scholarly journals Frege and the origins of model theory in nineteenth century geometry

Synthese ◽  
2019 ◽  
Author(s):  
Günther Eder
Keyword(s):  
Nineteenth Century ◽  
Projective Geometry ◽  
Model Theory ◽  
Substantial Role ◽  
Principle Of Duality ◽  
Transfer Principles

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.

scholarly journals HILBERT, DUALITY, AND THE GEOMETRICAL ROOTS OF MODEL THEORY

2017 ◽  
Vol 11 (1) ◽  
pp. 48-86 ◽  
Author(s):  
GÜNTHER EDER ◽  
GEORG SCHIEMER
Keyword(s):  
Nineteenth Century ◽  
19Th Century ◽  
Projective Geometry ◽  
Model Theory ◽  
Euclidean Geometry ◽  
Point Of View ◽  
Logical Theory ◽  
Foundations Of Geometry ◽  
Theoretic Point

AbstractThe article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations.

scholarly journals Frege on intuition and objecthood in projective geometry

Synthese ◽  
2021 ◽  
Author(s):  
Günther Eder
Keyword(s):  
Nineteenth Century ◽  
Projective Geometry ◽  
Mathematical Background ◽  
Center Stage ◽  
Proper Perspective

AbstractIn 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 which occupied center stage in nineteenth century geometry, namely, projective geometry.

scholarly journals Transfer Principles, Klein’s Erlangen Program, and Methodological Structuralism

2020 ◽  
pp. 106-141
Author(s):  
Georg Schiemer
Keyword(s):  
Present Article ◽  
Transformation Groups ◽  
Theoretic Approach ◽  
Structural Identity ◽  
Erlangen Program ◽  
Structural Equivalence ◽  
Mathematical Structuralism ◽  
Principle Of Duality ◽  
Philosophical Debates ◽  
Transfer Principles

The present article investigates Felix Klein’s mathematical structuralism underlying his Erlangen program. The aim here is twofold. The first aim is to survey the geometrical background of his 1872 article, in particular, work on the principle of duality and so-called transfer principles in projective geometry. The second aim is more philosophical in character and concerns Klein’s structuralist account of geometrical knowledge. The chapter will argue that his group-theoretic approach is best characterized as a kind of “methodological structuralism” regarding geometry. Moreover, one can identify at least two aspects of the Erlangen program that connect his approach with present philosophical debates, namely (i) the idea to specify structural properties and structural identity conditions in terms of transformation groups and (ii) an account of the structural equivalence of geometries in terms of transfer principles.

У пошуках гармонії: Vals mèlancolque в контексті малої прози О. Кобилянської кінця ХІХ ст.

2017 ◽  
Vol 5 (5) ◽  
pp. 0-0
Author(s):  
Olga Yablonska
Keyword(s):  
Nineteenth Century ◽  
Late Nineteenth Century ◽  
Female Characters ◽  
Spiritual Quest ◽  
Substantial Role ◽  
Late Nineteenth ◽  
Women's Emancipation ◽  
Relationship Of ◽  
The Relationship

This paper analizes O. Kobylanska’s story “Vals mèlancolque” as the epicenter of the writers’ refl ections on the category of harmony and happiness. The relationship of O. Kobylanska’s spiritual quest in «Diary» and short prose of the late nineteenth century is observed (“Nature”, “Rose”, “Ignorant”, “Vals mèlancolque”, “Humility”, etc.). The author’s vision of the substantial role of art and words in the story “Vals mèlancolque” is highligted. This paper also investigates the symbolist nature of a text. The writer emphasizes the understanding of the actual idea of women’s emancipation. The paper shows that female characters embody the author’s conscious distinction of such categories as “love” (Martha), “cold art” (Anna) and a harmonious combination of “pieces” and “love” (Sofi a). It is concluded that in the work of Kobylanska the text is a landmark, being both a kind of life and artistic credo.

scholarly journals Cassirer and the Structural Turn in Modern Geometry

2018 ◽  
Vol 6 (3) ◽  
Author(s):  
Georg Schiemer
Keyword(s):  
Projective Geometry ◽  
Concept Formation ◽  
Mathematical Practice ◽  
Specific Character ◽  
Erlangen Program ◽  
Geometrical Concept ◽  
Principle Of Duality ◽  
Structural Methods ◽  
Geometrical Knowledge

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff (1910). The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice.

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