Between Euclid, Kant, and Lobachevsky: On the Construction of Geometrical Objects in Pure Intuition

Abstract

Kant’s theory of geometry is compatible with non-Euclidean, hyperbolic geometry. That is, on Kant’s theory, the a priori forms of space and time together with the categories of the understanding ground the possibility of hyperbolic constructions in pure intuition. To show this we first develop an interpretation of Kant’s theory of geometry to the extent that it concerns the construction of geometrical objects in pure intuition. Thus we show how the a priori forms and the categories make possible Euclidean constructions in pure intuition. We then proceed to the main result. The latter is independent from some of the details of the interpretation. Under minor assumptions the result can be strengthened to the following: if Kant’s theory is compatible with Euclidean geometry, it is compatible with hyperbolic geometry as well.