Mathematical Representation: playing a role

The primary justification for mathematical structuralism is its ca-pacity to explain two observations about mathematical objects, typi-cally natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that at-tributing 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 inter-nal realism. I describe and argue for an alternative explanation for these features which instead explains the attributes them to the math-ematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on.