Article ID: CBB000930656

The Contributions of Hilbert and Dehn to Non-Archimedean Geometries and Their Impact on the Italian School (2007)

unapi

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. Keywords : David Hilbert, Max Dehn, Federico Enriques, Roberto Bonola, Non-Archimedean geometry

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Authors & Contributors
Ciocci, Argante
Genna, Caterina
Freitas, Pedro J.
Lugaresi, Maria Giulia
Giannini, Giulia
Villaggio, Piero
Journals
Bollettino di Storia delle Scienze Matematiche
Physis: Rivista Internazionale di Storia della Scienza
Perspectives on Science
Paedagogica Historica: International Journal of the History of Education
Kwartalnik Historii Nauki i Techniki
Historia Scientiarum: International Journal of the History of Science Society of Japan
Publishers
Guaraldi
Pavia University Press
Leo S. Olschki
Franco Angeli
Birkhäuser Basel
Bibliopolis
Concepts
Mathematics
Geometry
Physics
Biographies
Correspondence and corresponding
Formalization (philosophy)
People
Enriques, Federigo
Hilbert, David
Archimedes
Ptolemy, Claudius
Leonardo da Vinci
Euclid
Time Periods
20th century, early
19th century
Renaissance
Ancient
20th century
Early modern
Places
Italy
Europe
Portugal
North America
Japan
Greece
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