“A contribution to mathematics and/or science does not consist of a picture, but rather of a picture combined with a description. Without words and formulas, a picture can, at best, be praised for artistic quality. Without an interest in pictures and other aspects of ‘reality,’ pictures can play no role whatsoever.” (p. 26) #Mandelbrot #mathematics #picture
“Toute la Géométrie de Descartes est destinée à résoudre par une méthode nouvelle, analytique et non plus synthétique”. (p. 99) #Vuillemin #Descartes #méthode
“Autrement dit, les progrès scientifiques sont toujours subordonnés à la possibilité d’un instrument mental qui permette d’exprimer les correspondances, les régularités des choses.” (p. 96) #Thom #ProgrèsScientifique
“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