Philosophy of Mathematics: Structure and Ontology

Stewart Shapiro9780195094527, 0-19-509452-2

Do 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.

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Table of contents :
Contents……Page 10
Introduction……Page 14
PART I: PERSPECTIVE……Page 30
1 Mathematics and Its Philosophy……Page 32
1 Slogans……Page 47
2 Methodology……Page 49
3 Philosophy……Page 55
4 Interlude on Antirealism……Page 62
5 Quine……Page 63
6 A Role for the External……Page 68
PART II: STRUCTURALISM……Page 80
1 Opening……Page 82
2 Ontology: Object……Page 88
3 Ontology: Structure……Page 95
4 Theories of Structure……Page 101
5 Mathematics: Structures, All the Way Down……Page 108
6 Addendum: Function and Structure……Page 117
1 Epistemic Preamble……Page 120
2 Small Finite Structure: Abstraction and Pattern Recognition……Page 123
3 Long Strings and Large Natural Numbers……Page 127
4 To the Infinite: The Natural-number Structure……Page 129
5 Indiscernibility, Identity, and Object……Page 131
6 Ontological Interlude……Page 137
7 Implicit Definition and Structure……Page 140
8 Existence and Uniqueness: Coherence and Categoricity……Page 143
9 Conclusions: Language, Reference, and Deduction……Page 148
1 When Does Structuralism Begin?……Page 154
2 Geometry, Space, Structure……Page 155
3 A Tale of Two Debates……Page 163
4 Dedekind and ante rem Structures……Page 181
5 Nicholas Bourbaki……Page 187
PART III: RAMIFICATIONS AND APPLICATIONS……Page 190
1 Dynamic Language……Page 192
2 Idealization to the Max……Page 194
3 Construction, Semantics, and Ontology……Page 196
4 Construction, Logic, and Object……Page 200
5 Dynamic Language and Structure……Page 204
6 Synthesis……Page 209
7 Assertion, Modality, and Truth……Page 214
8 Practice, Logic, and Metaphysics……Page 222
1 Modality……Page 227
2 Modal Fictionalism……Page 230
3 Modal Structuralism……Page 239
4 Other Bargains……Page 241
5 What Is a Structuralist to Make of All This?……Page 246
1 Structure and Science—the Problem……Page 254
2 Application and Structure……Page 258
3 Borders……Page 266
4 Maybe It Is Structures All the Way Down……Page 267
References……Page 274
C……Page 284
E……Page 285
I……Page 286
N……Page 287
R……Page 288
T……Page 289
Z……Page 290