Structural Stability , Catastrophe Theory , and Applied Mathematics : The John von

We give a brief description of catastrophe theory, and of its applications; to my view, it is a fundamentally qualitative, interpretative theory, and, by itself, it has no ability to predict. Examples are given of interpretations of singularities in statistics and in geophysics (plate tectonics). It is perhaps ironical that I am now assessing the importance of such ideas and theories as structural stability and catastrophe theory in front of an audience of applied mathematicians. Why? Because in its very intention catastrophe theory emphasizes the qualitative aspect of empirical situations, whereas applied mathematics is fundamentally devoted to computation. Of course, applied mathematics cannot exclude qualitative thinking, as its problems are originally given in ordinary language-in a qualitative way. But most applied mathematicians would say-(I believe)-that "modelization" is nothing but translating this qualitative problem into a quantitative model, which then has to be confronted with experiment. On the contrary, catastrophe theory would say that quantitative studies-inasmuch as they are possible and reliablemay help in defining local morphological elements (singularities), from which a global qualitative construction may be built. Moreover, the truth is that I do not have myself a very clear picture of the activity of a professional applied mathematician. Hence it is quite possible that some of the ideas I express here may seem to you a bit out of the field, as I never had the opportunity of working myself on a very quantitative basis-ven in pure mathematics. But as many people-specially in popularization articles-have expressed tremendous hopes about the pragmatic possibilities of catastrophe theory, I think it is time to come back to a more sober appreciation of its impact. Perhaps a very important cause of the ambiguity, when dealing with catastrophe theory, is its radically novel epistomological status. You read frequently, in these popular articles about catastrophe theory (here abbreviated C.T.), that "Catastrophe theory is a mathematical theory". The truth is that C.T. is not a mathematical theory, but a "body of ideas", I daresay a "state of mind". As soon as the ideas developed by C.T. have reached a very rigorous mathematical status, then these ideas have been incorporated in specific branches of mathematics: singularities of smooth mappings, stratified spaces, singularities of differential forms, bifurcation theory, qualitative dynamics, etc. Hence, strictly speaking, C.T. is not a mathematical theory. Of course, C.T. arose from mathematics, and it has led to important progress in mathematics itself; and we may hope that this * The fifteenth John von Neumann Lecture delivered at the 1976 National Meeting of the Society for Industrial and Applied Mathematics, held at Chicago, Illinois, June 16-18,1976. Received by the editors June 1, 1976. ? Institut des Hautes Etudes Scientifiques, Bures-Sur-Yvette, France.