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 Chaos frees the Universe

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PostSubject: Chaos frees the Universe   Thu Mar 19, 2009 9:35 am

I thought this was an interesting take on the question of "free-will".

Quote :
Chaos seems to provide a bridge between the deterministic laws of physics and the laws of chance, implying that the Universe is genuinely creative and that the notion of free will is real.

by Paul Davies

All science is founded on the assumption that the physical world is ordered. The most powerful expression of this order is found in the laws of physics. Nobody knows where these laws come from, nor why they apparently operate universally and unfailingly, but we see them at work all around us: in the rhythm of night and day, the pattern of planetary motions, the regular ticking of a clock.

The ordered dependability of nature is not however, ubiquitous. The vagaries of the weather, the devastation of an earthquake, or the fall of a meteorite seem to be arbitrary and fortuitous. Small wonder that our ancestors attributed these events to the moodiness of the gods. But how are we to reconcile these apparently random "acts of God" with the supposed underlying lawfulness of the Universe?

The ancient Greek philosophers regarded the world as a battleground between the forces of order, producing cosmos, and those of disorder,which led to chaos. They believed that random or disordering processes were negative, evil influences. Today, we don't regard the role of chance in nature as malicious, merely as blind. A chance event may act constructively, as in biological evolution, or destructively, such as when an aircraft fails from metal fatigue.

Though individual chance events may give the impression of lawlessness, disorderly processes, as a whole, may still display statistical regularities. Indeed, casino managers put as much faith in the laws of chance as engineers put in the laws of physics. But this raises something of a paradox. How can the same physical processes obey both the laws of physics and the laws of chance?

Following the formulation of the laws of mechanics by Isaac Newton in the 17th century, scientists became accustomed to thinking of the Universe as a gigantic mechanism. The most extreme form of this doctrine was strikingly expounded by Pierre Simon de Laplace in the 19th century. He envisaged every particle of matter as unswervingly locked in the embrace of strict mathematical laws of motion. These laws dictated the behaviour of even the smallest atom in the most minute detail. Laplace argued that, given the state of the Universe at any one instant, the entire cosmic future would be uniquely fixed, to infinite precision, by Newton's laws.

The concept of the Universe as a strictly deterministic machine governed by eternal laws profoundly influenced the scientific world view, standing as it did in stark contrast to the old Aristotelian picture of the cosmos as a living organism. A machine can have no "free will"; its future is rigidly determined from the beginning of time. Indeed, time ceases to have much physical significance in this picture, for the future is already contained in the present. As Ilya Prigogine, a theoretical chemist at the University of Brussels, has eloquently expressed it, God is reduced to a mere archivist, turning the pages of a cosmic history book that is already written.

Implicit in this somewhat bleak mechanistic picture was the belief that there are actually no truly chance processes in nature. Events may appear to us to be random but, it was reasoned, this could be attributed to human ignorance about the details of the processes concerned.
Take, for example, Brownian motion. A tiny particle suspended in a fluid can be observed to execute a haphazard zigzag movement as a result of the slightly uneven buffeting it suffers at the hands of the hands of the fluid molecules that bombard it. Brownian motion is the archetypical random, unpredictable process. Yet, so the argument ran, if we could follow in detail the activities of all the individual molecules involved, Brownian motion would be every bit as predictable and deterministic as clockwork. The apparently random motion of the Brownian particle is attributed solely to the lack of information about the myriads of participating molecules, arising from the fact that our senses are too coarse to permit detailed observation at the molecular level.

For a while, it was commonly believed that apparently "chance" events were always the result of our ignoring, or effectively averaging over, vast numbers of hidden variables, or degrees of freedom. The toss of a coin or a die, the spin of a roulette wheel-these would no longer appear random if we could observe the world at the molecular level. The slavish conformity of the cosmic machine ensured that lawfulness was folded up in even the most haphazard events, albeit in an awesomely convoluted tangle.

Two major developments of the 20th century have, however, put paid to the idea of a clockwork universe. First there was quantum mechanics. At the heart of quantum physics lies Heisenberg's uncertainty principle, which states that everything we can measure is subject to truly random fluctuations. Quantum fluctuations are not the result of human limitations or hidden degrees of freedom; they are inherent in the workings of nature on an atomic scale. For example, the exact moment of decay of a particular radioactive nucleus is intrinsically uncertain. An element of genuine unpredictability is thus injected into nature.

Despite the uncertainty principle, there remains a sense in which quantum mechanics is still a deterministic theory. Although the outcome of a particular quantum process might be undetermined, the relative probabilities of different outcomes evolve in a deterministic manner. What this means is that you cannot know in any particular case what will be the outcome of the "throw of the quantum dice" but you can know completely accurately how the betting odds vary from moment to moment. As a statistical theory, quantum mechanics remains deterministic. Quantum physics thus builds chance into the very fabric of reality, but a vestige of the Newtonian-Laplacian world view remains.

Along came chaos
Then along came chaos. As the previous articles in this series have discussed, the essential ideas of chaos were already present in the work of the mathematician Henri Poincaré at the turn of the century, but it is only in recent years, especially with the advent of fast electronic computers, that people have appreciated the full significance of chaos theory.

The key feature of a chaotic process concerns the way that predictive errors evolve with time. Let me first give an example of a non-chaotic system: the motion of a simple pendulum. Imagine two identical pendulums swinging in exact synchronism. Suppose that one pendulum is slightly disturbed so that its motion gets a little out of step with the other pendulum. This discrepancy, or phase shift, remains small as the pendulums go on swinging.

Faced the task of predicting the motion of a simple pendulum, one could measure the position and velocity of the bob at some instant, and use Newton's laws to compute the subsequent behaviour. Any error in the initial measurement propagates through the calculation and appears as an error in the prediction. For the simple pendulum, a small input error implies a small output error in the predictive computation. In a typical non-chaotic system, errors accumulate with time. Crucially, though, the errors grow only in proportion to the time (or perhaps a small power thereof), so they remain relatively manageable.

Now let me contrast this property with that of a chaotic system. Here a small starting difference between two identical systems will rapidly grow. In fact, the hallmark of chaos is that the motions diverge exponentially fast. Translated into a prediction problem, this means that any input error multiplies itself at an escalating rate as a function of prediction time, so that before long it engulfs the calculation, and all predictive power is lost. Small input errors thus swell to calculation-wrecking size in very short order.

The distinction between chaotic and non-chaotic behaviour is well illustrated by the case of the spherical pendulum, this being a pendulum free to swing in two directions (see New Scientist, "Chaos in the swing of a pendulum", 24 July 1986). In practice, this could be a ball suspended on the end of a string. If the system is driven in a plane by a periodic motion applied at the pivot, it will start to swing about. After a while, it may settle into a stable and entirely predictable pattern of motion, in which the bob traces out an elliptical path with the driving frequency. However, if you alter the driving frequency slightly, this regular motion may give way to chaos, with the bob swinging this way and then that, doing a few clockwise turns, then a few anticlockwise turns in an apparently random manner.

The randomness of this system does not arise from the effect of myriads of hidden degrees of freedom. Indeed, by modelling mathematically only the three observed degrees of freedom (the three possible directions of motions), one may show that the behaviour of the pendulum is nonetheless random. And this is in spite of the fact that the mathematical model concerned is strictly deterministic.

It used to be supposed that determinism went hand in hand with predictability, but we can now see that this need not be the case. A deterministic system is one in which future states are completely determined, through some dynamical law, by preceding states. There is thus a one-to-one association between earlier and later states. In computational terms, this suggests a one-to-one association between the input and the output of a predictive calculation. But now we must remember that any predictive computation will contain some because we cannot measure physical quantities to unlimited precision. Moreover, computers can handle only finite quantities of data anyway...
Read on here.

"Enjoy every sandwich" ~ Warren Zevon
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PostSubject: Re: Chaos frees the Universe   Thu Mar 19, 2009 10:01 pm

Quote :
The concept of the Universe as a strictly deterministic machine governed by eternal laws profoundly influenced the scientific world view, standing as it did in stark contrast to the old Aristotelian picture of the cosmos as a living organism.

This may be one of the key ways in which Deists and Atheists differ in their worldviews. Modern science, until recently, had bet the house on the deterministic, clockwork universe*. Consequently, rationalistic Atheists tend to interpret the world according to the conventional scientific Machine Metaphor.

By contrast Deists, among others, have noticed---perhaps intuitively rather than rationally---that the real world is not as cut & dried as Classical Determinism had pictured it. Even before the science of Chaos and Complexity put randomness into a computable form, some observers (e.g. gamblers and insurance appraisers) had long ago noted that Chance was predictably unpredictable---sort of like a living organism.

According to my current reading of "Intelligent" Evolution, I imagine that an eternally creative First Cause could have precisely pre-destined the Omega Point of this creation. But as a Promethean gift to the more promising of the creatures, the point-to-point path of evolution was determined only in a statistical sense. Thus, the built-in uncertainty and unpredictability of our machine-world leaves some room for freewill choices by creatures smart enough to predict the future a few hours or days ahead.

What are the odds of Freewill? Very Happy

* But as Davies noted, a clock's pendulum is subject to the mechanical Three Body Problem (see Wiki). Any more than two variables in a calculation is essentially incalculable.
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