How structuralism can solve the ‘access’ problem

Abstract

According to mathematical structuralism, the subject matter of mathematics is not the study of mathematical objects, but of mathematical structures. By moving away from objects, the structuralist claims to be in a position to solve the ‘access’ problem: structuralism explains the possibility of mathematical knowledge without requiring any access to mathematical objects. Fraser MacBride has challenged the structuralist response, and argued that the structuralist faces a dilemma in the attempt to solve that problem (MacBride, 2004). In the present paper, I argue that MacBride’s dilemma can be resisted, and that, particularly in the version articulated by Michael Resnik (Resnik, 1997), structuralism can solve the ‘access’ problem. I show exactly how MacBride’s dilemma fails, and argue that this failure provides an opportunity to highlight a significant feature of structuralism: the way in which it articulates a fundamentally different picture of mathematical epistemology than traditional epistemology would suggest.