Ian Stewart - Visions of infinity: The great mathematical problems

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 Youve Got Me Into

Fearful Symmetry (with Martin Golubitsky)

Natures Numbers

From Here to Infinity

The Magical Maze

Lifes Other Secret

Flatterland

What Shape Is a Snowflake?

The Annotated Flatland

Math Hysteria

The Mayor of Uglyvilles Dilemma

Letters to a Young Mathematician

Why Beauty Is Truth

How to Cut a Cake

Taming the Infinite/The Story of Mathematics

Professor Stewarts Cabinet of Mathematical Curiosities

Professor Stewarts 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: Darwins 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)

The Great Mathematical Problems

IAN STEWART

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Library of Congress Control Number: 2012924095

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Printed in Great Britain in 2013 by Profile Books Ltd.

10 9 8 7 6 5 4 3 2 1

We must know. We shall know.
David Hilbert

Speech about mathematical problems in 1930, on the occasion of his honorary citizenship of Knigsberg.

M athematics is a vast, ever-growing, ever-changing subject. Among the innumerable questions that mathematicians ask, and mostly answer, some stand out from the rest: prominent peaks 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. Fermats last theorem was an enigma for 350 years until Andrew Wiles dispatched it after seven years of toil. The Poincar conjecture stayed open for over a century until it was solved by the eccentric genius Grigori Perelman, who declined all academic honours and a million-dollar prize for his work. The Riemann hypothesis continues to baffle the worlds 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.

A basic aim of mathematics is to uncover the underlying simplicity of apparently complicated questions. This may not always be apparent, however, because the mathematicians 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 worlds research mathematicians number about a hundred thousand, and they produce more than two million pages of new mathematics every year. Not new numbers, which are not what the enterprise is really about. Not new sums like existing ones, but bigger though we do work out some pretty big sums. One recent piece of algebra, carried out by a team of some 25 mathematicians, was described as a calculation the size of Manhattan. That wasnt quite true, but it erred on the side of conservatism. The answer was the size of Manhattan; the calculation was a lot bigger. Thats impressive, but what matters is quality, not quantity. The Manhattan-sized calculation qualifies on both counts, because it provides valuable basic information about a symmetry group that seems to be important in quantum physics, and is definitely important in mathematics. Brilliant mathematics can occupy one line, or an encyclopaedia whatever the problem demands.

When we think of mathematics, what springs to mind is endless pages of dense symbols and formulas. However, those two million pages generally contain more words than symbols. The words are there to explain the background to the problem, the flow of the argument, the meaning of the calculations, and how it all fits into the ever-growing edifice of mathematics. As the great Carl Friedrich Gauss remarked around 1800, the essence of mathematics is notions, not notations. Ideas, not symbols. Even so, the usual language for expressing mathematical ideas is symbolic. Many published research papers do contain more symbols than words. Formulas have a precision that words cannot always match.

However, it is often possible to explain the ideas while leaving out most of the symbols. Visions of Infinity takes this as its guiding principle. It illuminates what mathematicians do, how they think, and why their subject is interesting and important. Significantly, it shows how todays mathematicians are rising to the challenges set by their predecessors, as one by one the great enigmas of the past surrender to the powerful techniques of the present, which changes the mathematics and science of the future. Mathematics ranks among humanitys greatest achievements, and its great problems solved and unsolved have guided and stimulated its astonishing power for millennia, both past and yet to come.

Coventry, June 2012

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Carles Sim. From: European Congress of Mathematics, Budapest 1996, Progress in Mathematics 168, Birkhuser, Basel.

Pablo Mininni.

University College, Cork, Ireland.

Wolfram MathWorld.

T ELEVISION PROGRAMMES ABOUT MATHEMATICS are rare, good ones rarer. One of the best, in terms of audience involvement and interest as well as content, was Fermats last theorem. The programme was produced by John Lynch for the British Broadcasting Corporations flagship popular science series Horizon in 1996. Simon Singh, who was also involved in its making, turned the story into a spectacular bestselling book. On a website, he pointed out that the programmes stunning success was a surprise: