Hilbert and his Grundlagen der Geometrie

Abstract

AbstractAfter a brief biography of David Hilbert, we look at how he came to take up elementary geometry and to write his Grundlagen der Geometrie (1899). In this book he gave a careful exposition of axiom systems for elementary geometry that, in Hurwitz’s opinion, created the subject of axiomatics. Hilbert was able to show how different geometries may be studied and found to be inter-related, and how they may be established rigorously in terms of arithmetic. His book ran to 10 editions and opened the way to the axiomatic method in other branches of mathematics. The place of Desargues’ theorem in projective geometry is considered in some detail, using Forest Ray Moulton’s presentation (adopted by Hilbert in the 2nd and subsequent editions).More extracts from Hilbert’s Grundlagen der Geometrie are given at the end of Chapter 28.KeywordsEuclidean GeometryProjective GeometryAxiom SystemWinter SemesterParallel AxiomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.