Recursive functions and constructive mathematics
pp. 159-167
in: Jacques Dubucs, Michel Bourdeau (eds), Constructivity and computability in historical and philosophical perspective, Berlin, Springer, 2014Abstract
The goal of this paper is to discuss the following question: is the theory of recursive functions needed for a rigorous development of constructive mathematics? I will try to present the point of view of constructive mathematics on this question. The plan is the following: I first explain the gradual loss of appreciation of constructivity after 1936, clearly observed by Heyting and Skolem, in connection with the development of recursivity. There is an important change in 1967, publication of Bishop's book, and the (re)discovery that the theory of recursive functions is actually not needed for a rigorous development of constructive mathematics. I then end with a presentation of the current view of constructive mathematics: mathematics done using intuitionistic logic, view which, surprisingly, does not rely on any explicit notion of algorithm.