Is string theory beautiful? Its promoters think so. Smolin and Woit believe that its recent absorption into a richer conjecture called M-theory has turned the former beauty of strings into mathematical structures as ugly as the epicycles Ptolemy invented to explain the orbits of planets as they circle the earth.

We are back to the mystery of Keats's notorious lines. In my opinion, John Simon is right. Even beautiful mathematical proofs can be wrong. In 1879 Sir Alfred Kempe published a proof of the four-color map theorem. It was so elegant that for 10 years it was accepted as sound. Alas, it was not. Henry Dude¿ney, England's great puzzle maker, published a much shorter and even prettier false proof.

In The New Ambidextrous Universe I write about the vortex theory of atoms. This popular 19th-century conjecture had an uncanny resemblance to superstrings. It maintained that atoms are not pointlike but are incredibly tiny loops of energy that vibrate at different frequencies. They are minute whirlpools in the ether, a rigid, frictionless substance then believed to permeate all space. The atoms have the structure of knots and links, their shapes and vibrations generating the properties of all the elements. Once created by the Almighty, they last forever.

In researching vortex theory, I came across many statements by eminent physicists, including Lord Kelvin and James Clerk Maxwell, suggesting that vortex theory was far too beautiful not to be true. Papers on the topic proliferated, books about it were published. Scottish mathematician Peter Tait's work on vortex atoms led to advances in knot theory. Tait predicted it would take several generations to develop the theory's mathematical foundations. Beautiful though it seemed, the vortex theory proved to be a glorious road that led nowhere.

Stewart concludes his book with two maxims. The first: "In physics, beauty does not automatically ensure truth, but it helps." The second: "In mathematics beauty must be true--because anything false is ugly." I agree with the first statement, but not the second. We have seen how lovely proofs by Kempe and Dudeney were flawed. Moreover, there are simply stated theorems for which ugly proofs may be the only ones possible.

Let me cite two recent examples. Proof of the four-color map theorem required a computer printout so vast and dense that it could be checked only by other computer programs. Although there may be a beautiful proof recorded in what Paul Erd¿s called "God's book"--a book that, he suggested, included all the theorems of mathematics and their most beautiful proofs--it is possible that God's book contains no such proof. The same goes for Andrew Wiles's proof of Fermat's last theorem. It is not computer-based, but it is much too long and complicated to be called beautiful. There may be no beautiful proof for this theorem. Of course, mathematicians can always hope and believe otherwise.

Because symmetry is the glue and tape that binds the pages of Stewart's admirable history, a stanza from Lewis Carroll's immortal nonsense ballad The Hunting of the Snark could serve as an epigraph for the book:

You boil it in sawdust: you salt it in glue:
You condense it with locusts and tape:
Still keeping one principal object in view--
To preserve its symmetrical shape.

This article was originally published with the title Is Beauty Truth and Truth Beauty?.

1 Comments

1. 1. PeterROwen 07:22 PM 3/15/10

   I've always found Professor Stewart's books highly readable and would recommend them even to non-mathematicians, or maybe especially to non-mathematicians.

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