On interrelation of mathematical knowledge and objective reality

The problem of the ontological basis of mathematics is one of the central problems of math philosophy.Mathematics ontology is a special philosophical discipline aimed at revealing of general laws of mathematical object's being andmathematical reality as a whole. Its problem field is defined, first of all, by questions: "what is the mathematical object?" and "how doesit exist?" The review of the leading philosophical doctrines, each of which defends its own point of view on mathematical reality andways of its description, shows that the basic ideological opponents throughout the history of thought, since Antiquity, are representativesof realism (Plato, Leibniz, Frege, Russell, etc.) and constructivism (Kant, Brouwer, Weyl, etc.). It is shown that dispute of the considered competing research traditions can be considered in the context of the dialectic contradictions presented by the relation of pairs ofontological and modal categories ("individual-general", "subject-object", "essence-phenomenon", "final-infinite", "possibleimpossible",etc.). It is important to notice that the article does not deal with dialectic materialism, but with dialectics as a whole powerfulmethodology of rational knowledge. Some results reflected in the work show that dialectics is not only a reliable method of comprehensionof mathematical object's being in its development - it is a basis of the analysis of categorically represented system of extrahistoricalinvariant math bases. In work it is shown that interrelated categories of essence, phenomenon, relation, property, quality,quantity, measure, individual, plural, discrete, infinite, etc. enrich and constitute multidimensional mention of mathematical object'sbeing. Modal categories "necessary", "real" and "possible" play a special role in this process. The content of key mathematical conceptsreveals in their deep interrelation with basic philosophical categories. At the same time the author makes an attempt of estimation of therole of practice and experience in formation and development of mathematical knowledge. He comes to a conclusion that practice is themajor moment of the rational knowledge (including mathematical thinking) ontologisation. The perspective direction of research ofinterrelation between mathematical knowledge and objective reality is the "activity approach", in its non-metaphysical version removingcontradictions between extreme measures of apriorism and Platonism. According to it, the mathematics basis is absolute representationsreflecting universal requirements to objects of a reality from the point of view of human activity. Thus, possibility of practice is definedby pre-experience of mathematical thinking in its real interrelation with primary structures of activity. In preparing the article the authoraddressed to the works of Russian (V.V. Tselishchev, V.Ya. Perminov, I.N. Burova, A.G. Barabashev, etc.) and foreign (N. Mouloud, F.Kitcher, S. Kripke, etc.) experts engaged in the given problems.

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