by Ian Stewart
Basic Books, 2007
The title of Ian Stewart's book (he has written more than 60 others) is, of course, taken from the enigmatic last two lines of John Keats's "Ode on a Grecian Urn":
"Beauty is truth, truth beauty,"--that is all Ye know on earth, and all ye need to know.
But what on earth did Keats mean? T. S. Eliot called the lines "meaningless" and "a serious blemish on a beautiful poem." John Simon opened a movie review with "one of the greatest problems of art--perhaps the greatest--is that truth is not beauty, beauty not truth. Nor is it all we need to know." Stewart, a distinguished mathematician at the University of Warwick in England and a former author of this magazine's Mathematical Recreations column, is concerned with how Keats's lines apply to mathematics. "Euclid alone has looked on Beauty bare," Edna St. Vincent Millay wrote. To mathematicians, great theorems and great proofs, such as Euclid's elegant proof of the infinity of primes, have about them what Bertrand Russell described as "a beauty cold and austere," akin to the beauty of great works of sculpture.
Stewart's first 10 chapters, written in his usual easygoing style, constitute a veritable history of mathematics, with an emphasis on the concept of symmetry. When you perform an operation on a mathematical object, such that after the operation it looks the same, you have uncovered a symmetry. A simple operation is rotation. No matter how you turn a tennis ball, it does not alter the ball's appearance. It is said to have rotational symmetry. Capital "H" has 180-degree rotational symmetry because it is unchanged when turned upside down. It also has mirror reflection symmetry because it looks the same in a mirror. A swastika has 90-degree rotational symmetry but lacks mirror reflection symmetry because its mirror image whirls the other way.
Associated with every kind of symmetry is a "group." Stewart explains the group concept in a simple way by considering operations on an equilateral triangle. Rotate it 60 degrees in either direction, and it looks the same. Every operation has an "inverse," that cancels the operation. Imagine the corners of the triangle labeled A, B and C. A 60-degree clockwise rotation alters the corners' positions. If this is followed by a similar rotation the other way, the original positions are restored. If you do nothing to the triangle, this is called the "identity" operation. The set of all symmetry transformations of the triangle constitutes its group.
Stewart's history begins with Babylonian and Greek mathematics, introducing their basic concepts in ways a junior high school student can understand. As his history proceeds, the math slowly becomes more technical, especially when he gets to complex numbers and their offspring, the quaternions and octonions. The history ends with the discoveries of Sophus Lie, for whom Lie groups are named, and the work of a little-known German mathematician, Joseph Killing, who classified Lie groups. Through this historical section, Stewart skillfully interweaves the math with colorful sketches of the lives of the mathematicians involved.
Not until the book's second half does Stewart turn to physics and explain how symmetry and group theory became necessary tools. A chapter on Albert Einstein is a wonderful blend of elementary relativity and details of Einstein's life. Next comes quantum mechanics and particle theory, with several pages on superstrings, the hottest topic in today's theoretical physics. Stewart is a bit skeptical of string theory, which sees all fundamental particles as inconceivably tiny filaments of vibrating energy that can be open-ended or closed like a rubber band. He does not mention two recent books (reviewed in the September 2006 issue of Scientific American) that give string theory a severe bashing. Lee Smolin's The Trouble with Physics denounces string theory as "not a theory at all," only a mishmash of bizarre speculations in search of a viable theory. Peter Woit's book is entitled Not Even Wrong, a quote from the great Austrian physicist Wolfgang Pauli. He once described a theory as so bad it was "not even wrong."[break]
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.