The Independence of the Parallel Postulate and Development of Rigorous Consistency Proofs

History and Philosophy of Logic 28 (1):19-30 (2007)
I trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. I analyze the kinds of arguments offered by Jules Hoüel in 1860-1870 for the unprovability of the parallel postulate and for the existence of non-Euclidean geometries, especially his reaction to the publication of Beltrami’s seminal papers, showing that Beltrami was much more concerned with the existence of non-Euclidean objects than he was with the formal consistency of non-Euclidean geometries. The final step towards rigorous consistency proofs is taken in the 1880s by Henri Poincaré. It is the formal conception of geometry, stripping the geometric primitive terms of their usual meanings, that allows the introduction of a modern fully rigorous consistency proof.
Keywords Beltrami  geometry  consistency
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DOI 10.1080/01445340600758800
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Philosophical and Mathematical Correspondence.Gottlob Frege, Gottfried Gabriel, Brian Mcguinness & Hans Kaal - 1982 - Revue Philosophique de la France Et de l'Etranger 172 (1):64-64.
Cantorian Set Theory and Limitation of Size. [REVIEW]John Mayberry - 1986 - Philosophical Quarterly 36 (144):429-434.
Frege and Hilbert on Consistency.Patricia A. Blanchette - 1996 - Journal of Philosophy 93 (7):317-336.
The Foundations of Geometry.David Hilbert - 1899 - Open Court Company (This Edition Published 1921).

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