Chapter

Structure

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/10.1093/0195139305.003.0004
 Structure

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This chapter articulates structuralism, with focus on ontological matters. One group of issues concerns the status of structures themselves and another concerns the status of mathematical objects, the places within structures. The most perspicuous view takes structures to be like ante rem universals, existing independent of any systems that exemplify them. Other in re or eliminative views deny that structures exist independent of any systems or, invoking modality, possible systems that exemplify them. Strictly speaking, according to such views, structures do not exist. An axiomatization of ante rem structuralism is provided, in order to compare it to standard, set‐theoretic foundations of mathematics. Structuralism is compared to functionalism in the philosophy of mind.

Keywords: ante rem; axiomatization; foundations; functionalism; modality; object; ontology; platonism; structure; structuralism

Chapter.  21632 words. 

Subjects: Philosophy of Mathematics and Logic

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