Category Archives: Le Monde

At school, “fundamentals” aren’t everything: the aim is to learn to think.

“Why all these “accessories” that we value so highly, that we group around the fundamental and traditional teaching of “reading, writing and arithmetic”: object lessons, drawing lessons, natural history notions, school museums, gymnastics, school walks, manual labor, singing, choral music (…)? Because in our eyes, they are the main thing, because in them lies educational virtue. These were the words of Jules Ferry in 1880, a few months before the major laws establishing free, compulsory primary schooling for all French children, boys and girls alike, aged 6 to 13.

Thus, Jules Ferry never wished to focus primary education on the famous “fundamentals” – French and mathematics – contrary to what we often hear today. It was at this time that the physical and natural sciences entered the primary school curriculum, in perfect harmony with mathematics and French.

The aim was no longer, as it had been under the Ancien Régime, to give the children of the people the rudiments needed for work, but rather to “give children the habit of thinking early on”. These are fine intentions, but we shouldn’t forget that the reality of primary schooling in the Third Republic was very different. The school system was two-tiered: a tiny minority from privileged backgrounds benefited from a parallel elementary education (for which fees were charged), giving access to a secondary education that was essentially inaccessible to the majority of “communal” pupils. The mythical Certificat d’Etudes Primaires was not easy, and around a quarter of pupils passed it by 1900.

In 2023, it would be ridiculous to draw inspiration from the Ferry laws, many aspects of which would make no sense in today’s society. At the very least, we can retain the desire to avoid an excessive focus on fundamentals. However, this is what has been practiced for the past ten years, without any improvement in French or maths results.

Mathematics, a science among others

The result, on the other hand, is that access to culture (history, geography, science, languages, music, sport, etc.) is increasingly limited to children from privileged backgrounds, who have access to family dialogues and high-quality extracurricular activities. It is certainly difficult to approach physics without a minimum of practice in mathematics and French. But the reverse is also true: how can you understand the mathematical concept of volume, for example, without having decanted liquids from one container into another? Or that of the perimeter of a circle without having wrapped a string around a bottle? Reading, writing and counting” should be seen as a means to an end, but not as an end in itself, which is to think.

The French education system often forgets that mathematics is one science among others, and that the sciences should be studied as a whole, at least in elementary school. Many school teachers have understood the need for a global approach: they make sure that children write their science homework in good French, and don’t hesitate to offer dictations with a scientific content.

Will Le Monde readers be able to answer this question posed to the 1922 “certif”: “A paper casserole. For folding, the object must be exactly 8 centimeters high. Candidates will have to determine the dimensions of the square of paper to be used”? This example shows how mathematics, drawing and manual work could be combined in those days.

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.

In the summer of 1654, Blaise Pascal laid the foundations of the laws of chance

An exceptional scientist, Blaise Pascal, in collaboration with Pierre de Fermat, laid the foundations of probability theory in the summer of 1654. A geometry of chance with far-reaching consequences for contemporary science and philosophy, explains mathematician Etienne Ghys, in his Carte blanche for “Le Monde”.

An exceptional scientist, Blaise Pascal, in collaboration with Pierre de Fermat, laid the foundations of probability theory in the summer of 1654. A geometry of chance with far-reaching consequences for contemporary science and philosophy, explains mathematician Etienne Ghys, in his Carte blanche for Le Monde.

Blaise Pascal was born four hundred years ago, on June 19, 1623. Famous for his Pensées and Provinciales, he was also an exceptional scientist. I’d like to mention just one aspect of his work here: the introduction of probability, in collaboration with Pierre de Fermat, in the summer of 1654.

This theory “joins the rigorous demonstrations of science to the uncertainty of chance, and reconciling these apparently contrary things can, drawing its name from both, rightly claim this astounding title: the geometry of chance”, as he explained in Adresse à l’académie parisienne. The consequences for contemporary science, but also for philosophy, are considerable: chance has its own laws, which we understand better and better.

The starting point consists of two seemingly innocuous questions put to Pascal by a socialite, the Chevalier de Méré. The first is easy enough. If you throw two dice, it’s clearly more likely that you won’t get a double six than that you will. On the other hand, if we allow ourselves to repeat the experiment several times, we understand that after a certain number of throws, it becomes more probable to obtain a double six at least once than to obtain none at all. The question is to determine this number. The problem could be asked today at the baccalaureate, and the answer is 25.

The second question is much more subtle and was solved in a fascinating exchange of letters between Pascal and Fermat. Two people play heads or tails, and it’s agreed that whoever wins three games first will receive, say, 100 pistoles. After three games, the first player has won twice and the second once. Due to an unforeseen event, the game has to be interrupted. It’s clear that the first player has the advantage, but the second could still catch up.

“The spirit of geometry

How do you divide up the 100 pistols so that nobody feels cheated? I’ll leave it to Le Monde readers to ponder this problem. Pascal’s and Fermat’s approaches are distinct and complementary. Of course, they don’t stop there and discuss situations in which the number of players and the number of games are arbitrary.

In truth, neither of them discusses probabilities in the true sense of the word, and this problem is often considered to be as much the source of probability theory as of decision theory, in which agents must make “rational” choices. We recognize Pascal’s “heart and reason”, and the famous “spirit of finesse” alongside the “spirit of geometry”.

Some time after this exchange of letters, on November 23, 1654, “from about half past ten in the evening until about half past midnight”, Pascal experienced his second conversion and his “night of fire”. He turned his back on science to devote himself entirely to religion. He wrote to Fermat: “For, to speak frankly to you about geometry, I find it the highest exercise of the mind: but at the same time I know it to be so useless that I make little difference between a man who is only a geometer and a skilled craftsman.”

Pascal did, however, find a surprising use for geometry in 1657. While suffering from an excruciating toothache, he sought to occupy his mind with a question that would divert his thoughts from the pain. He pondered a geometry problem that dated back at least to Galileo, concerning a curve that was then called a “roulette wheel”. He thought about it so intensely that he not only found a solution, but also laid the foundations for integral calculus. The next day, the toothache had disappeared. I wouldn’t dare advise my reader to prefer the geometer’s roulette wheel to the dentist’s!

ChatGPT, an imaginary colleague for mathematicians

Like everyone else, mathematicians are wondering whether ChatGPT will radically transform their work. Initial trials have been inconclusive, encouraging some to reject the tool altogether. It’s true that there are some worrying errors.
When I asked GPT to give me a few examples of important theorems, he first explained that the question was a delicate one, as not everyone agreed on the meaning of the word “important” in this context, but that he would give me three examples that seemed consensual. That was an excellent start.
Read also: Article reserved for our subscribers “ChatGPT, an affabulatory intelligence made all the more formidable by the fact that it produces quite remarkable pastiches of science”.

The first two examples were quite relevant, but the third startled me. Admittedly, it was a fundamental theorem, but GPT attributed it to Jean-Pierre Serre in 1974, whereas any mathematician, even a beginner, knows that it was due to Evariste Galois in… 1832. It all sounded serious, and an uninformed reader would have been fooled.

When I asked for a demonstration of the Pythagorean theorem, I received a perfectly written proof, like a rigorous demonstration. GPT lowered one height of the right-angled triangle to break it down into two smaller right-angled triangles, and then applied… the Pythagorean theorem to each of them!
A vicious circle in a demonstration is, of course, unacceptable. How could an artificial “intelligence” “imagine” such a fallacy? Perhaps by getting its “ideas” from an Internet site, somewhere on the Web, which would contain lists of false proofs, intentional or otherwise. Let’s teach our students not to be fooled by these perfectly written but completely false demonstrations, sometimes in more subtle ways.
Read also: Article reserved for our subscribers Beyond artificial intelligence, the chatbot ChatGPT owes its oratorical talents to humans

We shouldn’t, however, throw the baby out with the bathwater. On the one hand, there’s no doubt that GPT will make rapid progress. For example, I was quick to criticize his proof of the Pythagorean theorem, and I hope he won’t make that mistake again. But, above all, we must learn to use him as an assistant, who knows a lot of things.
Looking for analogies
Mathematical literature is becoming so immense that it’s almost impossible to find one’s way around. Avalanches of longer and more technical articles flood the prepublication databases every day. GPT could help us to summarize works so as to select those that merit closer examination. Above all, it will soon enable us to look for analogies.

Today, the researcher navigates the scientific literature by sight, passing from one article to another, which he or she has found in the bibliography of the previous one. Fortunately, discussions between colleagues often lead us down new and promising avenues, even if they can sometimes turn out to be dead ends.
Similarly, couldn’t we regard GPT as an imaginary colleague who has read everything, including the wrong things, and can suggest some interesting ideas? Of course, this doesn’t mean we can let our guard down on the relevance and veracity of what he whispers in our ear.
GPT can even imitate the offbeat humor of mathematicians. I had to give a talk on April 1 and was looking for an idea for an April fool’s joke. Here’s GPT’s proposal: “You announce that you have solved the Riemann hypothesis [one of the most famous open problems], that the solution is so short that you were able to write it down on a small piece of paper and hold it up to the audience, but that you don’t want to make it public. Then you swallow the paper.” Does GPT have a sense of humor?

André Haefliger, the contagious passion for math

I was invited for the first time to present my work at an international conference in July 1981. I was nervous, facing impressive specialists. After my conference, an old man came to see me (I know now that he was 52 years old). He congratulated me warmly for my presentation, but an important point in my demonstration had escaped him and he asked me for explanations. Panic: did he want to point out with delicacy an error of reasoning? I answered that I was using a result that I had read in André Haefliger’s thesis.

He then introduced himself, and I understood that I was facing André Haefliger, one of the founding fathers of the theory of layering that I cherished at the time. He had not forgotten his own thesis, but he tried to convince me (unsuccessfully) that I had gone further than him in the interpretation of his result. It is always impressive for a young scientist to come face to face with one of his heroes.

André Haefliger died on March 7, at the age of 95, near Geneva, where he was a professor from 1962 until his retirement in 1995. Since our first meeting, he has been a source of inspiration and a model of mathematician, both as a researcher, as a leader of the scientific community, as a teacher and as a friend. His thesis, defended in Strasbourg in 1958, had an esoteric title, to say the least: “Laminated structures and value cohomology in a bundle of groupoids”, enough to scare off more than one interlocutor, so much so that it was expressed in an abstract language, so common at the time.

In the Swiss mountains

A puff pastry structure is actually very similar to puff pastry. The idea is to fill the space with sheets, like the pages of a book. This theory was born a few years earlier and was motivated by the understanding of the structure of dynamic systems. In 1969 he invented the “Haefliger classifier”, a concept that excited the young community working on these questions. Older geometers now fondly recall their memories of the presentation of his discovery, at the top of Mount Aigoual in the Cévennes, huddled in a weather station that housed a small conference center. It was reportedly very cold, but the atmosphere was warm.

When he arrived as a professor at the University of Geneva, there was no research department in mathematics. André always said that he did not have an office and that he had to go to a public phone booth. He contributed greatly to the foundation of the remarkable mathematics section in Geneva, which today houses two Fields Medalists. The greatest mathematicians came from all over the world to visit him, to share their discoveries with him and receive his advice. He knew how to listen to young people and encourage them to work together, which is not so common in this field.

I was not present at Mont Aigoual, but I no longer count the weeks spent in the Swiss mountains, in groups of about twenty young researchers, working on the understanding of this or that mathematical novelty. All this in a simple, relaxed and friendly atmosphere where the very idea of competition was excluded.
His influence at the international level is considerable: one does not count any more his students, the students of his students, etc. Vaughan Jones, one of his former students, was awarded the Fields Medal in 1990. It can be said of André, as was said of Monge, that “he was not satisfied with making discoveries, he also made students, which is sometimes better”.

Pierre Varignon, bridge between math and physics

Do you know Varignon’s theorem? The four middles of the sides of any quadrilateral always form a parallelogram. Did we really have to wait until the beginning of the 18th century to prove such an elementary result, which we sometimes find today in college mathematics textbooks? It is a simple result, and not very interesting, we must admit. It is a kind of curse: mathematicians are often credited with results that do not illustrate their work in any way. Arnold’s theorem even states that no theorem with a proper name is due to this person (and this theorem applies to himself).

Pierre Varignon was a mathematician born in 1654 and died in 1722. A colloquium, organized from January 17 to 19, on the occasion of the tercentenary of his death, allowed us to take a look at this fascinating period in the history of science. It is not about Isaac Newton (1642-1727) or Gottfried Wilhelm Leibniz (1646-1716), whose works have been extensively studied, but about a secondary character who nevertheless played an important role in French mathematics.

Varignon was both a teacher and a researcher, serving also as an intermediary between the great thinkers of his time. The polemic was raging: Newton and Leibniz both claimed the differential calculus, which they presented in very different ways. Varignon played the role of “translator” between the two variants of the same language. His posthumous work Eclaircissemens sur l’analyse des infiniment petits, published in 1725, allowed the introduction of this new differential calculus in France, at the origin of a true scientific revolution.

Premises of the vector calculus

In mechanics, we owe him clear statements on the composition of forces, only glimpsed before by Leonardo da Vinci and Galileo. His book Nouvelle Mécanique ou Statique, the draft of which was given in 1687, contains some admirable plates. We see weights suspended from cables in all sorts of configurations and the conditions of equilibrium are described. With a little imagination, one can guess the premises of the calculation of vectors, so important today, both in mathematics and in physics.

Varignon was perhaps the first professional teacher-researcher in France. He was the first professor of mathematics at the Collège Mazarin, in 1688, in the palace that would later house the Institut de France. He taught there until his death with great interest. His book Elémens de mathématique, published in 1731, contains his teaching and in particular Varignon’s theorem.

He was a true geometer, as testified by the comment of a contemporary who wrote that “he had all the difficulties in the world to say his breviary, because of the habit he had contracted of mathematical figures (…) and that it was necessary even that what he read, to be able to retain it, was susceptible of figures”. The Mazarine Library of the Institute is hosting, until April 15, a remarkable exhibition entitled “Pierre Varignon (1654-1722). Practice and transmission of mathematics at the dawn of the Enlightenment”. A visit not to be missed by lovers of old books, in an exceptional setting.

In 1694, he was elected to the Collège royal, which became the Collège de France. The new professors gave their inaugural lectures, but at the time, they were called “entrance speeches”. Here is the title of Varignon’s lecture (translated from Latin, of course): “On the mutual help that mathematics and physics provide each other: physics is uncertain without mathematics, mathematics is hardly useful without physics”. Here is a title that the writers of today’s school programs could meditate on.

World Cup 2022: geometry and speed of the tournament ball

Millions of viewers are watching the 2022 World Cup matches, but how many have actually watched the ball? It is called Al Rihla, which means “the journey” in Arabic. Mathematicians prefer to call it an icosidodecahedron… from the Greek ico for “twenty”, dode for “twelve” and èdre for “face”. Each Cup is the occasion of a new geometry. In Mexico in 1970, the ball was called Telstar. We all know it with its white and black leather panels.

I asked children from 6 to 12 years old to draw a Telstar ball that was facing them. It’s not easy and some of the drawings are… imaginative. The twelve black pieces are pentagons and the twenty white pieces are hexagons.

Since Greek antiquity, five regular polyhedra have been known, all of whose faces are identical: they play an important role in Plato’s philosophy, each being associated with an “element”. The one that is the most “round” is the icosahedron, with its twenty faces in the shape of equilateral triangles. To make it even rounder, we truncate its twelve vertices, we inflate the whole, and we obtain… the Telstar.

Archimedes, for his part, looked for semi-regular polyhedra, whose faces are still regular polygons, but not necessarily with the same number of sides. He found thirteen, including our icosidodecahedron, with twelve pentagons and twenty triangles. The Adidas engineers simply sewed each of the pentagons with a triangle to obtain Al Rihla. Beyond the aesthetics, it is the symmetries of the object that interest the mathematician and we no longer count the appearances of the icosahedron in contemporary mathematics.

Study of the resistance

The engineer has many other concerns than aesthetics, even if he must not forget it. Symmetries are also important to prevent the balloon from going in uncontrolled directions. The physics of soccer ball flight is complex and requires theoretical and experimental studies.

One of the pioneers was Gustave Eiffel, who was of course more interested in early aviation than in soccer. He began by observing the fall of balls of various sizes from the second floor of “his” tower, before continuing his research in one of the first wind tunnels. In 1912, he discovered a phenomenon he did not believe in at first, which is now called the drag crisis.

When a ball flies, the air exerts a resistance that tends to slow it down. It seems obvious that this force is weaker the lower the speed. Yet, when a ball slows down gradually and a certain speed is reached, there is suddenly a significant increase in resistance. This critical speed depends on the size of the balloon but also on the roughness of its surface. For a Telstar or Al Rihla ball, it is about 10 meters per second. When a player strikes the ball, the initial speed is often much higher than this value before gradually decreasing.

When the ball reaches the critical speed, the resistance suddenly increases and the trajectory seems to break: a phenomenon that goalkeepers know well. For smoother balls, such as the Jabulani of the World Cup in South Africa, the crisis occurs around 14 meters per second: this is one of the reasons why players did not like this ball at all. The Brazilian goalkeeper Julio Cesar said: “It’s terrible, horrible, it looks like one of those balls you buy in the supermarket.” According to striker Robinho: “The guy who designed this ball has certainly never played soccer.”

What will the Blues think of Al Rihla? Their opinion will prevail over that of the physicists.