This article discusses the role and place of intuition in mathematics, the ratio of intellectual thinking and intuitive contemplation. This study involves the relation of logical and intuitive knowledge, as well as determining the objectivity of truth obtained without evidence, then, with the help of intuition. The most important aim of the work, including the novelty, is to attempt a philosophical explanation of new discoveries, identifying logical and intuitive process of "enlightenment" of mathematical creativity. Contemporary Mathematics retains many features of previous epochs, but has its own characteristics. It is an extremely high level of abstraction and idealization of science, of considerable complexity of new mathematical models. Intuition peculiarity is detected when forming mathematical theories. We can give as specific examples four theories of mathematics, which have already proved this fact. We mean the theory of categories, non-standard analysis, the theory of odd sets and catastrophe theory. The algebraic theory of categories appearance, its heuristic potential testified the fact that mathematical knowledge is carried out in accordance with the basic principles and laws of materialist dialectics. On the one hand, being dependent on the set theory, it serves its further generalization, releasing mathematics from the individual (specific) form. As a result, the categories theory itself acquires the ability to be suitable for the study of a wide class of mathematical theories. Category theory destroyed dogmatic notions of set theory universality and uniqueness to be the foundation of all mathematics. It has laid a new foundation for both the theory of sets, and mathematics as a whole. Its heuristic opportunity was expressed in the ability to describe the properties and relations of mathematical objects from various areas of science. New horizons were opened for algebraic geometry, mathematical logics; it became possible for the theory of sets to construct a new type of models. Nowadays it is undoubtedly possible to use the fuzzy sets theory in computer systems in automatic control systems (eg, trains, banking industry), in medical diagnosis, in chemical engineering, psychology, sociology and other fields of human activity. Catastrophe theory is considered as a set of mathematical and physical ideas that have "exit" in geometry, algebra, analysis, topology, singularity theory, bifurcation theory, nonequilibrium thermodynamics, synergetic, dynamical systems theory and other fields of science. The author emphasizes the specificity of mathematical intuition to understand the world. Interpretation of mathematical knowledge in the tradition of non-classical epistemology, especially through intuition greatly expands the horizons of knowledge. The intellectual intuition plays a special role in explaining and justifying of the latest discoveries. The history of the mathematical knowledge development shows how important the role of intuition in mathematics is than in any other area of scientific knowledge. The intuition regularities identified on the basis of mathematical discoveries can be useful beyond basic sciences themselves.
