René F Thom - Collected Quotes

"Everything considered, mathematicians should have the courage of their most profound convictions and thus affirm that mathematical forms indeed have an existence that is independent of the mind considering them. […] Yet, at any given moment, mathematicians have only an incomplete and fragmentary view of this world of ideas." (René F Thom, "Modern Mathematics: An Educational and Philosophical Error?", American Scientist Vol. 59, 1971) 

"Any mathematician endowed with a modicum of intellectual honesty will recognise then that in each of his proofs he is capable of giving a meaning to the symbols he uses." (René F Thom, "Modern mathematics, does it exist?", 1972)

"One of the central problems studied by mankind is the problem of the succession of form. Whatever is the ultimate nature of reality (assuming that this expression has meaning). it is indisputable that our universe is not chaos. We perceive beings, objects, things to which we give names. These beings or things are forms or structures endowed with a degree of stability: they take up some part of space and last for some period of time." (René F Thom, "Structural Stability and Morphogenesis", 1972)

"The fact that we have to consider more refined explanations - namely, those of science - to predict the change of phenomena shows that the  determinism of the change of forms is not rigorous, and that the same local  situation can give birth to apparently different outcomes under the influence of unknown or unobservable factors." (René F Thom, "Structural Stability and Morphogenesis", 1972)

"The real problem which confronts mathematics is not that of rigour, but the problem of the development of ‘meaning’, of the ‘existence’of mathematical objects.'' (René F Thom, "Modern mathematics, does it exist?", 1972)

"There seems to be a time scale in all natural processes beyond which structural stability and calculability become incompatible." (René F Thom, "Structural Stability and Morphogenesis", 1972)

"This distinction between regular and catastrophic points is obviously somewhat arbitrary because it depends on the fineness of the observation used. One might object, not without reason, that each point is catastrophic to sufficiently sensitive observational techniques. This is why the distinction is an idealization, to be made precise by a mathematical model, and to this end we summarize some ideas of qualitative dynamics." (René F Thom, "Structural Stability and Morphogenesis", 1972)

"The catastrophe model is at the same time much less and much more than a scientific theory; one should consider it as a language, a method, which permits classification and systematization of given empirical data [...] In fact, any phenomenon at all can be explained by a suitable model from catastrophe theory." (René F Thom, 1973)

"Catastrophe Theory is-quite likely-the first coherent attempt (since Aristotelian logic) to give a theory on analogy. When narrow-minded scientists object to Catastrophe Theory that it gives no more than analogies, or metaphors, they do not realise that they are stating the proper aim of Catastrophe Theory, which is to classify all possible types of analogous situations." (René F Thom," La Théorie des catastrophes: État présent et perspective", 1977)

"Algebra is rich in structure but weak in meaning." (René F Thom) 

"If we admit a priori that science is just acquisition of knowledge, that is, building an inventory of all observable phenomena in a given disciplinary domain - then, obviously, any science is empirical.” (René F Thom) 

"The spirit of geometry circulates almost everywhere in the immense body of mathematics, and it is a major pedagogical error to seek to eliminate it." (René F Thom) 

"Topology is precisely that mathematical discipline which allows a passage from the local to the global." (René F Thom)