Early explicit examples of non-desarguesian plane geometries

Abstract

We trace the history of geometries where Desargues' theorem is not valid. Roughly we cover the time from the middle of the nineteenth century until the first decade of the twentieth century, discussing work by Beltrami, Klein, Wiener, Peano, Moulton, and of course Hilbert.Mathematics Subject Classification (2010). 51A35, 51-03, 01A55, 01A60.The purpose of these notes is to collect information about the earliest examples of non-desarguesian planes, or, more generally plane geometries (where lines need not meet, and the parallel axiom need not be satisfied). Roughly since the third decade of the twentieth century (definitely starting with Moufang's work [18], cf. [4]), non-desarguesian projective planes are considered as objects in their own right, and their systematic study has lead to an abundance of examples. For more information, consult Hall's seminal paper [6] and the monographs [13,22,25]. The present notes concentrate mainly on the very first (known) examples. At the time of their construction, these examples were considered as pathological counterexamples showing the need for the embedding of the plane in a space of higher dimension or other additional structure in the axiomatic treatment of projective (or affine) planes. I became aware of the interesting historical examples through various discussions with Benno Artmann. His knowledge of the literature and unpublished results has been very inspiring and helpful in the preparation of an early version of this paper.The presentation will not be strictly chronological but starts with the first explicit treatment that actually occurred in print (as far as I know).

On the Relationship Between Plane and Solid Geometry

“…We maintain that the relevant notion of content is what we have called "informal" content, the one available to ordinary practitioners. Schneider & Stroppel (2007) and Stroppel (2010). We now offer a series of reasons against construing purity in terms of formal content.…”

“…For detailed technical investigations of Hilbert's models in both the lectures and the Grundlagen, including both his affine and nonaffine models, cf. Schneider & Stroppel (2007) and Stroppel (2010).…”