Kolmogorov's Calculus of Problems and Its Legacy

TL;DR

This paper explores Kolmogorov's interpretation of the Calculus of Problems, contrasting it with Heyting's, analyzing its philosophical implications, and discussing its influence and justification within modern Univalent Mathematics.

Contribution

It clarifies the differences between Kolmogorov's and Heyting's interpretations and reconstructs Kolmogorov's philosophical views and their impact on the development of mathematical logic.

Findings

Kolmogorov's interpretation diverges from Heyting's in treating problems and propositions separately.

The paper reconstructs Kolmogorov's philosophical stance on mathematics.

It proposes a justification of Kolmogorov's distinction using Univalent Mathematics.

Abstract

Kolmogorov's Calculus of Problems is an interpretation of Heyting's intuitionistic propositional calculus published by A.N. Kolmogorov in 1932. Unlike Heyting's intended interpretation of this calculus, Kolmogorov's interpretation does not comply with the philosophical principles of Mathematical Intuitionism. This philosophical difference between Kolmogorov and Heyting implies different treatments of problems and propositions: while in Heyting's view the difference between problems and propositions is merely linguistic, Kolmogorov keeps the two concepts apart and does not apply his calculus to propositions. I stress differences between Kolmogorov's and Heyting's interpretations and show how the two interpretations diverged during their development. In this context I reconstruct Kolmogorov's philosophical views on mathematics and analyse his original take on the Hilbert-Brouwer controversy. Finally, I overview some later works motivated by Kolmogorov's Calculus of Problems and propose a justification of Kolmogorov's distinction between problems and propositions in terms of Univalent Mathematics.