Visions of infinity : the great mathematical problems

353 Pages·2013·2.18 MB·English

by  Ian Stewart

VISIONS OF INFINITY
Also by Ian Stewart
Concepts of Modern Mathematics
Game, Set, and Math
The Problems of Mathematics
Does God Play Dice?
Another Fine Math You’ve Got Me Into
Fearful Symmetry(with Martin Golubitsky)
Nature’s Numbers
From Here to Infinity
The Magical Maze
Life’s Other Secret
Flatterland
What Shape Is a Snowflake?
The Annotated Flatland
Math Hysteria
The Mayor of Uglyville’s Dilemma
Letters to a Young Mathematician
Why Beauty Is Truth
How to Cut a Cake
Taming the Infinite/The Story of Mathematics
Professor Stewart’s Cabinet of Mathematical Curiosities
Professor Stewart’s Hoard of Mathematical Treasures
Cows in the Maze
Mathematics of Life
In Pursuit of the Unknown
with Terry Pratchett and Jack Cohen
The Science of Discworld
The Science of Discworld II: The Globe
The Science of Discworld III: Darwin’s Watch
with Jack Cohen
The Collapse of Chaos
Figments of Reality
Evolving the Alien/What Does a Martian Look Like?
Wheelers(science fiction)
Heaven(science fiction)
VISIONS OF INFINITY
The Great Mathematical Problems
IAN STEWART
A Member of the Perseus Books Group
New York
Copyright © 2013 by Joat Enterprises
Published by Basic Books,
A Member of the Perseus Books Group
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Library of Congress Control Number: 2012924095
978-0-465-02240-3 (hardcover ISBN)
978-0-465-06599-8 (e-book ISBN)
Printed in Great Britain in 2013 by Profile Books Ltd.
British ISBN: 978-1-84668-1998
10 9 8 7 6 5 4 3 2 1
Contents
Preface/ix
1 Greatproblems/1
2 Primeterritory&GoldbachConjecture/16
3 Thepuzzleofpi&SquaringtheCircle/40
4 Mapmakingmysteries&FourColourTheorem/58
5 Spherefulsymmetry&KeplerConjecture/80
6 Newsolutionsforold&MordellConjecture/102
7 Inadequatemargins&Fermat’sLastTheorem/115
8 Orbitalchaos&Three-BodyProblem/136
9 Patternsinprimes&RiemannHypothesis/153
10 Whatshapeisasphere?&Poincare´ Conjecture/176
11 Theycan’tallbeeasy&P/NPProblem/203
12 Fluidthinking&Navier-StokesEquation/214
13 Quantumconundrum&MassGapHypothesis/228
14 Diophantinedreams&Birch–Swinnerton-DyerConjecture/245
15 Complexcycles&HodgeConjecture/258
16 Wherenext?/277
17 Twelveforthefuture/282
Glossary/294
Furtherreading/305
Notes/307
Index/325

We must know. We shall know.
David Hilbert
Speech about mathematical problems in 1930, on the
occasion of his honorary citizenship of Königsberg.1

Preface
M
athematicsisavast,ever-growing,ever-changingsubject.
Among the innumerable questions that mathematicians
ask,andmostlyanswer,somestandoutfromtherest:prominentpeaks
that tower over the lowly foothills. These are the really big questions,
the difficult and challenging problems that any mathematician would
give his or her right arm to solve. Some remained unanswered for
decades, some for centuries, a few for millennia. Some have yet to be
conquered. Fermat’s last theorem was an enigma for 350 years until
Andrew Wiles dispatched it after seven years of toil. The Poincaré
conjecture stayed open for over a century until it was solved by the
eccentric genius Grigori Perelman, who declined all academichonours
and a million-dollar prize for his work. The Riemann hypothesis
continues to baffle the world’s mathematicians, impenetrable as ever
after 150 years.
Visions of Infinity contains a selection of the really big questions
that have driven the mathematical enterprise in radically new
directions. It describes their origins, explains why they are
important, and places them in the context of mathematics and
science as a whole. It includes both solved and unsolved problems,
which range over more than two thousand years of mathematical
development, but its main focus is on questions that either remain
open today, or have been solved within the past fifty years.
Abasicaimofmathematicsistouncover theunderlyingsimplicity
ofapparentlycomplicatedquestions.Thismaynotalwaysbeapparent,
however, because the mathematician’s conception of ‘simple’ relies on
many technical and difficult concepts. An important feature of this
book is to emphasise the deep simplicities, and avoid – or at the very
least explain in straightforward terms – the complexities.
Mathematics is newer, and more diverse, than most of us imagine.
At a rough estimate, the world’s research mathematicians number