A Mathematician Comes of Age

i i “master” — 2011/11/9 — 15:21 — page i — #1 i i A Mathematician Comes of Age i i i i i i “master” — 2011/11/9 — 15:21 — page ii — #2 i i c 2012bytheMathematicalAssociationofAmerica,Inc. (cid:13) LibraryofCongressCatalogCardNumber2011943144 PrinteditionISBN:978-0-88385-578-2 ElectroniceditionISBN:978-1-61444-511-1 PrintedintheUnitedStatesofAmerica CurrentPrinting(lastdigit): 10987654321 i i i i i i “master” — 2011/11/9 — 15:21 — page iii — #3 i i A Mathematician Comes of Age Steven G. Krantz Washington University in St. Louis PublishedandDistributedby TheMathematicalAssociationofAmerica i i i i i i “master” — 2011/11/16 — 16:55 — page iv — #4 i i CouncilonPublicationsandCommunications FrankFarris,Chair CommitteeonBooks GeraldBryce,Chair SpectrumEditorialBoard GeraldL.Alexanderson,Editor RobertE.Bradley SusanS.Epp RichardK.Guy KeithM.Kendig ShawneeL.McMurran JeffreyL.Nunemacher KennethA.Ross FranklinF.Sheehan JamesJ.Tattersall RobinWilson SPECTRUMSERIES TheSpectrumSeriesoftheMathematicalAssociationofAmericawassonamedto re?ectitspurpose:topublishabroadrangeofbooksincludingbiographies,accessi- bleexpositionsofoldornewmathematicalideas,reprintsandrevisionsofexcellent out-of-printbooks,popularworks,andothermonographsofhighinterestthatwill appealtoabroadrangeofreaders,includingstudentsandteachersofmathematics, mathematicalamateurs,andresearchers. 777MathematicalConversationStarters,byJohndePillis 99PointsofIntersection:Examples—Pictures—Proofs,byHansWalser.Translated fromtheoriginalGermanbyPeterHiltonandJeanPedersen AhaGotchaandAhaInsight,byMartinGardner AlltheMathThat’sFittoPrint,byKeithDevlin BeautifulMathematics,byMartinErickson CalculusGems:BriefLivesandMemorableMathematics,byGeorgeF.Simmons CarlFriedrichGauss:TitanofScience,byG.WaldoDunnington,withadditional materialbyJeremyGrayandFritz-EgbertDohse TheChangingSpaceofGeometry,editedbyChrisPritchard Circles:AMathematicalView,byDanPedoe ComplexNumbersandGeometry,byLiang-shinHahn Cryptology,byAlbrechtBeutelspacher TheEarlyMathematicsofLeonhardEuler,byC.EdwardSandifer TheEdgeoftheUniverse:Celebrating10YearsofMathHorizons,editedbyDeanna HaunspergerandStephenKennedy EulerandModernScience,editedbyN.N.Bogolyubov,G.K.Mikhailov,andA.P. Yushkevich.TranslatedfromRussianbyRobertBurns. Eulerat300:AnAppreciation,editedbyRobertE.Bradley,LawrenceA.D’Antonio, andC.EdwardSandifer i i i i i i “master” — 2011/11/9 — 15:21 — page v — #5 i i ExpeditionsinMathematics,editedbyTatianaShubin,DavidF.Hayes,andGerald L.Alexanderson FiveHundredMathematicalChallenges,byEdwardJ.Barbeau,MurrayS.Klamkin, andWilliamO.J.Moser TheGeniusofEuler:Re?ectionsonhisLifeandWork,editedbyWilliamDunham TheGoldenSection,byHansWalser.TranslatedfromtheoriginalGermanbyPeter Hilton,withtheassistanceofJeanPedersen. TheHarmonyoftheWorld:75YearsofMathematicsMagazine,editedbyGeraldL. AlexandersonwiththeassistanceofPeterRoss AHistorianLooksBack:TheCalculusasAlgebraandSelectedWritings,byJudith Grabiner History of Mathematics: Highways and Byways,by Amy Dahan-Dalme´dicoand JeannePeiffer,translatedbySanfordSegal HowEulerDidIt,byC.EdwardSandifer IsMathematicsInevitable?AMiscellany,editedbyUnderwoodDudley IWanttoBeaMathematician,byPaulR.Halmos JourneyintoGeometries,byMartaSved JULIA:alifeinmathematics,byConstanceReid TheLighterSideofMathematics:ProceedingsoftheEuge`neStrensMemorialCon- ferenceonRecreationalMathematics&Its History,editedbyRichardK.Guy andRobertE.Woodrow LureoftheIntegers,byJoeRoberts MagicNumbersoftheProfessor,byOwenO’SheaandUnderwoodDudley MagicTricks,CardShuf?ing,andDynamicComputerMemories:TheMathematics ofthePerfectShuf?e,byS.BrentMorris MartinGardner’sMathematicalGames:TheentirecollectionofhisScienti?cAmer- icancolumns TheMathChatBook,byFrankMorgan MathematicalAdventuresfor StudentsandAmateurs,editedbyDavidHayesand TatianaShubin.WiththeassistanceofGeraldL.AlexandersonandPeterRoss MathematicalApocrypha,byStevenG.Krantz MathematicalApocryphaRedux,byStevenG.Krantz MathematicalCarnival,byMartinGardner Mathematical Circles Vol I: In Mathematical Circles QuadrantsI, II, III, IV, by HowardW.Eves Mathematical Circles Vol II: Mathematical Circles Revisited and Mathematical CirclesSquared,byHowardW.Eves MathematicalCirclesVolIII:MathematicalCirclesAdieuandReturntoMathemat- icalCircles,byHowardW.Eves MathematicalCircus,byMartinGardner MathematicalCranks,byUnderwoodDudley MathematicalEvolutions,editedbyAbeShenitzerandJohnStillwell MathematicalFallacies,Flaws,andFlim?am,byEdwardJ.Barbeau MathematicalMagicShow,byMartinGardner i i i i i i “master” — 2011/11/9 — 15:21 — page vi — #6 i i MathematicalReminiscences,byHowardEves MathematicalTreks:FromSurrealNumberstoMagicCircles,byIvarsPeterson AMathematicianComesofAge,byStevenG.Krantz Mathematics:QueenandServantofScience,byE.T.Bell MathematicsinHistoricalContext,,byJeffSuzuki MemorabiliaMathematica,byRobertEdouardMoritz Musings of the Masters: An Anthology of Mathematical Re?ections, edited by RaymondG.Ayoub NewMathematicalDiversions,byMartinGardner Non-EuclideanGeometry,byH.S.M.Coxeter NumericalMethodsThatWork,byFormanActon NumerologyorWhatPythagorasWrought,byUnderwoodDudley OutoftheMouthsofMathematicians,byRosemarySchmalz Penrose Tiles to Trapdoor Ciphers ...and the Return of Dr. Matrix, by Martin Gardner Polyominoes,byGeorgeMartin PowerPlay,byEdwardJ.Barbeau ProofandOtherDilemmas:MathematicsandPhilosophy,editedbyBonnieGold andRogerSimons TheRandomWalksofGeorgePo´lya,byGeraldL.Alexanderson RemarkableMathematicians,fromEulertovonNeumann,byIoanJames TheSearchforE.T.Bell,alsoknownasJohnTaine,byConstanceReid ShapingSpace,editedbyMarjorieSenechalandGeorgeFleck SherlockHolmesin BabylonandOtherTalesof Mathematical History,editedby MarlowAnderson,VictorKatz,andRobinWilson StudentResearchProjectsinCalculus,byMarcusCohen,ArthurKnoebel,Edward D.Gaughan,DouglasS.Kurtz,andDavidPengelley Symmetry,byHansWalser. Translatedfrom theoriginal GermanbyPeterHilton, withtheassistanceofJeanPedersen. TheTrisectors,byUnderwoodDudley TwentyYearsBeforetheBlackboard,byMichaelStuebenwithDianeSandford Who GaveYouthe Epsilon?andOther Talesof Mathematical History,editedby MarlowAnderson,VictorKatz,andRobinWilson TheWordsofMathematics,byStevenSchwartzman MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 800-331-1622 FAX301-206-9789 i i i i i i “master” — 2011/11/9 — 15:21 — page vii — #7 i i For Charles Paine, a goodfriendanda greatteacher. He wasthe ?rsttogive mea glimpse ofmathematicalmaturity. i i i i i i “master” — 2011/11/9 — 15:21 — page viii — #8 i i i i i i i i “master” — 2011/11/9 — 15:21 — page ix — #9 i i Contents Preface xiii Acknowledgements xvii 1 IntroductoryThoughts 1 1.0 ChapterOverview . . . . . . . . . . . . . . . . . . . . . . 1 1.1 BackStory . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 FirstPrinciples . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 NextSteps . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 TheMathematicallyNaiveView . . . . . . . . . . . . . . 8 1.5 HowDoesMathematicsDifferfromOtherFields? . . . . 8 1.6 ALittleHistory . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 ExamplesofMathematicalMaturity . . . . . . . . . . . . 11 1.8 MathematicalMaturityandFearofFailure. . . . . . . . . 14 1.9 LevelsofMathematicalMaturity . . . . . . . . . . . . . . 16 1.10 TheChangingNatureofMathematicalMaturity . . . . . . 18 1.11 OnProofandProgressinMathematics . . . . . . . . . . . 19 1.12 MoreonBrahe,Kepler,andNapier. . . . . . . . . . . . . 23 2 MathConcepts 27 2.0 ChapterOverview . . . . . . . . . . . . . . . . . . . . . . 28 2.1 ProblemsthatCanExhibitMathematicalMaturity. . . . . 28 2.2 ApproximateSolutions . . . . . . . . . . . . . . . . . . . 31 2.3 ComputersandCalculators . . . . . . . . . . . . . . . . . 33 2.4 ProvingLittleTheorems,ProvingBigTheorems . . . . . 35 2.5 Mistakes. . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6 MathematicalBeauty . . . . . . . . . . . . . . . . . . . . 39 2.7 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 TeachingTechniques 41 3.0 ChapterOverview . . . . . . . . . . . . . . . . . . . . . . 41 3.1 TeachingReform . . . . . . . . . . . . . . . . . . . . . . 42 ix i i i i