Kantian Journal

2016 Issue №1(55)

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A transcendental analysis of mathematics: The constructive nature of mathematics

DOI
10.5922/0207-6918-2016-1-2
Pages
16-33

Abstract

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 the “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]. Unlike specific ‘physical’ objects, mathematical objects are of abstract nature and they are introduced using Hume’s principle of abstraction. Based on the doctrine of schematism, Kant develops an original theory of abstraction: Kant’s schemes serve as a means to construct mathematical objects, as an “action of pure thought" [CPR, B 81]. A ‘constructive’ understanding of mathematical acts going back to Euclid’s genetic method is an important innovation introduced by Kant. This understanding is at the heart of modern mathematical formalism, intuitionism, and constructivism. Within Kant’s constructivism, mathematics can be described as a two-tier system, which suggests a “shift” from the level of concepts of the understanding to the level of sensual intuition, where mathematical acts are performed, followed by a subsequent return to the initial level. On this basis, the author develops a theory of transcendental constructivism (pragmatism). In particular, Kant's ‘intuitionism’ of mathematics can be understood as structural properties of mathematical language or its ‘logical space’ (Wittgenstein; cf. mathematical structuralism). In his theory, Kant distinguishes between two types of constructing — ostensive (geometric) and symbolic (algebraic). The paper analyses these types and shows that modern mathematical structures are a combination and intertwining of both. The author also identifies a third type — logical constructing [in proving theorems], which inherits the features of both Kant's types.

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