Modern Debates on the Nature of Mathematics (Comments by V. A. Uspenskii)

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 Gödel 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 [1] to [21]. 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.