Excursions in the History of Mathematics PDF
Preview Excursions in the History of Mathematics
Israel 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 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper www.birkhauser-science.com 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.
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