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The contributions of Hilbert and Dehn to non-archimedean geometries and their impact on the Italian school. (English) Zbl 1223.01026

Summary: 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 in order to prove the independence of the continuity axiom and we will study the interrelations between Archimedes’ axiom and Legendre’s theorems. Some of these interrelations were also studied by Bonola, who was one of the very few Italian scholars to appreciate Dehn’s work. We will see that, if Archimedes’ 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 of a triangle. Hilbert himself returned to this problem giving a very interesting model of a non-Archimedean geometry in which there are infinitely many lines parallel to a fixed line through a point while the sum of the inner angles of a triangle is equal to two right angles.

MSC:

01A70 Biographies, obituaries, personalia, bibliographies
01A60 History of mathematics in the 20th century
51A35 Non-Desarguesian affine and projective planes
05B35 Combinatorial aspects of matroids and geometric lattices
17D05 Alternative rings
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