Who Gave you the Epsilon?: & Other Tales of Mathematical History PDF
Preview Who Gave you the Epsilon?: & Other Tales of Mathematical History
P1:JZP/... P2:JZP fm MABK005/Anderson February2,2009 16:5 Who Gave You the Epsilon? and Other Tales of Mathematical History i Copyright 2009 by The Mathematical Association of America Library of Congress number 2008935288 Print ISBN 978-0-88385-569-0 Electronic ISBN 978-1-61444-504-3 Printed in the United States of America Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 P1:JZP/... P2:JZP fm MABK005/Anderson February2,2009 16:5 Who Gave you the Epsilon? and Other Tales of Mathematical History Editedby Marlow Anderson ColoradoCollege Victor Katz UniversityoftheDistrictofColumbia Robin Wilson OpenUniversity ® PublishedandDistributedby TheMathematicalAssociationofAmerica iii P1:JZP/... P2:JZP fm MABK005/Anderson February2,2009 16:5 CoordinatingCouncilonPublications PaulZorn,Chair SpectrumEditorialBoard GeraldL.Alexanderson,Editor WilliamDunham EdwardW.Packel RichardK.Guy KennethA.Ross MichaelA.Jones MarvinSchaefer KeithM.Kendig SanfordSegal JeffreyL.Nunemacher FranklinF.Sheehan iv P1:JZP/... P2:JZP fm MABK005/Anderson February2,2009 16:5 SPECTRUMSERIES The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publishabroadrangeofbooksincludingbiographies,accessibleexpositionsofoldornewmathematicalideas, reprintsandrevisionsofexcellentout-of-printbooks,popularworks,andothermonographsofhighinterest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs,andresearchers. 777MathematicalConversationStarters,byJohndePillis 99PointsofIntersection:Examples—Pictures—Proofs,byHansWalser.TranslatedfromtheoriginalGerman byPeterHiltonandJeanPedersen AlltheMathThat’sFittoPrint,byKeithDevlin CarlFriedrichGauss:TitanofScience,byG.WaldoDunnington,withadditionalmaterialbyJeremyGray andFritz-EgbertDohse TheChangingSpaceofGeometry,editedbyChrisPritchard Circles:AMathematicalView,byDanPedoe ComplexNumbersandGeometry,byLiang-shinHahn Cryptology,byAlbrechtBeutelspacher FiveHundredMathematicalChallenges,EdwardJ.Barbeau,MurrayS.Klamkin,andWilliamO.J.Moser FromZerotoInfinity,byConstanceReid TheGoldenSection,byHansWalser.TranslatedfromtheoriginalGermanbyPeterHilton,withtheassistance ofJeanPedersen. IWanttoBeaMathematician,byPaulR.Halmos JourneyintoGeometries,byMartaSved JULIA:ALifeinMathematics,byConstanceReid R.L.Moore:MathematicianandTeacher,byJohnParker TheLighterSideofMathematics:Proceedings oftheEuge`neStrensMemorialConferenceonRecreational Mathematics&ItsHistory,editedbyRichardK.GuyandRobertE.Woodrow LureoftheIntegers,byJoeRoberts TheMagicNumberoftheProfessor,byOwenO’SheaandUnderwoodDudley MagicTricks,CardShuffling,andDynamicComputerMemories:TheMathematicsofthePerfectShuffle,by S.BrentMorris MartinGardner’sMathematicalGames:TheEntireCollectionofHisScientificAmericanColumns TheMathChatBook,byFrankMorgan Mathematical Adventures for Students and Amateurs, edited by David Hayes and Tatiana Shubin. With the assistanceofGeraldL.AlexandersonandPeterRoss MathematicalApocrypha,byStevenG.Krantz MathematicalApocryphaRedux,byStevenG.Krantz MathematicalCarnival,byMartinGardner MathematicalCirclesVolI:InMathematicalCirclesQuadrantsI,II,III,IV,byHowardW.Eves MathematicalCirclesVolII:MathematicalCirclesRevisitedandMathematicalCirclesSquared,byHoward W.Eves Mathematical Circles VolIII:Mathematical CirclesAdieu and Return toMathematical Circles,by Howard W.Eves MathematicalCircus,byMartinGardner MathematicalCranks,byUnderwoodDudley v P1:JZP/... P2:JZP fm MABK005/Anderson February2,2009 16:5 MathematicalEvolutions,editedbyAbeShenitzerandJohnStillwell MathematicalFallacies,Flaws,andFlimflam,byEdwardJ.Barbeau MathematicalMagicShow,byMartinGardner MathematicalReminiscences,byHowardEves MathematicalTreks:FromSurrealNumberstoMagicCircles,byIvarsPeterson Mathematics:QueenandServantofScience,byE.T.Bell MemorabiliaMathematica,byRobertEdouardMoritz MusingsoftheMasters:AnAnthologyofMathematicalReflections,editedbyRaymondG.Ayoub NewMathematicalDiversions,byMartinGardner Non-EuclideanGeometry,byH.S.M.Coxeter NumericalMethodsThatWork,byFormanActon NumerologyorWhatPythagorasWrought,byUnderwoodDudley OutoftheMouthsofMathematicians,byRosemarySchmalz PenroseTilestoTrapdoorCiphers...andtheReturnofDr.Matrix,byMartinGardner Polyominoes,byGeorgeMartin PowerPlay,byEdwardJ.Barbeau TheRandomWalksofGeorgePo´lya,byGeraldL.Alexanderson RealityConditions:ShortMathematicalFiction,byAlexKasman RemarkableMathematicians,fromEulertovonNeumann,IoanJames TheSearchforE.T.Bell,alsoknownasJohnTaine,byConstanceReid ShapingSpace,editedbyMarjorieSenechalandGeorgeFleck SherlockHolmesinBabylonandOtherTalesofMathematicalHistory,editedbyMarlowAnderson,Victor Katz,andRobinWilson StudentResearchProjectsinCalculus,byMarcusCohen,ArthurKnoebel,EdwardD.Gaughan,DouglasS. Kurtz,andDavidPengelley Symmetry,byHansWalser.TranslatedfromtheoriginalGermanbyPeterHilton,withtheassistanceofJean Pedersen. TheTrisectors,byUnderwoodDudley TwentyYearsBeforetheBlackboard,byMichaelStuebenwithDianeSandford Who Gave You the Epsilon? and Other Tales of Mathematical History, edited by Marlow Anderson, Victor Katz,andRobinWilson TheWordsofMathematics,byStevenSchwartzman MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 800-331-1622 FAX:301-206-9789 vi P1:JZP/... P2:JZP fm MABK005/Anderson February2,2009 16:5 Introduction For over one hundred years, the Mathematical Association of America has been publishing high- quality articles on the history of mathematics, some written by distinguished historians and math- ematicians such as J.L.Coolidge, B.L.van der Waerden, Hermann Weyl and G.H.Hardy. Many well-knownhistoriansofthepresentdayalsocontributetotheMAA’sjournals,suchasIvorGrattan- Guinness,JudithGrabiner,IsraelKleinerandKarenParshall. Some years ago, we decided that it would be useful to reprint a selection of these papers and to settheminthecontextofmodernhistoricalresearch,sothatcurrentmathematicianscancontinue toenjoythemandsothatnewerarticlescanbeeasilycomparedwitholderones.Theresultwasour MAAvolumeSherlockHolmesinBabylon,whichtookthestoryfromearliesttimesuptothetime ofEulerintheeighteenthcentury.Thecurrentvolumeisasequeltoourearlierone,andcontinues with topics from the nineteenth and twentieth centuries. We hope that you will enjoy this second collection. Acarefulreadingofsomeoftheolderpapersshowsthatalthoughmodernresearchhasintroduced some new information or has fostered some new interpretations, in large measure they are neither dated or obsolete. Nevertheless, we have sometimes decided to include two or more papers on a singletopic,writtenyearsapart,toshowtheprogressinthehistoryofmathematics. We wish to thank Don Albers, Director of Publications at the MAA, and Gerald Alexanderson, former chair of the publications committee of the MAA, for their support for the history of math- ematics at the MAA in general, and for this project in particular. We also want to thank Beverly Ruediforhertechnicalexpertiseinpreparingthisvolumeforpublication. vii P1:JZP/... P2:JZP fm MABK005/Anderson February2,2009 16:5 viii P1:JZP/... P2:JZP fm MABK005/Anderson February2,2009 16:5 Contents Introduction vii Analysis Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 WhoGaveYoutheEpsilon?CauchyandtheOriginsofRigorousCalculus,JudithV.Grabiner . . . . .5 EvolutionoftheFunctionConcept:ABriefSurvey,IsraelKleiner . . . . . . . . . . . . . . . . . . . 14 S.Kovalevsky:AMathematicalLesson,KarenD.Rappaport . . . . . . . . . . . . . . . . . . . . . 27 HighlightsintheHistoryofSpectralTheory,L.A.Steen . . . . . . . . . . . . . . . . . . . . . . . . 36 AlanTuringandtheCentralLimitTheorem,S.L.Zabell . . . . . . . . . . . . . . . . . . . . . . . . 52 WhydidGeorgeGreenWritehisEssayof1828onElectricityandMagnetism?, I.Grattan-Guinness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ConnectivityandSmoke-Rings:Green’sSecondIdentityinitsFirstFiftyYears, ThomasArchibald . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 TheHistoryofStokes’Theorem,VictorJ.Katz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 TheMathematicalCollaborationofM.L.CartwrightandJ.E.Littlewood, ShawneeL.McMurranandJamesJ.Tattersall . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Dr.DavidHaroldBlackwell,AfricanAmericanPioneer,NkechiAgwu,LuellaSmithand AissatouBarry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Geometry, Topology and Foundations Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 GaussandtheNon-EuclideanGeometry,GeorgeBruceHalsted . . . . . . . . . . . . . . . . . . . 115 HistoryoftheParallelPostulate,FlorenceP.Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . 120 TheRiseandFallofProjectiveGeometry,J.L.Coolidge . . . . . . . . . . . . . . . . . . . . . . . 125 NotesontheHistoryofGeometricalIdeas,DanPedoe . . . . . . . . . . . . . . . . . . . . . . . . 133 AnoteonthehistoryoftheCantorsetandCantorfunction,JulianF.Fleron . . . . . . . . . . . . 137 EvolutionoftheTopologicalConceptof“Connected”,R.L.Wilder . . . . . . . . . . . . . . . . . 142 ABrief,SubjectiveHistoryofHomologyandHomotopyTheoryinthisCentury,PeterHilton . . . 148 TheOriginsofModernAxiomatics:PaschtoPeano,H.C.Kennedy . . . . . . . . . . . . . . . . . 157 C.S.Peirce’sPhilosophyofInfiniteSets,JosephW.Dauben . . . . . . . . . . . . . . . . . . . . . 161 OntheDevelopmento√fLogi√csbetw√eenthetwoWorldWars,I.Grattan-Guinness . . . . . . . . . 172 Dedekind’sTheorem: 2× 3= 6,DavidFowler. . . . . . . . . . . . . . . . . . . . . . . . . 185 Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 ix
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