The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead explains the (...) attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on. (shrink)

The aim of the paper is to answer some arguments raised against mathematical structuralism developed by Michael Resnik. These arguments stress the abstractness of mathematical objects, especially their causal inertness, and conclude that mathematical objects, the structures posited by Resnik included, are inaccessible to human cognition. In the paper I introduce a distinction between abstract and ideal objects and argue that mathematical objects are primarily ideal. I reconstruct some aspects of the instrumental practice of mathematics, such as symbolic manipulations or (...) ruler-and-compass constructions, and argue that instrumental practice can secure epistemic access to ideal objects of mathematics. (shrink)