An Intuitionist Reasoning Upon Formal Intuitionist Logic: Logical Analysis of Kolmogorov’s 1932 Paper

Two dichotomies are considered as the foundations of a scientific theory: the kind of infinity—either potential or actual-, and the kind of organization of the theory—axiomatic or problem-based. The original intuitionist program relied on the choices of potential infinity and the problem-based organization. I show that the logical theory of Kolmogorov’s 1932 paper relied on the same choices. A comparison of all other theories sharing the same foundational choices allows us to characterize their common theoretical development through a few logical steps. The theory illustrated by Kolmogorov’s paper is then rationally re-constructed according to the steps of this kind of development. One obtains a new foundation of intuitionist logic, which is of a structural kind since it is based on and developed according to the structure of the above mentioned two fundamental choices. In addition, Kolmogorov’s illustration of his theory of intuitionist logic is an instance of rigorous reasoning of the intuitionist kind.