Structural Stability and Morphogenesis: An Outline of a General Theory of Models
Catastrophe Theory: Selected Papers 1972-1977
Catastrophe Theory and Its Applications
If white and black blend, soften, and unite
A thousand ways, is there no black and white?
—Pope, An Essay on Man
“All things,” said Charles Peirce, “swim in continua.” At what wave length does blue become green? When does a child become a grown-up? Are viruses alive? Do cows think? It is also obvious that there are discrete “things” that swim in these spectrums, and sometimes jump from one part of a spectrum to another. Day fades into night, but a flicked switch produces instant darkness. One can imagine a hippopotamus changing by imperceptible degrees into a violet, but, as Charles Fort once asked, who would send a lady a bouquet of hippopotami?
The abstract world of pure mathematics displays the same crazy mixture of continua and discreteness. The counting numbers rise by jumps, but the real numbers form a continuum so dense that it is meaningless to ask what number comes next after any counting number. Between 2 and 2.000…1, where the dots represent, say, a billion zeros, there is an uncountable infinity of other numbers. Continuous functions often graph as curves with well-defined maxima and minima, and with singularities that can be as sharp as spearheads.
The continuities and discontinuities of mathematics, for reasons that trouble philosophers, fit the real world with incredible accuracy. Add two cats to two cats. Lo and behold, you get four cats. Apply calculus to the smooth motions of the earth, sun, and moon, and the abrupt start of an eclipse can be predicted with fantastic accuracy.
It is a naïve error to suppose that calculus is concerned only with smoothness. Like the real world, it too is riddled with abruptness. Toss a ball from here to there. At the top of its trajectory it enters a singularity where it makes an abrupt change from one type of behavior (rising) to another (falling). In the language of calculus, the derivative that measures its rate of vertical change goes to zero. More complicated events may involve many variables, each changing smoothly, yet the system may reach critical points at which it suddenly flips from one state to another. For centuries mathematicians applied calculus to such singularities, but it was not until the mid-Sixties that René Thom, a distinguished French topologist at the Institut des Hautes Études Scientifiques near Paris, hit on a startling new insight. Thom called his discontinuities “catastrophes.” His work in this field, and the work of his followers, quickly became known as “catastrophe theory,” or CT for short.
Thom’s fundamental discovery was that under certain precisely defined conditions there are just seven types of elementary catastrophes. Each involves no more than four variables, and can be modeled in what physicists call a “phase space” (in CT it is a “behavioral space”) of two through six dimensions. In these abstract spaces the change of a system is diagrammed by the path of a single point that moves over a smooth “behavior surface.” The catastrophe occurs when…
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