Early nineteenth century developments in mathematics as challenge to the Kantian doctrine of pure intuition

Ernesto Giusti

It is usually assumed that non-euclidian geometries posed a challenge to Kantian philosophy of mathematics, eventually leading to its downfall, and its abandonment as an adequate theory of mathematical knowledge. However, when looking at the developments in mathematics shortly after the publication of the Critique of Pure reason, another picture emerges. In fact, the renewal of projective geometry and the introduction of complex numbers constituted a challenge to the main tenets of Kant´s theory of mathematics as a kind of synthetic a priori knowledge long before the non-euclidian geometries. In this presentation, I will show how projective geometry can hardly be accommodated within the Kantian view of mathematics, and that its earlier proponents, although being primarily mathematicians, were acutely aware of this fact. This led them to engage directly with foundational researches in mathematics, announcing the intense cross-fertilization between philosophy and mathematics (and, ultimately, science in general) that will characterize the origins of contemporary philosophy in the second half of the nineteenth century.