Catastrophe Theory IV

"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 (in particular its essential concept of structural stability) is really a paradigm rather than a theory. It has attracted so much attention and generated so much argument because its scope and application appear to be virtually unlimited." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Is catastrophe theory correct? In its mathematics, yes; in the natural philosophy that inspired it and the scientific applications that flow from it, the only possible answer is that it's too soon to say. There is always a chance of error whenever we try to capture any aspect of reality in mathematical symbols, and another chance of error when (after working with the symbols) we use them to generate descriptions or predictions of reality." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Is the [catastrophe] theory useful? in the rigorous applications, yes; in the illustrations, sometimes; in the 'invocations', both yes and no. Yes, because catastrophe theory provides a common vocabulary for features of many different processes. Someday it may be as natural to speak of a 'cusp situation' or a 'butterfly compromise' as it is today to speak of the 'point of diminishing returns' or of a 'quantum jump'. No, because when the theory is invoked for the suggestiveness of its images, it cannot usually tell us anything we did not know before (although it can make explicit certain features that other models tend to neglect)." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"The most widely used mathematical tools in the social sciences are statistical, and the prevalence of statistical methods has given rise to theories so abstract and so hugely complicated that they seem a discipline in themselves, divorced from the world outside learned journals. Statistical theories usually assume that the behavior of large numbers of people is a smooth, average 'summing-up' of behavior over a long period of time. It is difficult for them to take into account the sudden, critical points of important qualitative change. The statistical approach leads to models that emphasize the quantitative conditions needed for equilibrium-a balance of wages and prices, say, or of imports and exports. These models are ill suited to describe qualitative change and social discontinuity, and it is here that catastrophe theory may be especially helpful." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"The major strength of catastrophe theory is to provide a qualitative topology of the general structure of discontinuities. Its major weakness is that it frequently is not associated with speciific models allowing precise quantitative prediction, although such are possible in principle." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)

"The key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)