Geometry as an object of experience: Kant and the missed debate between Poincar\'e and Einstein

Understand

Poincar\'e held the view that geometry is a convention and cannot be tested experimentally.

• This position was apparently refuted by the general theory of relativity and the successful confirmation of its predictions; unfortunately, Poincar\'e did not live to defend his thesis.
• In this paper, I argue that: 1) Contrary to what many authors have claimed, non-euclidean geometries do not rule out Kant's thesis that space is a form of intuition given {\it a priori}; on the contrary, Euclidean geometry is the condition for the possibility of any more general geometry.
• 2) The conception of space-time as a Riemannian manifold is an extremely ingenious way to describe the gravitational field, but, as shown by Utiyama in 1956, general relativity is actually the gauge theory associated to the Lorentz group.