How Mathematicians Think ❋ This page intentionally left blank How Mathematicians Think USING AMBIGUITY, CONTRADICTION, AND PARADOX TO CREATE MATHEMATICS ❋ ILLIAM YERS W B PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright©2007byPrincetonUniversityPress PublishedbyPrincetonUniversityPress,41WilliamStreet, Princeton,NewJersey08540 IntheUnitedKingdom:PrincetonUniversityPress,3MarketPlace, Woodstock,OxfordshireOX201SY Requestsforpermissiontoreproducematerialfromthisworkshouldbesent toPermissions,PrincetonUniversityPress. AllRightsReserved LibraryofCongressCataloging-in-PublicationData Byers,William,1943– Howmathematiciansthink:usingambiguity,contradiction,and paradoxtocreatemathematics/WilliamByers. p. cm. Includesbibliographicalreferencesandindex. ISBN-13:978-0-691-12738-5(acid-freepaper) ISBN-10:0-691-12738-7(acid-freepaper) 1.Mathematicians—Psychology.2.Mathematics—Psychologicalaspects. 3.Mathematics—Philosophy.I.Title. BF456.N7B942007 510.92—dc22 2006033160 BritishLibraryCataloging-in-PublicationDataisavailable ThisbookhasbeencomposedinPalatino Printedonacid-freepaper.∞ press.princeton.edu PrintedintheUnitedStatesofAmerica 1 3 5 7 9 10 8 6 4 2 ❋ Contents ❋ Acknowledgments vii NTRODUCTION I Turning on the Light 1 SECTION I THELIGHTOFAMBIGUITY 21 HAPTER C 1 Ambiguity in Mathematics 25 HAPTER C 2 The Contradictory in Mathematics 80 HAPTER C 3 Paradoxes and Mathematics: Infinity and the Real Numbers 110 HAPTER C 4 More Paradoxes of Infinity: Geometry, Cardinality, and Beyond 146 SECTION II THELIGHTASIDEA 189 HAPTER C 5 The Idea as an Organizing Principle 193 HAPTER C 6 Ideas, Logic, and Paradox 253 HAPTER C 7 Great Ideas 284 SECTION III THELIGHTANDTHEEYEOFTHEBEHOLDER 323 HAPTER C 8 The Truth of Mathematics 327 v CONTENTS HAPTER C 9 Conclusion: Is Mathematics Algorithmic or Creative? 368 Notes 389 Bibliography 399 Index 407 vi Acknowledgments ❋ ❋ I TISAPLEASURE toacknowledgetheassistanceofmanypeople who helped and supported me in the writing of this book. First of all, thanks to my wife, Miriam, and my children, Joshua and Michele. I would like to thank David Tall and Joe Auslander, who read earlier versions of this manuscript and made many valuablesuggestions.ThankstoHershyKisilevskyforthemany stimulating discussions about mathematics, to Monique Du- montforreadingapreliminaryversionofsomechapters,andto John Seldin for his help. I am also pleased to acknowledge the helpofGiannaVenettacciforthetransferringofthediagramsto the computer and the general organization of the manuscript, and to my editor at Princeton, Vickie Kearn, for her valuable help. I appreciate the support I received from David Graham, Dean of the Faculty of Arts & Science, and to John Capobianco, Vice-Dean Research, at Concordia University. I would especially like to thank Albert Low. A fascinating man, Low is a veritable fount of strikingly original ideas about, amongotherthings,creativityanditsrelationtothehumancon- dition.Inthecourseofmanyyearsnow,wehavehadfascinating discussions over lunch about mathematics and about life. The bookowesagreatdealtothoseconversations.ItwasLow’sway oflookingatambiguitythatledmetoaskthequestion,“Isthere ambiguityinmathematics?”whichultimatelyfloweredintothis book. His faith in this project, and his encouragement, have meant a great deal to me. vii This page intentionally left blank ❋ INTRODUCTION ❋ Turning on the Light A FEW YEARS AGO the PBS program Nova featured an inter- view with Andrew Wiles. Wiles is the Princeton mathematician who gave the final resolution to what was perhaps the most fa- mous mathematicalproblem ofall time—theFermat conjecture. The solution to Fermat was Wiles’s life ambition. “When he re- vealed a proof in that summer of 1993, it came at the end of sevenyearsofdedicatedworkontheproblem,adegreeoffocus and determination that is hard to imagine.”1 He said of this pe- riod in his life, “I carried this thought in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about itwhen I went tosleep. Without distractionI would have the same thing going round and round in my mind.”2 In the Novainterview,Wilesreflectsontheprocessofdoingmathemat- ical research: Perhaps I can best describe my experience of doing mathe- matics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completelydark.Youstumblearoundbumpingintothefur- niture, but gradually you learn where each piece of furni- ture is. Finally after six months or so, you find the light switch,youturniton,andsuddenlyit’sallilluminated.You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re mo- mentary, sometimes over a period of a day or two, they are the culmination of—and couldn’t exist without—the many monthsofstumblingaroundinthedarkthatprecedethem. This is the way it is! This is what it means to do mathematics at the highest level, yet when people talk about mathematics, theelementsthatmakeupWiles’sdescriptionaremissing.What ismissingisthecreativityofmathematics—theessentialdimen- 1