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

...More
Citation URI
https://data.isiscb.org/isis/citation/CBB000930656/

Similar Citations

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

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

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

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

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

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

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

Chapter Giulia Giannini; (2009)
Poincaré, la nozione di gruppo e il Programma di Erlangen di F. Klein (/isis/citation/CBB227666735/)

Book Alberto Cogliati; (2024)
La geometria non euclidea. Una breve storia dall’antichità a Poincaré (/isis/citation/CBB276091257/)

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

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

Book Gabriele Lolli; (2016)
Tavoli, sedie, boccali di birra: David Hilbert e la matematica del Novecento (/isis/citation/CBB100106060/)

Authors & Contributors
Giannini, Giulia
Ciocci, Argante
Genna, Caterina
Freitas, Pedro J.
Villaggio, Piero
Toscano, Marco
Concepts
Mathematics
Geometry
Physics
Biographies
Logic
Non-euclidean geometry
Time Periods
20th century, early
19th century
20th century
Renaissance
Ancient
Early modern
Places
Italy
Europe
Portugal
North America
Japan
Greece
Comments

Be the first to comment!

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

Log in or register to comment