René Frédéric Thom | Biography

René Frédéric Thom is best known as the founder of catastrophe theory, a field with numerous applications in the exact and social sciences. Catastrophe theory provides models for the description of continuous processes that experience abrupt change. Thom is the recipient of the Fields Medal and the Grand Prix Scientifique de la Ville de Paris. He is a member of the French Académie Royale des Sciences and has been named a Knight of the Legion of Honor in France.

Thom was born on September 2, 1923, at Montbéliard, France, to Gustav Thom, a pharmacist, and Louise Ramel. He married Suzanne Heimlinger on April 9, 1949, and they have three children, Françoise, Elizabeth, and Christian. Thom's formal education took place at the Collége Cuvier in Montbeliard and later in Paris at the Lycée Saint-Louis. After earning a master's degree in mathematics and a doctorate in science from the École Normale Supérieure in Paris, Thom went to Princeton University in 1951. He taught in Grenoble from 1953 to 1954 and in Strasbourg for the next decade.

Achieves Early Acclaim

In 1958, at the age of 35, Thom received the Fields Medal, the most prestigious prize in mathematics. When awarding the prize, the International Mathematical Union (IMU) cited Thom's 1954 invention of the theory of cobordism in algebraic topology. Cobordism is a classification scheme for manifolds(multidimensional topological surfaces) based on homotopy (a continuous deformation of one function into another). The IMU recognized Thom's use of this technique as a "prime example of a general cohomology theory." In a 1991 address to the Symposium on the Current State and Prospects of Mathematics, Thom recalled that after he received the Fields Medal, he experienced doubts about his ability to continue developing meaningful mathematical results. He decided to turn his attention from the more algebraic types of work and concentrate on singularities of differentiable maps, a topic he considered "more flexible and more concrete."

While he was teaching at Strasbourg University, Thom collaborated with a physicist on an investigation of caustics in optical geometry (i.e., rays reflected or refracted by curved solid surfaces), studying both their singularities and their deformations. He told the 1991 symposium that "to my surprise, I found that in caustics, organized by some very simple optical instruments like spherical mirrors and rectilinear diopters, one may find a singularity that should not theoretically exist." He eventually connected this phenomenon to Pierre de Fermat's principle of optimality.

Since 1964, Thom has been a professor at the Institut des Hautes Etudes Scientifiques (IHES) at Bures-sur-Yvette, where he concentrates on research, with no teaching or administrative duties. In this environment, he wrote a carefully reasoned article opposing the popular movement to discard geometry from general mathematics education. His argument concerning the importance of geometry centered on the potential value of studying singularities of function maps. As a result of this exercise, Thom developed the theory of singularities.

Formulates Catastrophe Theory

During the 1960s, Thom turned his attention from optical problems in physics to the biological topic of embryology. His objective was to apply transversality theory to natural science. Out of his mathematical analysis of current theories of cellular differentiation, he developed what he later called "my famous list of seven elementary catastrophes in space-time," a result that fit naturally into singularity theory. Applying a very abstract theorem about universal deformations of analytic sets proved by Alexander Grothendieck, a colleague at the IHES, Thom wrote a book attempting to explain the origin of natural forms. Delayed by the bankruptcy of the original publishing house, Stabilité Structurelle et Morphogénése (Structural Stability and Morphogenesis) finally appeared in print in 1972, although a few copies had already circulated in the mathematical community. E. Christopher Zeeman of the University of Warwick was fascinated with the contents of the book. Thom later told the 1991 symposium that "[Zeeman's] reflections on this subject led to a grandiose extension of the theory," extending it from four-dimensional space-time to any locally Euclidean space. Zeeman gave a couple of lectures on what he called "catastrophe theory," and wrote a fascinating account about it for Scientific American magazine.

Generally classified as a branch of geometry because variables and results are shown as curves or surfaces, catastrophe theory attempts to explain predictable discontinuities in output for systems characterized by continuous inputs--a task that cannot be done using differential calculus. Contrary to the implications of the theory's name, the "catastrophes" studied are not necessarily negative in nature; the term simply refers to sudden change. For example, when a balloon is steadily filled with air, it expands and changes its shape in a relatively uniform manner until the pressure in the balloon's interior reaches a critical value. Then, when the balloon undergoes more abrupt, but predictable, change, it pops. More complex phenomena, such as the refraction of light through moving water, the amount of stress that can be placed on a bridge, and the synergistic effects of the ingredients in drugs can also be effectively studied using catastrophe theory. The theory provides a universal method for the study of all jump transitions, discontinuities, and sudden qualitative changes.

Thom recalled, "as a result [of Zeeman's publicity], catastrophe theory . . . took off like a rocket, propelled by the principal media all over the world. This glory was short-lived, however, and the brief success of [catastrophe theory] soon fell to the slings and arrows of trans-Atlantic criticism." In fact, some scientists have hailed Thom's catastrophe theory as a tool more valuable to humanity than Newtonian theory, which considers only smooth, continuous processes. However, the theory became something of a fad in the 1970s and 1980s and was used in applications that the theory does not support. As a result of such indiscriminate application, the theory has at times been unjustly criticized as a cultural phenomenon or a metaphysical view rather than a legitimate branch of mathematics. Although it has been presented in a metaphysical vein as proof of the deterministic nature of the universe, catastrophe theory does not purport to abolish the indeterminacy that is central to nuclear physics. Nevertheless, catastrophe theory has been used to study abrupt systems changes in such diverse fields as hydrodynamics, geology, particle physics, industrial relations, embryology, economics, linguistics, civil engineering, and medicine.

Turns to Philosophy

By his own account, Thom has been primarily a mathematical philosopher since the early 1970s, largely due to the controversy surrounding catastrophe theory. A major criticism of the theory was that it allows only qualitative rather than quantitative predictions. That is, given a specific application, the type of abrupt change could be predicted, but it could not be quantified in numerical terms. While admitting that this is a legitimate concern, Thom still sees value in the modeling techniques catastrophe theory makes available to fields such as psychology. A classical example described by Zeeman, for instance, involved predicting the action of a dog experiencing a combination of the emotions of rage and fear--emotions that (if acting alone) would cause the dog to attack or flee, respectively. What catastrophe theory provides is a mathematical framework for clearly describing the situation being modeled, much as algebra provides a shorthand for expressing relationships between numbers and variables in a way that facilitates manipulating them to obtain results.

Speaking from the perspective of 20 years of philosophical reflection, Thom said in his 1991 symposium address: "What is important in science is not the distinction between true and false. This might seem strange to mathematicians, but I will say that if I had the choice between an error which has an organizing power of reality (this could exist) and a truth which is isolated and meaningless in itself, I would choose the error and not the truth. There are many examples of errors which are scientifically important, and there are many, many examples of meaningless truths in science."

Thom died on October 25, 2002, in Bures-sur-Yvette, France, of vascular disease. He was 79.