Outline of a Dynamical Inferential Conception of the Application of Mathematics

Abstract

We outline a framework for analyzing episodes from the history of science in which the application of mathematics plays a constitutive role in the concep-tual development of empirical sciences. Our starting point is the inferential conception of the application of mathematics, recently advanced by Bueno and Colyvan (2011). We identify and discuss some systematic problems of this approach. We propose refinements of the inferential conception based on theoretical considerations and on the basis of a historical case study. We demonstrate the usefulness of the refined, dynamical inferential conception using the well-researched example of the genesis of general relativity. Specif-ically, we look at the collaboration of the physicist Einstein and the math-ematician Grossmann in the years 1912–1913, which resulted in the jointly published “Outline of a Generalized Theory of Relativity and a Theory of Gravitation, ” a precursor theory of the final theory of general relativity. In this episode, an independently developed mathematical theory, the theory of differential invariants and the absolute differential calculus, was applied in the process of physical theorizing aiming at finding a relativistic theory of gravitation. We argue that the dynamical inferential conception not only provides a natural framework to describe and analyze this episode, but it also generates new questions and insights. We comment on the mathemat-ical tradition on which Grossmann drew, and on his own contributions to mathematical theorizing. We argue that the dynamical inferential concep