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What is mathematics?

Liberal Arts in Russia. 2015. Vol. 4. No. 5. Pp. 358-367.
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Antakov S. M.
Nizhny Novgorod State University
23 Gagarin Ave. 2 Bldg., 603950 Nizhny Novgorod, Russia


This article does not give the answer to the title question, but is only limited to studying the possibility of giving it. In particular, the author defends that it is legitimate to pose the fundamental question of the philosophy of mathematics and offers several criteria for such a question. As a first approach we propose the question which is incorrect and requires rectification, but is understandable: “What is Mathematics?”. We consider three groups of strategies of responding to it: 1) the question is naive and does not require an answer; 2) the answer is contained in the mathematical knowledge itself; 3) only philosophy can give the answer. Further rectifications to the basic question are questions about the essence and existence of mathematics. The fundamentalist and cultural-historical philosophy of mathematics gave their not fully satisfactory answers to this question in the past century. The dualism problem of philosophy of mathematics is fundamentally resolved through a certain kind of dialectics, which requires the philosophy of mathematics to submit to metaphysics. The unity of existence and essence is considered in the unity of the subject of dialectic - the mathematician who generates the knowledge as self-realization in a series of existential choices, as a means of saving transcendence - the existence “toward being” rather than “toward nothing”. Positive view of the mathematics turns out to be the explicit expression of negative (non-objective) unity i.e. the subject.


  • • the fundamental question of philosophy
  • • metaphysics
  • • fundamentalist philosophy of mathematics
  • • cultural-historical philosophy of mathematics
  • • positive philosophy
  • • dialectics
  • • essence and existence
  • • existential choice
  • • salvation


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