Reification, Curry-Howard Correspondence, and Didactical Consequences

Abstract

The roles of languages, processes, and objects in mathematical thinking have led to many theories, yet no consistent big picture has evolved from this. This paper puts forth the hypothesis that the Curry-Howard correspondence from computer science and the theories it is built on provide a unification framework. This correspondence asserts that (formal) proofs and programs (in functional programming languages) do not only have some similarities, but can, at least if formalized in an appropriate way, be mapped to each other by an isomorphism such that proofs are programs and vice versa. Moreover, objects can be realized as function evaluation strategies, and this provides a model of the reification process. The paper explores all this and discusses the didactical relevance; especially, the reification theories are revisited. Computer-based realizations of concepts are used as a tool to show the consistency of ideas and the practicability of concepts.