Gödel, Escher, Bach: An Eternal Golden Braid (19 page)

FIGURE 21. Tower of Babel, by M. C. Escher (woodcut, 1928).

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 I 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 unconsoled, 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 torn 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 t1 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 tI 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", ìf ", "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 pqsystem, 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.