Colin McCullough-Benner

According to standard accounts of mathematical representations of physical phenomena, positing structure-preserving mappings between a physical target system and the structure(s) picked out by a mathematical theory is essential to such representations. In this paper, I argue that these accounts fail to give a satisfactory explanation of scientific representations that make use of inconsistent mathematical theories and present an alternative, robustly inferential account of mathematical representation that provides not just a better explanation of applications of inconsistent mathematics, but also a compelling explanation of mathematical representations of physical phenomena in general. 1Inconsistent Mathematics and the Problem of Representation 2The Early Calculus 3Mapping Accounts and the Early Calculus 3.1Partial structures 3.2Inconsistent structures 3.3Related total consistent structures 4A Robustly Inferential Account of the Early Calculus in Applications 4.1The robustly inferential conception of mathematical representation 4.2The robustly inferential conception and inconsistent mathematics 4.3The robustly inferential conception and mapping accounts 5Beyond Inconsistent Mathematics.

Several philosophers have argued that to capture the generality of certain scientific explanations, we must count mathematical facts among their explanantia. I argue that we can better understand these explanations by adopting a more nuanced stance toward mathematical representations, recognizing the role of mathematical representation schemata in representing highly abstract features of physical systems. It is by picking out these abstract but nonmathematical features that explanations appealing to mathematics achieve a high degree of generality. The result is a rich conception of the role of mathematics in scientific explanations that does not require us to treat mathematical facts as explanantia.