Automorphisms of Hilbert's non-desarguesian affine plane and its projective closure

Schneider, Thomas; Stroppel, Markus

doi:10.1515/advgeom.2007.032

Cited by 2 publications

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References 11 publications

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“…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.…”

Section: Against Construing Purity In Terms Of Formal Contentmentioning

confidence: 99%

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On the Relationship Between Plane and Solid Geometry

Arana

Mancosu

2012

The Review of Symbolic Logic

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Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.In this paper our major concern is with methodological issues of purity and thus we treat the connection to other areas of the planimetry/stereometry relation only to the extent necessary to articulate the problem area we are after.Our strategy will be as follows. In the first part of the paper we will give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane and solid geometry. The sketch is given in broad strokes and only with the intent of acquainting the reader with some of the mathematical context against which the problem emerges. In the second part, we will look at a debate (on “fusionism”) in which for the first time methodological and foundational issues related to aspects of the mathematical practice covered in the first part of the paper came to the fore. We conclude this part of the paper by remarking that only through a foundational and philosophical effort could the issues raised by the debate on “fusionism” be made precise. The third part of the paper focuses on a specific case study which has been the subject of such an effort, namely the foundational analysis of the plane version of Desargues’ theorem on homological triangles and its implications for the relationship between plane and solid geometry. Finally, building on the foundational case study analyzed in the third section, we begin in the fourth section the analytic work necessary for exploring various important claims about “purity,” “content,” and other relevant notions.

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Section: Against Construing Purity In Terms Of Formal Contentmentioning

confidence: 99%

“…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).…”

Section: Against Construing Purity In Terms Of Formal Contentmentioning

confidence: 99%

On the Relationship Between Plane and Solid Geometry

Arana

Mancosu

2012

The Review of Symbolic Logic

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“…Hilbert's example of 1899 has found recent interest, in particular, its automorphisms are studied in [1] and [26]. We note that reprints of the original "Festschrift" (i.e., the first edition [9] of Hilbert's "Grundlagen der Geometrie") are available, see [7,Ch.…”

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Early explicit examples of non-desarguesian plane geometries

Stroppel

2011

J. Geom.

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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).

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Automorphisms of Hilbert's non-desarguesian affine plane and its projective closure

Cited by 2 publications

References 11 publications

On the Relationship Between Plane and Solid Geometry

On the Relationship Between Plane and Solid Geometry

Early explicit examples of non-desarguesian plane geometries

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