Frege Against Hilbert on the Foundations of Geometry

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Columbia University, 2006 - Geometry - 225 pages
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This dissertation is a close study of the Frege-Hilbert dispute over the foundations of geometry. The dispute has been the subject of active debate recently, with opinion divided as to the merits of Frege's position. In this dissertation I aim at a comprehensive assessment of Frege's position, its motivations, and its major consequences. I find that: (1) Frege's objections to Hilbert's Foundations of Geometry do not represent a mere misunderstanding of Hilbert's work, but stem from considerations of serious philosophical interest; (2) The same considerations that motivated Frege's objections suggest a conception of geometry---and a reading of the history of geometry---radically different from Hilbert's; (3) That conception of geometry---and reading of the history of geometry---are not obviously wrong, and indeed merit further investigation; (4) Part of Frege's objection to Hilbert's Foundations is that he gives no philosophical analysis of geometry, analogous to Frege's analysis of number in Foundations of Arithmetic; (5) The basic framework for such an analysis can be found in Frege's philosophical work, although it is far from obvious whether and how it can be carried through. The principal contributions of this dissertation lie in its clarification of the import of the Frege-Hilbert dispute for our understanding of the history of geometry, in particular the emergence of non-Euclidean and projective geometries; in its clarification of Frege's objections to Hilbert's independence proofs; and in its outline of a Fregean analysis of geometry, analogous to the analysis of number in Foundations of Arithmetic.

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