The Enormous Theorem and the jigsaw of numbers.
Mar 19, 2007, Vol. 12, No. 26 • By DAVID GUASPARI
Symmetry and the Monster
The earliest example of "modern" and "abstract" mathematics--and one of the most beautiful--is the theory of groups. It grew out of the study of equations, but can be understood as a theory of symmetry.
You might cryptically define a "symmetry" as a transformation or rearrangement of something that leaves it unchanged. For example, a square looks unchanged if you rotate it 90 degrees, or examine its reflection in a suitably positioned mirror. Collectively, the rotations and reflections that leave a square looking the same make up "the symmetry group of the square." Symmetries abound not only in mathematics, but also in physics. The theory of relativity says that the transformation from one observer's point of view to another's may alter the values of some observations but will leave the laws relating those observations unchanged.
The principle that any fundamental physical theory must possess certain kinds of symmetry has become a scientific axiom, and a tool of discovery.
In Symmetry and the Monster, Mark Ronan guides the lay reader through one of the major mathematical achievements of the 20th century--officially known as "the classification of finite simple groups," but often called "the enormous theorem," because its proof occupies more than 10,000 pages of dense reasoning spread through several hundred journal articles (and never-published manuscripts), with significant contributions from perhaps a hundred authors. For posterity's sake, a project is under way to simplify and unify this body of argument and present it in full detail. Estimated completion date: 2010. Anticipated length: 12 volumes.
The enormous theorem identifies and characterizes the infinitely many finite simple groups, which Ronan calls "atoms of symmetry" because their role in group theory is something like that of the elements in chemistry or the prime numbers in arithmetic: Building blocks from which all the groups can be constructed. Eventually, a kind of periodic table emerged into which almost all the known simple groups could be fit. Immense labor then proved that there are precisely 26 exceptions--simple groups not in the table and fitting no known pattern.
The existence of the largest and strangest of these, the eponymous Monster, was conjectured in the early 1970s and confirmed 10 years later. Here are some of its Babe Ruthian statistics: The Monster can be represented as a collection of rotations in a space of 196,883 dimensions. The size of that collection, which has been calculated exactly, is comparable to the number of elementary particles in the planet Jupiter. To write it in decimal form takes 54 digits. One is amazed, if not aghast, that the human mind can deal with such complexity.
The Monster also has spooky connections with seemingly remote branches of mathematics and with string theory, an ongoing attempt to unify all the fundamental forces of physics in a single theory. Glimpses of these mysterious affinities can make the world slip out of focus, as the boundary between sober scientific fact and numerological superstition seems to dissolve before our eyes.
It is an understatement to say that Ronan has set himself a formidable task, but to a reader willing to meet him part way, he offers an absorbing tale of discovery. It begins in 1830 with the great variste Galois, and chronicles both spectacular intellectual virtuosity and outsize personalities. Galois himself--killed in a duel at age 20--was a combination of Mozart and James Dean who introduced whole new fields of mathematical enquiry in a testament hastily composed the night before his trip to the field of honor.
Ronan's presentation naturally skews toward things that are more easily explained. So we get a great deal about the eccentricities of Sophus Lie, but only the most glancing account of the deep subject called (in his honor) Lie Theory, from which the periodic table of simple groups derives. Lacking a single theme, the narrative loses some momentum halfway through, but revs up when it reaches the 1960s and remains in high gear as "The Classification" evolves from a pipe dream to thinkability and then inevitability, and the quest to complete it culminates in a kind of international treasure hunt for the mysterious exceptions.
Many of the principals are still working mathematicians, and Ronan makes good use of their own words to describe the circumstances and excitement surrounding key discoveries. A reader may glimpse what it feels like to do mathematics at a high level, the sense of exploring a terrain that is mysterious, surprising, and completely unforgiving.
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.