Frege on intuition and objecthood in projective geometry

Eder, Günther

doi:10.1007/s11229-021-03080-0

Cited by 4 publications

(4 citation statements)

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“…Considered arithmetically, if one "adds" a perfect fourth to a perfect fifth, a full octave results. 31 However, when considered as intervals in a geometric sequence, a complete octave results from the multiplication of the two intra-octaval intervals, just as 3 2 × 4 3 = 2. 32 We should not be surprised, then, that the "musical tetraktys" turns out to be an instance of the principal invariant in projective geometry.…”

Section: The Role Of the Musical Ratios In Projective Geometrymentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Projective Geometry as a Model for Hegel’s Logic

Redding

2024

Logics

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Recently, historians have discussed the relevance of the nineteenth-century mathematical discipline of projective geometry for early modern classical logic in relation to possible solutions to semantic problems facing it. In this paper, I consider Hegel’s Science of Logic as an attempt to provide a projective geometrical alternative to the implicit Euclidean underpinnings of Aristotle’s syllogistic logic. While this proceeds via Hegel’s acceptance of the role of the three means of Pythagorean music theory in Plato’s cosmology, the relevance of this can be separated from any fanciful “music of the spheres” approach by the fact that common mathematical structures underpin both music theory and projective geometry, as suggested in the name of projective geometry’s principal invariant, the “harmonic cross-ratio”. Here, I demonstrate this common structure in terms of the phenomenon of “inverse foreshortening”. As with recent suggestions concerning the relevance of projective geometry for logic, Hegel’s modifications of Aristotle respond to semantic problems of his logic.

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Section: The Role Of the Musical Ratios In Projective Geometrymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Projective Geometry as a Model for Hegel’s Logic

Redding

2024

Logics

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“…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].…”

Section: 2mentioning

confidence: 99%

Peano’s structuralism and the birth of formal languages

Millán

2022

Synthese

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Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano's understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind's construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language.

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The Frege–Hilbert controversy in context

Rohr

2023

Synthese

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This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the developments in 19th-century geometry posed for the traditional Kantian philosophy of mathematics. Frege maintains that Anschauung is a source of knowledge by which we acknowledge geometrical axioms as true. For Hilbert, in contrast, axioms provide one of several correct “pictures” of reality. Hilbert’s position is thereby deeply influenced by epistemological ideas from Hertz and Helmholtz, and, in turn, influenced the neo-Kantian Cassirer.

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Frege on intuition and objecthood in projective geometry

Cited by 4 publications

References 54 publications

Projective Geometry as a Model for Hegel’s Logic

Projective Geometry as a Model for Hegel’s Logic

Peano’s structuralism and the birth of formal languages

The Frege–Hilbert controversy in context

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