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

TL;DR

This paper explores the philosophical and scientific debates on the nature of geometry, emphasizing that Euclidean geometry underpins all geometrical understanding and that general relativity can be viewed as a gauge theory, challenging conventional interpretations.

Contribution

It clarifies the relationship between Kantian a priori geometry, non-Euclidean geometries, and the gauge theory perspective of general relativity, highlighting overlooked historical debates.

Findings

Euclidean geometry is fundamental for understanding any geometry.

General relativity can be interpreted as a gauge theory, not necessarily requiring curved spacetime.

Non-Euclidean geometries do not contradict Kant's view of space as an a priori intuition.

Abstract

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. Utiyama's approach does not rely on the assumption that space-time is curved, though the equations of the gauge theory are identical to those of general relativity. Thus, following Poincar\'e, it can be claimed that it is only a matter of convention to describe the gravitational field as a Riemannian manifold or as a gauge field in Euclidean space.