Chapter

Epistemology and Reference

Stewart Shapiro

in Philosophy of Mathematics

Published in print October 2000 | ISBN: 9780195139303
Published online November 2003 | e-ISBN: 9780199833658 | DOI: http://dx.doi.org/10.1093/0195139305.003.0005
 Epistemology and Reference

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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

Chapter.  19195 words. 

Subjects: Philosophy of Mathematics and Logic

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