Chapter 2
THE
UNFOLDING OF
CATASTROPHE
THEORY
The imaginative ideas from which scientific work originates depend on a sensitive receptiveness to the oddity of nature, essentially similar to that of the artist. When they are first proposed they often have the same quality of unexpectedness, and perhaps wrongheadedness, as say, cubism, abstract art, or atonal music.
C. H. WADDINGTON
A new idea doesn't necessarily meet with a warmer reception in science than in art or any other human activity. Like most of us, scientists are generally content with established ideas unless a specific problem demands new mathematical or conceptual tools. They rarely have contact with pure mathematicians, and most of them are no more concerned with the philosophy of science than most lawyers are with the philosophy of law.
Although Rene' Thom's creativity in pure mathematics earned him honor among his colleagues twenty years ago, it was familiar only to a small group. And although his natural philosophy was profound and original, it would not by itself have made a stir in science. Yet combined in catastrophe theory, the mathematics and philosophy are having a wide impact. Thom's book, Structural Stability and Morphogenesis, was published only a few years ago,
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THE UNFOLDING OF CATASTROPHE THEORY 15
but today catastrophe theory is being used to describe phenomena as diverse as psychological crises and chemical reactions. At the same time, critics are denying that the theory can or should be applied at all. It's beautiful mathematics, they say, but has nothing to do with the real world, and the claims that have been made for its wide range and possible predictive value are "the height of scientific irresponsibility." In an exchange of letters in Science, mathematician Marjorie Senecbal wrote: "I am fascinated by the sense of mission that drives some scientists to denounce heretics and heresies, and by the ease with which others abandon their objectivity and jump on bandwagons."
How did Thom's abstract new vision of the world so quickly become a heresy, and the opposition to it a bandwagon? And bow was his work in an esoteric branch of mathematics turned to detailed applications? To answer these questions, we must look back at the theory's roots in mathematics and science; at its growth from the early 1960s onward, which brought it to the attention of many scientists even before the publication of Thom's book; and finally, at its divergent evolution in Thom's thinking and that of E. Christopher Zeeman, the English mathematician who contributed to catastrophe theory's birth, has been its most active advocate, and stands at the center of the current controversy.
THE ROOTS OF THE THEORY
Thom has long bad a reputation for the ambition, even the riskiness, of his mathematical ideas‑and for their success. In 1946, at twenty‑three, he graduated from the prestigious ‑rcole Normalee Supgrieure. This college, long a pinnacle of France's national school system, awards only a few dozen degrees each year in science and mathematics. In 1951, Thom wrote his doctorate in topology. "He was full of ideas and enthusiasm," recalls a contemporary. "There were great topologists in the older generation but Thom wasn't anyone's disciple‑he had more in
g
common with Darboux and Poincare" (Gaston Darboux was France's leading geometer at the end of the nineteenth century ‑a century in which Gauss, Bolyai, and Lobacbevsky bad created alternatives to classical, three‑dimensional, Euclidean geometry,
16 CATASTROPHE THEORY
and Riemann had generalized their work into a theory that paved the way for Einstein. Henri Poincare Darboux's contemporary, has been called "the last universalist": the last mathematician to do first‑rate work in all areas of pure and applied mathematics, and to write for a lay audience as well. Many believe that be very nearly anticipated Einstein in stating the theory of relativity.)
Thom published few papers, although it is said that they contained many of the most important ideas in topology of the 1950s. His colleagues prodded him to get his work on paper, and be gained a reputation for preferring intuition to technical rigor. "Some mathematicians go at their work like engineers building a six‑lane highway through the jungle," says mathematician Tim Poston, "laying out surveying lines, clearing the underbrush, and so on. But Thom is like some creature of the mathematical jungle, blazing a trail and leaving just a few marks on his way to the next beautiful clearing."
In a 1954 paper, Thom set forth the concept of transversality. To squeeze it into a nutshell, transversality concerns the ways in which the smooth curves of analysis (the abstract descendant of calculus) can intersect or "cut" each other. These cuts can be clean, or they can be mathematically messy in a variety of interesting ways; transversality demonstrated that most cuts are clean, and gave mathematicians a "handle" on those that aren't. Many problems considered as part of analysis were illuminated by Thom's topological approach, and transversality has stimulated and unified a number of mathematical advances since 1954. It made possible Thom's theory of co‑bordism, for example, for which be won the 1958 Fields Medal (mathematics' highest international honor), and other developments such as Milnor's "exotic sphere" ‑ a seven‑dimensional form with properties as surprising to a topologist as, say, those of a fire‑breathing dragon would be to a biologist.
Thom taught at the University of Grenoble until 1957, then moved to the University of Strasbourg. He pursued the implications of transversality and linked them to the work of others such as Hassler Whitney, an American topologist at Princeton's Institute for Advanced Studies. Whitney bad studied the singularities of mappings‑phenomena that occur when the points of
THE UNFOLDING OF CATASTROPHE THEORY 17
one surface are projected onto another as the surfaces are topologically distorted.
Besides its meaning in topology, "singularity" has another meaning in calculus and analysis There, it is a point on a graphic curve where the direction or quality of curvature changes. In Figure 1, the four singularities (a local maximum, two local minima, and a point of inflection) are labeled: you can see that at each one, the slope of the curve is momentarily level. The maximum and the two minima are called "local" because they are not necessarily the highest or lowest points on the curve, merely higher or lower than their immediate neighbors. These points are of interest in many practical applications of calculusfor example, if x represents the temperature at which a fuel is burned and y the amount of pollution; or if x represents the amount of pressure used to forge metal and y its resulting strength. There is a discipline known as the calculus of variations, in which mathematicians have developed general techniques for locating these singularities (given the equation corresponding to the curve). Thom became interested in the relationship between the calculus of variations and Whitney's topological singularities.
He was not the first to see such a connection. In the 1880s and 1890s, Henri Poincare bad linked calculus and topology (then called "analysis situs," analysis of location) to create qualitative Figure 1. Singularities of a curve.
y A
CATASTROPHE THEORY
dynamics and apply it to unsolved problems of planetary motion. This may seem strange; after all, dynamics had been a firmly quantitative field since Newton. But Newton's methods yield explicit solutions only for the interaction of two bodies‑for example, the sun and the earth, or the earth and the moon. When three or more bodies are involved, the equations of motion cannot be solved directly, and even approximate solutions require tedious, complex procedures. Around 1800 Pierre Simon de Laplace, the great mathematical physicist known as "the Newton of France," had tried at length‑but without success‑to show that all the two‑body attractions of the solar system added up to a stable dynamic system, a grand perpetual‑motion machine that would run forever.
Poincare set out to show that even if quantitative solutions were impossible, it was still possible to make progress on important questions: does a complex, many‑body system return periodically to the same arrangement? Does a slight perturbation simply "nudge" the whole system, or does it lead eventually to qualitatively different behavior, such as a planet spiraling into the sun or colliding with another planet? Though he did not fulfill Laplace's earlier hopes, Poincare inaugurated a valuable new approach. His fellow mathematicians saw its value, but thought it arbitrary because it was adapted to a particular physical problem rather than being part of a general method: they called it "so new, so perfect and so hard to follow up." Gaston Darboux, Poincare's biographer, explained: "Poincare was an intuitionist. Having once arrived at the summit he never retraced his steps. He was satisfied to have crashed through the difficulties, and left to others the pains of mapping the royal roads destined to lead more easily to the end." As E. T. Bell, a historian of mathematics, summed it up in 1937:
"Much of Poincare's work in his astronomical researches was qualitative rather than quantitative, as befitted an intuitionist, and this characteristic led him, as it had led Riemann, to the study of analysis situs. On this he published six famous memoirs which revolutionized the subject as it existed in his day . . . He modernized the attack [on planetary motions]; indeed his campaign was so extremely modern to the majority of experts ... (continues, but you get the drift)