Arche Papers on the Mathematics of Abstraction PDF
Preview Arche Papers on the Mathematics of Abstraction
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 AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress. ISBN 978–1–4020–4264–5(HB) ISBN 978–1–4020–4265–2(e-book) PublishedbySpringer, P.O.Box17,3300AADordrecht,TheNetherlands. www.springer.com Printedonacid-freepaper AllRightsReserved (cid:1)c 2007Springer Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmitted inanyformorbyanymeans,electronic,mechanical,photocopying,microfilming,recording orotherwise,withoutwrittenpermissionfromthePublisher,withtheexception ofanymaterialsuppliedspecificallyforthepurposeofbeingentered andexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. 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). xi
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