Chapter 7 Mathematical Concepts and Physical Objects

The notions of “construction principles” is proposed as a complementary notion w.r. to the familiar “proof principles” of Proof Theory. The aim is to develop a parallel analysis of these principles inMathematics and Physics : common construction principles, in spite of different proof principles, justify the effectiveness ofMathematics in Physics. The very “objects” of these disciplines are grounded on commun genealogies of concepts : there is no trascendence of concepts nor of objects without their contingent and shared constitution. A comparative analysis of Husserl’s and Gödel’s philosophy is hinted, with many references to H. Weyl’s reflections on Mathematics and Physics. 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; 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, 2006]). When proposing a mathematical structure, e.g., the integers or the real numbers, the Cartesian space or . . . a Hilbert space, we use a plurality of concepts often stemming from different conceptual experiences: the construction of the integers evokes the generalised successor operation, but at the same time we make sure they are “well-ordered,” in space or time, to obtain this well-ordered “line of integer numbers” which we easily “see,” within a mental space. And we construct the rationals, as ratios of integers modulo ratio equivalence, and then the real numbers,