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license plate, by which it can be referred to. And this coding trick enables statements of

number theory to be understood on two different levels: as statements of number theory,

and also as statements about statements of number theory.

Once Godel had invented this coding scheme, he had to work out in detail a way of

transporting the Epimenides paradox into a numbertheoretical formalism. His final

transplant of Epimenides did not say, "This statement of number theory is false", but

rather, "This statement of number theory does not have any proof". A great deal of

confusion can be caused by this, because people generally understand the notion of

"proof" rather vaguely. In fact, Godel's work was just part of a long attempt by

mathematicians to explicate for themselves what proofs are. The important thing to keep

in mind is that proofs are demonstrations within fixed systems of propositions. In the case

of Godel's work, the fixed system of numbertheoretical reasoning to which the word

"proof" refers is that of Principia Mathematica (P.M.), a giant opus by Bertrand Russell

and Alfred North Whitehead, published between 1910 and 1913. Therefore, the Godel

sentence G should more properly be written in English as:

This statement of number theory does not have any proof in the system of Principia

Mathematica.

Incidentally, this Godel sentence G is not Godel's Theorem-no more than the Epimenides

sentence is the observation that "The Epimenides sentence is a paradox." We can now

state what the effect of discovering G is. Whereas the Epimenides statement creates a

paradox since it is neither true nor false, the Godel sentence G is unprovable (inside

P.M.) but true. The grand conclusion% That the system of Principia Mathematica is

"incomplete"-there are true statements of number theory which its methods of proof are

too weak to demonstrate.

Introduction: A Musico-Logical Offering

18

But if Principia Mathematica was the first victim of this stroke, it was certainly not the

last! The phrase "and Related Systems" in the title of Godel's article is a telling one: for if

Godel's result had merely pointed out a defect in the work of Russell and Whitehead, then

others could have been inspired to improve upon P.M. and to outwit Godel's Theorem.

But this was not possible: Godel's proof pertained to any axiomatic system which

purported to achieve the aims which Whitehead and Russell had set for themselves. And

for each different system, one basic method did the trick. In short, Godel showed that

provability is a weaker notion than truth, no matter what axiomatic system is involved.

Therefore Godel's Theorem had an electrifying effect upon logicians, mathematicians,

and philosophers interested in the foundations of mathematics, for it showed that no fixed

system, no matter how complicated, could represent the complexity of the whole

numbers: 0, 1, 2, 3, ... Modern readers may not be as nonplussed by this as readers of

1931 were, since in the interim our culture has absorbed Godel's Theorem, along with the

conceptual revolutions of relativity and quantum mechanics, and their philosophically

disorienting messages have reached the public, even if cushioned by several layers of

translation (and usually obfuscation). There is a general mood of expectation, these days,

of "limitative" results-but back in 1931, this came as a bolt from the blue.

Mathematical Logic: A Synopsis

A proper appreciation of Godel's Theorem requires a setting of context. Therefore, I will

now attempt to summarize in a short space the history of mathematical logic prior to

1931-an impossible task. (See DeLong, Kneebone, or Nagel and Newman, for good

presentations of history.) It all began with the attempts to mechanize the thought

processes of reasoning. Now our ability to reason has often been claimed to be what

distinguishes us from other species; so it seems somewhat paradoxical, on first thought,

to mechanize that which is most human. Yet even the ancient Greeks knew that reasoning

is a patterned process, and is at least partially governed by statable laws. Aristotle

codified syllogisms, and Euclid codified geometry; but thereafter, many centuries had to

pass before progress in the study of axiomatic reasoning would take place again.

One of the significant discoveries of nineteenth-century mathematics was that there are

different, and equally valid, geometries-where by "a geometry" is meant a theory of

properties of abstract points and lines. It had long been assumed that geometry was what

Euclid had codified, and that, although there might be small flaws in Euclid's

presentation, they were unimportant and any real progress in geometry would be

achieved by extending Euclid. This idea was shattered by the roughly simultaneous

discovery of non-Euclidean geometry by several people-a discovery that shocked the

mathematics community, because it deeply challenged the idea that mathematics studies

the real world. How could there be many differ

Introduction: A Musico-Logical Offering

19

ent kinds of "points" and "lines" in one single reality? Today, the solution to the dilemma

may be apparent, even to some nonmathematicians-but at the time, the dilemma created

havoc in mathematical circles.

Later in the nineteenth century, the English logicians George Boole and Augustus De

Morgan went considerably further than Aristotle in codifying strictly deductive reasoning

patterns. Boole even called his book "The Laws of Thought"-surely an exaggeration, but

it was an important contribution. Lewis Carroll was fascinated by these mechanized

reasoning methods, and invented many puzzles which could be solved with them. Gottlob

Lrege in Jena and Giuseppe Peano in Turin worked on combining formal reasoning with

the study of sets and numbers. David Hilbert in Gottingen worked on stricter

formalizations of geometry than Euclid's. All of these efforts were directed towards

clarifying what one means by "proof".