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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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(p.71) 3 Structure
Philosophy of Mathematics

Stewart Shapiro (Contributor Webpage)

Oxford University Press

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

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