ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 4, pp. 379–389.
Pleiades Publishing, Inc., 2006.
Original Russian Text
A.N. Kolmogorov, 1929, V.A. Uspenskiy, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 4, pp. 129–141.
Contemporary Debates on the Nature of Mathematics
From the Editors of Problems of Information Transmission
Kolmogorov’s article “Contemporary Debates on the Nature of Mathematics” was published
in 1929 in Nauchnoe slovo journal (no. 6, pp. 41–54) and has not been republished since then.
At the end of the 19th and at the beginning of the 20th century, abstractions of a very high
order appeared and were rooted in mathematics, so that their correlation with reality became a
crucial question. A number of great mathematicians of the time were interested in this question,
and Kolmogorov’s article was quite up to date. The goal of the republication of this article,
which is currently diﬃcult to access, is not only to acquaint the reader with a scarcely known
article of the great scientist but to invite him to observe the process of scientiﬁc progress.
It should be noted that, only a year after the publication of the article, some astonishing
results of G¨odel appeared that gave answers to a number of questions in Kolmogorov’s article
and brought mathematical strictness in the very formulation of these questions. Therefore, the
article is accompanied by external comments in order to give an insight into the current state
of the art. In the text the comments are indicated by numbers in brackets, from  to .
As for the rest, Kolmogorov’s text is published in the same way as it was published in Nauchnoe
slovo, without any alterations of the sense. Accompanying comments given to the article by
the editors of Nauchnoe slovo (footnotes and introduction to the article) are omitted.
The claim of mathematics for immutability and general signiﬁcance of its issues had never before
undergone such severe ordeals as today. It was not by coincidence that the French mathematician
Hadamard  made a conjecture that the reason for some mathematical debates is the diﬀerence
of osmotic pressure in brain cells or some other diﬀerence that is as hard to rule out by logical
evidence. Even considering this conjecture to be somewhat jocular, the impossibility to agree on
some questions is itself acutely felt by many. Even in 1905 in “Five Letters on the Set Theory” ,
several French mathematicians (including Hadamard and Borel) gave opinions that were exactly
opposite to the so-called “principle of arbitrary choice” suggested by Zermelo not long before.
Things evident and not needing any proof from the point of view of Hadamard seemed not evident
at all and even senseless to Borel. Lebesgue and Baire in their letters expressed a new kind of opinion
on the same question. All the opinions above remain without agreement up to the moment.
However, inﬁnitesimal calculus in the ﬁrst stage of its development resulted in debates and
disagreement as well. However, at that time the reason was only in the lack of suﬃciently rigorous
deﬁnitions. The above-mentioned deﬁciency was realized by allies of the method themselves and
was solved in the 19th century. Presently, inﬁnitesimal calculus is deﬁned as strictly as older areas
of mathematics, and there is no misunderstanding about the sense of its main notions. In order
to attain this, it was suﬃcient to do purely mathematical work, more precisely, to give good
deﬁnitions and formulate an exhaustive system of admissions on which logical constructions could
Comments 3, 5, and 6, in which the way of Cyrillic writing of foreign family names is considered, are
omitted in the English version, as well as comment 18, which concerns Russian terminology usage.