The Topology of Everyday Life

A SPECTRE IS haunting science--the spectre of catastrophe theory.

It is with this sense of drama that Alexander Woodcock and Monte Davis hail the arrival of catastrophe theory, a methodology as different from traditional science as Marxism was from existing economic thought. Since its formulation around 1964, catastrophe theory has emerged as one of few mathematical breakthroughs in recent times to arouse public interest. The controversy lies in its claim to have broadened the scope of science to include the social sciences and humanities, uniting such diverse phenomena as the collapse of a bridge, the crash of the stock market, and the fall of the Roman empire. Yet its subject is not always "catastrophic" in the literal sense: optical scattering, embryonic growth, prison riots, aggressive behavior in dogs, and the rise of the nouveau riche also fall within its domain.

What allows the theory such wide-ranging applications is its emphasis on qualitative rather than quantitative analysis. What matters is not when or not to what extent something will happen, but whether it will take place at all. Thus catastrophe theorists can claim to understand phenomena other mathematical approaches cannot explain: naturally-occuring discontinuities or "jumps." Since the time of Newton and Leibniz, founders of the calculus three centuries ago, mathematical models in science have been concerned with the regular rotation of planets, the gradual increase in pressure of a gas being heated and the continuously-changing velocity of a falling object. But what about the suddent collapse of a beam, abrupt transition from water to ice or bursting of a bubble? Because they are discontinuous, catastrophists say, these phenomena have remained outside the scope of mathematical inquiry--until now.

Woodcock and Davis promote this viewpoint in what is perhaps the inevitable consequence of the catastrophe theory controversy, a book designed for the layman. Except for an occasional article in Scientific American or Newsweek, literature on this new methodology has been highly technical--and few members of the general public are sufficiently adept at differential topology to wade through such formidable math.

The authors bypass the math and cut to the core, relating the theory's history, fundamental concepts, applications and elements of the current controversy. Although swallowing the theory without the math requires some suspension of disbelief, Woodcock and Davis manage to present a cogent summary and the reader is left with the feeling that he has at least a handhold on the material.

The explanation is so simplified, though, that one must take care not to relax his critical faculties. Although the authors attempt to offer an objective analysis, as advocates of the theory, they tend to be overenthusiastic. It's not that they do not present opposing views (which they do), it's just that Woodcock and Davis always get the last word. Every objection is countered, and for a while it seems as if catastrophe theory really is the successor to the calculus--until the authors present a series of applications of their own device. The reader's reaction to these examples will most likely determine whether he becomes an advocate or opponent of the theory--a catastrophic jump to extremes, as some theorists have already noted.

IN THE DISCUSSION of catastrophe theory's history, the writers trace the influence of 19th-century mathematician Henri Poincare and 20th-century biologist D'Arcy Thompson on the thought of Rene Thom, a leading differential topologist at the French Institute for Advanced Scientific Studies. Thom's vision of an underlying geometric order to natural processes led to his publication of Structural Stability and Morphogenesis in 1972, eight years after he had formulated the models which were to become the foundation of catastrophe theory.

Although Thom described only limited applications to biology and linguistics, his ideas immediately took hold of the scientific community. Led by E.C. Zeeman of Cambridge University, mathematicians everywhere began to apply the theory to discontinuities ranging from shock waves in physics to schizophrenic cycles in psychology.

Thom's theory possesses the simplicity and elegance that is so appealing to a mathematician. His model is based on principles of topology, a field ofter described as "rubber-sheet geometry" because it concerns forms that may be stretched or distorted without changing their fundamental, qualitative properties. Thom contends that for a wide range of mathematical structures; including almost all natural processes, only seven stable "unfoldings" can occur. By varying the number and arrangement of factors controlling these structures, he determined that apart from the seven "elementary" structures, all others are doomed to degenerate into unstable configurations.

Even opponents concede that Thom's mathematics is impeccable; the trouble begins only when he applies the theory to phenomena outside the realm of pure mathematics. It turns out that most of the elementary structures are not quite so elementary: although the simplest of these, the "fold" catastrophe, may be depicted by a parabola, the complex "parabolic umbilic" model cannot be represented in fewer than six dimensions. In addition, some of the structures have such a narrow range of stable states that they are practically useless as real-life models.

THE MOST COMMONLY-USED model in applications is the "cusp," a three-dimensional structure dependent upon two variables or "control factors" with a third dimension, the "behavior axis," displaying the subject's reaction to these two factors.

In the accompanying diagram consider a point constrained to move along the surface subject to the value of both variables. A change in either control factor from (a) to (b), (a) to (c) or (b) to (e) produce only a continuous change in behavior, represented by a gradual rise or fall along the vertical axis. An increase in factor 2 from (c) to (d), however, results in a sudden drop to (a) on the surface below, once the point crossed the edge of the fold at (d). This dramatic plunge is a catastrophe and signifies a discontinuous "jump" in behavior from one stable state to another through an unstable intermediary state, A slight decrease in factor 2 from (e) yields an abrupt ascent to (c) in the same manner.

Translating this theoretical model into a practicable application involves isolating a particular type of behavior and identifying the control factors affecting it. One use in the social sciences is especially appropriate in view of last week's rally in Washington against nuclear power and the mishap at Three Mile Isle last month. The author's model for determining the rate of power plant construction pits the pro-nuclear lobby against the "ecology" lobby and predicts that at a certain point a nuclear disaster will cause a catastrophic cutback in construction.

Are we near to reaching that point? There is no way of telling since the evidence is not at all tangible or quantifiable. It seems as if the only hope of verifying the model is to wait for a nuclear disaster to occur and then analyze the effects with hindsight. If we are truly at a "critical turning point" as the authors claim, then it should be apparent very soon whether we are to have many power plants or none at all.

A compromise position may be possible if the cusp model is replaced by a "butterfly" catastrophe allowing the input of two additional control factors. Woodcock and Davis suggest that the absorption of the Atomic Energy Commission by the Energy Research and Development Agency may permit both lobbying groups to reconcile their demands.

The authors caution that if this and other applications seem suspect, that is not a justifiable reason for dismissing catastrophe theory altogether. The theory's lasting acceptance may have to await the development of mathematical techniques permitting more extensive applications and better predictions. Just as Newton's mechanics did not receive immediate acclaim, they maintain, neither should catastrophe theory's dubious reputation at present be seen as a sign of its ultimate success or failure.

Woodcock and Davis do, however, give ample consideration to objections to the theory: that it is incapable of making useful predictions; that it is so general and qualitative as to reveal nothing we don't already know; that alternative mathematical models already exist; and that its proponents have based their claims of its wide applicability on a few phenomena well-suited to the model. Finally, two of the harshest critics have charged that in substituting pure theory for "the hard work of learning the facts about the world," idealistic mathematicians have used the theory "deduce the world by thought alone."

WHILE THE authors try to deflect these criticisms, their own position, especially in light of some questionable applications, is not entirely convincing. Thom writes that "our use of local models...implies nothing about the 'ultimate nature of reality'." His catastrophe theory purported not to "explain" phenomena but merely to describe them--a crucial distinction the authors, as well as other proponents, refuse to make. If the mark of a science is both to explain and to predict phenomena, and catastrophe theory often does neither, a re-evaluation of its worth may be in order.

For their clear presentation of a difficult and controversial new theory, though, Woodcock and Davis deserve considerable credit. Their book is a fascinating introduction to the theory that may haunt science for a long time to come.