Article ID: CBB000930656

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


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

Citation URI

Similar Citations

Book Caterina Genna; (2021)
Federigo Enriques matematico e filosofo (/isis/citation/CBB703132259/)

Article Nastasi, Tina; (2008)
La storia del pensiero scientifico e il suo significato nell'opera di Federigo Enriques (/isis/citation/CBB001024085/)

Article Argante Ciocci; (2015)
Luca Pacioli e l'Archimede latino (/isis/citation/CBB290781922/)

Book Enriques, Federigo; Simili, Raffaella; (2000)
Per la scienza: Scritti editi e inediti (/isis/citation/CBB000400942/)

Chapter Arcangelo Rossi; (2017)
Federigo Enriques between popularization and scientific criticism (/isis/citation/CBB951087422/)

Article Stillwell, John; (2014)
Ideal Elements in Hilbert's Geometry (/isis/citation/CBB001213909/)

Article Sieg, Wilfried; (2014)
The Ways of Hilbert's Axiomatics: Structural and Formal (/isis/citation/CBB001213914/)

Book Enrico Giannetto; Giulia Giannini; (2009)
Da Archimede a Majorana: la fisica nel suo divenire (/isis/citation/CBB945473809/)

Chapter Villaggio, Piero; (2006)
On Enriques's Foundations of Mechanics (/isis/citation/CBB000774505/)

Book Palladino, Franco; Palladino, Nicla; (2006)
Dalla “moderna geometria” alla “nuova geometria italiana”. Viaggiando per Napoli, Torino e dintorni (/isis/citation/CBB000953048/)

Article Pedro J. Freitas; (2019)
The Correspondence from Ernesto Cesàro to Francisco Gomes Teixeira (/isis/citation/CBB711366439/)

Article Maria Giulia Lugaresi; (2024)
La biblioteca di Fabio Conforto (/isis/citation/CBB426073135/)

Book Elena Anne Corie Marchisotto; Francisco Rodriguez-Consuegra; James T. Smith; (2021)
The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics (/isis/citation/CBB763111584/)

Authors & Contributors
Ciocci, Argante
Genna, Caterina
Freitas, Pedro J.
Lugaresi, Maria Giulia
Giannini, Giulia
Villaggio, Piero
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
Pavia University Press
Leo S. Olschki
Franco Angeli
Birkhäuser Basel
Correspondence and corresponding
Formalization (philosophy)
Enriques, Federigo
Hilbert, David
Ptolemy, Claudius
Leonardo da Vinci
Time Periods
20th century, early
19th century
20th century
Early modern
North America

Be the first to comment!

{{ comment.created_by.username }} on {{ comment.created_on | date:'medium' }}

Log in or register to comment