What We Cannot Know: Explorations at the Edge of Knowledge

Born a hundred years after Laplace, Poincaré believed, like his compatriot, in a clockwork universe, a universe governed by mathematical laws and utterly predictable. ‘If we know exactly the laws of nature and the situation of the universe at the initial moment, we can predict exactly the situation of the same universe at a succeeding moment.’

Understanding the world was Poincaré’s prime motivation for doing mathematics. ‘The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law.’

Although Newton’s laws of motion had spawned an array of mathematical equations to describe the evolution of the physical world, most of them were still extremely complicated to solve. Take the equations for a gas. Think of the gas as made up of molecules crashing around like tiny billiard balls, and theoretically the future behaviour of the gas was bound up in Newton’s laws of motion. But the sheer number of balls meant that any exact solution was well beyond reach. Statistical or probabilistic methods were still by far the best tool to understand the behaviour of billions of molecules.

There was one situation where the number of billiard balls was reasonably small and a solution seemed tractable. The solar system. Poincaré became obsessed with the question of predicting what lay in store for our planets as they danced their way into the future.

Because the gravitational pull of a planet on another planet at some distance from the first planet is the same as if all the mass of the planet is concentrated at its centre of gravity, to determine the ultimate fate of the solar system one can consider planets as if they are just points in space, as Newton had done. This means that the evolution of the solar system can be described by three coordinates for each planet that locate the centre of mass in space together with three additional numbers recording the speed in each of the three dimensions of space. The forces acting on the planets are given by the gravitational forces exerted by each of the other planets. With all this information one just needs to apply Newton’s second law to map out the course of the planets into the distant future.

The only trouble is that the maths is still extremely tricky to work out. Newton had solved the behaviour of two planets (or a planet and a sun). They would follow elliptical paths, with their common focal point being the common centre of gravity. This would repeat itself periodically to the end of time. But Newton was stumped when he introduced a third planet. Trying to calculate the behaviour of a solar system consisting, say, of the Sun, the Earth and the Moon seemed simple enough, but already you are facing an equation in 18 variables: 9 for position and 9 for the speed of each planet. Newton conceded that ‘to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind’.

The problem got a boost when King Oscar II of Norway and Sweden decided to mark his sixtieth birthday by offering a prize for solving a problem in mathematics. There are not many monarchs around the world who would choose maths problems as their way to celebrate their birthdays, but Oscar had always enjoyed the subject ever since he had excelled at it when he was a student at Uppsala University.

His majesty Oscar II, wishing to give a fresh proof of his interest in the advancement of mathematical science has resolved to award a prize on January 21, 1889, to an important discovery in the field of higher mathematical analysis. The prize will consist of a gold medal of the eighteenth size bearing his majesty’s image and having a value of a thousand francs, together with the sum of two thousand five hundred crowns.

Three eminent mathematicians convened to choose a number of suitable mathematical challenges and to judge the entries. One of the questions they posed was to establish mathematically whether the solar system was stable. Would it continue turning like clockwork, or, at some point in the future, might the Earth spiral off into space and disappear from our solar system?

To answer the question required solving the equations that had stumped Newton. Poincaré believed that he had the skills to win the prize. One of the common tricks used by mathematicians is to attempt a simplified version of the problem first to see if that is tractable. So Poincaré started with the problem of three bodies. This was still far too difficult, so he decided to simplify the problem further. Instead of the Sun, Earth and Moon, why not try to understand two planets and a speck of dust? The two planets won’t be affected by the dust particle, so he could assume, thanks to Newton’s solution, that they just repeated ellipses round each other. The speck of dust, on the other hand, would experience the gravitational force of the two planets. Poincaré set about trying to describe the path traced by the speck of dust. Some understanding of this trajectory would form an interesting contribution to the problem.

Although he couldn’t crack the problem completely, the paper he submitted was more than good enough to secure King Oscar’s prize. He’d managed to prove the existence of an interesting class of paths that would repeat themselves, so-called periodic paths. Periodic orbits were by their nature stable because they would repeat themselves over and over, like the ellipses that two planets would be guaranteed to execute.

The French authorities were very excited that the award had gone to one of their own. The nineteenth century had seen Germany steal a march on French mathematics, so the French academicians excitedly heralded Poincaré’s win as proof of a resurgence of French mathematics. Gaston Darboux, the permanent secretary of the French Academy of Sciences, declared:

From that moment on the name of Henri Poincaré became known to the public, who then became accustomed to regarding our colleague no longer as a mathematician of particular promise but as a great scholar of whom France has the right to be proud.

A SMALL MISTAKE WITH BIG IMPLICATIONS

Preparations began to publish Poincaré’s solution in a special edition of the Royal Swedish Academy of Science’s journal Acta Mathematica. Then came the moment every mathematician dreads. Every mathematician’s worst nightmare. Poincaré thought his work was safe. He’d checked every step in the proof. Just before publication, one of the editors of the journal raised a question over one of the steps in his mathematical argument.

Poincaré had assumed that a small change in the positions of the planets, a little rounding up or down here or there, was acceptable as it would result in only a small change in their predicted orbits. It seemed a fair assumption. But there was no justification given for why this would be so. And in a mathematical proof, every step, every assumption, must be backed up by rigorous mathematical logic.

The editor wrote to Poincaré for some clarification on this gap in the proof. But as Poincaré tried to justify this step, he realized he’d made a serious mistake. He wrote to Gösta Mittag-Leffler, the head of the prize committee, hoping to limit the damage to his reputation:

The consequences of this error are more serious than I first thought. I will not conceal from you the distress this discovery has caused me … I do not know if you will still think that the results which remain deserve the great reward you have given them. (In any case, I can do no more than to confess my confusion to a friend as loyal as you.) I will write to you at length when I can see things more clearly.

Mittag-Leffler decided he needed to inform the other judges:

Poincaré’s memoir is of such a rare depth and power of invention, it will certainly open up a new scientific era from the point of view of analysis and its consequences for astronomy. But greatly extended explanations will be necessary and at the moment I am asking the distinguished author to enlighten me on several important points.

As Poincaré struggled away he soon saw that he was simply mistaken. Even a small change in the initial conditions could result in wildly different orbits. He couldn’t make the approximation that he’d proposed. His assumption was wrong.

Poincaré telegraphed Mittag-Leffler to break the bad news and tried to stop the paper from being printed. Embarrassed, he wrote:

It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible.

Mittag-Leffler was ‘extremely perplexed’ to hear the news.

It is not that I doubt that your memoir will be in any case regarded as a work of genius by the majority of geometers and that it will be the departure point for all future efforts in celestial mechanics. Don’t therefore think that I regret the prize … But here is the worst of it. Your letter arrived too late and the memoir has already been distributed.

Mittag-Leffler’s reputation was on the line for not having picked up the error before they’d publicly awarded Poincaré the prize. This was not the way to celebrate his monarch’s birthday! ‘Please don’t say a word of this lamentable story to anyone. I’ll give you all the details tomorrow.’

The next few weeks were spent trying to retrieve the printed copies without raising suspicion. Mittag-Leffler suggested that Poincaré should pay for the printing of the original version. Poincaré, who was mortified, agreed, even though the bill came to over 3500 crowns, 1000 crowns more than the prize he’d originally won.

In an attempt to rectify the situation, Poincaré set about trying to sort out his mistake, to understand where and why he had gone wrong. In 1890, Poincaré wrote a second, extended paper explaining his belief that very small changes could cause an apparently stable system suddenly to fly apart.

What Poincaré discovered, thanks to his error, led to one of the most important mathematical concepts of the last century: chaos. It was a discovery that places huge limits on what we humans can know. I may have written down all the equations for my dice, but what if my dice behaves like the planets in the solar system? According to Poincaré’s discovery, if I make just one small error in recording the starting location of the dice, that error could expand into a large difference in the outcome of the dice by the time it comes to rest on the table. So is the future of my Vegas dice shrouded behind the mathematics of chaos?

The chaotic path mapped out by a single planet orbiting two suns.

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