The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time. Keith Devlin. xii + 237 pp. Basic Books, 2002. $26.
Mathematicians solve problems: This is in some basic sense the essence of mathematical practice. We love a clever puzzle, and when a new one appears, it spreads like hot gossip. Every mathematical publication has a problem section. We organize problem-solving competitions within schools, between schools, between countries. The William Lowell Putnam Competition pits North American college students against each other in an all-day contest that awards four-figure cash prizes and lifelong notoriety to the winners. Russian, Chinese and American high school students annually square off in the International Mathematical Olympiad. The late Paul Erd"os, the eccentric, itinerant Hungarian mathematician, habitually offered cash prizes for the solution of problems that confounded him. (Most of the time he didn't even have to pony up for the eventual answers—solvers would frame his checks rather than cash them!) Posing and solving puzzles and problems is an integral part of the mathematical life. It's not just a game with us, it's the way mathematics advances.
The great German mathematician David Hilbert (1862–1943) outdid us all in problem setting (and problem solving too, but that's a different story). In August of 1900 he marked the turning of the century by posing 23 problems "from the discussion of which an advancement of science may be expected." A remarkable proportion of the subsequent half-century of mathematics was related to these problems. (See The Honors Class, by Benjamin Yandell [AK Peters, 2002]. Editor's note: Reviewed in the May–June 2002 issue.)
The turn of the millennium inspired several would-be Hilbert imitators to form lists of outstanding mathematics problems whose solution we might hope for in this century. The Millennium Problems discusses in lay terms the seven problems for the solution of which the Clay Mathematics Institute has offered million-dollar prizes.
Math is different from the other sciences. In a very real sense the problems, motivations and verification of mathematics come from inside the discipline itself, whereas the other sciences look to the world of phenomena for problems and affirmation. The chemist whose experiment yields a result within six decimal places of his theoretical prediction has good reason to feel pretty pleased with his theorizing. A mathematician rarely finds herself in such an empirically happy place vis-à-vis her theories. Usually a mathematician has only the cold reassurance of logic for comfort; the universe does not deign to validate our work except indirectly, when the work proves useful as a model in another science.
We have only ourselves to blame—we made the choice two millennia ago to opt for logic over phenomena when we accepted the irrational numbers. Nonrepeating, nonterminating decimals have no existence in the world of sense perception and scientific measurement, but they make for a nicer mathematics. In the intervening time we have piled abstraction upon abstraction: The counting numbers gave birth to the real numbers, which sired the complex numbers, which spawned the quaternions, which inspired vectors . . . and today we have a whole zooful of number systems taxonomically organized into the fields and rings and vector spaces of abstract algebra. The difficulty in learning mathematics consists almost entirely in becoming familiar enough with one level of this abstraction that the passage to the next level seems natural. In a sense, the lower level of abstraction must become concrete enough to support the next step up.
That summarizes Devlin's problem. The concepts of modern mathematics have been so abstracted from the concepts of school mathematics that no layperson can make the leap. Even professional mathematicians are going to be hard-pressed to really understand more than a few of these seven problems. Devlin knows this, and to his credit he even admits it; in his chapter on the Hodge Conjecture he says he is "doomed to fail" at making the problem at all meaningful. Despite his considerable expository skills, he is similarly doomed in nearly every chapter. Readers are not really going to understand what these problems are, why they're difficult or why they're important. What they will get is some interesting scientific history and, perhaps, a glimpse of how far research mathematics has progressed from school mathematics.
For me, the book raises two interesting questions: Why has the Clay Institute offered million-dollar prizes for the solutions of mathematics problems? and Is this good for mathematics?
The prizes are essentially a publicity stunt—one that worked. Of the many lists of millennial mathematics problems, the Clay Institute's is the only one that was reported by The New York Times, Le Monde, National Public Radio's All Things Considered, and Nature; it's the only list nonmathematicians are likely to have heard of. The folks at the Clay Institute surely realize that these prizes are not going to do much to advance us toward a solution for any of these problems. The solvers are going to have motivations other than a desire for wealth; money can be obtained in any number of ways much more easily than by tackling one of these problems. No, surely the motivation for offering the prizes was to attract attention to mathematics.
The difficulty is that once the attention was attracted, the public was told in effect, "Well, you've really got no chance of understanding what we're talking about, but trust us, it's important." Couldn't we have attracted some attention and then actually communicated some of the sublime and beautiful ideas of mathematics? Maybe we could have a million-dollar prize for the best book of mathematics popularization aimed at a layperson. Surely that would get attention and get people reading mathematics. Or a public million-dollar competition for the most stimulating popular mathematics lecture, à la American Idol. How about a TV game show in which contestants compete for cash by solving logic puzzles? Attracting attention is easy; rewarding it is hard.
It is not clear to me whether the Millennium Prizes are a good thing for mathematics, but I don't have time to think about that just now. I've got an idea about P = NP, and if I can just push this lemma . . . —Steve Kennedy, Mathematics, Carleton College, Northfield, Minnesota