"This isn't magic. There's a reason this stuff works," my high school math teacher used to say. Of course, there are some contentions, hypotheses, in math where we don't know if they work, if they are true.
For professional mathematicians, one of the most important of these is the Riemann Hypothesis. Everlasting fame amongst mathematicians, and, incidentally, a million dollars is waiting for the person who can nail the truth of the "RH" down.
Unlike some famous math problems, the gist of the RH is not readily apparent to most non-mathematicians. Derbyshire has to spend some time explaining what is meant by "All non-trivial zeros of the zeta function have real part one-half." And, as someone whose formal math instruction ended with four years of high school math and who reads the very occasional popular math book by Gleick, Peterson, or Paulos, I'm pretty much the target audience Derbyshire pitches that explanation to.
The book's style reminded me of the science histories of James Burke. But where Burke's work is a pinball version of history, caroming from person to person, theory to theory, Derbyshire's is a train of mathematical explanation covering the work leading up to, and proceeding from, the RH. Occasionally, Derbyshire stops at some station, pulls up the blind, and looks at some area of tangential interest: famous mathematicians including Gauss, Hilbert, Russell, Dyson, and Turing (who thought RH untrue and attempted to build a computing device to disprove it); German educational reforms of the early 19th century; the Cambridge Five spies; and, most often, since this book is ostensibly a biography of him, the life of Bernhard Riemann. But it's not long before we're back on that math train again. This is not to shortchange the non-math interludes of the book. Derbyshire's quick asides gave me a lot of ideas for further reading. And, if less than half of the book's 422 pages cover Riemann's life, you still get some idea of his protean mind so important not only to mathematics but modern physics.
Derbyshire's claim that, if you don't understand the RH after he explains it you never will, seems credible. I won't claim I immediately followed his chain of explanations the first time around. But that had more to do with trying to read this book in 15 minute intervals over a week rather than Derbyshire's prose. Upon reviewing many sections again, things became clearer.
The book briefly notes some of the consequences of RH, practical and theoretical. A lot of math is based on the assumption it's true. And the RH may have some mysterious relation to the world of quantum physics. In the commercial and military worlds, where encryption methods based on prime numbers are important, the RH, which has to do with the distribution of primes, may have significant importance if proved true.
I think one of the best things about this book is that, briefly, in a simple way, a non-mathematician like me can get some small idea of the excitement mathematicians feel upon discovering some curious pattern in the world of numbers.
The only complaint I have with this book is its format. Is it too much to ask that, in the age of computerized typesetting and with an author whose footnotes are all worth reading, that we put those footnotes at the bottom of the relevant page and not at the end of the book?