Curry's Formalism as Structuralism

In 1939, Curry proposed a philosophy of mathematics he called formalism. He made this proposal in two works originally written in 1939. These are the two philosophical works for which Curry is known, and they have left a false impression of his views. In this article, I propose to clarify Curry’s views by referring to some of his later writings on the subject. I claim that Curry’s philosophy was not what is now usually called formalism, but is really a form of structuralism. 1 Curry’s early philosophy of mathematics In his [1939], which is a shortened form of the original manuscript of his [1951], Curry proposed a philosophy of mathematics he called formalism. These two works, which represent Curry’s views in 1939, early in his career, are often the only works by which Curry’s philosophical ideas are known; see, for example, [Shapiro, 2000, Chapter 6, §5], where [Curry, 1951] is mistakenly identified as a mature work. ∗This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.