Citation: Zahler, Raphael S., and Hector J. Sussmann. "Claims and accomplishments of applied catastrophe theory." Nature 269.5631 (1977): 759-763.
Web: https://www.nature.com/articles/269759a0
Tags: Mathematical
The goal of this paper is to highlight the ways in which catastrophe theory, when applied to areas outside of pure mathematics, has failed to be a successful theory. I like this paper a lot - it seems to have served as a good conversation-starter to reassess the role of applied catastrophe theory in the 70s. The paper is phrased in a very blunt, aggressive, accusative way. As with most blunt, aggressive, accusative papers, I found myself at several points dismayed by the tone, but the article was short and on-topic enough that it was fine. I agreed with most of the points, and found myself disagreeing with others.
An example of a point where I disagree is the subsection about "confusion about continuity". Here, they say that in biological systems catastrophe theory predicts discontinuity but in reality the physical systems are continuous and have bounds on their derivative. To me, there is an obvious retort to this. The premise of catastrophe theory is often that a system will relax to a critical point. The discontinuity is in the limiting behavior at the critical point; the relaxation times will be finite so catastrophe theory predicts finite-rate changes. This is just like in Zeeman's catastrophe machine, where catastrophe theory manifestly applies but the wheel spins at a finite speed.
Another point where I disagree is the subsection about "Use of Thom's theorem to justify extrapolation". They state that "Mathematically, it is simply not true that Thom's theorem forces M to look like Fig. 1. There might be many cusps or folds, and they need not be located at the origin nor oriented as shown" and then "any surface is arbitrarily close to a surface which satisfies the conclusion of Thom's theorem, and therefore (since all observations have a finite error) invoking this theorem can tell us nothing new about behavior". I think the authors are missing the point of how catastrophe theorists do extrapolation. First, you observe some data. Then you argue that the "simplest" (least number of folds) surface must look like something, and the possible candidates you're picking from are dictated by Thom's theorem. You're trying to use Occam's razor, and Thom's theorem helps you with that.
At other times I agree very much with the authors - it seems like the way that catastrophe theorists try to argue is at times quite flawed.
As a place to learn catastrophe theory, I really appreciated the blog post by Job Feldbrugge: https://jfeldbrugge.github.io/Catastrophe-Theory/.