What is Catastrophe Theory about?

Abstract

When we start evaluating what is Catastrophe theory, it is very easy to be led into wrong conclusions just for terminological reasons. Of course, many people know already that in “Catastrophe theory”, the word “Catastrophe” does not have the sinister meaning it has in every day language. But few people realize that the word “theory” itself has here to be understood in a very peculiar sense. It would be completely wrong to equate Catastrophe theory with one of the standard scientific theories, like the Newtonian theory of gravitation, or the Darwinian theory of evolution. In such cases, one has to expect that the theory has to get some experimental confirmation, it has to be founded (or at least may be “falsified” in Popper’s sense) by experiment. The plain fact is that C. theory escapes this criterion: it cannot be “proved”, nor “falsified” by experiment. I owe this fundamental remark to a British Biologist, L. Wolpert, who once - in a meeting - told me: “Your theory seems to be able to explain anything, but a theory which explains everything explains nothing”. After some time of thinking, I had to accept this objection as perfectly valid. Being aware of this fact, we could try to find a way out by saying that Cat. Theory is a “mathematical theory”. Here again, I am afraid this statement is not correct. Of course, Cat. theory deals with mathematical theories, and mathematical objects. It has led already to many interesting results in Differential Analysis, theory of smooth maps, in bifurcation theory, qualitative Dynamics, etc.