David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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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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References found in this work BETA
Gottlob Frege, Gottfried Gabriel, Brian Mcguinness & Hans Kaal (1982). Philosophical and Mathematical Correspondence. Revue Philosophique de la France Et de l'Etranger 172 (1):64-64.
John Mayberry (1986). Cantorian Set Theory and Limitation of Size. [REVIEW] Philosophical Quarterly 36 (144):429-434.
Patricia A. Blanchette (1996). Frege and Hilbert on Consistency. Journal of Philosophy 93 (7):317-336.
Henri Poincaré (1946). The Foundations of Science: Science and Hypothesis, the Value of Science, Science and Method. Lancaster, Pa.,The Science Press.
Citations of this work BETA
Katherine Dunlop (2009). Why Euclid's Geometry Brooked No Doubt: J. H. Lambert on Certainty and the Existence of Models. Synthese 167 (1):33 - 65.
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