Philosophy of Mathematics

Philosophy of Mathematics : Structure and Ontology

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Do numbers, sets and so forth, exist? What do mathematical statements really 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 anti-realist 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 structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, and tautly argued, Shapiro's work is noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics. This book should be of interest to philosphers and to mathematicians.show more

Product details

  • Hardback | 289 pages
  • 150 x 230 x 25mm | 651g
  • Oxford University Press Inc
  • New York, United States
  • English
  • 0195094522
  • 9780195094527

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