The architecture of modern mathematics: essays in history and philosophy

455 Pages·2006·1.811 MB·English

by  J. Ferreirós, J. J. Gray

the architecture of modern
mathematics
This page intentionally left blank
The Architecture of
Modern Mathematics
Essays in History and
Philosophy
Editedby
J. FERREIRÓS
and
J. J. GRAY
1
3
GreatClarendonStreet,Oxfordox26dp
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwidein
Oxford NewYork
Auckland CapeTown DaresSalaam HongKong Karachi
KualaLumpur Madrid Melbourne MexicoCity Nairobi
NewDelhi Shanghai Taipei Toronto
Withofficesin
Argentina Austria Brazil Chile CzechRepublic France Greece
Guatemala Hungary Italy Japan Poland Portugal Singapore
SouthKorea Switzerland Thailand Turkey Ukraine Vietnam
OxfordisaregisteredtrademarkofOxfordUniversityPress
intheUKandincertainothercountries
PublishedintheUnitedStates
byOxfordUniversityPressInc.,NewYork
©OxfordUniversityPress,2006
Themoralrightsoftheauthorshavebeenasserted
DatabaserightOxfordUniversityPress(maker)
Firstpublished2006
Allrightsreserved.Nopartofthispublicationmaybereproduced,
storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,
withoutthepriorpermissioninwritingofOxfordUniversityPress,
orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate
reprographicsrightsorganization.Enquiriesconcerningreproduction
outsidethescopeoftheaboveshouldbesenttotheRightsDepartment,
OxfordUniversityPress,attheaddressabove
Youmustnotcirculatethisbookinanyotherbindingorcover
andyoumustimposethesameconditiononanyacquirer
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressCataloginginPublicationData
Dataavailable
TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India
PrintedinGreatBritain
onacid-freepaperby
BiddlesLtd.,King’sLynn,Norfolk
ISBN 0–19–856793–6 978–0–19–856793–6
1 3 5 7 9 10 8 6 4 2
Contents
Acknowledgements vii
Listof Contributors ix
Introduction 1
J.FerreirósandJ.J.Gray
REINTERPRETATIONS IN THE HISTORY AND
PHILOSOPHY OF FOUNDATIONS 45
FregeandtheRoleof HistoricalElucidation:Methodologyandthe
Foundationsof Mathematics 47
MichaelBeaney
Riemann’sHabilitationsvortragattheCrossroadsof Mathematics,Physics,
andPhilosophy 67
JoséFerreirós
TheRiemannianBackgroundtoFrege’sPhilosophy 97
JamieTappenden
Axiomatics,Empiricism,andAnschauunginHilbert’sConceptionof
Geometry:BetweenArithmeticandGeneralRelativity 133
LeoCorry
EXPLORATIONS INTO THE EMERGENCE
OF MODERN MATHEMATICS 157
Methodologyandmetaphysicsinthedevelopmentof Dedekind’stheoryof
ideals 159
JeremyAvigad
EmmyNoether’s‘SetTheoretic’Topology:FromDedekindtotheRiseof
Functors 187
ColinMcLarty
Tarskionmodelsandlogicalconsequence 209
PaoloMancosu
APathtotheEpistemologyof Mathematics:HomotopyTheory 239
Jean-PierreMarquis
v
Contents
ALTERNATIVE VIEWS AND PROGRAMS IN THE
PHILOSOPHY OF MATHEMATICS 261
FelixHausdorff’sConsideredEmpiricism 263
MoritzEpple
Practice-relatedsymbolicrealisminH.Weyl’smatureviewof
mathematicalknowledge 291
ErhardScholz
FromKanttoHilbert:Frenchphilosophyof conceptsinthebeginningof
thetwentiethcentury 311
HouryaBenisSinaceur
RelativeConsistencyandAccessibleDomains 339
WilfriedSieg
CODA 369
Modernmathematicsasaculturalphenomenon 371
JeremyGray
References 397
NameIndex 433
SubjectIndex 439
vi
Acknowledgements
Wewishtothankthecontributorstothisvolumefortheircommentsonaprevious
version of our introduction. This collective volume was arranged and prepared
in the course of the last four years, especially at two events that served us to
discusscommonviewpoints,choosecontributors,andselecttopics–aconference
onthehistoryandphilosophyof modernmathematics(May2002)attheOpen
University(CentrefortheHistoryof MathematicalSciences)andUniversityCollege
London; and an international meeting on the same topic (September 2003) at
the Universidad de Sevilla (Spain). The Seville meeting was financed thanks to
an‘acciónespecial’of theSpanishgovernment(BFF2002-10381-E)andcomple-
mentaryfinancialmeansofferedbytheJuntadeAndaluciaandVicerrectorado
deInvestigación(UniversidaddeSevilla).
vii
This page intentionally left blank
List of Contributors
Jeremy Avigad is an Associate Professor of Philosophy at Carnegie Mellon Uni-
versity, with research interests in mathematical logic, proof theory, automated
deduction,andthehistoryandphilosophyofmathematics.Hewassupportedbya
NewDirectionsFellowshipfromtheAndrewW.MellonFoundationin2003–2004.
MichaelBeaneyisReaderinPhilosophyattheUniversityof York,withresearch
interestsinthehistoryof analyticphilosophy,methodologyandthefoundations
of reasoning.Heisauthorof Frege:MakingSense(Duckworth,1996)andAnalysis
(Acumen, forthcoming), and editor of The Frege Reader (Blackwell, 1997) and
(withErichReck)of GottlobFrege: CriticalAssessmentsof LeadingPhilosophers(4
vols.,Routledge,2005).
HouryaBenisSinaceurhastaughtPhilosophyof ScienceattheUniversityParis
I-Sorbonne. SheisatpresentDirectorof ResearchattheCNRS(France), work-
ingattheInstitutd’HistoireetdePhilosophiedesSciencesetdesTechniquesin
Paris. Sheistheauthorof CorpsetModèles. Essaisurl’histoiredel’algèbreréelle,
Paris, Vrin, 1991 (second edition : 1999) and Jean Cavaillès. Philosophie Math-
ématique, Paris, Presses Universitaires de France, 1994. She edited A. Tarski’s
‘Address at the Princeton University Bicentennial Conference on Problems of
Mathematics’(TheBulletinof SymbolicLogic, March2000)andshetranslated
intoFrenchP.Bernays’‘AbhandlungenzurPhilosophiederMathematik’, Paris,
Vrin, 2003. She has published many papers, in particular on Bolzano, Cauchy,
Sturm, Poincaré, Hilbert, andTarski. She is presently preparing a new French
editionof R.Dedekind’s‘WassindundwassollendieZahlen?’and‘Stetigkeitund
irrrationaleZahlen’.
LeoCorryisDirectorof theCohnInstituteforHistoryandPhilosophyof Science
and Ideas, Tel-Aviv University, where he has taught history and philosophy
of mathematics for more than twenty years. He is coeditor of the interna-
tional journal Science in Context published by Cambridge University Press. His
recently published books are: Modern Algebra and The Rise of Mathematical
Structures (Basel: Birkhauser – 2nd, softcover edition: 2003) and David Hilbert
andtheAxiomatizationof Physics(2004).
MoritzEppleisProfessorof theHistoryof ScienceattheUniversityof Frankfurt
am Main. His interests include the history of topology, the history of math-
ematicalphysics,philosophicalaspectsof modernmathematics,andtherelations
betweenthemathematicalsciencesandsociety.Recentpublicationsinclude:Die
ix