**abstract:**
In our presentation, we will have to explain the connection Klein, E. Cartan and H. Weyl did between the Erlanger Program (1872) and the special theory of relativity. The main principles of the latter were clearly formulated by Einstein in June 1905. This connection is not self-evident for three reasons.
Firstly, The Erlanger Program belongs to pure mathematics; Klein’s aim was to classify and to organize the different geometries into a hierarchy based on projective geometry. On the other hand, the emergence of the special theory of relativity requires a critique of certain presuppositions regarding the interpretation of phenomena. For instance, Einstein had to disprove the existence of a privileged frame; then, according to the principle of relativity, the law of the constancy of the velocity of the light has to be the same in any inertial system.
Secondly, Klein did the connection between his Program and the special theory of relativity afterwards in an article published in 1910 and inspired by Minkowski’s works. One can wonder if this connection is not due to a retrospective illusion.
Thirdly, we have to concede that the structure of transformations group occurs in the Erlanger Program and in the formalization of special theory of relativity. Nevertheless, the special Lorentz group is not sufficient in order to deduce the special theory of relativity. Indeed, Poincaré and Einstein invented almost simultaneously the special Lorentz group. But, at the same time, their interpretations of phenomena diverged noticeably. This example shows that physical principles can not be derived merely from mathematical concepts.

Fri 11 Jul, 9:30 - 10:10, Aula Bianchi Scienze

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