Kantian Journal

2015 Issue №1(51)

Kant and new mathematics 100 years later

Abstract

Cassirer’s critique of Russell’s philosophy of mathematics and the Neo-Kantian philosophy of science and mathematics as a whole is of special relevance in the context of modern mathematics and mathematical physics. The fact that the modern standard axiomatic architecture of mathematical theories does not take into account the object-based character of mathematical knowledge, which was stressed after Kant by Cassirer, complicates the application of new mathematical theories in natural sciences and technology. In particular, this can explain why modern physical string theory is empirically unverifiable; it can be adjusted to accommodate a wide range of possible outcomes of observations and experiments. At the same time, there are reasons to believe that certain recent ap-proaches in foundations of mathematics such as category theory, topos theory, and Univalent Foundations may help to improve the situation in the near future. The problem of applicability of new mathematical knowledge in science and technology shows that the Kantian approach in phi-losophy of mathematics is at least partly relevant to today’s mathematics.

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Immanuel Kant and the problem of communicative constructivism

Abstract

This article focuses on the relation between the subject and object in theory of cognition. Special attention is paid to the role of social communication in the construction of the real world. The author stresses that the language communication plays an important role in this process. Thereby, it affects human cognition as well as all existence. When discussing this phenomenon, the author uses the methodology that was symbolically described by Kant as the Copernican turn. Wilhelm von Humboldt was one of the first to apply and develop Kant’s method in the theory and philosophy of language. He assumed that language had a priori roots and that it was energeia rather than ergon. The active theory of language (social communication) was further developed in the 20th century within the concept of language symbolism, namely, Ernst Cassirer’s system of symbolic forms. It suggests that communicative structures are independent from real facts, since they participate in the construction of these very objective facts. In the conclusion, the author examines several American theories of language (social communication), namely, relativism (Edward Sapir, Benjamin L. Whorf) and generative grammar (Noam Chomsky). Both systems presuppose language activism.

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