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Yearning for the
Impossible
The Surprising Truths
of Mathematics
Second Edition
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Yearning for the
Impossible
The Surprising Truths
of Mathematics
Second Edition
John Stillwell
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CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2018 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
Version Date: 20180404
International Standard Book Number-13: 978-1-1385-9621-4 (Hardback)
International Standard Book Number-13: 978-1-1385-8610-9 (Paperback)
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ToElaine
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Preface to the Second Edition
Forthepast12yearsIhavetaughtacoursebasedonthisbookatthe
UniversityofSanFrancisco. Itisafirst-yearseminar,intendedmainly
forstudentsinnon-mathematicalcourses. Inkeepingwiththetheme
ofthebook,theseminarisentitled"MathematicsandtheImpossible,"
andithasbeenpitchedasfollows:
This course is a novel introduction to mathematics and
its history. It puts the difficulties of the subject upfront
byenthusiasticallytacklingthemostimportantones: the
seeminglyimpossibleconceptsofirrationalandimaginary
numbers, thefourthdimension, curvedspace, andinfin-
ity. Similar"impossibilities"ariseinmusic,art,literature,
philosophy,andphysics—aswewillsee—butmathhasthe
precisiontoseparateactualimpossibilitiesfromthosethat
are merely apparent. In fact, "impossibility" has always
beenaspurtothecreativityofmathematicians,andama-
jor influence on the development of math. By focusing
reason and imagination on several apparent impossibili-
ties,thecourseaimstoshowinterestingmathtostudents
whose major may be in another field, and to widen the
horizons of math students whose other courses are nec-
essarilyrathernarrowlyfocused.
Thus the aim of the seminar is to introduce students to the ideas of
mathematics, rather than to drill them in mathematical techniques.
But(asallmathematiciansknow)mathematicsisnotaspectatorsport.
Soithasbeenmypracticetogiveexercisesalongsideclassroomdiscus-
sionofthebook,toensurethatstudentsgrapplewithinterestingideas
for themselves. Since other teachers may also decide to give courses
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viii
basedonthebook,Ihavedecidedtoproducethissecondedition,which
includestheexercisesIhaveusedwithmyclassandmanymore.
Theexercisesaredistributedinsmallbatchesaftermostsections,
andabatchshouldbeattemptedbystudentsafterthecorresponding
section has been discussed in class. Some of the exercises are quite
routine—intendedtoreinforcetheideasjustdiscussed—butotherex-
ercisesservetoextendtheideasanddevelopthemininterestingdirec-
tions. Therearealsoexercisesthatfillgapsbysupplyingproofsofre-
sultsclaimedwithoutproofinthetext,orbyansweringquestionsthat
are likely to arise. The more advanced exercises are accompanied by
commentarythatexplainstheircontextandbackground. Inthisway
Ihavebeenabletocoverseveraltopicsthatwillbeofinteresttomore
ambitiousreaders.
Ofcourse,itisbynomeansnecessarytousethebookinthecontext
ofacourse.Ihopethatitwillcontinuetobeusedforrecreationalread-
ingorself-study,andthateventhecasualreaderwillbetemptedtotry
someoftheexercises.However,forthoseteacherswhohavewishedto
giveacoursebasedonthebook,Ihopethatteachingfromithasnow
becomemucheasier.
JohnStillwell
SanFrancisco
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Preface
The germ of this book was an article I called (somewhat tongue-in-
cheek)“MathematicsAcceptstheImpossible.”IwroteitfortheMonash
University magazine Function in 1984 and its main aim was to show
thatthe“impossible”figureshownabove(thePenrosetribar)isactu-
ally not impossible. The tribar exists in a perfectly reasonable space,
differentfromtheonewethinkwelivein,butneverthelessmeaningful
andknowntomathematicians. Withthisexample,Ihopedtoshowa
generalaudiencethatmathematicsisadisciplinethatdemandsimag-
ination,perhapsevenfantasy.
Therearemanyinstancesofapparentimpossibilitiesthatareim-
portanttomathematics, andothermathematicianshavebeenstruck
bythisphenomenon. Forexample,PhilipDaviswroteinTheMathe-
maticsofMatrices of1965:
Itisparadoxicalthatwhilemathematicshasthereputation
ofbeingtheonesubjectthatbrooksnocontradictions,in
realityithasalonghistoryofsuccessfullylivingwithcon-
tradictions. Thisisbestseenintheextensionsoftheno-
tionofnumberthathavebeenmadeoveraperiodof2500
years ... eachextension,initsway,overcameacontradic-
torysetofdemands.
Mathematical language is littered with pejorative and mystical terms
suchasirrational,imaginary,surd,transcendentalthatwereonceused
toridiculesupposedlyimpossibleobjects.Andthesearejusttermsap-
pliedtonumbers.Geometryalsohasmanyconceptsthatseemimpos-
sible to most people, such as the fourth dimension, finite universes,
andcurvedspace—yetgeometers(andphysicists)cannotdowithout
them.Thusthereisnodoubtthatmathematicsflirtswiththeimpossi-
ble,andseemstomakeprogressbydoingso.
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