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Article

Трансцендентальный анализ математики: конструктивный характер математической деятельности

Кантовский сборник. 2016. Т. 55. № 1. С. 16-34.
Катречко С. Л.

Kant’s transcendental philosophy (transcendentalism) focuses on both the human method of cognition in general [CPR, B 25] and certain types of cognition aimed at justifying their objective significance. This article aims to explicate Kant’s understanding (resp. justification) of the abstract nature of mathematical knowledge (cognition) as “construction of concepts in intuition” (see: “to construct a concept means to exhibit a priori the intuition corresponding to it”; [CPR, A 713/В 741], which is “thoroughly grounded on definitions, axioms, and demonstrations” [CPR, A 726/В 754]. Mathematical objects, unlike specific ‘physical’ objects, are of abstract nature and are introduced (defined) within Hume’s principle of abstraction. Based on his doctrine of schematism, Kant develops an original theory of abstraction: Kant’s scheme serve as a means to construct mathematical objects, as an “action of pure thought" [CPR, B 81]. “Constructive” understanding of mathematical acts is a significant innovation of Kant. In this mathematical activity is considered as a two-level system, which supposes a “descent” from the level of concept understanding to the level of sensual intuition, where mathematical acts are performed, and a return “rise”. On this basis, we are developing theory of transcendental constructivism (pragmatism). In particular, Kant's "contemplation/intuition" of mathematics can be understood as the structural properties of mathematical language or its "logical space" (Wittgenstein; mathematical structuralism). In his theory Kant highlights ostensive (geometric) and symbolic (algebraic) constructing. The paper 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. In the paper we also highlight as a third type of construction — the logical constructing  [in proving theorems], which inherits the features of both Kant's types.