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The main types of mathematical structuralism that have been proposed and developed to the point of permitting systematic and instructive comparison are four: structuralism based on model theory, carried out formally in set theory (e.g., first- or second-order Zermelo–Fraenkel set theory), referred to as STS (for set-theoretic structuralism); the approach of philosophers such as Shapiro and Resnik of taking structures to be *sui generis* universals, patterns, or structures in an *ante rem* sense (explained in this article), referred to as SGS (for *sui generis* structuralism); an approach based on category and topos theory, proposed as an alternative to set theory as an overarching mathematical framework, referred to as CTS (for category-theoretic structuralism); and a kind of eliminative, quasi-nominalist structuralism employing modal logic, referred to as MS (for modal-structuralism). This article takes these up in turn, guided by few questions, with the aim of understanding their relative merits and the choices they present.

*Keywords: *mathematical structuralism;
model theory;
set theory;
sui generis structuralism;
category-theoretic modal-structuralism structuralism

*Article.*
*12457 words.*

*Subjects: *Philosophy
; Philosophy of Mathematics and Logic
; Metaphysics

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