Abstract

The author of the work proposes a philosophical and methodological interpretation of the mathematical objects, using the system triad of the main directions of substantiation of mathematics: the formalism of Hilbert, Brouwer’s intuitionism and Godel’s Platonism. The need for these directions in the concept of substantiation of mathematics from the point of view of the current state of the philosophy of mathematics is shown on the mathematical examples. The philosophical and methodological analysis of objects of mathematics has never been unambiguous, therefore in this paper the results of studies of philosophers, logicians and mathematicians, in which the problem of substantiation is explicated in the context of trends in the development of mathematics, are used. Their professional view on philosophical characteristics of the objects of mathematics contributes to the identification of the unity of all mathematical knowledge maintaining the initial mathematical base of knowledge and revealing new ways of integrating the directions of substantiation in the philosophy of mathematics. The practical problem of substantiation of mathematics is realized through the elaboration of metatheoretical knowledge under the paradigm shift in the philosophy of mathematics to the productive direction from analysis to synthesis