Mathematical Concepts and Physical Objects

Introduction (with F. Bailly) With this text, we will first of all discuss a distinction, internal to mathematics, between "construction principles" and "proof principles" (see [Longo, 1999], [Longo, 2002]). In short, it will be a question of grasping the difference between the construction of mathematical concepts and structures and the role of proof, more or less formalised. The objective is also to analyse the methods of physics from a similar viewpoint and, from the analogies and differences that we shall bring to attention, to establish a parallel between the foundations of mathematics and the foundations of physics. The paper is introduced by a joint reflection with a physicist, F. Bailly, coauthor of the complete French version of this work, originally a dialogue in two parts ([Bailly, Longo, 2005]).