Visions of Infinity: The Great Mathematical Problems

Front Cover
Basic Books, 2013 - Mathematics - 340 pages
3 Reviews
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.

  

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Review: Visions of Infinity: The Great Mathematical Problems

User Review  - Sam Pagel - Goodreads

I haven't actually finished this yet, and I'm not sure I'm going to. Although the various problems are interesting to read about, the level of knowledge required to understand this book is steep. This ... Read full review

Review: Visions of Infinity: The Great Mathematical Problems

User Review  - Ben - Goodreads

I had hoped this book would be "pop math" - that is, would describe the great mathematical problems throughout history with the level of detail and accessibility of a Malcolm Gladwell book. But it got ... Read full review

Contents

Prime territory I Goldbach Conjecture
16
The puzzle of pi I Squaring the Circle
40
Mapmaking mysteries I Four Colour Theorem
58
Sphereful symmetry I Kepler Conjecture
80
New solutions for old I Mordell Conjecture
102
Inadequate margins I Fermats Last Theorem
115
Orbital chaos I ThreeBody Problem
136
Patterns in primes I Riemann Hypothesis
153
They cant all be easy I PNP Problem
203
Quantum conundrum I Mass Gap Hypothesis
228
Diophantine dreams I BirchSWinnertonDyer Conjecture
245
Complex cycles I Hodge Conjecture
258
Where next?
277
Glossary
294
Notes
307
Index
325

What shape is a sphere? I Poincaré Conjecture
176

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About the author (2013)

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.

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