Organisers: Sylvain Cabanacq,
Emmylou Haffner,
David Rabouin
No prerequisites are expected from participants. On the contrary, the idea is to share historical knowledge, mathematical skills, philosophical insights and points of view. Our starting points will go from recent papers, discussed together, to mathematical theories (for example, group representation) or more “classical” historical and philosophical writings (Weyl, Lautman, …). This year, the group will be organized in series of sessions united by a theme. A large part of each session will be devoted to questions and discussion.
To current year and archives 2011– |
PROGRAMME 2013–2014 : twice a month, Thursdays, 10:00–13:00, Room Rothko (412B), University Paris Diderot, Building Condorcet, 75013 Paris – plan d’accès.
1. Forcing (2 sessions)
26 Sept.
Giorgio Venturi (Team of Logic of Univ. Paris Diderot/ Scuola Normale di Pisa)
Set Theory
Au début des années 1960, Paul Cohen tranchait une question centrale de la Théorie des Ensembles, formulée par Cantor et devenue le premier problème de Hilbert : l’existence d’un cardinal entre le cardinal des entiers naturels et celui des réels ne peut être décidée par la théorie ZFC. La portée fondationnelle de ce résultat est évidente, et a été largement commentée. Il s’agira plutôt, lors de cette première séance, de s’intéresser à la méthode même développée par Cohen, afin de mettre en lumière son originalité et de cerner ses implications épistémologiques : le forcing.
En quel sens, par exemple, peut-on affirmer que le forcing établit un nouveau rapport entre les objets et l’univers mathématique? Peut-on déceler une philosophie des mathématiques implicite dans la stratégie de Cohen (formalisme, réalisme...) ?
Une seconde séance, le 24 octobre, reviendra sur la reformulation catégorique de cette méthode, et le rôle de la notion de faisceau dans ce contexte.
24 Oct.
Sylvain Cabanacq (SPHERE)
Topos Theory
2. Focus on some problems occurring in Hausdorff (3 sessions)
14 Nov.
Pascal Bertin (SPHERE)
Le "Spielraum" de Hausdorff/Mongré
5 Dec.
Exposé de Patrick Dehornoy (Université de Caen Basse-Normandie)
La théorie des ensembles cinquante ans après Cohen.
On présentera quelques résultats de théorie des ensembles contemporaine autour de l’Hypothèse du Continu et on discutera la possibilité de résoudre la question après les résultats négatifs de Gödel et Cohen.
19 Dec.
Exposé de Baptiste Mélès (Univ. Blaise Pascal)
La philosophie des langages de programmation : des codes sources aux programmes.
J’expliquerai ce qu’est la programmation informatique en partant de l’architecture de l’ordinateur et des langages binaire et assembleur, puis en expliquant les notions de compilateur et d’interprète, de paradigmes de programmation (programmation impérative, fonctionnelle, logique, orientée objet, orientée aspect, etc.). Nous verrons à cette occasion en quoi les langages de programmation théoriques (par exemple les lambda calculs et les machines de Turing) manquent certaines propriétés particulièrement "intéressantes" des langages de programmation concrets. Enfin, comme l’objectif ultime de tout langage est de fonder une littérature, je présenterai les programmes particuliers que sont les systèmes d’exploitation en décrivant le code source d’Unix v6 comme créateur d’une ontologie de second niveau. L’idée générale serait ainsi d’appuyer une philosophie des langages de programmation sur les langages concrets (par opposition aux langages théoriques) et sur les programmes qu’ils servent à écrire.
3. Wittgenstein, unique session !!! session postponed !!!
9 Jan.
Jean-Philippe Narboux (Univ. Michel de Montaigne Bordeaux 3)
L’enjeu du jeu.
23 Jan.
Sebastien Gandon (PHIER, Univ. Blaise Pascal)
Y a-t-il de l’indexicalité en mathématique ?
Dans les années soixante-dix, les travaux de Kripke, de Kaplan, de Putnam, de Lewis (pour ne citer qu’eux) ont non seulement profondément et durablement renouvelé la philosophie du langage, mais également la philosophie des sciences. La philosophie des mathématiques est cependant restée à l’écart de ces bouleversements. Différents auteurs se sont interrogés depuis une quinzaine d’années sur cette singularité, soit pour affirmer que les travaux des philosophes du langage inspirés par la sémantique modale trouvent leurs limites lorsque l’on se penche sur le discours mathématique (Corfield), soit pour suggérer, au contraire, qu’il y aurait des usages pertinents de ces recherches en philosophie des mathématiques (Tappenden, Smadja, Baker). Mon intervention prolonge ce questionnement. Plus précisément, en partant d’une interprétation de la méthode axiomatique hilbertienne proposée par Hodges et Demopoulos, je m’interrogerai sur la pertinence de l’idée selon laquelle les constantes non-logiques d’une théorie formalisée doivent être traitées comme des indexicaux.
4. Visit of Jeremy Gray (3 sessions)
6, 13, 20 Feb.
1rst session, Thursd. 6 Feb.
The first talk will be focused on Poincaré’s work, and I will consider the three following topics:
- Poincaré’s idea of a transformation group
- Poincaré’s philosophy of mathematics
- Poincaré’s philosophy of science
In the first part, I will review Poincaré’s use of transformation groups as a consistent and unifying idea in mathematics. The second part will develop these ideas in the context of his geometric conventionalism and his complicated attitudes to Einstein’s special theory of relativity. The third part will present two reworkings of Poincaré’s philosophy of science: structural realism and a Wittgensteinian interpretation.
The second and third talks will be on aspects of the ideas of Hermann Weyl, and will have a more open character as a work in progress. I will concentrate on two publications, Das Kontinuum (1917) and Philosophie der Mathematik und Naturwissenschaft (1927) – later published in an English translation (with many revisions) as Philosophy of Mathematics and Natural Science (1949). (I will refer to them as Philosophie and PMNS respectively).
2nd session, Thursd. 13 Feb.
The first talk on Weyl’s work will be divided as follows:
- Weyl’s Das Kontinuum and the Philosophie
- Weyl’s Raum—Zeit—Materie and the Philosophie
- Weyl’s popular essays.
The first part will often follow the presentation of Solomon Feferman (at Paris Diderot and elsewhere) on the content of Das Kontinuum, but with a view to illuminating some underlying philosophical themes. The second will take up Husserlian themes in Weyl’s presentation of Einstein’s general theory of relativity, along the lines of work by Scholz and Ryckman, but also with an eye to how they are handled in the Philosophie. The third part will move forward chronologically to the late 1940s and early 1950s and Weyl’s retrospective account of aspects of his work in geometry and physics.
3rd session, Thursd. 20 Feb.
The third talk will explore further into Weyl’s work, and look at the roles he assigned to philosophy (specifically Husserlian transcendental phenomenology) in science, and to science in relation to mathematics. I hope to include some general remarks about the writing of the history of these ideas in what might be called a prospective and retrospective modes. The talk will be given as follows:
- Weyl’s philosophy of science
- Weyl’s philosophy of mathematics
- Weyl’s Philosophy of Mathematics and Natural Science (1949) and its cultural contexts.
The first two of these will be selective surveys of his positions in the 1920s and the 1950s. The third will look at the differences between the Philosophie (1927) and its English edition of 1949 as a way to explore some of the difficulties Weyl faced, and, perhaps, some that we face today.
5. Focus on the theory of algebric equations of Kronecker
7 and 20 March, 17 April
Sessions led by Cedric Vergnerie
For thirty years, since his election to the Academy of Sciences in Berlin in 1861 to his death in 1891, Kronecker taught at the University of Berlin a course called Vorlesungen über die Theorie der algebraischen Gleichungen. These "lessons on the theory of algebraic equations", which, unlike most of his other courses, have never been published, provide a better understanding of some parts of his algebraic work. During the three sessions devoted to it, we will seek to enter this course from one side, the general context of writing, and on the other hand a detailed description of one of its major headings: theory of characteristics.
7 March, except. Room Gris, 734A
(session 1)
Cédric Vergnerie (Sphere) will introduce, first to the talk of Norbert Schappacher (IRMA, University of Strasburg), a prerequisite to this theory of characteristics that Kronecker designed by Fortgangsprincip.
13 March, except. Room Gris, 734A
Michael Detlefsen (University of Notre Dame)
Axiomatic Method and Consistency.
5. Focus on the theory of algebraic equations of Kronecker (session 2)
20 March
Sessions led by Cedric Vergnerie
David Rowe (Johannes Gutenberg Universität, Mainz)
Kummer on ray systems and quartic surfaces.
3 April
Kenneth Manders (Pittsburgh University State)
Expressive Means and Mathematical Conceptualization.
Mathematics shapes special-purpose contents, by the expressive means it deploys (and avoids) in distinctive contexts (modes) especially suited to certain aspects of problems. The intellectual power of mathematics resides not only in rigorous proof but also in coordinating suitable modes to overall purposes.
5. Focus on the theory of equations of Kronecker (session 3)
17 April
Cedric Vergnerie (SPHERE)
About the Kronecker theory of characteristics.
I leave here prerequisites developed in the first session to present in detail the theory of characteristics Kronecker in particular cases. One goal of this talk is to show how, by going back and forth between the course and research articles, one can achieve an understanding of this "difficult" theory.
24 April
Mathieu Anel (ETH Zurich) {}
!! Wed. 30 April, exceptionnally 14:00– 17:00, Room Klein, 612B !!
Jean-Philippe Narboux
Les nombres naturels de Wittgenstein à Kripke.
Dans ses conférences « Logicism, Wittgenstein, and De Re Beliefs About Numbers » (1992, 2011), Kripke revient sur les analyses logicistes du concept de nombre naturel et leur critique par Wittgenstein dans le livre III des Remarques sur les fondements des mathématiques. Au moyen de la distinction entre désignateur ‘rigide’ (rigid designator) et désignateur ‘butoir’ (buck-stopper), Kripke s’attache à élucider les rapports du naturel et du conventionnel dans l’analyse du concept de nombre naturel et propose sa propre analyse de celui-ci. Cette analyse s’articule autour de la thèse que la notation décimale conventionnelle usuelle est logiquement privilégiée. On exposera les conférences de Kripke dans leurs grandes lignes en les replaçant dans le contexte de l’œuvre publiée de Kripke, puis on explicitera les filiations entre Kripke et Wittgenstein afin de circonscrire la signification et la portée de cette thèse.
References:
- Cavell S. (1979), The Claim of Reason. Oxford: Clarendon. Chapter V.
- Floyd J. (2007), Wittgenstein’s Philosophy of Mathematics, in Shapiro S. (ed.), The Oxford Handbook of Philosophy of Mathematics, Oxford, Oxford University Press.
- Kaplan D. (1968), “Quantifying In”. Synthese 19: 178-214.
- Kripke S. (1980), Naming and Necessity. Cambridge, MA: Harvard University Press. Lecture I.
- Kripke S. (1992), “Logicism, Wittgenstein, and De Re Beliefs About Numbers”, The Whitehead Lectures, Harvard University.
- Kripke S. (2011), “Wittgenstein, Logicism, and Buck-Stopping Identifications of Numbers”, unpublished lecture, Tel Aviv University/ The Van Leer Jerusalem Institute. (available on www.youtube.com)
- Kripke S. (2011), “Unrestricted Exportation and Some Morals for the Philosophy of Language”, in Philosophical Troubles, Collected Papers, volume 1. Oxford: Oxford University Press.
- Kripke S. (2013), Reference and Existence. The John Locke Lectures, 1973. Oxford: Oxford University Press. Lecture III.
- Putnam H. (2004), Ethics Without Ontology. Cambridge, MA: Harvard University Press. Part I.
- Steiner M. (1975), Mathematical Knowledge. Ithaca: Cornell University Press.
- Steiner M. (2011), “Kripke on Logicism, Wittgenstein, and De Re Beliefs about Numbers”, in Berger A. (ed.), Kripke. Cambridge, Cambridge University Press.
- Whitehead A. (1992) ), An Introduction to Mathematics. Chapter V.
- Wittgenstein L. (1976), Lectures on the Foundations of Mathematics, Cambridge, 1939. Ithaca: Cornell University Press.
- Wittgenstein L. (1978), Remarks on the Foundations of Mathematics. Oxford: Blackwell. Book III.
!! Wed. 4 June, room Kandinsky, 631B !!
10:00 –11:30 Ethan Galebach (U.C. Irvine) Visualization and O-minimality.
11:30 –13:00 Sundar Sarukkai (Manipal University)
The Writing of Mathematics.
This talk will explore some relations between the writing (and discursive) strategies of mathematics and the epistemological and ontological consequences of the same.
!! Fri. 13 June, 10:00 – 13:00 and 14:30 – 16:00 !!
Concepts of genericity in mathematics
Session organised by Brice Halimi (Univ. Paris Ouest-Nanterre) and Chris Porter
Chris Porter (LIAFA, Univ. Paris Diderot)
Effective notions of typicality.
In the investigation of various concepts of genericity in mathematics and the roles that
these concepts play, effective notions of typicality provide a particularly interesting case
study. More specifically, various definitions of algorithmic randomness and of effective
genericity have a number of striking features:
– they have proven to be useful in understanding the effective content of certain theorems
of classical mathematics;
– they can be seen as "miniaturizations" of set-theoretic notions of generic objects obtained
by forcing constructions; and
– they reflect the duality between measure and category.
In this talk, I will discuss these features of effective notions of typicality and explain how
these notions can be motivated by a specific problem facing Kit Fine’s account of arbitrary
objects.
Graham Leach-Krouse (Kansas State University)
Conceptualizing Generic Sets.
Generic sets, of the type employed by set theorists in forcing constructions, have occasionally been regarded as conceptually problematic. For example, Mostowski claimed that “No such clear interpretation has as yet emerged from Cohen’s models because we possess as yet no intuition of generic sets”, while Gödel is reported to have remarked that “Forcing is a method to make true statements about something of which we know nothing.” In this talk, after a short technical introduction to genericity and forcing, I describe some philosophical problems connected with the concept of a generic set. In so doing, I hope to shed light on these puzzling assertions.
Michel Vaquié (IMT, Univ. Toulouse)
Genericity in algebraic geometry.
The development of modern algebraic geometry, as represented in particular by the works of E. Noether and B. L. van Waerden, required to distinguish the preexisting notion of "sufficiently general point," on the one hand, and the new notion of "generic point," on the other. The introduction of the latter notion lies at the very foundations of algebraic geometry, but both notions continued to coexist. I will show how important and complementary to each other they are, and will study their interplay.