This is a request for serious mathematicians or professional philosophers who specialize in Kant. I’ve written a paper that takes on a considerably more challenging target than Darwin et al, and I’d like to get the benefit of review by a team of top-notch human reviewers before I publish it anywhere. It’s already been through the Red Team stress test, so most of the obvious flaws should already have been detected and addressed. Shoot me an email if you’re ready, willing, and able to read through and review the paper.
The Mathematical Refutation of Kant: The Irrational, the Imaginary, and the Infinite
Kant’s account of mathematical cognition, presented in the Doctrine of Method and elaborated through the Transcendental Aesthetic and the Prolegomena, holds that synthetic a priori mathematical knowledge is possible because mathematical objects are constructed by the finite cognitive subject in pure spatial or temporal intuition. Kant’s account grounds the application of mathematics to nature that the rest of the Critique presupposes, licenses the contrast between mathematical and philosophical method that organizes his epistemology, and sets the boundary between constitutive mathematical objects and the regulative ideas of reason. The inferential chain that delivers the unknowability of the noumenal begins at this point: if mathematical cognition is bounded by what the cognitive subject can construct in pure intuition, then mathematics cannot reach beyond the phenomenal, and the phenomenal restriction has its central case. This paper argues that the construction account is fundamentally false. Modern mathematics contains three large classes of objects that the construction account forbids and that working mathematicians treat as fully legitimate: the irrational numbers, the imaginary numbers, and the completed mathematical infinite. Each class has determinate properties, does indispensable work in mathematics and physics, and is required for the consistency of standard analysis and standard physical theory. The construction account is not strained by these three classes, but is refuted by them. The defenses available to the Kantian are systematic but self-undermining. Every retreat and redefinition that limits “construction” or “intuition” enough to accommodate the modern classes expands what counts as cognitively reachable and broadens the phenomenal correspondingly, until the noumenal becomes an empty residue that cannot carry the unknowability claim it was supposed to anchor. The Kantian doctrine can only be saved by being emptied of content.