Philosophy of Mathematics: Structure and Ontology - Hardcover

Shapiro, Stewart

4.17

24 ratings by Goodreads

9780195094527: Philosophy of Mathematics: Structure and Ontology

Hardcover

ISBN 10: 0195094522 ISBN 13: 9780195094527

Publisher: Oxford University Press, 1997

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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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About the Author:

Stewart Shapiro, Professor of Philosophy, Ohio State University at Newark.

Review:

"This book is an important contribution...presenting an original, structuralist philosophy and axiomatic framework in comprehensive detail, placing it in broad philosophical and historical perspective, and comparing it systematically with other approaches seen as leading structuralist
alternatives to the one set forth by Shapiro himself....this is an interesting, important, and thought-provoking book."--Journal of Symbolic Logic

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