Kant's transcendentalism is associated with the study and substantiation of objective value as a human way of cognition as whole and individual kinds of our knowledge. This article is devoted to Kant’s understanding (substantiation) of mathematics as cognition via constructing of concepts. Unlike the natural sciences the mathematics is an abstract – formal cognition, “[its] thoroughness is based on definitions and axioms” [B754].
The article consequently analyzes each of these components. Mathematical objects are determined by the principle of Hume. Transcendentalism considers the question of genesis and ontological status of concepts. To solve them Kant suggests the doctrine of schematism, which is compared with the modern concepts of mathematics. We develop the dating back to Kant original concept of the transcendental constructivism (pragmatism) as the programme of substantiation of mathematics. We also give a brief comparison of the axiomatic method of Kant and Hilbert.
“Constructive” understanding of mathematical constructions (calculations) is a significant innovation of Kant. Thus mathematical activity is considered as a two-level system, which supposes a “descent” from the level of rational understanding to the level of sensual contemplation and a return “rise”. The article also examines the difference between the mathematical structures and logical proofs.
In his concept Kant highlights ostensive (geometric) and symbolic (algebraic) design. The article analyses each of them and shows that it is applicable to modern mathematics, in activity of which both types of Kant's constructing are intertwined.
The monograph "Philosophy of Physics: towards New Principles of New Knowledge" is intended both for those who are just discovering the world of philosophy of science in the most fundamental field – physics, and for specialists who are professionally engaged in the topic and are interested in the most relevant research. The author analyzes a number of key problems of modern physics and cosmology, offers original interpretations and solutions, and also discusses the prospects for the development of science in the context of attempts to create a "Theory of Everything". The book focuses on the physical theories of the multiverse, the new principles of scientific knowledge derived from these theories, and the connection between consciousness and concrete physical reality.
The significance of the education in the field of philosophy of mathematics as the part of both philosohpy and mathematics at the universities is the subject of the article.