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

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.

Frege and the origins of model theory in nineteenth century geometry

“…Hesse explicitly compares his Übertragungsprinzip to the principle of duality (Hesse 1866, 31 ff.). For further discussion, see Tappenden (2000Tappenden ( , 2005. the axioms shall be correspondingly the same for the transformed things.…”

Frege on Formality and the 1906 Independence-Test

“…26 Along these lines we find, for example, Frege's 1879 definition of the lying of a point A on a line BC in terms of the congruence of pairs of points. 27 Let's return to the case of arithmetic. Frege's view is that when we have fully analyzed the contents of (α) and (β), we will be left with considerably more-complex sentences (α * ) and (β * ), sentences which cash out those thoughts in terms of considerably simpler functions and objects, as is done in Frege's Grundlagen and Grundgesetze.…”

“…is that given by the reinterpretation of F 25. For helpful discussion of this point see[27].UnauthenticatedDownload Date | 6/14/16 12:19 PM…”

Consistency, Models, and Soundness

“…Frege claims that the fundamental error of Hilbert's axiomatic definitions lies in the fact that he mixes first-and second-level concepts. 25 See, for example, Blanchette (1996Blanchette ( , 2007, Tappenden (2000), Antonelli and May (2000).…”