Today the standard interpretation of intuitionistic logic is the Brouwer-Heyting-Kolmogorov interpretation which was presented independently by Arend Heyting and Andrei Nikolajewitsch Kolmogorow. While Heyting was a student of Brouwer and an intuitionist himself, to me it does not look like Kolmogorov was an intuitionist.

Is there a text where Kolmogorov describes his own view in the philosophy of mathematics? Or, more general, is it known which views he was holding?


2 Answers 2


Kolmogorov was not exactly free to express his views considering the situation in the Soviet Union. Philosophical issues, even concerning mathematics, were ideologically sensitive, and everyone had to express allegiance, in one form or another, to the dialectical materialism of Marx and Engels. It went beyond that, as the only grand philosophy available it had genuine impact on the thinking of people immersed in Soviet life.

#1) "Extremes" of formalism and intuitionism

The closest we have is Kolmogorov's 1929 article Современные споры о природе математики, published before the total ideologization of 1930s. (English version Contemporary debates on the nature of mathematics, trans. V.A. Uspensky, 2006.) In it, he gives a very lucid presentation of the state of affairs in Hilbert's formalism vs Brouwer's intuitionism (platonism is not even considered as a serious contender), and criticizes both as "extreme", although with a clear sympathy to intuitionism. Still, his own position seems to be not intuitionism, but a form of empiricism, with "intuition" distilling ways of dealing with material objects through "idealization":

"The appearance of these extreme points of view is explained by the fact that the combination of both aspects of set-theoretic mathematics led to great difficulties and even contradictions. A common source of these difficulties is the following. Mathematicians are accustomed to dealing with numbers, functions, sets as as if they were things of the real world, in everything similar to material ones.

[...] From the epistemological side, Hilbert's point of view reduces to a strict limitation to the finite; all mathematical sentences in which the infinity enters one way or the other are declared devoid of any meaning. True, with a brilliant skill, Hilbert recovers rejected mathematical theories in the form of a formal consistent game of symbols. Yet, this way out, giving no explanation of what sustained mathematics to date, of why, while expressing judgments about infinity that have no meaning, mathematicians understood each other, is dictated only by inability to find a more satisfactory way out.

This makes us pay special attention to Brouwer, who, without shying away from the problem, promises to find out the nature of the infinite. But it is possible to doubt that intuition and the construction of new images, proceeding from a natural number, will turn out to be reliable guides. In particular, Brouwer studies the continuum in the form of infinite sequences of natural numbers, since only in this form it is natural to obtain it by purely logical means. Historically, the idea of ​​a continuum was created through idealization of truly observable continuous media; so far it is hard to imagine how this can provide a basis for the development of mathematical theory, but only this would be a direct path to understanding the nature of the mathematical continuum."

#2) Kolmogorov's "dialectical materialism"

In Conception of Mathematics of A.N. Kolmogorov (Russian) Baranets and Veryovkin try to disentangle Kolmogorov's own views from ideological declarations in his articles written for the Great Soviet Encyclopedia in 1930s and 50s. Ostensibly, he copies Engels's definition of mathematics from Dialectics of Nature, which is also empiricist in spirit, albeit outdated:

"Pure mathematics has as its subject matter spatial forms and quantitative relations of the real world, i.e. a very real content... to study these forms and relations in pure form, one should detach them from their content, eliminate it as something irrelevant to the matter".

To this Kolmogorov adds from himself:

"We will see that this very definition is fraught with development opportunities, acquiring a new, more broad meaning with the growth of science. We will also note narrower definitions, which mathematics has already outgrown... Both as a result of M.'s internal needs, as well as of new demands of natural science, the range of quantitative relations and spatial forms studied by M. is greatly expanded".

Moreover, Kolmogorov included mathematical logic, which does not exactly fit even into "expanded" Engels's definition, in his list of modern mathematical disciplines.

#3) Interpretation of intuitionism

On intuitionism specifically we also have Kolmogorov's works of a more technical character. In On the Principle "Tertium non Datur" (1925), where he gives interpretation (later rediscovered by Gödel) of classical theorems as intuitionistic theorems when their terms are replaced by their double negations, and concludes that "along with the Brouwerian presentation of mathematics without the tertium non datur principle it is necessary to retain the ordinary presentation, albeit only as a presentation of pseudo-truth mathematics", which is apparently his view of the classical mathematics. Later commenting on this paper in a letter to Heyting he remarks "I believe it is possible to go even further in this direction and so prove, from an intuitionistic point of view, the consistency of a large part of classical mathematics."

These and other remarks show that Kolmogorov's attitude was the opposite to that of Gödel with his platonism. While Gödel saw the classical interpretation of intuitionism as a limited validation of the latter, it was the opposite for Kolmogorov. In Zur Deutung der intuitionistischen Logik (1932) Kolmogorov also gives his own, non-Brouwerian, interpretation of intuitionism as a calculus not of statements but of "problems", expressing "the intention to find a certain construction", as he put it in another letter to Heyting. Intuitionism/constructivism emerges not as a logic of statements, let alone as self-sufficient foundation of mathematics, but as a logic of problems.

Uspenskii assembled a comprehensive bibliography of Kolmogorov's works with relation to philosophy (in Russian). Kolmogorov and mathematical logic is his commentary on the more technical works. Translations from Russian above are lightly edited Google translations.


Kolmogovov expressed his views in this paper:

MR2278817 Kolmogorov, A. N. Modern debates on the nature of mathematics. (Russian). With a commentary by V. A. Uspenskiĭ. Reprinted from Nauchnoe Slovo 1929, no. 6, 41–54. Problemy Peredachi Informatsii 42 (2006), no. 4, 129–141; translation in Probl. Inf. Transm. 42 (2006), no. 4, 379–389.

It is a reprint of his paper of 1929 (that is before Goedel theorem) with modern comments.

His views were not intuitionist. Formalization of intuitionist logic for him was a pure mathematical exercise. He actually wrote several papers on philosophy and foundation, expressing his general views on mathematics. The most famous is his entry "Mathematics" in the Great Soviet Encyclopedia:

MR2236304 Kolmogorov, Andrei Nikolaievich, Mathematics (Spanish). Translated from the Russian original (Great Soviet Encyclopedia of 1936). Gac. R. Soc. Mat. Esp. 9 (2006), no. 1, 108–141.

(I don't know whether an English translation exists. I was surprised that Encyclopedia Britannica (1960) had no entry with this title:-)

Remark. Soviet authorities did not permit any free discussion of philosophy by unauthorized scientists. In the Stalin period and some time later, the word "intuitionism" was banned in Soviet Union. Not as strictly as genetics, and they did not shoot people for intuitionism, but anyone using this word would have problems with the authorities and would be unable to publish. So soviet intuitionists (A. A. Markov Jr., N. A. Shanin, Esenin-Volpin) used an euphemism: "constructive mathematics". This remark may be helpful when you search the literature.


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