andrei rodin Andrei Rodin's blog about History and Philosophy of Mathematics

May 19, 2016

Axiomatic Method between Logic and Geometry: SPHERE, 23 Mai and 6 June 2016

Filed under: Uncategorized — Andrei Rodin @ 7:42 am

Chers collègues,

Nous avons le plaisir de vous annoncer deux séances supplémentaires du GDT « Mathématiques XIXe-XXe, Histoire et Philosophie » :

*Andrei Rodin* (Institute of Philosophy of the Russian Academy of Sciences), chercheur invité à SPHERE, donnera deux conférences sur le thème
« Axiomatic Method between Logic and Geometry » :

1. *Lundi 23 mai, de 10h à 13h, en salle Klein (371A)* :   « Axiomatic Geometry according to Euclid and according to Hilbert. »

2. *Lundi 6 juin, de 10h à 13h, en salle Rothko (412B)* :
« Logic and Geometry in Topos theory and in Homotopy Type theory »

à l’Université Paris 7 Diderot, bâtiment Condorcet (4 rue Elsa Morante, 75013 Paris).

Voici un résumé et quelques éléments de bibliographie.

The modern notion of axiomatic theory derives from Hilbert’s /Grundlagen der Geometrie/ of 1899. It essentially differs from the more traditional idea of axiomatic theory presented in Euclid’s Elements. More surprisingly, Hilbert’s account of axiomatic thinking does not seem to do full justice to some recent axiomatic theories such as the axiomatic Topos theory due to Lawvere and Homotopy Type theory due to Voevodsky and his collaborators. In this series of two talks I provide a historical perspective on the axiomatic method and on this basis develop an up-to-date concept of axiomatic theory, which I suggest to call *constructive*. The constructive axiomatic method, which is best represented today by Homotopy Type theory, is conceptually rooted in Euclid’s mathematics, geometric works by Grassmann and Peano in the 19th century, and Kolmogorov’s approach in constructive mathematics.

Talk 1: Axiomatic Geometry according to Euclid and according to Hilbert.

Bibliography:

H. Freudental, The main trends in the foundations of geometry in the 19th century, in: Logic, Methodology and Philosophy of Science: Proceedings of the 1960 International Congress. Standford University Press 1962, pp. 613-621.

A.N. Kolmogorov, On the Interpretation of Intuitionistic Logic, in: V.M.  Tikhomirov (ed.) Selected Works of A.N. Kolmogorov, Springer 1991, pp.  151-158 (Original publication in German: Zur Deutung der Intuitionistischen Logik. Math. Ztschr., 35  (1932), S. 58-65)

I. Mueller, Greek Mathematics and Greek Logic, in: Ancient Logic and Its Modern Interpretations (Synthese Historical Library vol.9), Reidel
Publishing 1974, pp. 35-70

A. Rodin, Axiomatic Method and Category Theory (Synthese Library vol. 364), Springer 2014,  Chapters 2-4.

Talk 2: Logic and Geometry in Topos theory and in Homotopy Type theory.

Bibliography:

F.W. Lawvere, Quantifiers and sheaves, in: M. Berger, J. Dieudonne et al.  (eds.), Actes du congres international des mathématiciens, Nice 1970, pp.  329 – 334

C. McLarty, Elementary Categories, Elementary Toposes, Oxford: Clarendon Press 1992, Chapters 13-18

Andrei Rodin, Axiomatic Method and Category Theory (Synthese Library vol.  364), Springer 2014. Chapters 5, 7, 10

Andrei Rodin, On Constructive Axiomatic Method, arXiv:1408.3591, to appear in Logique et Analyse

Univalent Foundations Program, Homotopy Type Theory: Univalent Foundations of Mathematics, Institute for Advanced Study (Princeton) 2013; available at http://homotopytypetheory.org/book/}, Chapters 1-3

En espérant vous retrouver nombreux,

Cordialement,

Les organisateurs.

Venus Homotopically: SPHERE 26 Mai 2016

Filed under: Uncategorized — Andrei Rodin @ 7:38 am

NEW: full paper

 

Cher(è)s collègues,

Vous êtes cordialement invité(e)s à la prochaine séance du séminaire de recherche L’(id)entité :: L’(id)entification, laquelle se tiendra le jeudi 26 mai de 15h à 17h dans la salle 646A-Mondrian de l’Université Paris Diderot (Bâtiment Condorcet, 4 Rue Elsa Morante, 75013). Pour cette séance, nous aurons le plaisir d’accueillir Andrei Rodin (Institute of Philosophy, Russian Academy of Science – Saint-Petersburg State University), qui fera l’exposé suivant:

Venus Homotopically

The identity concept developed in the Homotopy Type Theory (HoTT) supports an analysis of Frege’s famous Venus example, which explains how empirical evidences justify judgements about identities. In the context of this analysis we consider the traditional distinction between the extension and the intension of concepts as it appears in HoTT, discuss an ontological significance of this distinction and, finally, provide a homotopical reconstruction of a basic kinematic scheme (which is used in Classical Mechanics) and discuss its relevance in Quantum Mechanics.

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Bien cordialement,
Gabriel Catren
Laboratoire SPHERE – Sciences, Philosophie et Histoire (UMR 7219)
CNRS – Université Paris Diderot

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