: 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.