LEADER 05308nam 2200601Ia 450 001 9910810174403321 005 20200520144314.0 010 $a1-280-45096-7 010 $a9786610450961 010 $a0-19-802545-9 010 $a1-4237-6518-4 035 $a(CKB)1000000000460559 035 $a(StDuBDS)AH24085063 035 $a(SSID)ssj0000221045 035 $a(PQKBManifestationID)12059909 035 $a(PQKBTitleCode)TC0000221045 035 $a(PQKBWorkID)10160222 035 $a(PQKB)10330997 035 $a(MiAaPQ)EBC3052385 035 $a(Au-PeEL)EBL3052385 035 $a(CaPaEBR)ebr10212170 035 $a(CaONFJC)MIL45096 035 $a(OCoLC)922953120 035 $a(EXLCZ)991000000000460559 100 $a19960709d1997 uy 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt 182 $cc 183 $acr 200 10$aPhilosophy of mathematics $estructure and ontology /$fStewart Shapiro 205 $a1st ed. 210 $aOxford $cOxford University Press$d1997 215 $a1 online resource (x, 279 p.) 300 $aOriginally published: 1997. 311 $a0-19-509452-2 320 $aIncludes bibliographical references and index. 327 $aIntro -- Contents -- Introduction -- PART I: PERSPECTIVE -- 1 Mathematics and Its Philosophy -- 2 Object and Truth: A Realist Manifesto -- 1 Slogans -- 2 Methodology -- 3 Philosophy -- 4 Interlude on Antirealism -- 5 Quine -- 6 A Role for the External -- PART II: STRUCTURALISM -- 3 Structure -- 1 Opening -- 2 Ontology: Object -- 3 Ontology: Structure -- 4 Theories of Structure -- 5 Mathematics: Structures, All the Way Down -- 6 Addendum: Function and Structure -- 4 Epistemology and Reference -- 1 Epistemic Preamble -- 2 Small Finite Structure: Abstraction and Pattern Recognition -- 3 Long Strings and Large Natural Numbers -- 4 To the Infinite: The Natural-number Structure -- 5 Indiscernibility, Identity, and Object -- 6 Ontological Interlude -- 7 Implicit Definition and Structure -- 8 Existence and Uniqueness: Coherence and Categoricity -- 9 Conclusions: Language, Reference, and Deduction -- 5 How We Got Here -- 1 When Does Structuralism Begin? -- 2 Geometry, Space, Structure -- 3 A Tale of Two Debates -- 4 Dedekind and ante rem Structures -- 5 Nicholas Bourbaki -- PART III: RAMIFICATIONS AND APPLICATIONS -- 6 Practice: Construction, Modality, Logic -- 1 Dynamic Language -- 2 Idealization to the Max -- 3 Construction, Semantics, and Ontology -- 4 Construction, Logic, and Object -- 5 Dynamic Language and Structure -- 6 Synthesis -- 7 Assertion, Modality, and Truth -- 8 Practice, Logic, and Metaphysics -- 7 Modality, Structure, Ontology -- 1 Modality -- 2 Modal Fictionalism -- 3 Modal Structuralism -- 4 Other Bargains -- 5 What Is a Structuralist to Make of All This? -- 8 Life Outside Mathematics: Structure and Reality -- 1 Structure and Science-the Problem -- 2 Application and Structure -- 3 Borders -- 4 Maybe It Is Structures All the Way Down -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q. 327 $aR -- S -- T -- U -- V -- W -- Z. 330 8 $aThis text argues that both realist and anti-realist accounts of mathematics are problematic. It articulates a structuralist approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers.$bDo numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. 606 $aMathematics$xPhilosophy 606 $aMathematics 615 0$aMathematics$xPhilosophy. 615 0$aMathematics. 676 $a510/.1 700 $aShapiro$b Stewart$f1951-$0447519 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910810174403321 996 $aPhilosophy of mathematics$9105548 997 $aUNINA