Jean Petitot

Jean Petitot
École des Hautes Études en Sciences Sociales | EHESS · Centre d'analyse et de mathématique sociales (CAMS)

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99
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Publications

Publications (99)
Chapter
In the history of 20th-century mathematical structuralism, the figure of Bourbaki is prominent and sometimes even identified with the philosophical doctrine of structuralism. However, the Bourbaki group consisted of pure mathematicians, among them the greatest of their generation, most of whom seemed little inclined to, and even hesitant about, phi...
Article
Full-text available
Resumo O texto sumariza os longos anos de pesquisa efetuadas no contexto da semiótica estrutural de A.J. Greimas para chegar a desenvolver o modelo dinâmico introduzido por R. Thom. Ele sintetiza a contribuição desses modelos explicando como eles permitem desenvolver uma interpretação matemática coerente dos conceitos fundamentais do estruturalismo...
Chapter
Dynamical structuralism was a theoretical innovation in the 1970s, born of a synthesis between the mathematical works of René Thom that applied singularity theory to structuralism in biology and linguistics (Thom in Stabilité structurelle et Morphogenèse, New York, Benjamin, Paris, Ediscience, 1972), and the work of structuralists such as Roman Jak...
Chapter
This chapter discusses receptive fields and receptive profiles of visual neurons, starting with the photoreceptors and the ganglion cells of the retina, and proceeding via the neurons of the lateral geniculate nucleus to those of V1. It explains how they act on the optical signal as filters, and broaches the problem of their linearity or nonlineari...
Chapter
This final chapter describes the transition to the mathematical themes that will be developed in the second volume. (i) The explicit calculation of the elements of the sub-Riemannian geometry of the \(\mathbb {V}_\mathrm{J}\) model using the tools of control theory: geodesics, unit sphere, wave front, caustic, cut locus, conjugate points, and so on...
Chapter
This chapter studies the second component of the functional architecture of primary visual areas, namely what are called cortico-cortical ‘horizontal’ connections. They join together neurons that detect orientations at different points which are parallel, and even approximately coaxial. This coaxial ‘parallel transport’ has been confirmed by psycho...
Chapter
This chapter presents a first set of experimental data on the functional architectureof area V1 and, in particular, on what is referred to as its ‘pinwheel’ structure. Most of the so-called simple neurons in V1 detect positions and orientations in the visual field, with those detecting the various orientations for each position being grouped togeth...
Book
This book describes several mathematical models of the primary visual cortex, referring them to a vast ensemble of experimental data and putting forward an original geometrical model for its functional architecture, that is, the highly specific organization of its neural connections. The book spells out the geometrical algorithms implemented by thi...
Article
Full-text available
Large sets of elements interacting locally and producing specific architectures reliably form a category that transcends the usual dividing line between biological and engineered systems. We propose to call them morphogenetically architected complex systems (MACS). While taking the emergence of properties seriously, the notion of MACS enables at th...
Article
This is a compilation of some well known propositions of Alain Connes concerning the use of noncommutative geometry in mathematical physics.
Article
The paper (in French) exemplifies graphically a solution of the heat equation which is a 1-dimensional unfolding of an elliptic umbilic catastrophe. The example is due to James Damon and adapts Thom-Mather's singularity theory to multiscale models of scale-space analysis in image processing.
Article
One of the main difficulty concerning the nature of the continuum is to do justice, inside the set theoretical Cantorian framework, to the classical conception (from Aristotle to Thom, via Kant, Peirce, Brentano, Husserl and Weyl) according to which the continuum is a non-compositional, cohesive, primitive, and intuitive datum. This paper investiga...
Article
The paper (in French) presents a survey of Hilbert's Epsilon operator focusing on the intensional aspects of its semantics. It comments on some epistemological problems, from Albert Lautman in the 1930s to John Bell, Grigori Mints, Barry Hartley Slater, Richard Zach or Edward Zalta in the 1990s.
Article
We comment on Alan Turing's celebrated paper "The Chemical Basis of Morphogenesis" published in 1952 in the "Philosophical Transactions of the Royal Society of London". It is a typical example of a pioneering and inspired work in the domain of mathematical modelling.
Chapter
We present a historical survey of the way Neurogeometry links the modelling of the functional architectures of primary visual areas with sub-Riemannian geometry.
Article
The term “neurogeometry” denotes the geometry of the functional architecture of visual areas. The paper reviews some elements of the neurogeometry of the functional architecture of the first visual area V1 and explains why contact geometry, sub-Riemannian geometry, and noncommutative harmonic analysis are brought in as natural tools. It emphasizes...
Article
Full-text available
The paper summarizes the history of the morphodynamical turn in cognitive semio-linguistics. From the earliest Thom’s models introduced in the late 1960s to the recent Cognitive Morphodynamics (2011), it deals with forty years of research covering (i) topological and dynamical models of structural « actantial » syntax, (ii) connectionist models, (i...
Chapter
At the end of the 1960s, René Thom was the first scientist to develop a general mathematical theory of morphogenetic processes. This chapter presents the fundamental principles of that theory.
Article
Classically, a reductionist thesis posits that complex high level phenomena, structures, and processes can be reduced, as far as their scientifi c explanation is concerned, to underlying lower level phenomena, structures, and processes. The most paradigmatic and best-investigated example is the reduction of macroscopic thermodynamics to microscopic...
Article
We present a geometrical model of the functional architecture of the primary visual cortex. In particular we describe the geometric structure of connections found both in neurophysiological and psychophysical experiments, modeling both co-axial and trans-axial excitatory connections. The model shows what could be the deep structure for both boundar...
Chapter
In a 1999 article written as a tribute to Claude Lévi-Strauss for a special issue of the French journal Critique, I surveyed Lévi-Strauss’s references to Goethe, and this led me to refine the thesis that - as I had already argued in the mid-seventies with regard to René Thom’s morphodynamic structuralism - there exists an authentic morphological ge...
Chapter
In our neo-transcendental approach, physical theories are built up from a categorial structure that is mathematically interpreted (what Kant called the “mathematical construction of categories”). The interpretation of physical categories provided by noncommutative geometry is presented in this perspective.
Article
In recent years, many philosophers of modern physics came to the conclusion that the problem of how objectivity is constituted (rather than merely given) can no longer be avoided, and therefore that a transcendental approach in the spirit of Kant is now philosophically relevant. The usual excuse for skipping this task is that the historical form gi...
Book
In recent years, many philosophers of modern physics came to the conclusion that the problem of how objectivity is constituted (rather than merely given) can no longer be avoided, and therefore that a transcendental approach in the spirit of Kant is now philosophically relevant. The usual excuse for skipping this task is that the historical form gi...
Article
We propose to model the functional architecture of the primary visual cortex V1 as a principal fiber bundle where the two-dimensional retinal plane is the base manifold and the secondary variables of orientation and scale constitute the vertical fibers over each point as a rotation-dilation group. The total space is endowed with a natural symplecti...
Chapter
1. The more mathematics and physics unify themselves in the physico-mathematical modern theories, the more an objective epistemology becomes necessary. Only such a transcendental epistemology is able to thematize correctly the status of the mathematical determination of physical reality. 2. There exists a transcendental history of the synthetic...
Article
Full-text available
The debate between Jean-Pierre Changeux and Alain Connes is one of the most, interesting to take place in recent years. It re-frames in a very up-to-date context a whole series of traditional and difficult questions from the standpoint of the knowledge and experience of two of the leading protagonists of contemporary science. To the choice presente...
Article
We propose a novel dynamical system approach to cognitive linguistics based on cellular automata and spiking neural networks. How can the same relationship 'in' apply to containers as different as 'box', 'tree' or 'bowl'? Our objective is to categorize the infinite diversity of schematic visual scenes into a small set of grammatical elements and el...
Conference Paper
We propose a spiking neural network for mapping the infinity of schematic visual scenes to a small set of semantic symbols. According to cognitive linguistics, spatial prepositions such as 'in' or 'above' are neutral toward the shape and size of objects. We suggest that they correspond to morphodynamical transforms erasing details and creating virt...
Chapter
Due to the very strong geometrical structure of the functional architecture (hyper-columns, pinwheels, horizontal connections), the neural implementation of Kanizsa’s contours is deeply linked with sophisticated structures belonging to what is called contact geometry and with variational models analogue to models already well known in physics.
Article
RÉSUMÉ. Cet article aborde la question de la naturalisation de la phénoménologie à partir d'exemples précis concernant les descriptions husserliennes des relations de dépendance entre extension spatiale et qualités sensibles, de la perception par esquisses, des anticipations perceptives et des liens entre géométrie, vision et contrôle kinesthésique...
Article
We present a neuro-geometrical model for generating the shape of Kanizsa's modal subjective contours which is based on the functional architecture of the primary areas of the visual cortex. We focus on V1 and its pinwheel structure and model it as a discrete approximation of a continuous fibration π: R P → P with base space the space of the retina...
Article
We present a geometrical model of the functional architecture of the primary visual cortex (V1) and, more precisely, of its pinwheel structure. The problem is to understand from within how the internal "imminent" geometry of the visual cortex can produce the "transcendent" geometry of the external space. We use first the concept of blowing up to mo...
Article
Some papers of this special issue concern recent results on mathematical models of segmentation. As they are rather technical we propose here a pedagogical introduction for the non-mathematical reader. We briefly present the variational model of image segmentation proposed by David Mumford and we summarize some fundamental results of De Giorgi's sc...
Article
Full-text available
With Francisco Varela’s death, one of themost beautiful minds I ever met has gone toheaven. The inspired way he succeeded insynthesizing cognitive neurosciences,dynamical models of global brain activityand the phenomenological dimension ofconsciousness was truly amazing andwonderful. His scientific production wasconstitutive of a spiritual vocation...
Chapter
This paper develops an updated transcendentalist perspective concerning the epistemological status of objectivity. The main point is that objectivity is neither an ontology nor a mere description of a phenomenal state of affairs. It is instead a principled (categorial) “legalization” and a mathematical reconstruction of the phenomena. As the very c...
Article
Ce livre est la version française de l'ouvrage "Naturalizing phenomenology : issues in contemporary phenomenology and cognitive science", Stanford Univ. Press, 1999
Article
This ambitious work aims to shed new light on the relations between Husserlian phenomenology and the present-day efforts toward a scientific theory of cognition—with its complex structure of disciplines, levels of explanation, and conflicting hypotheses. The book’s primary goal is not to present a new exegesis of Husserl’s writings, although...
Article
This work presents some variational models for the cortical algorithms processing Kanizsa modal subjective contours . These models are based on the geometric concepts of fibration and contact structure. The retinoptic structure of the orientation hypercolumns in the visual area V1 is a functionnal architecture which can be mathematically idealized...
Chapter
Contemporary research concerning the cognitive links between perception, language, and action — see for instance Talmy’s works — have revolutionized the dominating traditions of formal semantics. They have led to what Husserl already called in Erfahrung and Urteil a genealogy of logic, a perceptive genealogy of cognitive symbolic structures. Such a...
Article
After a historical sketch of the dynamical hypothesis, we stress that it is a functionalist hypothesis. We then tackle the point of a dynamical approach to constituent structures and emphasize that dynamical modeling must be coupled with morphological analysis.
Article
Partant de la phenomenologie de la perception et de la conception structurale-dynamique des discontinuites geographiques dans les travaux de Gilles Ritchot et de Gaetan Desmarais sur la forme urbaine, l'article aborde le probleme fondamental de la segmentation spatiale et de l'emergence des discontinuites qualitatives dans des espaces substrats. Il...
Article
The paper tackles the problem of the links between the formal (logical) syntax and semantics of perceptive judgements and their perceptive (geometrica) content. In the most elementary situations, the perceptive content is reducible to a qualitative filling-in of spatial domains (objects' extensions) by sensible qualities (colors, textures, etc.). T...
Article
A l'occasion de ce Symposium du Centenaire, j'aimerais revenir sur les modèles morphodynamiques de perception phonétique que j'avais présentés au début des années 80 dans Les Catastrophes de la Parole, ouvrage dont le soustitre était «de Roman Jakobson à René Thom» et qui, en hommage à La Charpente phonique du langage, était précisément dédié à la...
Article
Full-text available
The paper challenges the assumption, common amongst philosophers, that the reality described in the fundamental theories of microphysics is all the reality we have. It will be argued that this assumption is in fact incompatible with the nature of such theories. It will be shown further that the macro-world of three-dimensional bodies and of such qu...
Article
Amit's “Attractor Neural Network” perspective on cognition raises difficult technical problems already met by prior dynamical models. This commentary sketches briefly some of them concerning the internal topological structure of attractors, the constituency problem, the possibility of activating simultaneously several attractors, and the different...
Article
This paper begins with Husserl's phenomenological distinction between formal ontology (analytic theory of general objects) and "material" regional ontologies (types of "essences" of objects which prescribe "synthetic a priori" rules). It then shows that, as far as its "ontological design" is concerned, transcendental phenomenology can be seen as an...
Article
The study of “form-bearing elements” presupposes the possibility of developing a specifically morphological analysis of sound forms. For that, a physico-mathematical theory of morphogenesis is required. Its role is to suggest morphodynamic models of natural morphologies. In order to comprehend sound forms (and, more generally, perceptual forms) it...
Article
Analyzing phonetic phenomena of categorical perception enables one to understand how the perceptual treatment of linguistic sounds segment and pattern the continuous audio-acoustic flux, and how this continuum can thus be transformed into strings of discrete elements from which a code can be built. These phonetic phenomena of categorical perception...
Article
L'article expose les contenus tant mathematiques que philosophiques des travaux d'Albert Lautman sur l'unite des mathematiques au sens de Bourbaki. En mathematiques, il aborde les fonctions analytiques, les groupes, les equations differentielles, la topologie algebrique, la theorie des nombres, la metamathematique. En philosophie il traite du local...
Article
Résumé La syntaxe structurale de Tesnière repose sur le concept central de connexion. Ce concept renvoie à une conception organiciste et dynamique (non logique) de l'energeia et de la performance linguistiques. A ce titre, il est d'une essence foncièrement différente de celle des règles formelles gouvernant les descriptions symboliques et logiques...

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