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Understanding in mathematical science

Liberal Arts in Russia. 2017. Vol. 6. No. 1. Pp. 33-39.
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Sultanova L. B.
Bashkir State University
32 Zaki Validi Street, 450076 Ufa, Republic of Bashkortostan, Russia


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.


  • • humanitarian cognition
  • • knowledge of natural science
  • • philosophy of science
  • • subject of knowledge
  • • understanding of mathematical terms and symbols
  • • triad implicit knowledge
  • • illumination
  • • verbalization and explication of implicit elements
  • • deductive mathematics
  • • programs of mathematics justification


  1. Doich D. Struktura real'nosti [The structure of reality]. Izhevsk: Regulyarnaya i khaoticheskaya dinamika, 2001.
  2. Gusev S. S., Tul'chinskii G. L. Problema ponimaniya v filosofii: filosofsko-gnoseologicheskii analiz [The problem of understanding in philosophy: philosophical-epistemological analysis]. Moscow: Politizdat, 1985.
  3. Veil' G. Matematicheskoe myshlenie. Moscow: Nauka. Pp. 24-41.
  4. Lolli G. Filosofiya matematiki: nasledie dvadtsatogo stoletiya [Philosophy of mathematics: the legacy of the twentieth century]. N. Novgorod: Izd-vo Nizhegorodskogo gosuniversiteta im. N. I. Lobachevskogo, 2012.
  5. Sultanova L. B. Liberal Arts in Russia. 2014. No. 4. Pp. 237-245.
  6. Adamar Zh. Issledovanie psikhologii protsessa izobreteniya v oblasti matematiki. Moscow: Sovet-skoe radio, 1970.
  7. Geiting A. Intuitsionizm [The intuitionism]. Moscow: Librokom, 2010.