Chaotic Musings

Chaos has deep roots — it is said that in the beginning, there was chaos and some god or another wrested a semblance of order from that primal chaos. But being the stuff that matter is made out of, chaos couldn’t really be exiled, and she hid herself in the interstices. And hid herself well.

While theologians and philosophers banished chaos from their utopias of thought, chaos hid herself deep inside the order of physical laws.

One of the leaders of natural philosophy — that olden name of physics — felt bold enough to state:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past could be present before its eyes.

— Pierre Simon Laplace, A Philosophical Essay on Probabilities (1814)

But is such an ‘intellect’ possible? A version of Douglas Adams’ Deep Thought that can compute a billiard game the size of the universe in its mind? Standing on the shoulders of Newton, the giant, Laplace thought so. It turns out that there are extremely simple systems where chaos rears its unexpected head to ruinous effect for Laplace’s dream of ultimate determinism.

Numbers that represent physical quantities are rarely known exactly or even knowable. In physics, the meaning of a number is contextual. The universe is 13.7 billion years old, cosmologists claim, give or take 200 million years, and so the last eight zeros in 13,700,000,000 mean that we don’t know exactly what those digits really are. They are simply placeholders for our ignorance. More experiments may improve our knowledge of the universe’s age, but we can never know the exact age of the universe.

Every number in the universe that represents some observable quantity has some uncertainty associated with it. Similarly, when we measure the fine-structure constant, which is a measure of the strength of the electromagnetic interaction, it turns out to be known extremely precisely: 0.0072973525693(11), where the notation means that the last two digits are uncertain to +/- 11. But we don’t know what it is to infinite precision.

(What complicates things even further is that, according to quantum field theory, the fine structure constant changes depending on the energy scale at which it is measured. It is supposed to ‘run’ with the energy scale of the process involved.)

And it is unknowable with infinite precision. Most numbers, from the realm of pure mathematics, are also unknowable, although they are known as uncomputable numbers. The numbers that we can write down precisely are the rational numbers, numbers that can be expressed as a/b for a and b whole numbers. Irrational numbers, which are numbers with non-repeating patters after the decimal point, are simply infinitely more common than rational numbers. This means that if we were to throw a dart at the set of all numbers the probability that the dart would hit a rational number was zero and an irrational number one.

But the strangness continues… it turns out that most real numbers are in principle uncomputable, meaning there are no algorithms that can compute them. Thus the probabilty of the description of all the particles of the universe at any given point with exact accurcy is zero.

So what happens when we accept that, as humans, we can’t measure with exactitude the numbers that describe reality? Laplace would say that we are not his ideal intellect, the Demon of Laplace, but we can still have approximate control over the prediction of the universe’s future. And Laplace would be correct, but there remains the possibility that the Daemon may exist, and therefore, all have been preordained. We may not know the future with perfect certainty, but the Daemon will.

[Is the Daemon God? No, by definition, God is beyond the laws of physics, but the Daemon is not. This will have significant consequences for our musings.]

The character I would like to introduce next in this ahistoric history of chaos is Alan Turing, who first formalized the nature of a computer. Turing’s computer, strangely called a Turing machine, is a simple device with a long, perhaps indefinitely long tape on which it can jot down or replace 0s and 1s. It is programmable, and it can simulate any computing device. In other words, it is a universal computer. The paper in which Turing introduced his machine is curiously titled ‘On Computable Numbers.’ With his thought machine, Turing proved a version of a famous result that says that mathematics is incomplete. But the essential moral for us is that Turing showed that there are numbers that are not computable. There are no algorithms that can tell us what those numbers are. These numbers are unknowable — and, this is the kicker, the vast majority of the inifinity of numbers are not computable.

So what Turing is telling us is that there are numbers not even Laplace’s Daemon can store in its memory or its Turing machine. So the best we can hope for is pitifully limited human beings or some higher-order improvement (say, a Macbook Pro with an M3 chipset).

The Poincaré Paradox

Henri Poincaré was a Renaissance man. A philosopher, a mathematician and a physicist, he mused about generalizing the Kepler problem. Kepler, who was an ardent believer in the beauty of mathematics, found out, to his disappointment, that the orbits of the planets of the solar system do not move on spheres that are shells for nested Platonic solids but that they move on certaintly-not-so-beautiful-to-behold-at-first-sight ellipses. Kepler formulated his empirical laws about the motion of the planets based on the data extracted from the original party animal astronomer Tycho Brahe. The Kepler problem explains why planets move on ellipses around the sun, with their orbits tracing out constant areas from the sun.

Sir Isaac Newton, who came after Kepler, found an elegant solution to that problem by marrying his famous laws of motion and his equally famous law of universal gravitation. He showed that two astronomical bodies must orbit each other so that the relative motion of these two bodies must trace out an ellipse around their centre of mass. When one of the bodies is much heavier than the other, as in the case of the earth and the sun, the centre of mass lies deep inside the heavier object, giving the impression that the lighter body is travelling in an elliptical orbit.

It is a problem that is ‘integrable,’ meaning it has a closed-form solution. The solution is an ellipse. Poincaré wanted to know if there is an integrable solution to the three-body problem, I.e., the general solution to the problem of three astronomical bodies moving under their mutual gravitational interaction. Although Poicaré and others found solutions to the three-body problem for very special initial conditions, no general solutions were found. A Finish mathematician found a formal series solution, but it is so slowly converging that it is utterly useless.

Poincaré found, after much denial, that the trajectory of the three-body problem is highly sensitive to the initial conditions of the three particles. If you change the position or momentum of one of the initial bodies even slightly, the evolution of the three bodies changes radically after a short time. Not only that, but Poincaré could not find an integrable solution for the general problem, that is, the problem in which the three particles have arbitrary positions and momenta. [Footnote: An integrable solution does exist, but in the form of a very slowly converging series that makes the solution useless.]

This has extremely important real-world implications. As we know, our solar system is not a two-body problem, so the orbits of the planets we observe around the sun may not be predictable and stable. This sensitivity to initial conditions is an important aspect of what we now call chaos: Many systems exist in which just a tiny variation in the initial conditions results in very different states of the system as time progresses. This is known as the butterfly effect.

Now, we come back to the Daemon. As we now know, there is no way to store the vast infinity of uncomputable numbers possibly. If we arbitrarily choose the initial conditions of a physical problem, it is almost certainly possible that the ‘real’ initial condition will be an uncomputable number. Thus, the best we can do is take an approximation to it. This, in a chaotic universe, means that our predictive model will break down after a while.