A workshop organized by Gabriel Catren:
Sur la notion d’identification en physique et en mathématiques
Thursday, November 13, 2014
l’amphithéâtre Pierre-Gilles de Gennes, bâtiment Condorcet
Université Paris Diderot
(10, rue Alice Domon et Léonie Duquet, 75013 Paris).
9h30 ? 11h – Mathieu Anel (Laboratoire SPHERE, Université Paris Diderot)
We shall explain how the actual construction and manipulation of mathematical objects (opposed to their axiomatization and idealization) leads to a re-foundation of mathematics on groupoids instead of sets.
Concrete manipulation of mathematical objects necessitates choices (basis, presentation, paths, order…) but the definition of an object neglects those choices and focus on a list of properties that do not depend on any presentation (and are often assuming a lot of implicit). However, staying close to actual constructions leads to more regular constructions than the one depending only on the definitions. It is often possible to improve the definition of an object by defining it as an equivalence class of presentations rather than by a list of properties. Such process is often called ‘derivation’. We shall give examples and illustrate how this can be understood as changing the paradigm of sets for groupoids.
11h15 - 12h45 – Eric Finster (Ecole Polytechnique Fédérale de Lausanne)
We give an overview of the foundational point of view advocated by Voevodsky’s Univalent Foundations program and explain how these ideas are realized by Martin-Löf type theory with identity types. In particular, we focus on the role of the univalence axiom as an invariance principle, embedded in type theory, which is absent from traditional set-theoretic foundations, and explain how this point of view leads to a unification of certain logical and geometric principles.
14h30 - 16h – Urs Schreiber (invited researcher, Laboratoire SPHERE, Université Paris Diderot)
The observation that the concept of identity has to be relaxed to one of equivalence is a central insight of 20th century physics – known as the gauge principle. This talk gives an exposition of how combining gauge equivalence with the locality principle in non-perturbative quantum field theory implies that gauge fields do not from sets, but form groupoids, in fact that they form smooth groupoids (smooth stacks). One consequence is that the field bundles of local non-perturbative gauge fields are really 2-bundles (gerbes) in higher differential geometry. I close with a brief outlook on implications for the formulation of higher dimensional Chern-Simons-type field theories.
(talk notes pdf)
16h15 à 17h45 – David Corfield (University of Kent)
Looking through Robert Torretti?s book ?Philosophy of Geometry from Riemann to Poincare? (1978), it is natural to wonder why, at least in the Anglophone community, we currently have no such subject today. By and large it is fair to say that any philosophical interest in geometry shown there is directed at the appearance of geometric constructions in physics, without any thought being given to the conceptual development of the subject within mathematics itself. Within the Anglophone philosophy of mathematics, the only relevant issue concerns the role of spatial intuition in diagrammatic proofs.
This is a result of a conception we owe to the Vienna Circle and their Berlin colleagues that one should sharply distinguish between mathematical geometry and physical geometry. With mathematical geometry being taken as the study of the consequences of axiom systems which happened to retain the epithet ?geometric?, there was no point in treating geometry separately from the rest of mathematics. In view of the enormous expansion through the 20th century of manifestations of the mathematical concept of space, it may have seemed hopeless to find anything interesting to say of what is common to them.
In recent years, however, there are strong indications that a modern general account of geometry is possible. It starts with homotopy type theory as the syntax for theories which can be interpreted in ∞-toposes. where the basic shapes of mathematics are now the so-called ?homotopy n-types?. It then adds further ?cohesive? structure, integrating the ideas of Bill Lawvere, motivated in turn by philosophical re?ection on geometry and physics. Claims have been made by Urs Schreiber that differential cohomology, which lies at the heart of so much current theorising in physics and mathematics, and which when viewed from set theoretic foundations appear elaborate and unprincipled, can now be seen as natural through the universal constructions of higher category theory. The question to be addressed in this talk is what philosophical sense should be made of this new approach which aims to provide a language for all fundamental physics.
(talk notes pdf)
Last revised on November 14, 2014 at 07:56:19. See the history of this page for a list of all contributions to it.