witnessing the limits of the subject itself, which is what true philosophy is all about. I love to tick off French literary intellectuals by naming Poincare as my favorite French philosopher.
I am looking at Poincare’s picture. He was a bearded, portly and imposing, well-educated patrician gentleman of the French Third Republic, a man who lived and breathed general science, looked deep into his subject, and had an astonishing breadth of knowledge. He was part of the class of mandarins that gained respectability in the late nineteenth century: upper middle class, powerful, but not exceedingly rich. His father was a doctor and professor of medicine, his uncle was a prominent scientist and administrator, and his cousin Raymond became a president of the republic of France. These were the days when the grandchildren of businessmen and wealthy landowners headed for the intellectual professions.
However, I can hardly imagine him on a T-shirt, or sticking out his tongue like in that famous picture of Einstein. There is something non-playful about him, a Third Republic style of dignity.
In his day, Poincare was thought to be the king of mathematics and science, except of course by a few narrow-minded mathematicians like Charles Hermite who considered him too intuitive, too intellectual, or too “hand-waving”. When mathematicians say “hand-waving”, disparagingly, about someone’s work, it means that the person has: a) insight, b) realism, c) something to say, and it means that d) he is right because that’s what critics say when they can’t find anything more negative. A nod from Poincare made or broke a career. Many claim that Poincare figured out relativity before Einstein – and that Einstein got the idea from him – but that he did not make a big deal out of it. These claims are naturally made by the French, but there seems to be some validation from Einstein’s friend and biographer Abraham Pais. Poincare was too aristocratic in both background and demeanor to complain about the ownership of a result.
Poincare is central to this chapter because he lived in an age when we had made extremely rapid intellectual progress in the fields of prediction – think of celestial mechanics. The scientific revolution made us feel that we were in possession of tools that would allow us to grasp the future. Uncertainty was gone. The universe was like a clock and, by studying the movements of the pieces, we could project into the future. It was only a matter of writing down the right models and having the engineers do the calculations. The future was a mere extension of our technological certainties.
Poincare was the first known big-gun mathematician to understand and explain that there are fundamental limits to our equations. He introduced nonlinearities, small effects that can lead to severe consequences, an idea that later became popular, perhaps a bit too popular, as chaos theory. What’s so poisonous about this popularity? Because Poincare’s entire point is about the limits that nonlinearities put on forecasting; they are not an invitation to use mathematical techniques to make extended forecasts. Mathematics can show us its own limits rather clearly.
There is (as usual) an element of the unexpected in this story. Poincare initially responded to a competition organized by the mathematician Gosta Mittag-Leffer to celebrate the sixtieth birthday of King Oscar of Sweden. Poincare’s memoir, which was about the stability of the solar system, won the prize that was then the highest scientific honor (as these were the happy days before the Nobel Prize). A problem arose, however, when a mathematical editor checking the memoir before publication realized that there was a calculation error, and that, after consideration, it led to the opposite conclusion – unpredictability, or, more technically, nonintegrability. The memoir was discreetly pulled and reissued about a year later.
Poincare’s reasoning was simple: as you project into the future you may need an increasing amount of precision about the dynamics of the process that you are modeling, since your error rate grows very rapidly. The problem is that near precision is not possible since the degradation of your forecast compounds abruptly – you would eventually need to figure out the past with infinite precision. Poincare showed this in a very simple case, famously known as the “three body problem”. If you have only two planets in a solar-style system, with nothing else affecting their course, then you may be able to indefinitely predict the behavior of these planets, no sweat. But add a third body, say a comet, ever so small, between the planets. Initially the third body will cause no drift, no impact; later, with time, its effects on the two other bodies may become explosive. Small differences in where this tiny body is located will eventually dictate the future of the behemoth planets.
Explosive forecasting difficulty comes from complicating the mechanics, ever so slightly. Our world, unfortunately, is far more complicated than the three body problem; it contains far more than three objects. We are dealing with what is now called a dynamical system – and the world, we will see, is a little too much of a dynamical system.
Think of the difficulty in forecasting in terms of branches growing out of a tree; at every fork we have a multiplication of new branches. To see how our intuitions about these nonlinear multiplicative effects are rather weak, consider this story about the chessboard. The inventor of the chessboard requested the following compensation: one grain of rice for the first square, two for the second, four for the third, eight, then sixteen, and so on, doubling every time, sixty-four times. The king granted this request, thinking that the inventor was asking for a pittance – but he soon realized that he was outsmarted. The amount of rice exceeded all possible grain reserves!
This multiplicative difficulty leading to the need for greater and greater precision in assumptions can be illustrated with the following simple exercise concerning the prediction of the movements of billiard balls on a table. I use the example as computed by the mathematician Michael Berry. If you know a set of basic parameters concerning the ball at rest, can compute the resistance of the table (quite elementary), and can gauge the strength of the impact, then it is rather easy to predict what would happen at the first hit. The second impact becomes more complicated, but possible; you need to be more careful about your knowledge of the initial states, and more precision is called for. The problem is that to correctly compute the ninth impact, you need to take into account the gravitational pull of someone standing next to the table (modestly, Berry’s computations use a weight of less than 150 pounds). And to compute the fifty-sixth impact, every single elementary particle of the universe needs to be present in your assumptions! An electron at the edge of the universe, separated from us by 10 billion light-years, must figure in the calculations, since it exerts a meaningful effect on the outcome. Now, consider the additional burden of having to incorporate predictions about
FIGURE 2: PRECISION AND FORECASTING
One of the readers of a draft of this book, David Cowan, gracefully drew this picture of scattering, which shows how, at the second bounce, variations in the initial conditions can lead to extremely divergent results. As the initial imprecision in the angle is multiplied, every additional bounce will be further magnified. This causes a severe multiplicative effect where the error grows out disproportionately.
Note that this billiard-ball story assumes a plain and simple world; it does not even take into account these crazy social matters possibly endowed with free will. Billiard balls do not have a mind of their own. Nor does our example take into account relativity and quantum effects. Nor did we use the notion (often invoked by phonies) called the “uncertainty principle”. We are not concerned with the limitations of the precision in measurements done at the subatomic level. We are just dealing with billiard balls!
In a dynamical system, where you are considering more than a ball on its own, where trajectories in a way depend on one another, the ability to project into the future is not just reduced, but is subjected to a fundamental limitation. Poincare proposed that we can only work with qualitative matters – some property of systems can be
