In Fraser MacBride (ed.), Identity and Modality. Oxford University Press. pp. 34--69 (2006)

Stewart Shapiro
Ohio State University
According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of criticisms of the idea of ante rem structures have appeared. Some argue that it is impossible to give identity conditions for places in homogeneous ante rem structures, invoking a version of the identity of indiscernibles. Others raise issues concerning the identity and distinctness of places in different structures, such as the the natural number 2 and the real number 2. The purpose of this paper is to take the measure of these objections, and to further articulate ante rem structuralism to take them into account.
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What Are Structural Properties?†.Johannes Korbmacher & Georg Schiemer - 2018 - Philosophia Mathematica 26 (3):295-323.
Two Types of Abstraction for Structuralism.Øystein Linnebo & Richard Pettigrew - 2014 - Philosophical Quarterly 64 (255):267-283.
How to Be a Minimalist About Sets.Luca Incurvati - 2012 - Philosophical Studies 159 (1):69-87.
The Structuralist Thesis Reconsidered.Georg Schiemer & John Wigglesworth - 2019 - British Journal for the Philosophy of Science 70 (4):1201-1226.

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