A Brief Guide to the Great Equations Read online

  Robert P. Crease is a professor in and chairman of the Department of Philosophy at Stony Brook University in New York, and historian at Brookhaven National Laboratory. He writes a monthly column, ‘Critical Point’, for Physics World magazine. His previous books include The Prism and the Pendulum: The Ten Most Beautiful Experiments in Science; Making Physics: A Biography of Brookhaven National Laboratory; The Play of Nature: Experimentation as Performance; The Second Creation: Makers of the Revolution in Twentieth-Century Physics (with Charles C. Mann); and Peace & War: Reminiscences of a Life on the Frontiers of Science (with Robert Serber). Crease’s translations include American Philosophy of Technology: The Empirical Turn and What Things Do: Philosophical Reflections on Technology, Agency, and Design. He lectures widely, and his articles and reviews have appeared in The Atlantic Monthly, The New York Times Magazine, The Wall Street Journal, Science, New Scientist, American Scientist, Smithsonian, and elsewhere. He lives in New York City.

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  A BRIEF GUIDE TO

  THE GREAT

  EQUATIONS

  The hunt for cosmic beauty in numbers

  Robert P. Crease

  Constable & Robinson Ltd

  55–56 Russell Square

  London WC1B 4HP

  www.constablerobinson.com

  First published in the USA 2008 as The Great Equations:

  Breakthroughs in Science from Pythagoras to Heisenberg

  by W.W Norton & Company, Inc.,

  500 Fifth Avenue, New York, NY 10110

  This edition published by Robinson, an imprint of Constable & Robinson Ltd, 2009

  Copyright © Robert P. Crease 2008, 2009

  The right of Robert P. Crease to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act, 1988.

  All rights reserved. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out or otherwise circulated in any form of binding or cover other than that in which it is published and without a similar condition being imposed on the subsequent purchaser.

  A copy of the British Library Cataloguing in Publication Data is available from the British Library.

  ISBN: 978-1-84529-281-2

  eISBN: 978-1-47210-017-7

  1 3 5 7 9 8 6 4 2

  Printed and bound in the EU

  For Stephanie,

  beyond category

  CONTENTS

  Illustration Credits

  Introduction

  1 ‘The Basis of Civilization’: The Pythagorean Theorem

  Interlude: Rules, Proofs, and the Magic of Mathematics

  2 ‘The Soul of Classical Mechanics’: Newton’s Second Law of Motion

  Interlude: The Book of Nature

  3 ‘The High Point of the Scientific Revolution’: Newton’s Law of Universal Gravitation

  Interlude: That Apple

  4 ‘The Gold Standard for Mathematical Beauty’: Euler’s Equation

  Interlude: Equations as Icons

  5 The Scientific Equivalent of Shakespeare: The Second Law of Thermodynamics

  Interlude: The Science of Impossibility

  6 ‘The Most Significant Event of the Nineteenth Century’: Maxwell’s Equations

  Interlude: Overcoming Anosognosia; or Restoring the Vitality of the Humanities

  7 Celebrity Equation: E = mc2

  Interlude: Crazy Ideas

  8 The Golden Egg: Einstein’s Equation for General Relativity

  Interlude: Science Critics

  9 ‘The Basic Equation of Quantum Theory’: Schrödinger’s Equation

  Interlude: The Double Consciousness of Scientists

  10 Living with Uncertainty: The Heisenberg Uncertainty Principle

  Interlude: The Yogi and the Quantum

  Conclusion: Bringing the Strange Home

  Notes

  Acknowledgements

  Index

  ILLUSTRATION CREDITS

  Excerpts from the column, ‘Critical Point’, by Robert P. Crease, are reprinted from Physics World with the kind permission of the publisher.

  Page 24: Columbia University.

  Pages 25, 28, 30, 39, 73, 99, 101, 104, 201: John McAusland.

  Page 43: Robert P. Crease.

  Page 58: AIP Galilei Galileo A9.

  Page 59: AIP Newton Isaac A6.

  Page 83: AIP Newton Isaac H5.

  Page 94: AIP Euler A1.

  Page 112: Boltzmann Ludwig A4; AIP Clausius Rudolf A3; AIP Helmoltz Hermann A2; AIP Carnot Sadi A1; AIP Joule James A3; AIP Maxwell James Cleak A5; AIP Rumford Benjamin H2; AIP Emilio Segre Visual Archives, E. Scott Barr Collection; AIP Kelvin William Thomson A16; AIP Planck Max A14; Wien, Wilhelm A1.

  Page 134: AIP Maxwell James Clerk A5.

  Page 140: Maxwell, James Clerk. A Treatise on Electricity and Magnetism, vol. 1. Oxford: 1873.

  Page 157: Sidney Harris.

  Page 167: Physics Today, January 2006.

  Page 179: Time magazine.

  Page 221: AIP Schrodinger, Erwin A10.

  Page 237: Heisenberg Werner A15.

  INTRODUCTION

  The first equation that most of us learn is a synonym for simplicity:

  1 + 1 = 2

  So elementary, yet so powerful! It imparts the very definition of addition: one unit plus one unit equals two units. It is powerful, too, because it exhibits the format for every other equation: in arithmetic, mathematics as a whole, physics, and other branches of science. It shows an arrangement of terms that asserts a particular kind of relationship among them. This little but fundamental equation opens so many doors that it seems like a magic wand. It is virtually the entrée into knowledge itself – the first little step, the basis for each of thousands of steps to follow. Richard Harrison, a poet and English teacher at Mount Royal College in Calgary, Canada, once wrote to me of this profoun
d expression:

  1 + 1 = 2 is the fairy tale of mathematics, the first equation I taught my son, the first expression of the miraculous power of the mind to change the real world. I remember my son holding up the index finger – the ‘one finger’– of each hand as he learned the expression, and the moment of wonder, perhaps his first of true philosophical wonder, when he saw that the two fingers, separated by his whole body, could be joined in a single concept in his mind... [W]hen I saw my son’s mind open in understanding that ‘1 + 1’ was more than ‘1 + 1’ I saw that small equation as my child’s key not to what was wonderful in the outside world, but what was wonderful in him and all of us.

  Harrison’s description reminds us that learning an equation, at least of the kind as fundamental as 1 + 1, is in effect a kind of journey. It is a journey that takes place in three stages. We begin naively without knowing the equation. We are led by schooling or accident or curiosity or intent to comprehend it, often accompanied by dissatisfaction and frustration. Finally, the experience of learning it transforms the way we experience the world, which fills us – naturally, if sometimes only momentarily – with wonder.

  This book is about those journeys.

  The first human beings lived without equations, and had no need for them. There were no equations in the Garden of Eden, not even on the Tree of Knowledge. None were present in the Sumerian paradise Dilmun, nor in the cosmic egg in which some Chinese believe that P’an Ku hatched the world, nor in any of the other places where various divine creation myths say that the first humans dwelt. Human beings did not even have the idea of equations. That idea is a human invention, the result of our efforts to make sense of the world. Even so, human beings did not wake up one day and suddenly decide to invent equations. They acquired the need to over time, and the idea of an equation in the scientific-technical sense first appeared late in human history.

  The Latin word aequare means to make level or even. Many modern English words spring from this root, including adequate, equanimity, equality, equilibrium, egalitarian, equivalence, and equivocation. The word ‘equation’ at first simply meant a partitioning into equal groupings. The ‘equator’, for instance, is the imaginary line drawn by geographers to separate the earth in two roughly equal halves. Medieval astrologers used the word ‘equation’ to refer to their practice of arbitrarily dividing up the path followed by the sun and planets into equal areas, each allegedly governed by a different constellation.1

  Meanwhile, numbers and counting were becoming important in human life. Businessmen used them in bookkeeping, finance, and budgets; religious authorities used them for record-keeping of years, seasons, and occasions such as births, deaths, and marriages; and in government officials used them in census, and for surveying and taxes.2 This generated the need to develop symbols to stand for numbers and quantities.3 In the third century BC, the Greek mathematician Diophantus took another step, using symbols to stand for unknown quantities, and providing some rules for operating on such quantities, including subtraction and addition. He showed not only how to use symbols to describe an unknown number so that it could be determined from known numbers (what is called a determinate equation) but also how symbols could describe something with an infinite set of solutions (a Diophantine or indeterminate equation). It was still a long route to the modern notion of equations. Even Galileo and Newton express their important results – Galileo’s law of falling bodies, and Newton’s laws of motion – in the form of ratios expressed in words, not in the familiar equation form known to science students. Not until the eighteenth century did natural scientists routinely express their conclusions in the form of equations as we know them today.

  A long historical and conceptual journey was required, therefore, to write even the simplest of equations. In 1910, Alfred North Whitehead and Bertrand Russell, two of the greatest mathematicians in history, published the Principia Mathematica, a famous, three-volume systematic textbook that derives the foundations of mathematics from the ground up in a purely logical way. When does the equation 1 + 1 = 2 first make its appearance? Well over halfway through volume one!4

  Thanks to this long journey, the word ‘equation’ eventually came to have a technical meaning as part of a specially constructed language – to refer to a statement that two measurable quantities, or sets of measurable quantities, are the same. (Strictly speaking, then, statements expressing inequalities are not equations.) In this codelike constructed language, indispensable to modern mathematics and science, symbols stand for sets of other things on which various operations (addition, subtraction, multiplication and division being the simplest) can be carried out.5

  Ever since this special technical language was developed, each individual equation has had two different types of discoveries. It was originally discovered by the first person to come across it – by the person or persons who introduced it into human culture. And it is rediscovered by each person who learns it since.

  The journey to a particular equation has a different kind of setting than that of other historical turning points. The appearance of equations is not framed by bloody battlefields or by clashes of titanic political forces. Equations tend instead to emerge in quiet locations, such as studies and libraries, removed from distractions and encroachments. Maxwell wrote down his world-transforming equations in his study; Heisenberg began to piece together his on an isolated island. Such environments allow scientists to address their dissatisfactions, to explore the gnawing sense that the pieces at hand are not fitting together well and need some adjustment or the addition of something new. Scientists then can focus on some problem that often can be articulated with deceptive simplicity: What is the length of this side of a right-angled triangle? What is the strength of the force between celestial objects? How does electricity move? Can a given pair of seemingly contradictory theories be made to fit together? Does this make sense?

  When the solution comes, it seems logical and even inevitable. This work is ‘universally received’, writes Roger Cotes, who contributed a Preface to the second edition of Newton’s famous masterwork, Mathematical Principles of Natural Philosophy.6 The discoverers often feel as if they’ve stumbled across something already there. Thus equations seem like treasures, spotted in the rough by some discerning individual, plucked and examined, placed in the grand storehouse of knowledge, passed on from generation to generation. This is so convenient a way to present scientific discovery, and so useful for textbooks, that it can be called the treasure-hunt picture of knowledge. It telescopes a difficult process and leaves us with an inventor, time, and place, and often a cause or purpose. An incident or moment, such as the fall of an apple, becomes a synecdoche that crystallizes the long discovery process. Generations of scholars then earn reputations criticizing the model and complicating this picture. The treasure-hunt picture is useful for everyone!

  The treasure-hunt picture of the world, however useful, promotes the view that equations are essential features of the world, not created by human beings. And indeed, we are born into a world that already has equations that ‘we’ did not create. This is why equations sometimes appear to be not really of human origin, around long before we humans got here: On the eighth day, God created the equations, as the blueprint for His work. Or, as Galileo wrote, the Book of Nature is written in mathematical symbols.

  But each and every equation had a human genesis. It was put together by a particular person at some specific place and time who felt a need – who was dissatisfied by what was at hand – and who wanted to make sense of things or sometimes merely wanted to make something that appeared hopelessly complicated easier to understand. Sometimes this creative process is buried in antiquity, as is true of the ‘Pythagorean’ theorem, whose principle was known long before Pythagoras. Sometimes the creative process is known in detail thanks to the correspondence, drafts, and notebooks of their inventors, as is the case with equations produced by Newton and Einstein. In each case, however, the equations cannot be said to be their work alone, fo
r these scientists – even when working alone – were involved in countless dialogues with other scientists in a shared process to make sense of nature.

  When British scientist Oliver Heaviside rearranged Maxwell’s work into what is essentially their now-famous form – into the form that today is known as ‘Maxwell’s equations’ – he remarked that he was simply trying to understand Maxwell’s work more clearly. That motivation – sensing that one can express better something that one already knows, but vaguely – might be said of all inventors of equations.

  After someone does come up with a new equation about some fundamental issue – when that person has answered his dissatisfaction – it changes both us and the world. Such equations thus do not simply instruct us how to calculate something, adding new tools to the same world, but do something ‘more’, as Harrison put it. In learning 1 + 1 = 2, his son did not merely input a new data point, but became transformed, possessing a new grip on the world. But along with this new grip comes new puzzles, and new dissatisfactions.7

  Harrison’s description, finally, reminds us that equations can inspire wonder. Science is not a robotic activity in which we manoeuvre in or gaze at the world indifferently, but a form of life with a highly nuanced affective dimension. There is, of course, the celebratory, cork-popping joy that is a natural concomitant to a new discovery or achievement. But if that were the only emotion involved in science – the pleasure of making a discovery that ensures fame and fortune – it would be a sorry profession, for such moments are few and far between. Fortunately, the emotions of science are much more diverse, and thicker, than that. Doing science is accompanied by unfolding feelings at every moment – puzzlement, bafflement, curiosity, desire, the urge to find the answer, boredom that nothing is happening, frustration at getting nowhere, the thrill of being on the right track. Such feelings are always present, not deeply hidden, often overlooked, but easy to notice once we decide to pay attention to them.