Philosophy of mathematics : structure and ontology / Stewart Shapiro.

Format:

Book

Author/Creator:

Shapiro, Stewart, 1951-

Language:

English

Subjects (All):

Mathematics--Philosophy.

Mathematics.

Physical Description:

1 online resource (x, 279 p.)

Edition:

1st ed.

Place of Publication:

Oxford : Oxford University Press, 1997.

Language Note:

English

Summary:

This 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. 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.

Contents:

Intro

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.

R

S

T

U

V

W

Z.

Notes:

Originally published: 1997.

Includes bibliographical references and index.

Description based on metadata supplied by the publisher and other sources.

ISBN:

1-280-45096-7

9786610450961

0-19-802545-9

1-4237-6518-4

OCLC:

922953120