Mathematics: Formation, Substantiation, Resolution of the Crisis

The article discusses the current situation in mathematics, which is interpreted as a crisis. The problems that allow making such a conclusion are considered: the question of the scientific status of mathematics, the possibility of its substantiation, the mental or experimental nature of mathematics, the time of its formation. The author presents the results of the genesis of geometry, the generalization of which makes it possible to assert that mathematics, on the one hand, can be considered a kind of empirical science, on the other, a constructive scientific discipline, in which more complex ideal objects are created on the basis of initial ideal objects and knowledge. Two different understandings of the foundations of mathematics are considered - general scientific and David Hilbert’s, as well as what paradoxes are in science and how they are removed. The author is inclined to believe that antinomies in mathematics cannot be eliminated once and for all, their source is attributing inconsistent characteristics to objects of mathematics, which is due to the very structure of mathematics. The fact is that the construction of ideal objects of mathematics proceeds under the influence of at least four areas: the empirical area, describing which mathematicians create initial ideal objects and knowledge; areas of design based on the original more complex ideal objects; areas of geometric theorem proofs; area of construction of the theory of geometry. In addition, here it is necessary to add the concept of the foundation of mathematics by D. Hilbert, and also take into account that there are different concepts of the foundation of mathematics. At the end of the article, the features of the current crisis in mathematics and the possible direction of its resolution are discussed.