05 September 2010

The rediscovery of cosmology

5.25 Introduction: ‘Newtonian’ limits to Newtonian physics

From the perspective of Newtonian physics, reality could be exhaustively understood in terms of particles moving in well-defined ways under the influence of certain forces. Of course it was recognised from the outset that real life was more complex than that. Many everyday situations involved too many factors to be amenable to such straightforward treatment. In such cases physicists had to be satisfied with approximations. Nevertheless it was assumed that, in principle, these awkward cases could be treated exactly.[1]

Take, for example, the motion of planets round the Sun. Using his laws of motion, Newton was able to provide an exact solution to the two-body problem – the case of two physical bodies interacting gravitationally but isolated from other influences. However, Newton’s successors were unable to create exact solutions for larger ensembles of bodies (e.g. the Solar System or even just the Sun, Earth and Moon considered in isolation from all the rest – the three-body problem). Instead they had to adopt a method of approximations – beginning with the simple case, they asked how the presence of an additional element might perturb the orbits of the two bodies, then they calculated the effect of that change on the third body, then corrected the original calculations in the light of that, and so on to higher and higher degrees of accuracy.

It was not until the end of the nineteenth century that astronomers finally abandoned the search for an exact solution to the three-body problem. In 1889 a young mathematical physicist, Henri Poincaré, won a competition sponsored by the King of Sweden with an essay demonstrating the impossibility of such a solution.

5.26 Recognising chaos

Poincaré may justly be called the father of chaos theory. In addition to demonstrating that there were physical systems which could not be precisely analysed using Newtonian physics, he was among the first physicists to comment on the extreme sensitivity of many physical systems to small variations in initial conditions. Little notice was taken of his remarks when he made them in 1903 but, since then, physicists have become much more conscious of the extent to which such chaotic behaviour is to be found in the physical world. This new awareness of chaos and complexity is not so much a recent discovery as a gradually changing perception resulting from a range of factors.

The research that has resulted from Poincaré’s own work on perturbation theory is one of these factors. This has revealed the existence of chaotic behaviour in simple isolated systems. Take, for example, the motion of balls on a snooker table. It can be shown that their motion is so sensitive to external factors that in order to predict the position of the cue ball after a minute of motion (and collisions), one would have to take into account the gravitational attraction of electrons on the far side of the galaxy! Even something as apparently simple as the tossing of a coin or the motion of a water droplet on a convex surface is so sensitive to minute variations in the environment as to be unpredictable.

A second area of research that has encouraged physicists to take chaos more seriously is that of turbulent flow in fluids. Its relevance to engineering and meteorology ensured that this was a growth area in research. Unlike the simple situations described above, turbulence is not merely a matter of uncertainties in the system created by random motion at the molecular level. That aspect of fluid dynamics can be handled statistically. The real issue is the sudden emergence of random motion on a macroscopic scale – eddies and currents involving large collections of molecules. Such situations are bounded but unstable – in many such cases we are now able to generate equations that tell us the boundaries within which the motion will take place. Inside those boundaries, however, the particles involved are subject to irregular fluctuations quite independent of any external perturbation.

A third aspect in the development of chaos theory has been the availability of more and more powerful electronic computers. They have allowed physicists to extend classical perturbation theory to situations that previously were too complicated to calculate. As a result more and more situations have been revealed to be chaotic.

Finally, the widespread acceptance of quantum theory may also have played an important part in changing the attitudes of physicists to unpredictable situations. This is not to suggest that quantum theory is directly relevant to chaos at an everyday leve1.[2] However, the acceptance of quantum uncertainties may have made it easier for physicists to accept a degree of unpredictability about the physical world at other levels.

5.27 Coming to terms with chaos

This new awareness of complexity implies a profound change in the way in which many physical scientists view the world, a new perception of the relation between freedom and necessity. This can be summarised in the apparently paradoxical statement that chaos is deterministic. The situations described above are not completely anarchic. On the contrary, we are dealing with ensembles of bodies moving at the everyday level where Newtonian laws of motion still hold sway. The behaviour of these chaotic situations is generated by fixed rules that do not involve any elements of chance. Many of the physicists of chaos would insist that, in principle, the future is still completely determined by the past in these situations. In an accessible introduction to the subject Alan Cook advocates that the term ‘deterministic chaos’ should always be used (Cook, 1998:31-41) However, these are situations which are so sensitive to the initial conditions that, in spite of the determinism of the associated physical laws, it is impossible to predict future behaviour. Deterministic physics no longer has the power to impose a deterministic outcome. According to one of the classic papers on the subject, ‘There is order in chaos: underlying chaotic behaviour there are elegant geometric forms that create randomness in the same way as a card dealer shuffles a deck of cards or a blender mixes cake batter’ (Crutchfield et al., 1995:35).

At first glance this may seem entirely negative. The admission that chaos is far more widespread than previously realised appears to impose new fundamental limits on our ability to make predictions. If prediction and control are indeed fundamental to science then chaos is a serious matter. On the other hand, the deterministic element in these chaotic situations implies that many apparently random phenomena may be more predictable than had been thought. The exciting thing about chaos theory is the way in which, across many different sciences, researchers have been able to take a second look at apparently random information and, while not being able to predict exact outcomes, nevertheless explain the random behaviour in terms of simple laws. This is true of meteorology. It can also be applied to dripping taps or to many biological systems (e.g. the mathematical physics of a heart-beat).

5.28 Implications for the philosophy of science

As we have just hinted, the emergence of a science of chaos has profound implications for our understanding of what science is and can do.

One may disagree with the notion that the raison d’être of science is prediction and control. Nevertheless, prediction still retains a central place in the scientific method. How else are we to test our scientific models? The classical approach is to make predictions from the model and devise experiments to test those predictions. Here, however, we are faced with situations in which such predictions seems inherently impossible. In fact, what is required is that we take a more subtle approach to prediction. What we observe may well be random (or pseudo-random). However, the deterministic element in mathematical chaos implies that the random observations will be clustered into predictable patterns.
A second important implication of chaos theory has to do with the continuing tendency to reductionism in the sciences (see 6.11-6.11.1). Chaos and complexity highlight the fact that only in the very simple systems that formed the backbone of classical physics is it true that the whole is merely the sum of the parts. Chaotic systems simply cannot be understood by breaking them down into their component parts and seeing how they fit together again.
Closely allied to this challenge to reductionism is a question about the possibility of completeness in physics. The reality of chaos undermines the hope that such completeness can be achieved by an increasingly detailed understanding of fundamental physical forces and constituents. It also provides a physical basis for the concept of emergence that is so important in philosophical and theological perspectives on the life and human sciences. The behaviour of chaotic systems suggests that interaction of components at one level can lead to complex global behaviour at another level – behaviour that is not predictable from a knowledge of the component parts. Indeed some chaos scientists suggest that ‘chaos provides a mechanism that allows for free will within a world governed by deterministic laws’ (Crutchfield et al., 1995:48).
However, we consider caution is needed at this point. Chaotic randomness is not complete randomness. True, we are unable to predict the detailed outcome of a chaotic scenario. However, the mathematics of chaos does permit us to predict the limits of the possible outcomes. This is randomness within constraints – deterministic constraints. In fact, chaos theory allows us to extend our physical understanding of the world into new areas specifically by applying deterministic covering laws to situations that were previously thought to be completely random. This could be taken as evidence that determinism really works.
On the other hand, the fact that the equations we use are deterministic does not necessarily mean that nature is deterministic. The equations are maps, not the reality. It could be that the apparent determinism is an artefact of the particular way in which we have chosen to map reality – in terms of mathematical physics. The idea that our equations are only approximations to the laws that govern the macroscopic world is an important part of Polkinghorne’s position on divine action [see 10.9(iv)(a) and Polkinghorne 1998a:64-66).

5.29 Conclusion

We have seen that, though Newtonianism remains very influential, in a number of areas modern physics has broken with the Newtonian paradigm. It has given rise to questions of interpretation which relate directly to theology – in the areas of quantum theory and chaos theory (re determinism), the Big Bang origin and final fate of the universe, and the question of evidence for design. A number of these areas will be considered again in our discussion of divine action in Chapter 10.


[1] See 1.17 on Laplace and the determinism of the Newtonian universe.
[2] Except in that Heisenberg’s Uncertainty Principle [5.11 (ii)] sets a limit on the precision of our knowledge of any system.

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