„On peut effectivement se poser le problème de la morphogénèse pour tout espèce de forme, et pas seulement pour les formes vivantes. Ce qui se passe, c’est que la plupart des formes des objets inanimés ont un déterminisme difficile qui, en principe, ne ressort pas franchement de la théorie des catastrophes.“ (p. 111f.) #Thom #morphogénèse #ThéorieDesCatastrophes
„Let f: U –> V be a map between two differential manifolds, and to each point u of U associate a variation δv of the image v=f(u). Taking all possible variations δv(u), depending differentiably on u, gives all maps g close to f, and these variations (regarded as vectors tangent to f) form an infinite-dimensional vector space. The function space L(U, V) of maps from U to V can thus be considered as an infinite-dimensional manifold. I refer to books on functional analysis for definitions and properties of the topologies (Hilbert space, Banach space, Frechet space, etc.) that are possible on infinite-dimensional vector spaces and, in turn, on infinite- dimensional manifolds. Here we are interested only in a closed subspace L of finite codimension, namely, the bifurcation set H. In the study of subspaces of this type, the actual choice of topology on the space of tangent vectors δv(u) is, in practice, irrelevant.“ (p. 339) #Thom #infinite-dimensionalVectorSpaces
„We propose the following general model to parameterize the local states of a system: the space of observables M contains a closed subset K, called the catastrophe set, and as long as the representative point m of the system does not meet K, the local nature of the system does not change. The essential idea introduced here is that the local structure of K, the topological type of its singularities and so forth, is in fact determined by an underlying dynamic defined on a manifold M which is in general impossible to exhibit. The evolution of the system will be defined by a vector field X on M, which will define the macroscopic dynamic. Whenever the point m meets K, there will be a discontinuity in the nature of the system which we will interpret as a change in the previous form, a morphogenesis. Because of the restrictions outlined above on the local structure of K we can, to a certain extent, classify and predict the singularities of the morphogenesis of the system without knowing either the underlying dynamic or the macroscopic evolution defined by X. In fact, in most cases we proceed in the opposite direction: from a macroscopic examination of the morphogenesis of a process and a local and global study of its singularities, we can try to reconstruct the dynamic that generates it. Although the goal is to construct the quantitative global model (M, K, X), this may be difficult or even impossible. However, the local dynamical interpretation of the singularities of the morphogenesis is possible and useful and is an indispensable preliminary to defining the kinematic of the model; and even if a global dynamical evolution is not accessible, our local knowledge will be much improved in the process.“ (p. 7f.) #Thom #system #discontinuity #morphogenesis
„History gives another reason for the physicist’s attitude toward the qualitative. The controversy between the followers of the physics of Descartes and of Newton was at its height at the end of the seventeenth century. Descartes, with his vortices, his hooked atoms, and the like, explained everything and calculated nothing; Newton, with the inverse square law of gravitation, calculated everything and explained nothing. History has endorsed Newton and relegated the Cartesian constructions to the domain of curious speculation. The Newtonian point of view has certainly fully justified itself from the point of view of its efficiency and its ability to predict, and therefore to act upon phenomena.“ (p. 5) #Thom #Descartes #Newton #physics
„In play the mind is given over to an eminently combinatorial activity; it constructs and uses a semantic model according to rules that it knows and respects.“ (p. 317) #Thom #play #model #rules