Visions of Infinity

December 10, 2012
Popular mathematics writer and researcher Stewart (The Mathematics of Life) delivers an entertaining history of mathematics and a fresh look at some of the most challenging problems and puzzles in the history of the field. The usual suspects are all present and accounted for, including the infamous algebraic muddle of Fermat’s Last Theorem, the quintessential prime number puzzler of the Goldbach Conjecture, the cartographical conundrum of the Four-Colour Theorem, and the topological intricacies of the Poincaré Conjecture, as well as some fascinatingly cryptic modern ones. An emeritus professor of mathematics at the University of Warwick, Stewart proceeds chronologically, offering historical insights as he discusses the multiple disciplines touched on by each problem and the decades or centuries during which obsessive mathematicians have searched for their solutions. Stewart’s loquacious yet lucid style makes the most complex mathematics accessible, even when discussing esoteric concepts like homology (used to measure and categorize topological surfaces) or the quantum physics behind the still-unsolved Mass Gap Hypothesis. Capping the discussion is a quick chapter detailing some of the problems that may give mathematicians fits and nightmares into the next century, including quaintly named perfect cuboids, Langton’s Ant, and mysterious constructs called Thrackles. Once again, Stewart delivers an intriguing book that rewards random reading as much as dedicated study. Agent: George Lucas, Inkwell Management.

January 1, 2013
An aggressively unsimplified account of 14 great problems, emphasizing how mathematicians approached but did not always solve them. Fermat's Last Theorem, 350 years old and solved by Andrew Wiles in 1995, produced headlines because laymen were amazed that mathematicians could make new discoveries. In fact, mathematics is as creative as physics, writes prolific popularizer Stewart (Mathematics Emeritus/Univ. of Warwick; The Mathematics of Life, 2011): "Mathematics is newer, and more diverse, than most of us imagine." Goldbach's Conjecture--that every even number can be written as the sum of two prime numbers (250 years old, probably true but not proven)--provides the background for a chapter on the unruly field of prime numbers: those divisible only by one and itself (3, 5, 7, 11, 13...). Squaring the Circle--constructing a square with an area identical to a given circle (2,500 years old; proven impossible)--introduces pi. Schoolchildren learn that pi is the ratio of the circumference of a circle to its diameter, but it's a deeply important number that turns up everywhere in mathematics. Most readers know that Newton's laws precisely predict motions of two bodies, but few know that they flop with three. The Three-body Problem (330 years old, unsolved) continues to worry astronomers since it hints that gravitational forces among three or more bodies may be unstable, so the planets may eventually fly off. Stewart's imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating many but not all of some very difficult ideas. It will enchant math enthusiasts as well as general readers who pay close attention.

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Starred review from February 15, 2013
Few of us share Stewart's mathematical skills. But we relish the intellectual stimulation of joining him in exploring mathematical problems that have pushed even genius to the limit. We thrill, for instance, to the ingenuity of a great Chinese mathematician coming tantalizingly close to proving the centuries-old Goldbach Conjecture. And we feel the human meaning of mathematical achievement when a triumphant British analyst weeps before television cameras after finally proving a seventeenth-century algebraic theorem. We feel that meaning again when a brilliant Russian mathematician retreats into reclusive isolation, distressed because of initial skepticism toward his groundbreaking work on a nineteenth-century riddle. But high-level mathematics stirs deep emotions largely because it taps into the mind's deepest impulses. Stewart repeatedly shows how a trivial mathematical curiosity can open up vital new conceptual insights. Readers learn, for example, that the apparently inconsequential four-color problem has led investigators deep into theoretical physics and has compelled fundamental rethinking of what constitutes a mathematical proof in a computerized age. Proofs incorporating computer-generated calculations may strike old-school mathematicians as unsatisfying, but Stewart assures readers that mathematics still depends on human investigators and that such investigators will not soon run out of daunting mathematical problems. A bracing mental workout for amateur mathematicians.(Reprinted with permission of Booklist, copyright 2013, American Library Association.)