The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940 PDF
Preview The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940
The Adventure of Reason This page intentionally left blank The Adventure of Reason Interplay between Philosophy of Mathematics and Mathematical Logic, 1900–1940 PAOLO MANCOSU 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 ©inthisvolumePaoloMancosu2010 Themoralrightsoftheauthorhavebeenasserted DatabaserightOxfordUniversityPress(maker) Firstpublished2010 Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization.Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove Youmustnotcirculatethisbookinanyotherbindingorcover andyoumustimposethesameconditiononanyacquirer BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressCataloginginPublicationData Dataavailable TypesetbySPIPublisherServices,Pondicherry,India PrintedinGreatBritain onacid-freepaperby MPGBooksGroup,BodminandKing’sLynn ISBN 978–0–19–954653–4 1 3 5 7 9 10 8 6 4 2 AdElena, Compagnanell’avventuradellavita PREFACE In the last ten years I have extensively investigated a variety of topics in the history of logic,thefoundationsofmathematics,andthephilosophyoflogic.Theperiodofinvesti- gationgoesroughlyfromthebeginningofthetwentiethcenturytotheearly1940s.This period corresponds to the ‘golden age’ of mathematical logic including the discovery of Go¨del’s incompleteness theorems and the development of Tarskian semantics, to mention perhaps the two most well-known achievements. These and other results in mathematicallogicbothinfluencedandwereinfluencedbycontemporarydevelopments inthephilosophyofmathematics,thephilosophyoflogic,andgeneralphilosophy.While thegeneraloutlineofsuchinteractionsarebynowwellknown,amorenuancedpicture ofsuchdevelopmentscanbegatheredbyexploiting,inadditiontothepublishedsources, therichcollectionofmaterialsfoundinseveralarchives.Thepapersgatheredhereexploit untapped archival sources and thus make available a wealth of new information that deepensinsignificantwaysourunderstandingoftheabove-mentionedareas.Inpursuing thisresearch,Iworkedatlengthin,amongothers,theHilbertarchive(Go¨ttingen),the BernaysandWeylarchives(Zurich),theCarnaparchive(PittsburghandKonstanz),the Neurath archive (Haarlem and Konstanz), the Tarski archive (Berkeley), the Woodger archive (London), the Behmann archive (Erlangen, now Berlin), the Mahnke archive (Marburg),theKaufmannarchive(Konstanz),andtheQuinearchive(Harvard). Whiletheresultsofthisresearchwerepublishedseparately,Iampleasednowtobring these papers together in a single volume. The essays are closely linked by the fact that the subject matter is homogeneous and were written with a single major aim, namely that of reaching a deeper understanding of the interaction between developments in mathematicallogicandthefoundationsofmathematicsandlogicfrom1900to1940. To the discerning reader, the developments described in these pages will justify the title of this book TheAdventureofReason. I borrowed this beautiful image from a letter from Becker to Mahnke, which also gives the title to Chapter 12. The adventure of reason reflects itself in the ambitious mathematical and philosophical aims that drive theconstitutionofmathematicallogic,thephilosophicalandmathematicalaimsthatit is made to serve, and the unexpected surprises lying in wait. That’s the way I like to think of the ambitious goals by Russell and Whitehead, of Hilbert’s program, and of Husserl’sphenomenologicalambitions,tonameonlyafewmajordevelopmentstreated in the book. But just like Columbus had set sail to reach the Indies and found other unexpected—notlessrichandexciting—lands,inthesamewaytheadventureofreason inlogicandthefoundationsofmathematicsledtosurprisingnewfindsthatdramatically reshapedtheoriginalgoals. Theessayscontainedinthisvolume,allpreviouslypublishedexceptChapters17and 18, fall naturally into the five parts that make up the book. The first part contains a Preface vii lengthyaccountofthehistoryofmathematicallogicfromtheearlypartofthetwentieth century to the middle of the 1930s. On account of its length, I recommend using this chapter as needed while reading the other chapters in the book. Part II contains seven chaptersthatdealwithfoundationalissuesinmathematics,namelytheRussellianinflu- enceinGo¨ttingen,constructivism,andGo¨del’sincompletenesstheorems.Thethirdpart detailsadramaticmomentinthedevelopmentofphenomenologythroughthethreefig- ureswho,besidesHusserl,hadthebestunderstandingofthenewdevelopmentsinlogic, mathematics,andphysics.IamreferringtoHermannWeyl,OskarBecker,andDietrich Mahnke.Inspiteoftheirdifferentsolutionstotheproblemoftherelationshipbetween phenomenologyandtheexactsciences,theyallagreedontheunprecedentedstrainput by the new scientific developments on the Husserlian phenomenological grounding of the sciences described in Ideas. Part IV focuses on the discussion of nominalism as a foundation for mathematics and science in the early stages of such discussions, that is duringthemeetingsamongCarnap,Quine,andTarskiinHarvardin1940-41and,more broadly in Quine’s and Tarski’s trajectory on such issues. Finally, part V both discusses theimpactofTarski’stheoryoftruthandlogicalconsequenceinthephilosophicalmilieu oftheViennaCircleandtreatsanumberofissuesrelatedtothepropercharacterization oflogicalconsequenceinTarskiduringthatperiod.Inanarchivalappendix,Chapter18, IpublishahithertounpublishedlecturebyTarski,whichcontainsimportantinformation onthenotionoflogicalconsequenceandonthedevelopmentofcentralnotionsinthe semantics of theories formulated in higher order logic. I would like to thank professor JanTarskithesonandManagerofAlfredTarski’sliteraryestate,forhavinggenerously grantedmepermissiontopublishthistext. Each part is prefaced by a summary of the contents of the articles and in addition contains a bibliographical update. These updates do not strive at completeness but Ihopetheyarecomprehensiveenoughtoindicatethemajornewcontributionsthathave appeared since the original publication of the articles. As a rule I have tried to repro- duce the articles without major modifications. When I felt, as in the case of Chapter2 and Chapter 16, that a substantial comment needed to be added, I have done so by appendinganAddendumtothosearticles.Other,veryminor,modificationsareindicated by an insertion in the text within double brackets, as in [[modified from the original introduction]].InafewcasesIhaveremovedinaccuraciesfromtheoriginallypublished version and replaced reference to my published articles by the corresponding chapters ofthisbookwithoutflaggingthisexplicitly.Thearticlesarereproducedintheirentirety with the only exception being Chapter 11 from which the first four sections have been removed.ThesesectionsareidenticalincontenttothefirstfoursectionsofChapter10, thus no loss in content ensues. The details of the original publications with permission to republish are contained in the Acknowledgments at the beginning of the book. The original acknowledgements, if any, appear as an uncued footnote at the bottom of the firstpageoftherelevantchapter.Thus,Iwillhereacknowledgeonlythosescholarswho helpedmesinceconceivingthisbook.IwouldliketothankPeterMomtchiloff,theeditor in Philosophy at OUP, Oxford, for having encouraged me to embark on this project starting with a conversation in Vancouver in November 2006 and for having followed viii Preface the project through with the Press. I would like to thank Solomon Feferman, Marcus Giaquinto, Donald Gillies, and Daniel Isaacson for having been so supportive of the project from its very inception and for having provided many excellent suggestions on avarietyofmatters.TomRyckmanandRichardZachhavehelpedmeimmenselywith comments on the introductions and the bibliographical updates. Fabrizio Cariani has takenon,onceagain,theungratefultaskofresettingtheoriginalarticlesintoabeautiful LATEX format. His work was partially supported by a COR grant at U.C. Berkeley; but only partially and I am very grateful for his unfailing dedication to the project. In this connection,IwouldalsoliketothankMarcoDettweilerandVolkerPeckhausforhaving provided us with the source files of Chapters 11 and 12 which saved Fabrizio from the drudgeryofresettingthepapersinLATEX.Duringthelateststagesofthepreparationof thebookIwasalsofortunatetoreceivethehelpofRebeccaMillsopwhosavedmefrom thepainfultaskofinsertinganalmostuncountablenumberoftinycorrectionsintothe source files. It is also a pleasure to acknowledge my co-authors for Chapter 1 (Richard ZachandCalixtoBadesa),Chapter6(MathieuMarion),andChapters10and11(Thomas Ryckman), for having allowed me to reprint our joint papers and, much more, for the intellectualstimulusandpleasurewesharedwhilewritingthem. Work on the final stages of the book has been made possible by a Guggenheim FellowshipandbyamembershipattheInstituteforAdvancedStudyinPrincetoninthe SpringTerm2009.Icannotthinkofbetterconditionsforbringingabooktocompletion thantheonesIwasfortunatetohave.IamgratefultotheGuggenheimFoundationand totheInstituteforAdvancedStudyfortheirgeneroussupportandtothelatterforhaving providedsuchidealworkingconditions. ThisbookisdedicatedtomywifeElenaRusso,forhavingbeenclosetome,through- outthecompositionofthisbook,inspirit,ifnotalwaysinperson. Princeton, InstituteforAdvancedStudy, March27,2009 ACKNOWLEDGMENTS Chapter 1 was originally published in L. Haaparanta (ed.), The Development of Modern Logic(Oxford:OxfordUniversityPress,2009),pp.318–470. Chapter2wasoriginallypublishedinP.Mancosu(ed.),FromBrouwertoHilbert(Oxford: OxfordUniversityPress,1998),pp.149–88. Chapter 3 was originally published in The Bulletin of Symbolic Logic, 5(3) (1999), pp.303–30.ReprintedwithkindpermissionoftheAssociationofSymbolicLogic. Chapter 4 was originally published in Synthese, 137 (2003), pp. 59–101. Reprinted with kind permission from Springer Science + Business Media, ©2003 Kluwer Academic Publishers. Chapter 5 was originally published in Wilfried Sieg, Richard Sommer, and Carolyn Talcott (eds.), Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman,AssociationforSymbolicLogic(2002),pp.349–71.Reprintedbykindpermis- sionoftheAssociationofSymbolicLogic. Chapter6wasoriginallypublishedinFriedrichStadler(ed.),TheViennaCircleandLogical Empiricism:Re-EvaluationandFuturePerspectives(Dordrecht:Kluwer,2003),pp.171–88. Chapter 7 was originally published in History and Philosophy of Logic, 20 (1999), pp.33–45.ReprintedwithkindpermissionofTaylorandFrancis. Chapter 8 was originally published in Notre Dame Journal of Formal Logic, 45(2) (2004), pp. 109–25. ©2004, Yale University. Reprinted by permission of the publisher, Duke UniversityPress. Chapter9wasoriginallypublishedinP.Mancosu(ed.),FromBrouwertoHilbert(Oxford: OxfordUniversityPress,1998),pp.65–85. Chapter 10 was originally published in Philosophia Mathematica, 10(3)(2002), pp.130–202. Chapter11wasoriginallypublishedinOskarBeckerunddiePhilosophiederMathematik,ed. Volker Peckhaus (Munich: Wilhelm Fink Verlag, 2005) (Neuzeit & Gegenwart Philoso- phie in Wissenschaft und Gesellschaft), pp. 153–228. Reprinted by kind permission of FinkVerlag. Chapter 12 was originally published in Oskar Becker und die Philosophie der Mathematik, ed. Volker Peckhaus (Munich: Wilhelm Fink Verlag, 2005) (Neuzeit & Gegenwart: PhilosophieinWissenschaftundGesellschaft),pp.229–43.Reprintedbykindpermission ofFinkVerlag. Chapter 13 was originally published in History and Philosophy of Logic, 26 (2005), pp.327–57.ReprintedwithkindpermissionofTaylorandFrancis.
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