**Amazon.com Product Description** (ISBN 023111639X, Paperback)

*Mathematics: The New Golden Age* offers a glimpse of the extraordinary vistas and bizarre universes opened up by contemporary mathematicians: Hilbert's tenth problem and the four-color theorem, Gaussian integers, chaotic dynamics and the Mandelbrot set, infinite numbers, and strange number systems. Why a "new golden age"? According to Keith Devlin, we are currently witnessing an astronomical amount of mathematical research. Charting the most significant developments that have taken place in mathematics since 1960, Devlin expertly describes these advances for the interested layperson and adroitly summarizes their significance as he leads the reader into the heart of the most interesting mathematical perplexities -- from the biggest known prime number to the Shimura-Taniyama conjecture for Fermat's Last Theorem.

Revised and updated to take into account dramatic developments of the 1980s and 1990s, *Mathematics: The New Golden Age* includes, in addition to Fermat's Last Theorem, major new sections on knots and topology, and the mathematics of the physical universe.

Devlin portrays mathematics not as a collection of procedures for solving problems, but as a unified part of human culture, as part of mankind's eternal quest to understand ourselves and the world in which we live. Though a genuine science, mathematics has strong artistic elements as well; this creativity is in evidence here as Devlin shows what mathematicians do -- and reveals that it has little to do with numbers and arithmetic. This book brilliantly captures the fascinating new age of mathematics.

(retrieved from Amazon Mon, 30 Sep 2013 13:57:23 -0400)

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This is perhaps best illustrated by considering the 4-colour problem, one of those to which Devlin devotes a chapter. It's one that pops up in many, many popular mathematics books over the years because its description is accessible even to children but its solution eluded many of mathematics' finest minds for 150 years. Before 1976 it was an interesting unsolved problem; a popular exposition might describe both the unsuccessful attempts of the past and speculate about how it might be solved in the future. After 1976, that question was settled. The problem had been solved by an unexpected means of attack - an exhaustive enumeration of 10,000 or so special cases and a mixture of hand and computer analysis which proved both that this set was complete and that each case consisted of an 'irreducible configuration.' It was a proof of a different and unexpected kind and some would argue that it transformed mathematics. Devlin was writing after this proof and so was able to consider its effect on mathematics more widely with nearly 10 years of hindsight. The same is not true of Fermat's Last Theorem, another of the topics he tackles. Unsolved at the time of writing, it was finally dealt with by Andrew Weil some 10 years later. Devlin make one successful prediction regarding the proof - he says that "if one is ever found, it will involve a great deal more than elementary considerations."

Enough about historical perspective. In addition to the two topics mentioned, Devlin also covers prime numbers and factoring, infinite sets and undecidable propositions, the class number problem, chaos theory and fractals, simple groups, Hilbert's tenth problem, knots, algorithmic efficiency and a collection of 'hard problems' in complex numbers including the Riemann hypothesis. You don't need to recognise all of these topics for this book to be interesting and accessible. But I suspect that if you have not heard of any of them you'll find the book very hard going. Devlin tries not to assume much knowledge on the part of the reader - he gives an explanation of complex numbers in chapter 3, for instance - but he does assume a familiarity with some basics of algebra and elements of mathematical notation. In some chapters he also moves rapidly from these basic explanations to some challenging concepts, a number of which defeated me on first reading despite having a degree in the subject.

These moments are rare, however. Overall this book does a good job of explaining the history of the problems discussed and describing many aspects of them which will be new even to those who may have encountered the topics in many similar books. It was new to me, for instance, to discover that the 4-space manifold has unique properties with regard to differentiation, something which has significant impact on much of theoretical physics given that this manifold describes space-time. The existence and accuracy of Heawood's formula (which places an upper limit on the number of colours needed for a map on a surface of a particular genus) was also new to me, as was its accuracy for every surface except the Klein Bottle.

Further reading is provided at the end of every chapter should you wish to investigate any of the topics in more depth. The book also has both an author index and subject index, an unusual but helpful division. Worth reading by the aspiring student of mathematics, those like me who studied it but moved elsewhere and those with interest and ability in the topic but without formal education beyond secondary school. ( )