Gödel, Escher, Bach: An Eternal Golden Braid (18 page)

FIGURE 20. Visual rendition of the principle underlying Gödel’s Theorem: two back-to-back mappings which have an unexpected boomeranging effect. The first is from groove
patterns to sounds, carried out by a phonograph. The second-familiar, but usually
ignored -- is from sounds to vibrations of the phonograph. Note that the second mapping
exists independently of the first one, for any sound in the vicinity, not just ones produced
by the phonograph itself, will cause such vibrations. The paraphrase of Gödel’s Theorem
says that for any record player, there are records which it cannot play because they will
cause its indirect self-destruction. [Drawing by the author.

Implicit Meanings of the Contracrostipunctus

What about implicit meanings of the Dialogue? (Yes, it has more than one of these.) The simplest of these has already been pointed out in the paragraphs above-namely, that the events in the two halves of the dialogue are roughly isomorphic to each other: the phonograph becomes a violin, the Tortoise becomes Achilles, the Crab becomes the Tortoise, the grooves become the etched autograph, etc. Once you notice this simple isomorphism, you can go a little further. Observe that in the first half of the story, the Tortoise is the perpetrator of all the mischief, while in the second half, he is the victim.

What do you know, but his own method has turned around and backfired on him!

Reminiscent of the backfiring of the records' muusic-or the goblet's inscription-or perhaps of the Tortoise's boomerang collection? Yes, indeed. The story is about backfiring 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 Gödel’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 Gödel’s Theorem, which, as I said in the Introduction, heavily on two different levels of meaning of statements of number t1

Each of the two halves of the Contracrostipunctus is an "isomorphic co Gödel’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 Gödel’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 Gödel’s theorem and the Contracrostipunctus, but it is the core of it. You need not if you don't fully grasp Gödel’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 Gödel’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 Gödel’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 Gödel’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 Gödel 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