The contributions of Hilbert and Dehn to non-archimedean geometries and their impact on the italian school

Cinzia Cerroni

Revue d'histoire des mathématiques (2007)

  • Volume: 13, Issue: 2, page 259-299
  • ISSN: 1262-022X

Abstract

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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 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.

How to cite

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Cerroni, Cinzia. "The contributions of Hilbert and Dehn to non-archimedean geometries and their impact on the italian school." Revue d'histoire des mathématiques 13.2 (2007): 259-299. <http://eudml.org/doc/274904>.

@article{Cerroni2007,
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 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.},
author = {Cerroni, Cinzia},
journal = {Revue d'histoire des mathématiques},
keywords = {David Hilbert; Max Dehn; Federico Enriques; Roberto Bonola; non-archimedean geometry},
language = {eng},
number = {2},
pages = {259-299},
publisher = {Société mathématique de France},
title = {The contributions of Hilbert and Dehn to non-archimedean geometries and their impact on the italian school},
url = {http://eudml.org/doc/274904},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Cerroni, Cinzia
TI - The contributions of Hilbert and Dehn to non-archimedean geometries and their impact on the italian school
JO - Revue d'histoire des mathématiques
PY - 2007
PB - Société mathématique de France
VL - 13
IS - 2
SP - 259
EP - 299
AB - 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 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.
LA - eng
KW - David Hilbert; Max Dehn; Federico Enriques; Roberto Bonola; non-archimedean geometry
UR - http://eudml.org/doc/274904
ER -

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