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The Predictability of Chaos

By Robin Craig

May 7, 2015

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The enthusiasm with which mystics seize on any alleged flaw [of Science] indicates how much they value human beings.

Understanding reality is vital for human life and happiness. Science, a systematization of our basic inquiry method (the interplay of senses, memory, reason, and bodily action), is our most powerful system for discovering the facts of reality. Since science has improved human life so remarkably, one would expect its demise to be depressing. So the enthusiasm with which mystics seize on any alleged flaw indicates how much they value human beings.

In  Holes in the Heart of Reason, I showed that attacks on reason and science from quantum physics and Gödel’s theorem are empty. But attacks from within do not stop there:

“And now chaos theory proves that unpredictability is built into our daily lives. It is as mundane as the rainstorm we cannot predict. And so the grand vision of science, hundreds of years old – the dream of total control – has died, in our century. And with it much of the justification, the rationale for science to do what it does. And for us to listen to it. Science has always said that it may not know everything now but it will know, eventually. But now we see that isn’t true. It is an idle boast. As foolish, and as misguided, as the child who jumps off a building because he believes he can fly.” (Michael Crichton, Jurassic Park).

So what is the truth about this remarkable chaos theory?

Fundamentals of Chaos

Chaos theory is a relatively new branch of mathematics concerned with complex and “unpredictable” systems. Curiously, its principles are actually the opposite of chaos’ normal meaning of “utterly without order or arrangement,”.I suppose that, as with “catastrophe theory” before it, a snappy and provocative name was a PR must! Chaos theory is wide-ranging, but has three basic features:

  1. Complex systems that are apparently random and unpredictable, can have an underlying order. Chaos theory can be used to determine such underlying rules. Some degree of prediction, and/or quantifying the degree of unpredictability, may then be possible.
  2. Conversely, simple rules can lead to complex systems. The most famous examples are the Mandelbrot sets of fractal geometry, in which simple basic formulae generate strikingly complex and beautiful patterns, repeated in an infinite progression of scales.
  3. Amplification of tiny effects can lead to major changes. This is often called the “butterfly effect.” The term is from computer models of weather, supposedly so sensitive to initial conditions that the outcome might be changed by the flapping of a butterfly’s wings.

The Limits of Chaos

It was not chaos theory which designed aeroplanes and put men on the moon.

Clearly, chaos mathematics applies to a particular set of systems: not life, the Universe, and everything. Non-chaos science and mathematics remain of major importance in understanding reality, from the subatomic to the cosmic scales. It was not chaos theory which designed aeroplanes and put men on the moon. Indeed, if chaos theory only concerns itself with the truly unpredictable, it admits that the job of improving human life belongs to non-chaos science!

Chaos theory occupies a middle ground. Some systems are amenable to direct calculation, such as the courses of the planets. Others are extremely complex, but system behaviour can be described from the statistical average of the components, as with the laws of thermodynamics and gases. Chaos theory is concerned with turbulent systems like the weather, whose behaviour is dependent on the detailed interactions of their components, but these are too complex for direct computation. Of course, as computers get better, more things shift from the chaotic to the calculable.

Chaos and Reality

To what extent does chaos mathematics have anything to do with the real world? Does it have any use besides making pretty pictures and giving science journalists something to write about? Let’s look at the three aspects noted above.

  1. It seems obvious that many apparently chaotic systems will have an underlying logic. If chaos theory can provide a new way of understanding such systems, whether predicting how they will evolve, calculating the range of possible outcomes, or merely quantifying probabilities, then it is both useful and testable.
  2. Fractal geometry may explain many patterns in nature. At least, those patterns can be generated by fractal rules. However, note that they are not truly fractal: the scales involved are limited, not infinite, in range. Nevertheless, patterns from crystal surface structures to animal coloration appear to be fractal over a certain range, and fractal geometry therefore appears to be a useful new perspective for studying such patterns, and discovering the rules which generate them. Fractals are more predictive and deterministic than the other chaos concepts.
  3. One must approach the “butterfly effect” with some caution. In many areas of science, a computer model which is too sensitive to initial conditions is simply regarded as a bad model (it can’t make any firm predictions)! However, this only applies to models of non-chaotic systems, and I am sure that there is some truth to the butterfly effect. What must be determined is exactly how much the mathematics describes reality: as with fractals, the real world is not as infinitely divisible as the maths.

Butterfly Power

To evaluate the butterfly effect, we need to consider a system’s inherent resistance to change. Resistance can be passive (inertia, friction, and buffering, named by analogy to physical qualities) or active (feedback control). These are all interrelated.

System inertia measures the amount of effort needed to change its state: a rock in a hollow has a much greater system inertia than a rock balanced on a pinnacle. System friction is its tendency to stop changing. Buffering is its tendency to “soak up” any disturbances.

Feedback may be negative (feedback inhibition) or positive (runaway effect). In the former, a change induces other changes which tend to push the system back to its initial state. In the latter, the induced changes tend to push the system even further away from its initial state.

The butterfly effect is simply one end of a continuum. Any system, even a very stable one, can be altered by a large enough catastrophe. The butterfly effect is merely the result of the system being so unstable that the required “catastrophe” is very small.

Thus, the butterfly effect can be constrained: any passive resistance restrains and dissipates it, and any negative feedback actively fights it. Basically, the effect is significant only if there is minimal inherent passive resistance, and if feedback is either absent or positive: that is, the system is unstable. Systems in the real world will often resist change to some degree, and will usually have feedback effects. So what must be determined for any system where a butterfly effect is suspected, is what the minimum size of the butterfly is! If it is the size of a whale, then the system will be much more amenable to prediction. Claiming that the butterfly effect prevents prediction, in the absence of hard data on system stability, is vacuous. It is a quantitative effect, not a magic charm: it does not prevent prediction absolutely, it simply limits how far ahead you can predict. All systems can be predicted some distance into the future, with confidence decreasing the further ahead you look. The smaller the “minimum effective butterfly,” the more rapidly your confidence falls off, so the less time ahead you can predict with any assurance of accuracy.

Thus, the butterfly effect is simply one end of a continuum. Any system, even a very stable one, can be altered by a large enough catastrophe. The butterfly effect is merely the result of the system being so unstable that the required “catastrophe” is very small.

Prediction and Control

There are two ways in which knowledge can benefit us. If we learn enough to predict what will happen, we can act accordingly. If we learn enough to control events, then we are even better off (given a touch of wisdom!). How does chaos theory affect these goals?

Chaos theory indicates that many complex systems cannot be predicted with any confidence beyond a certain point. We don’t really need chaos theory to tell us that, of course. Chaos theory’s main interest may well lie in quantifying this: to calculate, for any given system, how far into the future we can predict it with a given level of confidence. Or, from another angle, to calculate how fine the detail we must measure in order to predict a useful time ahead.

Control is another matter. Perhaps surprisingly, control of chaotic systems is often much easier than prediction. Predicting the exact course of a driverless car moving through a level field would be extremely difficult. The car’s aerodynamics, mass distribution, suspension, tyres and steering system quirks; the field’s exact topography and surfaces; wind direction and speed; would all have to be taken into account. Yet to steer it where you want is childishly simple. This is the butterfly effect again: small variations can add up to major changes in the final state of an unstable system; but add a little negative feedback (by a driver), or system inertia (make it a train on rails), and it takes major events to disturb the system in any significant way.

Jumping several orders of complexity, precise prediction of the course of natural evolution of living things is impossible, but humans have been able to influence that evolution in desired ways from long before the laws of evolution were understood.

Indeed, the more unstable the system, the (potentially) more controllable it can be: simply because such little force is required to change its state. Hence, with the advent of small, fast computers, high-performance jet aircraft are actually designed to be very unstable: with the computer providing feedback control as required, the instability translates into extremely good manoeuvrability!

Chaos and Science

Unpredictable systems almost certainly exist in reality. I find it hard to believe that, 800 million years ago, an alien race could have predicted the details of life on Earth today. The potential value of chaos theory lies in quantifying such things: determining the possible outcomes, and calculating how much detail we need to predict how far ahead with what confidence, the size of disturbances which would derail the predictions, and the precise amount of data and speed of calculation required to control any given system. Thus, we could develop a powerful science of prediction and control. Whether chaos theory is up to this challenge remains to be seen, but it is rather overrated if it isn’t!

To see in this (Chaos Theory) the end of science is absurd.

The purpose of science is to understand the natural world: its value does not depend on being able to predict and control everything. For the sake of human life, we do need to predict, and we do need to control: and through science, we can predict and control many things. If reality also contains unpredictable systems, then that is what is: it is not a flaw in science per se, it is something about reality that we need to know so we can act accordingly. Indeed, chaos theory shows that even in those cases, a significant degree of understanding and control can be possible. To see in this the end of science is absurd.

 

 

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