Structuralism Reconsidered

Fraser MacBride

in The Oxford Handbook of Philosophy of Mathematics and Logic

Published in print June 2007 | ISBN: 9780195325928
Published online September 2009 | e-ISBN: 9780199892082 | DOI:

Series: Oxford Handbooks

Structuralism Reconsidered

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The properties and relations that perform a role in mathematical reasoning arise from the basic relations that obtain among mathematical objects. It is in terms of these basic relations that mathematicians identify the objects they intend to study. The way in which mathematicians identify these objects has led some philosophers to draw metaphysical conclusions about their nature. These philosophers have been led to claim that mathematical objects are positions in structures or akin to positions in patterns. This article retraces their route from (relatively uncontroversial) facts about the identification of mathematical objects to high metaphysical conclusions. Beginning with the natural numbers, how are they identified? The mathematically significant properties and relations of natural numbers arise from the successor function that orders them; the natural numbers are identified simply as the objects that answer to this basic function. But the relations (or functions) that are used to identify a class of mathematical objects may often be defined over what appear to be different kinds of objects.

Keywords: structuralism; mathematical reasoning; mathematical objects; metaphysical conclusions; natural numbers; mathematical properties

Article.  13097 words. 

Subjects: Philosophy of Mathematics and Logic ; Metaphysics

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