Structural realists typically appeal to the explanatory and predictive success of science to suggest that the mathematical structure of scientific theory, which is continuous across theory change, provides an accurate description of some aspect of the structure of the world. In this paper, I present a challenge to this claim that concerns how the relevant structure in nature is identified and represented in the context of a physical theory. I argue that the structures, on which many structural realists base the historical support for their position, can only be taken to represent "physical structures" in the context of a broader theoretical framework and that this framework is not necessarily preserved through theory change.

Structuralists typically appeal to some variant of the widely popular 'mapping' account of mathematical representation to suggest that mathematics is applied in modern science to represent the world's physical structure. However, in this paper, I argue that this realist interpretation of the 'mapping' account presupposes that physical systems possess an 'assumed structure' that is at odds with modern physical theory. Through two detailed case studies concerning the use of the differential and variational calculus in modern dynamics, I show that the formal structure that we need to assume in order to apply the mapping account is inconsistent with the way in which mathematics is applied in modern physics. The problem is that a realist interpretation of the 'mapping' account imposes too severe of a constraint on the conformity that must exist between mathematics and nature in order for mathematics to represent the structure of a physical system.