How Mathematicians Think

424 Pages·2010·1.32 MB·English

by  Byers, William

How Mathematicians Think
❋
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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,
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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
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Printedonacid-freepaper.∞
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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
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❋ 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