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.Author's Profile
DOI
10.1093/philmat/9.2.184
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Citations of this work
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Identity, indiscernibility, and Ante Rem structuralism: The tale of I and –I.Stewart Shapiro - 2008 - Philosophia Mathematica 16 (3):285-309.
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References found in this work
The Foundations of Arithmetic.Gottlob Frege - 1884/1950 - Evanston: Ill., Northwestern University Press.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press.
Counterpart theory and quantified modal logic.David K. Lewis - 1968 - Journal of Philosophy 65 (5):113-126.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press USA.