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.
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Webb, J. (1995). Tracking Contradictions in Geometry: The Idea of a Model from Kant to Hilbert. In: Hintikka, J. (eds) From Dedekind to Gödel. Synthese Library, vol 251. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8478-4_1
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