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 the point is forced by the structure to jump from one sheet of the surface to another.

Thom’s seven types are defined topologically, which means that the actual magnitudes of each variable are irrelevant. Topology is a branch of geometry concerned only with properties that remain the same regardless of how a figure is deformed in a continuous way. Think of the figure as drawn on a rubber sheet. If you stretch, shrink, or bend the sheet, the figure’s topological properties are unaltered; for example, if points on a line are labeled 1,2,3,4,…, they cannot lose this ordering. There is no way to deform the sheet so that 3 is no longer between 2 and 4. The seven elementary catastrophes are like scaleless maps drawn on rubber surfaces. In their own way they are as beautiful as the five Platonic solids.

The tossed ball illustrates the simplest of the seven. Thom called it a “fold” catastrophe because it can be graphed on an ordinary sheet of paper creased along a line that goes through the curve’s singularity. The ball’s behavior is shown by a point that moves along this curve. When it crosses the crease it jumps from going up to going down.

The next simpler catastrophe, the “cusp,” is less trivial. Its model is three-dimensional: a two-dimensional surface pleated in three-dimensional space so that when it is projected on a plane the pleat maps as a cusp-shaped curve. (A familiar example of a cusp is the bright line, shaped like the top of a heart, that you sometimes see on the surface of a cup of coffee under strong slanting light. In the drawing on page 32 the cusp is shown at the bottom of the diagram.) The “swallowtail” catastrophe is modeled in four dimensions. The “butterfly,” “hyperbolic umbilic,” and “elliptic umbilic” catastrophes are five-dimensional. The “parabolic umbilic” requires six dimensions. There are infinitely many catastrophes in higher spaces, but all with four or fewer variables are topologically equivalent to one of the seven.

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Thom discusses his work and its biological implications in a remarkable book, *Structural Stability and Morphogenesis*, published in France in 1972 with a foreword by the late C.H. Waddington, a British geneticist who was the first top scientist to greet CT with enthusiasm. E. Christopher Zeeman, who heads the Mathematics Institute at the University of Warwick, was another British friend of Thom who became an enthusiast. Slightly younger than Thom, he too had specialized in topology; his doctorate was in knot theory. In a few years, after much writing and lecturing on CT, Zeeman became the theory’s foremost fugleman.

And now a funny thing happened in the history of modern mathematics. The event can be modeled—Zeeman himself in a playful mood so modeled it—as a catastrophe. What normally would have been a minor technical discovery in topology abruptly flowered into a crusading cult.

It all started in England in 1975 with a burst of publicity. The BBC’s television program *Horizon* hailed CT as a big scientific breakthrough. *The New Scientist*, a popular weekly, featured the theory on a cover that Alexander Wood-cock and Monte Davis describe in their book as resembling an advertisement for a Hollywood disaster film. In the US the theory got its first big boost in *Newsweek* with a two-page article by Charles Panati, a journalist of the paranormal who gave us that great scientific anthology, *The Geller Papers*. Panati stressed the application of CT to human behavior, quoted Zeeman’s ringing pronouncement “Catastrophe theory is a major step toward making the inexact sciences exact,” and declared that CT was the most important development in mathematics since calculus. In April 1976 Zeeman’s *Scientific American* article on CT added more prestige to the movement.

As I see it, three leading variables underlay this extraordinary surge of interest. “In the beginning,” as Tim Poston and Ian Stewart put it, “was Thom.” His book is a provocative blend of mathematics, wide-ranging science, misty metaphysics, impenetrable speculation, and purple propaganda. Although modest in his expectations of immediate applications, Thom offered CT as a new “language” for science, a new “paradigm” that eventually would have revolutionary consequences.

The second factor in CT’s abrupt rise was the charismatic, witty personality of Zeeman. He delighted his students by showing them his “catastrophe machine”—a rotating cardboard disk with two attached rubber bands—that demonstrates the behavior of a cusp. In 1975 he precipitated the first major battle over the theory by applying it to a 1972 riot in England’s Gartree prison. His enthusiasm was as catching as prisoner discontent. Soon a small band of followers, some of them his students, were beating the catastrophe drums with the zeal of a Salvation Army street band.

The third factor was the interplay between CT’s terminology and the temper of the times. The word “catastrophe” resonates with apocalyptic hopes and fears. One thinks of atom bombs, cosmological big bangs, black holes, political terrorism, airplane crashes, earthquakes, floods, fires, revolutions, third encounters, and the Second Coming. Catastrophe theory! What PR expert could have devised a better name? Had Thom called it “discontinuity theory” it is likely that only mathematicians would have learned of it.

Because any abrupt change that springs from a confluence of smoothly changing variables can be described by a catastrophe model, it follows that CT can invade any branch of science. In optics it has been most successfully applied to caustics—curves produced by reflection or refraction such as the coffee-cup cusp. A rainbow is a color caustic modeled by the trivial fold catastrophe. The rippling white lines you see at the bottom of swimming pools are caustics. One of the early triumphs of CT was a proof that these fluctuating lines are not like mudcracks but are elongated triangles with cusped corners.

CT applies to any kind of sudden bending such as the buckling of girders when a bridge collapses. A familiar novelty called a jumping disk is presented by Woodcock and Davis as an example of a cusp catastrophe. The slightly buckled metal disk is warmed, then pressure of the thumb forces the buckle to go the other way. Placed on the floor, the disk is stable until cooling causes the buckle to flip back again and send the disk several meters into the air. Magicians have a similar trick with a tennis ball. Make a tiny air hole so that pressure with the thumb will produce a small dimple in the ball’s surface. Set the ball on a slight incline, balanced on the invisible concavity. As air slowly reenters the ball, the dimple disappears and the ball suddenly rolls down.

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Any phase transition or simple threshold phenomenon in physics lends itself to CT: the abrupt freezing or boiling of water, the sudden shock wave produced by the crack of a whip or a plane breaking the sound barrier, and so on. An earthquake is an obvious example. Biological phenomena to which Zeeman and others have tried to apply CT are all over the place: the alteration of protein when you boil an egg, the sudden division of a cell, the firing of a nerve impulse, the beat of a heart, mutations, insect plagues, the rapid evolution of a new species or the vanishing of an old one, the differentiation of embryonic cells, the orgasm.

Animal behavior swarms with catastrophes. Zeeman’s favorite example is the cusp behavior of a slowly provoked dog. The conflicting drives of rage and fear do not cancel to produce neutral behavior. According to Zeeman and the cusp model, the dog’s behavior diverges in one of two directions that lead either to abrupt attack or to sudden flight. (See box on following page.)

Catastrophists are busy investigating psychological phenomena of the sort stressed by the Gestalt school: reversals of perspective in optical illusions, explosions of anger, bursts of tears, quick decisions to get married or divorced, nervous breakdowns, the “aha” reactions in creative thinking. When Archimedes ran naked down the street shouting “Eureka!” he was celebrating a cerebral catastrophe. Zeeman has applied the butterfly to the psychosis called *anorexia nervosa* (compulsive fasting). John A. Paulos, at Temple University, is writing a book about CT and humour—the slow build of a joke to a punch line that triggers the explosive guffaw. Going to sleep and waking up are abrupt transitions with cusp models. I cannot resist a juicy quote that I found in William James’s *Principles of Psychology* on the catastrophic decision to get up on a cold morning:

We know what it is to get out of bed on a freezing morning in a room without a fire, and how the very vital principle within us protests against the ordeal. Probably most persons have lain on certain mornings for an hour at a time unable to brace themselves to the resolve. We think how late we shall be, how the duties of the day will suffer; we say, “I

mustget up, this is ignominious,” etc.; but still the warm couch feels too delicious, the cold outside too cruel, and resolution faints away and postpones itself again and again just as it seemed on the verge of bursting the resistance and passing over into the decisive act. Now how do weeverget up under such circumstances? If I may generalize from my own experience, we more often than not get up without any struggle or decision at all. We suddenly find that wehavegot up. A fortunate lapse of consciousness occurs; we forget both the warmth and the cold; we fall into some revery connected with the day’s life, in the course of which the idea flashes across us, “Hollo! I must lie here no longer”—an idea which at that lucky instant awakens no contradictory or paralyzing suggestions, and consequently produces immediately its appropriate motor effects. It was our acute consciousness of both the warmth and the cold during the period of struggle, which paralyzed our activity then and kept our idea of rising in the condition ofwishand not ofwill. The moment these inhibitory ideas ceased, the original idea exerted its effects.

The social sciences have not escaped. CT models have been applied to stock market crashes, union decisions to strike, abrupt shifts of public opinion about anything, panic behavior of crowds, revolutions, alterations of social status by marriage, the fall of Rome. The sudden rise of the stock market last April 14 was an economic catastrophe. Robert Holt, a political scientist at the University of Minnesota, believes that the beginning and end of World War I can be usefully analyzed with catastrophe models. Next September’s issue of *Behavioral Science* will be devoted to CT.

No mathematician, it is important to realize, disputes the elegance of Thom’s models or denies their value as descriptive metaphors. It is one thing, however, to describe nature in a novel way, quite another to apply models that lead to significant explanations and predictions. It is not hard to understand why the absence of such results, especially in CT applications to the “soft” sciences, led to a second catastrophe—a bitter backlash of opposition on the part of both mathematicians and scientists.

In this country the first report on the backlash was Gina Bari Kolata’s hardhitting article in *Science* last year. Writing on “Catastrophe Theory: The Emperor Has No Clothes,” Ms. Kolata found that large numbers of eminent mathematicians held low opinions about applied CT. There are, they pointed out, well-established branches of mathematics, such as the theory of shock waves, quantum mechanics, and bifurcation theory, that handle complex natural discontinuities quite well. Applying CT to biology and human behavior is little more, they said, than describing a familiar structure in a colorful new terminology. The new descriptions are too vague and nonquantitative to lead to worthwhile insights. They tell us, said the critics, nothing we don’t already know.

One thinks of Kurt Lewin, the German Gestalt psychologist who became so enamored in the Thirties with topological diagrams that he applied them to hundreds of human behavioral events. Like CT, Lewin’s “topological psychology” made temporary converts, and there was even a school of topological sociology. I recently reread some of the pros and cons of this debate and was surprised by how much the rhetoric resembles that of today’s CT controversy. Even the behavior space of CT has its analogue in Lewin’s “life space.” His diagrams seemed promising at the time, but it soon became evident that they were little more than sterile restatements of the obvious.

This is exactly the criticism now being leveled at the more complicated topological diagrams of CT. “I do not see that fitting the surface of a cusp to a phenomenon involving discontinuity is an enormous conceptual breakthrough” is how topologist John Guckenheimer, one of the milder critics, has put it. Stephen Smale (like Thom he has won the Fields Medal, the highest honor a mathematician can receive) declared: “In a sense I reject catastrophe theory completely. It is more of a philosophy than mathematics, and even as a it doesn’t apply to the real world.” Marc Kac, another top mathematician, called Zeeman’s *Scientific American* article “the height of scientific irresponsibility.”

Hector Sussmann and Raphael Zahler, both of Rutgers, are CT’s most outspoken critics. They do not deny that CT has many useful applications in the physical sciences, but they are dubious about its applications to the biological and social sciences. “The proponents of catastrophe theory,” they declared recently, “have overwhelmed the public with a mass of claims, but reality fails to substantiate them. The great achievements of catastrophe theory are no more than delusions…. We find that the proponents…systematically make fundamental mathematical errors, reach conclusions that are either false, or vague, or meaningless, or trivial, and often misrepresent empirical evidence.” A long and detailed attack on applied CT, by Sussmann and Zahler, will appear in the forthcoming (Volume 37) issue of *Synthèse*.

Thesis, antithesis. It is too early to know what form the synthesis (if any) will take, but most mathematicians are now on the side of the critics. Laymen with no knowledge of CT but interested in this acrimonious fracas can do no better than read *Catastrophe Theory*, the little book by biologist Woodcock and science writer Davis. It is non-technical, and although the authors are strong defenders of applied CT they try to give a fair account of the opposition. They agree that CT has so far accomplished little except in physics, but they are optimistic. “Someday,” they write, “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 a ‘quantum jump.”‘

In contrast to the popularly written Dutton book, the volume by Poston and Stewart is a ponderous textbook of 491 pages intended mainly for scientists who are firmly grounded in calculus and interested in applying CT to their own fields of research. The book is handsomely illustrated, clearly written, and astonishingly up-to-date considering the speed with which papers on CT are proliferating. A valuable bibliography lists more than 400 references.

Half the book is on the mathematics of CT, half on applications. The emphasis is on applications in physics, with special attention to areas where CT is moving from its earlier, purely qualitative analysis toward quantitative methods. There are sections on the application of CT to the stability of ships (with some recent results on vertical-sided vessels such as oil drilling platforms), mirages, sonic booms, fluid mechanics, ocean waves, thermodynamics, magnetism, and laser physics. Scant attention is paid to biological applications, although the authors believe that CT will play a major role in this area. The behavioral sciences rate only a final paragraph. “If *any* mathematical methods can aid the growth of such wisdom,” the book’s last sentence predicts, “then catastrophe theory will be part of them.”

Poston and Stewart make no effort to reply in detail to the criticisms of Sussmann and Zahler. Neither appears in the index, although their forthcoming paper in *Synthèse* is mentioned as having “enjoyed a certain notoriety, but its usefulness is seriously marred by repeated errors.” Throughout the volume CT is treated as a powerful new tool that eventually will be useful in all the sciences. The book is dedicated “To Christopher Zeeman, at whose feet we sit, on whose shoulders we stand”—a curious position that seems not very stable.

For anyone interested in applications of CT to biology and the social sciences, the collection of papers by Zeeman is indispensable. The book is a photooffset of the original articles, plus a bibliography and general index. Thom’s book, which started it all, is best read only after one is familiar with CT’s basic ideas. Its idiosyncratic mix of technical mathematics, science, and philosophical musings is likely to inspire either admiration, scorn, or frustration over its opacities.

Although Thom is not easy to understand, the center of his vision is clear enough. It is the opposite of a famous statement by Paul Dirac which Thom quotes as follows: “The main object of physical systems is not the provision of pictures, but the formulation of laws governing phenomena, and the application of these laws to the discovery of new phenomena. If a picture exists, so much the better; but whether a picture exists or not is a matter of only secondary importance.”

For Thom the picture is of primary importance. “I am certain,” he writes, “that the human mind would not be fully satisfied with a universe in which all phenomena were governed by a mathematical process that was coherent but totally abstract. Are we not then in wonderland?” Simply to describe a paradoxical state of affairs, as for example in so many areas of quantum mechanics, and leave it at that is, for Thom, to “sink into resigned incomprehension”—a habit that stifles scientific progress because it leads to indifference. We must, Thom urges, constantly seek geometrical models. “The dilemma posed by all scientific explanation is this: magic or geometry.” On a deeper level, even successful geometry is magic in the sense that it miraculously fits the external world. And all successful magic, Thom adds, is geometry.

CT is thus viewed by Thom as a new way of modeling what may be an infinitely complex and basically unknowable reality. By breaking away from quantitative models, and by making use of the topology of higher- and even infinite-dimensional spaces, he believes that for the first time in history we have a method that ultimately can model everything. “There is no doubt it is on the philosophical plane that these models have the most immediate interest,” Thom writes. “They give the first rigorously monistic model of the living being, and they reduce the paradox of the soul and the body to a single geometric object. Likewise on the plane of the biological dynamic, they combine causality and finality into one pure topological continuum, viewed from different angles.”

“Whither away?” as Poston and Stewart head their final chapter. It is conceivable that CT, like information theory and game theory, will become a valuable tool for the behavioral sciences. It is also possible that, like Lewin’s topological pictures, it is fated to die away as another premature effort to apply geometry in areas where it is either trivial or wrong. More likely its trajectory will take a duller road that winds somewhere topologically in between.

This Issue

June 15, 1978