Philosophy of Mathematics: Structure and Ontology

Framsida
Oxford University Press, 7 aug. 1997 - 296 sidor
0 Recensioner
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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Innehåll

Mathematics and Its Philosophy
21
Object and Truth A Realist Manifesto
36
2 Methodology
38
3 Philosophy
44
4 Interlude on Antirealism
51
5 Quine
52
6 A Role for the External
57
Structure
71
3 A Tale of Two Debates
152
4 Dedekind and ante rem Structures
170
5 Nicholas Bourbaki
176
Practice Construction Modality Logic
181
2 Idealization to the Max
183
3 Construction Semantics and Ontology
185
4 Construction Logic and Object
189
5 Dynamic Language and Structure
193

Object
77
Structure
84
4 Theories of Structure
90
Structures All the Way Down
97
Function and Structure
106
Epistemology and Reference
109
Abstraction and Pattern Recognition
112
3 Long Strings and Large Natural Numbers
116
The Naturalnumber Structure
118
5 Indiscernability Identity and Object
120
6 Ontological Interlude
126
7 Implicit Definition and Structure
129
Coherence and Categoricity
132
Language Reference and Deduction
137
How We Got Here
143
2 Geometry Space Structure
144
6 Synthesis
198
7 Assertion Modality and Truth
203
8 Practice Logic and Metaphysics
211
Modality Structure Ontology
216
2 Modal Fictionalism
219
3 Modal Structuralism
228
4 Other Bargains
230
5 What Is a Structuralist to Make of All This?
235
Life Outside Mathematics Structure and Reality
243
2 Application and Structure
247
3 Borders
255
4 Maybe It Is Structures All the Way Down
256
References
263
Index
273
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Sidan 23 - It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence.

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