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  WHY BEAUTY IS TRUTH

  WHY BEAUTY IS TRUTH

  A History of Symmetry

  IAN STEWART

  A Member of the Perseus Books Group

  New York

  Copyright © 2007 by Joat Enterprises

  Published by Basic Books

  A Member of the Perseus Books Group

  All rights reserved. Printed in the United States of America. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address Basic Books, 387 Park Avenue South, New York, NY 10016-8810.

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  Designed by Jeff Williams

  Library of Congress Cataloging-in-Publication Data

  Stewart, Ian, 1945-

  Why beauty is truth : a history of symmetry / Ian Stewart.

  p. cm.

  ISBN-13: 978-0-465-08236-0

  ISBN-10: 0-465-08236-X

  1. Symmetry—History. I. Title.

  Q172.5.S95S744 2007

  539.7'25—dc22

  2006038274

  10 9 8 7 6 5 4 3 2 1

  When old age shall this generation waste,

  Thou shalt remain, in midst of other woe

  Than ours, a friend to man, to whom thou say’st,

  “Beauty is truth, truth beauty,”—that is all

  Ye know on earth, and all ye need to know.

  —JOHN KEATS, Ode on a Grecian Urn

  CONTENTS

  Preface

  1 The Scribes of Babylon

  2 The Household Name

  3 The Persian Poet

  4 The Gambling Scholar

  5 The Cunning Fox

  6 The Frustrated Doctor and the Sickly Genius

  7 The Luckless Revolutionary

  8 The Mediocre Engineer and the Transcendent Professor

  9 The Drunken Vandal

  10 The Would-Be Soldier and the Weakly Bookworm

  11 The Clerk from the Patent Office

  12 A Quantum Quintet

  13 The Five-Dimensional Man

  14 The Political Journalist

  15 A Muddle of Mathematicians

  16 Seekers after Truth and Beauty

  Further Reading

  Index

  PREFACE

  The date is 13 May 1832. In the dawn mist, two young Frenchmen face each other, pistols drawn, in a duel over a young woman. A shot is fired; one of the men falls to the ground, fatally wounded. He dies two weeks later, from peritonitis, aged 21, and is buried in the common ditch—an unmarked grave. One of the most important ideas in the history of mathematics and science very nearly dies with him.

  The surviving duelist remains unknown; the one who died was Évariste Galois, a political revolutionary and a mathematical obsessive whose collected works fill a mere sixty pages. Yet Galois left a legacy that revolutionized mathematics. He invented a language to describe symmetry in mathematical structures, and to deduce its consequences.

  Today that language, known as “group theory,” is in use throughout pure and applied mathematics, where it governs the formation of patterns in the natural world. Symmetry also plays a central role at the frontiers of physics, in the quantum world of the very small and the relativistic world of the very large. It may even provide a route to the long-sought “Theory of Everything,” a mathematical unification of those two key branches of modern physics. And it all began with a simple question in algebra, about the solutions of mathematical equations—finding an “unknown” number from a few mathematical clues.

  Symmetry is not a number or a shape, but a special kind of transformation—a way to move an object. If the object looks the same after being transformed, then the transformation concerned is a symmetry. For instance, a square looks the same if it is rotated through a right angle.

  This idea, much extended and embellished, is fundamental to today’s scientific understanding of the universe and its origins. At the heart of Albert Einstein’s theory of relativity lies the principle that the laws of physics should be the same in all places and at all times. That is, the laws should be symmetric with respect to motion in space and the passage of time. Quantum physics tells us that everything in the universe is built from a collection of very tiny “fundamental” particles. The behavior of these particles is governed by mathematical equations—“laws of nature”—and those laws again possess symmetry. Particles can be transformed mathematically into quite different particles, and these transformations also leave the laws of physics unchanged.

  These concepts, and more recent ones at the frontiers of today’s physics, could not have been discovered without a deep mathematical understanding of symmetry. This understanding came from pure mathematics; its role in physics emerged later. Extraordinarily useful ideas can arise from purely abstract considerations—something that the physicist Eugene Wigner referred to as “the unreasonable effectiveness of mathematics in the natural sciences.” With mathematics, we sometimes seem to get more out than we put in.

  Starting with the scribes of ancient Babylon and ending with the physicists of the twenty-first century, Why Beauty Is Truth tells how mathematicians stumbled upon the concept of symmetry, and how an apparently useless search for what turned out to be an impossible formula opened a new window on the universe and revolutionized science and mathematics. More broadly, the story of symmetry illustrates how the cultural influence and historical continuity of big ideas can be brought into sharp relief by occasional upheavals, both political and scientific.

  The first half of the book may seem at first sight to have nothing to do with symmetry and precious little to do with the natural world. The reason is that symmetry did not become a dominant idea by the route one might expect, through geometry. Instead, the profoundly beautiful and indispensable concept of symmetry that mathematicians and physicists use today arrived via algebra. Much of this book, therefore, describes the search for solutions of algebraic equations. This may sound technical, but the quest is a gripping one, and many of the key players led unusual and dramatic lives. Mathematicians are human, even though they are often lost in abstract thought. Some of them may let logic rule their lives too much, but we shall see time and again that our heroes could in fact be all too human. We will see how they lived and died, read of their love affairs and duels, vicious priority disputes, sex scandals, drunkenness, and disease, and along the way we will see how their mathematical ideas unfolded and changed our world.

  Beginning in the tenth century BCE and reaching its climax with Galois in the early nineteenth century, the narrative retraces the step-by-step conquest of equations—a process that eventually ground to a halt when mathematicians tried to conquer the so-called “quintic” equation, involving the fifth power of the unknown. Did the methods break down because there was something fundamentally different about the quintic equation? Or might there be similar, yet more powerful methods that would lead to formulas for its solution? Were mathematicians stuck because of a genuine obstacle, or were they just being obtuse?

  It is important to understand that solutions to quintic equations were known to exist. The question was, can they always be represented by an algebraic formula? In 1821 the young Norwegian Niels Henrik Abel proved that the quintic equation cannot be solved by algebraic means. His proof, however, was rather mysterious and indirect. It proved that no general solution is possible,
but it did not really explain why.

  It was Galois who discovered that the impossibility of solving the quintic stems from the symmetries of the equation. If those symmetries pass the Galois test, so to speak—meaning that they fit together in a very specific way, which I will not explain just yet—then the equation can be solved by an algebraic formula. If the symmetries do not pass the Galois test, then no such formula exists.

  The general quintic equation cannot be solved by a formula because it has the wrong kind of symmetries.

  This epic discovery created the second theme of this book: that of a group—a mathematical “calculus of symmetry.” Galois took an ancient mathematical tradition, algebra, and reinvented it as a tool for the study of symmetry.

  At this stage of the book, words like “group” are unexplained jargon. When the meaning of such words becomes important to the story, I will explain them. But sometimes we just need a convenient term to keep track of various items of baggage. If you run into something that looks like jargon but is not immediately discussed, then it will be playing the role of a useful label, and the actual meaning won’t matter very much. Sometimes the meaning will emerge anyway if you keep reading. “Group” is a case in point, but we won’t find out what it means until the middle of the book.

  Our story also touches upon the curious significance of particular numbers in mathematics. I am not referring to the fundamental constants of physics but to mathematical constants like π (the Greek letter pi). The speed of light, for instance, might in principle be anything, but it happens to be 186,000 miles per second in our universe. On the other hand, π is slightly larger than 3.14159, and nothing in the world can change that value.

  The unsolvability of the quintic equation tells us that like π, the number 5 is also very unusual. It is the smallest number for which the associated symmetry group fails the Galois test. Another curious example concerns the sequence of numbers 1, 2, 4, 8. Mathematicians discovered a series of extensions of the ordinary “real” number concept to complex numbers and then to things called quaternions and octonions. These are constructed from two copies of the real numbers, four copies, and eight copies, respectively. What comes next? A natural guess is 16, but in fact there are no further sensible extensions of the number system. This fact is remarkable and deep. It tells us that there is something special about the number 8, not in any superficial sense, but in terms of the underlying structure of mathematics itself.

  In addition to 5 and 8, this book features appearances by several other numbers, most notably 14, 52, 78, 133, and 248. These curious numbers are the dimensions of the five “exceptional Lie groups,” and their influence pervades the whole of mathematics and much mathematical physics. They are key characters in the mathematical drama, while other numbers, seemingly little different, are mere bit players.

  Mathematicians discovered just how special these numbers are when modern abstract algebra came into being at the end of the nineteenth century. What counts is not the numbers themselves but the role they play in the foundations of algebra. Associated with each of these numbers is a mathematical object called a Lie group with unique and remarkable properties. These groups play a fundamental role in modern physics, and they appear to be related to the deep structure of space, time, and matter.

  That leads to our final theme: fundamental physics. Physicists have long wondered why space has three dimensions and time one—why we live in a four-dimensional space-time. The theory of superstrings, the most recent attempt to unify the whole of physics into a single coherent set of laws, has led physicists to wonder whether space-time might have extra “hidden” dimensions. This may sound like a ridiculous idea, but it has good historical precedents. The presence of additional dimensions is probably the least objectionable feature of superstring theory.

  A far more controversial feature is the belief that formulating a new theory of space and time depends mainly on the mathematics of relativity and quantum theory, the two pillars on which modern physics rests. Unifying these mutually contradictory theories is thought to be a mathematical exercise rather than a process requiring new and revolutionary experiments. Mathematical beauty is expected to be a prerequisite for physical truth. This could be a dangerous assumption. It is important not to lose sight of the physical world, and whatever theory finally emerges from today’s deliberations cannot be exempt from comparison with experiments and observations, however strong its mathematical pedigree.

  At the moment, though, there are good reasons for taking the mathematical approach. One is that until a really convincing combined theory is formulated, no one knows what experiments to perform. Another is that mathematical symmetry plays a fundamental role in both relativity and quantum theory, two subjects where common ground is in short supply, so we should value whatever bits of it we can find. The possible structures of space, time, and matter are determined by their symmetries, and some of the most important possibilities seem to be associated with exceptional structures in algebra. Space-time may have the properties it has because mathematics permits only a short list of special forms. If so, it makes sense to look at the mathematics.

  Why does the universe seem to be so mathematical? Various answers have been proposed, but I find none of them very convincing. The symmetrical relation between mathematical ideas and the physical world, like the symmetry between our sense of beauty and the most profoundly important mathematical forms, is a deep and possibly unsolvable mystery. None of us can say why beauty is truth, and truth beauty. We can only contemplate the infinite complexity of the relationship.

  1

  THE SCRIBES OF BABYLON

  Across the region that today we call Iraq run two of the most famous rivers in the world, and the remarkable civilizations that arose there owed their existence to those rivers. Rising in the mountains of eastern Turkey, the rivers traverse hundreds of miles of fertile plains, and merge into a single waterway whose mouth opens into the Persian Gulf. To the southwest they are bounded by the dry desert lands of the Arabian plateau; to the northeast by the inhospitable ranges of the Anti-Taurus and Zagros Mountains. The rivers are the Tigris and the Euphrates, and four thousand years ago they followed much the same routes as they do today, through what were then the ancient lands of Assyria, Akkad, and Sumer.

  To archaeologists, the region between the Tigris and Euphrates is known as Mesopotamia, Greek for “between the rivers.” This region is often referred to, with justice, as the cradle of civilization. The rivers brought water to the plains, and the water made the plains fertile. Abundant plant life attracted herds of sheep and deer, which in turn attracted predators, among them human hunters. The plains of Mesopotamia were a Garden of Eden for hunter-gatherers, a magnet for nomadic tribes.

  In fact, they were so fertile that the hunter-gatherer lifestyle eventually became obsolete, giving way to a far more effective strategy for obtaining food. Around 9000 BCE, the neighboring hills of the Fertile Crescent, a little to the north, bore witness to the birth of a revolutionary technology: agriculture. Two fundamental changes in human society followed hard on its heels: the need to remain in one location in order to tend the crops, and the possibility of supporting large populations. This combination led to the invention of the city, and in Mesopotamia we can still find archaeological remains of some of the earliest of the world’s great city-states: Nineveh, Nimrud, Nippur, Uruk, Lagash, Eridu, Ur, and above all, Babylon, land of the Hanging Gardens and the Tower of Babel. Here, four millennia ago, the agricultural revolution led inevitably to an organized society, with all the associated trappings of government, bureaucracy, and military power. Between 2000 and 500 BCE the civilization that is loosely termed “Babylonian” flourished on the banks of the Euphrates. It is named for its capital city, but in the broad sense the term “Babylonian” includes Sumerian and Akkadian cultures. In fact, the first known mention of Babylon occurs on a clay tablet of Sargon of Akkad, dating from around 2250 BCE, although the origin of the Babylonian people probably goes ba
ck another two or three thousand years.

  We know very little about the origins of “civilization”—a word that literally refers to the organization of people into settled societies. Nevertheless, it seems that we owe many aspects of our present world to the ancient Babylonians. In particular, they were expert astronomers, and the twelve constellations of the zodiac and the 360 degrees in a circle can be traced back to them, along with our sixty-second minute and our sixty-minute hour. The Babylonians needed such units of measurement to practice astronomy, and accordingly had to become experts in the time-honored handmaiden of astronomy: mathematics.

  Like us, they learned their mathematics at school.

  “What’s the lesson today?” Nabu asked, setting his packed lunch down beside his seat. His mother always made sure he had plenty of bread and meat—usually goat. Sometimes she put a piece of cheese in for variety.

  “Math,” his friend Gamesh replied gloomily. “Why couldn’t it be law? I can do law.”

  Nabu, who was good at mathematics, could never quite grasp why his fellow students all found it so difficult. “Don’t you find it boring, Gamesh, copying all those stock legal phrases and learning them by heart?”

  Gamesh, whose strengths were stubborn persistence and a good memory, laughed. “No, it’s easy. You don’t have to think.”

  “That’s precisely why I find it boring,” his friend said, “whereas math—”

  “—is impossible,” Humbaba joined in, having just arrived at the Tablet House, late as usual. “I mean, Nabu, what am I supposed to do with this?” He gestured at a homework problem on his tablet. “I multiply a number by itself and add twice the number. The result is 24. What is the number?”

  “Four,” said Nabu.

  “Really?” asked Gamesh. Humbaba said, “Yes, I know, but how do you get that?”