The concept of implicit knowledge in the context of rational reconstruction of the history of mathematics

In the article, questions from the field of philosophy of mathematics are studied. The author is driven by the need to achieve a balance between the philosophy of science and the history of science in formation of concepts of the science development. In this regard, the author justifies the reliance on the methodology of implicit knowledge, combined with the epistemology principle of criticism in studying the development of mathematics as the most expedient and effective. The author expresses the necessity of criticizing the methodology of counterexamples typical for mathematical empiricism declared by I. Lakatos. In the article, an explanation of the concept of implicit knowledge, as well as the examples from mathematics are given. The author believes that application of the concept of implicit knowledge in philosophical and mathematical terms allows us to reveal the unique specificity of mathematical science deeper. The author concludes that it is necessary to recognize the fundamental importance of the methodology of implicit knowledge for the epistemology, as well as for the development of philosophy and methodology of science in general.