CHAOS THEORY

CHAOS THEORY . In the Principia (1687), Isaac Newton gave an account of mechanics formulated in terms of precise equations of motion. Given the initial conditions of a system, it was possible to predict completely its future behavior and to retrodict its past. Newton himself did not take a purely mechanical view of the world. There was the mysterious force of gravity, concerning whose origin and nature he declined to frame a hypothesis, and he also believed that the maintenance of the stability of the solar system would require occasional angelic intervention. Newton's eighteenth-century successors, however, had different opinions, and they celebrated the triumph of mechanical thinking. Julien de La Mettrie (1709–1751) wrote his book Man the Machine (1748), and Pierre-Simon de Laplace (1749–1827), in his great work on celestial mechanics, believed that he had established the natural stability of the solar system, so that appeal to the hypothesis of divine assistance in its preservation was no longer necessary.

The nineteenth-century development of field theories, inspired by the insights of Michael Faraday (1791–1867) and James Clerk Maxwell (1831–1879), did not essentially change the picture. The partial differential equations of a field theory are as deterministic in consequence as are the ordinary differential equations of Newtonian mechanics. Classical physics, as this whole body of theory is called, appeared to present the image of a clockwork universe, whose Creator could be no more than the Cosmic Clockmaker.

Unpredictabilities

Twentieth-century physics, however, saw the death of a merely mechanical understanding of the world. This came about through the discovery of widespread intrinsic unpredictabilities present in physical process, of a kind resulting from the way things actually are and not simply from deficiencies in experimental or calculational techniques. These unpredictabilities first manifested themselves in quantum theory's account of atomic and subatomic phenomena. In considering, for instance, the decay of a radioactive nucleus, scientists could do no more than assign a certain probability that such decay might occur in a given period of time. Newton's predictive style of reasoning had to be replaced by a purely probabilistic approach. Quantum effects, however, were only directly observable at the level of microscopic process, remote from the macroscopic experience of everyday reality. The latter was still the realm of classical physics.

In the 1960s many scientists began to realize that even classical physics was not as tame and controllable as had been supposed. There are certainly many systems that behave as if they were reliably predictable "clocks," but there are also other systems that behave like "clouds," that is to say their behavior is so sensitive to the fine detail of their circumstances that the slightest disturbance will radically alter their future behavior. One of the pioneers in making this discovery was the theoretical meteorologist Edward Lorenz (1917–). Lorenz had been studying certain equations that corresponded to a highly simplified model of a weather system. He determined that very slight changes in the input of initial conditions into his equations would totally change the character of their output predictions. This exquisite sensitivity to detail has come to be expressed through a serious scientific joke, the butterfly effect : the Earth's weather systems can be in so sensitive a state that a butterfly, stirring the air with its wings in the Amazonian jungle today, could produce effects that escalate until they result in a storm over New York City in about three weeks time! Obviously, according to this model, long-term weather forecasting is never going to work; future storms are intrinsically unpredictable because no one can know about all those butterflies.

Lorenz's surprising discovery had been anticipated by the French mathematician, Jules-Henri Poincaré (1854–1912). In 1889 he published a study of the gravitational three-body problem in which he showed that it did not always possess smoothly predictable solutions of the kind that Laplace had assumed.

Chaos Theory

The analysis of the sensitive and unpredictable behavior of various systems has been called chaos theory. Typically its occurrence is found to arise from equations that have the properties of reflexivity (they turn back upon themselves) and nonlinearity (doubling the input does not double the output, but it changes it in a much more radical way). In principle, the equations are exactly deterministic, in the sense that an absolutely precise input will yield an absolutely precise output, but in practice, because in the real world no initial conditions can be known with arbitrary accuracy, unpredictability results from the uncontrollable effects of residual ignorance. Deterministic chaos, as it is often called, gives rise to apparently random behavior.

In fact, the term chaos was somewhat ill chosen. Systems of this kind exhibit a kind of interlacing of order and disorder, which can best be illustrated by the case of dissipative systems that feature a degree of friction acting in the process. In this case, the future behavior is not totally haphazard; rather, its possibilities are contained within an extensive though limited portfolio of options called a strange attractor. The word "attractor" expresses the system's rapid convergence onto this range of possibilities, and "strange" refers to the mathematically intricate shape of this form of possibility. It turns out that the latter is characterized by fractal geometry, a "jagged" range of possibilities in striking contrast to the smoothly varying expectations of conventional classical physics. A fractal presents a pattern that appears essentially the same on whatever scale it is sampled—one might think of them as saw-teeth, themselves saw-toothed; the pattern continues to infinity. The interlacing of order and disorder in chaos theory connects with an important scientific insight. It has come to be recognized that the emergence of novelty requires a state of affairs that can be characterized as being "at the edge of chaos." To be too much on the orderly side of that border would correspond to a situation possessing a degree of rigidity that permitted only rearrangements to occur but did not allow the emergence of genuine novelty. On the other hand, to be too far on the haphazard side of the border would correspond to a situation so unstable that no novelty could persist. Fruitfulness requires a subtle balance between order and openness.

Metaphysics

The unpredictability in chaos theory is an epistemological property, telling us that we cannot know beforehand what the future behavior of a chaotic system will prove to be. There is no inescapable connection between epistemology (what we know) and ontology (what is the case). No logical entailment links the two together. Instead, the relationship is a matter for metaphysical decision (and so, for philosophical argument).

Immanuel Kant (1724–1804) maintained that the appearances of phenomena are no guide to the nature of noumena, or the character of things in themselves. On the other hand, most scientists, either consciously or unconsciously, assume a realist position, believing that scientific knowledge gives access to the nature of the physical world.

In the case of the intrinsic unpredictabilities of quantum theory, almost all physicists have adopted a realistic interpretation of phenomena. Werner Heisenberg's uncertainty principle is not regarded simply as an epistemological principle of ignorance, but it is taken to be an ontological principle of actual indeterminacy. The fact that the work of quantum physicist David Bohm (1917–1992) has provided an alternative option, of equal empirical adequacy, which offers a deterministic interpretation of quantum phenomena, shows that the majority position is indeed a matter of metaphysical choice rather than physical necessity.

In the case of the intrinsic unpredictabilities of chaos theory, however, the majority decision has gone the other way. Most physicists disconnect epistemology and ontology, concluding that the theory shows that deterministic equations are consistent with the appearance of random behavior. "Deterministic chaos" is indeed the way they think about the theory. This absence of any willingness to question the assumption of underlying determinism seems to have been influenced by the feeling of deep respect accorded to the historic equations of classical physics.

Yet a different strategy is metaphysically possible. This would involve interpreting the unpredictabilities as signs of ontological openness. In turn, this strategy would necessitate a reinterpretation of the equations of "deterministic chaos." The sensitivity of chaotic systems to the details of their circumstances provides a way in which such a reinterpretation can be accomplished. Because of such sensitivity, chaotic systems can never properly be treated in isolation. Their vulnerability to the slightest disturbance means that they are intimately linked to their environment. Yet the experimental support that "verifies" the laws of classical physics has all been obtained through investigations of situations in which the assumption of isolatability is an acceptable idealization. Otherwise, analysis of what was going on would have been too complex to be feasible. It is perfectly possible, therefore, to make the metaphysical conjecture that the supposed laws of classical physics are actually no more than approximations of the behavior of what may be conceived actually to be a more subtle and supple physical reality. The approximation involved could be called downward emergence, because it would relate to behavior observable only in the idealized circumstance of an isolated system.

In this metaphysical scheme, epistemological unpredictability is the sign of an ontological openness. Such openness is not meant to imply that the future is some sort of random lottery, but rather that the causal principles involved are more than those described simply by the conventional physical picture of the exchange of energy between constituents. The unisolatibility of chaotic systems means that though such systems may be made up of components, they also must be considered holistically, in the context of their totality. The variety of the different possible patterns of their future behavior, represented by the different ways in which a chaotic system might traverse its strange attractor, are not discriminated from each other by energy differences, but by differing patterns of dynamical behavior, characterized by different expressions of "information." These considerations led to the metaphysical conjecture that chaos theory should be interpreted as affording scope for a new kind of causal agency, having the character of being top-down (influence of the whole upon the parts) and corresponding to an input of information (specification of patterned dynamical behavior). One might summarize the proposal as suggesting the concept of holistic causality through active information.

Two general considerations may be offered in support of such a metaphysical project. One is that the stance taken accords with the realist strategy followed by almost all physicists in the case of quantum theory. There is no apparent reason to treat quantum theory and chaos theory differently in this respect. Second, human experience of agency, of the willed execution of the intentions of the whole person, encourages the belief that an account of the causal nexus of the world is needed, an account that goes beyond simple constituent notions. The concept of top-down causality, operating through the input of information, offers the glimmer of a prospect of how one might begin the task of reconciling the scientific account of process with the human experience of agency.

The theologian may also find these ideas to be of use in thinking about divine providential interaction with creation. If the causal grain of nature is open in the way suggested, there seems to be no difficulty in believing that the Creator also interacts with the unfolding history of creation through the input of active information.

Quantum Chaology

At first sight it might seem that the metaphysical problems discussed above might readily be solved by combining the insights of quantum physics and chaos theory.

The behavior of chaotic systems quite rapidly comes to depend upon the details of circumstances lying at the level of Heisenberg uncertainty or below. It might seem attractive, therefore, to appeal to the generally accepted indeterministic character of quantum process to induce openness in the behavior of chaotic systems. Yet this approach faces serious difficulties, resulting from a lack of understanding of how one might consistently combine quantum theory and chaos theory. Indeed, these theories appear to be mutually incom-patible.

Quantum theory possesses an intrinsic scale, set by Max Planck's fundamental constant. As a result, in quantum thinking it is possible to give a meaning to terms such as large or small. We have seen that chaos theory, on the contrary, is scale-free because of its fractal character, implying that everything looks roughly the same on whatever scale it is surveyed. Clearly some significant modification of thinking would be required to bring the two theories together. Modern physics has only a rather patchy picture of the causal nature of reality, and is far from being able to offer an integrated account, applicable at all levels.

The complexity of the considerations involved can be illustrated by the behavior of Hyperion, one of the moons of Saturn. It is an irregularly shaped piece of rock about the size of New York City, which is observed to be tumbling chaotically. Quantum effects, with their imposition of scale, would be expected to suppress this chaotic motion very effectively for so large an object. Calculations made on this basis indicate that tumbling should last for only about thirty-seven years. However another effect, called decoherence, due to the environmental influence of the radiation that bathes Hyperion, in its turn suppresses the quantum effects and explains why the moon's chaotic behavior can be expected to continue almost indefinitely. The causal nexus of the world is very complex, with a variety of effects interlacing.

See Also

Physics and Religion.

Bibliography

Gleick, James. Chaos: Making a New Science. London, 1988. An excellent introduction to chaos theory for the general reader.

Peitgen, Heinz-Otto, and Peter Richter. The Beauty of Fractals. Berlin, 1986. Lavishly illustrated account of fractal geometry.

Polkinghorne, John. Belief in God in an Age of Science. New Haven, Conn., 1998. Chapter 3 gives an account of divine action based on an ontological interpretation of chaos theory.

Prigogine, Ilya. The End of Certainty: Time, Chaos, and the New Laws of Nature. New York, 1997. A Nobel laureate's account of the openness of physical process.

Ruelle, David. Chance and Chaos. Princeton, N.J., 1991. Accessible survey by a distinguished mathematical physicist.

Russell, Robert, Nancey Murphy, and Arthur Peacocke, eds. Chaos and Complexity: Scientific Perspectives on Divine Action. Vatican City, 1995. Conference proceedings reporting a variety of points of view and focusing on questions of divine action.

Saunders, Nicholas. Divine Action and Modern Science. Cambridge, UK, 2002. A careful and comprehensive survey of current issues.

John Polkinghorne (2005)