|
The millennium problems: the seven greatest
unsolved mathematical puzzles of our time This ambitious book seeks to explain to the interested layman what the seven Millennium problems - the Riemann hypothesis; Yang-Mills theory; P versus NP; the Navier-Stokes equations; the Poincaré conjecture; the Birch-Swinnerton Dyer conjecture; and the Hodge conjecture - are all about. Such a project may raise eyebrows among mathematicians, who know they themselves have a limited grasp of most of these problems. But the author is well aware of the dangers: ‘To read my book, all you need by way of background is a good high school knowledge of mathematics. You will also need sufficient interest in the topic. This second prerequisite is more important than the first ... no matter how hard I tried I could not make this book an easy read.’ The book contains eight chapters: chapter zero (introducing the overall theme) and one chapter for each problem. The overall treatment is inevitably impressionistic: the true flavour of each problem cannot be conveyed in such a book. Yet the author struggles to bring the interested layman as close to the action as one could reasonably hope. The task may be impossible, yet Devlin does not shirk his commitment to try to explain. Each problem is exploited in order to teach, or review, some relevant elementary mathematics. This elementary platform is then used as a springboard to move on to the more serious content - trying to remain true to the spirit of the problem while avoiding inaccessible technical subtleties. The intended effect is to convey a sense of wonder and awe rather than detailed clarity. For example, the chapter on the Birch-Swinnerton Dyer conjecture includes a discussion of the structure of the group of rational points of an elliptic curve, the rank of such a group, reduction mod p, and the Hasse-Weil L-function L(E,s) - all on an elementary level. Devlin knows that the typical reader may understand a few of the easiest ideas here, but will soon be struggling. And he admits it: having explained the relevant terms and why they matter the chapter ends ‘…according to the conjecture, the rank of E gives an exact measure of the degree to which L(E,1) = 0. So now you know.’ The author appears to have been well-served by the specialists he consulted on each of the problems. Unfortunately, there is no discussion of the dangers of attaching a price tag to the Millennium problems. Money and mathematics are unnatural bedfellows, and here is a golden opportunity to discuss the issues openly. In the short term the money provides publicity, as in the public airing of recent work towards solving the Poincaré conjecture. But the advantage may be slight: non-Millennium problems - such as the recent solution by Ben Green and Terry Tao of the Hardy-Littlewood conjecture that the primes contain arbitrarily long arithmetic progressions - also gain significant publicity (New Scientist, 8 May 2004). And the drawbacks may be more serious. Like Hilbert’s 23 problems in 1900, the seven ‘Millennium problems’ look backwards and forwards: they represent central pillars of unfinished business from the previous century, while being chosen because of their potential fruitfulness for future progress. The criteria for the Millen-nium problems required that a jury be able to assess whether a particular contribution constitutes a ‘solution’. One can imagine problems (e.g. Moon-shine) or programmes (e.g. Grothendieck or Langlands), which at a particular time might have been excluded purely because the relevant infrastructure of language and conjectures had not been sufficiently well organised. If so, then this particular link between mathematics and Mammon may exert a narrowing, conservative influence on development which deserves to be more openly questioned. Like most popular mathematics books, Devlin’s would benefit from being proof-read by a sensitive historian of mathematics, to ensure that ‘historical distortions’ are deliberate! For example, it is sometimes defensible to write as though the ancient Greeks worked with ‘irrational numbers’; but it is more questionable to repeat the canard that their discovery of irrationals came as ‘such a shock that their progress in mathematics came to a virtual halt’ p.12. And there are many instances where a simple check would eliminate unnecessary errors (as in giving Gauss’ first name as Karl rather than Carl, p.24; or in the claim that Euclid proved the Fundamental Theorem of Arithmetic, p.20; or in the confusing assertion that Euclid showed that, if N is the product of all prime numbers from 2 up to the largest prime P, then N+1 must itself be a prime, p.53). In presenting elementary material, each populariser has his own preferred style. The elementary material in this book is especially important, since it is the only mathematics most readers are likely to really understand. Given the huge effort which the author has expended in making the harder material accessible, the overall impact could be improved by polishing some of this easier material for subsequent editions. Nevertheless, this is a well-constructed popular survey for which the author deserves our thanks. Tony Gardiner
|
BOOK
REVIEW