The Rise and Fall of Catastrophe Theory

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The Rise and Fall of Catastrophe Theory

Overview

In the 1960s a French mathematician named René Thom (1923- ) developed a mathematical tool known as catastrophe theory. Thom used his theory to study and make predictions of processes involving sudden changes. His ideas became popular with mathematicians and scientists in a variety of fields during the 1970s. However, catastrophe theory was sometimes applied to areas outside its scope, and for this reason it eventually became somewhat discredited.

Background

Scientists have found that there are two basic types of processes in nature: continuous and discontinuous. An example of a continuous process is the increase in temperature of a gas as it is heated. As one variable is changed at a constant rate (heat is added to the gas), a second variable also changes at a constant rate (the temperature of the gas increases). Because continuous processes are "smooth," they are relatively easy to predict. The branch of mathematics used to study continuous processes is called calculus and was developed by Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716) more than 300 years ago.

Discontinuous processes, on the other hand, are "abrupt" rather than smooth. An example of a discontinuous process involves an arched bridge to which more and more weight is added. At first, little effect is seen as the weight on the bridge is increased—the bridge begins to bend almost imperceptibly. At a certain point, however, enough weight is added to the bridge that it collapses. A steady change in one variable (the amount of weight on the bridge) results in almost no change in a second variable (the shape of the bridge distorts slightly) followed by a sudden change to a very different state (the bridge collapses).

A sudden change in a discontinuous process is called a catastrophe. In mathematics catastrophes can include sudden disasters, such as a bridge collapse or an earthquake, but they can also include much less dramatic events, such as the boiling of water. As room temperature water is slowly heated, it remains a liquid. Once it reaches its boiling point, however, the water suddenly begins to change state, from a liquid to a gas. In other words, a catastrophe has occurred. The values of the variables for which a catastrophe occurs are called the catastrophe set. For the boiling-water catastrophe, there is only one variable, that of temperature, and the catastrophe set consists of only one temperature, that of 100°C. Most discontinuous processes, however, involve more than one variable, and the catastrophe set may be quite large.

Because discontinuous processes involve sudden changes, they are usually much more difficult to predict than continuous processes. In the 1960s René Thom developed a way of studying discontinuous processes, which he called catastrophe theory. Thom became interested in catastrophes because he hoped to apply mathematics to the "inexact" science of biology. (Biology and sociology are said to be inexact sciences because they involve primarily discontinuous processes.) Thom presented his ideas in two books: Structural Stability and Morphogenesis, which was published in 1972, and Catastrophe Theory in Biology, which appeared in 1979.

As he was developing his theory, Thom collected data relating to the variables involved in sudden changes. When he then plotted these data on three-dimensional graphs, the result was a curved surface representing a catastrophe in mathematical form. Therefore, catastrophe theory allowed mathematicians to study not only numerical data from discontinuous processes but also visual data in the form of three-dimensional shapes. For this reason, catastrophe theory is considered to be a branch of geometry.

Thom showed that even though the number of discontinuous processes in nature is essentially infinite, the graphs of these processes could be categorized into a few basic shapes. For processes involving four variables, he discovered that there are seven basic types of catastrophes. They are named for the shapes formed when their variables are graphed: fold, cusp, swallowtail, butterfly, wave, hair, and fountain. To picture the graph of a fold catastrophe, for example, imagine taking a sheet of a paper and bending it into the shape of the letter C. The upper curve of the C would represent one stable state, and the lower half of the C would represent a second stable state. A catastrophe would be represented by a jump from the upper curve to the lower or vice versa. Thom also found that if a catastrophe depends on more than six variables, its graph becomes too complicated and results in no clear shapes that can be studied.

Impact

The ultimate goal of catastrophe theory was to produce a model of a discontinuous process that could then be used to make predictions. First, a scientist or mathematician would select variables related to the process being studied. For a chemical process, for example, these variables might be temperature and the concentration of reactants. Next, the scientist would collect as much data as possible about the effects of different combinations of temperature and concentration on the process. With the application of complex calculations and the aid of computer software, the scientist could transform the data into a three-dimensional graph, which could then be used as a model.

One scientist who used catastrophe theory to examine a discontinuous biological process—that of the fight-or-flight response in dogs—was E. Christopher Zeeman. Zoologist Konrad Lorenz (1903-1989) had studied aggressive behavior in dogs and believed that it was controlled by two variables: anger and fear. Zeeman drew upon Lorenz's work and defined a dog's mood as the combination of its anger, as indicated by the degree to which its teeth are bared, and its fear, as indicated by the degree to which its ears are flattened. He then proceeded to collect data on how a dog's mood affects its behavior. If a dog is angry, but not afraid, it will attack (the fight response). When the dog is afraid, but not angry, it will flee (the flight response). If the dog is neither afraid nor angry, it is unlikely to either flee or attack. What Zeeman was interested in, however, was what would happen (and how to predict what would happen) when a dog is angry and afraid at the same time.

Zeeman used the data he gathered to create a three-dimensional graph with anger plotted on one axis, fear plotted on a second, and the dog's behavior plotted on a third. The result was a curved surface. This surface matched one of the basic catastrophes described by Thom: a cusp catastrophe. The graph predicted that an angry dog that is slowly made afraid will continue to behave aggressively until its fear increases to a point that a catastrophe occurs. At this point, the dog suddenly flees—a flight catastrophe. Similarly, the opposite can also occur. A fearful dog that is slowly made angry will eventually attack—an attack catastrophe. Therefore, Zeeman predicted that the effects of frightening an angry dog would be different from those of angering a frightened dog.

Zeeman's use of catastrophe theory to study flight-or-fight responses showed that it could be used to approach certain problems in a new way. However, catastrophe theory and the models produced by it soon became somewhat of a fad. It was quickly embraced by mathematicians and scientists in diverse fields, and several national magazines, including Newsweek and Scientific American, published articles that explained it to the public in broad terms. One reason for the popularity of catastrophe theory was the belief that it could be applied to every branch of science. Some hoped that it would play the same role for inexact sciences as calculus had for the more exact sciences of physics and chemistry.

Despite the initial acceptance of the theory, it eventually became controversial. The number of variables involved in a discontinuous process must be small in order for catastrophe theory to model it with any accuracy. In the real world, however, especially in inexact sciences such as biology and sociology, these conditions rarely occur. One less than practical application of catastrophe theory involved its use to model the escalation of hostilities between nations. The variables used were threat and cost. It was argued that catastrophes—in this case, sudden attacks or surrenders—would occur when threat and cost were both high. Although such a model might be used to describe theoretical nations in very general terms, many more variables come into play when real people and real nations are involved. Therefore, such a model could not be used to make predictions of any practical value. Catastrophe theory was also applied with varying degrees of success and failure to social topics ranging from the stock market to prison riots to eating disorders.

Almost all biological and sociological systems are infinitely more complex than can be described adequately by catastrophe theory. In other words, they are essentially impossible to predict by this method. Therefore, catastrophe theory has turned out to be most useful in the exact sciences of physics, engineering, and chemistry even though Thom had originally intended it as a tool for studying the inexact sciences. One problem that catastrophe theory can be used to effectively study, for example, is whether light will reflect from or pass through moving water. Even in the arena of exact science, however, other mathematicians soon pointed out that many of the most useful ideas of catastrophe theory had already been developed under other names.

STACEY R. MURRAY