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.
Many scientists and philosophers have argued that the possibility of non-Euclidean geometries contradicts Kant's thesis that geometric axioms are synthetic judgments a priori (Kant 1929(Kant , 1986)). Accordingly, it became a common place to assert that the general theory of relativity, by revealing the non-Euclidean nature of physical space, refuted Kant's doctrine on the transcendental nature of space. However, Poincaré (1952) sustained that geometry is a convention and cannot be the object of any experience, but his point of view fall into oblivion due to the commonly held belief that the Riemannian nature of space-time is testable.
The plan of the present paper is as follows: In Section 1, a brief historical introduction to the problem is given and then it is shown that the existence of Riemannian geometry not only does not refute, but confirms Kant’s doctrine. In Section 2, the geometric conventionalism of Poincaré is reviewed. Section 3 is devoted to the alternative formulation of general relativity, as a gauge theory, given by Utiyama (1956). Finally, Section 4 compares the ideas of Poincaré and Einstein on the physical nature of space following a text written by the latter in 1949.
1 How can Riemannian geometry be possible?
The argument that non-Euclidean geometries contradict Kant’s doctrine on the nature of space apparently goes back to Helmholtz (1995) and was retaken by several philosophers of science such as Reichenbach (1958) who devoted much work to this subject.
In a essay written in 1870, Helmholtz (1995) argued that the axioms of geometry are not a priori synthetic judgments (in the sense given by Kant), since they can be subjected to experiments. Given that Euclidian geometry is not the only possible geometry, as was believed in Kant’s time, it should be possible to determine by means of measurements whether, for instance, the sum of the three angles of a triangle is 180 degrees or whether two straight parallel lines always keep the same distance among them. If it were not the case, then it would have been demonstrated experimentally that space is not Euclidean. Thus the possibility of verifying the axioms of geometry would prove that they are empirical and not given a priori.
Helmholtz developed his own version of a non-Euclidean geometry on the basis of what he believed to be the fundamental condition for all geometries: “the possibility of figures moving without change of form or size”; without this possibility, it would be impossible to define what a measurement is. According to Helmholtz (1995, p. 244): “the axioms of geometry are not concerned with space-relations only but also at the same time with the mechanical deportment of solidest bodies in motion.” Nevertheless, he was aware that a strict Kantian might argue that the rigidity of bodies is an a priori property, but “then we should have to maintain that the axioms of geometry are not synthetic propositions… they would merely define what qualities and deportment a body must have to be recognized as rigid”.
At this point, it is worth noticing that Helmholtz’s formulation of geometry is a rudimentary version of what was later developed as the theory of Lie groups, which I will mention in Section 3. As for the transport of rigid bodies, it is well known nowadays that rigid motion cannot be defined in the framework of the theory of relativity: since there is no absolute simultaneity of events, it is impossible to move all parts of a material body in a coordinated and simultaneous way. What is defined as the length of a body depends on the reference frame from where it is observed. Thus, it is meaningless to invoke the rigidity of bodies as the basis of a geometry that pretend to describe the real world; it is only in the mathematical realm that the rigid displacement of a figure can be defined in terms of what mathematicians call a congruence.
Arguments similar to those of Helmholtz were given by Reichenbach (1958) in his intent to refute Kant’s doctrine on the nature of space and time. Essentially, the argument boils down to the following: Kant assumed that the axioms of geometry are given a priori and he only had classical geometry in mind, Einstein demonstrated that space is not Euclidean and that this could be verified empirically, ergo Kant was wrong.
However, Kant did not state that space must be Euclidean; instead, he argued that it is a pure form of intuition. As such, space has no physical reality of its own, and therefore it is meaningless to ascribe physical properties to it. Actually, Kant never mentioned Euclid directly in his work, but he did refer many times to the physics of Newton, which is based on classical geometry. Kant had in mind the axioms of this geometry which is a most powerful tool of Newtonian mechanics. Actually, he did not even exclude the possibility of other geometries, as can be seen in his early speculations on the dimensionality of space (Kant 1986).
The important point missed by Reiche
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