Seduced by calculus

The 2010 Fields Medal was won by a French mathematician captivated by the crowning mathematical achievement of the Enlightenment. Alex Bellos explains.

Jeffrey Phillips

The French mathematician Cédric Villani is no ordinary looking university professor. Handsome and slender, with a boyish face and a wavy, neck length bob, he looks more like a dandy from the Belle Epoque, or a member of an avant garde student rock band.

He always wears a three-piece suit, starched white collar, lavaliere cravat – the kind folded extravagantly in a giant bow – and a sparkling, tarantula-sized spider brooch. “Somehow I had to do it,” he said of his appearance. “It was instinctive.”

I first met Villani in Hyderabad, India, at the 2010 International Congress of Mathematicians, or ICM, the four-yearly gathering of the tribe. Of the 3,000 delegates, Villani was the focus of most attention, not because he was the most elaborately dressed, but because he received the Fields Medal at the opening gala.

The Fields is the highest honour in maths and is awarded at each ICM to two, three or four mathematicians under the age of 40. The age rule recognises the original motivation behind the prize, which was conceived by the Canadian mathematician J. C. Fields. He wanted not only to recognise work already done, but also to encourage future success. Such is the acclaim afforded by a Fields Medal, however, that since the first two were awarded in 1936, they have helped establish a cult of youth, implying that once you hit 40 you’re past it. This is unfair. Many mathematicians produce their best work after the age of 40, although Fields medallists can struggle to regain focus, since fame brings with it other responsibilities.

Mathematicians gather at the ICM to take stock of their achievements, and the Fields Medal citations provide the clearest snapshot of the most exciting recent work. Unlike the citations for the other three winners in 2010, which were impenetrable to me and even to many of the mathematicians present, Villani’s citation was understandable to the non-specialist. He won “for his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation”.

The Boltzmann equation, devised by the Austrian physicist Ludwig Boltzmann in 1872, concerns the behaviour of particles in a gas, and is one of the best known equations in classical physics. Not only is Villani a devotee of the 19th century’s neckwear, he is also a world authority on its applied mathematics.

The Boltzmann equation is what is known as a partial differential equation, or PDE, and it looks like this:

The equation is written in the vocabulary of calculus. Shortly, I’ll explain the symbols. Calculus was the crowning intellectual achievement of the Enlightenment, and Villani’s Fields Medal demonstrates that it remains a rich area of advanced mathematical study. But before we return to the flamboyantly attired Frenchman, we first need to transport ourselves from southern India in 2010 to Sicily in around the third century BCE.

On the front of the Fields Medal is the bearded portrait of Archimedes, basking in the glow of his reputation as the most illustrious mathematician of antiquity. Archimedes, however, is usually remembered for his contributions to physical science, such as the screw that raises water when turned by hand. Yet Plutarch wrote that geometry was his true love. At bath times “while (his servants) were anointing of him with oils and sweet savours, with his fingers he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry”.

The initial task of geometry was the calculation of area. (According to Herodotus, geometry began as a practice devised by Egyptian tax inspectors to calculate areas of land destroyed by the Nile’s annual floods.) As we all know, the area of a rectangle is the width multiplied by the height, and from this formula we can deduce that the area of a triangle is half the base times the height. The Greeks devised methods to calculate the areas of more complicated shapes. Of these, the most impressive achievement was Archimedes’s “quadrature of the parabola”, by which is meant calculation of the area bounded by a line and a parabola, which is a specific type of U-shaped curve. Archimedes first drew a large triangle inside the parabola, as illustrated below, then on either side of this he drew another triangle. On each of the two sides of these smaller triangles, he drew an even smaller triangle, and so on, such that all three points of each triangle were always on the parabola. The more triangles he drew, the closer and closer their combined area was to the area of the parabolic section. If the process was allowed to carry on forever the infinite number of triangles would perfectly cover the desired area.

The quadrature of the parabola.

Archimedes’ quadrature of the parabola is the most sophisticated example from the classical age of the method of exhaustion, the technique of adding up a sequence of small areas that converge towards a larger one. The proof is considered his finest moment because it represents the first “modern” view of mathematical infinity. Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit. This was important not only for conquering the areas of shapes significantly more exotic than the parabola, but also for starting on the conceptual path towards calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first.

Infinity is a number bigger than any other. It has a twin concept, the infinitesimal, which is a number smaller than any other, yet still larger than zero.

In the 17th century, mathematicians realised how useful the infinitesimal was, even though it was a concept that didn’t make much sense – it was the mathematical equivalent of having your cake and eating it. The infinitesimal was both something and nothing: large enough to be of mathematical use, but small enough to disappear when you needed it to.

Calculating the area of a circle with infinitesimals.

For example, consider the circle illustrated here. Inside is a dodecagon, a 12-sided shape made up of 12 identical triangles sharing a common vertex, or point. The combined area of the triangles is approximately the area of the circle. If I drew a polygon with more sides within the circle, containing more, thinner triangles, their combined area would approximate the circle more closely. And if I kept on increasing the number of sides, in the limit I would have a polygon with an infinite number of sides containing an infinite number of infinitely thin triangles. The area of each triangle is infinitesimal, yet their combined area is the area of the circle, as illustrated below left.

Here's another way the infinitesimal was useful in determining gradients. For readers who have forgotten what a gradient is, it is the measure of the slope, calculated bydividing the distance moved up by the distance moved along. So, in the illustration below right, the gradient of the road is 1/4 because the distance moved up is 100m and the distance along is 400m. Mathematicians, however, wanted to find a method to calculate the gradient of tangents, which are those lines that touch a curve at a single point.

A gradient

the tangent

The trick to finding the gradient of a tangent at point P is to make an approximation of the tangent, and then to improve the approximation until it coincides with the desired line. We do this by drawing a line through P that cuts the curve at nearby point Q, and then we bring Q closer and closer to P. When Q hits P, the line is the tangent.

The gradient of the line through P and Q is ∆y/∆x. (The Greek letter delta, ∆, is a mathematical symbol meaning a small increment). As Q closes in on P, the value ∆y/∆x approaches the gradient of the tangent at P. But we have a problem. If we let Q actually reach P, then ∆y = 0 and ∆x = 0, meaning that the gradient of the curve at P is 0/0. Bad maths alert! The rules of arithmetic prohibit division by zero! The solution is to keep Q at an infinitesimal distance from P. If we do, we can say that when Q becomes infinitesimally close to P, the value ∆y/∆x is infinitesimally close to the gradient of the curve at P.

Approximating a tangent.

In 1665, Isaac Newton, recently graduated from Cambridge, returned to live with his mother in their Lincolnshire farmhouse. The Great Plague was devastating towns across the country. The university had closed down to protect its staff and students. Newton made himself a small study and started to fill a giant jotter he called the Waste Book with mathematical thoughts. Over the next two years the solitary scribbler, undistracted, devised new theorems that became the foundations of the Philosophiæ Naturalis Principia Mathematica, his 1687 treatise that, more than any work before or since, transformed our understanding of the physical universe. The Principia established a system of natural laws that explained why objects, from apples falling off trees to planets orbiting the Sun, move as they do. Yet Newton’s breakthrough in physics required an equally fundamental breakthrough in maths. He formalized the previous half-century’s work on infinity and infinitesimals into a general system with a unified notation. He called it the method of fluxions, but it became better known as the “calculus of infinitesimals’, and now, simply, “calculus’.

A body that moves changes its position, and its speed is the change in position over time. If a body is travelling with a fixed speed, it changes its position by a fixed amount every fixed period. A car with constant speed that covers 60 miles between 4pm and 5pm is travelling at 60 miles per hour. Newton wanted to solve a different problem: how does one calculate the speed of a body that is not travelling at a constant speed? For example, let’s say the car above, rather than travelling consistently at 60mph, is continually slowing down and speeding up because of traffic. One strategy to calculate its speed at, say, 4.30pm, is to consider how far it travels between 4.30pm and 4.31pm, which will give us a distance per minute. (We just need to multiply the distance by 60 to get the value in mph.) But this figure is just the average speed for that minute, not the instantaneous speed at 4.30pm. We could aim for a shorter interval – say, the distance travelled between 4.30pm and 1 second later, which would give us a distance per second. (We’d then multiply by 3,600 to get the value in mph). But again this value is the average for that second. We could aim for smaller and smaller intervals, but we are never going to get the instantaneous speed until the interval is tinier than any other – when it is zero, in other words. But when the interval is zero, the car does not move at all!

This line of reasoning should sound familiar, because I used it two paragraphs ago when explaining how to calculate the gradient of a tangent. To find the gradient we divide an infinitesimally small quantity (length) by another infinitesimally small quantity (another length). To get the instantaneous speed we also divide an infinitesimally small quantity (distance) by another infinitesimally small quantity (time). The problems are mathematically equivalent. Newton’s method of fluxions was a method to calculate gradients, which enabled him to calculate instantaneous speeds.

Calculus allowed Newton to take an equation that determined the position of an object, and from it devise a secondary equation about the object’s instantaneous speed. It also allowed him to take an equation determining the object’s instantaneous speed, and from it devise a secondary equation about position, which, as it turned out, was equivalent to the calculation of areas using infinitesimals! Calculus, therefore, gave him the mathematical tools to develop his laws of motion. In his equations, he called the variables x and y “fluents” and the gradients “fluxions’, written by the “pricked letters” ẋ and ẏ.

When Newton returned to Cambridge after two years avoiding the plague in Lincolnshire, he did not tell anyone about the method of fluxions. On the continent, Gottfried Leibniz was developing an equivalent system. Leibniz was German by birth but a man of the world – a lawyer, diplomat, alchemist, engineer and philosopher. Leibniz was also the mathematician most obsessed with notation. The symbols he used for his system of calculus were clearer than Newton’s, and are the ones we use today.

Leibniz introduced the terms dx and dy for the infinitesimal differences in x and y. The gradient, which is one infinitesimal difference divided by the other, he wrote dy/dx. Thanks to his use of the word “difference’, the calculation of gradient became known as “differentiation’. Leibniz also introduced the distinctive stretched “s’, ∫, as the symbol for the calculation of area. It’s an abbreviation of summa, or sum, since the calculation of area is based on infinite sums of infinitesimals. On the suggestion of his friend Johann Bernoulli, Leibniz called his technique calculus integralis, and the calculation of area became known as “integration’. Leibniz’s ∫ is the most majestic symbol in maths, reminiscent of the f-hole of a cello or violin.

Calculus comprises differentiation (computation of gradient) and integration (computation of area). In general terms, gradient is the rate of change of one quantity over another, and area is the measure of how much one quantity accumulates with respect to another. Calculus thus provided scientists with a way to model quantities that varied in relation to each other. It is a formidable instrument to explain the physical world because everything in the universe, from the tiniest atoms to the largest galaxies, is in a state of permanent flux.

When we know the relationship between two varying quantities, we can describe them in an equation using the symbols for differentiation and integration. An equation in x and y that includes the term dy/dx is called a “simple differential equation’. If there are more than two variables, say x, y and t, the rates of change are written ∂y/∂x, or ∂y/∂t, with the rounded ∂. The equation is called a “partial differential equation’, or PDE, because terms like ∂y/∂x tell us how one variable changes with respect to another one, but not to all of them. PDEs dominate applied mathematics. They allow scientists to make predictions. If we know how two quantities vary over time, then we can predict exactly what state they will be in at any time in the future. Maxwell’s equations, which explain the behaviour of magnetic and electric fields, the Schrödinger equation, which underlies quantum mechanics, and Einstein’s field equations, which are the basis of general relativity, are all PDEs.

The first important PDE described the behaviour of a violin string when bowed, a problem that had tormented scientists for decades. It was discovered in 1746 by Jean le Rond d’Alembert, the celebrity mathematician of his day. D’Alembert, the product of a brief liaison between an artillery general and a lapsed nun, was abandoned after he was born and left on the steps of the church Saint Jean Le Rond, next to Notre-Dame Cathedral in Paris, from which he took his name. Brought up by the wife of a glazier, he rose against the odds to become the permanent secretary of the Académie Française. As well as being a serious mathematician, he was also a vociferous apologist for the values of the Enlightenment. He was a public figure, a sought-after guest at aristocratic salons and one of the editors of the landmark Encyclopédie, for which he wrote the preliminary discourse and more than a thousand articles.

D’Alembert was the prototype French scientific intellectual, a role now occupied with gusto by Cédric Villani.

The second time I met Villani was in Paris. Since 2009 he has been director of the Institut Henri Poincaré, France’s elite maths institute, which is situated among the universities of the Latin Quarter. His office is a comfortable clutter of books, paper, coffee mugs, awards, puzzles and geometrical shapes.

Villani’s appearance was unchanged since we met in India at the International Congress of Mathematicians: burgundy cravat, blue three-piece suit, and a metal spider glistening on his lapel. He said his look emerged when he was in his twenties. He wore shirts with large sleeves, then with lace, then a top hat… “It was like a scientific experiment, and gradually it was ‘this is me’.” And the spider? He enjoys its ambiguity. “Some people think the spider is a maternal symbol. Others think that the web is a symbol for the universe, or that the spider is the big architect of the world, like a way to personify God. Spiders don’t leave people indifferent. You immediately have a reaction.” The spider is an archetype rich with interpretations, I thought, just like mathematics is an abstract language with innumerable applications. Villani’s field is PDEs. Even though PDEs have been around for almost three centuries, he says they are “for a large part still poorly understood. Each PDE seems to have a theory of its own. You have many sub-branches of PDEs with only a small common basis and no general classification. People have tried to classify them, but even the best specialists have failed.” The PDE that has occupied most of Villani’s time is the Boltzmann equation. It was the subject of his PhD and formed part of the subsequent work that led to his Fields Medal. He now views it with tenderness and devotion. “It’s like the first girl you fall in love with,” he confided. “The first equation you see – you think it is the most beautiful in the world.” Feast your eyes on her again:

The Boltzmann equation belongs to the field of statistical mechanics: the branch of mathematical physics that investigates how the behaviour of individual molecules in a cloud of gas influences macroscopic properties like temperature and pressure. The equation describes how a gas disseminates by considering the likelihood of any of its molecules being in any particular spot, with a particular speed, at a particular time. [The f is a “probability density function’, that gives the probability of particles having a position near x and a speed near v at time t.] The model assumes that particles in a gas bounce around according to Newton’s laws, but in random directions, and describes the effects of their collisions using the maths of probability. Villani pointed at the left side of the equation: “This is just particles going in straight lines.” He pointed to the right side of the equation: “And this is just shock. Tik-ding! Ting-dik!” He bumped his fists together several times. “Often in PDEs, you have tension between various terms. The Boltzmann equation is the perfect case study because the terms represent completely different phenomena and also live in completely different mathematical worlds.”

If you filmed a single gas particle bouncing off another gas particle, and showed it to a friend, there is no way he or she would know whether you were playing the film forwards or backwards, since Newton’s laws are time-reversible. But if you filmed a gas spreading from a beaker to its surroundings, a viewer would instantly be able to tell which way the film was being played, since gases do not suck themselves back into beakers. Boltzmann established a mathematical foundation for the apparent contradiction between micro- and macroscopic behaviour by introducing a new concept, entropy. This is the measure of disorder – in theoretical terms the number of possible positions and speeds of the particles at any time. Boltzmann then showed that entropy always increases. Villani’s breakthrough paper concerned just how fast entropy increases before reaching the totally disordered state.

The Boltzmann equation has straightforward applications, such as in aeronautical engineering, to determine what happens to planes when they fly through gases. Its usefulness is what first appealed to Villani when he embarked on his PhD. But as he became more intimate with the equation, its beauty seduced him. He compares it to a Michelangelo sculpture: “Not pure and ethereal and elegant, but very human, very tortured, with the strength of the energy of the world. In the equation you can hear the roar of the particles, full of fury.” He added that he prefers to spend years studying well-known equations, trying to find new insights into them, rather than inventing new concepts. “It’s what I like, and it’s part of a general attitude that says, ‘Hey, guys! High-energy physics, the Higgs boson, string theory or whatever – it may all be fascinating, but remember we still don’t understand Newtonian mechanics.’ There are still many, many open problems.” He showed me a PDE in a book. “Does this equation have smooth solutions? Nobody in hell knows that!” He shrugged his shoulders, his forehead criss-crossed with lines.

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Alex Bellos is the author of Alex’s Adventures in Numberland and Alex Through the Looking Glass. He writes a maths blog for The Guardian.
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Jeffrey Phillips is an illustrator, storyboard artist and graphic designer.