It is difficult to think of any literary enterprise more challenging than the presentation of advanced mathematical topics to a general audience. It is not just that math is hard; there is, as Keith Devlin noted in a previous book, The Math Gene, (and as Bertrand Russell remarked in the introduction to Principia Mathematica), something deeply and fundamentally unnatural about mathematical thinking. It goes against the grain of everyday thought processes somehow, in a way that is hard for trained mathematicians to sympathize with. The mathematically sophisticated writer, when addressing a lay audience, must try to guess, as best he can, which points will baffle and confuse the ordinary reader; and this is very difficult to do, as the author’s own bafflement and confusion over those points, if he ever suffered any, is usually many years in the past.
A few days ago, I was in conversation with a friend, a highly intelligent and well-educated person retired from a successful business career. He asked about a book I have just completed, that deals with the prime numbers. “Is 1 a prime number?” my friend wanted to know. I said that mathematicians do not consider 1 to be a prime. He: “Well, is it, or isn’t it?” I said it was not. He: “Why not? It fits the definition, doesn’t it?” I said that while, strictly speaking, that was true, including 1 among the prime numbers is more trouble than it’s worth, and mathematicians, by mutual agreement, do not. He, scornfully: “Well, what kind of definition is that …?” You realize, at such moments, that you are never, ever going to get past this particular point with this particular person, and that the entire astonishing and beautiful field of prime number theory will be forever beyond this man’s grasp, for want of the ability to discriminate between what is, mathematically speaking, trivial (a word mathematicians are very fond of), and what is important.
With these reflections in mind, it is hard to feel anything but pity for the poor wretch who sets himself the task of bringing mathematics to the general reading public. Keith Devlin has been farming this rocky soil for years. His 1999 book Mathematics: The New Golden Age was one of the best pop-math books ever, in my opinion, with the clearest short explanation of topics like the Class Number Problem that you will see anywhere. In his new book The Millennium Problems, Devlin shows no sign of yet being broken in mind or spirit by the difficulties inherent in his chosen line of work. Here he gives a brief but vigorous survey of seven outstanding problems in higher mathematics, and displays much ingenuity in bringing the more abstruse points down to earth for a lay readership.
The inspiration for this book comes from the very beginning of the 20th century. On August 8, 1900, in a stuffy lecture hall at the Sorbonne, the great German mathematician David Hilbert gave an address to the Second International Congress of Mathematicians. In that address, he presented a list of challenges for the coming century. That list became known, not very accurately, as “Hilbert’s 23 problems.” Its effect was more or less what Hilbert intended: to concentrate the minds of researchers on a small number of key topics, and on a similar number of outstanding problems whose investigation was likely to yield — in addition, of course, to solutions — fruitful new insights.
Of those items on Hilbert’s list that were actual, well-defined problems, all but one have been solved, the single exception being the fabulous Riemann Hypothesis. During the course of the last century, though, some of the less specific topics on Hilbert’s list crystallized into particular problems, and of course some entirely new problems emerged. Though the year 2000 did not bring forth any pronouncement as definitive as Hilbert’s address, the temptation to produce problem lists and topic lists for this new century proved to be irresistible. Numerous prominent individuals and organizations in the world of mathematics offered suggestions.
One of the best-publicized of these ventures was undertaken by the Clay Mathematics Institute. In May of 2000, in a lecture hall at the Collège de France — plainly they had Hilbert in mind — the Institute announced a $7m prize fund, distributed as $1m each for the solution of seven outstanding problems in higher mathematics. The problems are:
- Riemann’s Hypothesis
- The Mass Gap Hypothesis
- The P-vs.-NP Problem
- The Navier-Stokes Equations
- Poincaré’s Conjecture
- The Birch and Swinnerton-Dyer Conjecture
- Hodge’s Conjecture
As Devlin points out, none of these is exactly a household name. We are long past the point at which gifted amateurs can make any contribution at the rarefied heights wherein dwell these “Millennium problems.” With the just-possible exception of the P-vs.-NP Conjecture, it can be said with certainty that a person who solves any of these problems will be a professional mathematician with years of experience and a good trail of published papers in the appropriate mathematical sub-discipline.
I have given the problems in Devlin’s order, which differs from the order on the Clay Institute web site. Devlin has re-arranged them from the more to the less approachable. Riemann’s Hypothesis can be understood, after some modest additional explanation, by anyone who completed high school math: Hodge’s Conjecture, by contrast, is so abstruse, the author has to struggle just to describe the general area of mathematics to which it applies. I don’t think my own ordering would have been exactly the same as Devlin’s — the Birch and Swinnerton-Dyer Conjecture (with all respect to the two gentlemen involved, can’t we find a handier name for the darn thing?) is not that difficult to grasp — but on the whole the book is well-structured with a good narrative pull. Devlin knows how to do this, if anyone does — The Millenium Problems is, I believe, his twenty-fourth book.
As well as being a good read for the interested non-mathematician, this book helps to promote the Clay Institute, a thing well worth doing. One of the most heartening developments in mathematics during recent years has been the rise of these independent research institutes financed by wealthy private enthusiasts. C.M.I. (established in 1998 by Boston investment banker Landon T. Clay) followed the American Institute of Mathematics, founded in 1994 by West Coast electronics retailer John Fry. These private foundations have both, in their brief existences, done wonders in promoting research into particular areas of math, and also in publicizing the more exciting topics to non-mathematical audiences. The Millenium Problems is a splendid contribution to these supremely useful and worthwhile efforts.