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+-1 DOUGLAS R. HOFSTA DTER 
+
+godel,escher,bach 
+
+I AN ETERNAL GOLDEN BRAID 
+
+A METAPHORICAL FUGUE ON MINDS AND MACHINES, 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 
+
+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 
+
+Birthday 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 Magnfierab, Indeed 549 
+
+Chapter XVII: Church, Turing, Tarski, and Others 559 
+
+SHRDFU, Toy of Man's Designing 586 
+
+Chapter XVIII: Artificial Intelligence: Retrospects 594 
+
+Contrafactus 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 
+
+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 
+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 
+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. 
+
+...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. 
+
+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). 
+
+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. 
+
+CHAPTER N: Introduction: A Musico-Logical Offering 
+
+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 
+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 
+
+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.
+
+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 
+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 . 
+
+("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- 
+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 
+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 
+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 
+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 
+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 
+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 
+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. 
+
+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. 
+
+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 different 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 
+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 
+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 
+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 challenge: 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, 
+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 
+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 
+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 
+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. 
+
+DIALOGUE I: 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. 
+
+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! 
+
+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. 
+
+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! 
+
+CHAPTER I: 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 MIU-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 MIU-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. 
+
+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: 
+
+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) MII from (1) by rule II 
+
+(3) MIII 
+from (2) by rule II 
+
+(4) MIIIIU 
+from (3) by rule I 
+
+(5) MUIU 
+from (4) by rule III 
+
+(6) MUIUUIU 
+from (5) by rule II 
+
+(7) MUIIU 
+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 
+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 
+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 
+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. 
+
+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 
+
+MI 
+
+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 
+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. 
+CHAPTER
+
+DIALOGUE II: 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 
+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. 
+
+“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!" 
+
+CHAPTER II: 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. 
+
+* 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: 
+
+* RULE: 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 
+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: 
+
+(la) Throw the simplest possible axiom (-p-q- ) into the bucket. 
+
+(1b) 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. 
+
+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 
+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: 
+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 interpretation 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 
+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 
+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 
+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 
+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); 
+
+* 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 
+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. 
+
+DIALOGUE III: 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"? 
+
+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. 
+
+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. 
+
+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: 
+
+* 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. 
+
+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) 
+
+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). 
+
+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. 
+
+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 
+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 
+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 
+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:
+
+* 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. 
+
+DIALOGUE IV: 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. 
+
+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... 
+
+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 - say
+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. 
+
+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. 
+
+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 y?u 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 
+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. 
+
+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 
+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. 
+
+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 
+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!" 
+
+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 
+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. 
+
+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 
+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 
+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 
+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, 
+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. 
+
+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: 
+
+* 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 
+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. 
+
+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 
+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 
+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?) 
+
+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 
+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. 
+
+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! 
+
+DIALOGUE V: 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 it.
+
+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? 
+
+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. 
+
+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? 
+
+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.) 
+
+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 
+
+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, 
+
+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.) 
+
+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. 
+
+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.) 
+
+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, o Genie? What is 
+your wish? 
+
+Genie: I have a special wish to make of you, o 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 of smoke, they can all make out a ghostly fore towering above 
+them.) 
+
+Meta-Meta-Genie: I am the MetaMeta-Genie. You summoned me, 
+
+o Meta-Genie? What is your wish? 
+
+Meta-Genie: I have a special wish to make of you, oh 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, o MetaGenie. 
+
+Meta-Genie: Thank you, o 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, o Genie. 
+
+Genie: Thank you, o Djinn, and GOD. 
+
+(And the Meta-Genie, as all the higher ones before her, 
+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, o 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, o 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. 
+
+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 
+
+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 
+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. 
+
+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.
+
+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? 
+
+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. 
+
+(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.) 
+
+Tortoise: Hey! Put that back! I want to hear the end of this piece! 
+
+(Confused, Achilles obeys, and the music resumes.) 
+
+Tortoise: 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? 
+
+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 
+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.) 
+
+Achilles: 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 
+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".) 
+
+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! 
+
+(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 
+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! 
+
+CHAPTER V: 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.
+
+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 
+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.; 
+
+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 - 
+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 
+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 
+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. 
+
+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 
+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, 
+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. 
+
+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 
+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). 
+
+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: 
+
+* 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. 
+
+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. 
+
+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
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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, 
+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 convoluted "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 
+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 different 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 
+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 
+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 
+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. 
+
+DIALOGUE VI: 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 
+"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 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 
+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 
+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 
+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." 
+
+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 
+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. 
+
+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 intricate 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. 
+
+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 
+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. 
+
+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 
+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 
+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 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 sequences 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. Successively 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. 
+
+DIALOGUE VII: 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?" Well, 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 
+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: 
+
+(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 
+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: 
+
+* [ push 
+
+* <PaQ> premise 
+
+* P separation 
+
+* Q separation 
+
+* <QaP> joining 
+
+* ] pop 
+
+* «PaQ>=><QaP» 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 
+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. 
+
+* 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: 
+
+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 
+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: 
+
+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 
+
+(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 
+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. 
+
+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, 
+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: 
+
+* <~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). 
+
+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 
+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 
+
+(11) <~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 
+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. 
+
+DIALOGUE VIII: 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. 
+
+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. 
+
+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. 
+
+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.
+
+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. 
+
+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: 
+
+* zero: O
+
+* one:  SO
+
+* two:  SSO
+
+* three: 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: 
+
+(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. 
+
+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 
+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: 
+
+* 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 
+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 
+(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 
+
+* ¥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. 
+
+NUMERALS
+
+* O is a numeral. 
+
+* A numeral preceded by S is also a numeral. 
+
+* examples : O SO S5O SSSO SSSSO SSSSSO 
+
+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-. O b SSa' (SO • (SSO+c)) S(Sa • (Sb • Sc)) 
+
+* TERMS may be divided into two categories: 
+
+(1) DEFINITE terms. These contain no variables. 
+
+* examples: O (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
+
+* If v and t are terms, then s = t is an atom. 
+
+* examples: SO=O (SSO+SSO)=5SSSO 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: ~SO=O ~3b:(b+b)=SO -<O=Oz>SO=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~-O=O> >b=bv~3c:c=b> 
+
+* <SO=O=>¥c:~3b:(b+b)=c> 
+
+* The quantification status of a variable doesn't change here. 
+
+QUANTIFICATIONS
+
+* 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: 5O=O ~Vd:d=O 3c:<Vb:b=bA~c=c> 
+
+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: 
+
+<SO=Ov~SO=O>*  
+
+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=O 
+
+* AXIOM 2: ¥a:(a+O)=a 
+
+* AXIOM 3: ¥a:¥b:(a+Sb)=S(a+b) 
+
+* AXIOM 4: ¥a:(a-O)=O 
+
+* 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." 
+
+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 -SO=O 
+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. 
+
+O Restriction : The term which re* places 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=O axiom 1*  
+
+* ~SO=O 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=O 
+
+* ~S(c+SSO)=O 
+
+There is another rule, the rule of generalization, which allows us to put 
+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+SSO)=O 
+
+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=O axiom 1 
+
+* ~3a:Sa=O 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: 
+
+* ¥a:-Sa=O 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) (SO+SO)=S(SO+O) specification (O for b) 
+
+(4) ¥a:(a+O)=a axiom 2 
+
+(5) (SO+O)=SO specification (SO for a) 
+
+(6) S(SO+O)=SSO 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 (O for b) 
+
+(4) ¥a:-¥b:(a+Sb)=S(a+b) axiom 3 
+
+(5) ¥b:((SO.O)+Sb)=S((5O O)+b) specification ((SO-O) for a) 
+
+(6) ((SO .O)+SO)=S((SO.O)+O) specification (O for b) 
+
+(7) ¥a:(a+O)=a axiom 2 
+
+(8) ((SO.O)+O)=(SO.O) specification ((SO.O) for a) 
+
+(9) ¥a:(a.O)=O axiom 4 
+
+(10) (SO-O)=O specification (SO for a) 
+
+(I1) ((SO.O)+O)=O transitivity (lines 8,10) 
+
+(12) S((SO.O)+O)=SO addS 
+
+(13) ((SO -O)+SO)=SO transitivity (lines 6,12) 
+
+(14) (SO.SO)=SO transitivity (lines 3,13) 
+
+Illegal Shortcuts 
+
+Now here is an interesting question: "How can we make a derivation for the string O=O?" 
+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+O)=a axiom 2 
+
+(2) ¥a:a=(a+O) 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=O premise 
+
+(3) ¥a:a=O generalization ( Wrong ) 
+
+(4) Sa=() specification 
+
+(5) ] pop 
+
+(6) <a=O=>Sa=O> fantasy rule 
+
+(7) ¥a:<a=O=>Sa=O> generalization 
+
+(8) <o=o^so=o> specification 
+
+(9) O=O previous theorem 
+
+(10) SO=O 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. 
+
+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 O, SO, SSO, and s have been stuffed in): 
+
+(O + O) = O 
+
+(O + SO) = SO 
+
+(O+SSO)=SSO 
+
+(O+SSSO)=SSSO 
+
+(O+SSSSO)=SSSSO 
+
+* 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:(O+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 
+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:(O+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:(O+a)=a 
+
+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 
+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 
+
+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) specification
+
+(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 
+
+(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 specification (line 20) 
+
+(51) ¥a:(0+a)=a previous theorem 
+
+(52) (0+c)=c specification 
+
+(53) c=(0+c) symmetry 
+
+(54) (c+0)=(0+c) transitivity (lines 50,53) 
+
+(55) ¥c:(c+0)=(O+c) 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? 
+
+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 principle, 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 
+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. 
+
+DIALOGUE IX: 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 
+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. 
+
+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? 
+
+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 
+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 
+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. 
+
+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 
+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. 
+
+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.:) 
+
+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? 
+
+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. 
+
+(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!) 
+
+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. 
+
+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. 
+
+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. 
+
+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:' - 
+
+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:
+
+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’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? 
+
+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?" 
+
+"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’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 
+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.] 
+
+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 
+
+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 
+(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." 
+
+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. 
+
+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. 
+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. 
+
+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 
+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. 
+
+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 operations 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 
+
+(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 
+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. 
+
+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. 
+
+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 
+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: 
+
+* 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':(SO+Sa') = S( SO+a') 
+
+* 362,123,666,112,123,666,323,1 11,123,362,123,666,112,666,323 specification 
+
+( SSO + 5O ) = S ( SO + O ) 
+
+* 626,262,636,362 262,112,666, 3 23,111,262 axiom 
+
+* ¥• a : ( a + O ) = a 
+
+* 362.123.666.112.666.323.111.123.666 specification 
+
+( SO + O ) = SO 
+
+* 123,362,123,666,112,666,323,11 1,123,123,666 insert’12; 
+
+* S (SO+O) = SSO 
+
+* 362.123.666.112.123.666.323.111.123.123.666 transitivity 
+
+(SO + SO) = 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 
+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. 
+
+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 
+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. 
+
+DIALOGUE X: 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 
+
+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 way.
+
+Crab: Has someone at last managed to resolve this celebrated question?
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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? 
+
+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. 
+
+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. 
+
+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?" 
+
+Tortoise: Listen carefully, and you will! 
+
+(The prelude ends. There is a moment of silence; and ... 
+
+* [ATTACCA] 
+
+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 
+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 
+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 
+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 
+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. 
+
+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.) 
+
+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. 
+
+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. 
+
+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 
+language.) 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 acomplished 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 
+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 
+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. 
+
+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 
+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, 
+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 
+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 
+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 
+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. 
+
+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 
+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 
+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. 
+
+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.
+
+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 
+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 
+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? 
+
+DIALOGUE XI: ...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, 
+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. 
+
+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 
+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 
+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 
+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 
+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 
+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 
+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. 
+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 
+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 
+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- 
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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 
+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”, 
+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 
+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. 
+
+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? 
+
+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. 
+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: 
+
+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. 
+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.) 
+
+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. 
+
+CHAPTER XI: 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 
+remixed 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 &amp; 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 
+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. 
+
+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 
+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.' 
+
+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 
+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 
+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 
+"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 
+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, 
+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 
+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? 
+
+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. 
+
+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 
+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. 
+
+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 
+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 - 
+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 
+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 
+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 
+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. 
+
+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
+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. 
+
+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.' 
+
+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 
+"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 
+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 
+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 
+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. 
+
+DIALOGUE XII: 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 
+
+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. 
+
+'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. 
+
+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 conclusion 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 
+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? 
+
+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 
+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 
+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 
+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 
+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 
+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 
+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 
+
+"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 programmer 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". 
+
+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 
+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 
+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-&amp;-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 
+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. 
+
+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 representations 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. 
+
+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 
+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 
+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. 
+
+DIALOGUE XIII: 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 
+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 
+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 = 2 + 19 = 4 + 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. 
+
+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 
+
+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 
+
+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 
+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 
+
+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? 
+
+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. 
+
+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 
+
+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 
+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 
+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 
+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 
+
+* Sierpinski 
+
+* 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. 
+
+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! 
+
+CHAPTER XIII: 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 
+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 
+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 
+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. 
+
+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 
+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: 
+
+* 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 
+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 
+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!) 
+
+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)
+
+* 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) 
+
+* 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 
+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 
+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: 
+
+* Blueprogram{#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 
+
+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 predicting 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, 
+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; 
+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 
+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 
+
+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 
+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 
+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: 
+
+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? 
+
+GlooP ... 
+
+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 
+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 divided 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. 
+
+DIALOGUE XIV: 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. 
+
+(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 
+"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? 
+
+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? 
+
+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 
+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. 
+
+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: 
+
+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 
+
+* MII 
+
+* 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 
+
+* MII 
+
+* MIII 
+
+* 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 
+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 
+
+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 } 
+
+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- PAIR{a,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. 
+
+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. 
+
+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: 
+
+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 
+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' 
+
+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.” 
+
+-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”. 
+
+"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 
+
+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: 
+
+* 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. 
+
+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” 
+
+-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 
+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. 
+
+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 
+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 
+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. 
+
+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 
+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! 
+
+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! 
+
+DIALOGUE XV: 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. 
+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 one thing.
+
+Achilles: What? That I Ask for no present? 
+
+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 
+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 ... 
+
+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, 
+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 
+
+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” 
+
+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. 
+
+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- 
+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. 
+
+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.... 
+
+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 
+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 
+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. 
+
+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. 
+
+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 
+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. 
+
+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 
+wondering why there were three characters participating. Simplico, Salviati and Sagredo. 
+Why wouldn’t two have 
+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. 
+
+DIALOGUE XVI: 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. 
+
+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 
+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. 
+
+* 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 
+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. 
+
+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. 
+
+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 
+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- 
+
+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. 
+
+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 
+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 
+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. 
+
+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 ... 
+
+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 
+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: 
+
+"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). 
+
+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 
+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. 
+
+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). 
+
+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 
+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. 
+
+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 
+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 
+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 
+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 C G 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 
+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 
+
+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 
+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 
+
+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 
+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-# 
+
+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. 
+
+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 
+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 
+
+* 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. 
+
+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 
+
+* G: 
+guanine 
+
+* C: 
+cytosine 
+
+* T: 
+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 
+
+* 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. 
+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. 
+
+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 
+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 
+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. 
+
+
+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 
+
+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 
+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] 
+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 
+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". 
+
+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! 
+
+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 
+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] 
+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 
+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 
+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. 
+
+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. 
+
+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. 
+
+* 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 
+
+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) 
+1 
+
+(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. 
+
+The God el Code. 
+
+In the following chart, the mapping between systems 2 and 3 is explained carefully 
+
+* 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 
+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. 
+
+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): 
+
+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 
+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 
+
+(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 
+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 
+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 
+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. 
+
+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 
+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 
+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. 
+
+DIALOGUE XVII: 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. 
+
+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: 
+"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.) 
+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 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, 
+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-
+
+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 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 
+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.) 
+
+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. 
+
+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 
+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. 
+
+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 
+
+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 
+
+FIGURE 105. Srinivasa Ramanujan and 
+one of his strange Indian melodies. + e 
+
+Tt7T 
+
+V5 ♦ 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 
+
+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- 
+
+mediately jumped to fourth powers. After all, there are several other reasonably natural 
+,3 3,3 
+
+u + v = x + y 
+
+along different dimensions. For instance, there is the question about representing a 
+,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 . 
+
+ 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 
+
+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. 
+
+"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.. 
+
+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. 
+
+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". 
+
+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 
+
+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 
+neural network/ 
+
+* isomorphism 
+
+* (substrate: brain) 
+as collection 
+of neurons 
+
+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. 
+
+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: 
+
+(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. 
+
+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 
+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 
+
+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 
+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. 
+
+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 
+
+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 
+
+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 
+
+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 
+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 
+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. 
+
+DIALOGUE XVIII: 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. 
+
+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? 
+
+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. 
+
+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. 
+
+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? 
+
+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- 
+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. 
+
+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. 
+
+CHAPTER XVIII: 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- 
+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. 
+
+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 
+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 
+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 
+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 
+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 
+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 
+
+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. 
+
+* 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 
+
+* 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 
+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 
+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. 
+
+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 
+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 
+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 
+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 
+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 
+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 
+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 
+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 
+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. 
+
+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 
+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 
+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 
+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 "diffused" 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, 
+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- 
+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 
+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. 
+
+(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. 
+
+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 
+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 
+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- 
+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. 
+
+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: 
+
+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 
+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 
+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 systern'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. 
+
+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. 
+
+DIALOGUE XIX: 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 
+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? 
+
+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? 
+
+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- 
+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! 
+
+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 
+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. 
+
+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. 
+
+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.
+
+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.
+
+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: 
+
+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 
+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 
+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 
+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. 
+
+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 
+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. 
+
+* 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 
+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. 
+
+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:- 
+
+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:- 
+
+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. 
+
+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 
+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 
+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 
+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: 
+
+* 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- 
+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 
+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; 
+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. 
+
+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 
+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 
+
+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 
+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? 
+
+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 
+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 
+
+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 
+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 
+
+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. 
+
+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 
+
+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. 
+
+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 
+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. 
+
+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. 
+
+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 
+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 
+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 
+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 
+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 
+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? 
+
+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? 
+
+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. 
+
+DIALOGUE XX: 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 
+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. 
+
+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 
+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 
+
+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 
+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 
+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." 
+
+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 
+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 
+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. 
+
+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. 
+
+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: 
+
+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. 
+
+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 
+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 
+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- 
+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 
+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). 
+
+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... 
+
+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 
+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 
+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 
+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). 
+
+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. 
+
+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? 
+
+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 
+
+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 
+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 
+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 
+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 
+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. 
+
+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 
+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. 
+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. 
+
+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 
+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. 
+
+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) . 
+
+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
+
+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 
+
+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 
+
+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 
+
+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
+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 
+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. 
+
+DIALOGUE XXI: 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! 
+
+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, 
+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 
+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. 
+
+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. 
+
+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! 
+
+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. 
+
+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 
+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! 
+
+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 
+
+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. 
+
+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! 
+
+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. 
+
+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. 
+
+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. 
+
+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. 
+
+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! 
+
+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? 
+
+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. 
+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. 
+
+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). 
+
+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 
+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.