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Tracking Contradictions in Geometry: The Idea of a Model from Kant to Hilbert

  • Judson Webb
Chapter
Part of the Synthese Library book series (SYLI, volume 251)

Abstract

This paper explores such questions as who actually discovered non-euclidean geometry, who actually believed in its consistency and why, and who can be said to have proved it to be free of contradiction. To this end I will analyze some views and results if ten or so philosophers and mathematicians from Kant to Hilbert. One main theme is that without some rudimentary idea of a model, the discovery and establishment of non-euclidean geometry would not have been possible. Another is that only the notion of a model enabled thinkers to conceive of properties of logical inference such as soundness and completeness of axioms and/or rules. These themes are surprisingly difficult to articulate clearly without compromising historical accuracy, but I believe that in most cases the attempt to do so leads to a better understanding of the writers involved.

Keywords

Euclidean Geometry Projective Geometry Hyperbolic Geometry Spherical Triangle Constant Negative Curvature 
These 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.

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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Judson Webb
    • 1
  1. 1.Boston UniversityUSA

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