Notre Dame Journal of Formal Logic

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

Jamie Tappenden

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.

Article information

Source
Notre Dame J. Formal Logic Volume 41, Number 3 (2000), 271-315.

Dates
First available: 26 November 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1038336845

Digital Object Identifier
doi:10.1305/ndjfl/1038336845

Mathematical Reviews number (MathSciNet)
MR1943496

Zentralblatt MATH identifier
1009.03003

Subjects
Primary: 01A55: 19th century
Secondary: 03-03: Historical (must also be assigned at least one classification number from Section 01) 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Keywords
geometry independence logic invariance Frege Hilbert duality axiom of parallels

Citation

Tappenden, Jamie. Frege on Axioms, Indirect Proof, and Independence Arguments in Geometry: Did Frege Reject Independence Arguments?. Notre Dame Journal of Formal Logic 41 (2000), no. 3, 271--315. doi:10.1305/ndjfl/1038336845. http://projecteuclid.org/euclid.ndjfl/1038336845.


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