Mind 123 (489):39-78 (2014)

Edward Zalta
Stanford University
We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts
Keywords philosophy of mathematics  structuralism  foundations
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DOI 10.1093/mind/fzu003
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References found in this work BETA

The Principles of Mathematics.Bertrand Russell - 1903 - Cambridge, England: Allen & Unwin.
Essence and Modality.Kit Fine - 1994 - Philosophical Perspectives 8 (Logic and Language):1-16.
What Numbers Could Not Be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.

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Citations of this work BETA

Abstract Objects.Gideon Rosen - 2008 - Stanford Encyclopedia of Philosophy.
The Semantic Plights of the Ante-Rem Structuralist.Bahram Assadian - 2018 - Philosophical Studies 175 (12):1-20.

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