Arche Papers on the Mathematics of Abstraction

482 Pages·2010·2.38 MB·English

by  Roy T. Cook

THEARCHÉPAPERSONTHEMATHEMATICS
OFABSTRACTION
THEWESTERNONTARIOSERIES
INPHILOSOPHYOFSCIENCE
ASERIESOFBOOKS
INPHILOSOPHYOFSCIENCE,METHODOLOGY,EPISTEMOLOGY,
LOGIC,HISTORYOFSCIENCE,ANDRELATEDFIELDS
ManagingEditor
WILLIAMDEMOPOULOS
DepartmentofPhilosophy,UniversityofWesternOntario,Canada
DepartmentofLogicandPhilosophyofScience,
UniversityofCalifornina/Irvine
ManagingEditor1980–1997
ROBERTE.BUTTS
Late,DepartmentofPhilosophy,UniversityofWesternOntario,Canada
EditorialBoard
JOHNL.BELL,UniversityofWesternOntario
JEFFREYBUB,UniversityofMaryland
PETERCLARK,StAndrewsUniversity
DAVIDDEVIDI,UniversityofWaterloo
ROBERTDiSALLE,UniversityofWesternOntario
MICHAELFRIEDMAN,IndianaUniversity
MICHAELHALLETT,McGillUniversity
WILLIAMHARPER,UniversityofWesternOntario
CLIFFORDA.HOOKER,UniversityofNewcastle
AUSONIOMARRAS,UniversityofWesternOntario
JÜRGENMITTELSTRASS,UniversitätKonstanz
JOHNM.NICHOLAS,UniversityofWesternOntario
ITAMARPITOWSKY,HebrewUniversity
VOLUME71
THE ARCHÉ PAPERS ON
THE MATHEMATICS OF
ABSTRACTION
Editedby
ROYT.COOK
123
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ForAlice,whohaskindlytoleratedthecompanyof
manyabstractionists,andoneinparticular
Contents
Foreword ix
NotesontheContributors xi
Acknowledgements xiii
Introduction xv
PartI ThePhilosophyandMathematicsofHume’sPrinciple
IsHume’sPrincipleAnalytic? 3
G.Boolos
IsHume’sPrincipleAnalytic? 17
C.Wright
Frege,Neo-LogicismandAppliedMathematics 45
P.Clark
FinitudeandHume’sPrinciple 61
R.G.Heck,Jr.
OnFiniteHume 85
F.MacBride
CouldNothingMatter? 95
F.MacBride
OnthePhilosophicalInterestofFregeArithmetic 105
W.Demopoulos
PartII TheLogicofAbstraction
“Neo-logicist”LogicisnotEpistemicallyInnocent 119
S.Shapiro&A.Weir
vii
viii Contents
AristotelianLogic,Axioms,andAbstraction 147
R.T.Cook
Frege’sUnofficialArithmetic 155
A.Rayo
PartIII AbstractionandtheContinuum
RealsbyAbstraction 175
R.Hale
TheStateoftheEconomy:Neo-logicismandInflation 197
R.T.Cook
FregeMeetsDedekind:ANeo-logicistTreatmentofRealAnalysis 219
S.Shapiro
Neo-FregeanFoundationsforRealAnalysis:SomeReflectionson
Frege’sConstraint 253
C.Wright
PartIV BasicLawVandSetTheory
NewV,ZFandAbstraction 275
S.Shapiro&A.Weir
Well-andNon-well-foundedExtensions 303
I.Jané&G.Uzquiano
Abstraction&SetTheory 331
BobHale
ProlegomenontoAnyFutureNeo-logicistSetTheory:Abstractionand
IndefiniteExtensibility 353
S.Shapiro
Neo-Fregeanism:AnEmbarassmentofRiches 383
A.Weir
IterationOneMoreTime 421
R.T.Cook
Foreword
InSeptember2000theArchéCentrelaunchedafive-yearresearchprojectenti-
tled the Logical and Metaphysical Foundations of Classical Mathematics. Its
goalwastostudytheprospects,philosophicalandtechnical,forabstractionist
foundations for the classical mathematical theories of the natural, real and
complex numbers and standard set theory. Funding was provided by the then
ArtsandHumanitiesResearchBoard(nowtheArtsandHumanitiesResearch
Council) for the appointment of full-time postdoctoral research fellows and
PhDstudentstocollaboratewithmoreseniorcolleaguesintheproject,andat
the same time the British Academy awarded the Centre additional resources
to establish an International Network of scholars to be associated with the
work. This was the beginning of the serial ‘Abstraction workshops’ of which
the Centre had staged no less than eleven by December 2006. We grate-
fully acknowledge the generous support of the Academy and Council, sine
quanon.
The project seminars and Network meetings generated—and continue to
generate—a large number of leading-edge research papers on all aspects of
the project agenda. The present volume is the first of what we hope will
be a number of anthologies of these researches. With two exceptions,—the
contribution by the late George Boolos and the co-authored paper by Gabriel
Uzquiano and Ignacio Jané,—the papers that Roy Cook has collected in the
present volume are all authored by sometime members of the project team or
oftheBritishAcademyNetwork.Theirbroadfocus,asheexplains,isonsome
of the more technical issues thrown up by the Abstractionist project, and it is
anticipatedthatsubsequentvolumesmayhaveamorepurelymetaphysicalor
epistemologicalemphasis.
I would like to thank Roy Cook for all his hard work putting the vol-
ume together, and Bill Demopoulos for sponsoring its publication in the
Western Ontario Series in Philosophy of Science. Special thanks go to the
members of the core team and the Network not just for their direct con-
tributions to the researches of the project but for their continuing affirma-
tion, by their active participation, of the wider interest and importance of
ix
x Foreword
the neo-Fregean enterprise in the landscape of contemporary philosophy of
mathematics.
CJGW
St.Andrews6/07
TheLogicalandMetaphysicalFoundationsofClassicalMathematics
Sometimeprojectteammembers:
Crispin Wright, Peter Clark, Roy Cook, Philip Ebert, Bob Hale, Fraser
MacBride, Paul McCallion, Darren McDonald, Nikolaj Jang Pedersen,
Agustin Rayo, Marcus Rossberg, Andrea Sereni, Stewart Shapiro, Chiara
Tabet,RobertWilliams
Auditor:KitFine
BritishAcademyInternationalNetworkmembers:
Alexander Bird, Robert Black, Robin Cameron, William Demopoulos,
RichardHeck,KeithHossack,DanielIsaacson,JohnMayberry,MichaelPot-
ter, Adam Rieger, Ian Rumfitt, Peter Simons, William Stirton, Peter Sullivan,
AlanWeir
Notes on the Contributors
George Boolos was Professor of Philosophy at Massachusetts Institute of
Technology,andtheco-authorofComputabilityandLogic(withRichardJef-
frey,Cambridge2007)andtheauthorofTheLogicofProvability(Cambridge
1995).
Peter J. Clark is Professor of the Philosophy of Science and Head of the
School of Philosophical and Anthropological Studies in the University of St
Andrews.Heworksprimarilyinthephilosophyofphysicalsciencesandmath-
ematics and was editor of the British Journal for the Philosophy of Science
1999–2005.
Roy Cook is Visiting Assistant Professor of Philosophy at Villanova Univer-
sity, and an associate fellow of Arché. He has published papers in the phi-
losophyoflanguage,logic,andmathematics,focusingprimarilyonsemantic,
soritical, and set-theoretic paradoxes, and Fregean and neo-Fregean philoso-
phiesofmathematics.
William Demopoulos is a member of the Department of Philosophy of the
University of Western Ontario and the Department of Logic and Philosophy
ofScienceoftheUniversityofCalifornia,Irvine.Hehaspublishedarticlesin
diversefieldsinthephilosophyoftheexactsciences,andonthedevelopment
ofanalyticphilosophyinthetwentiethcentury.
Bob Hale is Professor of Philosophy at the University of Sheffield, and an
Associate Director of Arché. He works mainly on topics in the epistemol-
ogy and metaphysics of mathematics and modality. His publications include
Abstract Objects (Blackwell 1987), and, together with Crispin Wright, The
Blackwell Companion to the Philosophy of Language (Blackwell 1997) and
The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of
Mathematics(Oxford2001).
Richard Heck is Professor of Philosophy at Brown University and an asso-
ciatefellowofArché.Hehaspublishedextensivelyonhistorical,conceptual,
and technical issues emerging from Frege’s philosophy of mathematics. Phi-
losophyoflanguageandphilosophyoflogicarehisothermainareasofinter-
est. He is now working on a book on philosophy of language and another on
thedevelopmentofFrege’smaturephilosophy(co-authoredwithRobertMay).
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