Frege and the origins of model theory in nineteenth century geometry

Abstract

The aim of this article is to contribute to a better understanding of Frege's views on semantics and metatheory by looking at his take on several themes in nineteenth century geometry that were significant for the development of modern model-theoretic semantics. I will focus on three issues in which a central semantic idea, the idea of reinterpreting non-logical terms, gradually came to play a substantial role: the introduction of elements at infinity in projective geometry; the study of transfer principles, especially the principle of duality; and the use of counterexamples in independence arguments. Based on a discussion of these issues and how nineteenth century geometers reflected about them, I will then look into Frege's take on these matters. I conclude with a discussion of Frege's views and what they entail for the debate about his stance towards semantics and metatheory more generally.

Projective Geometry as a Model for Hegel’s Logic

“…9, 10), I have drawn attention to further ways in which logic over the last century has seemed to reinvent ideas that are easily detectable in Hegel's logic, but here, I want to conclude by drawing attention to a Hegelian analogue of a feature of projective geometry that has been invoked in contrast to the static universalism of Frege's logic, which is itself seen as underlying many of the semantic problems that faced its initial formulations. This concerns the idea developed by Gunther Edel [2], that logical systems must make possible the reinterpretation of the terms of their initial object languages.…”

“…In contrast to algebra, the Frege-Russell strand looked to analysis and, moreover, conceived of logic not as mathematics but as an autonomous discipline providing its rational foundation. 1 Within the emerging analytic paradigm of philosophy in the early twentieth century, the Frege-Russell approach would triumph over the earlier "algebra of logic" tradition stemming from Boole, 2 as well as over traditional Aristotelian forms of logic thought to be ultimately tied to an inadequate "subject-predicate" conception of logical form. One victim in particular would be the logic of Hegel. Recently, historians of the early years of the modern classicist movement have broadened the mathematical context within which it developed beyond analysis and algebra, with a number of investigators looking to the role of the nineteenth-century discipline of projective geometry-a discipline that had been singled out in the 1930s by Ernest Nagel [1] as particularly relevant.…”

“…One victim in particular would be the logic of Hegel. Recently, historians of the early years of the modern classicist movement have broadened the mathematical context within which it developed beyond analysis and algebra, with a number of investigators looking to the role of the nineteenth-century discipline of projective geometry-a discipline that had been singled out in the 1930s by Ernest Nagel [1] as particularly relevant. In fact, before his turn to foundational and logical issues, Frege had worked in projective geometry, and the relevance of this discipline has been raised especially in relation to addressing various semantic shortcomings apparent in the early forms of classicism, e.g., [2,3]. 3 Another example of such a possible role for projective geometry has been suggested by Pablo Acuña [6] (p. 8) with the suggestion that Wittgenstein, in describing the perceptible sign of a proposition as a "projection of a possible state of affairs" [7] ( § 3.11) may have had in mind the specific status of projection in projective geometry.…”

Peano’s structuralism and the birth of formal languages

“…On the development of nineteenth-century geometry and its relation to independence proofs and the pre-history of model theory, see [Blanchette, 2017, pp. 47-48], [Eder, 2019], Schiemer, 2018], Eder [2021], [Tappenden, 1997] and [Webb, 1995].…”

Frege on intuition and objecthood in projective geometry

“…andLorenat (2015) for a discussion of the duality controversy. Some of the issues that are discussed in this section have been investigated inEder (2019), which in turn is partly based on joint work with Georg Schiemer [seeEder and Schiemer (2018)]. 28 Chasles later clarified that really all that matters is that incidence is preserved.…”