Intuition and Heuristics in Mathematics

The article is devoted to philosophy of mathematics. Mathematical heuristics, being a complex of methods for solving the non-standard problems of mathematics (such problems which have no known algorithms to be solved), is the main subject of the research. As a specific mechanism for thinking, generating elements of guesswork needed as the basis of mathematical heuristics, the author considers intuition. In the work, the author uses Descartes’s, Poincaré’s, Hadamard’s and Piaget’s findings. Based on Descartes’s concept of rational intuition, the author develops the concept of heuristic intuition. As a result, the author turns to the question of possibility of a complete translation of the user-derived mathematical statements in a discourse, in fact, that means a maximum depth of mathematical proof, i.e. its maximum rationalization. For this purpose, it is necessary to re-attract the intuition since it is able to transform the intuitive elements into the discourse ones. Therefore, from this point of view, the rationale is intuitively derived mathematical proof should be no more than a “multilayer” creative process. In general, the author, based on Poincaré’s research, proves that the essence of mathematical creativity is not to «sort out» and «choose». Referring to examples for illustration, the author reveals moments of «interference» of intuition, even in the process of solving school problems. Therefore, it is currently impossible to ignore the phenomenon of intuition and the results that have been historically derived a theory of knowledge in the study of creative mechanisms.