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Visions of Infinity: The Great Mathematical Problems

Basic Books

Publication date: March 2013

ISBN: 9780465065998

Digital Book format: ePub (Adobe DRM)

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There are 250,000 research mathematicians at work today, each of them grappling with the unknowns that lie beyond the frontiers of mathematics. Some of these problems are new, while others have puzzled and bewitched thinkers across the ages. It is one of the great wonders of mathematics that, for every proposition mathematicians successfully complete, there is another waiting to perplex and galvanize them. Such mathematical challenges offer a tantalizing glimpse of the unlimited promise and potential that mathematics holds, and keep mathematicians looking toward the horizons of intellectual possibility.

In [Title TK] , celebrated writer and mathematician Ian Stewart provides a fascinating overview of the most formidable problems mathematicians have vanquished, and those that vex them still. Fermat’s Last Theorem, for example, was first posited in 1630 by Pierre de Fermat, who claimed that it was impossible to write a perfect cube as the sum of two cubes, or indeed to do so for any power higher than the second power. Although Fermat claimed to have proved the theorem, however, no evidence was found, and the search for a proof plagued mathematicians for the next three and a half centuries. Their efforts had a major impact on the development of mathematics, and led to the creation of algebraic number theory and complex analysis.

Fermat’s Last Theorem was finally proved by Andrew Wiles in 1995, but many of the problems Stewart describes have never been solved. Indeed, even as mathematicians slowly chip away at their layers, bearing down on the elusive proof, some problems continue to baffle us. Indeed, in 2000 the Clay Mathematics Institute of Cambridge, Massachusetts established seven Millennium Prize problems (each of which carries a prize of one million dollars) to mark the most difficult problems with which mathematicians were grappling. Only onethe Poincaré Conjecturehas been resolved since, and even then, nearly a century passed between its formulation in 1904 and its solution by Grigori Perelman in 2002 and 2003. Other prize problems include the Riemann Hypothesis, which Stewart refers to as the Holy Grail of pure mathematics” and which deals with the frequency of prime numbers. It was posited by German mathematician Bernhard Riemann in 1859, and its proof would unlock mysteries of algebraic number theory and primality tests. The more recent P/NP problem, meanwhile, was formulated in 1971 by Stephen Cook and Leonid Levin, and straddles mathematics and computer science. It asks whether problems whose solutions can be efficiently verified by a computer can also be efficiently solved by a computer. As Stewart explains, mathematicians are nowhere near to resolving the P/NP problem, and it could easily remain unproved for another hundred years.An approachable and illuminating guide to fourteen of history’s greatest mathematical problems, [Title TK] reveals how mathematicians the world over are rising to the challenges set by their predecessors, bringing about astonishing breakthroughs in the field and profoundly shaping science, technology, and human understanding in the process.PROBLEMS IN [TITLE TK]: Goldbach Conjecture Squaring the Circle Four Color Theorem Kepler Conjecture Mordell Conjecture Fermat’s Last Theorem Three-Body Problem Riemann Hypothesis Poincaré Conjecture P/NP Problem Navier-Stokes Equation Mass Gap Hypothesis BirchSwinnerton-Dyer Conjecture Hodge Conjecture