What we talk about when we talk about numbers

Abstract
In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
Keywords mathematical structuralism  instrumental nominalism  foundations of mathematics  ontology of mathematics  philosophy of mathematics  ante rem structuralism  eliminative structuralism  set theory
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