Oxford Handbook of Philosophy of Math and Logic PDF
Preview Oxford Handbook of Philosophy of Math and Logic
The Oxford Handbook of Philosophy of Mathematics and Logic STEWART SHAPIRO, Editor OXFORD UNIVERSITY PRESS the oxford handbook of ....................................................................................................................................... PHILOSOPHY OF MATHEMATICS AND LOGIC ....................................................................................................................................... ....................................................................................................................................................... OXFORD HANDBOOKS IN PHILOSOPHY PAUL K. MOSER, general editor ....................................................................................................................................................... Series AdvisoryBoard ROBERT AUDI University ofNebraska MARTHA NUSSBAUM University ofChicago ALVIN PLANTINGA University ofNotre Dame ERNEST SOSA BrownUniversity the oxford handbook of ....................................................................................................................................... PHILOSOPHY OF MATHEMATICS AND LOGIC ....................................................................................................................................... Edited by STEWART SHAPIRO 1 2005 1 OxfordUniversityPress,Inc.,publishesworksthatfurther OxfordUniversity’sobjectiveofexcellence inresearch,scholarship,andeducation. 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 Copyright #2005 byOxford University Press, Inc. PublishedbyOxfordUniversityPress,Inc. 198MadisonAvenue,NewYork,NewYork10016 www.oup.com OxfordisaregisteredtrademarkofOxfordUniversityPress Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, electronic,mechanical,photocopying,recording,orotherwise, withoutthepriorpermissionofOxfordUniversityPress. LibraryofCongressCataloging-in-PublicationData TheOxfordhandbookofphilosophyofmathandlogic / editedbyStewartShapiro. p. cm.—(Oxfordhandbooksinphilosophy) Includesbibliographicalreferencesandindex. ISBN-13 978-0-19-514877-0 ISBN0-19-514877-0 1.Mathematics—Philosophy. 2.Logic,Symbolicandmathematical—Philosophy. I.Shapiro,Stewart,1951– II.Series. QA8.4.O94 2004 510'.1—dc22 2004044847 1 3 5 7 9 8 6 4 2 PrintedintheUnitedStatesofAmerica onacid-freepaper P r e f a c e .................................... This volume provides comprehensive and accessible coverage of the disciplines of philosophy of mathematics and philosophy of logic, including an overview of the majorproblems,positions,andbattlelines.Inlinewiththeunderlyingthemeofthe series,eachauthorwasgivenafreehandtodevelophisorherdistinctiveviewpoint. Thus, the various chapters are not neutral.Readerssee exposition and criticism, as well as substantial development of philosophical positions. I am pleased to report thateachchapterbreaksnewground.Thevolumenotonlypresentsthedisciplines of philosophy of mathematics and philosophy of logic, but advances them as well. Formanyofthemajor positionsinthephilosophyofmathematics andlogic, the book contains at least two chapters, at least one sympathetic to the view and onecritical.Ofcourse,thisdoesnotguaranteethateverymajorviewpointisgiven a sympathetic treatment. For example, one of my own pet positions, ante rem structuralism, comes in for heavy criticism in two of the chapters, and is not defendedanywhere(exceptbrieflyinchapter1).Inlightofthedepthandextentof thedisciplinestoday,nosinglevolume,orseriesofvolumes,canprovideextensive and sympathetic coverage of even the major positions on offer. And there would hardly be a point to such an undertaking, since the disciplines are ever evolving. Newpositions andnew criticisms ofold positionsemergewitheach issueofeach majorphilosophyjournal.Mostofthechapterscontainanextensivebibliography. In total, this volume provides a clear picture of the state of the art. There is some overlap between the chapters. This is to be expected in a work ofthisscope,anditwasexplicitlyencouraged.Authorsoftendrawinteresting,but distinctive, conclusions from the same material. There is, of course, no sharp separation between the philosophy of mathematics and the philosophy of logic. The main issues and views of either discipline permeate those of the other. Just about every chapter deals with matters mathematical and matters logical. After the Introduction (chapter 1), the book begins with a historical section, consisting of three chapters. Chapter 2 deals with the modern period—Kant and his intellectual predecessors; chapter 3 concerns later empiricism, including John StuartMillandlogicalpositivism;andchapter4focusesonLudwigWittgenstein. The volume then turns to the ‘‘big three’’ views that dominated the philoso- phyandfoundationsofmathematicsintheearlydecadesofthetwentiethcentury: logicism, formalism, and intuitionism. There are three chapters on logicism, one vi preface dealing with the emergence of the program in the work of Frege, Russell, and Dedekind (chapter 5); one on neologicism, the contemporary legacy of Fregean logicism (chapter 6); and one called ‘‘Logicism Reconsidered,’’ which provides a technical assessment of the program in its first century (chapter 7). This is fol- lowedbyalengthychapteronformalism,coveringitshistoricalandphilosophical aspects(chapter8).Twoofthethreechaptersonintuitionismoverlapconsiderably. Thefirst(chapter9)providesthephilosophicalbackgroundtointuitionism,through the work of L. E. J. Brouwer, Arend Heyting, and others. The second (chapter 10) takes a more explicitly mathematical perspective. Chapter 11, ‘‘Intuitionism Re- considered,’’ focuses largely on technical issues concerning the logic. The next section of the volume deals with views that dominated in the later twentieth century and beyond. Chapter 12 provides a sympathetic reconstruction of Quinean holism and indispensability. This is followed by two chapters that focusdirectlyonnaturalism.Chapter13laysouttheprinciplesofsomeprominent naturalists, and chapter 14 is critical of the main themes of naturalism. Next up are nominalism and structuralism, which get two chapters each. One of these is sympathetic to at least one variation on the view in question, and the other ‘‘reconsiders.’’ Chapter19isadetailedandsympathetictreatmentofapredicativeapproachto both the philosophy and the foundations of mathematics. This is followed by an extensive treatment of the application of mathematics to the sciences; chapter 20 lays out different senses in which mathematics is to be applied, and draws some surprising philosophical conclusions. The last six chapters of the volume focus more directly on logical matters, in three pairs. There are two chapters devoted to the central notion of logical consequence. Chapter 21 presents and defends the role of semantic notions and model theory, and chapter 22 takes a more ‘‘constructive’’ approach, leading to prooftheory.Thenexttwochaptersdealwiththeso-calledparadoxesofrelevance, chapter 23 arguing that the proper notion of logical consequence carries a notion of relevance, and chapter 24 arguing against this. The final two chapters concern higher-order logic. Chapter 25 presents higher-order logic and provides an over- viewofitsvarioususesinfoundationalstudies.Ofcourse,chapter26reconsiders. Throughout the process of assembling this book, I benefited considerably from the sage advice of my editor, Peter Ohlin, of Oxford University Press, USA, and from my colleagues and friends, at Ohio State, St. Andrews, and other institu- tions. Thanks especially to Penelope Maddy and Michael Detlefsen. C o n t e n t s ........................................... Notes on the Contributors, xi 1. Philosophy of Mathematics and Its Logic: Introduction, 3 Stewart Shapiro 2. ApriorityandApplication:PhilosophyofMathematicsinthe Modern Period, 28 Lisa Shabel 3. Later Empiricism and Logical Positivism, 51 John Skorupski 4. Wittgenstein on Philosophy of Logic and Mathematics, 75 Juliet Floyd 5. The Logicism of Frege, Dedekind, and Russell, 129 William Demopoulos and Peter Clark 6. Logicism in the Twenty-first Century, 166 Bob Hale and Crispin Wright 7. Logicism Reconsidered, 203 Agust(cid:1)ıın Rayo 8. Formalism, 236 Michael Detlefsen 9. Intuitionism and Philosophy, 318 Carl Posy 10. Intuitionism in Mathematics, 356 D. C. McCarty viii contents 11. Intuitionism Reconsidered, 387 Roy Cook 12. Quine and the Web of Belief, 412 Michael D. Resnik 13. Three Forms of Naturalism, 437 Penelope Maddy 14. Naturalism Reconsidered, 460 Alan Weir 15. Nominalism, 483 Charles Chihara 16. Nominalism Reconsidered, 515 Gideon Rosen and John P. Burgess 17. Structuralism, 536 Geoffrey Hellman 18. Structuralism Reconsidered, 563 Fraser MacBride 19. Predicativity, 590 Solomon Feferman 20. Mathematics—Application and Applicability, 625 Mark Steiner 21. Logical Consequence, Proof Theory, and Model Theory, 651 Stewart Shapiro 22. Logical Consequence From a Constructivist Point of View, 671 Dag Prawitz 23. Relevance in Reasoning, 696 Neil Tennant contents ix 24. No Requirement of Relevance, 727 John P. Burgess 25. Higher-order Logic, 751 Stewart Shapiro 26. Higher-order Logic Reconsidered, 781 Ignacio Jane´ Index, 811
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