Humanizing Mathematics and its Philosophy: Essays Celebrating the 90th Birthday of Reuben Hersh

357 Pages·2017·4.444 MB·English

by  Bharath Sriraman (eds.)

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
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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