Specialeforsvar: Lars Steen Hvelplund
Titel: Non-Euclidean Geometry, The development of the mathematical model
Abstract: The aim of this paper is to investigate the discovery of the mathematical models of non-Euclidean geometry in the 19th century. The paper presents the main geometric properties of the mathematical models, and furthermore it examines their significance concerning the acceptance of non-Euclidean geometry. Beside that, it discusses the role of these models in relation to the correspondence between geometry and psychical space. Structurally, the paper starts surveying the historical progress of the controversial parallel postulate from Euclid’s elements, to the work on non-Euclidean geometry by Lobachevsky. Moreover, what follows is an analysis of Beltrami’s paper Essay on the interpretation of Noneuclidean Geometry from 1868. The focus of this analysis is to detect the association to the work of Lobachevsky. Furthermore, the thesis explains the "real substrate" of Beltrami and explicates the importance of Beltrami’s contribution according to the acceptance of non-Euclidean geometry. In addition to that, an analysis of Poincaré’s Theory of Fuchsian Groups from 1881 is included.
It focusses on the connection between the theory of fuchsian functions and the models of non-Euclidean geometry. Regarding the mathematical models, this paper introduces the concepts of consistency and independence, related to geometric axiom systems. Especially the independence of the parallel postulate will be emphasized. Finally, the paper relates these models of Beltrami and Poincaré to the treatise Foundations of Geometry from 1899 by David Hilbert.
In connection to Hilbert’s work, the paper discusses the role of the models regarding the construction of consistent geometries as well as more philosophical aspects.
vejleder: Jesper Lützen
censor: Poul Hjorth, DTU