Differences in Conceptions of the Nature of Mathematical Propositions in Frege and Kant

The article presents a comparative analysis of the views of Frege and Kant on the nature of mathematical propositions. The author of the article puts the distinction between the epistemological and logico-semantic justification of knowledge as the basis for comparing Kant's transcendental approach and Frege's logicism. The first is carried out by referring to the process of thinking and to the subject's mind. The second considers the objective conditions for the truth of propositions and aims to make the truth of propositions as obvious as possible by the rigorous logical proof of the latter. In Kant's interpretation, the structure of the justification of mathematical knowledge contains an essential reference to the a priori faculties of our reason. Frege's interest lies in revealing the objective relations between propositions that lie in the foundation of mathematics. For Frege, the proof of mathematical propositions must be an invariable and objective sequence of truths that are fundamentally independent of the subject's thinking. However, Frege tends to refer exclusively to the laws of logic in the proof of arithmetic propositions. Such an explanation of mathematics through logic has its own epistemological motive. Frege hopes that mathematics, as well as logic, will have a priori laws of thought as its subject. If it were technically possible to prove that mathematics is an objective extension of logic, the epistemological question about the nature of mathematical knowledge would also be answered. Obviously, the method of transcendental philosophy is not applicable to such a narrow task. However, it is wrong to interpret Frege's logicism as overcoming Kant's transcendentalism. The transcendental method, within the framework of the problem of the nature of mathematics, probably has a place, but only as a solution to the epistemological question of how we know or understand mathematical judgments. Nevertheless, Frege's works consistently reject some of Kant's particular ideas. His criticism of the idea of justifying mathematical knowledge by appealing to space and time, and his demand to pay attention to the proof of mathematical propositions certainly modify and replace the definitions of a priori, a posteriori, analytical, and synthetic knowledge given by Kant.

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