Three varieties of mathematical structuralism

Philosophia Mathematica 9 (2):184-211 (2001)
Abstract
Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it is argued that the modality of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious.
Keywords structuralism   WAYS
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DOI 10.1093/philmat/9.2.184
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Structuralism and the Notion of Dependence.Øystein Linnebo - 2008 - Philosophical Quarterly 58 (230):59-79.
Category Theory as an Autonomous Foundation.Øystein Linnebo & Richard Pettigrew - 2011 - Philosophia Mathematica 19 (3):227-254.

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