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Philosophy of MathematicsStructure and Ontology$
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Stewart Shapiro

Print publication date: 2000

Print ISBN-13: 9780195139303

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0195139305.001.0001

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Construction, Modality, Logic

(p.181) 6 Practice
Philosophy of Mathematics

Stewart Shapiro (Contributor Webpage)

Oxford University Press

This chapter concerns a gap between the practice of mathematics and its current philosophical and semantic foundations. Mathematicians sometimes speak as if they perform constructions and dynamic operations: lines are drawn, functions are applied, choices are made, and sets are formed. In contrast, the standard realism in ontology, or platonism, and the current semantics, holds or presupposes that mathematics concerns an independent, static realm. Traditional intuitionism suggests the opposite view, that mathematics really is mental construction, and thus the static orientation is misleading and inappropriate. I show how structuralism provides a single conceptual umbrella for the various static/dynamic, classical/intuitionistic systems, as well as the relevant idealizations. Michael Dummett's argument that an assertabilist orientation, based on Heyting semantics, leads to intuitionistic revisionism is challenged. One is lead to revisionism only if assertabilist semantics is combined with a certain pessimism concerning the idealized epistemic powers; a more optimistic philosophy supports classical logic.

Keywords:   assertabilism, construction, Dummett, Heyting, idealization, intuitionism, logic, Platonism, realism, semantics, structuralism

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