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![Frege’s representation of the imaginary straight line by guide lines G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G} $$\end{document} and H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{H} $$\end{document}. A point P on an imaginary line is represented by a line that joins a point on G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G} $$\end{document} with a point on H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{H} $$\end{document} (or the corresponding pair of points in I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {I}$$\end{document} and R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}$$\end{document} that determine this line). (The diagram is an adaptation from Frege’s first diagram in his dissertation.)](publication/349933651/figure/fig3/AS:11431281115654846@1675050047790/Freges-representation-of-the-imaginary-straight-line-by-guide-lines.png)
Frege’s representation of the imaginary straight line by guide lines G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G} $$\end{document} and H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{H} $$\end{document}. A point P on an imaginary line is represented by a line that joins a point on G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G} $$\end{document} with a point on H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{H} $$\end{document} (or the corresponding pair of points in I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {I}$$\end{document} and R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}$$\end{document} that determine this line). (The diagram is an adaptation from Frege’s first diagram in his dissertation.)
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In recent years, several scholars have been investigating Frege’s mathematical background, especially in geometry, in order to put his general views on mathematics and logic into proper perspective. In this article I want to continue this line of research and study Frege’s views on geometry in their own right by focussing on his views on a field wh...
Citations
... La distinction entre les actants en constituants stables et les circonstants en figurants vue uniquement à partir de la boutade ''la phrase est un petit drame'' nous oriente vers les rouages épistémiquement inconscients de cette découvrabilité : chercher une voie d'issue au potentiel du signe comme une projection argumentale à la base d'un tout-narratif auquel obéissent les textes culturels. Or, c'est la même recherche que Frege a entreprise par les moyens de la géométrie projective pour obtenir sa logique prédicative et sémiotique (Eder, 2021). Il s'agit du remplacement de l'axe fonctionnel du signe par une définition de l'axe textuel du parcours mais sans les particules argumentales de la projection permise par les textes ''authentiques'' (lire : observés et observables, non-idéologiques/culturels, non-• Dont les notions 'intuitivement' inhérentes de signature mathématique ou les traces constantes et inertielles. ...
The present books is a step further in the theoretical definition of the ROAL-Hypothesis, in which new empirical and experimental dimensions and findings encounter a space hypothesis of language and language-in-individual-languages. The space hypothesis has been a necessary mode to strengthen mathematical postulates on text/field levels between verifiability/demonstration and events of individual languages.
In addition to past books, the present one discusses two difficult questions:
- Demonstration of translatability
- The relation between the linguistic object and it's semiotic projection.
Both face the challenge of immanence/relevance from both epistemic and mathematical modelling.
The result as such derives from a constant search of a scientific theory of language accounting for verifiability/demonstration and empirical observability beyond idealizations and idealization raising lacking complex relations between acts and facts.
... 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. ...
... 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. three means, a unity holding despite the incommensurability between the geometric mean, the calculation of which would have required square roots, and the other two. ...
... 2 Of course, Boolean logic would continue in the twentieth century in the new discipline of computer science which, developing in the 1920s, predated the actual birth of computers. 3 The role of the related discipline of topology has been similarly discussed in this context. See, for example, [4]. ...
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.
... 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 nineteenth thirties of Earnest 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]. 2 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 2 "projection of a possible state of affairs", Wittgenstein [7] ( § 3.11) may have had in mind the specific status of projection in projective geometry. ...
... Pappus, however, had sought to preserve work from earlier times and had taken structures at the heart of projective geometry from Apollonius of Perga , and hints at earlier associations with the Pythagorean music theory can be found in the name that would be eventually given to projective geometry's principal "invariant", the "harmonic crossratio". 3 Pappus's early steps in projective geometry had been revived and built on in the seventeenth century by Girard Desargues [8], a French mathematician and engineer and contemporary of Rene Descartes. Although Desargues had a few initial followers, notably the young Blaise Pascal, his work would fall into neglect, swamped by the success of the analytic geometry introduced by Descartes's Geometrie in 1637. ...
... Finally, in Section 5, distinctive features of Hegel's logic are considered as expressing a projective equivalent of Aristotle's more "Euclidean" syllogistic. 3 ...
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.
... Geometry is also a theory of space consisting of elements. Elements of all geometrical constructs refer to intuition in different and diverse domains of mathematics (Eder, 2021). Some experts argue that geometry requires intuition (Biagioli, 2014;Frege, 1980;Kant, 1998). ...
Geometrical intuition is the ability to visualize, construct, and manage geometrical shapes in the mind when solving geometry problems. Geometrical intuition requires four skills: the ability to construct and manage geometrical figures in mind, perceive geometrical properties, connect pictures to concepts and theories in geometry, and determine where and how to begin when solving geometry problems. This geometric intuition ability is important for developing problem-solving. Therefore, we need a task that can be used to identify and develop students' geometric intuition abilities. This research aims to design a geometric intuition task. We employ design research methods to design geometrical intuition tasks by conducting a literature review on geometric intuition and geometry tasks, creating geometrical intuition tasks, and estimating and noting the possible student responses. This study produced three types of tasks based on the four components of geometric intuition. We provide a list of possible responses that junior high school students may provide, as well as practical suggestions for teachers. We recommend research using our developed task to evaluate students' geometrical intuition.
... 7 Hilbert, on the other hand, did not break as radically the connection between geometry and Anschauung (in his Festschrift) as some interpretations (among others, that 2 For example, Blanchette writes, "Hilbert is clearly the winner in this debate, in the sense that roughly his conception of consistency is what one means today by 'consistency' in the context of formal theories, and a near relative of his methodology for consistency-proofs is now standard" Blanchette (2018). 3 Letter to Frege from December 29th, 1899 (Frege, 1980, p. 40). 4 More recent contributions have been made by Eder (2021), Schirn (2019), and Shipley (2015), to mention just a few. 5 Freudental claims, "Frege, rebuking Hilbert like a schoolboy, also joins the Boeotians. (I have never understood why he is so highly esteemed today)" (Freudenthal, 1962, p. 618). ...
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.
... 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),(Eder & Schiemer, 2018),(Eder, 2021),(Tappenden, 1997) and(Webb, 1995). ...
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.
