Mathematical Concepts and Physical Objects 1 Giuseppe Longo Labo. d'Informatique, CNRS – ENS et CREA, Paris 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]). When proposing a mathematical structure, for example 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.
- standard analysis
- between sensible
- breaks any epistemological
- cartesian space
- any cognitive
- sensible epistemology
- cognitive subject
- distinction between
- mathematics themselves