The contributions of Hilbert and Dehn to non-Archimedean geometries and their impact on the Italian school

Abstract

In this paper we investigate the contribution of Dehn to the development of non- Archimedean geometries. We will see that it is possible to construct some models of non- Archimedean geometries to prove the independence of the continuity axiom and we will study the interrelations between Archimede’s axiom and Legendre‘s theorems. Some of these interrelations were studied also by Bonola who was one of the very few italian scholars to really appreciate Dehn’s work. We will see that, if Archimede’s axiom does not hold, the hypothesis on the existence and the number of parallel lines through a point is not related to the hypothesis on the sum of the inner angles a triangle. Hilbert himself returned on this problem giving a very interesting model of a non-Archimedean geometry such that in it there are infinite parallel lines to a fixed line throught a point and the sum of inner angles of a triangle is equal a two right angles.