Mathematical Formalisms and their Realizations

Extract

In a short article, published in an earlier volume of Philosophy¹ under the title “Philosophy and Mathematics,” I tried to explain the current conception of pure mathematics as the study of abstract structure by construction and elaboration of appropriate axiomatic formalisms. In the present paper I propose to consider certain philosophical problems, of interest to philosophers and mathematicians alike, which have their origin in the relation between such formalisms and any applications to experience that they may possess. Consideration of problems of this kind is no new undertaking, and in Reichenbach's Wahrscheinlichkeitslehre, for instance, a considerable amount of space is devoted to the Anwendungsproblem or problem of application of the formal calculus under consideration. Most such discussions, however, are at bottom an appendage to an account of the formalism itself, and the author's interest is primarily mathematical. The result is that the philosophical issues involved are not given due weight; and it is these philosophical issues that I wish to discuss in the present paper.