Table Of ContentIsrael Kleiner
Excursions in the History
of Mathematics
IsraelKleiner
DepartmentofMathematics
andStatistics
YorkUniversity
Toronto,ONM3J1P3
Canada
[email protected]
ISBN978-0-8176-8267-5 e-ISBN978-0-8176-8268-2
DOI10.1007/978-0-8176-8268-2
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011940430
Mathematics Subject Classification (2010): 00-01, 00A30, 01Axx, 01A45, 01A50, 01A55, 01A60,
01A61,01A99,03-03,11-03,26-03
(cid:2)c SpringerScienceCBusinessMedia,LLC2012
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Withallmyloveto
Nava
Ronen, Leeor, Tania,Ayelet,Avy, Tamir
Tia,Jordana,Jake, Elise
Preface
The primary audience for this book, as I see it, is teachers of mathematics. The
bookmayalsobeofinteresttomathematiciansdesiringahistoricalviewpointona
numberofthe subject’sbasictopics.Anditmayproveusefultothoseteachingor
studyingthesubject’shistory.
Thebookcomprisesfiveparts.Thefirstthree(A–C)containtenhistoricalessays
onimportanttopics:numbertheory,calculus/analysis,andproof,respectively.(The
choiceoftopicsisdictatedbymyinterestsandisbasedonarticlesIhavepublished
overthepasttwenty-fiveyears.)PartDdealswithfourhistoricallyorientedcourses,
andPartEprovidesbiographiesoffivemathematicianswhoplayedmajorrolesin
thehistoricaleventsrelatedinPartsA–D.
Each of the first three parts – on numbertheory,calculus/analysis, and proof–
beginswithasurveyoftherespectivesubject(Chaps.1,4,and7),whichisfollowed
in more depth by specialized themes. In number theory these themes deal with
Fermatas the founderofmodernnumbertheory(Chap.2) andwith Fermat’sLast
Theorem from Fermat to Wiles (Chap.3). In calculus/analysis, the special topics
describevariousaspectsofthehistoryofthefunctionconcept,whichwasintimately
relatedtodevelopmentsincalculus/analysis(Chaps.5and6).Thethemesonproof
discuss paradoxes (Chap.8) and the principle of continuity (Chap.9), and offer a
historical perspective on a very interesting debate about proof initiated in a 1993
articlebyJaffeandQuinn(Chap.10).
The four chapters in Part D (Chaps.11–14) describe courses showing how a
teacher can benefit from the historical point of view. More specifically, each of
Chaps.11–14 describes a mathematics course inspired by history. Chapters 11
and12areaboutnumbersasasourceofideasinteaching.Chapters13and14deal,
respectively,withgreatquotationsandwithfamousproblems.Moreover,Chaps.4
and6(onanalysisandonfunctions)containexplicitsuggestionsforteachers,while
suchsuggestionsareimplicitthroughoutthebook.
Mathematics was discovered/invented by mathematicians. In each of the first
14 chapters the creators of the relevant mathematics are mentioned prominently,
but because of space constraints are given shorter shrift than they deserve. I have
thereforefounditusefultosetasideachapterthatwillgiveamuchfulleraccountof
vii
viii Preface
fivemathematicianswhohaveplayedimportantrolesinthedevelopmentsthatIam
recountinginthebook.TheyareDedekind,Euler,Gauss,Hilbert,andWeierstrass
(Chap.15).Ihopethesemini-biographieswillprovetobeinstructiveandinspiring.
(My choice is limited to five by space considerations, but other mathematicians
couldjustifiablyhavebeenpicked.)
There is considerablerepetitionamongthe variouschapters.This should make
possibleindependentreadingofeachchapter.Thebookhasmanyreferences,placed
attheendofeachofitsfifteenchapters(inthecaseofChap.15,attheendofeach
ofthefivebiographies).Thereferencesaremainlytosecondarysources.Theseare,
asarule,easiertocomprehendthanprimarysources,andmorereadilyaccessible.
(Many of the secondarysources contain references to primary sources, which are
ofteninGermanorFrench.)
Ihadtwomaingoalsinwritingthisbook:
(a) Toarousemathematicsteachers’interestinthehistoryofmathematics.
(b) Toencouragemathematicsteacherswithatleastsomeknowledgeofthehistory
ofmathematicstooffercourseswithastronghistoricalcomponent.
LetmeexplainwhyIviewtheseasimportantgoals.
I come to the history of mathematics from the perspective of a mathematician
ratherthanofahistorianofmathematics.Thetwoperspectivesare,ingeneral,not
thesame.Mylongstandinginterestinthehistoryofmathematicsstemslargelyfrom
tryingtoimprovemyteachingofmathematics.
EarlyinmyteachingcareerIbecamedissatisfiedwiththeexclusivefocusonthe
formal theorem-proof mode of instruction. I admired the elegance of the logical
structure of our subject, but over time I did not find it sufficient to sustain my
enthusiasmintheclassroom,perhapsbecausemostofmystudentsdidnotsustain
theirs.
InduecourseIfoundthatthehistoryofmathematicshelpedboostmyenthusiasm
for teaching by providing me with perspective, insight, and motivation – surely
importantingredientsinthemakingofagoodteacher.Forexample,whenItaught
calculus I was able to understand where the derivative came from, and how it
evolved into the form we see in today’s textbooks; and when I taught abstract
algebra,Iwasabletounderstandhowandwhytheconceptsofringandidealcame
into being, and the source of Lagrange’stheoremaboutthe order of subgroupsof
finitegroups.
Such examplescould be multipliedendlessly. Theysupplied insightand added
a new dimension to my appreciationof mathematics. I came to realize that while
itisimportanttohavetechnicalknowledgeofmathematicalconcepts,results, and
theories, it is also important to know where they came from and why they were
studied.ThefollowingquotationfromtheprefacetoC.H.Edwards’TheHistorical
DevelopmentoftheCalculusisapt:
Although the study of the history of mathematics has an intrinsic appeal of its own, its
chiefraisond’eˆtreissurelytheilluminationofmathematicsitself.Forexample,thegradual
unfolding oftheintegral concept –from thevolumecomputations ofArchimedes tothe
intuitiveintegralsofNewtonandLeibnizandfinallythedefinitionsofCauchy,Riemann,
Preface ix
andLebesgue–cannotfailtopromoteamorematureappreciationofmoderntheoriesof
integration.
Ihopetoachievemyfirstgoal–toarousemathematicsteachers’interestinthe
historyofmathematics–byfocusinginthisbookontwoimportantareas,number
theory and analysis, and on the fundamental notion of proof, a perennially hot
topicofdiscussionamongmathematicsteachers(Chaps.1–10).Itrustthatthefive
biographieswillalsocapturethereader’sinterest(Chap.15).
Ihavefoundhistoricaldigressionstobeausefuldeviceinteachingmathematics
courses. For example, when introducing infinite sets I will give a brief history,
startingwith Zeno andculminatingwith Cantor,of howandwhythey cameto be
studied; when discussing Pythagorean triples in a first course in number theory, I
willcommentonFermat’snoteinthemarginofDiophantus’Arithmeticaaboutthe
“marvelousproof” he had of what came to be known as Fermat’s Last Theorem;
and when appropriate I will briefly recount interesting stories of mathematicians
(Archimedesand Galois come to mind). Such historical departuresfrom standard
teachingpracticeshouldconvincestudentsthatmathematicsis ahumanendeavor,
thatitshistoryisinteresting,andthatitcangivethemsomeinsightintothegrandeur
ofthesubject.
I have taught upper-level undergraduate mathematics courses with a strong
historical orientation, dealing broadly with “mathematical culture,” and a grad-
uate course in the history of mathematics – a required course in an In-Service
Master’s Program for high school teachers of mathematics. (I was fortunate that
my colleagues recognized – not without a “battle” – that these courses could be
giveninadepartmentofmathematics,andthattheyformedadesirablecomponent
in the education of budding mathematicians.) Chapters 11–14, on numbers, great
quotations, and famous problems, describe courses of the above types. Suitable
material from Chaps.1–10 can be used in these courses when appropriate. For
example,thetheme“Algebraicnumbersanddiophantineequations”inSect.11.3.4
willbenefitfrommaterialinChap.3;thetheme“Changingstandardsofrigorinthe
evolutionof mathematics,”Sect. 14.2.3,can usefullydraw uponChaps. 7 and10;
andSect.14.4,(e)and(g),willfindChap.2useful.
Butaquestionpresentsitself:Whyshouldweteachsuchcoursesinamathemat-
ics department (or for that matter, why make historical digressions)? The answer
dependsonhowweviewtheeducationofmathematicsstudents.Thesecourses(or
digressions)maynotmakestudentsintobetterresearchersortheoremprovers,but
theycanhelpmakethem“mathematicallycivilized.”
ThelastphraseisthetitleofanotebyProfessorO.ShishaintheNoticesofthe
American Mathematical Society (vol. 30, 1983, p. 603). In it he briefly discusses
what it means for students to be mathematically civilized (or cultured). Among
otherdesiderata,suchstudentswillhave“goodmathematicaltasteandjudgment,”
andwillknow“howtoexpressmathematicalideas,orallyandinwriting,correctly,
rigorously, and clearly.” We can encourage mathematical culture, according to
ProfessorShisha, by (amongotherthings) “constantlypointingoutin our courses
the historical development of the subjects, their goals and relations with other
x Preface
subjects in and outside mathematics,” and by “requiring students to take courses
inthehistoryofmathematics.”
The implementation of the second important goal of this book – to encourage
teacherstooffermathematicscourseswithastronghistoricalcomponent(orcourses
inhistorywithastrongmathematicscomponent)–shouldresultinmathematically
cultured students. Such students might be able to discuss, for example, whether
therearerevolutionsinmathematics(whatissucharevolutionanyway?),andwhat
tomakeofCantor’sdictumthat“theessenceofmathematicsliesinitsfreedom.”
The following quotation, from an editorial about mathematics teaching by the
then editorsof The MathematicalIntelligencer,B. Chandlerand H. Edwards,is a
fittingconclusiontothesecomments(vol.1,1978,p.125):
Doletustrytoteachthegeneralpublicmoreofthesortofmathematicsthattheycanusein
everydaylife,butletusnotallowthemtothink–andcertainlyletusnotslipintothinking–
thatthisisanessentialqualityofmathematics.
Thereisagreatculturaltraditiontobepreservedandenhanced.Eachnewgenerationmust
learnthetraditionanew.Letustakecarenottoeducateagenerationthatwillbedeaftothe
melodiesthatarethesubstanceofourgreatmathematicalculture.
IwanttoexpressheartfeltgratitudetomyfriendandcolleagueHardyGrantfor
his kindness,support,andassistance overthe past40 years, and, in particular,for
hishelpwiththiswork.Ofcourseallremainingerrors(ofomissionorcommission)
are solely mine;I wouldbe gratefulif theywere broughtto my attention.Finally,
IwanttothankTomGrasso,KatherineGhezzi,andJessicaBelangerofBirkha¨user
fortheiroutstandingcooperationinseeingthisbooktocompletion.
