О соотношении логики и философии математики Канта

Abstract

For Kant’s theoretical philosophy, logic was not at all a separate or peripheral discipline. Quite the opposite, it can be argued that formal general logic became a kind of a paradigm of his transcendental philosophy, and its central ideas are due to the specific nature of Kant’s logical concept. Kant’s philosophy of mathematics is of particular interest, since one of the central questions of his Critique of Pure Reason is the question of how mathematics is possible as a science of universal and necessary truths. Kant’s theoretical philosophy cannot be understood without his mathematical concept, since the type of synthesis that, according to Kant, underlies mathematics is the same as for all other objects of perception. In addition, in his transcendental philosophy, the difference between the philosophical and mathematical methods turns out to be fundamental. Undoubtedly, Kant’s original logical concept, including the construction of a new and non-standard transcendental logic, could not but have a significant impact on the main characteristics of his philosophy of mathematics. Their mutual influence may be of interest to anyone who wants to find the key to understanding Kant’s philosophy in a more general context. The following central provisions of Kant’s philosophy of mathematics, formulated by him in “The Transcendental Aesthetic”, “The Transcendental Doctrine of Method” and other small fragments of the Critique of Pure Reason, can be distinguished: firstly, it is the idea of formality of both mathematical and any rational cognition (just as formal logic is the basis of “empty” logical forms, mathematics is formal, since it deals with pure a priori forms of intuitions); secondly, it is the doctrine of the synthetic a priori character of mathematical truths; thirdly, it is his idea that mathematical knowledge is realized through the construction of concepts, and, finally, it is the idea of the direct and necessary connection of mathematical knowledge with pure forms of intuitions, i.e. with extremely general areas of empirical experience. At the same time, the apparatus of traditional logic available to Kant had significant limitations that did not make it possible to adequately represent mathematical knowledge. On the one hand, the features of Kant’s interpretation of formal logic and the transcendental logic developed by him in his critical philosophy determine the constructive and synthetic nature of mathematical knowledge; on the other hand, the non-trivial philosophy of mathematics developed by him is a response to the limitations of traditional logic and an attempt to overcome them. The construction of mathematical concepts in pure a priori intuition allows one to go beyond the limits of these limitations in mathematical knowledge, which had exceptional consequences for the history of modern philosophy of mathematics and the history of the foundations of mathematics. It must be assumed that the key features of Kant’s logical concept and his philosophy of mathematics have not yet exhausted their heuristic possibilities for topical research in these areas of science and philosophy.