Frege on Axioms, Indirect Proof, and Independence Arguments in Geometry: Did Frege Reject Independence Arguments?

Notre Dame Journal of Formal Logic 41 (3):271-315 (2000)
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Abstract

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments

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Jamie Tappenden
University of Michigan, Ann Arbor

Citations of this work

Why Is a Valid Inference a Good Inference?Sinan Dogramaci - 2015 - Philosophy and Phenomenological Research 94 (1):61-96.
Frege on Consistency and Conceptual Analysis.Patricia A. Blanchette - 2007 - Philosophia Mathematica 15 (3):321-346.
Frege’s philosophy of geometry.Matthias Schirn - 2019 - Synthese 196 (3):929-971.

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References found in this work

What are logical notions?Alfred Tarski - 1986 - History and Philosophy of Logic 7 (2):143-154.
Compound thoughts.Gottlob Frege - 1963 - Mind 72 (285):1-17.
Logic and Truth in Frege.Thomas Ricketts & James Levine - 1996 - Aristotelian Society Supplementary Volume 70 (1):121 - 175.
About the law of inertia.Gottlob Frege - 1961 - Synthese 13 (4):350 - 363.
Frege and Hilbert on Consistency.Patricia A. Blanchette - 1996 - Journal of Philosophy 93 (7):317-336.

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