The Millennium Problems 1

The noble idea that advanced mathematics can be made comprehensible to laypeople is tested in this sometimes engaging but ultimately unsatisfying effort. Mathematician and NPR commentator Devlin (The Math Gene) bravely asserts that only""a good high-school knowledge of mathematics"" is needed to understand these seven unsolved problems (each with a million-dollar price on its head from the Clay Mathematics Institute), but in truth a Ph.D. would find these thickets of equations daunting. Devlin does a good job with introductory material; his treatment of topology, elementary calculus and simple theorems about prime numbers, for example, are lucid and often fun. But when he works his way up to the eponymous problems he confronts the fact that they are too abstract, too encrusted with jargon, and just too hard. He finally throws in the towel on the Birch and Sinnerton-Dyer Conjecture (""Don't feel bad if you find yourself getting lost... the level of abstraction is simply too great for the nonexpert""), while the chapter on the Hodge Conjecture is so baffling that the second page finds him morosely conceding that""the wise strategy might be to give up."" Nor does Devlin make a compelling case for the real-world importance of many of these problems, rarely going beyond vague assurances that solving them""would almost certainly involve new ideas that will... have other uses."" Sadly, this quixotic book ends up proving that high-level mathematics is beyond the reach of all but the experts.