# nLab Brouwer-Heyting-Kolmogorov interpretation

### Context

#### Constructivism, Realizability, Computability

intuitionistic mathematics

# Contents

## Idea

The Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic is a description of proofs of propositions in intuitionistic logic as functions, often computable functions, where it is also called the realizability interpretation.

This is otherwise known as the paradigm of propositions as types and proofs as programs, and in a precise form as the Curry-Howard correspondence. See there for more.

The name “Brouwer-Heyting-Kolmogorov” is due to Troelstra, and it is a matter of some dispute whether Brouwer’s name should be included. Brouwer never explicitly formulated any interpretation of this sort, and remained against all formalism his entire life. Moreover, Escardo-Xu have shown that Brouwer’s famous intuitionistic theorem “all functions $\mathbb{N}^{\mathbb{N}} \to \mathbb{N}$ are continuous” is actually inconsistent under a literal version of this interpretation (i.e. without including propositional truncation). Thus, perhaps it should only be called the “Heyting-Kolmogorov” interpretation.

## References

• Wikipedia, BHK interpretation

• L. E. J. Brouwer, Points and Spaces , CJM 6 (1954) pp.1-17. (pdf)

• H. Freudenthal , Zur intuitionistischen Deutung logischer Formeln , Comp. Math. 4 (1937) pp.112-116. (pdf)

• A. Heyting , Die intuitionistische Grundlegung der Mathematik , Erkenntnis 2 (1931) pp.106-115.

• A. Heyting , Bemerkungen zu dem Aufsatz von Herrn Freudenthal “Zur intuitionistischen Deutung logischer Formeln” , Comp. Math. 4 (1937) pp.117-118. (pdf)

• A. Kolmogoroff, Zur Deutung der intuitionistischen Logik , Math. Z. 35 (1932) pp.58-65. (gdz)

• G. Kreisel, Mathematical Logic , pp.95-195 in Saaty (ed.), Lectures on Modern Mathematics III , Wiley New York 1965.

• E. G. F. Díez, Five observations concerning the intended meaning of the intuitionistic logical constants , J. Phil. Logic 29 no. 4 (2000) pp.409–424 . (preprint)

• Jean-Yves Girard et al., Proofs and Types , CUP 1989.

• Anne Sjerp Troelstra, Principles of Intuitionism , Springer Heidelberg 1969. (§2)

• Anne Sjerp Troelstra, Aspects of Constructive Mathematics , pp.973-1052 in Barwise (ed.), Handbook of Mathematical Logic , Elsevier Amsterdam 1977.

• Anne Sjerp Troelstra, History of Constructivism in the Twentieth Century (1991). (preprint)

• Wouter Pieter Stekelenburg, Realizability Categories , (arXiv:1301.2134).

• Martin Escardo and Chuangjie Xu, The inconsistency of a Brouwerian continuity principle with the Curry–Howard interpretation . (pdf)

Links to many papers on realizability and related topics may be found here.