Frege’s ‘On the Foundations of Geometry’ and Axiomatic Metatheory

Abstract

In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert's methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a 'new science' with its own basic truths. This paper aims to provide a reconstruction of this New Science that meets modern standards and to examine possible problems surrounding Frege's original proposal. The paper is organized as follows: the first two sections summarize the main points of the Frege-Hilbert controversy and discuss some issues surrounding the problem of independence proofs. Section 3 contains an informal presentation of Frege's proposal. In section 4 a more detailed reconstruction of Frege's New Science is set out while section 5 examines what is left out. The concluding section is devoted to a discussion of Frege's strategy and its significance from a broader perspective.

On consistency and existence in mathematics

“…by asking after how hard it is to determine whether there exists a proof-theoretic interpretation of Γ in a theory Γ + satisfying various adequacy conditions on successful conceptual analyses. A related proposal is given by Eder (2015) in his reconstruction of Frege's unpublished (1906) attempt to understand Hilbert's independence proofs on his own terms.…”

Hilbert, Duality, and the Geometrical Roots of Model Theory

“…This rediscovery was to a large extent effected by Gaspard Monge, but it is his pupil Jean Victor Poncelet, who is usually credited with the honorific title 'father of modern projective geometry'. 7 If one had to choose the single most important book in the development of modern projective geometry, most historians of mathematics would probably cite Poncelet's Traité des propriétés projectives des figures of 1822. Though obscure in some respects, the book contains a number of ideas that were formative for 19th century projective geometry.…”

“…'It is', he claims 6 Desargues himself gave two proofs of the theorem named after him, one in a purely Euclidean setting, the other using points at infinity. For more details on Desargues' theorem and its history see e.g., [26, 7 See, e.g., [11]. 8 As Poncelet states in the Traité: 'One always reasons upon the magnitudes themselves which are always real and existing, and one never draws conclusions which do not hold for the objects of sense, whether conceived in imagination or presented to sight' (see [28, p. 153]).…”

“…On the Frege-Hilbert controversy see, in particular,[5] and the references listed there. For more recent and detailed studies of Frege's understanding of metatheoretical concepts see[4] and[7].…”