Giuseppe Longo - The geometric intelligibility of space and the foundation of mathematical knowledge

The problem of space and of its mathematical intelligibility has always been at the core of foundational issues in mathematics. We will first review the (distant) debate between Riemann and Frege which set a "bifurcation" in history: the approach to foundation by Logic, via language, far prevailed and marked the subsequent XX century, in spite of the success of Riemann's ideas in Physics.

Riemann, Poincare and Weyl's foundational views will be revitalized as well as the role of "geometric construction principles" in mathematics. Some remarks will be made concerning the geometric organisation of information, in general and in recent cognitive theories, vision in particular.

(The talk will develop some of the ideas presented in a recent research project "Geometrie et Cognition", directed by the speaker, which may be found here. As the project is just starting, the talk will present more problems than results.)