René Thom, catastrophe theorist, in the spotlight

To mark the centenary of the birth of the mathematician and 1958 Fields Medal winner, the Académie des Sciences and the Institut des Hautes Études Scientifiques paid tribute to him. Etienne Ghys, also a mathematician, looks back at the intellectual legacy of this extraordinary geometer, in his “carte blanche” with Le Monde.

“At a time when so many scientists are calculating all over the world, isn’t it desirable for those who can to dream? So concludes René Thom’s (1923-2002) book Stabilité structurelle et morphogenèse, written in 1972. The author is one of the most influential mathematicians of the 20th century. To mark the centenary of his birth, two symposia at the Académie des Sciences and the Institut des Hautes Études Scientifiques have just reviewed the intellectual legacy of this outstanding geometer.

A dreamer, no doubt. When he was preparing his thesis, his mentor, Henri Cartan, had a hard time channelling him. He wrote to him, for example: “Refrain from stating theorems that are not only unproven, but whose statement does not even have a clearly defined meaning.” A colleague of Henri Cartan’s explained that a dozen mathematicians could provide the missing demonstrations, but that only René Thom was capable of imagining such incredibly innovative statements. The thesis was defended in 1951, and René Thom was awarded the Fields Medal in 1958 for his discoveries in the field of differential topology, of which he was one of the pioneers. The medal was a great shock to him. He explained that he didn’t think he deserved it, which is not what medal winners usually think.

He decided to go in a different direction, one that could be described as “more applied”. He postulated that, in general, a system (e.g. physical or biological) is in a stable state, and that at certain very particular moments it goes through what he called “singular” or “catastrophic” situations, jumping very rapidly from one domain of stability to another. It was therefore necessary to understand the nature of these singularities, and this was the birth of “catastrophe theory”, which enjoyed immense success in the 1970s.

A question of boundary

Thom lists seven elementary catastrophes with poetic names: “fold”, “pucker”, “dovetail”, “butterfly”, “elliptical umbilicus”, “parabolic” or “hyperbolic”. The English mathematician Erik Christopher Zeeman extended the field of application to increasingly varied situations: prison riots, dog aggression, stock market crashes and so on.

Many criticisms were levelled at this theory, and Thom analyzed them seriously, often acknowledging their validity. For example, he was criticized for failing to take into account the existence of chaotic dynamics (the theory of which was then being developed), or for having built a tool for understanding but not for predicting. He questioned the need for experimental validation in science, which led to violent, but often justified, reactions from biologists. His book Prediction is not an explanation (1991) examines all these issues with honesty and serenity.

Today, the concepts introduced by Thom, such as structural stability, genericity (the fact that a property applies to the general case) and transversality, are so important that they have entered the subconscious of all mathematicians.

Later, he turned to the philosophy of Aristotle. He recognized many aspects of this philosophy, such as defining an object or an idea through its edge. His thesis had already developed a theory of “cobordism”, and his catastrophes are nothing more than crossings of the edge of a domain of stability. In truth,” he writes, “there is a real unity in my thinking. I can only see it now, after much reflection, on a philosophical level. And I find this unity in the notion of the edge.

Salvador Dali’s last four paintings, in 1983, are Hommages à René Thom.