Abstract. The article focuses on the problem of understanding a priori (pre-experimental) and a posteriori (empirical) knowledge, their relations and their role in the cognitive process, this problem has deep roots in the philosophy of knowledge: the opposite approaches to understanding the nature of knowledge are found in the philosophy of Ancient Greece (Plato and Aristotle) and continue their existence in modern philosophy (rationalism of R. Descartes and G. Leibniz and the empiricism of F. Bacon, T. Hobbes and J. Locke). The author points out the role of Kant in solving this problem, whose realization of the a priori as synthetic a priori together with the proposed “set” of so-called “a priori forms” of sensuality and reason further established in the philosophy of knowledge. It is shown that neokantian Marburg school (Cohen, Natorp, Cassirer, etc.), who shared the idea of Kant that “experience is given in mathematics and pure natural science” and had a particular influence on Husserl in his formulation of the concept of postural aprioristic-tion in his phenomenology. Special attention is paid to rendering of the a priori in the works of V. Y. Perminov, who linked the original idea in mathematics with the necessary activity (practical) orient-doing thinking. The work reveals some weaknesses of the a priori in mathematical knowledge (choice of only two of all the a priori forms of sensuality and reason by Kant for Mathematics; the limitation of the mathematical knowledge by arithmetic and Euclidean geometry; etc.). It is stated that they contributed to the emergence of “neoempiricism” in the philosophy of mathematics which to a certain extent is similar to the “moderate empiricism” – a version of empiricism that exists in contemporary epistemology along with another version – the “extreme” or “radical” empiricism and the same versions of apriorism. In this regard, there expressed an assumption that “hard” opposition of the a priori and the empirical in philosophy and science (in logic and mathematics, in particular) are unlikely to correspond with the actual ratio in real knowledge. Many fundamental scientific concepts (those of Mathematics and logics, in particular) that may seem nonexperimental, purely a priori, in fact arise as perception and are formed as concepts in the process of practical activities, in the process of socialization. This idea is confirmed by the work of proponents of the concept of physiological and neurophysiological interpretation of Mathematics, research in the field of cognitive Psychology, Ethno-mathematics and sociocultural Philosophy of Mathematics.
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Keywords: apriorism, priori, posteriori, mathematics, neokantians, empiricism, epistemology.
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