Salle du Théâtre, Grand Château.Université Côte d’Azur, Parc Valrose, Nice.
10.30 Welcome
11.00-12:15 Carlo Cellucci (Università Roma 1) Introducing Heuristic Philosophy of Mathematics
14:00 – 15:15 Jessica Carter (Aarhus University) Pragmatic realism and social ontology
15:30 – 16:45 Georg Schiemer (Wien Universität) Symmetries and implicit structure
17:00 – 18:15 Andrea Sereni (IUSS, Pavia) Chaos in Heaven. Fictionalism, Definitions and Mathematical Creation
Organisateurs: Paola Cantù (CNRS / Université Aix-Marseille), Frédéric Patras (CNRS/ Université Côte d’Azur
The workshop will not be broadcasted on the web.
Registration is free but mandatory. Please get registered by sending an email to paola dot cantu at univ-amu dot fr
Green pass required to access the conference room.
Résumés
Jessica Carter (Center for Science Studies at Aarhus University: Pragmatic realism and Social Ontology. According to some interpretations of the Philosophy of Mathematical Practice is a focus on human agents and their activities when formulating positions on the ontology and epistemology of mathematics. Recently J. Cole has formulated a position on the ontology of mathematics. Cole draws on tools from Social Ontology to say that mathematical domains are introduced by collective declarations and that they serve a representational function. He further argues that mathematical objects so introduced are objective, atemporal and exists by necessity. J. Ferreirós, on the other hand, formulates an epistemology of mathematics that is based on the view that mathematics is the outcome of various activities of human agents. In both accounts is the idea that mathematics consists of different “levels” or strata and that different levels interact in various ways. Mathematics dealing with ‘hypothetical states of things’ (Carter 2014) proposes a similar idea. In general, Mathematicians formulate hypotheses. Case studies from contemporary mathematics reveal that when new hypotheses (and the objects they refer to) are introduced they relate in various ways to previously studied domains. Inspired by readings of C.S. Peirce, I identify three types of “introduction processes” referred to as abstraction, generalization and abduction and argue that they conform to this description. In addition, a pragmatic view of reality is explored in the case of mathematics, that “the reality of a substance depends on the truth of statements concerning a more primary substance”. The main part of the talk will consist in an elaboration and critical assessment of the above formulated views: that it is possible to characterize the mathematical universe as consisting of different levels that are interconnected in various ways and the idea that there is a “bottom” level that is somehow related to the world of sensory appearances. In addition, I will express certain concerns about the historical arguments offered by Cole in support of his claim that mathematical domains exist by necessity and that they are atemporal.
Carlo Cellucci (Università Roma 1): Introducing Heuristic Philosophy of Mathematics. The purpose of the talk is to outline the main aspects of heuristic philosophy of mathematics, which is supposed to be an alternative to mainstream philosophy of mathematics, the philosophy of mathematics that has been dominant in the past century. I will argue that heuristic philosophy of mathematics must not be confused with the philosophy of mathematical practice, because the latter is essentially continuous with mainstream philosophy of mathematics. What is more important, I will argue that heuristic philosophy of philosophy is not subject to the shortcomings of mainstream philosophy of mathematics. As an example, I will compare the approaches of heuristic philosophy of mathematics and mainstream philosophy of mathematics to some aspects of mathematics.
Georg Schiemer Symmetries and implicit structure. According to a dominant view in modern philosophy of mathematics, mathematics can be understood as the study of the abstract structure of objects such as groups, number systems, graphs, or topological spaces. But what precisely is the relevant intrinsic struc- ture of such mathematical entities? How can we think about their structural content? In the present present talk, I will compare two general ways to think about the implicit structure of entities of pure mathematics. According to the first approach, the struc- tural properties of such objects are specified with reference to formal languages, usually based on some notion of definability. According to the second approach, structures are determined in terms of invariance under symmetries. For instance, the structural prop- erties of a given mathematical system are often said to be those properties invariant under certain structure-preserving mappings between similar systems (see, e.g., [3]). In the talk, we will investigate these two approaches by drawing to a particular mathe- matical example, namely the study of incidence structures in finite affine and projective geometry. Given this case study, I give a philosophical analysis of the conceptual differences be- tween the two methods to express implicit structure. The talk will focus on two issues. The first concerns the conceptual motivation for treating mathematical structures in terms of the notions of definability and invariance under symmetries. As will be argued, both methods capture some form of “topic neutrality” underlying the structuralist ac- count of mathematics. In the case of invariance, this is due to the fact that mathematics is indifferent to the intrinsic nature of mathematical objects and thus also indifferent to arbitrary switchings of such objects in a given system. In the case of definability-based approaches, the relevant topic neutrality is related to the the “formality” of logic and the fact that adequate logical definitions should be reducible to statements about the primitive mathematical structure (see [1]). Secondly, we discuss the relevance of the two ways to think about implicit structure for our understanding of mathematical structuralism. Here, in particular, the focus will be on the notion of the equivalence of mathematical structures. Building on the existing literature on the topic, I will discuss two notions of structural equivalence that take into account not only the (axiomatically defined) primitive structure, but also its implicit structural content. The first notion is motivated by the idea of definable implicit structure and based on the notion of interpretability (compare, e.g., [2]). The second notion, in turn, is motivated by the invariant approach and based on the concept of “transfer principles” between structures.
References
[1] D. Bonnay. Logicality and invariance. The Bulletin of Symbolic Logic, 14(1):29–68, 2008.
[2] T. Button and S. Walsh. Philosophy and Model Theory. Oxford: Oxford University Press, 2018.
[3] J. Korbmacher and G. Schiemer. What are structural properties? Philosophia Mathematica, https://doi.org/10.1093/philmat/nkx011, 2017.
Andrea Sereni. Chaos in Heaven. Fictionalism, Definitions and Mathematical Creation
Mathematical fictionalism is a welcome view for those who want the benefits of a literal reading of mathematical discourse while dispensing with its apparent problematic ontological commitments. Fictionalism can and has been argued for in a number of different ways, and remains a controversial position. We argue that among possible motivations – epistemic, attitudinal, ontological – the most plausible route should be considered to be semantical. Fiction-introducing principles are much alike (and sometimes are) mathematical definitions. While fiction-introducing principles may involve acts of creation, they have rarely been compared to creative definitions. The paper investigates what the debate on creative definitions, in particular as regards Frege’s concerns and Dedekind’s remarks on this score, can contribute to the assessment of contemporary mathematical fictionalism.
Salle E013, Bâtiment EGGER, Université Aix-Marseille, 29 Av. Schuman, Aix-en-Provence.
14h00-15h20 Gerhard Heinzmann (Archives Poincaré, Université de Lorraine), L’expérience de pensée en mathématique
15h30-16h50 Viviane Durand-Guerrier (IMAG, Université de Montpellier), Vérité et preuve en mathématiques
17h00-18h20 Francesca Biagioli (Dipartimento di Filosofia, Università di Torino), Cassirer and some recent discussions on the relativized a priori
Organisation : Paola Cantù (CGGG, CNRS et Aix-Marseille Université) (paola.cantu@univ-amu.fr)
The meeting will be followed by the 4th edition of PHILMATHMED. For further information, click here.
The program and the abstracts of the two meetings can be downloaded here.
]]>Aix-Marseille Université, Faculté des Lettres
En ligne sur la plateforme Zoom
To register, click on the following link :
https://univ-amu-fr.zoom.us/meeting/register/tJMsfuiprDsjGNLfHIql3mhyS3JETPsrr-__
After registering, you will receive a confirmation email including information for joining the meeting.]
PROGRAM
Mercredi 21 Octobre 2020
10h15-10h30 Welcome
10h30-11h15 Mario Castellana. Maximilien Winter: une contribution cachée à la philosophie des sciences
11h15-11h30 Pause
11h30-12h15 Pause
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”
11h45-12h00 Pause
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
10h45-11h00 Pause
11h00-11h45 Eduardo Giovannini and Georg Schiemer, Hilbert’s Early Views on Completeness and Categoricity
11h45-12h00 Pause
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
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.
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.
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.
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.
]]>Sponsors : CNRS, Université Aix-Marseille (Centre Gilles Gaston Granger), Vienna Universität (Institut Philosophie) and ERC Prehistory of Structuralism, Università degli Studi di Torino (Dipartimento di Matematica).
Day 1 Wednesday May 15, 2019
14:00-14:30 Opening
14:30 – 15:30 Georg Schiemer, Padoa, Carnap, and Tarski on deductive theories
15:30 – 16:30 Pierre Wagner, Schlick and Carnap on definitions
16:30 – 17:00 Break
17:00-18:00 Gabriel Giovannetti, The Theory of Relativity against Kantian epistemology: Schlick and Reichenbach
Day 2 Thursday May 16, 2019
9:00 – 10:00 Paola Cantù, The Peano’s school epistemology and its relation to logicism and structuralism
10:00 – 11:00 Erika Luciano, Between theory and practice: The contributions of the School of Peano to philosophy of mathematics education
11:00-11:30 Break
11:30 – 12:30 Livia Giacardi, Enriques’ s «Battle» for a Scientific Humanitas and its Effects on Mathematics Education
12:30-14:30 Lunch
14:30 – 15:30 Francesca Biagioli, Federigo Enriques and Felix Klein on mathematical concept formation
15:30 – 16:30 Massimo Ferrari, Federigo Enriques’ “Problems of Science”. A Chapter in the History of Scientific Philosophy
16:30 – 17:00 Break
17:00-18:30 Round table: Carnap and Enriques. Definitions and the role of logic in scientific philosophies . A discussion of Federigo Enriques, Problemi della scienza. Zanichelli, Bologna. Engl. trans. Problems of science. Open Court Publishing Company, Chicago, 1914, § IIIA. The problems of logic. Pure logic and of Rudolf Carnap, Eigentliche und uneigentliche begriffe. Symposion: Philosophische Zeitschrift für Forschung und Aussprache, 1(4) 1927:355–374.
Day 3 Friday May 17, 2019
9:00 – 10:00 Frédéric Patras, Leibniz, Kant et Frege sur les nombres et l’analyticité en arithmétique
10:00 – 11:00 Charles Alunni, Hegel chez Enriques
11:00-11:30 Break
11:30-12:30 Fabio Minazzi, The Problem of Historical Epistemology in the Italian Debate of the 20th Century
12:30-14:30 Lunch
December 12-14, 2018,
Place: Room 3A, Department of Philosophy, University of Viienna, Universitätsstraße7, 1010 Vienna.
Organized by: Georg Schiemer and Francesca Biagioli (Universityof Vienna) and PaolaCantù (Aix-Marseille Université and CNRS)
Day 1 | Wednesday | December 12, 2018 |
9:00 (s.t.) – 9:15 | Opening |
9:15 – 10:15 | Julien Bernard (Aix-Marseille Université) “Schlick, Weyl and Husserl on scientific philosophy”. |
10:15– 10:30 | Tea/Coffee Room C 0219, Kommunikationsraum |
10:30 – 11:30 | Christian Damböck (Universität Wien) “The Development of Carnap’s Aufbau”. |
11:30 – 12:30 | Elisabeth Nemeth (Universität Wien) “ On Otto Neurath’s Pictorial Statistics “. |
12:30 – 14:30 | Lunch: Restaurant Roth Währinger Str. 1, 1090 Vienna |
14:30 – 15:30 | Christoph Limbeck (Universität Wien) “Carnap´s Early Draft for the `Aufbau´ and the Erlangen Conference (1923)”. |
15:30 – 15:45 | Tea/Coffee Room C 0219, Kommunikationsraum |
15:45 – 16:45 | John Wigglesworth (Universität Wien) “Benacerraf’s identification problem revisited”. |
19:00 | Conference Dinner: Rebhuhn Berggasse 24, 1090 Vienna |
Da y 2 | Thursday | December 13, 2018 |
9:00 – 10:00 | Frédéric Patras (Université de Nice Sophia-Antipolis) “Carnap and mathematical structuralism”. |
10:00 – 10:15 | Tea/Coffee Room C 0219, Kommunikationsraum |
10:15 – 11:15 | Francesca Biagioli & Georg Schiemer (Universität Wien) “Structuralism and Ordnungsgefüge”. |
11:15 – 12:15 | Pierre Wagner (Université Paris 1 – Panthéon-Sorbonne) “Carnap on incompleteness and the unity of formal science”. |
12:15 – 14:15 | Lunch: Restaurant Stadtkind Universitätsstraße 11, 1010 Vienna |
14:15 – 15:15 | Gabriel Giovannetti (Aix-Marseille Université) “Scientific revolutions and scientific philosophy: does the claim that philosophy is scientific come from the revolutionary structure of scientific progress?”. |
15:15 – 15:30 | Tea/Coffee Room C 0219, Kommunikationsraum |
15:30 – 16:30 | Laurent Mazliak (Sorbonne Université, Paris) “Some considerations about the axiomatization of probability in USSR in the 1930s. A mathematical and political narrow path…”. |
Day 3 | Friday | December 14, 2018 |
9:00 – 10:00 | Massimo Ferrari (Università degli Studi di Torino) “Leibniz and the Vienna Circle“. |
10:00 – 10:15 | Tea/Coffee Room C 0219, Kommunikationsraum |
10:15 – 11:15 | Paola Cantù (Aix-Marseille Université and CNRS) “Scientific philosophies: the role of logic and history in conceptual analysis”. |
11:15 – 12:15 | Friedrich Stadler (Universität Wien) “Principle of Economy, Simplicity, and Parsimony – From Mach to the Vienna Circle”. |
12:15 – 14:15 | Lunch: Restaurant Roth Währinger Str. 1, 1090 Vienna |
15:00 – 17:00 | Roundtable on INTEREPISTEME project |
For further information visit: https://structuralism.phl.univie.ac.at/ or contact florian.kolowrat@univie.ac.at
the workshop is co-funded by:
CNRS International Project of Scientific Cooperation (PICS) N° PICS07887 : “The effect of interdisciplinary collaboration on early 20th century epistemologies.A comparison among the Peano School, the Vienna Circle, and the Editorial Boardof the Revue de Métaphysique et de Morale: protostructuralism andproto-pluralistic logicism.” (Austria) 2018 – 2020
]]>La réunion de lancement du projet INTEREPISTEME aura lieu
le mercredi 7 novembre 2018
dans la salle 16-26 113
du Laboratoire de Probabilités, Statistiques et Modélisation
Sorbonne Université. Campus Pierre et Marie Curie
4, Place Jussieu, Paris (5e)
de 17 heures à 19 heures
La réunion est ouverte à tous. Merci de bien vouloir nous confirmer votre participation (paola dot cantu at univ-amu dot fr).
Informations pratiques. Le campus Pierre et Marie Curie (métro Jussieu) est joignable par les lignes 7 et 10 du métro. Il faut monter au 1er étage de la tour 26. La salle 113 est au Laboratoire de Probabilités, Statistiques et Modélisation, au milieu du couloir 16-26 à droite.
]]>The project is funded by the National Center for Scientific Research (France) in collaboration with the University of Vienna (Austria).
It is co-directed by Paola Cantù (Université Aix-Marseille and CNRS / Centre Gilles Gaston Granger) and Georg Schiemer (Universität Vienna, Institut für Philosophie).
]]>