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The question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: (1) indispensability arguments that are aimed at justifying mathematics itself; (2) philosophical justifications of the successful application of mathematics to scientific theories; and (3) discussions on the application of real numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant’s position on the problem, several neo-Kantian perspectives (Helmholtz and Cassirer in particular, but also Otto Holder), early analytic philosophy (Frege), and late 19th century mathematicians (Grassmann, Dedekind, Hankel, and Bettazzi). Finally, the debate on the cogency of an application constraint in the definition of real numbers is discussed in relation to a contemporary debate in neo-logicism (Hale, Wright and some criticism by Batitksy), in order to suggest a comparison not only with Frege’s original positions, but also with the ideas of several neo-Kantian scholars, including Holder, Cassirer, and Helmholtz.

The Epistemological Question of the Applicability of Mathematics