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.