Geometry and Cognition from the Foundations of Mathematics to Theories of Knowledge and Cognition

In 1999, in close collaboration with Jean Petitot and Bernard Teissier, the author of this text set up a scientific project titled Geometry and Cognition (GeoCo) at Ecole Normale Supérieure in Paris (see the “Startup Conference Cycle”, February–April, 1999, in “ http://www.di.ens.fr/users/longo/geocogni.html ”). This text introduces our project that focused on a better understanding of the advances in natural sciences and cognition in the analysis of certain fundamental problems of mathematics traditionally considered philosophical as well as on the development of certain aspects of Differential Geometry and a new role of Geometry in Computer Science.