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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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Epistemology and Reference

Epistemology and Reference

Chapter:
(p.109) 4 Epistemology and Reference
Source:
Philosophy of Mathematics
Author(s):

Stewart Shapiro (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/0195139305.003.0005

This chapter deals with matters of epistemology and reference, resolving the well‐known problems with realism or platonism due to Paul Benacerraf. A series of speculative epistemic strategies is presented, to cover the various mathematical structures. Small, finite structures are apprehended by pattern recognition, a form of simple abstraction. Larger finite structures and small infinite structures are grasped via some plausible extensions of pattern recognition. For example, a subject might grasp a method for extending a series of finite structures. The next strategy employs a linguistic abstraction similar to that of Robert Kraut and the neologicists Bob Hale and Crispin Wright. The most speculative strategy is implicit definition, a technique employed in the more abstract branches of mathematics, the branches with large ontologies. I provide a brief account of reference to mathematical objects, which is consistent with the realist ontology and epistemology of ante rem structuralism.

Keywords:   abstraction, Benacerraf, epistemology, Kraut, logicism, pattern, platonism, reference, structure, Wright

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