2022 Theses Bachelor's

Unforeseen Asymmetries: A Contemporary Mathematical Reading of Kant’s Second Antinomy

Connell, Anna F.

In the Critique of Pure Reason, Kant proposes transcendental idealism as our reality’s metaphysical and epistemological framework, holding that all objects of possible experience have no existence outside of thought, but rather are mere appearances for cognizers. His project is to limit cognition “to make room for faith” by showing that boundless reason and transcendental realism—the notion that representations of our world are things in themselves—ultimately drives cognizers to contradictory pseudo-rational assertions (Kant, Bxxx). The Second Antinomy is one such case in which Kant demonstrates that unbounded reason and realism results in contradiction as a way to propose his own transcendental idealism as a solution. In the Second Antinomy, Kant works through how demands of reason drive cognizers to disprove both (defeasibly) exclusive and exhaustive options for a mereological account of the world, namely, atomism or infinite complexity of composite substances. Kant offers a thesis that every composite substance in the world is made up of simple parts, and an antithesis that no composite thing in the world is made up of simple parts, both of which he will reduce to absurdity in order to disprove transcendental realism. Yet through his arguments against the thesis and antithesis, Kant’s mereological account of space and bodies and conceptualization of reason’s infinite regress raises its own questions. These arguments on which the Second Antinomy rely, I will argue, suppose a decomposition operation that leaves open the possibility that the composite whole be continuous prior to the infinite regress of division of the body. Yet at the same time, the composition of particular parts can only result in a discrete, or at best, dense, whole. These mathematical distinctions regarding cardinalities of infinity and transfinite numbers were still largely controversial while Kant was writing the Critique, and thus, led him to mistake these mereological notions of a whole. Yet for our purposes, contemporary mathematical definitions can clarify the unforeseen asymmetry that Kant’s mistake creates in the Second Antinomy. This paper will use these contemporary mathematical distinctions to demonstrate that when Kant uses the concepts of decomposition to disprove the antithesis and composition to disprove the thesis, his asymmetric presuppositions of the structure of the body we are dealing with stifle his ability to use these two opposing assertions as a reductio of transcendental realism. In fact, if the proposed asymmetry holds, the entire project of the Second Antinomy collapses.