The Enormous Theorem and the jigsaw of numbers.
Mar 19, 2007, Vol. 12, No. 26 • By DAVID GUASPARI
Much of the mathematics created for the enormous theorem is highly technical, and understood only by group theory experts. As that leaves me out, I have no grounds for suggesting any improvements to Ronan's account of these higher flights. At the elementary level, however, he misses some opportunities by not expecting enough of his audience. Remarkably, he never defines group or simple group, even though these notions can easily be explained to a willing reader. So, in an important way, the book fails to say what it's about.
The publisher may have scared him off: Stephen Hawking has written that his publisher warned against including equations in a popular book--as each one would cut his readership in half. An author is, of course, entitled to aim for the audience of his choice, but I doubt Ronan would deny that he, writing not for money but for love, assumes a special responsibility toward those readers who are willing to do some work.
It is a difficult balancing act that, on balance, Ronan carries off. But a pedagogical slip in a key example is worth correcting. The text says that x2 - x - 1 has no factors. A lazy reader will nod "whatever," and carry on undisturbed; but the ideal reader, with high school algebra book dusted off and at the ready, will be flummoxed when the next page provides the polynomial's roots--since having roots and having factors amount to the same thing. A mathematician would immediately recognize that Ronan is using "no factors" as shorthand for "no factors over the field of rationals." Omitting that qualifier, in order to avoid the requisite explanation, was a misjudged kindness.
Ronan can also be criticized for introducing idiosyncratic terminology that eventually comes to seem affected and annoying. "Symmetry atom" is a helpful phrase, but by using it systematically, almost to the exclusion of "simple group," he (a) allows readers to remain ignorant of the subject's standard vocabulary, and (b) must edit quotation after quotation from the published literature and from participants' reminiscences to replace standard terminology with his own.
Another example: Instead of "decomposition" (the standard term for breaking a group down into simple groups), he uses the cant term "deconstruction"--a borrowing that reverses the usual procedure whereby intellectual snake oil salesmen appropriate the terminology of a legitimate science.
The ending of Ronan's story could not be better scripted. The quest succeeds, the summit is attained, and from it new and mysterious regions open out. In 1978, for example, a group theorist named John McKay was struck by the fact that the number 196,884 popped out of an important construction in the seemingly distant field of number theory. It was known at the time that the Monster, if it existed, could be represented as a group of rotations in 196,883 dimensions.
Could this near miss have any meaning? The Monster's character table--an array of numbers encoding information about its structure--had been conjecturally determined, and its leading entries were one and 196,883. The number that caught McKay's eye is their sum; and other coefficients appearing in that same number theoretic construction could also be teased out of entries in the Monster's character table by simple arithmetic maneuvers.
Did that mean anything? The well-known, and famously whimsical, mathematician J.H. Conway called the suggestion "moonshine," and immediately enlisted colleagues to join him in its pursuit. What followed might have made a dream sequence from Rain Man: First-rate mathematicians massaging, like numerologists, the statistics of the Monster in hopes of generating numbers that cropped up elsewhere. They hit pay dirt often enough to formulate a precise guess, known as the Monstrous Moonshine conjecture, that a far-reaching generalization of McKay's observation would be true.
In 1992 Richard Borcherds proved that the guess was right by calling on ideas from, of all places, string theory. For that, and related work, he was awarded the Fields Medal--an honor always described as "the mathematicians' Nobel Prize."
The Monster's mysteries are far from resolved. Its connections to fundamental physics are, as Ronan says, tantalizing. (Though he overstates the status of string theory. As the string theorist Brian Greene has acknowledged, many physicists will tell you that the jury is still out.) To bring us up to date on the evolving story of the Monster, Ronan says, would require another book. Should he undertake to write it, I have one piece of advice: include the definitions. Readers willing to take on a challenging book will do fine.
David Guaspari is a mathematician and computer scientist in Ithaca, New York.