Visions of Infinity: Great Mathematical Problems [PDF, ePub, Mobi]

It is one of the wonders of mathematics that, for every problem mathematicians solve, another awaits to perplex and galvanize them. Some of these problems are new, while others have puzzled and bewitched thinkers across the ages. Such challenges offer a tantalizing glimpse of the field’s unlimited potential, and keep mathematicians looking toward the horizons of intellectual possibility.

In Visions of Infinity, celebrated mathematician Ian Stewart provides a fascinating overview of the most formidable problems mathematicians have vanquished, and those that vex them still. He explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole.

The three-century effort to prove Fermat’s last theorem - first posited in 1630, and finally solved by Andrew Wiles in 1995 - led to the creation of algebraic number theory and complex analysis. The Poincaré conjecture, which was cracked in 2002 by the eccentric genius Grigori Perelman, has become fundamental to mathematicians’ understanding of three-dimensional shapes. But while mathematicians have made enormous advances in recent years, some problems continue to baffle us.

Indeed, the Riemann hypothesis, which Stewart refers to as the "Holy Grail of pure mathematics," and the P/NP problem, which straddles mathematics and computer science, could easily remain unproved for another hundred years.

An approachable and illuminating history of mathematics as told through fourteen of its greatest problems, Visions of Infinity reveals how mathematicians the world over are rising to the challenges set by their predecessors - and how the enigmas of the past inevitably surrender to the powerful techniques of the present.

"As a guide to the inner workings of the mathematical jungle, Stewart provides an engaging and informative experience. If you wish to intelligently discuss the Riemann hypothesis, P/NP problems or the Hodge conjecture, you ought to read this book first."
--Science News

"Anyone who has always loved math for its own sake or for the way it provides new perspectives on important real-world phenomena will find hours of brain-teasing and mind-challenging delight in the British professor’s survey of recently answered or still open mathematical questions.... Individual readers will dig deeply into certain chapters and skim others according to personal preference, but every one of them will be captivated by the technical achievements, loose ends and human insights that Stewart shares on his grand mathematical tour."
--Dallas Morning News

"Entertaining and accessible.... Ian Stewart belongs to a very small, very exclusive club of popular science and mathematics writers who are worth reading today."
--New York Journal of Books

"An entertaining history of mathematics and a fresh look at some of the most challenging problems and puzzles in the history of the field…. Stewart’s loquacious yet lucid style makes the most complex mathematics accessible, even when discussing esoteric concepts…. Once again, Stewart delivers an intriguing book that rewards random reading as much as dedicated study."
--Publishers Weekly

"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.... Stewart repeatedly shows how a trivial mathematical curiosity can open up vital new conceptual insights.... A bracing mental workout for armchair mathematicians."
--Booklist, starred review

"Stewart’s imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating…some very difficult ideas. It will enchant math enthusiasts as well as general readers who pay close attention."
--Kirkus

Ian Stewart is Emeritus Professor of Mathematics and active researcher at the University of Warwick. The author of many books on mathematics, he lives in Coventry, England.