21-23 Octobre 2020, 10h00-13h00
Aix-Marseille Université, Faculté des Lettres
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Mercredi 21 Octobre 2020
10h30-11h15 Mario Castellana. Maximilien Winter: une contribution cachée à la philosophie des sciences
12h00-12h45 Francesca Biagioli. Federigo Enriques and the philosophical background to the discussion of implicit definitions
Jeudi 22 Octobre 2020
10h00-10h45 Christoph Limbeck, Carnap´s `Aufbau´ project at the Erlangen Conference 1923
10h45-11h00 Julien Bernard, Reconstruction of Weyl’s history of geometry, following the philosophical guideline of “purification of the a priori”
12h00-12h45 Dewi Trebaul The Vienna Circle facing the possibility of metalanguage and metatheory: an incomplete debate in the early 1930’s.
Vendredi 23 Octobre 2020
10h00-10h45 Jessica Carter, Structuralism and Mathematical Practice
11h00-11h45 Eduardo Giovannini and Georg Schiemer, Hilbert’s Early Views on Completeness and Categoricity
12h00-12h45 Gabriel Giovannetti, Reichenbach and the criticism of conventionalism as an attempt to constitute an empiricist scientific philosophy.
Organisation: Paola Cantù, Gabriella Crocco et Georg Schiemer
Partenariat: Centre Gilles Gaston Granger (CGG UMR 7304), UFR ALLSHS, Université Aix-Marseille, CNRS.
Les langues du colloque sont le français et l’anglais. Inscription gratuite mais obligatoire. Le lien pour accéder au colloque en ligne sera envoyé aux participants enregistrés.
Contact : Paola dot Cantu at univ-amu dot fr
Dewi Trebaul. The Vienna circle facing the possibility of metalanguage and metatheory : an incomplete debate in the early 1930’s
We would like to investigate how the members of the Vienna circle embraced or rejected the adoption of a metalanguage in order to build metatheories in the early 1930’s. Under the impulse of Hilbert’s and Tarki’s achievements, Carnap promoted such an approach that culminated in his Logische Syntax der Sprache in 1934. This approach divided the circle : while Hahn and Carnap adopted the new methodological standpoint, Schlick and Waismann remained sceptical about it. At the same time they assisted to deep mutations in Wittgenstein’s thinking, who rejected altogether the possibility of a metatheory. However a fully explicit debate on this topic didn’t take place. A reconstruction of the opposite arguments shall show us that different conceptions of logic and of its relation to the other sciences are here at work.
Mario Castellana. Maximilien Winter: une contribution cachée à la philosophie des sciences
Une histoire critique de la philosophie des sciences nous oblige aussi de tenir compte des figures cachées et oubliées qui ont donné des contributions considérables au développement de ce savoir; en ses essais écrits dans les premières année du XXe siècle et apparus dans la ‘Revue de Métaphysique et de Morale’, dont a été cofondateur avec Xavier Léon e L. Brunschvicg, Maximilien Winter (1871-1935) a dessiné un parcours de recherche épistémologique, merci à des analyses profondes des oeuvres de Poincaré et d’autres savants, orienté à fournir ce qu’il a appelé ‘la méthode historico-critique’ en philosophie des mathématiques; pour tout cela son ouvrage ‘La Méthode dans la philosophie des mathématiques’ (1911) peut être considéré un des textes fondateurs de l’épistémologie française et aussi de la même philosophie des sciences.
Eduardo Giovannini and Georg Schiemer. Hilbert’s Early Views on Completeness and Categoricity
David Hilbert’s epochal Foundations of geometry, originally published in 1899, is usually conceived as a turning point in the introduction of “model-theoretic” ideas in (early) modern axiomatic geometry. His systematic application of the method of construction of ‘models’ of geometrical axioms, to prove metatheoretical properties such as consistency and independence, had a great significance for the emergence of the semantical conception in modern logic. Moreover, Hilbert’s remarkable axiom of completeness [Vollständigkeitsaxiom] (introduced in the first French edition of Foundations in 1900) is perhaps one of his most innovative and interesting contributions to the early development of the “model-theoretic” viewpoint in modern logic. Its application in the geometrical context had also notable consequences for the geometrical theory, namely by means of the axiom of completeness the usual analytic geometry over the real numbers became the unique model (up to isomorphism) of his axioms for elementary Euclidean geometry. More importantly, with Hilbert’s formulation of the axiom of completeness the notion of a maximal model was first conceived.
The present talk will provide a historical reconstruction and analysis of the vicissitudes around Hilbert’s inclusion of the axiom of completeness into his axiomatic system for Euclidean geometry. This task will be undertaken on the basis of a series of unpublished notes for lecture courses, corresponding to the period 1894–1905. On the one hand, we will argue that this historical and conceptual analysis not only sheds new light on how Hilbert originally conceived the nature of his geometrical axiom of completeness, but also it allows to clarify some misunderstandings concerning the relation between the axiom and several notions of ‘completeness’ of an axiomatic system. On the other hand, w will claim that the material included in his notes for lecture courses brings new insights into how Hilbert regarded the importance of metalogical properties like completeness and independence for his axiomatic investigations on the foundations of geometry.
The second aim is to provide a more systematic assessment of Hilbert’s initial understanding of several key “model-theoretic” notions, which were involved directly and indirectly in the formulation of the axiom of completeness. In particular, I will present formal reconstructions of Hilbert’s early insights into several central semantical concepts such as ‘isomorphism’, ‘categoricity’, different notions of ‘completeness’ of an axiom system, and specially, ‘model maximality’. This task will be carried out by examining a sketch of a categoricity proof of the axiom system for analysis, outlined by Hilbert in the notes for lectures course Logical Principles of Mathematical Thinking (1905). Finally, I will argue that these formal reconstructions prove to be relevant for a better understanding of the kind of ‘completeness’ that Hilbert aimed to achieve with the incorporation of the axiom of completeness, respectively, in the arithmetical and in the geometrical context. The talk is based on joint research with Eduardo Giovannini.
Francesca Biagioli, Federigo Enriques and the philosophical background to the discussion of implicit definitions
Implicit definitions have been much discussed in the history and philosophy of science in relation to logical positivism. Not only have the logical positivists been influential in establishing this notion, but they have addressed the main problems connected with the use of such definitions, in particular the question whether there can be such definitions of basic mathematical concepts, and the problem of delimiting their scope. This paper aims to draw further insights on implicit definitions from the development of this notion from its first occurrence in German language in Enriques’s “Principles of Geometry” (1907) to Schlick’s General Theory of Knowledge (1918). It will be argued that Enriques offers one way to counter some of the classical objections against the early twentieth-century conceptualization of implicit definitions. Enriques did not conflate the distinct notions that had been identified as implicit definitions in the recent history of mathematics, but he tried to offer an account of the process leading to structural definitions. The paper will point out, furthermore, that Enriques’s account differs significantly from Schlick’s. The scientific interpretations of implicit definitions in Schlick’s theory of knowledge depend on the coordination of the terms of abstract mathematical structures with physical realities. By contrast, Enriques addressed the problem of bridging the gap between abstract and concrete terms by identifying patterns within mathematics that provide a clarification of conceptual relations, and so also serve (indirectly) the purposes of applied mathematics.
Christoph Limbeck, Carnap´s `Aufbau´ project at the Erlangen Conference 1923
Carnap wrote a first draft of his future Aufbau in the summer of 1922 under the title “From Chaos to Reality” (“Vom Chaos zur Wirklichkeit”). In that draft, he sketched already how our concepts can be constructed by the application of Russell´s logic of relations to experience. He specified basic units of experience (“Bausteine”), relations among them and classes of such experiences which defined concept (e. g. of specific colors). The draft was followed by the plan to organize a conference about the application of Russell´s logic to epistemological problems, a conference which took place in 1923 in Erlangen. The conference was also intended as a forum for the discussion of Carnap´s draft. Carnap, together with Reichenbach, invited therefore philosophers, logicians (Heinrich Behmann) and psychologists (Kurt Lewin, Fritz Heider) in order to discuss the general aim of an application of logic to epistemology and more specifically in order to discuss his own draft. My talk will analyze how Carnap´s first draft of the Aufbau fitted into the more general aim of the Erlangen Conference.