René Thom > Quotes

“The essential point of view advanced here is this: the stability of every
living being, as of every structurally stable form, rests, in the last analysis, on a for-
mal structure—in fact, a geometrical object—whose biochemical realization is the
living being.”
― René Thom, Morphogenèse et imaginaire (Circé)

“At a time where so many scholars are calculating, is it not desirable that some, who can, dream?”
― René Thom, Structural Stability and Morphogenesis

“The problem of stability is then: Is the collection of structurally stable vector fields everywhere dense in the space B of all vector fields? A partial answer has been obtained recently: yes, when dimension m is less than or equal to 2 (Peixoto[18]), no when dimension m is greater than or equal to 4(Smale[22]), or 3 (Smale and Williams[24]).”
― René Thom, Structural Stability and Morphogenesis

“A continuous space given in the intuition always has some further structure, a complement so to speak, of properties; a topology, a metrics, a dimension . .They are complementary data which more often than not allow the operations and constructions to be defined.”
― René Thom

“Le domaine scientifique dans lequel on peut construire des modèles quantitatifs certains permettant la prévision et par suite l’action est beaucoup plus restreint qu’on ne le pense généralement. C’est un petit halo autour de la physique fondamentale, aux frontières d’autant plus imprécises que les considérations statistiques entrent plus en jeu.
[...] Cette dégénérescence relativement rapide des possibilités de l’outil mathématique lorsqu’on va de la physique vers la biologie est certes connue des spécialistes, mais il en est fait fort peu mention aux yeux du grand public.

"Mathématique et théorisation scientifique", in Penser les mathématiques (ouvrage collectif), Seuil, 1982, p. 263 et 264.”
― René Thom

“The subjects mentioned in his title, Structural Stability and Morphogenesis, have a much wider reference; and he relates his topological system of thought to physical and indeed to general philosophical problems.”
― René Thom, Structural Stability and Morphogenesis

“―“God created the integers and the rest is the work of man.” This maxim spoken by the algebraist Kronecker reveals more about his past as a banker who grew rich through monetary speculation than about his philosophical insight. There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity. If one has any consciousness at all, it is consciousness of time and space; geometric continuity is in some way inseparably bound to conscious thought.”
― René Thom

“The constructive and operational aspect of mathematics is necessarily linked to its discrete aspect, that is, to discontinuity. Since the operations cannot be carried out simultaneously, each of them takes place in a well-defined segment of time. The indefinite interation of operations creates objects for which it is often difficult, if not impossible, to obtain intuitive representations.”
― René Thom

“There seems to be a time scale in all natural processes beyond which structural stability and calculability become incompatible. In planetary mechanics this scale is of such an extent that the incompatibility is not evident, whereas in quantum mechanics it is so short that the incompatibility is immediately felt, and today the physicist sacrifices structural stability for computability. I hope that he will not have cause to regret this choice.”
― René Thom, Structural Stability and Morphogenesis

“A mathematical object, defined by an infinity of operations, can be considered to have a real existence only if this same object can be, as it were, immersed 'in a natural way' in the continuum, and the iteration itself has a continuous representation.”
― René Thom