Predictability, reproducibility and determinism in chaos

On a Peaceful Science thread I promised Chris Falter that I’d respond to his argument that chaotic systems are instrinsically indeterminate. The context, of course, as the thread title shows (Every Birth is a Statistical Impossibilty) has to do with the possibility of determination of events by God, as well as by us.

This distinction is crucial, because the presence of agency alters probability: a thrown ball that you calculate has a 0% chance of hitting a mark on the wall can have that impossibility transformed into a certainty by my catching the ball and walking with it to the spot, and the same can presumably be true of God, prejudices about whether he would being beside the point.

The immediate question, then, is whether chaotic systems are physically indeterminate, so that God could not predict their physical outcomes (though he might somehow alter them, if he chooses). All chaotic systems are, by definition, unpredictable and non-reproducible for humans. As Chris rightly says:

Because it requires energy to gather information, it could be that not enough energy is available to gather the data required to make a deterministic prediction.

The point can even be made more general than the question of energy – it takes resources of exact information, calculation and time to predict a chaotic system, in which small changes produce highly variable outcomes. Nobody can possibly muster those resources, even in theory. A classic example of this is the gravitational 3 body problem:

The rules of the game are Newton’s laws of motion, but the results are so dependent on exact initial conditions that one can only simulate it for one given instance. Our solar system being “on the edge of chaos,” rather than unstable as is the one in the video, multiple simulations based on the limitations of accurate measurement produce results showing a statistical likelihood that it will remain stable for the next billion years … except in 1% of cases where Mercury trashes the inner planets.

Incidentally, if chaotic systems were truly indeterminate, the usual argument that Isaac Newton used an invalid God of the Gaps argument is demolished. Newton predicted that God might have to adjust the course of the planets at some stage to prevent collisons, and Leibniz retorted that a competent God would use “a perpetual motion.” But the simulations show that a truly indeterminate solar system might, as Newton said, need to be adjusted by God to guarantee its survival. All this really shows is one reason for the poverty of the “God of the Gaps” accusation (especially when Newton is the villain) – but in fact, there is no reason for chaos to be physically indeterminate, as I will show from a relatively simple example – that of billiards.

Billiards is, in terms of its physics, surprisingly rich. But it seems that factoring out gross human imprecision, balls that are not perfect spheres, the losses due to friction and so on, even an idealised system of serial collisions is impossible to predict after, maybe, half-a dozen collisions.

The reason is easy to see: even the minutest variation in two spheres colliding will slightly change the angular momentum, and that will be exaggerated with each collision. I’ve been unable to find a neat video of that, but the effect can be seen from this simple simulation (take it from the 1 minute mark):

All that is needed to destroy the predictability of the result is a small change in 1 initial parameter, and mathematics and determinate law does the rest. The algorithm has actually predicted the result (of course, or the trace could not be made), but only by simulating a demon who knows the exact initial conditions. That is what the limitations of human measurement cannot do in a real situation, in which the individual outcome would therefore be unpredictable and non-reproducible, but entirely determinate, from the scientific point of view.

In my own “multiple hits” scenario, understanding is helped by breaking down the situation of 6 or more “cannons” into small stages. Just as a decent player could reproduce the first two or three stages of the progression quite nicely, he could also, starting from any point of a previous trace, reproduce the next two or three stages. Each “slice” of the action is to all intents and purposes predictable and reproducible. So how does the chaos get into the system as a whole? Simply by the multiplication of the miniscule errors at each stage. At no stage can any indeterminate cause be seen.

The important point is that nothing occurs that isn’t entirely comprehensible by classically exact physics, though in a real situation, additional imprecision is introduced by factors like the physiological inexactness of the human player, variations in the cue, balls and table, and even (so I’m told) the effects of the gravity not only or the earth but, after a few shots, the players and spectators around the table.

That last factor sounds dubious to me, but if my source is correct, gravity in any case would only become relevant because of the basic chaotic system of spheres rebounding off spheres, as already mentioned. What is more, all these factors, though unmeasurable humanly, are accountable in terms of known laws that, as far as we have any reason to know, are precise and constant. The system we have chosen is physically determinate, in other words.

No indeterminate quantum effects get a look in – though there is something in this scenario that, according to Werner Heisenberg, may be what also determines quantum “indeterminacy.” For the unpredictability of the real billiard table is largely due to the fact that it is not an idealised mathematical abstraction in which balls are points, tables are perfectly elastic and so on, but is a contingent reality affected by everything in the real world about it. Chaotic systems are non-reproducible because the real world is not reproducible.

It was Heisenberg’s view that quantum events are similarly unpredictable because, inevitably, they are always enmeshed in an experimental situation that involves the whole world, which cannot be controlled. I’m not aware that his view has been refuted.

This brings us to God’s ability to “predict” chaotic events. The whole problem with this concept is that it puts him in an analogous position to us, as we try to use one event to predict a second, similar, event. For example, we input measured data we obtain from the real solar system up to now, to run simulations which create, as it were, multiple new possible worlds. Lo and behold, we find that all of these worlds are different from each other, and we have no way of knowing which, if any, will correspond to the real solar system in the future, except statistically (thus putting a useful figure on our ignorance).

But God is not in the business of predicting possible worlds. He is creating just one real world, the world in which the solar system does what it will do – indeed, what he decides it will do. Speaking in terms of his power, he has only to determine exactly what the initial conditions will be, and the governing laws – he does not have to measure those conditions to some degree of accuracy. This does not, in any way, prevent him from showing forth his power, responding providentially to human actions, or directly acting in whatever manner he chooses to affect the outcome. But he does not have to do so because of inherent physical unpredictability.

I noted in the Peaceful Science thread that a surprising number of natural systems (the solar system, the weather, life) are in the special situation of being one the edge of chaos. Stable systems strongly resist change, and totally chaotic systems strongly resist control. But if one wanted to create a world that can be affected by its creatures, or governed peacefully by the Creator, we have just such a world.

(A theological note here, on that point: classical theology describes how God, as the sole cause of all things, knows all things by knowing himself as the creator. What he knows, he sustains in being. That’s why, in classical theology, the whole idea of causes unknown to God is incoherent. Another way of regarding this, drawing on Berkeley, is that what is real is only what is held in the mind of God – or perhaps, in Christian terms, what is spoken from his mind by his Logos, Christ.)

But if, abandoning good theology, we reduced God a sub-Christian deity like Laplace’s demon, just knowing all events accurately, then even so what we call chaotic events, which are in his eyes just single real events unfolding, not multiple mathematical ones, require him only to know the system exactly. Will we say that God is not able to know the exact angles, velocities, spins, gravitation effects and so on of a billiard shot, and so determine that event in order to produce a specific outcome, just as the algorithm on the video clip was able to determine the result?

Does he bother with such “micromanagement” though, some will ask? Given the nature of chaotic systems, you’d better hope he does. Chaos is not about the capsule landing off target, but about missing the world.

Chaos, then, is not a physical source of indeterminacy, but only of the epistemological uncertainty of predictability and reproducibility, neither of which affect God who is Creator, not Predictor, and of one reality, not of mathematical abstractions to calculate the future.

Personally (like Heisenberg) I believe that quantum events, too, are similarly determined by the world as a whole – but even if not, as we have often discussed, there seems as yet no good evidence that they are not simply averaged out in the classical world.

Accordingly, I have seen no evidence that any category of events in this world are caused by chance, in the sense of Epicurean indeterminacy. Chaotic systems do not need indeterminacy to work as they do, and nobody seems to have produced evidence that there are exceptions to this. And if quantum events were, strictly speaking, physically indeterminate, no clear evidence exists that it matters in the classical world.

On the other hand, that is not to say in any way that the events of the world are determined rigidly by physical laws. What actually matters is that they are determined by the choices of minds, for not only is God free to choose the outcomes of billiards, solar systems or anything else, but so are we, within our physical limits. What will happen in a complex billiard shot may be somewhat beyond our conscious control, and so deemed “chaotic.” But the truth is that the outcomes are not the result of some mystical “indeterminacy,” but of our choices and actions as billiards players.

Banish ontological chance, and you affirm not only divine freedom, but human freedom – rational human freedom. I like that, a lot more than I like “the slings and arrows of outrageous fortune” indeterminate even to God.

The world under “light control” management:

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About Jon Garvey

Training in medicine (which was my career), social psychology and theology. Interests in most things, but especially the science-faith interface. The rest of my time, though, is spent writing, playing and recording music.
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