Dimitry Gawronsky: Reality and Actual Infinitesimals

Abstract

The aim of this paper is to analyze Dimitry Gawronsky’s doctrine of actual infinitesimals. I examine the peculiar connection that his critical idealism establishes between transcendental philosophy and mathematics. In particular, I reconstruct the relationship between Gawronsky’s differentials, Cantor’s transfinite numbers, Veronese’s trans-Archimedean numbers and Robinson’s hyperreal numbers. I argue that by means of his doctrine of actual infinitesimals, Gawronsky aims to provide an interpretation of calculus that eliminates any alleged given element in knowledge. The author emphasizes not the mathematical, but the transcendental or metaphysical aspect of Gavronsky’s teaching. It follows from Gavronsky’s doctrine that infinitesimals are the key to a correct philosophical explanation of the relationship between thinking and being: mathematics, and differential calculus in particular, turns out to be the means by which pure thought constructs being. Thus, we are talking about the conception of transcendental mathematics, which solves the problem of the applicability of mathematics to nature. Thus, nature is understood as a product of thought, created in accordance with the infinitesimal method: since thought creates natural objects in accordance with mathematical methods, the latter have the necessary reliability in relation to the former. The relevance of infinitesimals turns out to be Gavronsky’s relevance of pure thought in the generation of being, and the first relevant product of pure thought is the reality of being.