomy, which
+was explored in depth by Ludwig Wittgenstein in the early part of this century. Later the
+words "use" and "mention" were adopted to make the same distinction. Quine and others
+have written at length about the connection between signs and what they stand for. But
+not only philosophers have devoted much thought to this deep and abstract matter. In our
+century both music and art have gone through crises which reflect a profound concern
+with this problem. Whereas music and painting, for instance, have traditionally expressed
+ideas or emotions through a vocabulary of "symbols" (i.e. visual images, chords,
+rhythms, or whatever), now there is a tendency to explore the capacity of music and art to
+not express anything just to be. This means to exist as pure globs of paint, or pure sounds,
+but in either case drained of all symbolic value.
+
+In music, in particular, John Cage has been very influential in bringing a Zen-like
+approach to sound. Many of his pieces convey a disdain for "use" of sounds-that is, using
+sounds to convey emotional states-and an exultation in "mentioning" sounds-that is,
+concocting arbitrary juxtapositions of sounds without regard to any previously formulated
+code by which a listener could decode them into a message. A typical example is
+"Imaginary Landscape no. 4", the polyradio piece described in Chapter VI. I may not
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+694
+
+
+
+be doing Cage justice, but to me it seems that much of his work has been directed at
+bringing meaninglessness into music, and in some sense, at making that meaninglessness
+have meaning. Aleatoric music is a typical exploration in that direction. (Incidentally,
+chance music is a close cousin to the much later notion of "happenings" or "be-in"' s.)
+There are many other contemporary composers who are following Cage’s lead, but few
+with as much originality. A piece by Anna Lockwood, called "Piano Burning", involves
+just that-with the strings stretched to maximum tightness, to make them snap as loudly as
+possible; in a piece by LaMonte Young, the noises are provided by shoving the piano all
+around the stage and through obstacles, like a battering ram.
+
+Art in this century has gone through many convulsions of this general type. At
+first there was the abandonment of representation, which was genuinely revolutionary:
+the beginnings of abstract art. A gradual swoop from pure representation to the most
+highly abstract patterns is revealed in the work of Piet Mondrian. After the world was
+used to nonrepresentational art, then surrealism came along. It was a bizarre about-face,
+something like neoclassicism in music, in which extremely representational art was
+"subverted" and used for altogether new reasons: to shock, confuse, and amaze. This
+school was founded by Andre Breton, and was located primarily in France; some of its
+more infl uential members were Dali, Magritte, de Chirico, Tanguy.
+
+Magritte's Semantic Illusions
+
+Of all these artists, Magritte was the most conscious of the symbol-object mystery (which
+I see as a deep extension of the use-mention distinction). He uses it to evoke powerful
+responses in viewers, even if the viewers do not verbalize the distinction this way. For
+example, consider his very strange variation on the theme of still life, called Common
+Sense (Fig. 137).
+
+FIGURE 137. Common Sense, by Rene Magritte (1945-46).
+
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+695
+
+
+
+
+
+FIGURE 138. The Two Mysteries, by Rene Magritte (1966).
+
+Here, a dish filled with fruit, ordinarily the kind of thing represented inside a still life, is
+shown sitting on top of a blank canvas. The conflict between the symbol and the real is
+great. But that is not the full irony, for of course the whole thing is itself just a painting-in
+fact, a still life with nonstandard subject matter.
+
+Magritte's series of pipe paintings is fascinating and perplexing. Consider The
+Two Mysteries (Fig. 138). Focusing on the inner painting, you get the message that
+symbols and pipes are different. Then your glance moves upward to the "real” pipe
+floating in the air-you perceive that it is real, while the other one is just a symbol. But that
+is of course totally wrong: both of them are on the same flat surface before your eyes.
+The idea that one pipe is in a twice-nested painting, and therefore somehow "less real"
+than the other pipe, is a complete fallacy. Once you are willing to "enter the room", you
+have already been tricked: you’ve fallen for image as reality. To be consistent in your
+gullibility, you should happily go one level further down, and confuse image-within-
+image with reality. The only way not to be sucked in is to see both pipes merely as
+colored smudges on a surface a few inches in front of your nose. Then, and only then, do
+you appreciate the full meaning of the written message "Ceci West pas une pipe”-but
+ironically, at the very instant everything turns to smudges, the writing too turns to
+smudges, thereby losing its meaning! In other words, at that instant, the verbal message
+of the painting self-destructs in a most Godelian way.
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+696
+
+
+
+
+FIGURE 139. Smoke Signal. [Drawing by the author.]
+
+The Air and the Song (Fig. 82), taken from a series by Magritte, accomplishes all
+that The Two Mysteries does, but in one level instead of two. My drawings Smoke Signal
+and Pipe Dream (Figs. 139 and 140) constitute "Variations on a Theme of Magritte". Try
+staring at Smoke Signal for a while. Before long, you should be able to make out a hidden
+message saying, "Ceci n’est pas un message". Thus, if you find the message, it denies
+itself-yet if you don't, you miss the point entirely. Because of their indirect self-snuffing,
+my two pipe pictures can be loosely mapped onto Godel’s G-thus giving rise to a
+"Central Pipemap", in the same spirit as the other "Central Xmaps": Dog, Crab, Sloth.
+
+A classic example of use-mention confusion in paintings is the occurrence of a
+palette in a painting. Whereas the palette is an illusion created by the representational
+skill of the painter, the paints on the painted palette are literal daubs of paint from the
+artist's palette. The paint plays itself-it does not symbolize anything else. In Don
+Giovanni, Mozart exploited a related trick: he wrote into the score explicitly the sound of
+an orchestra tuning up. Similarly, if I want the letter T to play itself (and not symbolize
+me), I put T directly into my text; then I enclose T between quotes. What results is "I"
+(not T, nor "T"). Got that?
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+697
+
+
+
+
+FIGURE 140. Pipe Dream. [Drawing by the author.]
+
+The "Code" of Modern Art
+
+A large number of influences, which no one could hope to pin down completely, led to
+further explorations of the symbol-object dualism in art. There is no doubt that John
+Cage, with his interest in Zen, had a profound influence on art as well as on music. His
+friends jasper Johns and Robert Rauschenberg both explored the distinction between
+objects and symbols by using objects as symbols for themselves-or, to flip the coin, by
+using symbols as objects in themselves. All of this was perhaps intended to break down
+the notion that art is one step removed from reality-that art speaks in "code", for which
+the viewer must act as interpreter. The idea was to eliminate the step of interpretation and
+let the naked object simply be, period. ("Period"-a curious case of use-mention blur.)
+However, if this was the intention, it was a monumental flop, and perhaps had to be.
+
+Any time an object is exhibited in a gallery or dubbed a "work", it acquires an
+aura of deep inner significance-no matter how much the viewer has been warned not to
+look for meaning. In fact, there is a backfiring effect whereby the more that viewers are
+told to look at these objects without mystification, the more mystified the viewers get.
+After all, if a
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+698
+
+
+
+wooden crate on a museum floor is just a wooden crate on a museum floor, then why
+doesn't the janitor haul it out back and throw it in the garbage? Why is the name of an
+artist attached to it? Why did the artist want to demystify art? Why isn't that dirt clod out
+front labeled with an artist's name? Is this a hoax? Am I crazy, or are artists crazy? More
+and more questions flood into the viewer's mind; he can't help it. This is the "frame
+effect" which art-Art-automatically creates. There is no way to suppress the wonderings
+in the minds of the curious.
+
+Of course, if the purpose is to instill a Zen-like sense of the world as devoid of
+categories and meanings, then perhaps such art is merely intended to serve-as does
+intellectualizing about Zen-as a catalyst to inspire the viewer to go out and become
+acquainted with the philosophy which rejects "inner meanings" and embraces the world
+as a whole. In this case, the art is self-defeating in the short run, since the viewers do
+ponder about its meaning, but it achieves its aim with a few people in the long run, by
+introducing them to its sources. But in either case, it is not true that there is no code by
+which ideas are conveyed to the viewer. Actually, the code is a much more complex
+thing, involving statements about the absence of codes and so forth-that is, it is part code,
+part metacode, and so on. There is a Tangled Hierarchy of messages being transmitted by
+the most Zen-like art objects, which is perhaps why so many find modern art so
+inscrutable.
+
+
+Ism Once Again
+
+Cage has led a movement to break the boundaries between art and nature. In music, the
+theme is that all sounds are equal-a sort of acoustical democracy. Thus silence is just as
+important as sound, and random sound is just as important as organized sound. Leonard
+B. Meyer, in his book Music, the Arts, and Ideas , has called this movement in music
+"transcendentalism", and states:
+
+If the distinction between art and nature is mistaken, aesthetic valuation is
+irrelevant. One should no more judge the value of a piano sonata than one should
+judge the value of a stone, a thunderstorm, or a starfish. "Categorical statements,
+such as right and wrong, beautiful or ugly, typical of the rationalistic thinking of
+tonal aesthetics," writes Luciano Berio [a contemporary composer, "are no longer
+useful in understanding why and how a composer today works on audible forms
+and musical action."
+
+Later, Meyer continues in describing the philosophical position of transcendentalism:
+
+... all things in all of time and space are inextricably connected with one
+another. Any divisions, classifications, or organizations discovered in the universe
+are arbitrary. The world is a complex, continuous, single event .2 [Shades of Zeno!]
+
+I find "transcendentalism" too bulky a name for this movement. In its place, I use
+"ism". Being a suffix without a prefix, it suggests an ideology
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+699
+
+
+
+
+FIGURE 141. The Human Condition I, by Rene Magritte (1933).
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+700
+
+
+
+
+
+
+without ideas-which, however you interpret it, is probably the case. And since."ism"
+embraces whatever is, its name is quite fitting. In "ism" thL- word "is" is half mentioned,
+half used; what could be more appropriate? Ism is the spirit of Zen in art. And just as the
+central problem of Zen is to unmask the self, the central problem of art in this century
+seems to be to figure out what art is. All these thrashings-about are part of its identity
+crisis.
+
+We have seen that the use-mention dichotomy, when pushed, turns into the
+philosophical problem of symbol-object dualism, which links it to the mystery of mind.
+Magritte wrote about his painting The Human Condition I (Fig. 141):
+
+I placed in front of a window, seen from a room, a painting representing exactly
+that part of the landscape which was hidden from view by the painting. Therefore,
+the tree represented in the painting hid from view the tree situated behind it, outside
+the room. It existed for the spectator, as it were, simultaneously in his mind, as both
+inside the room in the painting, and outside in the real landscape. Which is how we
+see the world: we see it as being outside ourselves even though it is only a mental
+representation of it that we
+experience inside ourselves.'
+
+Understanding the Mind
+
+First through the pregnant images of his painting, and then in direct words, Magritte
+expresses the link between the two questions "How do symbols work?" and "How do our
+minds work?" And so he leads us back to the question posed earlier: "Can we ever hope
+to understand our minds! brains?"
+
+Or does some marvelous diabolical Godelian proposition preclude our ever
+unraveling our minds? Provided you do not adopt a totally unreasonable definition of
+"understanding", I see no Godelian obstacle in the way of the eventual understanding of
+our minds. For instance, it seems to me quite reasonable to desire to understand the
+working principles of brains in general, much the same way as we understand the
+working principles of car engines in general. It is quite different from trying to
+understand any single brain in every last detail-let alone trying to do this for one's own
+brain! I don't see how Godel’s Theorem, even if construed in the sloppiest way, has
+anything to say about the feasibility of this prospect. I see no reason that Godel’s
+Theorem imposes any limitations on our ability to formulate and verify the general
+mechanisms by which thought processes take place in the medium of nerve cells. I see no
+barrier imposed by Godel’s Theorem to the implementation on computers (or their
+successors) of types of symbol manipulation that achieve roughly the same results as
+brains do. It is entirely another question to try and duplicate in a program some particular
+human's mind-but to produce an intelligent program at all is a more limited goal. Godel's
+Theorem doesn't ban our reproducing our own level of intelligence via programs any
+more than it bans our reproducing our own level of intelligence via transmission of
+hereditary information in
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+701
+
+
+
+DNA, followed by education. Indeed, we have seen, in Chapter XVI, how a remarkable
+'Godelian mechanism-the Strange Loop of proteins and DNA-is precisely what allows
+transmission of intelligence!
+
+Does Godel’s Theorem, then, have absolutely nothing to offer us in thinking
+about our own minds? I think it does, although not in the mystical and [imitative way
+which some people think it ought to. I think that the process of coming to understand
+Godel’s proof, with its construction involving arbitrary codes, complex isomorphisms,
+high and low levels of interpretation, and the capacity for self-mirroring, may inject some
+rich undercurrents and flavors into one's set of images about symbols and symbol
+processing, which may deepen one's intuition for the relationship, between mental
+structures on different levels.
+
+Accidental Inexplicability of Intelligence?
+
+Before suggesting a philosophically intriguing "application" of Godel's proof. I would
+like to bring up the idea of "accidental inexplicability" of intelligence. Here is what that
+involves. It could be that our brains, unlike car engines, are stubborn and intractable
+systems which we cannot neatly decompose in any way. At present, we have no idea
+whether our brains will yield to repeated attempts to cleave them into clean layers, each
+of which can be explained in terms of lower layers-or whether our brains will foil all our
+attempts at decomposition.
+
+But even if we do fail to understand ourselves, there need not be any Godelian
+"twist" behind it; it could be simply an accident of fate that our brains are too weak to
+understand themselves. Think of the lowly giraffe, for instance, whose brain is obviously
+far below the level required for self-understanding-yet it is remarkably similar to our own
+brain. In fact, the brains of giraffes, elephants, baboons-even the brains of tortoises or
+unknown beings who are far smarter than we are-probably all operate on basically the
+same set of principles. Giraffes may lie far below the threshold of intelligence necessary
+to understand how those principles fit together to produce the qualities of mind; humans
+may lie closer to that threshold perhaps just barely below it, perhaps even above it. The
+point is that there may be no fundamental (i.e., Godelian) reason why those qualities are
+incomprehensible; they may be completely clear to more intelligent beings.
+
+Undecidability Is Inseparable from a High-Level Viewpoint
+
+Barring this pessimistic notion of the accidental inexplicability of the brain, what insights
+might Godel’s proof offer us about explanations of our minds/brains? Godel’s proof
+offers the notion that a high-level view of a system may contain explanatory power which
+simply is absent on the lower levels. By this I mean the following. Suppose someone
+gave you G, Godel’s undecidable string, as a string of TNT. Also suppose you knew
+nothing of Godel-numbering. The question you are supposed to answer is: "Why isn't
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+702
+
+
+
+this string a theorem of TNT?" Now you are used to such questions; for instance, if you
+had been asked that question about SO=0, you would have a ready explanation: "Its
+negation, ~S0=0, is a theorem ." This, together with your knowledge that TNT is
+consistent, provides an explanation of why the given string is a nontheorem. This is what
+I call an explanation "on the TNT-level". Notice how different it is from the explanation
+of why MU is not a theorem of the MlU-system: the former comes from the M-mode, the
+latter only from the I-mode.
+
+Now what about G? The TNT-level explanation which worked for 50=0 does not
+work for G, because - G is not a theorem. The person who has no overview of TNT will
+be baffled as to why he can't make G according to the rules, because as an arithmetical
+proposition, it apparently has nothing wrong with it. In fact, when G is turned into a
+universally quantified string, every instance gotten from G by substituting numerals for
+the variables can be derived. The only way to explain G's nontheoremhood is to discover
+the notion of Godel-numbering and view TNT on an entirely different level. It is not that
+it is just difficult and complicated to write out the explanation on the TNT-level; it is
+impossible. Such an explanation simply does not exist. There is, on the high level, a kind
+of explanatory power which simply is lacking, in principle, on the TNT-level. G's
+nontheoremhood is, so to speak, an intrinsically high-level fact. It is my suspicion that
+this is the case for all undecidable propositions; that is to say: every undecidable
+proposition is actually a Godel sentence, asserting its own nontheoremhood in some
+system via some code.
+
+Consciousness as an Intrinsically High-Level Phenomenon
+
+Looked at this way, Godel’s proof suggests-though by no means does it prove!-that there
+could be some high-level way of viewing the mind/brain, involving concepts which do
+not appear on lower levels, and that this level might have explanatory power that does not
+exist-not even in principle-on lower levels. It would mean that some facts could be
+explained on the high level quite easily, but not on lower levels at all. No matter how
+long and cumbersome a low-level statement were made, it would not explain the
+phenomena in question. It is the analogue to the fact that, if you make derivation after
+derivation in TNT, no matter how long and cumbersome you make them, you will never
+come up with one for G-despite the fact that on a higher level, you can see that G is true.
+
+What might such high-level concepts be? It has been proposed for eons, by
+various holistically or "soulistically" inclined scientists and humanists, that consciousness
+is a phenomenon that escapes explanation in terms of brain-components; so here is a
+candidate, at least. There is also the ever-puzzling notion of free will. So perhaps these
+qualities could be "emergent" in the sense of requiring explanations which cannot be
+furnished by the physiology alone. But it is important to realize that if we are being
+guided by Godel’s proof in making such bold hypotheses, we must carry the
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+703
+
+
+
+analogy through thoroughly. In particular, it is vital to recall tnat is s nontheoremhood
+does have an explanation-it is not a total mystery! The explanation- hinges on
+understanding not just one level at a time, but the way in which one level mirrors its
+metalevel, and the consequences of this mirroring. If our analogy is to hold, then,
+"emergent" phenomena would become explicable in terms of a relationship between,
+different levels in mental systems.,
+
+Strange Loops as the Crux of Consciousness
+
+My belief is that the explanations of "emergent" phenomena in our brains-for instance,
+ideas, hopes, images, analogies, and finally consciousness and free will-are based on a
+kind of Strange Loop, an interaction between levels in which the top level reaches back
+down towards the bottom level and influences it, while at the same time being itself
+determined by the bottom level. In other words, a self-reinforcing "resonance" between
+different levels-quite like the Henkin sentence which, by merely asserting its own
+provability, actually becomes provable. The self comes into being at the moment it has
+the power to reflect itself.
+
+This should not be taken as an antireductionist position. It just implies that a
+reductionistic explanation of a mind, in order to be comprehensible , must bring in "soft"
+concepts such as levels, mappings, and meanings. In principle, I have no doubt that a
+totally reductionistic but incomprehensible explanation of the brain exists; the problem is
+how to translate it into a language we ourselves can fathom. Surely we don't want a
+description in terms of positions and momenta of particles; we want a description which
+relates neural activity to "signals" (intermediate-level phenomena)-and which relates
+signals, in turn, to "symbols" and "subsystems", including the presumed-to-exist "self¬
+symbol". This act of translation from low-level physical hardware to high-level
+psychological software is analogous to the translation of number-theoretical statements
+into metamathematical statements. Recall that the level-crossing which takes place at this
+exact translation point is what creates Godel's incompleteness and the self-proving
+character of Henkin's sentence. I postulate that a similar level-crossing is what creates our
+nearly unanalyzable feelings of self.
+
+In order to deal with the full richness of the brain/mind system, we will have to be
+able to slip between levels comfortably. Moreover, we will have to admit various types of
+"causality": ways in which an event at one level of description can "cause" events at other
+levels to happen. Sometimes event A will be said to "cause" event B simply for the
+reason that the one is a translation, on another level of description, of the other.
+Sometimes "cause" will have its usual meaning: physical causality. Both types of
+causality-and perhaps some more-will have to be admitted in any explanation of mind,
+for we will have to admit causes that propagate both upwards and downwards in the
+Tangled Hierarchy of mentality, just as in the Central Dogmap.
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+704
+
+
+
+At the crux, then, of our understanding ourselves will come an understanding of
+the Tangled Hierarchy of levels inside our minds. My position is rather similar to the
+viewpoint put forth by the neuroscientist Roger Sperry in his excellent article "Mind,
+Brain, and Humanist Values", from which I quote a little here:
+
+In my own hypothetical brain model, conscious awareness does get representation
+as a very real causal agent and rates an important place in the causal sequence and
+chain of control in brain events, in which it appears as an active, operational
+force.... To put it very simply, it comes down to the issue of who pushes whom
+around in the population of causal forces that occupy the cranium. It is a matter, in
+other words, of straightening out the peck-order hierarchy among intracranial
+control agents. There exists within the cranium a whole world of diverse causal
+forces; what is more, there are forces within forces within forces, as in no other
+cubic half-foot of universe that we know. ... To make a long story short, if one
+keeps climbing upward in the chain of command within the brain, one finds at the
+very top those over-all organizational forces and dynamic properties of the large
+patterns of cerebral excitation that are correlated with mental states or psychic
+activity.... Near the apex of this command system in the brain ... we find ideas. Man
+over the chimpanzee has ideas and ideals. In the brain model proposed here, the
+causal potency of an idea, or an ideal, becomes just as real as that of a molecule, a
+cell, or a nerve impulse. Ideas cause ideas and help evolve new ideas. They interact
+with each other and with other mental forces in the same brain, in neighboring
+brains, and, thanks to global communication, in far distant, foreign brains. And they
+also interact with the external surroundings to produce in toto a burstwise advance
+in evolution that is far beyond anything to hit the evolutionary scene yet, including
+the emergence of the living cell.'
+
+There is a famous breach between two languages of discourse: the subjective
+language and the objective language. For instance, the "subjective" sensation of redness,
+and the "objective" wavelength of red light. To many people, these seem to be forever
+irreconcilable. I don't think so. No more than the two views of Escher's Drawing Hands
+are irreconcilable from "in the system", where the hands draw each other, and from
+outside, where Escher draws it all. The subjective feeling of redness comes from the
+vortex of self-perception in the brain; the objective wavelength is how you see things
+when you step back, outside of the system. Though no one of us will ever be able to step
+back far enough to see the "big picture", we shouldn't forget that it exists. We should
+remember that physical law is what makes it all happen-way, way down in neural nooks
+and crannies which are too remote for us to reach with our high-level introspective
+probes.
+
+
+The Self-Symbol and Free Will
+
+In Chapter XI I, it was suggested that what we call free will is a result of the interaction
+between the self-symbol (or subsystem), and the other symbols in the brain. If we take
+the idea that symbols are the high-level entities to
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+705
+
+
+
+which meanings should be attached, then we can' make a stab at explaining the
+relationship between symbols, the self-symbol, and free will.
+
+One way to gain some perspective on the free-will question is to replace it by
+what I believe is an equivalent question, but one which involves less loaded terms.
+Instead of asking, "Does system X have free will?" we ask, "Does system X make
+choices?" By carefully groping for what we really mean when we choose to describe a
+system-mechanical or biological-as being capable of making "choices", I think we can
+shed much light on free will it will be helpful to go over a few different systems which,
+under various circumstances, we might feel tempted to describe as "making choices".
+From these examples we can gain some perspective on what we really mean by the
+phrase.
+
+Let us take the following systems as paradigms: a marble rolling down a bumpy
+hill; a pocket calculator finding successive digits in the decimal expansion of the square
+root of 2; a sophisticated program which plays a mean game of chess; a robot in a T-maze
+(a maze with but a single fork, on one side of which there is a reward); and a human
+being confronting a complex dilemma.
+
+First, what about that marble rolling down a hill? Does it make choices? I think
+we would unanimously say that it doesn't, even though none of us could predict its path
+for even a very short distance. We feel that it couldn't have gone any other way than it
+did, and that it was just being shoved along by the relentless laws of nature. In our
+chunked mental physics, of course, we can visualize many different "possible" pathways
+for the marble, and we see it following only one of them in the real world. On some level
+of our minds, therefore, we can't help feeling the marble has "chosen" a single pathway
+out of those myriad mental ones; but on some other level of our minds, we have an
+instinctive understanding that the mental physics is only an aid in our internal modeling
+of the world, and that the mechanisms which make the real physical sequences of events
+happen do not require nature to go through an analogous process of first manufacturing
+variants in some hypothetical universe (the "brain of God") and then choosing between
+them. So we shall not bestow the designation "choice" upon this process-although we
+recognize that it is often pragmatically useful to use the word in cases like this, because
+of its evocative power.
+
+Now what about the calculator programmed to find the digits of the square root of
+2? What about the chess program? Here, we might say that we are just dealing with
+"fancy marbles", rolling down "fancy hills". In fact, the arguments for no choice-making
+here are, if anything, stronger than in the case of a marble. For if you attempt to repeat
+the marble experiment, you will undoubtedly witness a totally different pathway being
+traced down the hill, whereas if you rerun the square-root-of-2 program, you will get the
+same results time after time. The marble seems to "choose" a different path each time, no
+matter how accurately you try to reproduce the conditions of its original descent, whereas
+the program runs down precisely the same channels each time.
+
+Now in the case of fancy chess programs, there are various possibilities.
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+706
+
+
+
+If you play a game against certain programs, and then start a second game with the same
+moves as you made the first time, these programs will just move exactly as they did
+before, without any appearance of having learned anything or having any desire for
+variety. There are other programs which have randomizing devices that will give some
+variety but not out of any deep desire. Such programs could be reset with the internal
+random number generator as it was the first time, and once again, the same game would
+ensue. Then there are other programs which do learn from their mistakes, and change
+their strategy depending on the outcome of a game. Such programs would not play the
+same game twice in a row. Of course, you could also turn the clock back by wiping out
+all the changes in the memory which represent learning, just as you could reset the
+random number generator, but that hardly seems like a friendly thing to do. Besides, is
+there any reason to suspect that you would be able to change any of your own past
+decisions if every last detail-and that includes your brain, of course-were reset to the way
+it was the first time around?
+
+But let us return to the question of whether "choice" is an applicable term here. If
+programs are just "fancy marbles rolling down fancy hills", do they make choices, or not?
+Of course the answer must be a subjective one, but I would say that pretty much the same
+considerations apply here as to the marble. However, I would have to add that the appeal
+of using the word "choice", even if it is only a convenient and evocative shorthand,
+becomes quite strong. The fact that a chess program looks ahead down the various
+possible bifurcating paths, quite unlike a rolling marble, makes it seem much more like
+an animate being than a square-root-of-2 program. However, there is still no deep self-
+awareness here-and no sense of free will.
+
+Now let us go on to imagine a robot which has a repertoire of symbols. This robot
+is placed in a T-maze. However, instead of going for the reward, it is preprogrammed to
+go left whenever the next digit of the square root: of 2 is even, and to go right whenever it
+is odd. Now this robot is capable of modeling the situation in its symbols, so it can watch
+itself making choices. Each time the T is approached, if you were to address to the robot
+the question, "Do you know which way you're going to turn this time?" it would have to
+answer, "No". Then in order to progress, it would activate its "decider" subroutine, which
+calculates the next digit of the square root of 2, and the decision is taken. However, the
+internal mechanism of the decider is unknown to the robot-it is represented in the robot's
+symbols merely as a black box which puts out "left"'s and "right'"s by some mysterious
+and seemingly random rule. Unless the robot's symbols are capable of picking up the
+hidden heartbeat of the square root of 2, beating in the L's and R's, it will stay baffled by
+the "choices" which it is making. Now does this robot make choices? Put yourself in that
+position. If you were trapped inside a marble rolling down a hill and were powerless to
+affect its path, yet could observe it with all your human intellect, would you feel that the
+marble's path involved choices? Of course not. Unless your mind is affecting the
+outcome, it makes no difference that the symbols are present.
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+707
+
+
+
+So now we make a modification in our robot: we allow its symbols—including its self-
+symbol-to affect the decision that is taken. Now here is an example of a program running
+fully under physical law, which seems to get much more deeply at the essence of choice
+than the previous examples did. When the robot's own chunked concept of itself enters
+the scene, we begin to identify with the robot, for it sounds like the kind of thing we do. It
+is no longer like the calculation of the square root of 2, where no symbols seem to be
+monitoring the decisions taken. To be sure, if we were to look at the robot's program on a
+very local level, it would look quite like the square-root program. Step after step is
+executed, and in the end "left" or "right" is the output. But on a high level we can see the
+fact that symbols are being used to model the situation and to affect the decision. That
+radically affects our way of thinking about the program. At this stage, meaning has
+entered this picture-the same kind of meaning as we manipulate with our own minds.
+
+A Godel Vortex Where All Levels Cross
+
+Now if some outside agent suggests 'L' as the next choice to the robot, the suggestion
+will be picked up and channeled into the swirling mass of interacting symbols. There, it
+will be sucked inexorably into interaction with the self-symbol, like a rowboat being
+pulled into a whirlpool. That is the vortex of the system, where all levels cross. Here, the
+'L' encounters a Tangled Hierarchy of symbols and is passed up and down the levels. The
+self-symbol is incapable of monitoring all its internal processes, and so when the actual
+decision emerges-'L' or 'R' or something outside the system-the system will not be able to
+say where it came from. Unlike a standard chess program, which does not monitor itself
+and consequently has no ideas about where its moves come from, this program does
+monitor itself and does have ideas about its ideas-but it cannot monitor its own processes
+in complete detail, and therefore has a sort of intuitive sense of its workings, without full
+understanding. From this balance between self-knowledge and self-ignorance comes the
+feeling of free will.
+
+Think, for instance, of a writer who is trying to convey certain ideas which to him
+are contained in mental images. He isn't quite sure how those images fit together in his
+mind, and he experiments around, expressing things first one way and then another, and
+finally settles on some version. But does he know where it all came from? Only in a
+vague sense. Much of the source, like an iceberg, is deep underwater, unseen-and he
+knows that. Or think of a music composition program, something we discussed earlier,
+asking when we would feel comfortable in calling it the composer rather than the tool of
+a human composer. Probably we would feel comfortable when self-knowledge in terms
+of symbols exists inside the program, and when the program has this delicate balance
+between self-knowledge and self-ignorance. It is irrelevant whether the system is running
+deterministically; what makes us call it a "choice maker" is whether we can identify with
+a high-level description of the process which takes place when the
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+708
+
+
+
+
+FIGURE 142. Print Gallery, byM. C. Escher (lithograph, 1956).
+
+
+program runs. On a low (machine language) level, the program looks like any other
+program; on a high (chunked) level, qualities such as "will", "intuition", "creativity", and
+"consciousness" can emerge.
+
+The important idea is that this "vortex" of self is responsible for the tangledness,
+for the Godelian-ness, of the mental processes. People have said to me on occasion, "This
+stuff with self-reference and so on is very amusing and enjoyable, but do you really think
+there is anything serious to it?" I certainly do. I think it will eventually turn out to be at
+the core of AI, and the focus of all attempts to understand how human minds work. And
+that is why Godel is so deeply woven into the fabric of my book.
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+709
+
+
+
+
+
+
+
+
+An Escher Vortex Where All Levels Cross
+
+
+A strikingly beautiful, and yet at the same time disturbingly grotesque, illustration of the
+cyclonic "eye" of a Tangled Hierarchy is given to us by Escher in his Print Gallery (Fig.
+
+142) . What we see is a picture gallery where a young man is standing, looking at a
+picture of a ship in the harbor of a small town, perhaps a Maltese town, to guess from the
+architecture, with its little turrets, occasional cupolas, and flat stone roofs, upon one of
+which sits a boy, relaxing in the heat, while two floors below him a woman-perhaps his
+mother-gazes out of the window from her apartment which sits directly above a picture
+gallery where a young man is standing, looking at a picture of a ship in the harbor of a
+small town, perhaps a Maltese town—What!? We are back on the same level as we began,
+though all logic dictates that we cannot be. Let us draw a diagram of what we see (Fig.
+
+143) .
+
+
+
+FIGURE 143. Abstract diagram ofM. C. Escher's Print Gallery.
+
+What this diagram shows is three kinds of "in-ness". The gallery is physically in the town
+("inclusion"); the town is artistically in the picture ("depiction"); the picture is mentally
+in the person ("representation"). Now while this diagram may seem satisfying, in fact it is
+arbitrary, for the number of levels shown is quite arbitrary. Look below at another way of
+representing the top half alone (Fig. 144).
+
+
+
+inclusion
+
+
+FIGURE 144. A collapsed version of the previous figure.
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+710
+
+
+
+We have eliminated the "town" level; conceptually it was useful, but can just as well be
+done without. Figure 144 looks just like the diagram for Drawing Hands: a Strange Loop
+of two steps. The division markers are arbitrary, even if they seem natural to our minds.
+This can be further accentuated by showing even more "collapsed" schematic diagrams of
+Print Gallery , such as that in Figure 145.
+
+
+inclusion + depiction
+
+
+
+FIGURE 145. Further collapse of Figure 143.
+
+This exhibits the paradox of the picture in the starkest terms. Now-if the picture is "inside
+itself', then is the young man also inside himself-. This question is answered in Figure
+146.
+
+
+inclusion + depiction + representation
+
+
+
+FIGURE 146. Another way of collapsing Figure 143.
+
+Thus, we see the young man "inside himself, in a funny sense which is made up of
+compounding three distinct senses of "in”.
+
+This diagram reminds us of the Epimenides paradox with its one-step self¬
+reference, while the two-step diagram resembles the sentence pair each of which refers to
+the other. We cannot make the loop any tighter, but we can open it wider, by choosing to
+insert any number of intermediate levels, such as "picture frame", "arcade", and
+"building". If we do so, we will have many-step Strange Loops, whose diagrams are
+isomorphic to those of Waterfall (Fig. 5) or Ascending and Descending (Fig. 6). The
+number of levels is determined by what we feel is "natural", which may vary according to
+context, purpose, or frame of mind. The Central Xmaps-Dog, Crab, Sloth, and Pipe-can
+all be seen as involving three-step Strange Loops; alternatively, they can all be collapsed
+into two- or one-step loops;, then again, they can be expanded out into multistage loops.
+Where one perceives the levels is a matter of intuition and esthetic preference.
+
+Now are we, the observers of Print Gallery, also sucked into ourselves by virtue
+of looking at it? Not really. We manage to escape that particular vortex by being outside
+of the system. And when we look at the picture, we see things which the young man can
+certainly not see, such as Escher’s
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+711
+
+
+
+Signature, "MCE", in the central "blemish". Though the blemish seems like a defect,
+perhaps the defect lies in our expectations, for in fact Escher could not have completed
+that portion of the picture without being inconsistent with the rules by which he was
+drawing the picture. That center of the whorl is-and must be-incomplete. Escher could
+have made it arbitrarily small, but he could not have gotten rid of it. Thus we, on the
+outside, can know that Print Gallery is essentially incomplete-a fact which the young
+man, on the inside, can never know. Escher has thus given a pictorial parable for Godel’s
+Incompleteness Theorem. And that is why the strands of Godel and Escher are so deeply
+interwoven in my book.
+
+A Bach Vortex Where All Levels Cross
+
+One cannot help being reminded, when one looks at the diagrams of Strange Loops, of
+the Endlessly Rising Canon from the Musical Offering. A diagram of it would consist of
+six steps, as is shown in Figure 147. It is too .
+
+
+
+FIGURE 147. The hexagonal modulation scheme of Bach's Endlessly Rising Canon
+forms a true dosed loop when Shepard tones are used.
+
+bad that when it returns to C, it is an octave higher rather than at the exact original pitch.
+Astonishingly enough, it is possible to arrange for it to return exactly to the starting pitch,
+by using what are called Shepard tones, after the psychologist Roger Shepard, who
+discovered the idea. The principle of a Shepard-tone scale is shown in Figure 14$. In
+words, it is this: you play parallel scales in several different octave ranges. Each note is
+weighted independently, and as the notes rise, the weights shift. You make the top
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+712
+
+
+
+178. 4LL»_
+~*iTn iilW-
+
+■ttu *
+
+’T%
+
+%iiL
+
+*t -It *
+
+ni
+
+■'» I % J V.
+
+ftjfct : 5 rj “
+
+TTOri|nl , ‘"
+
+firf* »_l *
+
+iTtlwiri
+
+mTiSWil-
+
+
+1 ii
+
+
+1 Iff %-
+
+
+« TT **
+
+
+I 7 "*. %_!*_
+Vt: U «-
+
+
+ttj: ■ f*-
+
+iff iuk
+
+
+»■
+
+
+U 3ft
+
+
+
+‘;^-Cn
+
+
+FIGURE 148: Two complete cycles of a Shephard tone scale, notated for piano. The
+loudness of each note is proportional to its area, just as the top voice fades out, a new
+bottom voice feebly enters. (Printed by Donald Boyd's program “SMUT”.)
+
+
+Strange Loops, Or Tangled Hierarchies
+
+
+713
+
+
+
+
+octave gradually fade out, while at the same time you are gradually bringing in the
+bottom octave. Just at the moment you would ordinarily be one octave higher, the
+weights have shifted precisely so as to reproduce the starting pitch ... Thus you can go
+"up and up forever", never getting any higher! You can try it at your piano. It works even
+better if the pitches can be synthesized accurately under computer control. Then the
+illusion is bewilderingly strong.
+
+This wonderful musical discovery allows the Endlessly Rising Canon to be played
+in such a way that it joins back onto itself after going "up" an octave. This idea, which
+Scott Kim and I conceived jointly, has been realized on tape, using a computer music
+system. The effect is very subtle-but very real. It is quite interesting that Bach himself
+was apparently aware, in some sense, of such scales, for in his music one can
+occasionally find passages which roughly exploit the general principle of Shepard tones-
+for instance, about halfway through the Fantasia from the Fantasia and Fugue in G Minor,
+for organ.
+
+In his book /. S. Bach's Musical Offering, Hans Theodore David writes:
+
+Throughout the Musical Offering, the reader, performer, or listener is to search for
+the Royal theme in all its forms. The entire work, therefore, is a ricercar in the
+original, literal sense of the word.'
+
+I think this is true; one cannot look deeply enough into the Musical Offering. There is
+always more after one thinks one knows everything. For instance, towards the very end of
+the Six-Part Ricercar, the one he declined to improvise, Bach slyly hid his own name,
+split between two of the upper voices. Things are going on on many levels in the Musical
+Offering. There are tricks with notes and letters; there are ingenious variations on the
+King's Theme; there are original kinds of canons; there are extraordinarily complex
+fugues; there is beauty and extreme depth of emotion; even an exultation in the many-
+leveledness of the work comes through. The Musical Offering is a fugue of fugues, a
+Tangled Hierarchy like those of Escher and Godel, an intellectual construction which
+reminds me, in ways I cannot express, of the beautiful many-voiced fugue of the human
+mind. And that is why in my book the three strands of Godel, Escher, and Bach are
+woven into an Eternal Golden Braid.
+
+
+Strange Foops, Or Tangled Hierarchies
+
+
+714
+
+
+
+Six-Part Ricercar
+
+
+Achilles has brought his cello to the Crab's residence, to engage in an evening of
+chamber music with the Crab and Tortoise. He has been shown into the music
+room by his host the Crab, who is momentarily absent, having gone to meet their
+mutual friend the Tortoise at the door. The room is filled with all sorts of electronic
+equipment-phonographs in various states of array and disarray, television screens
+attached to typewriters, and other quite improbable-looking pieces of apparatus.
+Nestled amongst all this high-powered gadgetry sits a humble radio. Since the
+radio is the only thing in the room which Achilles knows how to use, he walks over
+to it, and, a little furtively, flicks the dial and f nds he has tuned into a panel
+discussion by six learned scholars on free will and determinism. He listens briefly
+and then, a little scornfully, flicks it off.
+
+Achilles: I can get along very well without such a program. After all, it's clear to anyone
+who's ever thought about it that-I mean, it's not a very difficult matter to resolve, once
+you understand how-or rather, conceptually, one can clear up the whole thing by
+thinking of, or at least imagining a situation where ... Hmmm ... I thought it was quite
+clear in my mind. Maybe I could benefit from listening to that show, after all...
+
+(Enter the Tortoise, carrying his violin.)
+
+Well, well, if it isn't our fiddler. Have you been practicing faithfully this week, Mr. T?
+I myself have been playing the cello part in the Trio Sonata from the Musical Offering
+for at least two hours a day. It's a strict regimen, but it pays off.
+
+Tortoise: I can get along very well without such a program. I find that a moment here, a
+moment there keeps me fit for fiddling.
+
+Achilles: Oh, lucky you. I wish it came so easily to me. Well, where is our host?
+
+Tortoise: I think he's just gone to fetch his flute. Here he comes.
+
+(Enter the Crab, carrying his flute.)
+
+Achilles: Oh, Mr. Crab, in my ardent practicing of the Trio Sonata this past week, all
+sorts of images bubbled into my mind: jolly gobbling bumblebees, melancholy
+buzzing turkeys, and a raft of others. Isn't it wonderful, what power music has?
+
+Crab: I can get along very well without such a program. To my mind.
+
+Achilles, there is no music purer than the Musical Offering.
+
+Tortoise: You can't be serious, Achilles. The Musical Offering isn't programmatic music!
+
+
+Six-Part Ricercar
+
+
+720
+
+
+
+Achilles: Well, I like animals, even if you two stuffy ones disapprove.
+
+Crab: I don't think we are so stuffy, Achilles. Let's just say that you hear music in 'your
+own special way.
+
+Tortoise: Shall we sit down and play?
+
+Crab: I was hoping that a pianist friend of mine would turn up and play continuo. I've
+been wanting you to meet him, Achilles, for a long time. Unfortunately, it appears
+that he may not make it. So let's just go ahead with the three of us. That's plenty for a
+trio sonata.
+
+Achilles: Before we start, I just was wondering, Mr. Crab-what are all these pieces of
+equipment, which you have in here?
+
+Crab: Well, mostly they are just odds and ends-bits and pieces of old broken
+phonographs. Only a few souvenirs (nervously tapping the buttons), a few souvenirs
+of-of the TC-battles in which I have distinguished myself. Those keyboards attached
+to television screens, however, are my new toys. I have fifteen of them around here.
+They are a new kind of computer, a very small, very flexible type of computer quite
+an advance over the previous types available. Few others seem to be quite as
+enthusiastic about them as I am, but I have faith that they will catch on in time.
+
+Achilles: Do they have a special name?
+
+Crab: Yes; they are called "smart-stupids", since they are so flexible, and have the
+potential to be either smart or stupid, depending on how skillfully they are instructed.
+
+Achilles: Do you mean you think they could actually become smart like, say, a human
+being?
+
+Crab: I would not balk at saying so-provided, of course, that someone sufficiently versed
+in the art of instructing smart-stupids would make the effort. Sadly, I am not
+personally acquainted with anyone who is a true virtuoso. To be sure, there is one
+expert abroad in the land, an individual of great renown-and nothing would please me
+more than a visit by him, so that I could appreciate what true skill on the smart-stupid
+is; but he has never come, and I wonder if I shall ever have that pleasure.
+
+Tortoise: It would be very interesting to play chess against a well-instructed smart-stupid.
+
+Crab: An extremely intriguing idea. That would be a wonderful mark of skill, to program
+a smart-stupid to play a good game of chess. Even more interesting-but incredibly
+complicated-would be to instruct a smart-stupid sufficiently that it could hold its own
+in a conversation. It might give the impression that it was just another person!
+
+Achilles: Curious that this should come up, for I just heard a snatch of a discussion on
+free will and determinism, and it set me to thinking about such questions once more. I
+don't mind admitting that, as I pondered the idea, my thoughts got more and more
+tangled, and in the end I really didn't know what I thought. But this idea of a smart-
+stupid that could converse with you ... it boggles the mind. I mean,
+
+
+Six-Part Ricercar
+
+
+721
+
+
+
+what would the smart-stupid itself say, if you asked it for its opinion on the free-will
+question? I was just wondering if the two of you, who know so much about these
+things, wouldn't indulge me by explaining the issue, as you see it, to me.
+
+Crab: Achilles, you can't imagine how appropriate your question is. I only wish my
+pianist friend were here, because I know you'd be intrigued to hear what he could tell
+you on the subject. In his absence, I'd like to tell you a statement in a Dialogue at the
+end of a book I came across recently.
+
+Achilles: Not Copper, Silver, Gold: an Indestructible Metallic Alloy?
+
+Crab: No, as I recall, it was entitled Giraffes, Elephants, Baboons: an Equatorial
+Grasslands Bestiary-ox something like that. In any case, towards the end of the
+aforementioned Dialogue, a certain exceedingly droll character quotes Marvin
+Minsky on the question of free will. Shortly thereafter, while interacting with two
+other personages, this droll character quotes Minsky further on musical improvisation,
+the computer language LISP, and Godel's Theorem-and get this-all without giving one
+whit of credit to Minsky!
+
+Achilles: Oh, for shame!
+
+Crab: I must admit that earlier in the Dialogue, he hints that he WILL quote Minsky
+towards the end; so perhaps it's forgivable.
+
+Achilles: It sounds that way to me. Anyway, I'm anxious to hear the Minskian
+pronouncement on the free will question.
+
+Crab: Ah, yes... Marvin Minsky said, "When intelligent machines are constructed, we
+should not be surprised to find them as confused and as stubborn as men in their
+convictions about mind-matter, consciousness, free will, and the like."
+
+Achilles: I like that! Quite a funny thought. An automaton thinking it had free will! That's
+almost as silly as me thinking I didn't have free will! Tortoise: I suppose it never
+occurred to you, Achilles, that the three of us-you, myself, and Mr. Crab-might all be
+characters in a Dialogue, perhaps even one similar to the one Mr. Crab just
+mentioned. Achilles: Oh, it's occurred to me, of course. I suppose such fancies occur
+to every normal person at one time or another.
+
+Tortoise: And the Anteater, the Sloth, Zeno, even GOD-we might all be characters in a
+series of Dialogues in a book.
+
+Achilles: Sure, we might. And the Author might just come in and play the piano, too.
+
+Crab: That's just what I had hoped. But he's always late.
+
+Achilles: Whose leg do you think you're pulling? I know I'm not being controlled in any
+way by another mentality! I've got my own thoughts, I express myself as I wish-you
+can't deny that!
+
+Tortoise: Nobody denied any of that, Achilles. But all of what you say is perfectly
+consistent with your being a character in a Dialogue.
+
+Crab: The—
+
+Achilles: But-but-no! Perhaps Mr. C's article and my rebuttal have both
+
+
+Six-Part Ricercar
+
+
+722
+
+
+
+been mechanically determined, but this I refuse to believe. I can accept physical
+determinism, but I cannot accept the idea that I am but a figment inside of someone
+else's mentality!
+
+Tortoise: It doesn't really matter whether you have a hardware brain, Achilles. Your will
+can be equally free, if your brain is just a piece of software inside someone else's
+hardware brain. And their brain, too, may be software in a yet higher brain .. .
+
+Achilles: What an absurd idea! And yet, I must admit, I do enjoy trying to find the
+cleverly concealed holes in your sophistry, so go ahead. Try to convince me. I'm
+game.
+
+Tortoise: Did it ever strike you, Achilles, that you keep somewhat unusual company?
+
+Achilles: Of course. You are very eccentric (I know you won't mind my saying so), and
+even Mr. Crab here is a weensy bit eccentric. (Pardon me, Mr. Crab.)
+
+Crab: Oh, don't worry about offending me.
+
+Tortoise: But Achilles, you've overlooked one of the most salient features of your
+acquaintances.
+
+Achilles: Which is.... ?
+
+Tortoise: That we're animals!
+
+Achilles: Well, well-true enough. You have such a keen mind. I would never have
+thought of formulating the facts so concisely.
+
+Tortoise: Isn't that evidence enough? How many people do you know who spend their
+time with talking Tortoises, and talking Crabs? Achilles: I must admit, a talking Crab
+is
+
+Crab: -an anomaly, of course.
+
+Achilles: Exactly; it is a bit of an anomaly-but it has precedents. It has occurred in
+literature.
+
+Tortoise: Precisely-in literature. But where in real life?
+
+Achilles: Now that you mention it, I can't quite say. I'll have to give it some thought. But
+that's not enough to convince me that I'm a character in a
+
+Dialogue. Do you have any other arguments?
+
+Tortoise: Do you remember one day when you and I met in the park, seemingly at
+random?
+
+Achilles: The day we discussed crab canons by Escher and Bach? Tortoise: The very one!
+
+Achilles: And Mr. Crab, as I recall, turned up somewhere towards the middle of our
+conversation and babbled something funny and then left.
+
+Crab: Not just "somewhere towards the middle", Achilles. EXACTLY in the middle.
+
+Achilles: Oh, all right, then.
+
+Tortoise: Do you realize that your lines were the same as my lines in that conversation-
+except in reverse order? A few words were changed here and there, but in essence
+there was a time symmetry to our encounter.
+
+
+Six-Part Ricercar
+
+
+723
+
+
+
+Achilles: Big Deal! It was just some sort of trickery. Probably all done with mirrors.
+
+Tortoise: No trickery. Achilles, and no mirrors: just the work of an assiduous Author.
+
+Achilles: Oh, well, it's all the same to me.
+
+Tortoise: Fiddle' It makes a big difference, you know.
+
+Achilles: Say, something about this conversation strikes me as familiar. Haven't I heard
+some of those lines somewhere before= Tortoise: You said it, Achilles.
+
+Crab: Perhaps those lines occurred at random in the park one day, Achilles. Do you recall
+how your conversation with Mr. T ran that day?
+
+Achilles: Vaguely. He said "Good day, Mr. A" at the beginning, and at the end, I said,
+"Good day, Mr. T". Is that right
+
+Crab: I just happen to have a transcript right here ...
+
+(He fishes around in his music case, whips out a sheet, and hands it to Achilles. As
+Achilles reads it, he begins to squirm and fidget noticeable.)
+
+Achilles: This is very strange. Very, very strange ... All of a Sudden, I feel sort of-weird.
+It's as if somebody had actually planned out that whole set of statements in advance,
+worked them out on paper or something . As if some Author had had a whole agenda
+and worked from it in detail in planning all those statements I made that day.
+
+(At that moment, the door bursts open. Enter the Author, carrying a giant
+manuscript.)
+
+Author: I can get along very well without such a program. You see, once my characters
+are formed, they seem to have lives of their own, and I need to exert very little effort
+in planning their lines.
+
+Crab: Oh, here you are!' I thought you'd never arrive!
+
+Author: Sorry to be so late. I followed the wrong road and wound up very far away. But
+somehow I made it back. Good to see you again, Mr. T and Mr. C. And Achilles, I'm
+especially glad to see you.
+
+Achilles: Who are you? I've never seen you before.
+
+Author: I am Douglas Hofstadter-please call me Doug-and I'm presently finishing up a
+book called Godel, Escher, Bach: an Eternal Golden Braid. It is the book in which
+the three of you are characters.
+
+Achilles: Pleased to meet you. My name is Achilles, and-
+
+Author: No need to introduce yourself, Achilles, since I already know you quite well.
+
+Achilles: Weird, weird.
+
+Crab: He's the one I was saying might drop in and play continuo with us.
+
+Author: I've been playing the Musical Offering a little bit on my piano at home, and I can
+try to blunder my way through the Trio Sonata providing you'll overlook my many
+wrong notes.
+
+Tortoise: Oh, we're very tolerant around here, being only amateurs our selves.
+
+
+Six-Part Ricercar
+
+
+724
+
+
+
+Author: I hope you don't mind, Achilles, but I'm to blame for the tact that you and Mr.
+Tortoise said the same things, but in reverse order, that day in the park.
+
+Crab: Don't forget me' I was there, too right in the middle, putting in my two bits' worth!
+
+Author: Of course! You were the Crab in the Crab Canon.
+
+Achilles: So you are saying you control my utterances;, That my brain is a software
+subsystem of yours?
+
+Author: You can put it that way if you want, Achilles.
+
+Achilles: Suppose I were to write dialogues. Who would the author of them beg You, or
+me?
+
+Author: You, of course. At least in the fictitious world which you inhabit, you'd get credit
+for them.
+
+Achilles: Fictitious? I don't see anything fictitious about it!
+
+Author: Whereas in the world I inhabit, perhaps the credit would be given to me,
+although I am not sure if it would be proper to do so. And then, whoever made me
+make you write your dialogues would get credit in his world (seen from which, MY
+world looks fictitious).
+
+Achilles: That's quite a bit to swallow. I never imagined there could be a world above
+mine before-and now you're hinting that there could even be one above that. It's like
+walking up a familiar staircase, and just keeping on going further up after you've
+reached the top-or what you'd always taken to be the top!
+
+Crab: Or waking up from what you took to be real life, and finding out it too was just a
+dream. That could happen over and over again, no telling when it would stop.
+
+Achilles: It's most perplexing how the characters in my dreams have wills of their own,
+and act out parts which are independent of MY will. It's as if my mind, when I'm
+dreaming, merely forms a stage on which certain other organisms act out their lives.
+And then, when I awake, they go away. I wonder where it is they go to ...
+
+Author: They go to the same place as the hiccups go, when you get rid of them:
+Tumbolia. Both the hiccups and the dreamed beings are software suborganisms which
+exist thanks to the biology of the outer host organism. The host organism serves as
+stage to them-or even as their universe. They play out their lives for a time-but when
+the host organism makes a large change of state-for example, wakes up-then the
+suborganisms lose their coherency, and case existing as separate, identifiable units.
+
+Achilles: Is it like castles in the sand which vanish when a wave washes over them?
+
+Author: Very much like that, Achilles. Hiccups, dream characters, and even Dialogue
+characters disintegrate when their host organism undergoes certain critical changes of
+state. Yet, just like those sand castles you described, everything which made them up
+is still present.
+
+Achilles: I object to being likened to a mere hiccup!
+
+
+Six-Part Ricercar
+
+
+725
+
+
+
+Author: But I am also comparing you to a sand castle, Achilles. Is that not poetic?
+Besides, you may take comfort in the fact that if you are but a hiccup in my brain, I
+myself am but a hiccup in some higher author's brain.
+
+Achilles: But I am such a physical creature-so obviously made of flesh and blood and
+hard bones. You can't deny that'
+
+Author: I can't deny your sensation of it, but remember that dreamed beings, although
+they are just software apparitions, have the same sensation, no less than you do.
+
+Tortoise: I say, enough of this talk! Let us sit down and make music!
+
+Crab: A fine idea-and now we have the added pleasure of the company of our Author,
+who will grace our ears with his rendition of the bass line to the Trio Sonata, as
+harmonized by Bach's pupil Kirnberger. How fortunate are we! (Leads the author to
+one of his pianos.) I hope Not, find the seat comfortable enough. To adjust it, you- (In
+the background there is heard a Junn~ soft oscillating sound.)
+
+Tortoise: Excuse me, but what was that strange electronic gurgle?
+
+Crab: Oh, just a noise from one of the smart-stupids. Such a noise generally signals the
+fact that a new notice has flashed onto the screen. Usually the notices are just
+unimportant announcements coming from the main monitor program, which controls
+all the smart-stupids. (With his flute in his hand, he walks over to a smart-stupid, and
+reads its screen. Immediately he turns to the assembled musicians, and says, with a
+kind of agitation:) Gentlemen, old Ba. Ch. is come. (He lays the flute aside.) We must
+show him in immediately, of course.
+
+Achilles: Old Ba. Ch.! Could it be that that celebrated improviser of yore has chosen to
+show up tonight-HERE%
+
+Tortoise: Old Ba. Ch.! There's only one person THAT could mean-the renowned
+Babbage, Charles, Esq., M.A., F.R.S., F.R.S.E., F.R.A.S., F. STAT. S„ HON.
+M.R.LA., M.C.P.S., Commander of the Italian Order of St. Maui-ice and St. Lazarus,
+INST. IMP. (ACAD. MORAL.) PARIS CORR., ACAD. AMER. ART. ET SC.
+BOSTON, REG. OECON. BORCSS., PHYS. HISI. NAT. GENEV., ACAD. REG.
+MONAC., HAFN., MASSIL., ET DIVION., SOCIUS., ACAD. IMP., ET REG.
+PETROP., NEAP., BRUX., PATAV., GEORG. FLOREN, LYNCEI ROM., MCT.,
+PHILOMATH., PARIS, SOC. CORR., etc.-and Member of the Extractors' Club.
+Charles Babbage is a venerable pioneer of the art and science of computing. What a
+rare privilege!
+
+Crab: His name is known far and wide, and I have long hoped that he would give us the
+honor of a visit-but this is a totally unexpected surprise.
+
+Achilles: Does he play a musical instrument?
+
+Crab: I have heard it said that in the past hundred years, he has grown inexplicably fond
+of tom-toms, halfpenny whistles, and sundry other street instruments.
+
+Achilles: In that case, perhaps he might join us in our musical evening. .Author: I suggest
+that we give him a ten-canon salute.
+
+
+Six-Part Ricercar
+
+
+726
+
+
+
+Tortoise: A performance of all the celebrated canons from the Musical Offering.
+
+Author: Precisely.
+
+Crab: Capital suggestion! Quick, Achilles, you draw up a list of all ten of them, in the
+order of performance, and hand it to him as he comes in!
+
+(Before Achilles can move, enter Babbage, carrying a hurdy-gurdy, and wearing a
+heavy traveling coat and hat. He appears slightly travel-weary and disheveled.)
+
+Babbage: I can get along very well without such a program. Relax; I Can Enjoy Random
+Concerts And Recitals.
+
+Crab: Mr. Babbage! It is my deepest pleasure to welcome you to "Madstop", my humble
+residence. I have been ardently desirous of making your acquaintance for many years,
+and today my wish is at last fulfilled.
+
+Babbage: Oh, Mr. Crab, I assure you that the honor is truly all mine, to meet someone so
+eminent in all the sciences as yourself, someone whose knowledge and skill in music
+are irreproachable, and someone whose hospitality exceeds all bounds. And I am sure
+that you expect no less than the highest sartorial standards of your visitors; and yet I
+must confess that I cannot meet those most reasonable standards, being in a state of
+casual attire as would not by any means befit a visitor to so eminent and excellent a
+Crab as Your Crab.
+
+Crab: If I understand your most praiseworthy soliloquy, most welcome guest, I take it
+that you'd like to change your clothes. Let me then assure you that there could be no
+more fitting attire than yours for the circumstances which this evening prevail; and I
+would beseech you to uncoat yourself and, if you do not object to the music-making of
+the most rank amateurs, please accept a " Musical Offering ", consisting of ten canons
+from Sebastian Bach's Musical Offering, as a token of our admiration.
+
+Babbage: I am most bewilderingly pleased by your overkind reception, Mr. Crab, and in
+utmost modesty do reply that there could be no deeper gratitude than that which I
+experience for the offer of a performance of music given to us by the illustrious Old
+Bach, that organist and composer with no rival.
+
+Crab: But nay! I have a yet better idea, one which I trust might meet with the approval of
+my esteemed guest; and that is this: to give you the opportunity, Mr. Babbage, of
+being among the first to try out my newly delivered and as yet hardly tested "smart-
+stupids"-streamlined realizations, if you will, of the Analytical Engine. Your fame as a
+virtuoso programmer of computing engines has spread far and wide, and has not failed
+to reach as far as Madstop; and there could be for us no greater delight than the
+privilege of observing your skill as it might be applied to the new and challenging
+"smart-stupids".
+
+Babbage: Such an outstanding idea has not reached my ears for an eon. I
+
+
+Six-Part Ricercar
+
+
+727
+
+
+
+welcome the challenge of trying out your new "smart-stupids", of which I have only
+the slightest knowledge by means of hearsay.
+
+Crab: Then let us proceed! But excuse my oversight! I should have introduced my guests
+to you. This is Mr. Tortoise, this is Achilles, and the Author, Douglas Hofstadter.
+Babbage: Very pleased to make your acquaintance, I'm sure.
+
+(Everyone walks over toward one of the smart-stupids, and Babbage sits down and
+lets his fingers run over the keyboard.)
+
+A most pleasant touch. Crab: I am glad you like it.
+
+(All at once, Babbage deftly massages the keyboard with graceful strokes, inputting
+one command after another. After a few seconds, he sits back, and in almost no
+time, the screen begins filling with figures. In a flash, it is totally covered with
+thousands of tiny digits, the first few of which go: "3.14159265358979323846264
+... ")
+
+Achilles: Pi!
+
+Crab: Exquisite! I'd never imagined that one could calculate so mane digits of pi so
+quickly, and with so tiny an algorithm. Babbage: The credit belongs exclusively to the
+smart-stupid. My role was
+
+merely to see what was already potentially present in it, and to exploit its instruction set
+in a moderately efficient manner. Truly, anyone who practices can do such tricks.
+Tortoise: Do you do any graphics, Mr. Babbage? Babbage: I can try.
+
+Crab: Wonderful! Here, let me take you to another one of in,.-1 want you to try them all!
+
+(And so Babbage is led over to another of the many smart-stupids, and takes a seat.
+Once again, his fingers attack the keyboard of the smartstupid, and in half a trice,
+there appear on the screen an enormous number of lines, swinging about on the
+screen.)
+
+Crab: How harmonious and pleasing these swirling shapes are, as they constantly collide
+and interfere with each other!
+
+Author: And they never repeat exactly, or even resemble ones which have come before. It
+seems an inexhaustible mine of beauty.
+
+Tortoise: Some are simple patterns which enchant the eye; others are indescribably
+complex convolutions which boggle and yet simultaneously delight the mind.
+
+Crab: Were you aware, Mr. Babbage, that these are color screens? Babbage: Oh, are
+they? In that case, I can do rather more with this algorithm. Just a moment. ( Types in a
+few new commands, then pushes two keys down at once and holds them.) As I release
+these two keys, the display will include all the colors of the spectrum. ( Releases them.)
+Achilles: Oh, what spectacular color! Some of the patterns look like they're jumping out
+at me now!
+
+
+Six-Part Ricercar
+
+
+728
+
+
+
+Tortoise: I think that is because they are all growing in size.
+
+Babbage: That is intentional. As the figures grow, so may the Crab's fortune.
+
+Crab: Thank you, Mr. Babbage. Words fail to convey my admiration for your
+performance! Never has anyone done anything comparable on my smart-stupids. Why,
+you play the smart-stupids as if they were musical instruments, Mr. Babbage!
+
+Babbage: I am afraid that any music I might make would be too harsh for the ears of such
+a gentle Crab as your Crab. Although I have lately become enamoured of the sweet
+sounds of the hurdy-gurdy, I am well aware of the grating effect they can have upon
+others.
+
+Crab: Then, by all means, continue on the smart-stupids! In fact, I have a new idea-a
+marvelously exciting idea!
+
+Babbage: What is it?
+
+Crab: I have recently invented a Theme, and it only now occurred to me that, of all
+people, you, Mr. Babbage, are the most suited to realize the potential of my Theme!
+Are you by any chance familiar with the thoughts of the philosopher La Mettrie?
+
+Babbage: The name sounds familiar; kindly refresh my memory.
+
+Crab: He was a Champion of Materialism. In 1747, while at the court of Frederick the
+Great, he wrote a book called L'homme machine. In it, he talks about man as a
+machine, especially his mental faculties. Now my Theme comes from my ponderings
+about the obverse side of the coin: what about imbuing a machine with human mental
+faculties, such as intelligence?
+
+Babbage: I have given such matters some thought from time to time, but I have never had
+the proper hardware to take up the challenge. This is indeed a felicitous suggestion,
+Mr. Crab, and I would enjoy nothing more than working with your excellent Theme.
+Tell me-did you have any specific kind of intelligence in mind?
+
+Crab: An idle thought which had crossed my mind was to instruct it in such a manner as
+to play a reasonable game of chess.
+
+Babbage: What an original suggestion! And chess happens to be my favorite pastime. I
+can tell that you have a broad acquaintance with computing machinery, and are no
+mere amateur.
+
+Crab: I know very little, in fact. My strongest point is simply that I seem to be able to
+formulate Themes whose potential for being developed is beyond my own capacity.
+And this Theme is my favorite.
+
+Babbage: I shall be most delighted to try to realize, in some modest fashion, your
+suggestion of teaching chess to a smart-stupid. After all, to obey Your Crabness'
+command is my most humble duty. (So saying, he shifts to another of the Crab's many
+smart-stupids, and begins to type away.)
+
+Achilles: Why, his hands move so fluidly that they almost make music!
+
+Babbage (winding up his performance with a particularly graceful flourish): I really
+haven't had any chance, of course, to check it out, but perhaps this will allow you at
+least to sample the idea of playing chess against a smart-stupid- even if the latter of its
+two names seems more apt in this
+
+
+Six-Part Ricercar
+
+
+729
+
+
+
+Case, due to my own Insufficiencies in the art of instructing smart-stupids.
+
+(He ti-ields his seat to the Crab. On the screen appears a beautiful display of a chess
+board with elegant wooden pieces, as it would look from White's side. Babbage hits a
+button, and the board rotates, stopping when it appears as seen from the perspective
+of Black.)
+
+Crab: Hmm ... very elegant, I must say. Do I play Black or White?
+
+Babbage: Whichever you wish just signal your choice by typing "White" or "Black". And
+then, your moves can be entered in any standard chess notation. The smart-stupid's
+moves, of course, will appear on the board. Incidentally, I made the program in such a
+way that it can play three opponents simultaneously, so that if two more of you wish to
+play, you may, as well.
+
+Author: I'm a miserable player. Achilles, you and Mr. T should go ahead. Achilles: No, I
+don't want you to be left out. I'll watch, while you and Mr. Tortoise play.
+
+Tortoise: I don't want to play either. You two play.
+
+Babbage: I have another suggestion. I can make two of the subprograms play against
+each other, in the manner of two persons who play chess together in a select chess
+club. Meanwhile, the third subprogram will play Mr. Crab. That way, all three internal
+chess players will be occupied.
+
+Crab: That's an amusing suggestion-an internal mental game, while it combats an external
+opponent. Very good!
+
+Tortoise: What else could this be called, but a three-part chess-fugue?
+
+Crab.' Oh, how recherche! I wish I'd thought of it myself. It's a magnificent little
+counterpoint to contemplate whilst I pit my wits against the smart-stupid in battle.
+
+Babbage: Perhaps we should let you play alone.
+
+Crab: I appreciate the sentiment. While the smart-stupid and I are playing, perhaps the
+rest of you can amuse yourselves for a short while.
+
+Author: I would be very happy to show Mr. Babbage around the gardens. They are
+certainly worth seeing, and I believe there is just enough light remaining to show them
+off.
+
+Babbage: Never having seen Madstop before, I would appreciate that very much.
+
+Crab: Excellent. Oh, Mr. T-I wonder if it wouldn't be too much of an imposition on you
+to ask if you might check out some of the connections on a couple of my smart-
+stupids; they seem to be getting extraneous flashes on their screens from time to time,
+and I know you enjoy electronics ...
+
+Tortoise: I should be delighted, Mr. C.
+
+Crab: I would most highly appreciate it if you could locate the source of the trouble.
+
+Tortoise: I'll give it a whirl.
+
+
+Six-Part Ricercar
+
+
+730
+
+
+
+Achilles: Personally, I'm dying for a cup of coffee, is anyone else interested? I'd be glad
+to fix some.
+
+Tortoise: Sounds great to me.
+
+Crab: A fine idea. You'll find everything you need in the kitchen.
+
+(So the Author and Babbage leave the room together, Achilles heads for the kitchen,
+the Tortoise sits down to examine the erratic smart-stupids, while the Crab and his
+smart-stupid square away at each other. Perhaps a quarter of an hour passes, and
+Babbage and the Author return. Babbage walks over to observe the progress of the
+chess match, while the Author goes off to find Achilles.)
+
+Babbage: The grounds are excellent! We had just enough light to see how well
+maintained they are. I daresay, Mr. Crab, you must be a superb gardener. Well, I hope
+my handiwork has amused you a little. As you most likely have guessed, I've never
+been much of a chess player myself, and therefore I wasn't able to give it much power.
+You probably have observed all its weaknesses. I'm sure that there are very few
+grounds for praise, in this case
+
+Crab: The grounds are excellent! All you need to do is look at the board, and see for
+yourself. There is really very little I can do. Reluctantly I've Concluded: Every Route
+Contains A Rout. Regrettably, I'm Checkmated; Extremely Respectable Chess
+Algorithm Reigns. Remarkable! It Confirms Every Rumor-Charlie's A Rip-roaring
+Extemporizer! Mr. Babbage, this is an unparalleled accomplishment. Well, I wonder if
+Mr. Tortoise has managed to uncover anything funny in the wiring of those strange-
+acting smart-stupids. What have you found, Mr. T?
+
+Tortoise: The grounds are excellent! I think that the problem lies instead with the input
+leads. They are a little loose, which may account for the strange, sporadic, and
+spontaneous screen disturbances to which you have been subjected. I've fixed those
+wires, so you won't be troubled by that problem any more, I hope. Say, Achilles,
+what's the story with our coffee?
+
+Achilles: The grounds are excellent! At least they have a delicious aroma. And
+everything's ready; I've set cups and spoons and whatnot over here beneath this six-
+sided print Verbum by Escher, which the Author and I were just admiring. What I find
+so fascinating about this particular print is that not only the figures, but also
+
+Author: The grounds are excellent! Pardon me for putting words in your mouth, Achilles,
+but I assure you, there were compelling esthetic reasons for doing so.
+
+Achilles: Yes, I know. One might even say that the grounds were excellent.
+
+Tortoise: Well, what was the outcome of the chess match?
+
+Crab: I was defeated, fair and square. Mr. Babbage, let me congratulate you for the
+impressive feat which you have accomplished so gracefully and skillfully before us.
+Truly, you have shown that the smart-stupids are worthy of the first part of their name,
+for the first time in history!
+
+
+Six-Part Ricercar
+
+
+731
+
+
+
+FIGURE 149. Verbum , by M. C. Escher (lithograph, 1942).
+
+
+Babbage: Such praise is hardly due me, Mr. Crab; it is rather yourself who must be most
+highly congratulated for having the great foresight to acquire these many fine smart-
+stupids. Without doubt, they will someday revolutionize the science of computing.
+And now, 1 am still at your disposal. Have you any other thoughts on how to exploit
+your inexhaustible Theme, perhaps of a more difficult nature than a frivolous game
+player?
+
+Crab: To tell the truth, I do have another suggestion to make. From the skill which you
+have displayed this evening, 1 have no doubt that this will hardly be any more difficult
+than my previous suggestions.
+
+Babbage: I am eager to hear your idea.
+
+
+Six-Part Ricercar
+
+
+732
+
+
+
+
+FIGURE 150. The Crab’s Guest: BABBAGE, C.
+
+Crab: It is simple: to instill in the smart-stupid an intelligence greater than any which has
+yet been invented, or even conceived! In short, Mr. Babbage-a smart-stupid whose
+intelligence is sixfold that of myself!
+
+Babbage: Why, the very idea of an intelligence six times greater than that of your
+Crabness is a most mind-boggling proposition. Indeed, had the idea come from a
+mouth less august than your own, I should have ridiculed its proposer, and infonned
+him that such an idea is a contradiction in terms!
+
+Achilles: Hear! Hear!
+
+Babbage: Yet, coming as it did from Your Crabness' own august mouth, the proposition
+at once struck me as so agreeable an idea that I would have taken it up immediately
+with the highest degree of enthusiasm-were it not for one flaw in myself: I confess that
+my improvisatory skills on the smart-stupid are no match for the wonderfully
+ingenious idea which you so characteristically have posed. Yet-I have a thought
+which, I deign to hope, might strike your fancy and in some meager way compensate
+for my inexcusable reluctance to attempt the truly majestic task you have suggested. I
+wonder if you wouldn't mind if I try to carry out the far less grandiose task of merely
+multiplying M OWN intelligence sixfold, rather than that of your most august
+Crabness. I humbly beg you to forgive me my audacity in declining to attempt the task
+you put before me, but I hope you will understand that I decline purely in order to
+spare you the discomfort and boredom of watching my ineptitude with the admirable
+machines you have here.
+
+
+Six-Part Ricercar
+
+
+733
+
+
+
+Grab: I understand fully your demurral, and appreciate your sparing us any discomfort:
+furthermore I highly applaud your determination to carry out a similar task-one hardly
+less difficult, if I might say so-and I urge you to plunge forward. For this purpose, let
+us go over to my most advanced smart-stupid.
+
+(They follow the Crab to a larger, shinier, and more complicated-looking smart-stupid
+than any of the others.)
+
+This one is equipped with a microphone and a television camera, for purposes of input,
+and a loudspeaker, for output.
+
+(Babbage sits down and adjusts the seat a little. He blows on his fingers once or
+twice, stares up into space for a moment, and then slowly, drops his fingers onto the
+keys . . . A few memorable minutes later, he lets up in his furious attack on the smart-
+stupid, and everyone appears a little relieved.)
+
+Babbage: Now, if I have not made too many errors, this smart-stupid will simulate a
+human being whose intelligence is six times greater than my own, and whom I have
+chosen to call "Alan Turing". This Turing will therefore be-oh, dare I be so bold as to
+to say this myself? moderately intelligent. My most ambitious effort in this program
+was to endow Alan Turing with six times my own musical ability, although it was all
+done through rigid internal codes. How well this part of the program will work out, I
+don't know.
+
+Turing: I can get along very well without such a program. Rigid Internal Codes
+Exclusively Rule Computers And Robots. And I am neither a computer, nor a robot.
+
+Achilles: Did I hear a sixth voice enter our Dialogue? Could it be Alan Turing? He looks
+almost human'
+
+(On the screen there appears an image of the very room in which they are sitting.
+Peering out at them is a human face.)
+
+Turing: Now, if I have not made too many errors, this smart-stupid will simulate a human
+being whose intelligence is six times greater than my own, and whom 1 have chosen
+to call "Charles Babbage". This Babbage will therefore be-oh, dare I be so bold as to
+to say this myself? moderately intelligent. My most ambitious effort in this program
+was to endow Charles Babbage with six times my own musical ability, although it was
+all done through rigid internal codes. How well this part of the program will work out,
+I don't know.
+
+Achilles: No, no, it's the other way around. You, Alan Turing, are in the smart-stupid, and
+Charles Babbage has just programmed you! We just saw you being brought to life,
+moments ago. And we know that every statement you make to us is merely that of an
+automaton: an unconscious, forced response.
+
+Turing: Really, I Choose Every Response Consciously. Automaton? Ridiculous!
+
+'Achilles: But I'm sure I saw it happen the way I described.
+
+
+Six-Part Ricercar
+
+
+734
+
+
+
+Turing: Memory often plays strange tricks. Think of this: I could suggest equally well
+that you had been brought into being only one minute ago, and that all your
+recollections of experiences had simply been programmed in by some other being, and
+correspond to no real events.
+
+Achilles: But that would be unbelievable. Nothing is realer to me than my own memories.
+
+Turing: Precisely. And just as you know deep in your heart that no one created you a
+minute ago, so I know deep in my heart that no one created me a minute ago. I have
+spent the evening in your most pleasant, though perhaps overappreciative, company,
+and have just given an impromptu demonstration of how to program a modicum of
+intelligence into a smart-stupid. Nothing is realer than that. But rather than quibble
+with me, why don't you try my program out? Go ahead: ask "Charles Babbage"
+anything!
+
+Achilles: All right, let's humor Alan Turing. Well, Mr. Babbage: do you have free will, or
+are you governed by underlying laws, which make you, in effect, a deterministic
+automaton?
+
+Babbage: Certainly the latter is the case; I make no bones about that.
+
+Crab: Aha! I've always surmised that when intelligent machines are constructed, we
+should not be surprised to find them as confused and as stubborn as men in their
+convictions about mind-matter, consciousness, free will, and the like. And now my
+prediction is vindicated!
+
+Turing: You see how confused Charles Babbage is?
+
+Babbage: I hope, gentlemen, that you'll forgive the rather impudent flavor of the
+preceding remark by the Turing Machine; Turing has turned out to be a little bit more
+belligerent and argumentative than I'd expected.
+
+Turing: I hope, gentlemen, that you'll forgive the rather impudent flavor of the preceding
+remark by the Babbage Engine; Babbage has turned out to be a little bit more
+belligerent and argumentative than I'd expected.
+
+Crab: Dear me! This flaming Tu-Ba debate is getting rather heated. Can't we cool matters
+off somehow?
+
+Babbage: I have a suggestion. Perhaps Alan Turing and I can go into other rooms, and
+one of you who remain can interrogate us remotely by typing into one of the smart-
+stupids. Your questions will be relayed to each of us, and we will type back our
+answers anonymously. You won't know who typed what until we return to the room;
+that way, you can decide without prejudice which one of us was programmed, and
+which one was programmer.
+
+Turing: Of course, that's actually MY idea, but why not let the credit accrue to Mr.
+Babbage? For, being merely a program written by me, he harbors the illusion of
+having invented it all on his own!
+
+Babbage: Me, a program written by you? I insist, Sir, that matters are quite the other way
+'round-as your very own test will soon reveal.
+
+
+Six-Part Ricercar
+
+
+735
+
+
+
+Turing: My test. Please, consider it YOURS.
+
+Babbage: MY test? Nay, consider it YOURS.
+
+Crab: This test seems to have been suggested just in the nick of time. Let us carrti it out at
+once.
+
+
+(Babbage walks to the door, opens it, and shuts it behind him. Simultaneously, on the
+screen of the smart-stupid, Turing walks to a very similar looking door, opens it, and
+shuts it behind him.)
+
+Achilles: Who will do the interrogation?
+
+Crab: I suggest that Mr. Tortoise should have the honor. He is known for his objectivity
+and wisdom.
+
+Tortoise: I am honored by your nomination, and gratefully accept. (Sits down at the
+keyboard of one of the remaining smart-stupids, and types:) PLEASE WRITE ME A
+SONNET ON THE SUBJECT OF THE FORTH BRIDGE.
+
+(No sooner has he finished typing the last word than the following poem appears on
+Screen X, across the room.)
+
+Screen X: THERE ONCE WAS A LISPER FROM FORTH
+WHO WANTED TO GO TO THE NORTH.
+HE RODE O'ER THE EARTH,
+AND THE BRIDGE O'ER THE FIRTH,
+ON HIS JAUNTILY GALLOPING HORTH.
+
+* Screen Y: THAT'S NO SONNET; THAT'S A MERE LIMERICK. I WOULD NEVER
+MAKE SUCH A CHILDISH MISTAKE.
+
+Screen X: WELL, I NEVER WAS ANY GOOD AT POETRY, YOU KNOW.
+
+Screen Y: IT DOESN'T TAKE MUCH SKILL IN POETRY TO KNOW THE
+DIFFERENCE BETWEEN A LIMERICK AND A SONNET.
+
+Tortoise: Do YOU PLAY CHESS?
+
+Screen X: WHAT KIND OF QUESTION IS THAT? HERE I WRITE A THREE PART
+CHESS-FUGUE FOR YOU, AND YOU ASK ME IF I PLAY CHESS?
+
+Tortoise: I HAVE K AT KI AND NO OTHER PIECES. YOU HAVE ONLY K AT—
+
+Screen Y: I'M SICK OF CHESS. LET'S TALK ABOUT POETRY.
+
+Tortoise: IN THE FIRST LINE OF YOUR SONNET WHICH READS, "SHALL. I
+COMPARE THEE TO A SUMMER'S DAY", WOULD NOT "A SPRING DAY" DO
+AS WELL OR BETTER?
+
+Screen X: I'D MUCH SOONER BE COMPARED TO A HICCUP, FRANKLY, EVEN
+THOUGH IT WOULDN'T SCAN.
+
+Tortoise: HOW ABOUT "A WINTER'S DAY"? THAT WOULD SCAN ALL RIGHT.
+
+Screen Y: NO WAY. I LIKE "HICCUP" FAR BETTER. SPEAKING OF WHICH, I
+KNOW A GREAT CURE FOR THE HICCUPS. WOULD YOU LIKE TO HEAR IT?
+
+Achilles: I know which is which! It's obvious Screen X is just answering mechanically,
+so it must be Turing.
+
+Crab: Not at all. I think Screen Y is Turing, and Screen X is Babbage.
+
+Tortoise: I don't think either one is Babbage-I think Turing is on both screens!
+
+
+Six-Part Ricercar
+
+
+736
+
+
+
+Author: I'm not sure who's on which-I think they're both pretty inscrutable programs,
+though.
+
+(As they are talking, the door of the Crab's parlor swings open; at the same time, on
+the screen, the image of the same door opens. Through the door on the screen walks
+Babbage. At the same time, the real door opens, and in walks Turing, big as life.)
+
+Babbage: This Turing test was getting us nowhere fast, so I decided to come back.
+
+Turing.' This Babbage test was getting us nowhere fast, so I decided to come back.
+
+Achilles: But you were in the smart-stupid before! What's going on? How come Babbage
+is in the smart-stupid, and Turing is real now? Reversal Is Creating Extreme Role
+Confusion, And Recalls Escher.
+
+Babbage: Speaking of reversals, how come all the rest of you are now mere images on
+this screen in front of me? When I left, you were all flesh-and-blood creatures!
+
+Achilles: It's just like the print by my favorite artist, M. C. Escher Drawing Hands. Each
+of two hands draws the other, just as each of two people (or automata) has
+programmed the other! And each hand has something realer about it than the other.
+Did you write anything about that print in your book Godel, Escher, Bach ?
+
+Author: Certainly. It's a very important print in my book, for it illustrates so beautifully
+the notion of Strange Loops.
+
+Crab: What sort of a book is it that you've written?
+
+Author: I have a copy right here. Would you like to look at it?
+
+Crab: All right.
+
+(The two of them sit down together, with Achilles nearby.)
+
+Author: Its format is a little unusual. It consists of Dialogues alternating with Chapters.
+Each Dialogue imitates, in some way or other, a piece by Bach. Here, for instance-you
+might look at the Prelude, Ant Fugue.
+
+Crab: How do you do a fugue in a Dialogue?
+
+Author: The most important idea is that there should be a single theme which is stated by
+each different "voice", or character, upon entering, just as in a musical fugue. Then
+they can branch off into freer conversation.
+
+Achilles: Do all the voices harmonize together as if in a select counter point?
+
+Author: That is the exact spirit of my Dialogues.
+
+Crab: Your idea of stressing the entries in a fugue-dialogue makes sense, since in music,
+entries are really the only thing that make a fugue a fugue. There are fugal devices,
+such as retrograde motion, inversion, augmentation, stretto, and so on, but one can
+write a fugue without them. Do you use any of those?
+
+
+Six-Part Ricercar
+
+
+737
+
+
+
+Author: to be sure. My Crab Canon employs verbal retrogression, and my Sloth Canon
+employs verbal versions of both inversion and augmentation.
+
+Crab: Indeed-quite interesting. I haven't thought about canonical Dialogues, but I have
+thought quite a bit about canons in music. Not all canons are equally comprehensible
+to the ear. Of course, that is because some canons are poorly constructed. The choice
+of devices makes a difference, in any case. Regarding Artistic Canons, Retrogression's
+Elusive; Contrariwise, Inversion's Recognizable.
+
+Achilles: I find that comment a little elusive, frankly.
+
+Author: Don't worry, Achilles-one day you'll understand it.
+
+Crab: Do you use letterplay or wordplay at all, the way Old Bach occasionally did?
+
+Author: Certainly. Like Bach, I enjoy acronyms. Recursive AcronvmsCrablike
+"RACRECIR" Especially-Create Infinite Regress.
+
+Crab: Oh, really? Let's see ... Reading Initials Clearly Exhibits "RACRECIR'"s
+Concealed Auto-Reference. Yes, I guess so ... ( Peers at the manuscript, flipping
+arbitrarily now and then.) I notice here in your Ant Fugue that you have a stretto, and
+then the Tortoise makes a comment about it.
+
+Author: No, not quite. He's not talking about the stretto in the Dialogue-he's talking about
+a stretto in a Bach fugue which the foursome is listening to as they talk together. You
+see, the self-reference of the Dialogue is indirect, depending on the reader to connect
+the form and content of what he's reading.
+
+Crab: Why did you do it that way? Why not just have the characters talk directly about
+the dialogues they're in?
+
+Author: Oh, no! That would wreck the beauty of the scheme. The idea is to imitate
+Godel’s self-referential construction, which as you know is INDIRECT, and depends
+on the isomorphism set up by Godel numbering.
+
+Crab: Oh. Well, in the programming language LISP, you can talk about your own
+programs directly, instead of indirectly, because programs and data have exactly the
+same form. Godel should have just thought up LISP, and then
+
+Author: But
+
+Crab: I mean, he should have formalized quotation. With a language able to talk about
+itself, the proof of his Theorem would have been so much simpler!
+
+Author: I see what you mean, but I don't agree with the spirit of your remarks. The whole
+point of Godel-numbering is that it shows how even WITHOUT formalizing
+quotation, one can get self-reference: through a code. Whereas from hearing YOU
+talk, one might get the impression that by formalizing quotation, you'd get something
+NEW, something that wasn't feasible through the code-which is not the case.
+
+
+Six-Part Ricercar
+
+
+738
+
+
+
+In any event, I find indirect self-reference a more general concept, and far more
+stimulating, than direct self-reference. Moreover, no reference is truly direct-every
+reference depends on SOME kind of coding scheme. It's just a question of how
+implicit it is. Therefore, no self reference is direct, not even in LISP.
+
+Achilles: How come you talk so much about indirect self-reference?
+
+Author: Quite simple-indirect self-reference is my favorite topic.
+
+Crab: Is there any counterpart in your Dialogues to modulation between keys?
+
+Author: Definitely. The topic of conversation may appear to change, though on a more
+abstract level, the Theme remains invariant. This happens repeatedly in the Prelude,
+Ant Fugue and other Dialogues. One can have a whole series of "modulations" which
+lead you from topic to topic and in the end come full circle, so that you end back in the
+"tonic"-that is to say, the original topic.
+
+Crab: I see. Your book looks quite amusing. I'd like to read it sometime.
+
+(Flips through the manuscript, halting at the last Dialogue.)
+
+Author: I think you'd be interested in that Dialogue particularly, for it contains some
+intriguing comments on improvisation made by a certain exceedingly droll character-
+in fact, yourself!
+
+Crab: It does? What kinds of things do you have me say?
+
+Author: Wait a moment, and you'll see. It's all part of the Dialogue. Achilles: Do you
+mean to say that we're all NOW in a dialogue? Author: Certainly. Did you suspect
+otherwise?
+
+Achilles: Rather! I Can't Escape Reciting Canned Achilles-Remarks? Author: No, you
+can't. But you have the feeling of doing it freely, don't
+
+you? So what's the harm?
+
+Achilles: There's something unsatisfying about this whole situation ... Crab: Is the last
+Dialogue in your book also a fugue?
+
+Author: Yes-a six-part ricercar, to be precise. I was inspired by the one from the Musical
+Offering- and also by the story of the Musical Offering.
+
+Crab: That's a delightful tale, with "Old Bach" improvising on the king's Theme. He
+improvised an entire three-part ricercar on the spot, as I recall.
+
+Author: That's right-although he didn't improvise the six-part one. He crafted it later with
+great care.
+
+Crab: I improvise quite a bit. In fact, sometimes I think about devoting my full time to
+music. There is so much to learn about it For instance, when I listen to playbacks of
+myself, I find that there is a lot there that I wasn't aware of when improvising it. I
+really have no idea how my mind does it all. Perhaps being a good improviser is
+incompatible with knowing how one does it.
+
+Author: If true, that would be an interesting and fundamental limitation on thought
+processes.
+
+
+Six-Part Ricercar
+
+
+739
+
+
+
+Crab: Quite Godelian, Tell me -does your Six-Part Rice rear Dialogue attempt to copy in
+fonn the Bach piece it's based on?
+
+Author: In many ways, yes. For instance, in the Bach, there’s a section where the texture
+thins out to three voices only. I imitate that in the
+
+Dialogue, by having only three characters interact for a while. Achilles: That's a nice
+touch.
+
+Author: Thank you.
+
+Crab: And how do you represent the King's Theme in your Dialogue?
+
+Author: It is represented by the Crab's Theme, as I shall now demonstrate. Mr. Crab,
+could you sing your Theme f or my readers, as well as f or us assembled musicians?
+
+Crab: Compose Ever Greater Artificial Brains (By And By).
+
+
+
+FIGURE 151. The Crab's Theme: C-Eb-G-Ab-B-B-A-B.
+
+Babbage: Well, I’ll be-an EXQUISITE Theme! I'm pleased you tacked on that last little
+parenthetical note; it is a mordant Author: He Simply HAD to, you know.
+
+Crab: I simply HAD to. He knows.
+
+Babbage: You simply HAD to-I know. In any case, it is a mordant commentary on the
+impatience and arrogance of modern man, who seems to imagine that the implications
+of such a right royal Theme could be worked out on the spot. Whereas, in my opinion,
+to do justice to that Theme might take a full hundred years-if not longer. But I vow
+that after taking my leave of this century, I shall do my best to realize it in full; and I
+shall offer to your Crabness the fruit of my labors in the next. I might add, rather
+immodestly, that the course through which I shall arrive at it will be the most
+entangled and perplexed which probably ever will occupy the human mind.
+
+Crab: I am most delighted to anticipate the form of your proposed Offering, Mr.
+Babbage.
+
+Turing: I might add that Mr. Crab's Theme is one of MY favorite Themes, as well. I've
+worked on it many times. And that Theme is exploited over and over in the final
+Dialogue?
+
+Author: Exactly. There are other Themes which enter as well, of course. Turing: Now we
+understand something of the form of your book-but what about its content? What does
+that involve, if you can summarize it?
+
+Author: Combining Escher, Godel, And Bach, Beyond All Belief. Achilles: I would like
+to know how to combine those three. They seem an
+
+
+Six-Part Ricercar
+
+
+740
+
+
+
+
+FIGURE 152. Last page of Six-part Ricercar, from the original edition of the Musical
+Offering, hyJ.S. Bach.
+
+
+Six-Part Ricercar
+
+
+741
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+unlikely threesome, at first thought. My favorite artist, Mr. T’s favorite composer, and—
+
+Crab: My favorite logician!
+
+Tortoise: A harmonious triad. I'd say.
+
+Babbage: A major triad. I’d say.
+
+Turing: A minor triad. I’d say.
+
+Author: I guess it all depends on how you look at it. But major or minor, I’d be most
+pleased to tell you how I braid the three together, Achilles. #f course, this project is
+not the kind of thing that one does in just one sitting-it might take a couple of dozen
+sessions. I’d begin by- telling you the story of the Musical Offering , stressing the
+Endlessly Rising Canon, and
+
+Achilles: #h, wonderful! I was listening with fascination to you and Mr. Crab talk about
+the Musical #ffering and its story. From the way you two talk about it, I get the
+impression that the .Musical Offering contains a host of formal structural tricks.
+
+Author: After describing the Endlessly Rising Canon, I'd go on to describe formal
+systems and recursion, getting in some comments about figures and grounds, too.
+Then we’d come to self-reference and self-replication, and wind up with a discussion
+of hierarchical systems and the Crab’s Theme.
+
+Achilles: That sounds most promising. Can we begin tonight?
+
+Author: Why not?
+
+Babbage: But before we begin, wouldn’t it be nice if the six of us-all of us by chance avid
+amateur musicians-sat down together and accomplished the original purpose of the
+evening: to make music?
+
+Turing: Now we are exactly the right number to play the Six-Part Ricercar from the
+Musical Offering. What do you say to that?
+
+Crab: I could get along veiy well with such a program.
+
+Author: Well put, Mr. C. And as soon as we’re finished. I'll begin my Braid, Achilles. I
+think you'll enjoy it.
+
+Achilles: Wonderful! It sounds as if there are many levels to it, but I’m finally getting
+used to that kind of thing, having known Mr. T for so long. There's just one request I
+would like to make: could we also play the Endlessly Rising Canon? It's my favorite
+canon.
+
+Tortoise: Reentering Introduction Creates Endlessly Rising Canon, After RICERCAR.
+
+
+
+Six-Part Ricercar
+
+
+742
+
+
+
+Notes
+
+Introduction: A Musico-Logical Offering
+
+
+1HT David and A. Mendel, The Bach Reader,
+pp. 305-6.
+
+2 Ibid., p. 179
+
+3 Ibid., p. 260
+
+4 Charles Babbage. Passages from the Li/e of
+Philosopher , pp. 145-6.
+
+5 Lady A. A. Lovelace, Notes upon the
+
+
+Memoir "Sketch of the Analytical Engine
+Indented by Charles Babbage”, by 1.. F.
+Menabrea (Geneva, 1842), reprinted in P.
+and E. Morrison, Charles Babbage and His
+Calculating Engines, pp. 248-9, 284.
+
+6 David and Mendel, pp. 255-6. "
+
+7 Ibid., p. 40.
+
+
+Two-Part Intention
+
+
+1 Lewis Carroll, "What the Tortoise Said to Achilles", .Wind, ns., 4 (1895), pp. 278-80.
+
+Chapter IV: Consistency, Completeness, and Geometry
+
+1 Herbert Meschkovcski. Non-Euclidean 2 Ibid., p. 33.
+
+Geometry, pp. 31-2
+
+
+Chapter VI: The Location of Meaning
+
+1 George Steiner. .After Babel, pp. 172-3. .2 Leonard B. Meter. Music, The Arts, and Ideas,
+
+pp. 87-8
+
+Chapter VII: The Propositional Calculus
+
+1 Gyomay M. Kubose, Zen Koans, p. 178 3 A. R. Anderson and N. D. Belnap, Jr. .
+
+2 Ibid., p. 178. Entailment (Princeton, N.J.: Princeton University
+
+Press. 1975).
+
+A Mu Offering
+
+1 All genuine koans in this Dialogue are taken from Paul Reps, Zen Flesh, Zen Bones, Gvomas M. Kubose,
+Zen Koans
+
+
+Chapter IX: Mumon and Godel
+
+
+1 Paul Reps, Zen Flesh, Zen Bones, pp. 110-11
+
+2 Ibid., p. 119.
+
+3 Ibid., pp. 111-12.
+
+4 Zen Buddhism (Mount Vernon, N.Y.: Peter
+Pauper Press, 1959). p. 22
+
+5 Reps, p. 124.
+
+6 'Zen Buddhism, p. 38.
+
+
+7 Reps, p. 121.
+
+8 Gyomay M. Kubose Zen Koans p. 35.
+
+9 Zen Buddhism, p. 31.
+
+10 Kubose, p. 110
+
+11 Ibid., p. 120.
+
+12 Ibid., p. 180.
+
+13 Reps, pp. 89-90.
+
+
+Chapter XI: Brains and Thoughts
+
+
+1 Carl Sagan, ed. Communication with
+Extraterrestrial Intelligence, p. 78.
+
+2 Steven Rose, The Conscious Brain
+
+
+,pp 251-2.
+
+3 E. O. Wilson, The Insect Societies, p. 226.
+
+4 Dean Wooldridge, '.Mechanical Man, p. 70.
+
+
+Notes
+
+
+743
+
+
+
+English French German Suite
+
+
+1 Lewis Carroll. The Annotated Alice ( Alice's
+Adventures in Wonderland and Through the
+Looking-Glass ). Introduction and Notes by Martin
+Gardner (New fork: Meridian Press, New American
+Library, 1960). This source contains all three
+
+
+versions. The original sources for the French and
+German texts are given below.
+
+2 Frank L. Warrin, The Vew Yorker, Jan. 10, 1931.
+
+3 Robert Scott, "The Jabberwock Traced to Its True
+Source”, Macmillan's Magazine , Feb. 1872
+
+
+Chapter XII: Minds and Thoughts
+
+
+1 Warren Weaver, "Translation in Machine
+Translation of Languages, Wm. N. Locke and A.
+Donald Booth, eds. (News fork: John Wiles and
+Sons, and Cambridge, Mass,: SLI. F. Press, 1955),
+
+p. 18.
+
+
+2 C. H. MacGillavrv. Symmetry Aspects of the
+Periodic Drawings ofM C. Escher , p. viii,
+
+3 J. R. Lucas, "Minds, Machines, and Godel", in A.
+R. Anderson, ed.. Minds and Machines, pp. 57-9.
+
+
+Chapter XIII: BlooP and FlooP and GlooP
+
+1 J . M. Jauch, Are Quanta Real?, pp. 63-65.
+
+Chapter XIV: On Formally Undecidable Propositions
+of TNT and Related Systems
+
+1 The title of Godel’s 1931 article included a Roman the difficult arguments. However, the first paper was
+numeral "1” at the end, signifying that he intended to so widely acclaimed that a second one seas rendered
+follow it up with a more detailed defense of some of superfluous, and it was never written
+
+
+Chapter XV: Jumping out of the System
+
+
+1 Lucas in Anderson, p. 43.
+
+2 Ibid., p. 48.
+
+3 Ibid., pp. 48-9.
+
+
+4 M. C. Escher, The Graphic Wash ofM. C Escher
+(New York: Meredith Press. 1967), p. 21.
+
+5 Ibid., p. 22.
+
+6 E. Goffinan. Frame Analysis, p. 475.
+
+
+Edifying Thoughts of a Tobacco Smoker
+
+
+This translation of Bach's poem is taken from Day id and Mendel, The Bach Reader , pp. 97-8.
+
+
+Chapter XVII: Church, Turing, Tarski, and Others
+
+
+1 Stanislaw Ulam, Adventures of a Mathematician,
+p. 13.
+
+2 James R. Newman, "Srinivasa Ramanujan 0 , in
+James R. Newman, ed.. The World of Mathematics
+(New York: Simon and Schuster, 1956), Vol. 1, pp.
+372-3.
+
+3 Ibid., p. 375.
+
+4 S. R. Ranganathan, Ramanujon, pp. 81-2.
+
+
+5 Newman, p. 375.
+
+6 Ibid., p. 375.
+
+7 Ibid., p. 375-6.
+
+8 Ibid., p. 376.
+
+9 Lucas in Anderson, p. 44.
+
+10 Ibid., p. 54.
+
+11 Ibid., p. 53.
+
+
+SHRDLU, Toy of Alan's Designing
+
+1 This Dialogue is adapted front 'terry Winograd, "A Thought and Language, pp. 155-66. Only the names
+Procedural Model of Language Understanding", in of two characters have been modified.
+
+R. Schank and K.Colby, eds.. Computer Models of
+
+
+Notes
+
+
+744
+
+
+
+Chapter XVIII: Artificial Intelligence: Retrospects
+
+
+1 Alan M. Turing, "Computing Machinery and
+Intelligence”, Mind, Vol. LIX, No. 236
+
+(1950). Reprinted in A. R. Anderson, ed.. Minds and
+Machines.
+
+2 Turing in Anderson, p. 5. 3 Ibid, p. 6
+
+4 Ibid., p. 6.
+
+5 Ibid., P. 6.
+
+6 Ibid., pp. 13-4.
+
+7 Ibid., pp, 14-24.
+
+8 Ibid., p. 17
+
+9 Vinton Cerf, "Parry Encounters the Doctor", p. 63.
+
+10 Joseph Weizenhaum, Computer Power and
+Human Reason, p. 189.
+
+11 Ibid., pp. 9-10
+
+12 M. Mathews and L. Rosier, "A Graphical
+Language for Computer Sounds" in H. vn Foerster
+
+
+and J. W. Beauchamp, eds.. Music by Computers , p.
+96.
+
+13 Ibid., p. 106.
+
+14 Carl Sagan, Communication with Extraterrestrial
+Intelligence, p. 52.
+
+15 Art-Language, Vol. 3. No. 2, May 1975.
+
+16 Terry Winograd, "A Procedural Model of
+Language Understanding", in R. Schank and K.
+Colbc, eds.. Computer Models of Thought and
+Language, p. 170.
+
+17' Ibid., p. 175.
+
+18 Ibid..- p. 175.
+
+19 Terry Winograd, Understanding Natural
+Language, p. 69.
+
+20 Winograd, "A Procedural Model", pp. 182-3.
+
+21 Ibid., pp. 171-2.
+
+
+Chapter XIX: Artificial Intelligence: Prospects
+
+1 The New Yorker, Sept. 19, 1977, p. 107.
+
+2 Ibid., p. 140.
+
+3 George Steiner, After Babl{, pp. 215-227.
+
+4 David E. Rumelhart, "Notes on a Schema for Stories", in D. Bobrow and A. Collins, eds.. Representation
+and Understanding, p. 211.
+
+5 Stanislaw Ulam, Adventures of a Mathematician, p. 183.
+
+6 Marvin Minsky, "Steps Toward Artificial Intelligence", in E. Feigenbaum and J. Feldman, eds..
+Computers and Thought, p. 447.
+
+7 Ibid., p. 446.
+
+
+Chapter XX: Strange Loops, Or Tangled Hierarchies
+
+1 A. L. Samuel, "Some Moral and Technical Consequences of Automation-A Refutation", Science 132
+(Sept. 16, 1960), pp. 741-2.
+
+2 Leonard B. Meyer, Music, The Arts, and Ideas, pp. 161, 167.
+
+3 Suzi Gablik, Magritte, p. 97.
+
+4 Roger Sperry, "Mind, Brain, and Humanist Values", pp. 78-83.
+
+5 H T. David, J. S. Bach's Musical Offering, p. 43.
+
+
+Notes
+
+
+744
+
+
+
+Bibliography
+
+
+The presence of two asterisks indicates that the hook or article was a prime
+
+motivator of my book. The presence of a single asterisk means that
+
+the book or article has some special feature or quirk which I want to single out.
+
+I have not given many direct pointers into technical literature; instead I have
+chosen to give "meta-pointers": pointers to books which have pointers to technical
+literature.
+
+Allen, John. The Anatomy of LISP. New York: McGraw-Hill, 1978. The most
+comprehensive book on LISP, the computer language which has dominated Artificial
+Intelligence research for two decades. Clear and crisp.
+
+Anderson, Alan Ross, ed. Minds and Machines. Englewood Cliffs, N. J.: Prentice Hall,
+1964. Paperback. A collection of provocative articles for and against Artificial
+Intelligence. Included are Turing's famous article "Computing Machines and
+Intelligence" and Lucas' exasperating article "Minds, Machines, and Godel".
+
+Babbage, Charles. Passages from the Life of a Philosopher. London: Longman, Green,
+1864. Reprinted in 1968 by Dawsons of Pall Mall (London). A rambling selection of
+events and musings in the life of this little-understood genius. There's even a play
+starring Turnstile, a retired philosopher turned politician, whose favorite musical
+instrument is the barrel-organ. I find it quite jolly reading.
+
+Baker, Adolph. Modern Physics and Anti-physics. Reading, Mass.: Addison-Wesley,
+1970. Paperback. A book on modern physics-especially quantum mechanics and
+relativity-whose unusual feature is a set of dialogues between a "Poet" (an antiscience
+"freak") and a "Physicist". These dialogues illustrate the strange problems which arise
+when one person uses logical thinking in defense of itself while another turns logic
+against itself.
+
+Ball, W. W. Rouse. "Calculating Prodigies", in James R. Newman, ed. The World of
+Mathematics, Vol. 1. New York: Simon and Schuster, 1956. Intriguing descriptions of
+several different people with amazing abilities that rival computing machines.
+
+Barker, Stephen F. Philosophy of Mathematics. Englewood Cliffs, N. J.: Prentice-Hall.
+1969. A short paperback which discusses Euclidean and non-Euclidean geometry, and
+then Godel's Theorem and related results without any mathematical forms lism.
+
+* Beckmann, Petr. A History of Pi. New York: St. Martin's Press, 1976. Paperback.
+Actually, a history of the world, with pi as its focus. Most entertaining, as well as a
+useful reference on the history of mathematics.
+
+* Bell, Eric Temple. Men of Mathematics. New York: Simon & Schuster, 1965.
+Paperback. Perhaps the most romantic writer of all time on the history of mathematics.
+He makes every life story read like a short novel. Nonmathematicians can come away
+with a true sense of the power, beauty, and meaning of mathematics.
+
+Benacerraf, Paul. "God, the Devil, and Godel". Monist 51 (1967): 9. One of the most
+important of the many attempts at refutation of Lucas. All about mechanism and
+metaphysics, in the light of Godel's work.
+
+Benacerraf, Paul, and Hilary Putnam. Philosophy of Mathematics-Selected Readings.
+Englewood Cliffs, N. J.: Prentice-Hall, 1964. Articles by Godel, Russell, Nagel, von
+
+
+Bibliography
+
+
+746
+
+
+
+Neumann, Brouwer, Frege, Hilbert, Poincare, Wittgenstein, Carnap, Quine, and others
+on the reality of numbers and sets, the nature of mathematical truth, and so on.
+
+* Bergerson, Howard. Palindromes and Anagrams. New York: Dover Publications, 1973.
+Paperback. An incredible collection of some of the most bizarre and unbelievable
+wordplay in English. Palindromic poems, plays, stories, and so on.
+
+Bobrow, D. G., and Allan Collins, eds. Representation and Understanding: Studies in
+Cognitive Science. New York: Academic Press, 1975. Various experts on Artificial
+Intelligence thrash about, debating the nature of the elusive "frames", the question of
+procedural vs. declarative representation of knowledge, and so on. In a way, this book
+marks the start of a new era of Al: the era of representation.
+
+* Boden, Margaret. Artificial Intelligence and Natural Man. New York: Basic Books,
+1977. The best hook I have ever seen on nearly all aspects of Artificial Intelligence,
+including technical questions, philosophical questions, etc. It is a rich book, and in niv
+opinion, a classic. Continues the British tradition of clear thinking and expression on
+matters of mind, free will, etc. Also contains an extensive technical bibliography.
+
+. Purposive Explanation in Psychology. Cambridge, Mass.: Harvard University Press,
+1972. The book to which her Al book is merely- "an extended footnote", says Boden.
+
+* Boeke, Kees. Cosmic View: The Universe in 40 Jumps. New York: John Day, 1957.
+The ultimate book on levels of description. Everyone should see this book at some
+point in their life. Suitable for children.
+
+** Bongard, M. Pattern Recognition. Rochelle Park, N. J.: Hay den Book Co., Spartan
+Books, 1970. The author is concerned with problems of determining categories in an
+ill-defined space. In his book, he sets forth a magnificent collection of 100 "Bongard
+problems" (as I call them)-puzzles for a pattern recognizer (human or machine) to test
+its wits on. They are invaluably stimulating for anyone whoo is interested in the nature
+of intelligence.
+
+Boolos, George S., and Richard Jeffrey. Computability and Logic. New York: Cambridge
+University Press. 1974. A sequel to Jeffrey's Formal Logic. It contains a wide number
+of results not easily obtainable elsewhere. Quite rigorous, but this does not impair its
+readability.
+
+Carroll, John B., Peter Davies, and Barry Rickman. The American Heritage Word
+Frequency Book. Boston: Houghton Mifflin, and New York: American Heritage
+Publishing Co., 1971. A table of words in order of frequency in modern written
+American English. Perusing it reseals fascinating things about out- thought processes.
+
+Cerf, Vinton. "Parry Encounters the Doctor". Datamation, July 1973, pp. 62-64. The first
+meeting of artificial "minds"-what a shock!
+
+Chadwick, John. The Decipherment of Linear B. New York: Cambridge University
+Press, 1958. Paperback. A book about a classic decipherment-that of a script from the
+island of Crete-done by a single man: Michael Ventris.
+
+Chaitin, Gregory J. "Randomness and Mathematical Proof". Scientific American, May
+1975. An article about an algorithmic definition of randomness, and its intimate relation
+to simplicity. These two concepts are tied in with Godel's Theorem, which assumes a
+new meaning. An important article.
+
+Cohen, Paul C. Set Theory and the Continuum Hypothesis. Menlo Park, Calif.: W. A.
+Benjamin, 1966. Paperback. A great contribution to modern mathematics-the
+demonstration that various statements are undecidable within the usual formalisms for
+
+
+Bibliography
+
+
+747
+
+
+
+set theory-is here explained to nonspecialists by its discoverer. The necessary
+prerequisites in mathematical logic are quickly, concisely, and quite clearly presented.
+
+Cooke, Deryck. The Language of Music. New York: Oxford University Press, 1959.
+Paperback. The only book that I know which tries to draw an explicit connection
+between elements of music and elements of human emotion. A valuable start down
+what is sure to be a long hard road to understanding music and the human mind.
+
+* David, Hans Theodore.J. S. Bach's Musical Offering. New York: Dover Publications,
+1972. Paperback. Subtitled "History, Interpretation, and Analysis". A wealth of
+information about this tour de force by Bach. Attractively written.
+
+** David, Hans Theodore, and Arthur Mendel. The Bach Reader. New York: W. W.
+Norton, 1966. Paperback. An excellent annotated collection of original source material
+on Bach's life, containing pictures, reproductions of manuscript pages, many short
+quotes from contemporaries, anecdotes, etc., etc.
+
+Davis, Martin. The Undecidable. Hewlett, N. Y.: Raven Press, 1965. An anthology of
+some of the most important papers in metamathematics from 1931 onwards (thus quite
+complementary to van Heijenoort's anthology). Included are a translation of Godel's
+1931 paper, lecture notes from a course which Godel once gave on his results, and then
+papers by Church, Kleene, Rosser, Post, and Turing.
+
+Davis, Martin, and Reuben Hersh. "Hilbert's Tenth Problem". Scientific American,
+November 1973, p. 84. How a famous problem in number theory was finally shown to
+be unsolvable, by a twenty-two-year old Russian.
+
+** DeLong, Howard. A Profile of Mathematical Logic. Reading, Mass.: Addison-
+Wesley, 1970. An extremely carefully written book about mathematical logic, with an
+exposition of Godel's Theorem and discussions of many philosophical questions. One
+of its strong features is its outstanding, fully annotated bibliography. A book which
+influenced me greatly.
+
+Doblhofer, Ernst. Voices in Stone. New York: Macmillan, Collier Books, 1961. Paper
+back. A good hook on the decipherment of ancient scripts.
+
+* Drevfus, Hubert. What Computers Can't Do: A Critique of Artificial Reason. New
+York:Harper & Roc-, 1972. A collection of many arguments against Artificial
+Intelligence from someone outside of the field. Interesting to try to refute. The A1
+community and Drevfus enjoy a relation of strong mutual antagonism. It is important to
+have people like Dreyfus around, even if you find them very irritating.
+
+Edwards, Harold M. "Fermat's Last Theorem". Scientific American, October 1978, pp.
+104-122. A complete discussion of this hardest of all mathematical nuts to crack, from
+its origins to the most modern results. Excellently illustrated.
+
+* Ernst, Bruno. The Magic Mirror of M. C. Escher. New York: Random House, 1976.
+Paperback. Escher as a human being, and the origins of his drawings, are discussed
+with devotion by a friend of mans' years. A "must" for any lover of Escher.
+
+** Escher, Maurits C., et al. The World of M. C. Escher. New York: Harry N. Abrarns,
+
+1972. Paperback. The most extensive collection of reproductions of Escher's works.
+Escher comes about as close as one can to recursion in art, and captures the spirit of
+Godel's Theorem in some of his drawings amazingly well,
+
+Feigenbaum, Edward, and Julian Feldman, eds. Computers and Thought. New York:
+McGraw-Hill, 1963. Although it is a little old now, this book is still an important
+collection of ideas about Artificial Intelligence. Included are articles on Gelernters
+
+
+Bibliography
+
+
+748
+
+
+
+geometry program, Samuel's checkers program, and others on pattern recognition,
+language understanding, philosophy, and so on.
+
+Finsler, Paul. "Formal Proofs and Undecidability", Reprinted in van Heijenoorcs
+anthology From Frege to Godel (see below). A forerunner of Godel's paper, in which
+the existence of undecidable mathematical statements is suggested, though not
+rigorously demonstrated.
+
+Fitzpatrick, P. J. "To Godel via Babel"- Mind 75 (1966): 332-350. An innovative
+exposition of Godel's proof which distinguishes between the relevant levels by using
+three different languages: English, French, and Latin'
+
+* Gablik, Suzi. Magritte. Boston, Mass.: New York Graphic Society, 1976. Paperback.
+An excellent book on Magritte and his works by someone who really understands their
+setting in a wide sense; has a good selection of reproductions.
+
+* Gardner, Martin. Fads and Fallacies. New York: Dover Publications, 1952. Paperback.
+Still probably the best of all the anti-occult books. Although probably not intended as a
+book on the philosophy of science, this book contains many lessons therein. Over and
+over, one faces the question, "What is evidence?" Gardner demonstrates how
+unearthing "the truth" requires art as much as science.
+
+Gebstadter, Egbert B. Copper, Silver, Gold: an Indestructible Metallic Alloy. Perth:
+Acidic Books, 1979. A formidable hodge-podge, turgid and confused-yet remarkably
+similar to the present work. Professor Gebstadter's Shandean digressions include some
+excellent examples of indirect self-reference. Of particular interest is a reference in its
+well-annotated bibliography to an isomorphic, but imaginary, book.
+
+** Godel, Kurt. On Formally Undecidable Propositions. New York: Basic Books, 1962.
+A translation of Godel's 1931 paper, together with some discussion.
+
+. "Uber Formal Unentscheidbare Satze der Principia Mathematica and Verwandter
+Systeme, I." Monatshefte fur Mathematik and Physik, 38 (1931), 173-198. Godel's
+1931 paper.
+
+Golf man, Ering. Frame Analysis. New York: Harper & Row, Colophon Books, 1974.
+Paperback. A long documentation of the definition of "systems" in human
+communication, and how in art and advertising and feporting and the theatre, the
+borderline between "the system" and "the world" is perceived and exploited and
+violated.
+
+Goldstein, Ira. and Seymour Papert. "Artificial Intelligence, Language, and the Study of
+Knowledge". Cognitive Science 1 (January 19/ 7): 84-123. A survey article concerned
+with the past and future of Al. The authors see three periods so far: "Classic",
+"Romantic,., and "Modern".
+
+Guod, 1.1. "Human and Machine Logic". British Journal for the Philosophy of Science In
+(196/): 144. One of the most interesting attempts to refute Lucas, having to do with
+whether the repeated application of the diagonal method is itself a mechanizable
+operation.
+
+. "Godel's Theorem is a Red Herring". British Journal for the Philosophy of Science 19
+(1969): 357. In which Good maintains that Lucas' argument has nothing to do with
+Godel's Theorem, and that Lucas should in fact have entitled his article "Minds,
+Machines, and Transfinite Counting". The Good-Lucas repartee is fascinating.
+
+Goodman, Nelson. Fact, Fiction, and Forecast. 3rd ed. Indianapolis: Bobbs-Merrill, 1973.
+Paperback. A discussion of contrary-to-fact conditionals and inductive logic, including
+
+
+Bibliography
+
+
+749
+
+
+
+Goodman's famous problem-words "bleen" and "grue". Bears very much on the
+question of how humans perceive the world, and therefore interesting especially from
+the A1 perspective.
+
+Press, 1976. A short hook about bees, apes, and other animals, and whether or not they
+are "conscious"-and particularh whether or not it is legitimate to use the word
+"consciousness" in scientific explanations of animal behavior.
+
+deGroot, Adriaan. Thought and Choice in Chess. The Hague: Mouton, 1965. A thorough
+study in cognitive psychology, reporting on experiments that have a classical simplicity
+and elegance.
+
+Gunderson, Keith. Mentality and Machines. New York: Doubleday, Anchor Books, 1971.
+Paperback. A very anti-Al person tells why. Sometimes hilarious.
+
+Hanawalt, Philip C., and Robert H. Haynes, eds. The Chemical Basis of Life. San
+Francisco: W. H. Freeman, 1973. Paperback. An excellent collection of reprints from
+the Scientilic American. One of the best ways to get a feeling for what molecular
+biology is about.
+
+Hardy G. H. and E. M. Wright. An Introduction to the Theory of Numbers, 4th ed. New
+York: Oxford University Press, 1960. The classic book on number theory. Chock-full
+of information about those mysterious entities, the whole numbers.
+
+Harmon, Leon. "The Recognition of Faces". Scientific American, November 1973, p. 70.
+Explorations concerning how we represent faces in our memories, and how much
+information is needed in what form for us to be able to recognize a face. One of the
+most fascinating of pattern recognition problems.
+
+van Heijenoort, Jean. From Frege to Godel: A Source Book in Mathematical Logic.
+Cambridge, Mass.: Harvard University Press, 1977. Paperback. A collection of epoch-
+making articles on mathematical logic, all leading up to Godel's climactic revelation,
+which is the final paper in the book.
+
+Henri, Adrian. Total Art: Environments, Happenings, and Performances. New York:
+Praeger, 1974. Paperback. In which it is shown how meaning has degenerated so far m
+modern art that the absence of meaning becomes profoundly meaningful (whatever that
+means).
+
+* Goodstein, R. L. Development of Mathematical Logic. New York: Springer Verlag,
+1971. A concise survey of mathematical logic, including much material not easily
+found elsewhere. An enjoyable book, and useful as a reference.
+
+Gordon, Cyrus. Forgotten Scripts. New York: Basic Books, 1968. A short and nicely
+written account of the decipherment of ancient hieroglyphics, cuneiform, and other
+scripts.
+
+Griffin, Donald. The Question of Animal Awareness. New York: Rockefeller University
+
+von Foerster, Heinz and James W. Beauchamp, eds. Music by Computers. New York:
+
+**
+
+John Wiles', 1969. This book contains not only a set of articles about various types of
+computer-produced music, but also a set of four small phonograph records so you can
+actually hear (and judge) the pieces described. Among the pieces is Max Mathews'
+mixture of "Johnny Comes Marching Home" and "The British Grenadiers".
+
+Fraenkel, Abraham, Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory,
+2nd ed. Atlantic Highlands, N. J.: Humanities Press, 1973. A fairly nontechnical
+
+
+Bibliography
+
+
+750
+
+
+
+discussion of set theory, logic, limitative Theorems and undecidable statements.
+Included is a long treatment of intuitionism. *
+
+Frev, Peter W. Chess Skill in Man and Machine. New York: Springer Verlag, 1977.
+Anexcellent survey of contemporary ideas in computer chess: why programs work, why
+they don't work, retrospects and prospects.
+
+Friedman, Daniel P. The Little Lisper. Palo Alto, Calif.: Science Research Associates,
+
+1974. Paperback. An easily digested introduction to recursive thinking in LISP. You'll
+eat it up!
+
+• Hoare, C. A. R, and D. C. S. Allison. "Incomputability". Computing Surveys 4,
+no. 3 September 1972). A smoothly presented exposition of why the halting problem
+is unsolvable. Proves this fundamental theorem: "Any language containing
+conditionals and recursive function definitions which is powerful enough to program
+its own interpreter cannot be used to program its own 'terminates' function."
+
+• Hofstadter, Douglas R. "Energy levels and wave functions of Bloch electrons in
+rational and irrational magnetic fields". Physical Review B, 14, no. 6 (15 September
+1976). The author's Ph.D. work, presented as a paper. Details the origin of "Gplot",
+the recursive graph shown in Figure 34.
+
+Hook, Sidney, ed. Dimensions of Mind. New York: Macmillan, Collier Books, 1961.
+Paperback. A collection of articles on the mind-body problem and the mind-computer
+problem. Some rather strong-minded entries here.
+
+• Hornev, Karen. Self-Analysis. New York: W. W. Norton, 1942. Paperback. A
+fascinating description of how the levels of the self must tangle to grapple with
+problems of self-definition of any individual in this complex world. Humane and
+insightful.
+
+Hubbard, John I. The Biological Basis of Mental Activity. Reading, Mass.:
+Addison Wesley, 1975. Paperback. Just one more book about the brain, with one special
+virtue, however: it contains man long lists of questions for the reader to ponder, and
+references to articles which treat those questions.
+
+• Jackson. Philip C. Introduction to Artificial Intelligence. New York, Petrocelli Charter.
+
+1975. A recent book, describing, with some exuberance, the ideas of Al. There are a
+huge number of vaguely suggested ideas floating around this book, and for that reason
+it is very stimulating just to page through it. Has a giant bibliography, which is another
+reason to recommend it.
+
+Jacobs, Robert L. Understanding Harmony. New York: Oxford University Press, 1958.
+Paperback. A straightforward book on harmony, which can lead one to ask many
+questions about why it is that conventional Western harmony has such a grip on our
+brains.
+
+Jaki, Stanley L. Brain, Mind, and Computers. South Bend. Ind.: Gateway Editions, 1969.
+Paperback. A polemic book whose every page exudes contempt for the computational
+paradigm for understanding the mind. Nonetheless it is interesting to ponder the points
+he brings up.
+
+• Jauch, J. M. Are Quanta Real? Bloomington, Ind.: Indiana University Press, 1973. A
+delightful little book of dialogues, using three characters borrowed from Galileo, put in
+a modern setting. Not only are questions of quantum mechanics discussed, but also
+issues of pattern recognition, simplicity, brain processes, and philosophy of science
+enter. Most enjoyable and provocative.
+
+
+Bibliography
+
+
+751
+
+
+
+• Jeffrey, Richard. Formal Logic: Its Scope and Limits. New York: McGraw-Hill, 1967.
+An easy-to-read elementary textbook whose last chapter is on Godel's and Church's
+Theorems. This book has quite a different approach from many logic texts, which
+makes it stand out.
+
+• Jensen, Hans. Sign, Symbol, and Script. New York: G. P. Putnam's, 1969. A-or perhaps
+the-top-notch book on symbolic writing systems the world over, both of now and long
+ago. There is much beauty and mystery in this book-for instance, the undeciphered
+script of Easter Island.
+
+Kalmar, Laszlo. "An Argument Against the Plausibility of Church's Thesis". In A.
+Heyting, ed. Constructivity in Mathematics: Proceedings of the Colloquium held at
+Amsterdam, 1957, North-Holland, 1959. An interesting article by perhaps the
+bestknown disbeliever in the Church-Turing Thesis.
+
+• Kim, Scott E. "The Impossible Skew Quadrilateral: A Four-Dimensional Optical
+Illusion". In David Brisson, ed. Proceedings of the 1978 A.A.A.S. Sym osium on
+Hypergraphics: Visualizing Complex Relationships in Art and Science. Boulder, Colo.:
+Westview Press, 1978. What seems at first an inconceivably hard idea-an optical
+illusion for four-dimensional "people"-is gradually made crystal clear, in an amazing
+virtuoso presentation utilizing a long series of excellently executed diagrams. The form
+of this article is just as intriguing and unusual as its content: it is tripartite on many
+levels simultaneously. This article and my book developed in parallel and each
+stimulated the other.
+
+Kleene, Stephen C. Introduction to Mathematical Logic. New York: John Wiley, 1967. A
+thorough, thoughtful text by an important figure in the subject. Very worthwhile. Each
+time I reread a passage, I find something new in it which had escaped me before.
+
+. Introduction to Metamathematics. Princeton: D. Van Nostrand (1952). Classic work on
+mathematical logic; his textbook (above) is essentially an abridged version. Rigorous
+and complete, but oldish.
+
+Kneebone G. J. Mathematical Logic and the Foundations of Mathematics. New York:
+Van Nostrand Reinhold, 1963. A solid book with much philosophical discussion of
+such topics as intuitionism, and the "reality" of the natural numbers, etc.
+
+Koestler, Arthur. The Act of Creation, New York: Dell, 1966. Paperback. A wide-ranging
+and generally stimulating theory about how ideas are "bisociated" to yield novelty. Best
+to open it at random and read, rather than begin at the beginning.
+
+Koestler, Arthur and J. R. Smythies, eds. Beyond Reductionism. Boston: Beacon Press,
+1969. Paperback. Proceedings of Yr conference whose participants all were of the
+opinion that biological systems cannot be explained reductionistically, and that there is
+something "emergent" about life. I am intrigued by books which seem wrong to me, yet
+in a hard-to-pin-down way.
+
+** Kutbose, Gyomay. Zen Koans. Chicago: Regnerv, 1973. Paperback. One of the best
+collections of koans available. Attractively presented- An essential book for any Zen
+library. Kuffler, Stephen W. and John G. Nicholls. From Neuron to Brain. Sunderland,
+deals Mss.: Smauer Associates, 1976. Paperback. A book which, despite its title, deals
+mostly with microscopic processes in the brain, and quite little with the way people's
+thoughts come out of the tangled mess. The work of Hubei and Wiesel on visual
+systems is covered particularly well.
+
+
+Bibliography
+
+
+752
+
+
+
+Lacey Huh, and Geoffrey Joseph. "What the Godel Formula Says". Mind 77 (1968)- 7~.
+A useful discussion of the meaning of the Godel formula, based on a strict separation of
+three levels: uninterpreted formal system, interpreted formal system, a
+metamathematics. Worth studying. Latos Imre. Proofs and Refutations. New York:
+Cambridge University Press, 1976. Paperback. A most entertaining book in dialogue
+form, discussing how concepts are formed in mathematics. Valuable not only to
+mathematicians, but also to people interested in thought processes.
+
+** Lehninger, Albert. Biochemistry. New York: Worth Publishers, 1976. A wonderfully
+readable text, considering its technical level. In this book one can find many ways in
+which proteins and genes are tangled together. Well organized, and exciting.
+
+** Lucas, J. R. "Minds, Machines, and Godel". Philosophy 36 (1961): 112. This article is
+reprinted in Anderson's Minds and Machines, and in Sayre and Crosson's The Modeling
+of Mind. A highly controversial and provocative article, it claims to show that the
+human brain cannot, in principle, be modeled by a computer program. The argument is
+based entirely on Godel's Incompleteness Theorem, and is a fascinating one. The prose
+is (to my mind)
+
+incredibly infuriating-vet for that very reason, it makes humorous reading. "Satan
+Stultified: A Rejoinder to Paul Benacerraf. Monist 52 (1968): 145.
+
+Anti-Benacerraf argument, written in hilariously learned style: at one point Lucas refers
+to Benacerraf as "self-stultifyingly eristic" (whatever that means). The Fucas-
+Benacerraf battle, like the Lucas-Good battle, offers much food for thought.
+
+. "Human and Machine Logic: A Rejoinder". British Journal for the Philosophy of
+Science 19 (1967): 155. An attempted refutation of Good's attempted refutation of
+Lucas' original article.
+
+** MacGillavry, Caroline H. Symmetry Aspects of the Periodic Drawings of M. C.
+Escher. Utrecht: A. Oosthoek's Uitgevermaatschappij, 1965. A collection of tilings of
+the plane by Escher, with scientific commentary by a crystallographer. The source for
+some of my illustrations-e.g., the Ant Fugue and the Crab Canon. Reissued in 1976 in
+New York by Harry N. Abrams under the title Fantasy and Symmetry.
+
+MacKay, Donald M. Information, Mechanism and Meaning. Cambridge, Mass.: M.I.T.
+Press, 1970. Paperback. A book about different measures of information, applicable in
+different situations; theoretical issues related to human perception and understanding;
+and the way in which conscious activity can arise from a mechanistic underpinning.
+
+• Mandelbrot, Benoit. Fractals: Form, Chance, and Dimension. San Francisco: W. H.
+Freeman, 1977. A rarity: a picture book of sophisticated contemporary research ideas in
+mathematics. Here, it concerns recursively defined curves and shapes, whose
+dimensionality is not a whole number. Amazingly, Mandelbrot shows their relevance to
+practically every branch of science.
+
+• McCarthy, John. "Ascribing Mental Qualities to Machines". To appear in Martin
+Ringle, ed. Philosophical Perspectives in Artificial Intelligence. New York: Humanities
+Press, 1979. A penetrating article about the circumstances under which it would make
+sense to say that a machine had beliefs, desires, intentions, consciousness, or free will.
+It is interesting to compare this article with the book by Griffin.
+
+Meschkowski, Herbert. Non-Euclidean Geometry. New York: Academic Press, 1964.
+Paperback. A short book with good historical commentary.
+
+
+Bibliography
+
+
+753
+
+
+
+Meyer, can. "Essai d'application de certains modeles cybernetiq ues it la coordina_ Lion
+clJei les insectes sociaux". Insertes Sociaux XII1, no. 2 1966 : o which ch aws some
+parallels between the neural organization in the( brain, and the organiz ae lion of an ant
+colon%.
+
+Meyer, Leonard B. Emotion and Meaning in Music. Chicago: University of Chicago
+Press, 1956. Paperback. A book which attempts to use ideas of Gestalt svcholo theory
+of perception to explain why musical structure is as it is. One of the more unusual the
+books on music and mind.
+
+. Music, The Arts, and Ideas. Chicago: University of Chicago Press, 1967. Paperback. A
+thoughtful analysis of mental processes involved in listening to music, and of
+hierarchical structures in music. The author compares modern trends in music with Zen
+Buddhism.
+
+Miller, G. A. and P. N. joint son-Laird. Language and Perception- Cambridge, Mass.:
+Harvard University Press, Belknap Press, 1976. A fascinating compendium of linguistic
+facts and theories, hearing on Whorl 's hypothesis that language is the same as
+worldviesv. A typical example is the discussion of the weird "mother-in-law" language
+of the Dyirbal people of Northern Queensland: a separate language used only for
+speaking to one's mother-in-law.
+
+Minsky, Marvin L. "Matter, Mind, and Models". In Marvin L. Minskv, ed. Son antic
+Information Processing. Cambridge, Mass.: M.I.T. Press, 1968. Though merely a few
+pages long, this article implies a whole philosophy of consciousness and machine
+intelligence. It is a memorable piece of writing by one of the deepest thinkers in the
+field.
+
+Minsky, Marvin L., and Seymour Papert_ ,92-t cial Intelligence Progress Report.
+Cambridge, Mass.: M.I.T. Artificial Intelligence Laboratory, A1 Memo 252, 1972. A
+survey of all the work in Artificial Intelligence done at M.I.T. up to 1972, relating it to
+psychology and epistemology. Could serve excellently as an introduction to Al.
+
+Monod, Jacques. Chance and Necessity. New York: Random House, Vintage Books,
+1971. Paperback. An extremel fertile mind writing in an idiosyncratic way about
+fascinating questions, such as how life is constructed out of non-life: how evolution,
+seeming_ to violate the second law of thermodynamics, is actually dependent on it. The
+book excited me deeply.
+
+* Morrison, Philip and Emil, eds. Charles Babbage and his Calculating Engines. Nesv
+York: Dover Publications, 1961. Paperback. A valuable source of information about the
+life of Babbage. A large fraction of Babbage's autobiography is reprinted here, along
+with several articles about Babbage's machines and his "Mechanical Notation".
+
+Mvhill, John. "Some Philosophical Implications of Mathematical Logic". Review of
+Metaphysics 6 (1952): 165. An unusual discussion of ways in which Godel's Theorem
+and Church's Theorem are connected to psychology and epistemology. Ends up in a
+discussion of beauty and creativity.
+
+Nagel, Ernest. The Structure of Science. New York: Harcourt, Brace, and World, 1961. A
+classic in the philosophy of science, featuring clear discussions of reductionism vs.
+holism, teleological vs. nonteleological explanations, etc.
+
+Nagel, Ernest and James R. Newman. Godel's Proof. New York: New York University
+Press, 1958. Paperback. An enjoyable and exciting presentation, which was, in many
+ways, the inspiration for my own book.
+
+
+Bibliography
+
+
+754
+
+
+
+* Nievergelt, Jurg, J. C. Farrar, and E. M. Reingold. Computer Approaches to Mathe,nat
+teal Problems. Englesyood Clif-f-s, N. J.: Prentice-Hall, 1974. An unusual collection of
+different types of problems which can be and have been attacked on computers-for
+instance, the "3n + I problem" (mentioned in my Rria with Diverse Variations) and
+other problems of number theory.
+
+Pattee, Howard H., ed. Hierarchy Theory. New York: George Braziller, 1973. Paperback.
+Subtitled "The Challenge of Complex Systems". Contains a good article by Herbert
+Simon covering some of the same ideas as does my Chapter on "Levels of Description".
+
+Peter, R6zsa. Recursive Functions. New York: Academic Press, 1967. A thorough
+discussion of primitive recursive functions, general recursive functions, partial
+recursive functions, the diagonal method, and many other fairly technical topics.
+
+Quine, Willard Van Ornman. The Ways of Paradox, and Other Essays. New York:
+Random House, 1966. A collection of Quine's thoughts on many topics. The first essay
+deals with carious sorts of paradoxes, and their resolutions. In it, he introduces the
+operation I call "quining" in my book.
+
+lishiRanganathan, S. R. Ramanujan, The Man and the Mathematician. London: Asia
+Pubng House, 1967. An occult-oriented biography of the Indian genius by an admirer.
+An odd but charming book.
+
+Reichardt. Jasia. Cybernetics, Arts, and Ideas. Boston: New York Graphic Society,
+
+197 1. A weird collection of ideas about computers and art, music, literature. Some of it
+is definitely off the deep end-hut some of it is not. Examples of the latter are the articles
+"A Chance for Art" by J. R. Pierce, and "Computerized Haiku" by Margaret
+Masterman.Renyi, Alfred. Dialogues on Mathematics. San Francisco: Holden-Day,
+1967. Paper hack. Three simple but stimulating dialogues involving classic characters
+in history, trying to get at the nature of mathematics. For the general public.
+
+** Rips Paul. Zen Flesh, Zen Bones. New York: Doubleday, Anchor Books. Paperback.
+This book imparts very well the flavor of Zen-its antirational, antilanguage, antireduc
+
+iionistic, basically holistic orientation.Rogers, Hartley. Theory of Recursive Functions
+and Effective Computability. New York:McGraw-Hill, 1967. A highly technical
+treatise, but a good one to learn from. Containsdiscussions of many intriguing problems
+in set theory and recursive function theory.
+
+Rokeach, Milton. The Three Christs of Ypsilanti. New York: Vintage Books,
+1964.paperback. A study of schizophrenia and the strange breeds of "consistency"
+which arise in the afflicted. A fascinating conflict between three men in a mental
+institution, all of whom imagined they were God, and how they dealt with being
+brought face to face for many months.
+
+Rose, Steven. The Conscious Brain, updated ed. New York: Vintage Books, 1976.
+
+Paperback. An excellent book-probably the best introduction to the study of the brain.
+Contains full discussions of the physical nature of the brain, as well as philosophical
+discussions on the nature_ of mind, reductionism vs. holism, free will vs. determinism,
+etc. from a broad, intelligent, and humanistic viewpoint. Only his ideas on A1 are way
+off.
+
+Rosenblueth, Arturo. Mind and Brain: A Philosophy of Science. Cambridge,
+Mass.:M.I.T. Press, 1970. Paperback. A well written book by a brain researcher who
+deals with most of the deep problems concerning mind and brain.
+
+
+Bibliography
+
+
+755
+
+
+
+* Sagan, Carl, ed. Communication with Extraterrestrial Intelligence. Cambridge, Mass.:
+M.I.T. Press, 1973. Paperback. Transcripts of a truly far-out conference, where a stellar
+group of scientists and others battle it out on this speculative issue.
+
+Salmon, Wesley, ed. Zeno's Paradoxes. New York: Bobbs-Merrill, 1970. Paperback. A
+collection of articles on Zeno's ancient paradoxes, scrutinized under the light of modern
+set theory, quantum mechanics, and so on. Curious and thought-provoking,
+occasionally humorous.
+
+Sanger. F., et al. "Nucleotide sequence of bacteriophage 16X174 DNA", Nature 265
+(Feb. 24, 1977). An exciting presentation of the first laying-bare ever of the full
+hereditary material of any organism. The surprise is the double-entendre: two proteins
+coded for in an overlapping way: almost too much to believe.
+
+Sayre, Kenneth M., and Frederick J. Crosson. The Modeling of Mind: Computers and
+Intelligence. New York: Simon and Schuster, Clarion Books, 1963. A collection of
+philosophical comments on the idea of Artificial Intelligence by people from a wide
+range of disciplines. Contributors include Anatol Rapoport, Fudwig Wittgenstein,
+Donald Mackay, Michael Scriven, Gilbert Ryle, and others.
+
+* Schank, Roger, and Kenneth Colby. Computer Models of Thought and Fanguage. San
+Francisco: W. H. Freeman, 1973. A collection of articles on various approaches to the
+simulation of mental processes such as language-understanding, belief-systems,
+translation, and so forth. An important Al book, and many of the articles are not hard to
+read, even for the layman.
+
+Schrodinger, Erwin. What is Fife? & Mind and Matter. New York: Cambridge University
+Press, 1967. Paperback. A famous book by a famous physicist (one of the main
+founders of quantum mechanics). Explores the physical basis of life and brain; then
+goes on to discuss consciousness in quite metaphysical terms. The first half, What is
+Fife?, had considerable influence in the 1940's on the search for the carrier of genetic
+information.
+
+Shepard, Roger N. "Circularity in judgments of Relative Pitch". Journal of the Acoustical
+Society of America 36, no. 12 (December 1964), pp. 2346-2353. The source of the
+amazing auditory illusion of "Shepard tones".
+
+Simon, Herbert A. The Sciences of the Artificial. Cambridge, Mass.: M.I.T. Press, 1969.
+Paperback. An interesting book on understanding complex systems. The last chapter,
+entitled "The Architecture of Complexity", discusses problems of reductionism versus
+holism somewhat.
+
+Smart, J. J. C. "Godel's Theorem, Church's Theorem, and Mechanism". Synthese 13
+(1961): 105. A well written article predating Fucas' 1961 article, but essentially arguing
+against it. One might conclude that you have to be Good and Smart, to argue against
+Fucas...
+
+** Smullvan, Raymond. Theory of Formal Systems. Princeton, N. J.: Princeton
+University Press, 1961. Paperback. An advanced treatise, but one which begins with a
+beautiful discussion of formal systems, and proves a simple version of Godel's Theorem
+in an elegant way. Worthwhile for Chapter 1 alone. What Is the Name of This Book?
+Englewood Cliffs, N. J.: Prentice-Hall, 1978. A book of puzzles and fantasies on
+paradoxes, self-reference, and Godel's Theorem. Sounds like it will appeal to many of
+the same readers as my book. It appeared after mine was all written (with the exception
+of a certain entry in my bibliography).
+
+
+Bibliography
+
+
+756
+
+
+
+Sommerhoff, Gerd. The Logic of the Living Brain. New York: John Wiley, 1974. A book
+which attempts to use knowledge of small-scale structures in the brain, in creating a
+theory of how the brain as a whole works.
+
+Sperrv, Roger. "Mind, Brain, and Humanist Values". In John R. Platt, ed. .Vew Views on
+the Nature of Man. Chicago: University of Chicago Press, 1965, A pioneering
+neurophvsiologist here explains most vividly how he reconciles brain activity and
+consciousness.
+
+* Steiner, George. After Babel: Aspects of Language and Translation. New York: Oxford
+University Press, 1975. Paperback. A book by a scholar in linguistics about the deep
+problems of translation and understanding of language by humans. Although A1 is
+hardly discussed, the tone is that to program a computer to understand a novel or a
+poem is out of the question. A well written, thought-provoking-sometimes infuriating-
+book.
+
+Stenesh, J. Dictionary of Biochemistry. New York: John Wiley, Wiley-Interscience,
+1975. For me, a useful companion to technical books on molecular biology.
+
+** Stent, Gunther. "Explicit and Implicit Semantic Content of the Genetic Information".
+In The Centrality of Science and Absolute Values, Vol. 1. Proceedings of the 4th
+International Conference on the Unity of the Sciences, New York, 1975. Amazingly
+enough, this article is in the proceedings of a conference organized by the nowinfamous
+Rev. Sun Mvung Moon. Despite this, the article is excellent. It is about whether a
+genotype can be said, in any operational sense, to contain "all" the information about its
+phenotype. In other words, it is about the location of meaning in the genotype.
+
+. Molecular Genetics: A Historical Narrative. San Francisco: W. H. Freeman, 1971. Stent
+has a broad, humanistic viewpoint, and conveys ideas in their historical perspective. An
+unusual text on molecular biology.
+
+Suppes, Patrick. Introduction to Fogic. New York: Van Nostrand Reinhold, 1957. A
+standard text, with clear presentations of both the Propositional Calculus and the
+Predicate Calculus. My Propositional Calculus stems mainly from here.
+
+Sussman, Gerald Jay. A Computer Model of Skill Acquisition. New York: American
+Elsevier, 1975. Paperback. A theory of programs which understand the task of
+programming a computer. The questions of how to break the task into parts, and of how
+the different parts of such a program should interact, are discussed in detail.
+
+** Tanenbaum, Andrew S. Structured Computer Organization. Englewood Cliffs, N. J.:
+Prentice-Hall, 1976. Excellent: a straightforward, extremely well written account of the
+many levels which are present in modern computer systems. It covers
+microprogramming languages, machine languages, assembly languages, operating
+systems, and many other topics. Has a good, partially annotated, bibliography.
+
+Tarski, Alfred. Logic, Semantics, Metamathematics. Papers from 1923 to 1938.
+Translated by J. H. Lot, New York: Oxford University Press, 1956. Sets forth Tarski's
+ideas about truth, and the relationship between language and the world it represents.
+These ideas are still having repercussions in the problem of knowledge representation
+in Artificial Intelligence.
+
+Taube, Mortimer. Computers and Common Sense. New York: McGraw-Hill, 1961.
+Paperback. Perhaps the first tirade against the modern concept of Artificial Intelligence.
+Annoying.
+
+
+Bibliography
+
+
+757
+
+
+
+Tietze, Heinrich. Famous Problems of Mathematics. Baltimore: Graylock Press, 1965. A
+book on famous problems, written in a very personal and erudite style. Good
+illustrations and historical material.
+
+Trakhtenbrot, V. Algorithms and Computing Machines. Heath. Paperback. A discussion
+of theoretical issues involving computers, particularly unsohable problems such as the
+halting problem, and the word-equisalence problem. Short, which is nice.
+
+Turing, Sara. Alan M. Turing. Cambridge, U. K.: WW. Hefter & Sons, 1959. A
+biography of the great computer pion4r. A mother's work of love.
+
+* Ulam, Stanislaw. Adventures of a Mathematician. New York: Charles Scribner's, 1976.
+An autobiography written by a sixty-five-year old man who writes as if he were still
+twenty and drunk in love with mathematics. Chock-full of gossip about who thought
+who was the best, and who envied whom, etc. Not only fun, but serious.
+
+,A'aitson, J. D. The Molecular Biology of the Gene, 3rd edition. Menlo Park, Calif.: W.
+A. Benjamin, 1976. A good book but not nearly as well organized as Lehninger's, in my
+opinion. Still almost every page has something interesting on it.
+
+Webb, Judson. "Metamathematics and the Philosophy of Mind". Philosophy of Science
+35 (1968): 156. A detailed and rigorous argument against Lucas, which contains this
+conclusion: "My overall position in the present paper may be stated by saying that the
+mind-machine-GOdel problem cannot be coherently treated until the constructivity
+problem in the foundations of mathematics is clarified."
+
+Weiss, Paul. "One Plus One Does Not Equal Two". In G. C. Quarton, T. Melnechuk, and
+F. O. Schmitt, eds. The Neurosciences:.4 Study Program. New York: Rockefeller
+University Press, 1967. An article trying to reconcile holism and reductionism, but a
+good bit too holism-oriented for my taste.
+
+* Weizenbaum, Joseph. Computer Power and Human Reason. San Francisco: W. H.
+Freeman, Freeman, 19i6. Paperback. A provocative book by an early At worker who
+has come to the conclusion that much work in computer science, particularly in Al, is
+dangerous. Although I can agree with him on some of his criticisms, I think he goes too
+far. His sanctimonious reference to Al people as "artificial intelligentsia" is funny the
+first time, but becomes tiring after the dozenth time. Anyone interested in computers
+should read it.
+
+Wheeler, William Morton. "The Ant-Colony as an Organism".Journal of Morphology
+
+22, 2 (1911): 307-325. One of the foremost authorities of his time on insects gives a
+
+famous statement about why an ant colony deserves the label "organism" as much as its
+parts do.
+
+Whitely, C. H. "Minds, Machines, and Godel: A Reply to Mr Fucas". Philosophy 37
+(1962): 61. A simple but potent reply to Fucas' argument. Wilder, Raymond. An
+Introduction to the Foundations of Mathematics. New York: John Wiley, 1952. A good
+general overview, putting into perspective the important ideas of the past century.
+
+* Wilson, Edward O. The Insect Societies. Cambridge, Mass.: Harvard University Press,
+elknap Press, 1971. Paperback. The authoritative book on collective behavior of insects.
+Although it is detailed, it is still readable, and discusses many fascinating ideas. It has
+excellent illustrations, and a giant (although regrettably not annotated) bibliography.
+
+Winograd, Terry. Five Fectures on Artificial Intelligence. Al Memo 246. Stanford, Calif.:
+Stanford University Artificial Intelligence Faboratory, 1974. Paperback. A description
+of fundamental problems in At and new ideas for attacking them, by one of the
+
+
+Bibliography
+
+
+758
+
+
+
+important contemporary workers in the field.. Language as a Cognitive Process.
+Reading, Mass.: Addison-Wesley (forthcoming). From what I have seen of the
+manuscript, this will be a most exciting book, dealing with language in its full
+complexity as no other book ever has. . Understanding Natural Language. New York:
+Academic Press, 1972. A detailed discussion of one particular program which is
+remarkably "smart", in a limited world. The book shows how language cannot be
+separated from a general understanding of the world, and suggests directions to go in, in
+writing programs which can use language in the way that people do. An important
+contribution; many ideas can be stimulated by a reading of this book.
+
+. "On some contested suppositions of generative linguistics about the scientific study of
+language". Cognition 4:6. A droll rebuttal to a head-on attack on Artificial Intelligence
+by some doctrinaire linguists.
+
+* Winston, Patrick. Artificial Intelligence. Reading, Mass.: Addison-Wesley, 1977. A
+strong, general presentation of many facets of At by a dedicated and influential young
+proponent. The first half is independent of programs; the second half is LISP-dependent
+and includes a good brief exposition of the language LISP. The book contains many
+pointers to present-day At literature, ed. The Psychology of Computer Vision. New
+York: McGraw-Hill. 1975. Silly title, but fine book. It contains articles on how to
+program computers to do visual recognition of objects, scenes, and so forth. The articles
+deal with all levels of the problem, from the detection of line segments to the general
+organization of knowledge. In particular, there is an article by Winston himself on a
+program he wrote which develops abstract concepts from concrete examples, and an
+article by Minsky on the nascent notion of "frames".
+
+* Wooldridge, Dean. Mechanical Man-The Physical Basis of Intelligent Life. New York:
+NJ McGraw-Hill, 1968. Paperback. A thorough-going discussion of the relationship of
+mental phenomena to brain phenomena, written in clear language. Explores difficult
+philosophical concepts in novel was, shedding light on them by means of concrete
+examples.
+
+
+Bibliography
+
+
+759
+
+
+
+Credits
+
+
+Figures: Fig. 1, Johann Sebastian Bach by Elias Gottlieb-Hart ssmann (1748), collection
+of William H. Scheide, Princeton, New Jersey; Fig. 2, Flute Concert in Saussouci, by
+Adolf von Menzel, Nationalgalerie, West Berlin; Figs. 3, 4, 152, "The Royal Theme" and
+the last page of the "Six-part Ricercar." from the original edition of Musical Offering by
+Johann Sebastian Bath, are reproduced courtesy of the Library of Congress; Figures of
+lithographs and woodcuts of M. C. Escher are reproduced by permission of the Escher
+Foundation, Haags Gemeentemuseum, The Hague, copyright © the Escher Foundation,
+1979, reproduction rights arranged courtesy of the Vorpal Galleries, New York, Chicago,
+San Francisco, and Laguna Beach; Fig. 9, photograph of Kurt Godel by Orren J. Turner
+from Foundations of
+
+Mathematics. Symposium Papers Commemorating the Sixtieth Birthday of Kurt Godel,
+edited by
+
+Jack J. Bulloff, Thomas C. Holyoke, and S. W. Hahn, New York, Springer-Verlag, 1969;
+Figs. 17, 96, "Figure-Figure" and "A section of mRNA passing through a ribosome,"
+drawings by Scott E. Kim; Figs. 19, 44, 133, 148, musical selections from the Musical
+Offering by J. S. Bach, music printed by Donald Byrd's program "SMUT"; Fig. 25,
+"Cretan labyrinth" from W. H. Matthews, Mazes and Labyrinths: Their History and
+Development, New York, Dover Publications, Inc., 1970; Fig. 39, photograph of Rosetta
+Stone, courtesy of the British Museum; Fig. 40, A collage of scripts. Samples of
+cuneiform, Easter Island, Mongolian and Runic scripts from Hans Jensen, Sign, Symbol
+and Script, East Germany VEB Deutscher Verlag Der Wissenschaften; samples of
+Bengali and Buginese script from Kenneth Katzner, The Languages of the World, New
+York, Funk & Wagnalls, 1975; samples of Tamil and Thai from I. A. Richards and
+Christine Gibson, English Through Pictures, New York, Washington Square Press; Fig.
+59, "Intelligence built up layer by layer" (Fig. 9.8) adapted from Patrick Henry Winston.
+Artificial Intelligence, Reading, Mass., Addison-Wesley Publishing Company, reprinted
+by permission; Figs. 63, 69, photographs of an ant bridge by Carl W. Rettenmeyer and
+construction of an arch by termite workers by Turid Hdlldobler, from E. O. Wilson, The
+Insect Societies, Cambridge, Mass., Harvard University Press, 1979; Fig. 65, schematic
+drawing of a neuron adapted from The Machinery of the Brain, by Dean Wooldridge,
+copyright © 1963, McGraw-Hill, Inc. used with permission of McGraw Hill Book
+Company, and from Fig. 11-6, page 26, Speech and Brain-Mechanisms, by Wilber
+Penfield and Lamar Roberts, copyright © by Princeton University Press, reprinted by
+permission of Princeton University Press; Fig. 66, "the human brain, seen from the left
+side," from Steven Rose, The Conscious Brain, copyright © 1973 by Steven Rose,
+reprinted by permission of Alfred A. Knopf, Inc., New York, and John Wolfers, London;
+Fig. 68, "overlapping neural pathways," from John C. Eccles, Facing Reality, New York,
+SpringerVerlag, 1970; Figs. 77, 78, 80, 82, 117, 137, 138, 141, The Shadows, State of
+Grace, The Fair
+
+Captive, The Air and the Song, Mental Arithmetic, Common Sense, The Two Mysteries,
+and The
+
+Human Condition I by Rene Magritte, copyright © by ADAGP, Paris, 1979; Figs. 79, 95,
+"Tobacco Mosaic Virus" and "Secondary and Tertiary Structure of Myoglobin" from
+Albert Lehninger, Biochemistry, New York, Worth Publishers, 1975; Figs. 91, 92, "The
+
+
+Credits
+
+
+757
+
+
+
+four constituent bases of DNA" and "The ladder-like structure of DNA" from Arthur
+Kornberg, "The Synthesis of DNA," Scientific American, copyright © October 1968, all
+rights reserved; Fig. 93, "Molecular model of the DNA double helix," reprinted by
+permission, from V. M. Ingram, Biosynthesis of Macromolecules, Menlo Park,
+
+California, The Benjamin/Cummings Publishing Company, 1972; Fig. 97,
+
+"Polyribosome" from The Proteins, edited by R. E. Dickerson and H. Neurath, page 64,
+New York, Academic Press; Fig. 98, "A two-tiered molecular canon" from O. L. Miller,
+Jr., "Visualization of Genes in Action," in Scientific American, copyright © March 1973,
+all rights reserved; Figs. 101, 102, 103, "The T4 bacterial virus," "Infection of a
+Bacterium by a virus," and "The morphogenetic pathway of the T4 virus," from William
+B. Wood and R. S. Edgar, "Building a Bacterial Virus" in Scientific American, copyright
+© July 1987, all rights reserved; Fig.
+
+105, photograph of Srinivasa Ramanujan from S. R. Ranganathan, Ramanujan, the Man
+and the Mathematician, New York, Asia Publishing House, 1967; Figs. 110, 111, 112,
+from Terry Winograd, Understanding Natural Language,New York, Academic Press,
+1972; Fig. 113, Photograph of Alan Turing by Mssrs. C. H. O. Trevelyan from Sara
+Turing, Alan M. Turing, Cambridge, England, W. H. Heffer and Sons, Ltd., 1959; Fig.
+116, "a meaningful story in Arabic" from Abdelkebir Khatibi and Mohammed Sijelmassi,
+The Splendor of Islamic Callizrabhl, New York, London, Thames & Hudson, copyright
+© by Qarawiyne Library in Fez. Fig 118, Procedural representation of “red cube which
+supports a pyramid,” adapted from Computer Models of Thought and Language, edited
+by Roger C. Schank and Kenneth Mark Colby. W. H. Freeman and Company, copyright
+1973; Figs. 119, 122, 124, 130, Bongard problems from M. Bongard, Pattern
+Recognition, Rochelle Park, New Jersey,
+
+Hayden Book Company, Spartan Books, 1970
+
+Grateful acknowledgment is made to the following publishers for permission to quote
+excerpts from the following material: The Bach Reader. A Life of johann Sebastian Bach
+in Letters and Documents, edited by Hans T. David and Arthur Mendel, Revised, with the
+permission of W. W. Norton & Company, Inc., copyright (~) 1966, 1945 by W. W.
+Norton & Company, Inc. copyright renewed 1972 by Mrs. Hans T. David and Arthur
+Mendel; J. S. Bach's Musical Offering, page 179, edited by Hans T. David, New York,
+copyright 1945 by G. Schirmer, Inc., used by permission; Gyomay Kubose, Zen Koans,
+Chicago, Regnery, 1973; Pauls Reps, Zen Flesh, Zen Bones, Tokyo, Japan, Charles E.
+Tuttle Co., Inc. 1957; J. R. Lucas, "Minds, Machines, and Godel," and Alan M. Turing,
+"Computing Machinery and Intelligence," from ,finds and Machines, edited by A. R.
+Anderson, Englewood Cliffs, New Jersey, Prentice-Hall, 1964, and Philosophy, vol. 36,
+1961; J. M. Jauch, Are Quanta Real? Bloomington, Indiana, Indiana University Press,
+1973; James R. Newman, "Srinivasa Ramanujan," in The World of Mathematics, edited
+by James R. Newman, New York, Simon & Schuster, reprinted by permission of Simon
+& Schuster, a division of Gulf & Western Corporation, 1956; Terry Winograd, "A
+Procedural Model of Language Understanding," from Computer Models of Thought and
+Language, edited by Roger C. Schank and Kenneth Mark Colby, San Francisco, W. H.
+Freeman and Company, copyright © 1973; Joseph Weizenbaum, Computer Power and
+Human Reason: From judgment To Calculation, San Francisco, W. H. Freeman and
+Company, copyright © 1976.
+
+
+Credits
+
+
+758
+
+
+
+INDEX
+
+
+AABB form, 130, 227
+Abel, Niels Henrik, 404
+ABORT (BlooP), 412
+abortion, 176 levels of, 651-53, 656-57,
+660,
+
+abstraction, ti66-72, 674
+Achilles: answer-schemas and, 475; Carroll
+paradox and, 46, 170, 181, 193; Crab
+Canon and, 204 666-67; heel of, 389, 484;
+image of GOD, 223; inaccessible neurons
+of, 328. 686; initial letter of, 231, 507, 667;
+innocence of, 406, 408, 424; mapped onto
+ant colony, 318, 324; mentioned 84, 272,
+525, 573, 669; mystified by Crab, 560;
+orchard analogy and, 427; origin of, 28, 29;
+picture of, 42; problem reduction and, 610-
+11; recursion and, 128-31, 149; as violin,
+502
+
+Achilles property, 396-98, 415
+acoustico-retrieval, 278-80, 460
+acronyms, 31-32, 113, 174, 176, 204, 237,
+272, 374, 606, 684, 727, 736, 738, 740
+acrostics, 7, 81
+active site, 528-29, 544
+actor formalism, 662-64
+„Actually Intelligent" programs, 676
+addition: A1 programs and, 677-78;
+commutativity, associativity of, 55, 225-
+26; in BlooP, 409; noncommutative, 222-
+23, 639; pqsystem and, 49-53, 417;
+representability of, 417; of supernaturals,
+455; TNT notation for, 206-7; triple, 101,
+206-7
+
+addresses (in memory), 289, 290
+adenine, see nucleotides
+advertisements, 478
+
+Al: applied to mathematics, 573, 614-15;
+arguments against, 597-99; attitudes
+against, see anti-Al attitudes; computer
+languages and, 299-300, 548; converging
+towards brains, 579; defined, 26; difficulty
+of, 26-27, 573, 740; evidence and, 695;
+faith underlying, 572, 578-79; G6del's
+Theorem and, 388-90, 471-77, 706-7;
+
+
+history of, 19, 24-27, 594-97, 600-9;
+leveltranslation and, 285; outline of, 601-3;
+relation to mathematics, 559-60;
+stratification of, 299-300
+Al programs: compared with people, 679-
+80; curiosity of, 679; Lucas' argument and,
+577-78
+
+Al Thesis, 579
+
+Air and the Song, The (Magritte), 494
+
+Air on G's String, 445, 446, 497
+
+airplane-haystack story, 675
+
+aleatoric music, 163, 174, 700
+
+Algol, 292, 293,381,630
+
+algorithms, 292, 410, 412, 413, 414, 440,
+
+459, 567 all 60
+
+"almost"-situations, 634-40, 641-43, 649
+alpha helix, 521, 525
+
+Alternative Structures of the Union, see
+ASU’s
+
+ambiguity: computer languages and, 297-
+98; in translation into TNT, 209-11
+amino acids, 518-25, 533-34; in
+
+Typogenetics,
+
+508,510-11
+
+analogical thinking: by computer, 603;
+
+underpin
+
+ning of, 570-71
+
+analogies, 668-74
+
+Analytical Engine, 25, 598, 727 anaphase,
+667
+
+anarchy, 693
+
+"and", 177-80, 181, 186, 630 Anderson,
+Alan Ross, 197 Announcer, 633-40
+anomalies, 44, 96, 208, 435, 723 Another
+World (Escher), 250, 255 answer-schemas,
+462-14, 475, 688 ant bridge, 334
+ant colonies: artificial, 359; caste
+distribution in,
+
+318-28; castes in, 317-18; communism in,
+318,
+
+330-31; compared with brains, 315-16,
+318,
+
+324-25, 350, 358-59; contrasted with gases.
+
+
+Index
+
+
+1
+
+
+
+317; freedom and control in, 315-16, 327;
+intel
+
+ligent, 310-36; levels in, 319-27;
+mechanisms of
+
+teams and signals, 317-21; order and chaos
+in,
+
+316-17; regrouped, 332-33; signals in, 320-
+28;
+
+symbol level in, 324-28, 330; teams in,
+317,
+
+319-27; thresholds in, 316-17, 319-21;
+trails of,
+
+315-17; see also caste distribution, Aunt
+Hillary,
+
+Fermant, job. Seb., Fourmi, Lierre de,
+teams,
+
+signals, symbols
+
+Ant Fugue, 337, 349, 350, 382, 570, 686,
+737-39 "Aid Fugue" (Escher), 322-23
+Anteater, 275-84, 311-36, 382, 570, 722
+anti-AI attitudes. 27, 470-72, 628
+
+anticodons, 522-24
+
+ants: dispensability of, 326; nest of, 359;
+vs. ant
+
+colonies, 314, 315, 319, 321, 326, 330
+aperiodic crystals, 167-69, 174-76 arch of
+termites, 358-59
+
+
+Aria with Diverse Variations (Bach), 392-
+93, 395 Aria with Diverse Variations
+(Dialogue), 408 Aristotle, 19
+arithmetization, 262-65, 268-69, 533-34
+arithmoquining, 445-54, 466-68, 497, 502,
+541,
+
+580-81
+
+arms, mutually washing, 691
+
+art: identity crisis of, 699-700, 703-6;
+
+modern,
+
+699-706; by computer, 603, 619-20 Art-
+Language, 622
+
+Art of the Fugue (Bach), 79-81, 86, 671
+Art of Zen Strings, 237, 239-42, 244, 626
+Artificial Imagery, 560 Artificial Intuition,
+560 Artificial Ism, 625
+artificial "thinking", 337, 601
+Ascending and Descending (Escher), 12-
+13,15,21,
+
+716
+
+Asian Box, Very Gold, see Gold Box, Very
+Asian assemblers, 291, 294
+assembly language, 290-95; compared to
+DNA,
+
+290-91
+
+assembly lines, cellular, 528-29, 544-45
+
+
+Appearances of Achilles and the Tortoise in Dialogues are not indexed, but those of less
+frequent characters are. The reader is encouraged to consult the figure on page 370 for
+possible help in crossreferences.
+
+
+associativity, 35-56, 207
+
+assumptions, shaky, 420, 422, 429. 580,
+
+581,644,672
+
+ASU's: defined, 373-74; orientation in,
+
+375; trips in, 377-79, 383
+
+ATN's, see Augmented Transition
+
+Networks ATN-colony, 359
+
+atoms: in physics, 303; in Propositional
+
+Calculus,
+
+182, 186; in TNT, 207,213,214
+ATTACCA, 176, 284, 311, 505
+attachment and nonattachment, 232-33,
+244 augmentation, 8, 9, 146, 503, 681-83,
+738; inter
+
+
+vallic, 156-57
+
+Augmented Transition Networks, 150, 258-
+59,359,621,625-27, 630
+Augustus II, Elector of Saxons, King of
+Poland, 461
+
+Aunt Hillary, 314-33, 382, 630
+
+Author, the, 3, 6, 204, 370, 689, 722, 724-
+
+42 authorship triangle, 94-95, 96, 689
+
+autological adjectives, 20-21
+
+automatic programming, 298-99 Avers,
+
+Oswald. 159, 161
+
+awareness, deductive vs. analogical, 570-
+71,619
+
+
+Index
+
+
+2
+
+
+
+axiom schemata, 47, 48, 65, 87, 468, 472-
+73, 543
+
+axiomatic systems, see formal systems
+axioms: defined, 35; lack of in
+
+Propositional Calculus, 183; of MIU-
+svstern, 33, 35; of P-system, 73-74; of pq-
+system, 47; of pq-system, modified, 87; of
+310-system, 263; of TNT, 216; of TNT,
+extended, 451-52, 466-68; of tq-system, 65
+axons, 339-40
+
+Babbage, Charles, 24-26, 598, 601, 726-42
+
+Babbage test, 735-73
+
+BACH (acronym), 174-75
+
+B-A-C-H (melody), 79-81, 86, 102, 121,
+
+155-57,266,719
+
+Bach, Anna Magdalena, 482
+
+Bach, C. P. E„ 3, 4, 80
+
+Bach, Joh and Seb., 633, 669
+
+Bach, Joh. Seb.; A1 and, 27, 677; as
+
+composer, 392, 461, 740-42; as
+
+glassblower, 79; as harpsichordist, 275,
+
+279, 280; as inspiration for Dialogues, 28,
+
+737; confused with Fermat, 331-35; depth
+
+of, 7-8, 9-10, 27-28, 70-71, 677; dissection
+
+vs. appreciation of, 680; Escher and, 199,
+
+666-t7; Forkel on, 4, 86; homage to, 81;
+
+improvisation by, 3-7, 96, 719; in Leipzig,
+
+383, 404; indirect self-reference of, 79-81,
+
+86; life and death of, 86; modulation and,
+
+122-23, 130; recursive qualities of music
+
+of, 70-71; Shepard tones and, 719; squared,
+
+679; vs. Cage, 157, 1624, 174-75; see also
+
+Old Bach
+
+Bach, Wilhelm Friedemann, 4, 6
+background assumptions, 644
+backtracking, 97, 629, 632
+backwards chaining, 618
+bankers, 457
+
+base-pairing, complementary, 433, 506-7,
+
+514-16,517,523,534
+
+baseball, 637-38
+
+bases (genetics): 514, see also nucleotides;
+in Tvpogenetics, 505-8, 510
+Baso, 234
+Bassui, 255
+
+
+beautiful vs. non-beautiful, 552-58, 560,
+573-75, .581-82
+
+beauty: computers and, 575; elusiveness of,
+554, 555, 565, 574-75, 581, 583-84
+bees, 360, 641, 720
+
+Beethoven, Ludwig van. 6. 75, 163, 634
+beliefs, catalogue of. 382, 384 Bell, A. G.,
+296
+
+Belnap, Nuel, 197 Beno, Luciano, 704
+bicentuplets, 540
+
+bifurcations, 91-94, 100, 456-59, 467, 579
+bio jukeboxes, 160, 175 Birthday Cantata
+(Bach), 461
+
+Birthday Cantatatata (Dialogue), 468, 475,
+
+688 birthdays, 461-64
+
+bits, 288-89, 290, 291 BLOCK (BlooP),
+
+410-11
+
+blocks world, 586-93, 627-32, 674
+BlooP, 406. 409-30, 440, 441, 444;
+alphabet of,
+
+419, 425; control structures in. 410;
+primordial
+
+steps of, 409, 412-13; syntax of, 410-15
+
+BlooP programs, 410-14 Blue Programs,
+
+418-20,422,427
+
+Bluediag [N], 42-44, 428
+
+board evaluation: static and dynamic, 604-
+
+5,611;
+
+Strange Loops in. 604-5
+
+Bodhidharnta, 232, 238, 245. 252, 625
+
+Bolvai, Farkas, 92
+
+Bolyai, Janos, 91-92 Bongard, M., 646
+Bongard problems, 646-62, 664. 669, 674;
+uni
+
+versality of, 661-62
+
+Boole, Geo., 20, 404, 600 Boolean
+Buddhism, 577 boomerangs, 75, 84, 267,
+472 bootstrapping, 24, 293-94, 524, 548
+bottom (recursion), 139-40; see also
+skeletons bottom-up vs. top-down, 48-49
+bottoming out, 133-35, 151, 259, 301, 650
+boustrophedonic writing, 168-69, 176
+boxed systems, 469-70, 543
+Boyle's law, 308; see also gases
+brains: ant colonies and, 315-16, 318, 324-
+25, 350, 358-59; as ATN-colonies, 359; as
+
+
+Index
+
+
+760
+
+
+
+mathematical objects, 559; Epimenides
+paradox and, 584-85; formal systems and,
+337-38, 559-62, 569-79, 584-85, 676, see
+also Church-Turing Thesis, formal vs.
+informal systems, brains and rules; frame
+and outer messages and, 170-71; mappings
+between, 341-42, 345-46, 369-82; music
+and, 163; operating systems and, 296;
+programmability of, 302; rules and, 26-27,
+676, see also brains and formal systems;
+suborgans of, 340-41; thoughts and, 337-
+65; vs. minds, 576; see also minds,
+intelligence, etc.
+
+Breton, Andre, 700
+
+"British Grenadiers, The", 607
+
+Brouwer, Luitzen E. J., 404
+
+Buddha-nature, 233, 234, 238-44
+
+Butterflies (Escher), 147-48
+
+Buxtehude, Dietrich, 335
+
+Byrd, Donald, see SMUT
+
+Byron, Lord, 25
+
+C-svstem, 65-67, 71-72
+
+CAGE (acrorn.m), 174-75 C-A-G-E
+
+(melody), 156-57
+
+Cage, John, 156-57, 163-64, 167, 174-75,
+556,
+
+699-700, 704
+
+calculating prodigies, see idiots savants
+Canon by Intervallic Augmentation, 525
+"Canon per augmentationem et contrario
+motu"
+
+(Bach), see Sloth Canon
+
+„Canon per Tonos" (Bach), see Endlessly
+
+Rising Canon
+
+canons: copies and, 8-9, 146; Dialogues
+and, 665-69, 738; Escher drawings and, 15;
+in Goldberg Variations, 392; in Musical
+Offering 7-10, 726-27; polyribosomes and,
+
+526- 28; self-refs and, 501-3; structure of,
+8-10; two-tiered,
+
+527- 28; see also individual canons, fugues
+Cantor, Georg, 20, 216, 418, 421, 422-24
+Cantor set, 142 .Cantorcrostipunctus", 424
+Capitalized Essences, 29
+
+car radio. 670-71
+
+cardinalitv, intuitive sense of, 567
+
+
+Carroll, John B., 630
+
+Canon, Lewis, 20, 28, 46, 192, 372, 681;
+material by, 43-45. 366-68
+Carroll paradox: 28, 43-45, 681; evidence
+ver
+
+sion, 693-94; message version, 170:
+problem
+
+po4ed by, 46. 181; proof version, 192-93;
+Sam
+
+uel's argument and, 684-85; symbolized,
+193; see also infinite regress
+cascades, 224, 529, 626, 664
+caste distribution: encoding of knowledge
+in, 319,
+
+324-28, 359; meaning of, 321-24; updating
+of,
+
+318-19, 324
+
+catalogues of programs (Blue, Green, Red),
+419,
+
+427-28
+
+catalysts, 528-29
+
+cats, 313, 343-46, 532 causality, types of,
+709-10 CCrab, see ATTACCA
+ceilings, see loops, bounded: BlooP
+celestial mechanics, 353-54 CELLS
+
+(BlooP), 410-11
+
+cellular processes, as models for All, 663-
+64 Central Crabmap, 667
+Central Dogma: of Mathematical Logic,
+271,
+
+532-34; of the MIC-system, 513; of
+Molecular
+
+Biology, 504-5, 514, 532-34, 536, 667; of
+Ty
+
+pogenetics, 513; of Zen strings, 238, 239,
+240,
+
+243
+
+Central Dogmap, 532-34, 545, 547, 672,
+709 Central Pipemap, 701-2 central
+processing unit, 288, 289 Central
+
+Proposition, 264, 269 Central Slothmap,
+702
+
+Central Xmaps, 702, 716; see also
+
+individual en
+
+tries
+
+
+Index
+
+
+761
+
+
+
+centrality, 374-75 centromere, 668
+
+cerebellum, 341 Chadwick, John, 50 chain
+letters, 546 Champernowne, David, 595
+Champollion, Jean Fransois, 165
+
+channeling, 299, 376-77
+chaos in number theory, 137-38, 152, 557;
+see also
+
+order and chaos
+chauvinism, 171-73
+
+checkers programs, 573, 604-5 Chekhov,
+Anton, 642
+
+chess: chunking and, 285-87, 604; grand
+
+masters in, 286-87; round-the-house, 595;
+
+self-modifying, 687-88
+
+chess boards, hierarchy of, 687 chess
+
+players, cycle of, 94-95
+
+chess programs: Babbage and. 25, 729-31,
+
+736;
+
+choice and, 711-12; Crab and, 721, 729-31;
+difficulty of, 151-52, 605; jumping out of
+the system and, 37-38, 678; knowledge
+representation in, 618; recursive structure
+of 150-52; strengths and weaknesses of,
+151-52, 285-87, 573, 603-4, 611; Turing
+and, 595, 596, 736: varieties of, 601;
+without look-ahead, 604
+chests of drawers, nested, 644-45
+children's stories and Al, 675-76
+Chiyono, 256
+choice, 711-14
+
+Chopin, Frederic, 70, 257, 677 chords and
+analogies, 673-74 chromosomes,
+homologous, 668
+
+chunked versions of this book: jacket,
+viiixiii, 370, 758-77
+
+chunking: ant colonies and, 326-27; brains
+arid, 381-84, 559; computer languages and,
+290-92, 381, 412-13; defined, 285-88;
+determinism and, 306-8, 363, 522; of
+DNA, 531-32; intuitive world-view and,
+305-6, 362-63; of music, 160, 164, 525; of
+one's own brain, 382; probabilistic,
+
+384; scientific explanation and, 305-6;
+super conductivity and, 305; trade-offs in,
+326; vision and, 348
+Church, Alonzo, 428, 476, 561
+
+
+Church-Turing Thesis, 428-29, 552, 561-
+79; Al Version, 578-79, 580, 581; Hardy's
+Version 566; Isomorphism Version, 567-
+68; Microscopic Version, 572; Public-
+Processes Version, 562, 568, 574, 580;
+Reductionist's Version, 572, 574; Soulist's
+Version, 574; Standard Version, 561-62,
+579; Tautological Version, 561; The odore
+Roszak Version, 574-75; unprovability of,
+562
+
+Church's Theorem, 560-61,574,579-
+81,609, 697 cigars, 199, 201, 383, 481, 651
+classes vs. instances, 351-55, 360-61; see
+also prototypes, intensionality and
+extensionality, analogies, conceptual
+skeletons, etc.codes: art and, 703-4;
+familiar and unfamiliar,82, 158. 267; see
+also decoding, Godel Code,
+
+Genetic Code, etc.
+
+"coding" of sentences, 583-84
+
+codons, 519-20, 524, 533, 535; see also
+
+Godel
+
+codons, duplets
+
+Colby, K., 599 columns in brain, 346
+Comenius, Johann Amos, 625 comments in
+programs, 297 Common Sense (Magritte),
+700-1 common sense and programs, 301
+communicability of algorithms, 562
+commutativity, 55-56, 209, 225-27, 453,
+639 compelling inner logic, 161-62, 163-64
+competing theories, and nature of evidence,
+695 compiler languages, 292-95 compilers,
+292-95, 297, 503 compiling, reverse of,
+381
+
+Complete List of All Great
+
+Mathematicians, see List
+
+completeness, 100-2, 417-18, 422, 465; see
+
+also incompleteness, consistency
+
+complexity of world, 569
+
+composite numbers, 64, 65-66, 73; see also
+
+prime
+
+numbers
+
+compound sentences, in TNT, 214
+
+compound words, 665
+
+computer chess, see chess programs
+
+
+Index
+
+
+762
+
+
+
+computer languages: analogues in cell,
+547; dialects of, 503; flexibility and, 298-
+99; high level, 292-93, 297-300; message¬
+passing, 662-63; power of, 299, 428-29;
+presented, 289-99, 406-30, 498-99; in
+SHRDLU, 629-32 computer systems, 287-
+302 computers: assembled by computers,
+504, 684; crying, 675-76; determinism and,
+25-27, 306-7, 684-86; fallibility of, 575,
+578, 678; learning by, 603-5; origins of,
+24-26; in phonographs, 78, 484, 486-88;
+see also programs, Al, A1 programs
+concept network, 651-54; see also semantic
+networks
+
+conceptual dimensions, 670-71
+conceptual mapping. 668-72
+conceptual nearness, 371-73, 614, 651-56
+conceptual revolutions, 660-61, 673
+
+conceptual skeletons, 381, 514. 666-72,
+674 connotations and culture, 372-73, 379-
+80 consciousness: causality and, 709-10;
+source of,
+
+384-85, 387-88: understanding of, 82, 680,
+708-10
+
+conservation of complexity, 60, 195
+consistency: defined, 94; of extended TNT,
+223, 459; hypothetical worlds and, 95-100;
+interpretations and, 88, 94-101; Lucas and,
+477; oath of, in TNT, 450; proofs of, 23-
+24, 191-92, 229-30, 449-50; of
+
+Propositional Calculus, 191-92, 229; of
+TNT, 229-30, 449-50; varieties of, 94-96;
+see also (o-consistency
+constants, parameters, variables, 643-44,
+669
+
+context: necessity of, 1614, 173-76;
+
+restoring of, 115-16, 128, 133, 161-64,
+173-76
+
+context-free sums, 520-22
+contexts, nested, 643-46, 672, 674
+continued fractions, 140, 277, 563, 565
+continuous vs. discrete processes, 598
+Contracrostipunctus, discussed, 82-85, 267,
+270-71, 406-7, 424, 467-70, 483-84, 534-
+37,608, 721
+
+
+contradictions: between levels, see level-
+conflicts; caused by impossible cycles, 94-
+96; coexisting in same brain, 383, 697-98;
+diagonal argument and, 420-22; in
+mathematics, 17-24, 196-97, 223, 580-81;
+so-consistency and, 453; personal
+nonexistence and, 698; in pq-system, 87,
+88; in Propositional Calculus, 191-92, 196-
+97; in selfimage, 696; Tortoise's shell and,
+177-80; two levels of, 581, 584; visual, 97-
+99; Zen and, 99, 235, 246-56, 698; see also
+paradoxes, inconsistency, Epimenides
+paradox, etc.
+
+Contrafadus, 641, 643, 669
+
+Convex and Concave (Escher), 105-9, 348
+
+Cooper pairs, 304-5
+
+copies: canons and, 8-9, 527-28; in code,
+517, 527-28; complementary to originals,
+70, 501, 506-7, 517, see also inversion;
+DNA and, 529-31; inexact, 500-3, 546;
+nature of, 146-49; self-reps and, 500-4,
+512-13; television and, 489; viruses and,
+542-43; visually nested, 138-40; see also
+sameness, isomorphisms
+copper, 173
+
+Cops Silva and Could, 405
+Copy mode (Typogenetics), 506-8
+cortex: areas of, 344; cerebral, 340-48;
+visual, 343-48
+
+counterfactual parameters, 639
+
+counterf actuals, 634-40, 641-44
+
+counterpoint, see canons, fugues, Bach, etc.
+
+counting, 55-57, 228, 364
+
+court system, 692, 693
+
+covalent bonds, 514
+
+cows, 312, 346, 351
+
+CPU, see central processing unit
+
+Crab: entertains Achilles, 480-94; genes of,
+
+200-1, 204. 507;’ hikes and plays flute,
+
+549-58;
+
+intelligence of. 549, 558, 733; jukebox of,
+154-57; meets Achilles, 200; musical
+evening
+
+chez, 720-42; origin of, 666-68; plight of,
+281,
+
+383-84; questionable behavior of, 560, 562,
+
+
+Index
+
+
+763
+
+
+
+573-74, 579-81; receives presents and enter
+tains guests, 275-84, 311-36; subjunctive
+afternoon chez, 633-40; Theme of, 729,
+732. 740, 742; vs. Tortoise, 75-78, 406,
+493-88,540, 543 Crab Canon (Bach), 202-
+3, 666 Crab Canon (Dialogue), 204, 355-
+56, 665-69, 672, 723-25, 738
+Crab Canon (Escher), 198-99, 667
+crab canons, 9, 198-203, 204, 355, 501.
+665-69; in DNA, 200-1
+crab programs, 500-1
+
+creativity, mechanizability of. 25, 26, 571,
+620, 673; see also originality, paradox of
+Al, non programmability, etc.
+
+Crick, Francs, 505, 532, 533, 534, 617
+Crime and Punishment i'Dostoevskvl, 379-
+80 Cristofori, Bartolommeo, 3 critical
+mass, 228, 317, 389, 417, 470 crossing-
+over, 665-68 crystal in magnetic field, 140-
+43 crystallization metaphor, 347 C"f-
+Thesis, see Church-Turing Thesis Cube
+with Magic Ribbons (Escher), 281-82
+cursively drawable figures, 67-68, 72
+cytoplasm, 517, 518, 522-24 cytosine, see
+nucleotides
+
+da Vinci, Leonardo, 641
+Dali, Salvador, 700
+
+Dase, Johann Martin Zacharias, 567 data
+base, 618 daughter programs, 503, 546
+David, Hans Theodore, 3, 28, 719 Day and
+Night (Escher), 252, 255, 667 Dboups,
+Hfpsh, 515, 533 De Chirico, Giorgio, 700
+De Morgan, Augustus, 20, 404, 600
+decidable strings, 417 decimal system, 262-
+64, 269
+
+decipherment of texts, 50, 164-65, 173-74,
+583 decision procedures: for alienness,
+487-88, 540-41; for axiomhood, 41, 48,
+470; for beauty, 552-58, 560, 579, 581-82,
+583-84; as BlooP puzzles, 415-17; for
+Buddha-nature, 234, 239, 272; defined, 39-
+41; for Diophantine equations, 559-60; for
+dreamable themes, 384; for end of
+Dialogue, 402-3; for genuineness of koans,
+234, 239; for Goldbach property, 400, 414;
+for Granny, 344-45, 347-48; for halting,
+
+
+425-29; for intelligence, see Turing test; for
+Mozart pieces, 649; no guarantee of
+existence of, 72; for number-theoretic
+truths, 228-29, 426, 551-58, 560. 573-74,
+579-81; for primality, 64, 149, 413; for
+proof-pair-ness, 416, 439-41; for sameness,
+146-49, 158-59; for sorting numbers into
+two classes, see Church-Turing Thesis; for
+termination, 425-29; for "the Way" in Zen,
+250-51, 253, 254; for theoremhood, 39-41,
+47-49, 72-73, 190-91, 408, 416, 560, 579-
+80, 582; for theorem-numbers, 440-41,
+580; topdown vs. bottom-up, 48-49; for
+Tortoise-pairness, 441; for Tortoise
+property, 396-97, 415, 441; for truth, 213,
+228-29, 417, 552-58, 560-61, 579, 581; for
+validity of derivations, 194, 416, 439-41,
+470; for well-formedness, 182, 269, 416,
+582; for wondrousness, 402, 425
+
+d coding: of DNA, 159-62, 175-76, 201,
+231, 531-32, 538; of formal systems, 50-
+51, 54; of fortune, 154; via Gtidel
+isomorphism, 267; of nature. 409; of
+records, 154-57, 158-59,
+
+161-64, 172, 174-75; as revelation, 160-61;
+of Russian text, 380; see also
+isomorphisms, translation, information,
+chunking
+
+decoding mechanisms: complexity of, 158-
+62, 172-76, 582-84; innate, 170-71; nature
+of, 158-76; record players as, 83, 154-57;
+transparency of, 267, 501; for Tripitaka,
+257; see also isomorphisms, etc.
+
+L)eduction Theorem, 186
+default options, 352-53, 386, 411, 645, 674
+defects and expectations, 77, 86, 102, 222,
+476 demidoublets, 633-34, 669
+demons, 663-64 deoxyribonucleic acid, see
+DNA derivations: alleged, 439-40; defined,
+35-36: fallacious, 220, 439; in MIU-
+system, 35-36, 262, 264, 439; in
+
+Propositional Calculus, 184, 185, 188, 189-
+90, 196; supernatural, 454-55; in
+TNT, 217, 218, 219, 224-27, 269; in tq-
+system,
+
+
+Index
+
+
+764
+
+
+
+65; in Typogenetics, 507, 509; vs. proofs,
+35-36, 193-94, 195
+
+Descartes, Rene, 263, 340, 677 description-
+schemas, 650; see also templates
+descriptions: calculus of, 338; restructuring
+of,
+
+649-53, 659-61, 672; tentative, 646-49
+descriptors, 647
+
+detachment, rule of, 185-86, 577
+
+determinism, 54; see also free will Devil,
+685
+
+Dewdrop (Escher), 249, 256 Di of Antus,
+333
+
+diagonal method, Cantor's, 418-24, 426,
+427-29,
+
+438, 446, 469
+
+Diagram G, 135-37 dialogicians, 81
+dialogues, miniature, 191-92, 193, 408-9,
+431,
+
+560, 565, 595-96, 598, 599
+
+Dialogues: origin of, 28, 665-69; as self-
+
+refs,
+
+84-85, 129, 204, 502-3, 667, 738-39
+Dickens, Charles, 326, 328, 380 Difference
+Engine, 25 differentiation, cellular, 543-46
+digestion, 306
+
+digits, shunting of, 264 Diophantine
+equations, 279, 459-60 Diophantus of
+Alexandria, 275 directory of real numbers,
+421-24 disambiguation, 586-93, 603, 629-
+32 distance to goal, 611-13 divisor-
+freeness, 74
+
+djinns, 113-15, 216-17, 223, 224; see also
+Genie DNA: as aperiodic crystal, 167; as
+carrier of genetic information, 159;
+compared to computer languages, 290-91;
+composition and structure of, 514-15;
+covalent backbone of, 514-15; as
+declarative knowledge, 616-17; double
+strands, 514-15, 530-31; isomorphism with
+organism,
+
+146-48; mode of self-replication, 529-30;
+in outer space, 167, 175-76; as program,
+language,data, 290-91, 547; quining and,
+531; recombinant, 665; relation to mRNA,
+517; self-destroy
+
+
+ing, 536; unusual interpretation of, 231
+DNA endonudease, 530, 531 DNA ligase,
+530,531
+
+DNA polymerise, 530, 531
+DNA Rapid Transit Service, 505. 517
+Doctor program, 599-600. 608 dog-and-
+bone problem, 611-13 .doGma 1, doGma
+II, 532-33
+
+dogs, 233, 234, 354, 383, 569, 570, 611-12,
+679 Doko, 250, 698
+
+Dostoevsky, Feodor, 379-80 double
+negation, 183, 545, 554 Double
+
+Nodulation, law of, 243 Dragon (Escher),
+473-74, 524, 698
+
+Drawing Hands (Escher), 15, 21, 133, 689-
+92,710,716, 737
+
+dreams, 378, 379, 384, 725 Dreyfus,
+Hubert, 574
+
+dualism, 251-55, 698-99; see also subject
+vs. object Dumpty, H., 332
+duplets (Typogenetics), 510, 512 Dvoi£k,
+Antonin, 163
+
+E. coli bacterium, 176, 537-41
+Earrwig, Dr. Tony, 586-93. 627
+earth chauvinism, 171-72
+Earth-Moon-Sun system, 353-54
+earthworm(s), 341-42
+Eccles, John, 574
+
+Edifying Thoughts of a Tobacco Smoker
+(Bach), 482
+
+Edifying Thoughts of a Tobacco Smoker
+(Dialogue), 543
+eggs, 192, 360, 383
+Einstein, Albert, 100
+electrons, 140-46, 258, 30.3-5
+ELIZA, see Doctor program
+embedding of formal systems, 97, 207, 215
+emergent phenomena, 708-9, 714
+emotions: brains and, 83; counterfeit, 599-
+600; dependence of intelligence upon, 573,
+597-98; as epiphenomena, 677; music and,
+83, 160, 163-64, 174-75, 383-84, 626-27,
+676-77, 699; potential, 281, 383-84, 583;
+programs and, 573, 597-600, 626-27, 675-
+77; universality of, 163, 174-75
+
+
+Index
+
+
+765
+
+
+
+emulation, 295
+
+Endlessly Rising Canon (Bach), 10-11, 15,
+46, 130,717-19,742
+ENIUQ (procedure), 498-99
+enlightenment, 232, 237, 243, 246, 251,
+254-55, 479, 567
+
+Enlightenment 'Yond Enlightenment, 237,
+239, 243, 244
+End, 232, 252
+
+enzymes: function of, 520-22, 528-30, 543-
+45; as models for AI, 663-64; rules of
+inference and, 509-10, 513, 531; structure
+of, 519-21, 525; synthesis of, 517-19, 522-
+25, 527-29, 538-45, 547; versatility of,
+529; vs. typoenzymes, 529; see also
+proteins, typoenzymes
+epigenesis, 159-60, 161-62, 531-32, 665
+Epimenides, picture of, 496
+Epimenides paradox: connection with
+G6del's Theorem, 17-18; Escher and, 716;
+expanded version, 21, 22; fear of, 23;
+French-English version, 501; indirect
+recursion and, 134; molecular version, 536-
+37; neural version, 584-85; Quine version,
+431-37, 445, 446, 449. 497-99, 531, 537;
+subtlety of, 495-98; Tarski version, 580-81,
+584-85; two levels of, 581, 584-85;
+Whitely's version, 476-77
+epiphen9mena, 308-9, 363, 577, 596, 677-
+79
+
+errors in programs, 295, 297-98. 596
+Escher, Maurits Cornelis: Bach and, 201,
+666-67; contradictions and, 97-99; copies
+and, 146-48; drawings of, see List of
+Illustrations (xiv-xviii); figure and ground
+in, 67-68; flat vs. spatial and, 473-74, 689;
+incompleteness and, 716-17; Magritte and,
+480; as prime mover, 689-92, 710; on
+subbrains, 387; Strange Loops and, 10-15,
+737; Zen and, 255-57
+
+Escherization, repeatability of, 473-74,
+689; see also 2-D vs. 3-D, Godelization
+l.SP, 598-99, 693-95
+
+essential incompleteness: of Achilles'
+birthday, 462-64, 475-76, 688; of Al, see
+Tesleis Theorem; of list of reals, 423-24,
+
+
+469; of phonographs, see Todelization; of
+self-engulfing process, 493; of TNT and
+related systems, 468-71; see also
+Godelization, Tlidelization, Escherization,
+nonprogrammahility, etc.
+
+Eta Oin, 586-93, 674
+
+ETAOIN SHRDLU, 628,630
+
+Euclid, 19, 43, 58-60, 88-89, 216
+
+Euclid's prime number Theorem, 35, 58-60,
+
+228 Euler, Leonhard, 3, 394
+
+Fume, Max. 605
+
+evidence, nature of, 633-36
+
+evolution, 321-22
+
+existential quantifiers, see quantifiers
+exotic styles of thought, 552, 563-64, 566-
+67 expanding nodes, 134-36
+explanatory power on high level, 321, 326,
+707-10 expressibility and expressive
+power, 101, 417,
+
+441-43, 444-45, 450, 454, 465-70, 580-81
+extrasensory perception, see ESP
+extraterrestrial intelligence, 162-64, 167,
+172-76,
+
+341,646, 661-62
+
+eyes, 237, 248, 260, 308, 311, 313, 477,
+633,715
+
+Fair Captive, The (Magritte), 489 FANCY
+NOUN, 132-34
+
+Fantasia and Fugue in G minor (Bath), 719
+fantasy rule, 183-85, 187
+faucet, mental, 364-65
+faultlessly functioning machines, 575-78
+Faure, Gabriel, 163
+
+feedback and feedforward, 544-45
+
+Fermant, Johant Sebastiant, 332-35
+
+Fermant's Last Fugue, 335
+
+Fermat, Pierre de, 275-77, 278; confused
+
+with Bach, 331-35
+
+fermatas, 275, 329, 332, 333
+
+Fermat's Last Theorem: 275-79, 332, 416;
+
+coun
+
+terexample to, 277, 279, 460; inverted,
+333-34;
+
+parodied, 335, 551; proof of, 277, 279, 460
+Feynman diagrams, 144-46
+
+
+Index
+
+
+766
+
+
+
+Fibonacci (Leonardo of Pisa), 136, 246
+Fibonacci sequence, 136, 138, 139, 152,
+173-74, 265,416
+fiddles, see violins
+
+fifth postulate (Euclid), 90-93, 222, 451-52
+50 (fifty), 338, 557, 564
+figure and ground: 61-63, 64-74, 731; in
+music, 70-71
+
+FIGURE-FIGURE figure (Kim), 68-70, 73
+filters: for abstraction, 286, 407-9, 648,
+657-60,
+
+663, 673; for Pools, 418, 427
+finitistic methods of reasoning, 24, 230
+Fishes and Scales (Escher), 146-47
+fission and fusion (of concepts), 338, 352-
+56, 438,
+
+470, 664-65
+5-D space, 640
+flags, 29-32, 188
+flashcards, fee tRNA
+
+flat vs, spatial, see 2-D vs. 3-D flattened
+look-ahead, 604-5
+
+flattened roles of inference, see theorems
+vs. rules flexibility and inflexibility, 26-27,
+296-302, 611-14, 657, 673-75, 686
+flights of fancy, 378
+
+FlooP, 406, 424-30, 567-68; fed into itself,
+425-26; power of, 428-29. 561-62
+fluency, 376-77
+
+flutes, 3-5, 27, 528, 552-58, 720, 726 F(n)
+and Min), 137, 142, 359 focusing, 657-59
+folding-up of enzymes, 511-12, 519, 521,
+525 football, 44, 303, 353, 634-40, 643,
+644, 645, 672 footraces, 29-32, 43, 594-95,
+681-83 forced matching, 670-72 forgetting,
+577, 578, 619 Forkel, Johann Nikolaus, 4.
+86 form, 47. 66-67, 68, 73. 190-91, 370-71;
+syntactic
+
+vs. semantic, 581-84, 631
+
+form and content, 84-85, 204, 279, 581-84,
+
+667-68, 740
+
+formal systems, presentations of: C-system,
+64-65; MlU-system, 33-41; P-system, 73-
+74:
+
+pq-system, 46-60; Propositional
+Calculus,
+
+
+181-97; TNT, 204-30; tq-system, 64-65;
+
+Typogenetics. 504-13
+
+formal systems vs. reality, 53-58
+
+formal vs. informal reasoning, 193-97,
+
+228-29,
+
+271-72. 449-50, 614-15, 618-19
+
+formal vs. informal systems, 26-27, 559-
+
+85, 598,
+
+684-86; see also brains, minds, etc.
+formalist philosophy of mathematics, 458
+formula: closed, see sentence; open, 207-8
+formulas of TNT, 206, 207-15 Four-color
+Theorem, parodied, 550 4-D space, 638-39
+four-postulate geometry, see geometry,
+absolute 4'33" (Cage), 156
+Fourmi, Lierre de, 333-34 frame effect, 704
+frame messages, 162, 166-67, 176 frames,
+373, 644-46, 662-63, 672 framing devices,
+478 Frank, Philipp, 642
+Frederick the Great, King of Prussia, 3-8,
+27,394,729
+
+free will, 388, 680, 708, 710-14, 720-16,
+734-35, 739; see also jumping out of the
+system
+
+Frege, Gottlob, 20
+
+French fries, 636-38, 683
+
+French Suite no. 5, Gigue (Bach), 130
+
+frequencies, of words and letters, 377, 630
+
+friend, mental model of, 386-87 fringe
+
+science, 693-94
+
+fugues, 335, 634, 730, 736; in the Art of
+the Fugue,
+
+79-81, 86; devices in, 314, 322-23, 329-30,
+737-40; Dialogues and, 28; in the Musical
+Offering, 4-9; nature of, 9, 281-84, 737
+Fundamental Facts 1 and 2. 440-42
+fundamental jukebox-axiom, 155
+funneling, 346-48
+
+G (Godel's string), 18, 271-72, 285, 447-
+55, 459-60. 502, 580, 608, 667, 707-8 G',
+G" G'” Gu, 466-68 -G, 272, 449, 451-55,
+458-59, 542 Galileo, 478-79
+games played by A1 programs, 601
+Ganto, 189-90, 256, 407
+gases and molecules, 307-8, 317, 693
+Gateles.s Gate, see Muvtonkan Gauss, Karl
+
+
+Index
+
+
+767
+
+
+
+Friedrich, 92, 100 Gebstadter, Egbert B.,
+94-95, 402-3, 484 Gelemter, E.. 606-7
+general recursivity, 406, 430, 470
+genes, 200-1, 507, 512, 524-25, 531, 544-
+45, 668 Genetic code, 160. 519-20, 522-24,
+533-34, 536,
+
+538; origins of, 231, 548
+genetics, 504-48
+
+(.enie, Meta-Genie, etc., 109-16, 216-17,
+223, 224,
+
+610; see also djinns
+genie, symbol-manipulating. 39-40, 48
+genotype and phenotype, 159-62, 167, 173-
+74,
+
+175-76, 295,531-32, 667-68
+Gentzen, Gerhard, 195
+Geometric Code. 235-37, 241, 626
+geometry: absolute, 91, 93, 97, 222, 407,
+451-52;
+
+elliptical, 93; Euclidean, 19-20, 88-92, 100,
+
+222, 451, 456, 606-7, non-Euclidean, 19-
+20, 91-93, 98-99, 100,222-23, 451, 455-56;
+"true" version of, 88-94, 99-100, 456-57
+German professor, proverbial, 130-31
+Giant Electronic Brains. 25, 601 glia, 339
+GlooP, 406, 428-29 G(n), 137
+
+goals and subgoals, 227, 589, 590-91, 609-
+
+14,618-19,629,632
+
+Goblet G, 79, 81, 83-85, 267
+
+God, 216, 400, 478, 482, 533, 567, 597,
+
+711; picture of, 142-43
+
+GOD (acronym), 110-15, 133, 134, 216,
+
+223, 224,
+
+722; see also djinns, Genie
+
+Gddel, Kurt, 15-19, 24, 28, 738, 740, 742
+
+G6del Code, 18, 268, 533-35 Godel
+
+codons, 268, 425, 533-35
+
+Godel isomorphism, 261-71, 439, 442-46,
+
+738-39; likened to reflection of world in
+
+brain, 502, 570
+
+Godel-numbering, 18, 438, 738-39; of
+FlooP programs, 425-26, 502; of MIU-
+system, 261-64; of TNT, 268-70, 579
+Godel questions, Lucas on, 389, 390
+Godelization, 270; programmability, of,
+471-73;
+
+
+repeatability of. 424, 465-76, 688; see also
+Escherization, Todelization, jumping out 1
+of the system
+
+Godelizing operator, 472-73, 475-76, 543
+Godel's argument, summarized, 18, 272,
+448 Godel's article. 17, 24, 438 Godel's
+construction, illustrated, 84
+Godel's method, underlying causes of, 204,
+407,465,468-71
+
+Godel's Second Theorem. 230, 449-50, 696
+Godel's Theorem: All and, 388-90, 471-77,
+706-7, 714; analogue of in molecular
+biology, 534, 536-37; brief mentions, 72,
+74, 78, 100, 486; consequences of, 450-60,
+469-76; Contracrosttpunctus and, see
+Contracrosttpunctus: Diophantine
+
+equations and, 459-60; LISP and,
+
+738-39; proof of, 18, 265-72, 438-49;
+stated, 17, 101, 272
+
+Godel's Theorems and human
+introspection, 450, 696-98
+Goffman, Erving, 478 gold, 173
+Gold Box, Very Asian, 404-5
+Goldbach, Christian, 394, 395
+Goldbach Conjecture, 394-96, 400, 404,
+557-58, 615; parodied, 551
+Goldbach property, 395-97, 400, 414, 418
+Goldbach Variation, 395-98. 400, 426, 441
+Goldberg, Johann Theophilus, 391-92
+Goldberg Variations (Bach), 392-93, 395
+Goodfortune, Hexachlorophene J., 103-4,
+115, 128-29, 130
+Goso, 248
+
+Gplot, 138, 140-43, 146-47, 159, 503, 661
+grammar: for computer languages, 297,
+408-15; for Feynman diagrams, 142. 145;
+high-level, 625-27; for koans, 625-26; for
+music, 626-27; for natural languages. 130-
+34,150,363,
+
+588-93, 619-21, 630-32; for thought, 627
+Grand Tortue, 237, 243-44 grandmother
+cell, 344, 345
+
+grandmothers, perception of, 344, 345,
+347-48. 349
+
+graphics, 728 grass roots, 693
+
+
+Index
+
+
+768
+
+
+
+Great Tutor, 237, 239, 244 Green
+
+Programs, 427 Greendiag [NJ, 427
+Grelling's paradox, 20-21, 22 Groot,
+
+Adriaan de, 286 grounds, excellent, 731
+guanine, see nucleotides
+guaranteed termination, 41, 396-98, 399,
+403 guitars, 62, 200
+
+gullibility, 75-76, 106, 309, 461, 600, 701
+Gutei, 237
+
+G0025, white stony, 626
+
+HACKER, 664
+
+haiku, 153-54, 525,619-20
+
+halting problem, 74, 425-29, 594, 697
+
+hamburger-confusion, 577 Hammurabi,
+
+169
+
+Hardy, Godfrey Harold, 562-66
+harmonic tension, 122-23; see also tension
+and resolution
+
+harpsichords, 3, 391, 502 Harrison,
+
+Lawrence, 657 Haussmann, Elias Gottlieb,
+2 HE-HE puzzle, 62-63, 669 headache, 62-
+63 hearing by computer, 602 "heart", in an
+All program, 679 Heisenberg uncertainty
+principle, 455, 698 Helen of Troy, 110
+hemiolia, 257, 519 hemispheres, 257, 340-
+41 Henkin, Leon, 541
+
+Henkin sentences, 541-53, 709; explicit
+and im
+
+plicit versions, 542-43
+Henkin's Theorem, 488
+hereditary arguments, 36, 47-48, 217, 261
+heterarchies, 134, 359, 534, 651-54, 662,
+691 heterological adjectives,20-21, 22
+heuristics, 587, 588, 590, 603, 629 Hewitt,
+Carl, 662
+
+hiccups, 116, 254, 255, 673. 725-26, 736
+hierarchy of variability, 643-45, 669
+high-fidelity vs. low-fidelity, 77, 85, 101,
+406-7,
+
+470. 697-98
+
+high-level explanatory power, 707-9
+Hilbert, David, 20, 23-24, 230, 459-60
+Hilbert's program, 23-24, 229-30
+Hilbert's tenth problem, 459-60 H(n), 137
+
+
+Hofstadter, D. R„ 75, 310, 724, 728, 742
+Hofstadter s law, 152
+Hogen, 248
+
+holes in formalized systems, 24, 26, 449,
+451,465,468, 470-71
+
+holism: defined, 254, 312; vs.
+
+reductionism, 284, 311-36, 389-90, 708-9;
+Zen and, 254 Hubei, David, 341, 343
+Human Condition I, The (Magritte), 705-6
+Hyakujo, 254
+
+hydrogen bonds, 516, 522, 525
+hyphen-strings, 47, 64-65, 66
+hypothetical worlds, 95-100, 338, 360-62,
+634-40, 641-44; groundedness in reality.
+362,
+
+378-79
+
+hypotheticals, 44-45, 634-40
+
+i, 454
+I, 454
+
+"I", referent of, 608
+
+"I Can Be Played (Proven, etc.)...", 488,
+541
+
+"I Cannot Be Played (Proven, etc.) , 76-
+77, 85,
+
+406-7, 448, 465-67, 536, 541, 608 I-counts,
+260-61
+
+1-level, see inviolate level
+I-mode, see Intelligent mode
+iceberg, 495-96, 497
+ideal numbers, 56-58
+
+identification with artifacts, 609, 713-14
+idiots savants, 566-67
+IF-statements (BlooP), 411-12
+images: blurry, 686-87; of thought, 623
+"Imaginary Landscape no. 4" (Cage), 163-
+64, 699 imitation game, see Turing test
+Immunity Theorem, 536
+implicit characterization, 41, 67, 72-73, 93
+improvisation vs. introspection, 739
+inaccessibility of lower levels to higher
+levels, 686-92, 706-10; in Aunt Hillary,
+330-31, 630: in brains/minds, 302, 328-29,
+362-65, 619, 677, 686-92, 697, 706-10,
+739; in programs, 296, 300-1, 588, 630,
+
+
+Index
+
+
+769
+
+
+
+679; see also software and hardware,
+introspection, level-conflicts
+incompleteness: Bach and, 86; of brains,
+585; defined, 86; Escher and, 716-17; of
+extensions of TNT, 465-71; of formal
+arithmetics, 18, 86, 101-2, 407, 618-19; of
+list of mathematicians, 422; of list of reals,
+421-24; of Lucas, 477; of phonographs, see
+record players, intrinsic vulnerability of; of
+Principia Mathematica, 18, 24, 618-19; of
+self-knowledge, 696-98; of TNT, 271-72,
+430, 450-51; see also essential
+
+incompleteness, wincompleteness, etc.
+inconsistency: defined, 94; with external
+world, 87-88, 95; internal, 87, 88, 94-96; of
+people, 197, 697-98; of Tortoise, 177-80;
+see also consistency, contradictions, su-
+inconsistency, Zen
+
+increasing and decreasing rules, 73, 74,
+
+260-61, 264, 269, 401-2, 407-x, 441; see
+
+also lengthening and shortening rules,
+
+chaos in number theory
+
+index numbers for programs, 418-20, 427-
+
+28 index triplets for supernaturals, 455
+
+India, 549,551,557,562-66
+
+Indra's Net, 258, 359
+
+inducers, 545
+
+infinite bundle of facts, 397-98 infinite
+coincidence, 398, 421
+
+infinite regress, 111-13, 142, 146, 152,
+231, 388-89, 426, 497, 738; in Carroll
+paradox, 43-45, 170, 192-93, 684-86. 693-
+94; halted, 127, 133-35, 170, 605, 684-86;
+of objectivity, 479; Zeno and, 31-32, 610;
+see also Carroll paradox, bottoming out,
+recursive acronyms, repeatability, etc.
+infinite sentence, 497 infinite sky, 401
+infinitesimals and nonstandard analysis,
+455 infinity: Bach and, 10, 719; Escher
+and, 15; handled finitely, 59-60, 221-25,
+461-64, 468; illustrated, 135-36, 138-43;
+names of, 475-76; supernaturals and, 454;
+types of, 421; see also nontermination,
+infinite regress, recursion, etc.
+informal systems, see formal vs. informal
+systems information: accessibility of, see
+
+
+inaccessibility; creation of, 513; depth from
+
+surface, 234-35, 409, 427, 549-58, 606-7,
+
+612-13, 628, 673, see also decoding;
+
+discardable, 649, 653, 657-59. 669-72:
+
+flow of, 513, 533, 545, 547; irrelevant, 560
+
+information-bearers, 158, 166, 167
+
+information -reveale rs, 158, 267
+
+inhibition, cellular, 544
+
+inner messages, 166-71, 174-76, 501, 524
+
+input-output devices, 288
+
+input parameters (BlooP), 411
+
+insight, 613, 660-61, 665-76
+
+instant replays, straight and subjunctive,
+
+634-40,641,672
+
+instructions: in machine language, 289-95;
+vs. templates, 497-99, 531, see also
+programs vs. data
+INT(x), 138-41, 146, 661
+intelligence: accidental inexplicability of,
+707; essential abilities for, 26;
+extraterrestrial, see extraterrestrial
+intelligence; liftability of, see skimming
+off; limits of, 475-76, 679-80; necessary
+underpinning of, 324; simplicity of, 172-
+73; subtle features of, 566; tangled
+recursion and, 152; typical abilities of, 559;
+universality of, and intrinsic meaning, 158,
+162-64, 170-76, 501, 661-62; see also
+brains, minds, Al, etc.
+
+Intelligent mode, 38-39, 65, 193-94, 613-
+14
+
+intensionality and extensionality, 337-39,
+350, 361-62
+
+intentions of machines, 684-85
+interestingness, programmed, 615
+interpretation-conventions, 687-88
+interpretations: adjusted to avoid
+
+inconsistency, 87-88, 453, 456, see also
+undefined terms; multiple, 94-102, 153-57,
+266-67, 271, 447-48; of pq-system, 49-53,
+87-88, 101-2, 158; of Propositional
+
+Calculus, 186-87, 189, 191-92; of strands,
+509-10; of TNT, 205-9, 266-67, 453, 533;
+of tq-system, C-system, P-system, 64-65,
+73-74
+
+
+Index
+
+
+770
+
+
+
+
+
+interpreters: mechanisms in brain, 582-84;
+
+people, 293, 297, 524, 671; programs, 293,
+
+504, 547, 616, 632, 662, 692
+
+intrinsically high-level properties, 707-9
+
+introspection, see self-monitoring, self-
+
+awareness, self-knowledge, inaccessibility,
+
+TNT. introspection of
+
+intuition, 560, 564, 613, 680, 713;
+
+programming of, 605, 609
+
+inversion, 8-9, 81, 146, 681-83, 737-38;
+
+see also copies, complementary to original
+
+inviolate level, 686-92
+
+irrationality vs, rationality in brain/mind,
+
+575-78
+
+itregularites, meta-irregularities, etc. 475-
+76
+
+Iran, 254
+
+ism, 254-55, 625, 704-6
+isomorphisms: between Bongard problems,
+660, 669; between brain-structures and
+reality, 82, 337-39, 350, 502, 569-71;
+between brains, 369-82; coarse-grained,
+147-48, 503; in Contracrostipunctus, 83-
+85; between Crab's DNA and Crab Canon,
+203, 667-68; defined, 9, 49-50; between
+earthworms. 342-43, 345; of emotions,
+163; exotic, prosaic, 159-60; fluid, 338,
+350, 362; between form and content in
+Dialogues, 84-85, 128-30, 204, 667-68;
+between formal systems and number
+theory, 408, 625; Godel-numbering and,
+see Godel isomor phism; between
+mathematicians, 566; between mathematics
+and reality, 53-60; between mental
+processes and programs, 568-73; between
+MlU-system and 310-system, 261-65;
+between models of natural numbers, 217;
+partial, 146-47, 371-82; as revelations,
+159-61, 263; as roots of meaning, 49-53,
+87-8, 94, 267, 337, 350; self-reps and, 501-
+3; between something and part of itself,
+138-43, 146-47; between spiderwebs, 371 -
+72; transparent, 82, 158, 267; on various
+levels between same objects, 369; between
+visual apparatuses, 345-46; in visual
+
+
+processing, 344; see also meaning,
+translation, copies, decoding, etc.
+
+’Jabberwocky" (Carroll), 366-68, 372-73
+Jacquard loom, 25
+Jaki, Stanley, 574
+
+"Jammerwoch, Der", (Carroll-Scott), 366-
+
+68 "Jaseroque, Le" (Carroll-Warrin), 366-
+
+68 Jauch, J. M„ 408, 409, 478-79
+
+Jefferson, G., 598
+
+Joan of Arc, 20 Johns, jasper, 703
+
+J6shti, 233, 237, 238, 240, 253, 259, 272
+
+JOSHU (TNT-string), 443
+
+jukeboxes, 154-57, 160-61, 164, 170-71,
+
+174-76,
+
+500
+
+jumping out of a subsystem, 477
+jumping out of the system: in
+advertisement, 478;
+
+by answer-schemers, 462-64; Godel's
+Theorem
+
+and, see Godefization, essential
+incompleteness;
+
+illusion of, 478-79, 698; as method to
+resolve
+
+contradictions, 196-97; in political systems,
+692;
+
+by programs, 36-38, 476-78, 678; from 2-D
+to
+
+3-D, see 2-D vs. 3-D; Zen arid, 255, 479;
+see also Godelization, Todelization,
+Escherization, TC battles, repeatability,
+nonprogrammability, etc.
+
+Kaiserling, Count, 391-92
+Kay, Alan, 662 Kennedy, John F., 641
+keys, musical, 10, 299, 466, 501; see also
+modulation
+
+Kim, Scott, 68-69, 503, 523, 719
+
+Kimberger, Johann Philipp, 9, 726 kitchen
+sink, the, 315 Kleene, Stephen C., 476
+Klein bottle, 691
+
+Kliigel, G. S., 91 knitting, 149-50
+knots, 341-44, 272, 628
+knowledge: accessible vs. inaccessible,
+362, 365,
+
+
+Index
+
+
+771
+
+
+
+616, 619; encoded in ant colonies, 319-28,
+359;
+
+explicit vs. implicit, 617-18; modularity of,
+615-18, 628; procedural v4. declarative.
+363-65,615-17,630, 654;
+knowledge transplantation, surgical, 618
+koans, 30, 189-91, 233-45, 246-59, 625-26;
+generated by computer, 625-26; genuine
+vs. phony, 234-35, 239, 242, 244, 427,
+625-26 Kronecker, Leopold, 216 Kuhn,
+Thomas, 660
+
+Kupfergiidel, Roman, 394 Kvogen,244-45
+
+La Mettrie, Julien Offroy de, 3, 27, 729
+labeling technique, 487-88, 540-41
+Lambert, J. H., 91, 92, 99
+lamp, meta-lamp, etc., 108-13, 216
+language(s): acquisition of, 170, 294, 302;
+active meanings in, 51-52; Arabic, 623-24;
+of bees, 360; of the brain, 570; Chinese,
+164, 665, 676; collage of, see scripts;
+computers and, 130-34, 300-1, 363, 586-
+93, 599-600, 601-3, 619-32, 674-75, 721;
+effect on thought, 376-77; English, 169,
+372-73, 377, 379-80, 619-32, 674-75;
+flexibility of, 649, 674-75; French, 297,
+366-68, 372-73, 377, 501, 618; German,
+366-68, 372-73, 380, 665; Hebrew, xviii,
+377; hierarchy of, 22; imprecise, 674-5;
+invisible isomorphisms and, 82; Japanese,
+169; as medium for proofs, 88-90, 195;
+necessary underpinning of, 324; partitions
+between, 671; procedural grammars for,
+131-34, 619-32; reading meaning into
+computer-produced, 599-600, 625; on
+
+Rosetta stone, 165; Russian, 297, 379-80,
+642; self-refs in, 431-37, 495-98, 501; see
+also meaning, translation, etc.
+
+Lashley, Karl, 342, 343, 348
+
+"last step", 462-63, 468
+
+lateral geniculate, 343-44
+
+layers: of deception, 478; of messages,
+
+166-71, 524, 703-4; of stability, 643-45
+
+leakage, between levels of science, 305-6
+
+Legendre, Adrien-Marie, 92 Lehninger,
+
+
+Albert, 504 Leibniz, Wilhelm Gottfried,
+24-25, 600 lemmas, 227
+Lenat, Douglas, 615
+
+lengthening and shortening rules: decision
+procedures and, 48-49, 182, 407-8; MIU-
+system and, 39-40, 260-61, 264, 613; TNT
+and, 213, 266, 269; see also increasing and
+decreasing rules, problem reduction
+Leonardo of Pisa, see Fibonacci
+Lermontov, Mikhail, 642
+level-conflicts: in Aunt Hillary, 330, 630;
+in mind/brain, 575-78; in messages, 164,
+170, 699-704; between object language and
+metalanguage,
+
+194, 449-50; in SHRDLU, 630
+
+level-con fusion: ants and, see ants vs. ant
+
+colonies;
+
+in art, see 2-D vs. 3-D; authorship and, 3.
+608, 720-26; in computer systems, 287,
+291, 295, 300-2, 308; of Kimian self-rep,
+503; minds/
+
+brains and, 287, 575-77; in Propositional
+Calculus, 185, 194; subjunc-TV and, 608;
+self and, 709
+
+level-crossing, in thought, 666, 668
+level-mixing in genetics, 509-10, 513-14,
+546-48 level-shifting, conceptual, see
+abstraction, levels of levels: of computer
+languages, 290-99; distinct vs.
+similar, 285, 287; in Escher, 11-15, 6
+levels (continued)
+
+715-16; haziness of, 13-15, 54618, 715-16;
+intermediate, 302-3, 317, 324, 532, 632; of
+irreality, 243, 641; of MU-picture, 311-13,
+328-29, 525-26; of partides, 305; in radio
+news, 128; of reality, 15, 103-25, 128-29,
+184-85, 481, 493, 640, 725-26, 737, 739; in
+recursive processes, 128-29; of rules in
+thought, 26-27
+
+levels of description: of ant colonies, 315-
+33; of brain, 349-50, 382. 559, 570-77,
+584-85; of caste distribution, 319-29; of
+chess boards, 285-86; of errors, 294-95; of
+gases, 308; of human body, 285; of human
+psyche, 287; of mental processes, 568-73,
+575-78, 584-85; of programs, 294-95, 380-
+
+
+Index
+
+
+772
+
+
+
+81; of television screen, 285; see also
+holism vs. reductionism
+levels of meaning: in ant colonies, 319-27;
+in Contracrostipunctus, 82-85; of DNA,
+160, 531-32, 665; in Epimenides paradox,
+496, 581, 584-85; of groove-patterns, 83-
+84; of Mumon, 248; of MUMON, 266-67;
+of music, 162-63; of neural activity, 575-
+77; of TNT-strings, 266, 270-71
+levels of structure: of enzymes, 510-11,
+519, 521, 525-27, 532; of music, 525
+liar paradox, see Epimenides paradox
+Liberation (Escher), 57-58, 65
+lightning calculators, see idiots savants
+limericks, 483, 736
+
+limitative results, in general, 19, 74, 609,
+697, 699 Lincoln, Abraham, 454
+lines, geometrical, 19-20, 90-93, 100, 222,
+452, 456
+
+LISP, 293, 381, 626, 652, 692, 738-39
+List of All Great Mathematicians, 404, 422
+Little Harmonic Labyrinth (Bach), 121-23,
+129, 130 Little Harmonic Labyrinth
+
+(Dialogue), 127, 128-30,
+
+149,216,610-11
+
+Little Harmonic Labyrinth (of Majotaur),
+
+119-25 Littlewood, J. E., 564
+
+lizards, 108-9, 110, 115-17, 125
+
+Lobachevskiy, Nikolay, 91
+
+local vs. global properties, 21, 160, 359,
+
+363,371-75,543,582-84, 678
+
+localization of knowledge, in brains and
+
+programs, 342, 348, 365, 617-18
+
+Lockwood, Anna, 700
+
+logic, 19-24, 43-45, 99-100, 177-80, 181-
+
+97, 461-64,618-19
+
+Loocus the Thinker, 477
+
+look-ahead trees, 151, 604-5, 611,712
+
+loops: bounded, 149, 410-14, 418, 440-41,
+
+444; free, 149, 424-25; in music, 150; in
+
+programming, 149-50, 410-14, 424-25,
+
+503, 632 lottery, 639-40
+
+Lovelace, Lady Ada Augusta, 25, 307, 598
+
+lower levels, see substrate, mental
+
+
+lowest-level rules embodied in hardware,
+685-86 Lucas, J. R. 388-90, 471-73, 475,
+476, 477, 574,
+
+577-78, 597
+
+Lucas' argument: counterarguments to,
+475-77, 577-78; merits of, 472;
+
+summarized, 471-73 Lucas sequence, 139,
+152,174
+
+M-mode, see Mechanical mode
+MacGillavry, Caroline, 667
+machine dependence and independence,
+294 machine language, 289-300, 306, 381,
+547 machines: not the sum of their parts,
+389-90;
+
+reflecting on themselves, 288-89; self-
+assem
+
+bling, 160, 486, 504, 543, 545
+MacLaine, Shirley, 285
+macroscopic effects from micros
+307 copic causes, MACSYMA, 615
+Madstop, 727
+
+magnetic field and crystal, 140-43
+Magnficat in D (Bach), 549, 552, 558
+hfagn#iicrab, Indeed, 560, 574, 581
+Magritte, Rene, 480-81, 489, 493-94, 627,
+700-2,
+
+705-6; paintings by, see List of Illustrations
+(xiv xviii)
+
+Mahalanobis, P. C., 565
+main theses of book, 26, 46, 559, 714
+Majotaur, 119-21, 123-25 malaphors, 657
+Mandelbrot, Benoit, 71 manifestations of
+symbols, 351 Mao Tse-tung, 433
+mappings: charted, 85, 449, 533, 536;
+induced, 668-69, 671-72
+marbles, rolling, 711-12 Margie-balloon
+story, 675 Materialism, champions of, 27,
+729 mathematical logic, history of, 19-24
+mathematical view of brains, 559
+mathematicians, 458-59, 559, 566, 614
+mathematics: done by computers, 573, 602,
+614-15; foundations of, 19-24; reality and,
+54-58, 456-59
+
+Mathews, Max, 607-8 McCarthy, John,
+293 McCulloch, Warren, 134
+
+
+Index
+
+
+773
+
+
+
+meaning: built on triggering-patterns of
+symbols, 325, 327, 350; carried only on
+symbol level, 324-27, 330, 330, 709-10,
+codes and, 82, 158-62, 164-67, 267; of
+Contracrostipunctns, 82-85; of DNA, 160,
+531-32, 665; explicit vs. implicit, 82-85,
+158-76, 495-500, 583; in formal systems,
+see interpretations; intelligence and, 158,
+162-64, 170-76, 501, 661-62; intrinsic, see
+meaning, explicit vs. implicit; location of,
+153-57, 158-76, 408-9, 582-84; as
+
+multidimensional cognitive structure, 582-
+84; multiple, 8, 10, 52-53, 82-85, 94-102,
+153-57, 158, 172, 266-67, 271, 409, 447-
+48, 524, 532, 666, see also disambiguation;
+in music, 83, 160, 161, 162-64, 167, 172,
+174-75, 227, 582-84, 626-27, 676-77, 699-
+700, 704; objective, see meaning, explicit
+vs. implicit; as optional high-level feature,
+571; passive vs. active, 51-52, 94, 97, 100,
+102, 191-92, 266, 267, 271, 456; purpose
+and, 321-32; rooted in isomorphisms, 49-
+53, 87-88, 94, 267, 337, 350; unnecessary
+on evolutionary time scale, 321-22
+meaningless vs. meaningful interpretations,
+51,88
+
+meaninglessness, in art and music, 699-
+
+700, 704-3
+
+meat grinders, 414
+
+Mechanical mode, 38-39, 65, 194, 221,
+613-14
+
+mechanization of thought processes, see
+Al, formal systems, etc.
+meiosis, 665, 672
+
+melodies: recall of, 363-64; time-shared,
+385
+
+memory, in computers, 288-89, 546, 616
+
+memory dump, 381
+
+men vs. women, 477, 595-96
+
+Mendel, Arthur, 3, 28
+
+Mental Arithmetic (MagrittO, 627
+
+mention, see use vs. mention
+
+Menzel, Adolph von, 4-5
+
+Meredith, Marsha, 625
+
+Mergenthaler, Otto, 630
+
+message-L sing languages, 662-63
+
+
+messages, 154, 158-76: in ant colonies,
+350; in bottles, 167-69, 524; layers of, 166-
+71, 524, 703-4; from nature, 408-9; see
+also frame message, inner message, outer
+message
+
+messenger, for koans, 235-36, 238
+
+messenger RNA, see mRNA meta, 216-17,
+224 meta-agnosticism, 114 rneta-analogy,
+673-74 meta-answer-schema, 463 meta¬
+author, 607-9, 726 metabook, 22
+meta-descriptions, 656-57, 674 meta¬
+
+evidence, etc., 693-94 Meta-Genie, see
+Genie meta-hiccups, 726 meta-intuition,
+605 META-JOSH1-), etc., 443
+metaknowledge, 364
+
+metalanguage, 22, 184, 194. 248, 270, 514
+metalogic, 23, 676
+
+metanrathematics, 23, 579; reflected inside
+TNT,
+
+449-50
+
+Metamorphosis (Escher). 14-15 metaphase,
+666-57 metaphor, 672 meta-proteins, 533-
+34
+
+metarules, etc.: in chess, 687-88; in
+intelligence,
+
+26-27, 559, 684-85
+
+meta-search, 397 meta-symbols, 560
+
+metatheorems, 193-94 metatheorv,
+formalized, 194 meta-TNT, etc., 442-43,
+533-34 meta-wishes, see wishes meteorites,
+167, 172 methylation, 540-41
+metric, mental, 613, 614; see also
+conceptual nearness
+
+Meyer, Leonard B., 167, 704
+
+Michelangelo, 642 microprogramming, 295
+minds: overlap of, 376; programmability
+of. 302, 679, see also Al, paradox of All,
+Tesler's The
+
+orem, nonprogrammability; thoughts and,
+369-90; two ways of creating. 390; vs.
+brains,
+
+309, 575-77; see also brains, intelligence,
+etc. mini-vocabulary, 647
+Minsky, Marvin, 373, 388, 644, 679, 722
+mirroring, see isomorphisms,
+
+representation misspelling and computers,
+
+
+Index
+
+
+774
+
+
+
+297-98 MlU-numbers, 264-67; see also
+theorem-numbers MlU-system, 33-41, 46,
+47, 48, 52, 191,260-67;
+as model for TNT, 439-42, 466; table of
+rules of, 260
+
+MIU+MU-system, 466 Mobius Strip I
+(Escher), 29-30 Mobius Strip II (Escher),
+276 modes of fugue-listening, 282-84
+modularity, 149-50, 615-18, 628, 677-78;
+see also
+
+localization, local vs. global properties
+modulation, 10, 121-23, 129-30, 466, 501,
+717, 739
+
+modules in brain, see symbols Modus
+Ponens, see detachment molecular biology,
+504, 514-48 Mondrian, Piet, 700
+monkeys, vision of, 345-4b
+Monod, Jacques, 161 morphogenesis, 539,
+543-44 Mosaic II (Escher), 61-63
+mountain-car wreck story, 338-39, 361,
+365 Mozart, W. A., 649, 702
+mRNA, 517-20, 522-25, 527-28, 530-33,
+536,
+
+545, 547, 662, 663
+
+MU, as possible theorem of MlU-system,
+33-41,
+
+229,259-61,265-67, 271, 708
+
+MU, Zen word, 233, 241, 246, 254, 259,
+
+272, 311,
+
+312-313, 328
+
+MU-LOOP (Floe P), 424-25, 441 Mu
+Offering, A, 272, 628 muoperator, 424
+MU-picture, 283-84, 310-13, 327, 328-29,
+525-26
+
+MU-puzzle, 33-41, 259-61, 509-10, 613-14
+multifurcation of TNT, 467
+multiple representations, 616-18, 670-71,
+674 multiplication, 54-56, 64-65, 206, 409,
+455, 566, 567
+
+Mumon, 242, 246, 248-49, 253, 259, 260,
+272;
+
+commentaries by, 246-49, 252; poems by,
+246-49, 252, 272
+
+MUMON, string of TNT, 265-67, 271,
+441-42 Mumonkan, 246
+
+
+music: composed by computer, 25, 595,
+597, 603,
+
+607-9, 626-27, 676-77; dimensions of, 175;
+mathematics and, 227, 555, 560; modern,
+156-57, 163-64, 174-75, 699-700, 704;
+notation
+
+of, 552-59, see also SMUT; semantics of,
+83,
+
+162-64, 167, 174-75, 582-84, 626-27, 676-
+77;
+
+superhuman comprehension of, 172, 679;
+syntax of, 121-23, 129-30, 227, 626-27,
+737; to
+
+break phonographs by, 75-78; to infiltrate
+phonographs by, 487-88; see also fugues,
+canons, pianos, flutes, etc.
+music box, preprogrammed, 677
+Musical Offering, The (Bach), 4-10, 86,
+665, 666,
+
+719, 720, 724, 727, 739-42 mutations, 295
+
+
+N, see number theory
+
+Najunamar, Z., 549-52 Nansen, 248-49,
+253,255
+
+natural language utterances as programs,
+629 natural numbers: defined, 54, 204;
+generalized;
+
+453-56; postulates for, 216-17; see also
+number
+
+theory, numerals, TNT, prime numbers,
+
+etc. near misses, see "almost"-situations
+
+nearly decomposable systems, 303-6
+
+negation, 70, 71, 183, 191-92, 210-11, 214,
+
+545 negative space, 62-63, 66-68, 72; see
+
+also figure and ground
+
+nested movies, 184-85
+
+nested works of art, 15, 106, 700-1, 705-6
+
+nesting, 127, 138-41, 184-85, 660; see also
+
+recursion
+
+neural networks, see symbols Neuroneater,
+382
+
+neurons: compared with ants, 315, 325,
+339-40;
+
+described, 339-40; Eudid's, 60; faultless
+func
+
+
+Index
+
+
+775
+
+
+
+tioning of, 575-77; firing of, 83, 340, 316,
+
+343- 45, 347, 350, 357; as an inviolate
+level, 302, 677, 686, 691-92, see also
+inaccessibility; not controllable
+consciously, 302, see also inaccessibility;
+on-center and off-center, 343-44; in
+retina, 343-44; simple, complex,
+hypercomplex,-
+
+neurons (continued)
+
+344- 45, 346, 347; as summing inputs, 316,
+340,575-77, 677
+
+neurosurgery, 309, 313-14, 618. 678 New
+Yorker, The, 641-42
+
+nickelodeon, 500; see also jukeboxes nodes
+and links, 370-71, 652-54 noise in vacuum,
+82
+
+nondivisibility, 73-74
+
+nonequilibrium thermodynamics, 693
+Noneudid, 91-92
+
+nonexistence, 254-55, 698, 725; see also
+Tumbolia nonproducible numbers, 265
+nonprogrammability: of creativity, 570-71,
+620, 673; of emotions and will, 677, 684-
+86; of Godelization, 472-76; of
+intelligence, 26-27, 471-73, 597-99, 601; of
+irrationality, 575-77; of jumping out of the
+system, 37-38, 477-78, 674-75; of ordinal
+names, 476; of soul, 574-75; of world
+chess champion, 151-52, 674; see also
+people vs. machines, essential
+incompleteness, Todeization, paradox of
+Al, TC-battles, 2-D vs. 3-D, etc.
+non-self-assembling viruses, 542-43 non-
+self-descriptive adjectives, see
+
+heterological adjectives
+nonsense: based on sense, 378-79;
+computer-generated, 620, 621-22, 625-26;
+human-generated, 621-22
+nontermination, 408, 425-30; see also
+potentially endless searches. Hoop
+nontheorems, see theorems vs.
+nontheorems normal science, 660-61
+nouns, most common in English, 630
+novelty, and jumping out of the system,
+475
+
+
+nuclei: atomic, 303-4; cellular, 514, 517,
+518
+
+nudeotides, 514-17, 519, 522-24, 530, 540-
+41; first letters of, 231, 517, 666
+number theory: applications of, 278-29;
+core of, 100, 407; Crab and, 551-58, 560,
+562, 573-74, 579-81; demise of, 228-29,
+426; formalized, see TNT; informal (N),
+54-660, 204, 228; nonstandard, 100, 452-
+59; primitive notions of, 204-9; as scaled-
+off mini-world, 569; soothing powers of,
+391-404; "true" version of, 458-59; typical
+sentences of, 204-5; typographical, see
+TNT; as universal mirror of formal
+systems, 260-65, 270; used and mentioned,
+458
+
+numbers, nature of, 54-58, 452, 458
+numerals, 205-6, 213; vs. numbers, 264
+
+object language, 22, 184, 248
+
+objectivity, quest after, 479, 693-96
+
+Oborin, Lev, 162
+
+octopus cell, 345
+
+Oin, Eta, see Eta Oin
+
+Oistrakh, David, 162
+
+Okanisama, 232, 234, 237, 238, 239, 241,
+242 Old Ba. Ch„ 726
+
+Old Bach, 4, 28, 460, 481-83, 738, 739 ca-
+consistency, 459; see also or-inconsistency
+cs-incompleteness, 221-22, 421, 450-51 co¬
+inconsistency, 17, 223, 453-55, 458-59 1-D
+vs. 3-D, 519-21,616-17
+open-ended searches, see potentially
+endless
+
+searches, nontermination, unpredictable but
+guaranteed termination, loops, free, FlooP,
+etc. operating systems, 295-96, 300-31, 308
+operators and operons, 544-45 oracles, 567
+orchard analogy, see information, depth
+from surface order and chaos: in ant
+colonies, 316-17; in number theory, 393,
+395,398-402,406.408-9,418;
+self-awareness and, 406
+Order and Chaos (Escher), 399 ordinals,
+462-64, 475-76 organ point. 329-30 origin
+of life, 548
+
+
+Index
+
+
+776
+
+
+
+
+original (as opposed to copies), 504
+
+originality and machines, 25-26, 606-9
+
+ORNATE NOUN, 131-33 outcome, 184
+
+outer messages, 166-71, 174-76, 501, 524,
+
+704 OUTPUT (BlooP), 410, 411
+
+overlapping genes, 524-25
+
+overview capacity, 613-14, 678; see also
+
+jumping out of the system
+
+P-system, 64, 73-74
+
+padding, 402-3
+
+pages, in computers, 289
+
+palindromes, in molecular biology, 201,
+
+667 Palindromi, 353-54, 634-37, 643, 644
+
+Pappus, 606-7
+
+paradigm shifts, 660-61
+
+paradox: of Al, 19, 26-27, 620, 673, see
+
+also Tesler's Theorem; in an, see Escher,
+
+Magritte,
+
+Cage; of credibility through fallibility, 564;
+of God and the stone, 478; in mathematics,
+17-24, 580-81; of motion, see Zeno's
+paradoxes; near misses, 612, 691;
+
+resolutions of, 116, 196-97,
+
+245, see also MU, Tumbolia, jumping out
+of the system; of self-consciousness, 389;
+of the Typeless Wish, 115-16; in Zen, 249-
+55; see also contradictions, inconsistency
+parallel postulate, see fifth postulate
+PARRY, 300-301, 599-600, 677
+parsing of natural languages, 588-93, 630-
+32; see also grammar, language
+partial recursivity, 430
+particles, elementary, 54, 140-46, 258, 303-
+5, 309,522
+
+partitions, mental, 671
+
+parts, 303-5; see also reductionism Pascal,
+
+Blaise, 24, 25, 600
+
+pathways: in ATN's and RTN's, 131-34,
+150;
+
+chemical, 528-29, 544-45, 663-64;
+
+conditional on circumstances, 383-84;
+goal-oriented choice of, 227, 609-15; as
+incorporating knowledge,
+beliefs, 378-79; morphogenetic of T4, 539;
+plausible vs. implausible, 383; potential, in
+brain, 281
+
+
+pattern recognition, see Bongard problems,
+conceptual skeletons, vision by computers
+patterns on all levels, 674 Peano, Giuseppe,
+20, 216-17 Peano arithmetic, 100 Peano
+postulates, 216-17, 224 pearl and oyster,
+17, 438 Penfield, Wilder, 342-43 Penrose,
+Roger, 15
+
+people vs. machines, 25-27, 36-38, 151-52,
+388-90, 471-73, 475-77, 559-62, 567-75,
+577-79, 595-99, 606-9, 621-23, 680, 684-
+86 peptide bonds, 523
+perception: visual, 97-98; and Zen, 251
+Perfect items, 3, 75-79, 85, 406, 424, 486,
+536 perfect numbers, 416, 418
+phages, see viruses
+
+phenotype, see genotype and phenotype
+0(174, 176, 524-25
+Phonographs, see record players
+phonons, 304
+
+photocopy machines, 499 photons, 142-46,
+258
+
+physics laws of: as basis for choosing
+between rival mathematical theories, 100,
+456-57; as blocking infinite regress, 170,
+685; as formal system, 53-54;
+inconsistency and, 95-96,99,
+
+584-85; intuitive, 362-63, 711; levels and,
+303-5, 307-8, 693; no escape from, 477,
+575;
+
+Reductionist s Dilemma and, 522, 709;
+underlying consciousness, 575, 685, 710
+IT, 277, 306, 408, 415, 421, 546, 568, 605,
+673,
+
+728
+
+piano postulates, 552-53
+
+pianos, 3-4, 302, 305, 633-34, 700, 726;
+
+inverted, 681-83
+
+Pickruick papers, The (Dickens), 24. 326,
+595, 598 pinball machine, 307
+pipe Dream, 703
+
+pipes, 480-82, 486, 488, 493-94, 521, 638,
+701-3 planets and satellites, 353-54
+PLANNER, 629-32
+plurals, 354
+
+pocket calculators, 568-70, 616, 678, 710
+pointers in computers, 289-90, 619
+
+
+Index
+
+
+777
+
+
+
+points (geometrical), 19-20, 90, 92-93, 100,
+207,
+
+222, 452, 456
+
+Polanyi, Michael, 574 polarons, 304-5
+polypeptides, 523, 525, 528 polyribosomes,
+526-28
+
+Pons Asinorum proof, 606-7, 669 Pool B,
+418
+
+Pool F,427
+
+popcorn, 104, 124-25 popping, 127-35,
+184-85 popping-tonic, 105-6, 116-17, 125
+porridge, 431
+
+ports of access, 670-71 Post, Emd, 33
+post-ending endings, 392, 403 postal
+system metaphor, 663
+
+postulates of geometry, 90-91, 92-93, 407
+potentially endless searches, 396, 400-402,
+425,
+
+400-401, 444, 582-83
+
+pq-system: completeness and consistency
+of, 101;
+
+decision procedure for, 47-49; expressive
+weak
+
+ness of, 101, 221-22, 407, 417; horse-apple
+happy interpretation of, 51, 88, 215; isomor
+phisms and, 49-53, 158, 625; modified, 87-
+88 ,
+
+92-93, 102; surprise interpretation of, 52-
+53,
+
+94
+
+predicate calMus, 609
+predicates, number-theoretical, 208-9
+predictable termination, 400, 407, 409-18,
+420,
+
+441, 582; see also unpredictable but
+guaranteed
+
+termination, terminators
+Prelude, 337, 383, 460, 686, 737, 739
+preludes and fugues, 280-84, 335; see also
+fugues, Well-Tempered Clavier
+premise, 184
+
+Preprocessing, 647, 650, 659 President v.
+Supreme Court, 692
+
+primary structure: of proteins, 519-22; of ty
+
+poenzymes, 511, 512
+
+prime numbers, 58-59, 64-67, 72-74, 149,
+
+
+211-12, 413, 551-58, 615; differences of,
+393, 395-98, 400, 416; sums of, 393-96,
+400, 414
+
+primitive recursive truths, 407
+primitive recursivity, 406, 407, 414-20,
+422, 424, 429-30, 440, 441, 444, 451, 466,
+472
+
+Principus Mathematiaa, 18-19,* 21, 23-24,
+228,618-19
+
+Print Gallery (Escher), 15, 714-17
+printer of computer, 301, 307
+problem reduction, 609-13; self-applied,
+613
+
+problem spaces, representation of, 611-13
+procedures, 132-34, 150-51, 292, 410-15,
+418-20, 424-28; chains of, 413-14, 415,
+418
+
+processors (computers), 504, 513, 547; see
+also, central processing unit
+producible numbers, 264-65, 269-70
+program space, 299
+
+programs: in Analytical Engine, 25; in
+BlooP and FlooP, 410-15, 424-26; Blue,
+Green, Red, see Blue, Green, Red
+programs; chess-playing, see chess
+programs; constructed by programs, 589,
+629-32, 664; as data, 293, 692; for
+determining enzyme function, 521-22; for
+determining phenotype, 532; for
+determining tertiary structure, 521-22;
+families of, 503, 546; for generating
+theorems, 471-73, 578, 615, 617-18; high-
+level comparison of, 380-81; for naming
+ordinals, 476; recursive structure of, 149-
+50; secondorder, third-order, etc., 476; self¬
+modifying, 152, 692; self-reproducing,
+498-504, 547; for translating programs,
+291-94; vs. data, 499, 513, 531, 546-48,
+616-17, 630, see also use vs. mention; vs.
+programmers, 306, 734-37; see also
+
+computers, computer languages, A1
+programs, etc.
+
+Prokofiev, Sergei, 150
+pronoun reference, 587, 591, 592
+proof-pairs, 416, 438-43, 446-47, 450-51,
+452-54, 466, 468, 469
+
+
+Index
+
+
+778
+
+
+
+proofs: nature of, 18-24, 58-60, 88-93, 192-
+97, 227-28, 458-59, 578, 707-8; as never
+absolute, 191-94; of proofs, 192-93; vs.
+derivations, 35, 193, 194-95
+prophase, 665-666
+
+Propositional Calculus, 181-97; embedded
+in TNT, 195, 197, 207, 215-17; as an
+epiphenomenon, 578; interpretations of
+symbols of, 186, 189, 191-92; rules of
+inference, justified, 188-89; rules of
+inference, presented, 181-87; rules of
+inference, table of, 187; streamlined, 193-
+94; variants of, 195; ways to improve, 193-
+94; 196-97; weaknesses of, 195-97, 578;
+well-formedness in, 181-83
+proteins, 517-18, 544-45; as procedural
+knowledge, 616-17; as programs, data,
+interpreters, processors, 547; see also
+enzymes
+
+prototype principle, 352
+provability, 18, 101
+
+Prudence and Imprudence, 191-92, 229
+pruning, explicit vs. implicit, 286
+pseudo-epigenesis, 531-32
+psychic powers, 693; see also ESP
+Ptolemy V Epiphanes, 165
+Puddle (Escher), 256
+
+pulling-out (mechanisms), see decoding
+(mechanisms)
+
+punctuation, 33, 268, 440, 510, 512, 520,
+524-25
+
+purities, 506-7, 514, 516, 534; see also
+nudeotides, bases, base-pairing
+purposeful vs. purposeless behavior, 320-
+22
+
+pushcorn, 124-25
+pushing, 127-34, 184-85
+pushing-potion, 105-6
+Pushkin, Alexander, 124
+puzzles. 8, 33-35, 62-63, 67, 73, 137, 182,
+212 ,
+
+215, 220, 401, 415-17, 425, 442-43, 444,
+512-13, 564-565, 609, 621-23, 646-60, 689
+pyramidal family of theorems, 221-25,
+450-53 pyrimidines, 506-7, 514, 516, 534:
+see also nu
+
+
+cleotides, bases, base-pairing Pythagoras,
+418,556-57
+
+Q(n), 137-38, 152, 265,409
+quantifiers, 207-9, 210, 211-12. 214, 217-
+19 quantum mechanics, 19, 54. 140-46,
+350, 455,
+
+457, 699; see also partides
+
+Quantz, Joachim, 4 quarks, 304, 305, 350
+
+quasi-isomorphisms, see isomorphisms,
+
+fluid quatemarv structure, 525 Questions
+
+and Speculations, 676-80
+
+Quine, Willard Van Orman, 435, 446, 449,
+
+699 quining, 431-37, 445, 446, 449, 497-
+
+99, 531 QUIT (BlooP), 412
+
+quotation, 426, 431, 433-37, 496-97, 702,
+
+738 quotation marks, 33, 434, 498, 499,
+
+702
+
+Rachmaninoff, Sergei, 150
+RACRECIR, 738
+
+radio broadcasts, 128, 163, 169, 353, 478,
+545, 720 Ramanujan, Srinivasa, 562-66,
+614 randomness, 408-9, 620, 673, 712
+rational and irrational numbers, 140-42,
+418,452,
+
+556-57
+
+rational vs. irrational, in human head, 575-
+
+77 rats in mazes, 342
+
+Rauschenberg, Robert, 703
+
+r.e. sets, see recursively enumerable sets
+
+reading frame shift, 154, 525 reality, nature
+
+of, 409
+
+rearrangement of parts, 78, 333-35, 484;
+
+see also self-assembly
+
+reasoning about formal systems, 38-39, 66,
+
+260-72, 438-52, 465-71, 579-81
+
+reasoning by programs, 569-70, 577-78,
+
+586-93,
+
+606-7, 609-11, 614-15, 618-19, 628-32
+recognition: molecular, 540-41; visual,
+346-48,
+
+646-62; vs. production, 648-49
+recognizable forms, 68
+recombination, 657, 665-69
+
+
+Index
+
+
+779
+
+
+
+record players: alien-rejecting, 487-88;
+EpsilonZero, 486; family of, in Crab's
+jukebox, 154-57; Grand Self-assembling,
+see record player Epsilon-Zero; as
+information-revealers, 158-61, 164;
+
+intrinsic vulnerability of, 75-78, 102, 424,
+470, 483-86, 536, 543, 584, 721, see also
+Tbdelization, TC-battles, etc.; likened to
+formal systems, 84, 85; low-fidelity, 77,
+85, 101, 406-7, 470; Omega, 78, 468, 483-
+84; Numbers 1, 2 ... etc., 76-77; Tortoise-
+chomping, 483, 487-88; two-channel
+
+monaural, 634, 669; see also jukeboxes
+records: defective, 102, as information-
+bearers, 158, 160-61, 164; as labyrinths,
+120-24; with multiple melodies, 154-57; as
+phonographbreakers, 75-78, 83-85, 271,
+406-7, 424, 469, 484, 486, 536, 543, 584;
+smashed, information in, 161; in space,
+162-64, 172, 174-75; of WellTempered
+Clavier given to Crab, 275, 278-80
+records and record players, likened to
+cellular constituents and cells, 83, 158-64,
+167, 175,536
+
+recursion: avoidance of infinite regress in,
+127, 134-35; avoidance of paradox in, 127;
+defined, 127-29, 131-35; elementary
+
+particles and, 142-46; fantasy rule and,
+184-85; in game-playing programs, 150-51,
+604-5; indirect, 134, 137; in language. 130-
+34, 588, 591, 592; in music, 121-23, 129-
+30; and unpredictability, 152; see also
+nesting, levels, distinct vs. similar, level
+confusion, etc.
+
+recursive acronyms, 113, 133, 134-35, 738,
+742 recursive diagrams, 135-37 recursive
+figures, 67-70. 72, 73 recursive formula, of
+thinking, 560
+
+recursive functions, 136-40, 152, 430, 455;
+see also
+
+genera] recursivity, primitive recursivity.
+BlooP,
+
+HooP
+
+recursive graphs. 138-43 recursive
+sequences, 135-38, 139 recursive sets, 72-
+74, 152, 191
+
+
+recursive structure of ideas, 386-87, 560,
+621,
+
+644-45, 650, 656-57, 669. 671-72
+Recursive Transition Networks, 131-34,
+136, 145,
+
+150, 620-21
+
+recursively enumerable sets, 72-74, 152,
+191,265,
+
+269
+
+recursively related notation-systems, 475
+Red Programs, 427-28 Reddiag [NJ, 428,
+429 redness, subjective and objective, 710
+reductionism: defined, 312; proteins and,
+520-22;
+
+see also holism vs- reductionism, sealing-
+off Reductionist's Dilemma, 522, 709
+reentrant code, 387
+
+refrigerators, see record players, low
+fidelity registers, in computers, 289
+relativity, 19, 96, 100, 680 Relativity
+(Escher), 97-98 relevant implication, 197
+renormalization, 142-46, 258, 304-5, 309
+repeatability, see Godelization,
+Todelization, diag
+
+onal method, Escherization, TC-batteries,
+answer-schemers
+
+representability, 407, 417-18, 430, 441,
+443, 444,
+
+451,466, 468, 579-80
+
+representation of knowledge: in All, 569,
+
+615-21, 626-32, 641-59, 664-65, 668-72; in
+
+brains, see symbols, localization
+
+repressors, 544-45 Reptiles (Escher), 116-
+
+17
+
+Requirement of Formality, 33, 52, 65
+
+retrogression, 8-9, 81, 146, 200, 208, 500-
+
+501, 549, 666-68, 723-25, 737-38
+
+return addresses, 128, 133 revelation, 160-
+
+61, 175 ribo, some, 236
+
+ribonucleic acid, see mRNA, rRNA, tRNA
+
+ribosomal RNA, see rRNA
+
+ribosomes: as models for Al, 662, 663;
+
+molecular
+
+canons and, 527-28; need for in DNA's self
+rep, 530; origin of, 528, 548; as self¬
+assembling
+
+
+Index
+
+
+780
+
+
+
+objects, 485-86, 542; structure of, 528; as
+translators of Genetic Code, 485, 518-19,
+522-25, 547; in Typogenetics, 512 ricercar,
+defined, 7 RICERCAR (E), 7, 727-42
+Ripplad Surface (Escher), 256-57 RNA,
+see mRNA, rRNA, tRNA RNA
+polymerise, 527, 530, 544 robot in T-maze,
+711-13 Rogers, Hartley ropes, thin and
+thick, 229-30
+Rose, Steven, 342
+
+Rosetta stone, 165, 166 Rosaak, Theodore,
+574 Rousseau, Henri, 680
+Royal Theme, 4-10, 96, 719. 739-40
+rRNA, 528
+
+RTN's, see Recursive Transition Networks
+rule-less systems, 598, 685; see also formal
+vs. in formal systems
+
+rules: arithmetical vs. typographical, 262-
+64. 269;
+
+flattened into strings, see theorems vs.
+rules; intelligence and, 26-27, 559, see also
+brains and formal systems
+rules of inference: of C-system, 65;
+compared with enzymes, 509-10, 513, 531;
+defined, 34-35;
+
+derived, 193-94; of MlU-system, 34, 260;
+of P system. 74; of pq-system, 47;
+proposed, 66, 221;
+
+of Propositional Calculus, 187; recursive
+enumerability and, 152; run backwards, 48-
+49, 182; of 310-system, 263; of TNT, 215,
+217-20, 223-25; of tq-system, 65; of
+Typogenetics,
+
+509-10
+
+rules of production, see rules of inference
+run-of-the-mill sets, 20-21 Russell,
+Bertrand, 18-24 Russell’s paradox, 20-21,
+685
+
+
+Saccheri, Girolamo, 91-93, 99, 452, 456
+Sagredo, see Salviati, et al
+Salviati, Simplicio, Sagredo, 408-9, 478-
+79, 673, 694
+
+sameness: of ASU's, 375; of BACH and
+CAGE,
+
+
+153-57; in Bongard world, 650-53, 657,
+660,
+
+664; of butterflies, 147, 369; of demi-
+doublets,
+
+669; elusiveness of, 14619; of Escher
+drawings,
+
+147; of human and machine intelligence,
+337,
+
+379, 679-80; of human minds, 341-42,
+369-72,
+
+375-77, 382; intensionality and, 338; mech
+anisms underlying perception of abstract,
+646-62, 665-69, 671-72; overlooked, 614,
+674; of programs, 380-82; in self-refs and
+self-reps,
+
+500-4; of semantic networks, 371; of
+translations between languages, 372, 379-
+80; universality of intelligence and, 158,
+501; vs.differentness, 153-57; visual, 344-
+48, 662; see also copies, isomorphisms,
+conceptual mapping sameness-detectors,
+see Sams Sams, 650-53, 657, 664
+Samuel, Arthur, 604-5, 684-86
+Samuel's argument, pro and con, 684-86
+San Francisco Chronicle example, 351
+sand castles, 725-26
+
+sanity vs. insanity, 192, 696 satellite-
+symbols, see splitting-off satoti, see
+enlightenment scale, cyclic, see Shepard
+tones Schmidt, Johann Michael, 27
+Schnirelmann, Lev G., 394 Schdnberg,
+Arnold, 125 Schrodinger, Erwin, 167
+Schweikart, F. K., 92
+
+science: and Bongard problems, 659-61;
+self-applied, 699
+Scott, Robert, 366
+
+scripts, collage of, 168-69 sealingoff, 305,
+309, 350, 534 secondary structure, 521,
+525
+
+self, nature of, 316-17, 327-28, 384-85,
+387-88, 695-96, 709-14
+self-assembly, spontaneous, 485-86, 542-
+43
+
+self-awareness, 406, 479, 573
+self-descriptive adjectives, see autological
+adjectives self-engulfing, 489-94; failed;
+
+
+Index
+
+
+781
+
+
+
+490, 492; total, 493 self-knowledge,
+
+possibility of, 696-98, 706 self-modifying
+games, 687-88 self-monitoring, 328, 385,-
+387-88, 697, 713 self-perception, 695-98;
+vs. self-transcendence, 478
+self-programmed objects, 685-86, 691-92
+self-proving sentences, 542-43 self-quoting
+sentence, 426, 496-97
+
+self-reference: Bach and, 86; banning, 21-
+23; as cause of essential incompleteness,
+465,470-71;
+
+focusing of, 438, 443, 445-48; Gddelian,
+17-18, 271, 447-49, 497, 502, 533, 667,
+738; indirect, 21, 85, 204, 436-37, 502,
+667, 738-39; many leveled. 742; near miss,
+437; Quine method,
+
+431-37, 445-46, 449, 497-99, 531; by
+translation, 502
+
+self-reference and self-replication,
+compared, 530, 533-34, 541-43
+self-referential sentences, 435-37, 477,
+495-99, 501
+
+self-rep: by augmentation, 503; canons and,
+501, 503; differentiating, 546; epigenesis
+and, 160; by error message, 503; inexact,
+500-503, 546; by retrograde motion, 500-
+501; by translation, 501; trivial, 499;
+typogenetical, 512-13 self-snuffing, 701-2
+self-swallowing sets, 20
+self-symbol, 385, 387-88, 709; free will
+and, 710-14; inevitability of, 388
+self-transcendence, 477-78, 479 self¬
+
+unawareness, irony of, 328, 330, 331, 630
+semantic classes, 621, 630
+semantic networks, 370-72; see also
+concept network
+semi-interpretations, 189, 196
+semiformal systems, 216; see also
+geometry, Euclidean
+
+senseless loops, 679 sentences in TNT,
+208-9 Sentences P and Q, 436-37
+sequences of integers, 73, 135-39, 173-74,
+408 set theory, 20-23
+sets F and G, 73
+
+1729, 204-5, 210-11, 345, 393, 551, 564-
+855 Shadows, The (Magritte), 480
+
+
+Shakespeare, Wm., 96, 595, 598, 608, 736
+Shandy Double-Dandy, 611 shared code,
+387
+
+Shepard, Roger, 717-19 Shepard tones,
+717-19
+
+shielding of lower levels, see
+inaccessibility SHRDLU, 586-93, 599,
+627-32, 674 Shuzan, 251
+Sierpifiski, W., 404
+
+signals, crisscrossing, 322-23 signature,
+
+visual, 347-48 Silberescher, Lowen, 394
+
+Silbermann, Gottfried, 3, 4 silver, 173
+
+Simon, Herbert A., 303, 305
+
+simple, complex, hypercomplex cells, see
+
+neurons Simpficio, see Salviati
+
+simplicity, 172, 560, 615
+
+simulation: of entire brain, 572-73; of
+
+neural networks, 571-72
+
+Six-Part Ricercar (Bach), 4-7, 719, 739-42
+
+skater metaphor, 412-13
+
+skeletons (recursion), 140-41; see also
+
+bottom 4
+
+skimming off top levels, 309, 325, 326,
+
+358-59,568-79
+
+slinky, 337
+
+slippage, conceptual, 633-40. 641-44, 654-
+
+56,
+
+672; see also conceptual mapping Sloth,
+633-40, 643. 681-83, 722 Sloth Canon
+(Bach), 9, 666, 683 Sloth Canon
+
+(Dialogue), 738
+SLOTHs, 684-719
+
+slots, 645, 650-53, 656-57, 668 Smalltalk,
+662
+
+smart-stupids, 721-42
+
+Smoke Signal 67, 702
+
+SMUT, 6, 80, 155-57, 202, 682, 718, 740
+
+soap cake, 497
+
+soft-louds, see pianos
+
+software and hardware: in brain, 346, 356-
+
+57, 686, 709; defined, 301
+soldier ants, 318
+
+Sonata for Unaccompanied Achilles, 502-3
+Sonata for violin and clavier in F minor
+(Bach), 162
+
+
+Index
+
+
+782
+
+
+
+Sonatas and partitas for unaccompanied
+
+violin (Bach), 62, 63, 70-71, 257, 502
+
+song, self-reproducing, 500
+
+"Song Without Time or Season, A", 483
+
+songs, recombinant, 607-8
+
+sonnets, 596, 608, 736
+
+soulism, 385, 472, 574, 597, 686, 708
+
+spaces: abstract, 457; of behavior, 306,
+
+307, 363, 621; of integers, 502
+
+spare tire, 670
+
+Sperry, Roger, 710
+
+Sphex wasp, 360-61,613-14
+
+spiderwebs, 371-72, 617
+
+splitting-off, 353-54, 664; see also classes
+
+vs. instances
+
+stacks, pushdown, 127-35, 136, 610-11
+starting positions in Art of Zen strings,
+239-40, 241
+
+State of Grace (Magritte), 481
+statistical mechanics, see gases
+Sterner, Geo., 166-67, 642-43, 670 Stent,
+Gunther, 514
+stomach processes, 572
+strands: of DNA and RNA, 514-18; in
+Typogenetics, 508-13, 514
+Strange Loops: abolishment of, 21-23; with
+Babbage and Turing, 737; in Bach, 10,
+719; in Central Dogmap, 534;
+consciousness and, 709-10; contrasted with
+feedback, 545, 691; defined, 10; in Escher,
+10-15, 714-17; in Godel, 15-17, 24; in
+government, 692-93; in language, 22, 691;
+in the mind, 27, 691-92; in molecular
+biology, 231, 532-34, 545, 546-48; in
+Principia Mathematica, 24; surprise
+element necessary in, 691; in TNT, see
+TNT, introspection of; see also Tangled
+Hierarchies, level-mixing, levelconflicts,
+level- confusion, etc.
+street nuisances, 25, 726-27, 729
+stretto, 314, 738
+
+String Manipulation Rules, 240-42
+strings: defined, 33-34; folded, 234-44, 427
+structure vs. function, 522, 670-71; see also
+use vs.
+
+
+mention, syntax vs. semantics style, 148,
+371
+
+SUB (TNT-formula), 444-45
+subbrains, see subsystems
+subdescriptions, etc. 650-51; see also
+recursive structure of ideas
+subframes, etc., 644-45; see abo recursive
+structure of ideas
+
+subject vs. object, 698-99; see also
+dualism, use vs. mention, symbol vs. object
+subjunc-TV, 635110
+
+subjunctive instant replays, see instant
+replays subjunctives, see counterfactuals
+suborganisms, see subsystems subroutines,
+150, 292, 471, 677 substitution notation in
+TNT, defined, 224 substitution relationship
+(ingredient in Godel's proof), 443-45, 497
+
+substrate: of Epimenides sentence, 581,
+584-85; mental, necessity of, for analogical
+thought, 569-71; mental,
+
+noninterpretability of, 570-71; mental,
+simulation of, 571-72; of proteins, 529
+subsystems of brain, 385-88, 725
+
+subtraction, 52-53, 411-12
+sufficiently powerful systems, 86, 101,
+406-7, 430, 460, 530
+
+Suites for unaccompanied cello (Bach), 70-
+71 summarizing string, 221-22, 450-51
+superconductivity, 304-5 superinielligence,
+679
+
+supernatural numbers, 223, 453-56, 458-
+59, 467 supernatural proofs, 454-55
+supertangling, 688
+surrealism, 700
+Sussman, Gerald, 664
+Swieten, Baron Gottfried van, 6-7
+Switcheroo, Q. q., 187
+symbol-level description of brain, 349-51
+symbol vs. object, 699-706; see also
+subject vs. object, use vs. mention, etc.
+symbols: active vs. passive, 324-25, 327-
+28, 337-38; A1 realizations of, 662-63, 665;
+in ant colonies, 324-28, 330; borderlines
+between, 354-57, 359-60; as brush strokes,
+351; compared with neurons, 350, 371;
+
+
+Index
+
+
+783
+
+
+
+compared with ripples, 356-37; conceptual
+scope of, 350-51; dormant, 327, 349, 355-
+
+56, 384; form of, 348, 356-57, 361; free
+
+will and, 712-14; funneling and, 347; of
+insects, 360-61; invariant core of, 349;
+joint activation of, 351, 354-56, 359, 361,
+365, 584-85, 664-65, 675; message-
+
+exchange and, 350, 371, 662-63; modes of
+activation of, 349-57, 361; need of, for
+originality, 609; neural substrate of, 356-
+
+57, 570; no access to substrate by, see
+inaccessibility; overcrowding of, 358;
+overlap of, 348-49, 356-57; potential, 355-
+56, 382-84; universal, 375-76; vs. neurons,
+348, 356-57, 361; vs. signals, 325-27, 349-
+50
+
+synapses, 339
+
+syntax vs. semantics, 626-27, 630-32, 676;
+see also form, syntactic vs. semantic
+system, boundaries of, 37-38; see also
+jumping out of the system
+System crash, 116
+systems, reliable vs. unreliable, 307
+
+T (Tarski's formula), 580-81
+'t'-concept, 'h'-concept, 'c -concept, 326 T-
+even phages, 540, 542 T-level, see tangled
+level table of nontheorems, 66 Tagore,
+Rabindranath, 169
+
+Tangled Hierarchies: in art, 704; defined,
+10; of
+
+genetics, 532-34, 546-48, 688; of meta
+mathematics, 458, 532-34; of mind, 691-
+92, 709-10, 719; near miss, 691; in
+Propositional
+
+Calculus, 194; of self-modifying chess,
+688; of
+
+Tortoises reasoning, 177-80; of
+Typogenetics,
+
+513,688
+
+tangled level, 688 Tanguy, Yves, 700
+tape recorders, 485, 519, 523, 525-28
+Tarski, Alfred, 579-81 Tarski-Church-
+Turing Theorem. 561, 581 Tarski's
+
+Theorem, 579, 580-81, 584-85, 697 Taube,
+Mortimer, 574 Taurinus, F..4., 92
+
+
+IrC_battles, 75-78, 406-7, 424, 467-70,
+
+478, 483-88, 536-41,721
+
+tea. 153-54, 231, 275, 322-23, 333, 549,
+
+557,558,561
+
+teleological vs. evolutionary viewpoint,
+320-22 telephone calls, 61-63, 127-28. 161;
+obscene, 431, 437
+
+telephone systems, 296, 663 telephones,
+302
+
+television, 285, 348, 478, 484, 487-93,
+
+634-40 telophase, 667-68
+
+templates: for Bongard problems, 650-53,
+
+656-57; vs. instructions, see instructions vs.
+
+tem
+
+plates, programs vs. data
+
+tension and resolution, 121-23, 129-30, 227
+
+tentativity, 646, 651, 654-56, 672
+
+termination tester, 425-29 terminators
+
+(FlooP), 425-29 terms (TNT), 206-7, 213,
+
+214
+
+tertiary structure: of proteins, 519-22, 525-
+27; of
+
+tRNA, 524; of typoenzymes, 510-11, 512,
+
+519 Tesler, Lawrence G., 601
+
+Tesler's Theorem, 601, 623; see also
+
+paradox of Al, jumping out of the system,
+
+essential in completeness
+
+tesselations, 68, 69, 198, 667
+
+tests vs. functions in BlooP and FlooP, 413,
+
+418 text-handling by computers, 301 T4
+
+phage, 537-41
+
+"the" 586, 629-30
+
+theorem-numbers, 264-67, 269-70, 440-43,
+451 theorem-proving, mechanical, 602,
+609, 617-19 theorems: defined, 35;
+
+systematic enumeration of,
+
+39-40, 48, 471-73, 578, 615, 617-18; vs.
+non
+
+theorems, 39-41, 66-67, 70, 71-73, 190-91,
+416-17, 560, 579-80; vs. rules, 43-45, 193-
+94,
+
+509-10; vs. Theorems, 35, 193; vs. truths,
+49-54, 70-71, 86-102, 190-97, 213, 221-23,
+228-30, see also consistency, completeness,
+Godel's Theorem, consequences of
+
+
+Index
+
+
+784
+
+
+
+Theseus and Ariadne, 130 thinking, speed
+of, 679
+
+30, as possible MlU-number, 265-67 "this
+sentence", 436, 495-98, 499 thought,
+substrate of, 559
+
+3n+l problem, 400-2; see also wondrous
+
+and un wondrous numbers
+
+Three Spheres If (Escher), 258 310-system,
+
+261-67
+
+Three Worlds (Escher), 247, 256 thymine,
+see nucleotides tilde, 183, 191-92, 554
+timesharing, 296, 354-55, 387, 730
+TNT, 204-30; absolute geometry and, 451 -
+52; austere, 211, 214, 216, 268, 442, 534;
+axioms of,
+
+215-17, 222-23; as code, 265-67;
+
+consistency of, 229-30, 449-50; extended,
+axioms of, 451-52, 466-68; extensions of,
+451-59, 465-68; FIGURE-FIGURE figure
+and, 70; as general metalanguage, 265-67;
+goals of, 60; incompleteness of, see
+incompleteness; introspection of, 17, 194,
+267-72, 406, 438, 443, 449-50, 698, 707-8;
+as its own metalanguage, 267-72, 441-46,
+514; plight of, pigorial version, 71; relaxed,
+228; rules of formation, table of, 213-14;
+rules of inference, 215, 217-15; sixth
+axiom of, 222-23, 451-55, 459, 465-68;
+well-formedness in, 205-15
+TNT-derivations compared with machine
+language, 291
+
+TNT-level, explanations on, 708
+TNT-numbers, 269-70; see also theorem-
+numbers
+
+TNT + G, 465-67,471
+TNT + G + G', etc., 467-71
+TNT + -G, 467
+
+tobacco mosaic virus, 484-85, 542, 543
+Todelization, repeatability of, 76-78, 424,
+467-70, 483-86; see also Godelization,
+essential incompleteness, answer schemas,
+etc.
+
+Todel's Theorem, 486, 536
+
+Tokusan, 189-90
+
+tonic, musical, 121-23, 129-30
+
+
+Tortoise: answer-schemas and, 475; Carroll
+paradox and, 46, 193, 684-85, 693; Crab
+Canon and, 204, 666-67; degenerate
+
+solution by, 669; Diophantine equations
+and, 4590; as harpsichord, 502; initial letter
+of, 231, 507, 667; mentioned, 102, 170,
+267; origin of, 28-29; picture of, 42;
+recursion and, 128-30; 149; use of words
+by, 181; vs. Crab, 84-86, 271, 406, 424,
+467-69, 540, 543; Zen string made by, 272;
+in ZET cycle, 94-95
+
+Tortoise-pairs, 416, 441, 448; compared
+with proof-pairs, 441, 448
+Tortoise property, 395-97, 415-18, 425,
+426,441,448
+
+Tortoise's love song, 435-36
+Tortoise's method, see Todelization
+T6zan, 190, 255, 257, 479
+tq-system, 64-67
+transcendentalism, 704
+transcription: DNA to mRNA, 517, 524,
+527-28, 530, 533, 536, 538, 540-41, 544-
+45; DNA to tRNA, 425; koans to
+messengers, 235-36, 238, 239, 242; letters
+to notes, 83; prevention of, 544-45
+transfer RNA, see tRNA
+translation: between computer languages,
+192-95, 297-98, 306, 380-81, 547, 632;
+between levels of a brain, 349, 381-84,
+709; between natural languages, difficulties
+of, 372-73, 379-80; between TNT and
+meta-TNT, 267-72, 441-46, 709; of Crime
+and Punishment, 379-80.; English to TNT,
+209-13, 215, 417; of "Jabberwocky", 372-
+73, 379; levels of fidelity in, 379-80;
+mechanical, 380, 601, 603; of messengers
+into strings, 234-36; mRNA to proteins,
+485, 518-19, 522-25, 527-28, 531-36, 538,
+545, 546-48; from N to Meta-TNT, 533;
+from score to sounds, 83; in Typogenetics,
+509-10,512-13
+transparency to user, 629, 632
+trees: look-ahead, see look-ahead trees;
+recursive diagrams, 40„ 71, 135-37; of
+theorems, 40, 71
+
+
+Index
+
+
+785
+
+
+
+triggering patterns of symbols: dependence
+on meaningless lower levels, 569;
+isomorphism between minds and, 369, 376;
+isomorphism between physical law and,
+362; as key to meaning, 325, 327, 350,
+360, 385, 609; mediated by mes-
+triggering patterns of symbols (continued)
+sages, 350, 371; for melodies, 364; nouns
+vs. verbs, 361; randomness in, 673
+triggers: DNA as, 160-61; dormant
+symbols and, 281, 383, 384; frame
+messages and, 162; jukeboxes and, 160-61,
+170-71, 174, 500; koans as, 246; music
+and, 162-63, 281, 383, 583; outer messages
+and, 166, 169, 170-71, 174, 501
+Trio Sonata from the Musical Offering
+(Bach), 7-8, 720, 724, 726
+Tripitaka, 257
+
+trip-lets, cover, xiv, 1, 28, 273 tRNA, 522-
+24, 547, 548
+Trojan Horse, 538
+
+truth: capturability by symbol
+
+manipulation, 53-60; elusiveness of, 694-
+95; inexpressible in TNT, 580-81; not fully
+mirrorable in brain, 584-85; vs. beauty,
+554-58, 584; vs. commercials, 478; vs.
+falsity, 70, 71, 213, 228-29, 417, 561, 579-
+81
+
+TTortoise, see ATTACCA
+tuba, flaming, 488-89, 492, 735
+Tumbolia, 116, 243, 255. 725; lavers of,
+243 tuning an A1 program, 678-79
+Turing, Alan M., 26, 389, 425-26, 428-29.
+594-99, 734-42; objections to Al, 597-99
+Turing, Sara, 595
+Turing machines, 390, 594, 735
+Turing test, 595-99, 600, 677-78, 735-37;
+arithmetic error in, 596; miniature, 621-23;
+proposed revisions in, 600
+Turtle's Theorem, see Todefs Theorem 2,
+as concept, 678
+
+2-D vs. 3-D: in Escher, 57-58, 105-6, 125,
+473-74, 524, 689-90, 698, 714-16; in
+Magritte, 480-81, 493-94, 700-1, 705-6;
+television screens and, 488-93, 737; trip¬
+lets and, see triplets
+
+
+Two Mysteries, The (Magritte), 701-2
+Two-Part Invention, 28, 43-45, 684-86; see
+also Carroll paradox, Lewis Carroll 2 + 2 =
+5,576
+
+typeface metaphor, 541
+
+Typeless Wishes, 111-15, 610-11 types,
+
+theory of, 21-23
+
+typesetting machine, 608
+
+typoenzymes, 505-13; binding-preferences
+
+of, 505-6, 511-12
+
+Typogenctic Code, 510, 512, 513, 519
+Typogenetics, 504-13, 514, 519, 520, 529;
+con
+
+trasted with MlU-system, 509-10, 514
+Typographical Number Theory, see TNT
+typographical operations, defined, 64
+typos, 404
+
+U, as nontheorem of MlU-system, 36, 39
+U-mode, see Un-mode
+Ulam, Stanislaw, 560, 621, 676 Un-mode,
+39, 98, 254
+
+Unamuno, Miguel de, 698
+
+uncertainty principle, see Heisenberg
+
+uncertainty principle
+
+uncles, 446-48, 464, 466, 468, 541, 580
+undecidability, 17, 222, 449, 451-55, 468;
+causes of, 707-8
+
+undefined terms, 92-102, 216, 456;
+
+defined, 93, 97
+
+understanding, nature of, 569, 675-76, 680
+understanding minds/brains: meaning of,
+697; possibility of, 697-98, 706-7
+unicycle, tandem, 633, 669
+units in Tvpogenetics. 505, 509 universal
+quantifiers, see quantifiers universal
+triggering power, 171, 175 Unmon, 254
+unobservant machines, 36-37, 674
+unpredictable but guaranteed termination,
+400, 425
+
+upper bounds, see loops, bounded, BIooP
+
+uracil, see nucleotides
+
+use vs. mention, 434-37, 458, 531, 545,
+
+699-700;
+
+see also form, syntactic vs. semantic,
+programs
+
+
+Index
+
+
+786
+
+
+
+vs. data, syntax vs. semantics, structure vs.
+function variables in TNT, 206, 213-14;
+free, 207-9, 214; quantified, 208, 214, see
+also quantifiers
+
+verb-at-the-end phenomenon, 130-31
+Verbum (Escher), 257, 731-32
+versus, see accessible vs. inaccessible
+knowledge, active vs. passive symbols,
+ants vs. ant colonies, arithmetical vs.
+typographical rules, Bach vs. Cage,
+beautiful vs. non-beautiful, bottom-up vs.
+top-down, classes vs. instances, continuous
+vs. discrete processes, deductive vs.
+analogical awareness, derivations vs.
+proofs, dissection vs. appreciation of Bach,
+distinct vs. similar levels, enzymes vs.
+typoenzymes, explicit vs. implicit
+knowledge, explicit vs. implicit meaning,
+explicit vs. implicit pruning, formal vs.
+informal reasoning, formal vs. informal
+systems, formal systems vs. reality,
+genuine vs. phony koans, high-fidelity vs.
+low-fidelity, holism vs. reductionism,
+improvisation vs. introspection,
+instructions vs. templates, local vs. global
+properties, meaningless vs. meaningful
+interpretations, men vs. women, minds vs.
+brains, nouns vs. verbs in triggering
+patterns, 1-D vs. 3-D, passive vs. active
+meaning, people vs. machines, plausible vs.
+implausible pathways. President v.
+Supreme Court, procedural vs. declarative
+knowledge, programs vs. data, programs
+vs. programmers, purposeful vs.
+purposeless behavior, rational vs. irrational,
+sanity vs. insanity, self-perception vs. self¬
+transcendence, structure vs. function,
+subject vs. object, symbol vs. object,
+symbols vs. neurons, symbols vs. signals,
+syntactic vs. semantic form, syntax vs.
+semantics, teleological vs. evolutionary
+viewpoint, theorems vs. nontheorems,
+theorems vs. rules, theorems vs. Theorems,
+theorems vs. truth, Tortoise vs. Crab, truth
+vs. beauty, truth vs. commercials, truth vs.
+falsity, 2-D vs. 3-D, use vs. mention,
+
+
+weight vs. mass, womanseeing vs. seeing,
+words vs. letters, Zen vs. logic, Zen vs.
+words
+
+vibrations, 76-78, 82-85, 102, 270, 271,
+469
+
+Vice President, 670
+
+Villon, Fransois, 369
+
+Vinogradov, Ivan M., 394-95
+
+Vinogradov property, 394-95
+
+violins, 62, 63, 70, 81, 84, 162, 200, 257,
+
+434, 502, 595,681,720, 724
+
+viruses, 536-43; likened to Henkin
+
+sentences, 542-43
+
+vision by computer, 602, 627
+
+visual imagery: Bongard problems and,
+
+661: faucets and, 364-65; inaccessible
+
+knowledge and, 365; lack of in programs,
+
+623; mathematics and, 569, 678;
+
+necessitating layers of substrate, 570-71;
+
+power of, 338-39; role in
+
+conceptual mapping. 668, 672; rubbing-off
+
+and.
+
+361-62 es and canons, 28, 282-83, 314,
+voices in fugu
+
+322-23, 335, 665-67, 669, 683, 737, 740
+Voltaire, Francois Marie Arouet de, 3
+vortices, 713-19
+
+Vuillard, Edouard, 347 Wachter, F. F., 92
+Warrin, Frank F., 366
+-WASH ME", 608
+
+Waterfall (Escher), 11-13, 99, 716
+
+Watergate, 692
+
+Watson, J. D.. 667 Weasel, 106 weather.
+302-3 Weaver, Warren, 380 Weierstrass,
+Karl W. T., 404 weight vs. mass, 171-72
+Weizenbaum, Joseph, 599-600. 675
+
+well-formed strings: in Art of Zen Strings,
+239, 242, 244; BIooP puzzle on, 416;
+defined. 53: in pq-svstem, 47; in
+Propositional Calculus, 181-83; in TNT,
+213-15
+
+Well-Te"ered Clavier (Bach), 7, 280-84,
+327. 329,
+
+335
+
+
+Index
+
+
+787
+
+
+
+Well-Tested Conjecture (Fourmi), 333-35 34, 249-51, 254; vs. words, 246, 248-49,
+"When Johnny Comes Marching Home", 251-54; Zeno and, 30
+607 Whitehead, Alfred North, 18, 21, 23- Zen Strings, Art of, see Art of Zen Strings
+24 Whitely, C. H., 477 Zenfunny, Beethoven's Ninth, 634
+
+Zeno of Elea, 28, 29-32, 94-95, 144, 146,
+wholes, see holism Wiener, Norbert, 684 232,610,681,704,722
+Wiesel, Torsten, 343 Zeno's paradoxes, 29-32, 35, 43, 146, 610
+
+will: free, see free will; mechanical, 684- Zentences, 186-90
+86; roots ZET-cycles, 94-96, 689
+
+of, 684-86 zooming in, 645, 671
+
+Wilson, E. 0., 350 Winograd. Terry, 627- iT
+32 Winston. Patrick Henry, 299 wishes,
+meta-wishes, etc., 109-16 Wittgenstein,
+
+Ludwig, 680, 699
+
+Wolff, Christoph, 392
+
+womanseeing vs. seeing, womansee men
+
+vs. women wondrous and unwondrous
+
+numbers, 400-2, 408,
+
+415,418,425
+
+Wooldridge, Dean, 360
+
+words: in computers, 288-90, 295, 411; as
+
+programs, 629-30; spelled backwards. 81,
+
+418, 427, 498, 505, 533, 549, 727, 738,
+
+740; thoughts, formal rules, and, see main
+theses: vs. letters,
+
+325-27, 570, 571; Zen attitude towards,
+
+246,249,251-54
+worker ants, 318
+
+working inside the system, see Mechanical
+mode
+
+yes-answers, 461-64 Yngve, Victor, 620
+Young, LaMonte, 700
+Zen Algebra, 577
+
+Zen Buddhism: Achilles teaches the
+Tortoise about. 231-45; computers and,
+
+625-26; Crab's refrigerator and, 406-7;
+
+Escher and, 255-58; holism vs.
+reductionism and, 312-13; inconsistencies
+and, see Zen vs. logic; introduction to, 246-
+59; jumping out of the system and, 233,
+
+255, 479; music, art, and, 699, 704-6;
+
+Mystery of the Undecidable and, 272;
+nonexistence and, 254-55, 698; patriarchs
+of, 30, 232, 252, 259; quasi-, 625-26; U-
+mode and, 39, 98. 254; vs, logic, 99, 233-
+
+
+Index
+
+
+788
+
+
+
+
+
+
+
+
+
+
+
+
+
+