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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

(p.109) 4 Epistemology and Reference
Philosophy of Mathematics

Stewart Shapiro (Contributor Webpage)

Oxford University Press

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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