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Foundations of geometry after Friedrich Schur. (Les fondements de la géométrie selon Friedrich Schur.) (French. English summary) Zbl 1218.01010

Friedrich Schur’s 1909 Grundlagen der Geometrie is eclipsed in history by Hilbert’s work of the same title of 1899. This article argues that even though we may regard Schur’s work as essentially nineteenth-century in comparison with Hilbert’s more modern treatise, it played an important role as a work of synthesis and, at the very least, shows that Hilbert was not the only player in this area. Schur’s work was original and different from Hilbert’s; for example, it takes projective geometry as its point of departure, gives a central role to the notion of displacement, and provides simultaneously foundations of several geometries. Also, Schur, in contrast with Hilbert, makes use of the works of the Italian school: Peano, Veronese, Cremona, and Ingrami. (Surprisingly Pieri does not make an appearance.) Next to the Italian influence, Schur was mainly influenced by G. K. C. von Staudt and Moritz Pasch. An analysis is given of each section of the 1909 work with reference also to an earlier version published as a paper in 1901. This is followed by a description of Schur’s epistemology for foundations of geometry as revealed in the introduction to the Geometrie. Good use is made of the unpublished correspondence between Schur and Friedrich Engel who was a colleague at Leipzig in the 1880s.

MSC:

01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
51-03 History of geometry
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