Tracking contradictions in geometry
the idea of a model from Kant to Hilbert
pp. 1-20
in: Jaakko Hintikka (ed), From Dedekind to Gödel, Berlin, Springer, 1995Abstract
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.
