A transcendental analysis of mathematics: The abstract nature of mathematical knowledge

Abstract

Kant’s transcendental philosophy (transcendentalism) focuses on both the human method of cognition in general [CPR, B25] 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, A713/В 741], which is “thoroughly grounded on definitions, axioms, and demonstrations” [CPR, A726/В 754]. Mathematical objects, unlike specific ‘physical’ objects, are of abstract nature (a-obj¬ects vs. the-objects) 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, B81]. The article investigates the ontological status of mathematical objects/abstractions and describes three possible ontologies — the understanding of mathematical objects/abstractions as: 1 complete objects (the ontology of things; "full-blooded Platonism"); 2) a substantivized set of properties (ontology of properties; E. Zalta); 3) relations (the ontology of relations; category theory, structuralism).

Keywords

• Kant’s transcendental philosophy (transcendentalism)
• transcendental method
• transcendental pragmatism
• Kant's theory of construction of mathematical concepts (mathematical cognition as construction of concepts in intuition)