Humanizing Mathematics and its Philosophy: Essays Celebrating the 90th Birthday of Reuben Hersh PDF
Preview Humanizing Mathematics and its Philosophy: Essays Celebrating the 90th Birthday of Reuben Hersh
Bharath Sriraman Editor Humanizing Mathematics and its Philosophy Essays Celebrating the 90th Birthday of Reuben Hersh Bharath Sriraman Editor Humanizing Mathematics and its Philosophy Essays Celebrating the 90th Birthday of Reuben Hersh Editor BharathSriraman DepartmentofMathematicalSciences TheUniversityofMontana Missoula,MT,USA ISBN978-3-319-61230-0 ISBN978-3-319-61231-7 (eBook) DOI10.1007/978-3-319-61231-7 LibraryofCongressControlNumber:2017958868 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.com TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface – Reuben Hersh: Humanizing Mathematics and Its Philosophy Itisdifficulttofindwordstosuccinctlycaptureamathematicianwhohassystemati- callydeconstructedthecoldedificeoftheinstitutionofmathematics.ReubenHersh turns90onDecember9,2017,andhisnine-decadejourneytothepresentmoment offersglimpsesintotheworldofmathematicsaswellasthechangingnatureofthe Americanlandscape.BornintheBronxtoworking-classimmigrantparents,Reuben embracedtheidealsof the“workingclass” inspite of graduatingfromHarvardat theageof19.AfterastintatScientificAmerican,Reubenspentthe1950sworking asamachinist.WhenIaskedReubenaboutthis,hesaid“Beingsoyoungandnaive, Ijustfeltfrustratedathis(Svirsky’s)constantdissatisfactionwith mywork.Then therewerepoliticalreasonstoo.Iwasdeludedintothinkingthatonlytheworking class could save the world, so I ought to be part of the working class. Learning toruna lathewasinterestingandina waygratifyingwork”(see interviewin“An InterviewwithReubenHersh”). One could say that becoming a machinist created a dual identity, namely, that ofa“workingman”aswellasa“thinkingman”andonethatseemsdifferentwith thewayprofessionsarestructuredtoday,particularlyifonelooksattheivorytower of academia. However,being able to do many things was the hallmark of learned peopleforcenturies.Gauss’s“dayjob”wasthatofasurveyor;Eulerworkedasan engineer;vonHelmholtzstartedoutasaphysicianbutwentontomakeastonishing contributionstomathematicalphysics. Reuben went into mathematics after obtaining a degree in literature and then working as a machinist. He chose mathematics because he always enjoyed it. It should not be surprising then to know that he went on to complete a PhD under Peter Lax and had a fruitful career as a mathematician for many decades with researchinpartialdifferentialequations,randomevolutions,andoperatorequations. Ifoneimaginesthesedecadesofhislifeasthatofa“workingmathematician”with literaryleaningsanalogoustothe“workingmachinist”with literaryleanings,then these leanings came into full force when he started to write about the nature of mathematics,whatitmeanstobeamathematician,thesocialnatureofmathematics, the burdenof proof,andwhat it meansto questionthe status of mathematics.His first expository book The Mathematical Experience, cowritten with Phil Davis, v vi Preface–ReubenHersh:HumanizingMathematicsandItsPhilosophy won a National Book Award in Science in 1983. His subsequent book What Is Mathematics, Really? picked up where the Courant-Robbins classic ended, with a rhetorical question, but provided a more detailed exposition on what it means tobea mathematician,howmathematiciansthinkabouttheirwork(asopposedto “science” or “craft” or “art”), and philosophical problems that arise when doing mathematics. Many of us can “do” things, i.e., do mathematics, do physics, do biologyordogardenwork,dowoodwork,docooking,etc.,butveryfewofusare abletoarticulatewhatitreallymeanstodosomethinginawaythatwouldappeal and interest a layperson. Reuben’s expository books have been very impactful to many of us, and his position of mathematics as a human endeavor or humanistic allowsthefootsoldiersandthelaymathematiciansadoorwaythroughwhichthey canexaminetheirownmathematicalendeavors.Awell-knownmathematicianonce toldmethatthereareveryfewthatcanleapfromoneareaofresearchmathematics toanotherandthenbeabletoclearlyarticulatetheconnectionsthatledthemtomake these “creative leaps.” Reuben’sjourney as writer, machinist, mathematician, and philosopherovertheninedecadesofhislifecontainsmanysuchleaps,notsimply withinmathematicsbutbetweendisciplinesandnotsimplybetweendisciplinesbut between completely different “working lives.” His corpus of writings that range from technical mathematics to reviews to expository and philosophical writings offerclearglimpsesofReuben’smanyleaps. IconferredwithReubenaboutputtingtogetheraFestschriftforhis90thbirthday, and he agreed to it based on some conditions of course, namely, being able to shape this in order to break convention and be different! So he sent me a list of colleagues/friends/scholars whose work he has been influenced by, and I invitedthem to contributeto this book.When I asked Reuben whathe would like contributorstoaddress,hesaidthefollowing: Forty years ago, Paul Cohen enraged me by predicting that (at some unspecified future time)mathematicianswouldbereplacedbycomputers.SoInowaskyou,1.Canpracticing mathematicians, as such, contribute anything to philosophy of math? Can or should philosophersofmath,assuch,sayanythingtopracticingmathematicians?2.20or50years fromnow,whatwillbesimilar,andwhatwill,orcould,orshouldbealtogetherdifferent: Aboutthephilosophyofmath?Aboutmatheducation?Aboutmathresearchinstitutions? Aboutdataprocessingandscientificcomputing? The colorfuland eclectic essays in this Festschrift from numerouswell-known mathematicians,philosophers,logicians,andlinguistsofferin parthis colleagues’ attemptsatansweringReuben’squestionsandalsoinpartglimpsesintoReuben’s fertilemindandhisinfluenceonmanygenerationsanddecadesofmathematicallife. Inhis90thyear,hecontinuestoproducemathematicsandwritingsaboutitthatis accessibletousall.ReubenHershepitomizesthephrase“humanistmathematician andphilosopher,”andIhopethisFestschriftcelebrateshismanyaccomplishments andcontributionstothefield.Iamdeeplyhonoredtobeabletoeditthiscollection andjointheauthorsinthisbooktowishhimahappybirthday,andIhopeforanother decadeofcontributionsfromReuben. Missoula,MT,USA BharathSriraman Contents AnInterviewwithReubenHersh .............................................. 1 BharathSriraman NineDecades...................................................................... 11 BharathSriraman PluralismasModelingandasConfusion...................................... 19 ReubenHersh “Now”HasanInfinitesimalPositiveDuration................................ 31 ReubenHersh ReviewofHowHumansLearntoThinkMathematically:Exploring theThreeWorldsofMathematics ............................................... 39 ReubenHersh CanYouSayWhatMathematicsIs?........................................... 45 WilliamByers TheExactSciencesandNon-EuclideanLogic................................ 61 DavidA.Edwards Xenomath! ........................................................................ 69 IanStewart CognitiveNetworks:Brains,Internet,andCivilizations .................... 85 DmitriiYu.ManinandYuriI.Manin ReubenHershontheGrowthofMathematicalKnowledge:Kant, Geometry,andNumberTheory................................................ 97 EmilyGrosholz DoMathematiciansHaveResponsibilities?................................... 115 MichaelHarris vii viii Contents SchoolMathematicsand“Real”Mathematics................................ 125 BonnieGold WhatIsMathematicsandWhatShouldItBe?............................... 139 DoronZeilberger HumanismAboutAbstractObjects............................................ 151 JulianCole CanSomethingJustHappentoBeTrue?..................................... 167 ChandlerDavis The “Artificial Mathematician” Objection: Exploring the(Im)possibilityofAutomatingMathematicalUnderstanding........... 173 SvenDelarivièreandBartVan Kerkhove Wittgenstein,Mathematics,andtheTemporalityofTechnique ............ 199 PaulM.Livingston Gödel’sLegacy ................................................................... 215 MartinDavis VarietiesofMaverickPhilosophyofMathematics ........................... 223 CarloCellucci DoesReasonEvolve?(DoestheReasoninginMathematicsEvolve?)...... 253 JodyAzzouni MathematicalTheoriesasModels.............................................. 291 MichèleFriend MathematicsforMakersandMathematicsforUsers........................ 309 AlexandreV.Borovik ACaseStudyinReubenHersh’sPhilosophy:Bézout’sTheorem.......... 329 ElenaAnne CorieMarchisotto AGifttoTeachers................................................................ 347 NelNoddings ThePhilosophyofReubenHersh: ANontechnicalAssessment..................................................... 351 WilliamLabov FriendsandFormerComrades................................................. 355 ChandlerDavis OntheNatureofMathematicalEntities....................................... 361 ReubenHersh Contributors JodyAzzouni DepartmentofPhilosophy,TuftsUniversity,Medford,MA,USA AlexandreV.Borovik School of Mathematics, The University of Manchester, Manchester,UK WilliamByers Department of Mathematics & Statistics, Concordia University, Montreal,QC,Canada CarloCellucci Department of Philosophy, Sapienza University of Rome, Rome, Italy JulianCole DepartmentofPhilosophy,SUNYBuffaloState,Buffalo,NY,USA ChandlerDavis DepartmentofMathematics,UniversityofToronto,Toronto,ON, Canada MartinDavis Courant Institute of MathematicalSciences, New York University, NewYork,NY,USA SvenDelarivière VrijeUniversiteitBrussel,Brussels,Belgium DavidA.Edwards Department of Mathematics, University of Georgia, Athens, GA,USA MichèleFriend DepartmentofPhilosophy,GeorgeWashingtonUniversity,Wash- ington,DC,USA BonnieGold Department of Mathematics, Monmouth University, Long Branch, NJ,USA EmilyGrosholz Department of Philosophy, Pennsylvania State University, Uni- versityPark,PA,USA MichaelHarris ColumbiaUniversityandUniversitéParis-Diderot,Paris,France ReubenHersh DepartmentofMathematicsandStatistics,TheUniversityofNew Mexico,Albuquerque,NM,USA ix x Contributors WilliamLabov Departmentof Linguistics, Universityof Pennsylvania,Philadel- phia,PA,USA PaulM.Livingston DepartmentofPhilosophy,UniversityofNewMexico,Albu- querque,NM,USA DmitriiYu.Manin NewYork,NY,USA YuriI.Manin Max–Planck–InstitutfürMathematik,Bonn,Germany ElenaAnneCorieMarchisotto Department of Mathematics, California State University,Malibu,CA,USA NelNoddings StanfordGraduateSchoolofEducation,Stanford,CA,USA BharathSriraman Department of Mathematical Sciences, The University of Montana,Missoula,MT,USA IanStewart MathematicsInstitute,UniversityofWarwick,Coventry,UK BartVanKerkhove VrijeUniversiteitBrussel,Brussels,Belgium DoronZeilberger Department of Mathematics, Rutgers University, Piscataway, NJ,USA
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