Transcendental Analysis of Mathematics: The Transcendental Constructivism (Pragmatism) as the Program of Foundation of Mathematics

Abstract

Kant's transcendental philosophy (transcendentalism) is associated with the study and substantiation of objective validity both “a human mode of cognition” as whole, and specific kinds of our cognition (resp. knowledge) [KrV, B 25]. This article is devoted to Kant’s theory of the construction of mathematical concepts and his understanding (substantiation) of mathematics as cognition “through construction of concepts in intuition” [KrV, B 752] (see also: “to construct a concept means to exhibit a priori the intuition corresponding to it”; [KrV, Â 741]). Unlike the natural sciences the mathematics is an abstract – formal cognition (knowledge), its thoroughness “is grounded on definitions, axioms, and demonstrations” [KrV, B 754]. The article consequently analyzes each of these components. Mathematical objects, unlike the specific ‘physical’ objects, have an abstract character (a–objects vs. the–objects) and they are determined by Hume’s principle (Hume – Frege principle of abstraction). Transcendentalism considers the question of genesis and ontological status of mathematical concepts. To solve them Kant suggests the doctrine of schematism (Kant’s schemata are “acts of pure thought" [KrV, B 81]), which is compared with the contemporary theories of mathematics. We develop the dating back to Kant original concept of the transcendental constructivism (pragmatism) as the as the program of foundation of mathematics. “Constructive” understanding of mathematical acts 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 under-standing to the level of sensual contemplation and a return “rise”. In his theory Kant highlights ostensive (geometric) and symbolic (algebraic) constructing. 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