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

Author(s):  
Georg Schiemer

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.

Author(s):  
Georg Schiemer

This chapter investigates Carnap’s structuralism in his philosophy of mathematics of the 1920s and early 1930s. His approach to mathematics is based on a genuinely structuralist thesis, namely that axiomatic theories describe abstract structures or the structural properties of their objects. The aim in the present article is twofold: first, to show that Carnap, in his contributions to mathematics from the time, proposed three different (but interrelated) ways to characterize the notion of mathematical structure, namely in terms of (i) implicit definitions, (ii) logical constructions, and (iii) definitions by abstraction. The second aim is to re-evaluate Carnap’s early contributions to the philosophy of mathematics in light of contemporary mathematical structuralism. Specifically, the chapter discusses two connections between his structuralist thesis and current philosophical debates on structural abstraction and the on the definition of structural properties.


2020 ◽  
Vol 28 (3) ◽  
pp. 360-384
Author(s):  
Francesca Biagioli

Abstract It is well known that Felix Klein took a decisive step in investigating the invariants of transformation groups. However, less attention has been given to Klein’s considerations on the epistemological implications of his work on geometry. This paper proposes an interpretation of Klein’s view as a form of mathematical structuralism, according to which the study of mathematical structures provides the basis for a better understanding of how mathematical research and practice develop.


2008 ◽  
Vol 31 (4) ◽  
pp. 19-42
Author(s):  
François Rouget

According to contemporary observers, in the years from 1576 to 1579 Henry III brought together groups of orators-men and women, poets and courtiers-to discuss questions of moral philosophy. This happened both in Paris, and in provincial towns such as Blois and Ollainville. Several of the speeches given by these diverse orators remain extant. Among the spectators was Marguerite de Valois, who ordered the transcription of thirteen speeches that were pronounced during the January-February period of 1576. This album, preserved in a beautifully-bound manuscript, provides an interesting testimony to the intellectual curiosity of Marguerite and to the abiding interest of the king and his court in the domain of eloquence. The present article describes the context in which Marguerite de Valois participated in the first sessions of the Palace Academy, and presents the manuscript volume containing several, palpable traces of her reading of the texts. It also evaluates the contribution of some poets from Marguerite's entourage, such as Ronsard, Desportes, Baïf and Jamyn, and examines aspects of their philosophical debates as well as their oratorical skills.


Synthese ◽  
2019 ◽  
Author(s):  
Günther Eder

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.


Author(s):  
Georg Schiemer

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.


Author(s):  
Robert P. Apkarian

A multitude of complex ultrastructural features are involved in endothelial cell (EC) gating and sorting of lipid through capillaries and into steroidogenic cells of the adrenal cortex. Correlative microscopy is necessary to distinguish the structural identity of features involved in specific cellular pathways. In addition to diaphragmed fenestrae that frequently appear in clusters, other 60-80 nm openings; plasmalemma vesicles (PV), channels and pockets fitted with diaphragms of the same dimension, coexist on the thin EC surface. Non-diaphragmed coated pits (CP) (100-120 nm) involved in receptor mediated endocytosis were also present on the EC membrane. The present study employed HRSEM of cryofractured and chromium coated specimens and low voltage HRSTEM of 80 nm thick LX-112 embedded sections stained with 2.0% uranyl acetate. Both preparations were imaged at 25 kV with a Topcon DS-130 FESEM equipped with in-lens stage and STEM detector.HRSEM images of the capillary lumen coated with a lnm continuous fine grain Cr film, provided the ability to scan many openings and resolve (SE-I contrast) the fine structure of diaphragm spokes and central densities (Fig. 1).


2003 ◽  
Vol 34 (4) ◽  
pp. 219-226 ◽  
Author(s):  
Bart Duriez ◽  
Claudia Appel ◽  
Dirk Hutsebaut

Abstract: Recently, Duriez, Fontaine and Hutsebaut (2000) and Fontaine, Duriez, Luyten and Hutsebaut (2003) constructed the Post-Critical Belief Scale in order to measure the two religiosity dimensions along which Wulff (1991 , 1997 ) summarized the various possible approaches to religion: Exclusion vs. Inclusion of Transcendence and Literal vs. Symbolic. In the present article, the German version of this scale is presented. Results obtained in a heterogeneous German sample (N = 216) suggest that the internal structure of the German version fits the internal structure of the original Dutch version. Moreover, the observed relation between the Literal vs. Symbolic dimension and racism, which was in line with previous studies ( Duriez, in press ), supports the external validity of the German version.


Author(s):  
Odile Husain

Le présent article tente d’effectuer un rapprochement entre un article européen de Rossel et Merceron et un livre américain de Reid Meloy, tous deux consacrés à l’analyse des organisations psychopathiques. Si tous les auteurs s’entendent sur l’économie narcissique du psychopathe, le choix de la population d’étude diffère quelque peu, en raison de l’approche structurale des premiers et de l’approche symptomatique du second. Tandis que l’étude suisse ne retient que des psychopathes du registre des états-limites, l’étude américaine inclut également des psychopathes de niveau psychotique. Par contre, la mésentente règne au niveau des outils d’analyse du discours psychopathique: analyse statistique et échelles validées chez Meloy; approche qualitative chez Rossel et Merceron. Aux premiers, l’on reprochera un certain réductionisme et appauvrissement du discours, prix à payer pour le respect de la standardisation et de la cotation. Aux seconds, l’on reprochera l’absence de toute quantification qui pose problème lorsque l’on aborde la question de la validité des données. Néanmoins, Européens et Américains s’entendent sur la notion d’un fonctionnement psychopathique. La relation d’objet est marquée par la pulsion agressive et ses dérivatifs, par la recherche de pouvoir et de contrôle. La lutte contre la dépendance est déduite chez Meloy de l’absence de réponse de texture et chez Rossel et Merceron de l’absence de contenus de dépendance. La qualité narcissique des représentations d’objet est mise en évidence, chez Meloy, par le biais de l’investissement du paraître, chez Rossel et Merceron par l’importance du processus d’externalisation. La dévalorisation des objets est aussi décrite. Ni les uns ni les autres ne font réellement référence à l’angoisse car cette angoisse qualifiable d’anaclitique s’exprime justement sous des manifestations tout à fait opposées. Le vide intérieur est déduit, chez Meloy, à partir de l’ennui que vit le psychopathe et, chez Rossel et Merceron, à partir de la survalorisation de la référence au réel. Une grande convergence existe entre les deux écrits au sujet des mécanismes de défense. Tous les auteurs s’accordent sur la prépondérance du clivage et du déni, un déni par le mot et l’acte chez Meloy, un déni hypomaniaque chez Rossel et Merceron. De part et d’autre de l’Atlantique, on s’accorde également pour attribuer une place importante à l’identification projective et à l’identification à l’agresseur. Par ailleurs, Rossel et Merceron démontrent comment à travers les caractéristiques de l’énonciation et les nuances de la verbalisation du psychopathe, il est possible d’inférer son non-investissement de la mentalisation et du savoir au profit d’un surinvestissement de l’agir. La complémentarité, voire la similarité, des commentaires dans les deux ouvrages devrait réconforter certains cliniciens, désarmés devant le fossé qui semble parfois régner entre la littérature des deux continents et confirmer, qu’indépendamment du type de méthodologie et de validation choisi, l’observation clinique du psychologue expérimenté demeure la pierre angulaire de toute recherche en psychopathologie.


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