:                   GEOMETRY AND COGNITION
:                  "a two ways relation"

        Participants & English Summary

Durée du projet:  24 mois

Mots clés: architecture fonctionnelle, perception, fondements cognitifs de la géométrie.

Responsable scientifique:

Giuseppe Longo,  Directeur de Recherches (DR1), CNRS
 Laboratoire d'Informatique (CNRS) et DMI, Ecole Normale Superieure
 45, Rue D'Ulm,  75005  Paris   (France)
 http://www.di.ens.fr/users/longo       E-mail: longo@di.ens.fr
 tel. ++33-1-4432-3328, secr.-3216; FAX -2080

Composition des autres équipes participant au programme de recherche :

Bailly Francis, DR1 - CNRS, Physique, LPSB-CNRS, 1, Place Aristide Briand
 92195 Meudon Cedex, bailly@cnrs-bellevue.fr       Temps:  1 mois

Frégnac Yves, DR1 - CNRS, Unité de Neurosciences Intégratives et Computationnelles, Institut A. Fessard, Av. de La Terrasse, Gif/Yvette.    E-mail : yves.fregnac@iaf.cnrs-gif.fr    Temps:  1 mois

Goubault Eric, DEIN/SLA, CEA Saclay, 91191 Gif-sur-Yvette, email: Eric.Goubault@cea.fr    Temps:  1 mois

Longo Giuseppe (voir dessus); DeJaeger F., Santini G. (thésards), 10 mois, 1mois, 1 mois

Lorenceau Jean,  DR2 - CNRS, Unité de Neurosciences Intégratives et Computationnelles, Institut A. Fessard, Av. de La Terrasse, Gif/Yvette.    E-mail : lorenceau@iaf.cnrs-gif.fr    Temps:  2 mois

Morel Jean-Michel, Prof., Maths.; Desolneux A., Moisan L., CMLA, Ecole Normale Supérieure de Cachan, 61 Av. du President Wilson, 94235 Cachan Cedex   Temps: 1, 1, 1 mois

Nadal Jean-Pierre, DR2 - CNRS;  Ninio Jacques, DR2 - CNRS, Psychophysique;  Brunel Nicolas, CR1 - CNRS;  Turiel Antonio, Post-doc., Laboratoire de Physique Statistique, ENS, ninio@lps.ens.fr  Temps:  2, 8, 12 mois.

O'Regan J.Kevin, DR2 - CNRS, Laboratoire de Psychologie Expérimentale, Centre Universitaire de Boulogne, 71, avenue Edouard Vaillant, Boulogne-Billancourt, http://nivea.psycho.univ-paris5.fr, oregan@ext.jussieu.fr. Temps:  1 mois.

Petit Jean-Luc, Prof., Dépt. de Philosophie, Université Marc Bloch, UFR PLISE, 14, rue Descartes, 67000 Strasbourg,  petit@ushs.u-strabg.fr Temps:  10 mois

Petitot Jean, Directeur d'Etudes, Deglas Michel, DR2 - CNRS,  Centre de Maths. - EHESS,
 54 Bld Raspail, 75006 Paris,   E-mail : petitot@ehess.fr    Temps:  6, 4 mois

Teissier Bernard, DR1 - CNRS, Laboratoire de Mathématiques (CNRS) et DMI, ENS,
 45, Rue D'Ulm,  75005  Paris   E-mail: teissier@di.ens.fr    Temps:  4 mois

Victorri Bernard, DR2 - CNRS; Visetti Yves-Marie, CR1- CNRS, Linguistique, Lab. LTM, ENS, 1 rue M. Arnoux 92120 Montrouge   E-mail : Bernard.Victorri@ens.fr   Temps:  2, 2 mois.

Background and Objectives / Situation du sujet et objectifs généraux:

The aim of this project is to focus on the connection between geometry and cognition, with a two-fold approach:
 1) from geometry to cognition: the mathematical analysis of human cognition (especially vision);
 2) from cognition to geometry: the cognitive foundations of mathematics (in ''cognitive'' we include the historical evolution of conceptual constructions).
This projet is grounded in an ongoing working group activity
(see http://www.di.ens.fr/users/longo/geocogni.html).

The expected results are both technical and epistemological, and hopefully also pedagogical in the long run.
 1) The development of certain aspects of differential geometry and of a new role for geometry in computer science.
 2) A better understanding of the role played by these advances of the natural sciences and cognition in the analysis of certain problems which traditionally have been considered philosophical.

It is becoming possible to analyse, in the enormous complexity of the neural and cerebral system, the connections and functions which are relevant for visual, spatial, auditive perception. The relationship between the geometric organization of data and functions in the visual system and the mathematical construction of geometry is at the core of our project. This connection requires to blend our respective scientific competences: for example, there are fascinating analogies between neuro-physiological structures in the visual cortex and advanced constructions of differential geometry.  Similarly, the combination of ideas from signal analysis and differential geometry seems very helpful in the modelling of visual functions.
 On the other hand, the last few decades have seen the emergence of two important trends in the philosophy of mathematics: the growing call for a ``return to meaning'' in mathematical logic, and the ``cognitivist revolution'' made possible by the remarkable progress of integrative neuroscience.  Mathematicians know well from experience that the sources of discovery and certainty are found on the side of meaning at least as much as on the side of rigor; moreover, learning mathematics is extremely difficult without the support of intuition and meaning, a support which has been rejected for too long because of the fear of being led astray.  There is a need now to give a solid, cognitive, foundation of the "vague" notion of intuition, as this is grounded on constructed meanings, by our action as living and historical beings.
 To launch oneself into such an endeavour, after a century of magnificent expansion of a logicist approach, based partly on the axiomatic method, is certainly not without danger, but seems essential to us. The meaning of a mathematical object or an operation seems to be a fuzzy concept, which can be studied only through a discourse referring to other fuzzy concepts.  We believe instead that this meaning is rooted in our cognitive functions.
 In view of this two ways connection, our project is, in one direction, to study and develop mathematical models for the organization of cognitive functions and, conversely, to explore to what extent these functions are correlated with the intuitive meaning one gives to mathematics, especially geometric structures, and contribute to provide foundations for mathematics and structuring forces in their development.

Program and Timetable / Programme et échéancier des travaux

The projet described in Annexe 2 summarizes our long term objectives.  In the 24 months of the expected financial support, we plan to make a strong start towards these objectives.
 Our plan for the first year is to develop the system of lectures and working meetings which we started in the February-June 1999 term (see
http://www.di.ens.fr/users/longo/GeoCo-fold/conf-fev-avril-1999.html), but with the addition of (foreign) visiting lecturers and redactions of the lectures.  This will allow us to strengthen the common language and shared knowledge, which are necessary for the cooperation of researchers in the different disciplines involved.   During the second year we plan to continue these activities and possibly begin to co-direct theses of students of the DEA ''Sciences cognitives''.  Moreover, we are starting to appreciate the impact on teaching, in particular in primary and secondary schools, of our approach: contacts with several groups in educational research are being established, in France and in Italy. The scientific organizer of the "Palais de la Decouverte" in Paris, who attended some of our lectures, has asked us to organize a one- or two-days conference in September/October 2000, to end with a "lecture for the general public" on the theme of our research progam.
 Scientific publications are of course expected.  After two years we will submit a detailed report. We also plan to maintain a web page on the topics of our project.