Understanding in mathematical science

In the article, the phenomenon of understanding in mathematics is studied. This topic is relevant in contemporary philosophy of science, in which the classic dichotomy of “understanding-explanation”, characteristic of classical science, undergoes serious transformations. One reason for this transformation is a quantitative increase in the flow of information in modern science. It is clear that in such a situation you want to hold a serious understanding of this information in the context of modern scientific picture of the world. As the modern science is inextricably linked to mathematical science, it is required to rely on the results of studies of the phenomenon of understanding in mathematics. Such studies are already under way in the modern philosophical and scientific literature, although this issue is still relatively new. The author identifies the most important, in his view, types of understanding in math, taking into account the situation of the teaching of mathematics, the situation of mathematical discovery, and “mathematical symbolism”. A study of the main types of understanding in math and the identification of its specificity would have been impossible without the support of the concept of tacit knowledge taken as a methodological tool and author’s previously published works taken as a basis. In this article, the author relies on the classic work on the philosophy of mathematics (J. Hadamard, H. Weyl, A. Heyting), as well as the works of famous contemporary Russian and foreign scientists and philosophers (S. Gusev, D. Deutsch, G. Lolli, Tul’chinsky GL). The results of the study are summarized in the form of conclusions.