ON A CARIBBEAN ISLAND, a butterfly flutters briefly to the left instead of to the right. As a result, swirls of air produced by its wings move in a slightly different way than they would have. A few days later, a hurricane gradually building force in the Caribbean veers into the Florida coast instead of heading out to sea.
Could the hurricane's change in course have been caused by the altered flight of the butterfly? According to Edward N. Lorenz of the Massachusetts Institute of Technology, the answer is yes. ** (E. N. Lorenz, "Deterministic Non-periodic Flow," Journal of Atmospheric Science, vol. 20, pp. 130-141, 1963. R. E. May, "Simple Mathematical Models with Very Complicated Dynamics," Nature, vol. 261, pp. 459-467, 1976.) Computer simulations Lorenz has carried out suggest that the flow of air in the atmosphere may display the property of "exponential instability."
Because of this property, extremely small changes in the flow of air can quickly be amplified until they have a major impact on weather. This, says Lorenz, makes short-range weather hard to forecast, because small changes, difficult to monitor, could result in large-scale effects.
This unpredictability has been called "deterministic chaos" because it arises in systems that, from a mathematical point of view, should be strictly deterministic and predictable. The idea of deterministic chaos gives new ways of looking at the question of God's ability to act within the framework of physical laws.
During the seventeenth century, at the time of Isaac Newton, many European naturalists and philosophers held that the material world was not only created but directly controlled by God. But the publication of Newton's Principia heralded the soon to be established view that everything in nature happens in a rigidly deterministic fashion, in accordance with fixed mathematical laws.
Newton's work confirmed an image of reality that had been growing prominent in Europe since the late Middle Ages—the picture of the universe as a machine like a vast clock. According to this picture, there are only three ways in which an agency transcendental to the material world could influence the behavior of matter. These are (1) by directly interfering with mechanical cause and effect in the world-machine at various times, (2) by programming the world-machine in the beginning of time to unfold future history automatically, and (3) by simply allowing events to happen according to the mechanical laws of cause and effect.
In case (3) the elements of interference and preprogramming are both absent, and the only role of the transcendental agency is to keep things going by the laws. The viewpoint summed up by option (3) is often called deism. It contrasts with atheism, which holds that events unfold according to physical laws that require no God to sustain them.
Newton himself felt that divine intervention of type (2) was necessary for the creation of the solar system. The solar system, he thought, would not have arisen on its own. Newton also believed that intervention of type (1) was necessary to keep the solar system operating smoothly.
Many scientists and philosophers rejected Newton's theistic arguments. Newton's rival Leibniz, for example, thought of God as a perfect clockmaker who created the universal machine, set it in motion, and then had no need to intervene further in its operation.
In due course, the research of Pierre de Laplace and J. L. Lagrange buttressed the views of Leibniz. Newton had argued that the gravitational influence of the planets on one another would eventually perturb their orbits to a degree demanding divine correction. Otherwise, the solar system would fly apart. But Laplace and Lagrange showed mathematically that in an idealized Newtonian model of the planetary system the orbits would oscillate within fixed limits. So the stability of the solar system would be preserved.
As time went on, divine interventions of type (2) also began to seem more and more implausible. Newton had argued that the regular arrangement of the nearly circular planetary orbits required the "divine arm." ** (S. L. Jaki, The Relevance of Physics (Chicago: University of Chicago Press, 1966) p. 429.)(2) But Laplace suggested that this regular pattern may have formed naturally as the planets condensed by physical processes from a primordial nebula. Even though Laplace did not fully work out the mathematics of his hypothesis, it carried the day and further limited the scope allowed for God's activity within the universe.
God in the World of Modern Science
At present, as in the past, people hold a wide range of opinions on the relation between God and the material world. But many theologians in mainline denominations of Christianity tend to embrace some version of deism, which holds that God's role in the universe is limited to creating and maintaining the laws of physics.
We will explore this view by examining the teachings of John Polkinghorne, formerly a professor of theoretical physics at Cambridge University. Polkinghorne, now president of Cambridge and an Anglican priest, is an articulate spokesman for deism.
For Polkinghorne, the world is "the expression of the will of a Creator, subtle, patient, and content to achieve his purposes by the slow unfolding of process inherent in those laws of nature which, in their regularity, are but pale reflections of his abiding faithfulness." ** (J. C. Polkinghorne, One World: The Interaction of Science and Theology (Princeton, N. J.: Princeton University Press, 1987) p. 80.)
Everything works by the laws of physics, but God sustains these laws, and indeed the universe would cease to exist if He were to stop doing so.
Yet Polkinghorne also takes religious experiences seriously and accepts God as a "persuading, sustaining, transforming presence in the depths of our being." ** (Ibid., p. 70.) Here one encounters a serious inconsistency in Polkinghorne's views.
What is the meaning of Polkinghorne's statement that God acts as a "transforming presence" in the depths of our being? One might say the transformations are simply internal. But if the transformations brought about by God's presence do not affect our words and actions in any measurable way, then in what sense are they real or significant. Admitting that God's transforming presence does affect our behavior, Polkinghorne would say that these behavioral changes are governed by the laws of physics. But if they are governed by the laws of physics, in what sense are they caused by a divine presence?
If God can play a meaningful role in people's lives, it seems we must also allow God to direct the flow of material processes on a regular day-to-day basis. And this would violate the laws of physics.
Polkinghorne says of the laws of atomic physics, "I could literally write them down on the back of an envelope. Yet the fact that they have such remarkable consequences as you and me speaks of the amazing potentiality contained in their structure." ** (Ibid., p. 54.) Polkinghorne holds that God created the laws of physics with this potentiality.
Granting this, one might suppose that God also endowed these laws with the potential to generate human brains programmed to produce religious experience.
One might then say that the "transforming presence" in the depths of our being is a neural process ordained by God when He first selected the terms in the differential equations of physics.
But is it meaningful to attribute such physically generated "religious experiences" to the presence of God? If the interactions of charged particles and electromagnetic fields are enough to bring forth all these manifestations, why bother to bring God into the picture at all? It seems simpler and more plausible to adopt the atheistic view: religious experience is just another natural outcome of the underlying laws of nature; and ideas about God, though generated by nature, are false.
A Positive Solution
Polkinghorne is grappling with an impossible problem. He is committed to two contradictory ideas: (1) that material events unfold according to deterministic physical laws and (2) that a transcendental supreme being is at work within the world. It is simply not possible for both of these ideas to be correct.
The root of this predicament is the belief that the laws of physics are deterministic. This belief began in Newton's era, and it persists today. Of course, the quantum theory is famous for introducing an element of indeterminism into physics. But as Polkinghorne points out, this indeterminism does not allow for nonphysical agencies to interact with living organisms. So the problem still lies with the determinism that seems inherent in classical physics.
Classical determinism, however, is an essentially illusory idea. It is not an inherent feature of the mathematics and empirical predictions of classical physics. Rather, it is an error arrived at by generalizing from a limited class of machines (such as clocks). So Polkinghorne's predicament, and more generally the predicament of Western spirituality during the last three centuries, need never have arisen.
This is where deterministic chaos comes in. In classical physics, this phenomenon is sufficiently widespread to render the behavior of many important systems completely unpredictable. And this inherent unpredictability, as we will show, simultaneously allows such systems to (1) obey the classical equations of motion within the limits of observational accuracy and (2) exhibit patterns of behavior freely selected from a wide range of options.
As a result, it is possible to re-formulate classical physics so that (1) the fundamental laws of dynamics stay essentially the same, (2) the predictions of the theory stay the same within the limits of observational accuracy, and (3) the theory has an element of indeterminism that allows extensive control over the course of events. In this form, the theory does not conflict with the idea that the phenomenal world is directly controlled by a transcendental being.
How Deterministic Chaos Works
To lay the groundwork for our model, we must first explain how deterministic chaos works. The key to understanding this is a phenomenon called exponential instability, and this can be most readily explained by an example.
Imagine firing a bullet at a solid metal sphere a few inches in diameter, and imagine that the bullet will bounce off the sphere. If you're firing from a distance of several yards, slightly changing your aim can greatly affect the direction in which the bullet bounces. If you aim directly at the center of the sphere, the bullet will bounce back towards you. But if you aim slightly off center it will bounce off to one side. The curvature of the sphere amplifies the slight change in your aim, and this greatly changes the direction in which the bullet bounces.
If there are several spheres and the bullet bounces from one to another, the change grows greater from bounce to bounce and will quickly become enormous. This is called exponential amplification.
Figures 1 and 2 show how such amplification can generate preplanned patterns in ordinary matter. In these figures we consider a simple two-dimensional model of a gas. The gas molecules are represented by disks of equal mass, which elastically collide with one another and with the rigid walls of the square chamber that encloses them.
It is easy to see intuitively why this system should exhibit exponential instability. If two disks are about to collide, the curvature of one will magnify even a slight change in direction by the other, just as we saw with the bullet bouncing from the sphere. After N collisions this magnification will rise to the Nth power. In this way, small changes in direction can quickly have large effects.
To see what can happen, consider Figure 1. The four frames show the system of bouncing disks at four successive times. In each frame the arrangement of the disks seems random, as we would expect for the molecules of a gas.
In Figure 2 we start with an initial arrangement that looks the same as in Figure 1. But it's slightly different: the directions in which the disks are moving have been systematically rotated by angles less than a millionth of a degree (too small to measure by observation).
As a result, by frame c the arrangement of disks is significantly different from that in Figure 1, and in frame d the disks have lined up in an orderly pattern we would not expect to find in a system of randomly colliding objects.
Now, let's assume that the disks in the model are about the size of air molecules (some 2 angstrom units in diameter). And let's say the disks are moving at the same average speed as air molecules at room temperature. Then the time that would elapse for the four frames in Figures 1 and 2 would be 2.75 X 10-11seconds. From this we can see that when suitable, extremely small changes are made in the motion of the disks, planned patterns of organization can develop quickly in this system. There is every reason to think that similar effects could come about in real systems of molecules.
The key to the exponential amplification in our example is the nonlinearity in motion caused by the curvature of the disks. It turns out that in real physical systems such nonlinearity is the rule rather than the exception. We can therefore expect exponential instability and unpredictable behavior to show up in a wide variety of physical systems. Examples reported thus far include:** (P. Cvitanovic, ed., Universality in Chaos (Bristol: Adam Hilger, 1984).) the flow of air and fluids in a wide variety of situations; oscillating chemical reactions; the beating of heart cells; a large number of electrical and mechanical oscillators; the dripping of faucets; the Newtonian three-body problem; models of nerve cells and glial cells in the brain; and models of epidemics, animal populations, and economics.
The Unmanifest And the Immeasurably Small
Deterministic chaos renders the laws of classical physics flexible instead of rigid and deterministic. So without producing measurable deviations from these laws, an unlimited intelligence with direct control over matter on a submicroscopic level could guide the course of events freely.
We can speak of this submicroscopic level as the "unmanifest," since it involves phenomena we cannot directly perceive or measure. In the classical models we are considering here, the unmanifest is the domain of immeasurably small changes in the position and velocity of particles. In other physical models (including quantum mechanical systems) one can also speak of an unmanifest level involving extremely small changes in the state of a system.
Our proposed change in the laws of physics is simply this: on the manifest, or measurable, level, leave the laws as they are, but at the unmanifest, or unmeasurable, level, allow for intelligently directed changes.
The All-Pervading Supersoul
To effectively control material phenomena through action on the unmanifest level, a controlling agency would have to make minute but precisely coordinated adjustments in the course of events at many points in space and time. Traditional conceptions of God found in cultures all over the world seem to allow for such coordinated control. One of the great perennial ideas of mankind is that God, or some aspect of God, is present everywhere in space and able to perceive and act at all locations simultaneously.
Polkinghorne retains a modified form of this idea. According to Polkinghorne, physical interactions occur in the same way everywhere because of the "abiding faithfulness" of God, who acts everywhere to sustain the laws of physics. Of course, one might argue, if all God does is sustain the laws of physics, why not simply accept the regularity of physical behavior as axiomatic? Then there would be no need to bring in the greater mystery of God to explain it. Nonetheless, Polkinghorne reflects traditional ideas by proposing that not a single electron interacts without the will of God.
Newton, at the beginning of the modern mechanistic era, held a somewhat stronger view of God's all-pervading nature. He regarded absolute space and time as the sensorium of God, through which God could perceive all phenomena, sustain the laws of physics, and also modify the course of events according to His will. ** (R. Palter, ed., The Annus Mirabilis of Sir Isaac Newton, 1666-1966 (Cambridge, Mass.: M.I.T. Press, 1970) p. 340.) Such modifications, he thought, were gross violations of natural law but would be needed from time to time to keep things running smoothly.
The Christian thinker Augustine, writing in the fourth century, described God this way: "God so fills all things as to be not a quality of the world, but the very creative being of the world, governing the world without work, sustaining it without effort. … Unconfined to any place, He is in Himself everywhere wholly."** (M. T. Clark, Saint Augustine, Bishop of Hippo (N.Y.: Paulist Press, 1984) p. 409.)
Here the phrases "without work" and "without effort" could be taken to mean that God simply sustains a completely autonomous world system. But Augustine was writing in a pre-mechanistic era, and this suggests that he used these phrases to refer to God's power to control events personally, from moment to moment.
The Vedic literature of India provides further insight into this point. The Bhagavad-gita explains that all material phenomena are created, maintained, and annihilated by a single part of Krsna, the supreme transcendental person. This part, or amsa, is known as the Supersoul, or Paramatma, and is described in the Thirteenth Chapter of the Gita. There it is stated that the Supersoul: ** (A. C. Bhaktivedanta Swami Prabhupada, Bhagavad-gita As It Is (Los Angeles: The Bhaktivedanta Book Trust, 1983) Chapter 13, texts 13-17.) (1) lies beyond material cause and effect; (2) has transcendental senses of perception and action at all locations in space and time but has no material senses; (3) is the master of material nature yet is beyond it; (4) is present inside and outside of all living beings; (5) is situated as one, even though appearing to be divided.
This description more elaborately presents some of the ideas mentioned by Augustine. First of all, it indicates that the Supersoul can perceive and act in a unified way at all points in space and time. This is precisely what our model requires. A being fully present at all locations would be uniquely suited to gather the needed data and make the needed calculations to guide the nondeterminate flux of events in any desired direction.
Of course, we do not wish to suggest that God operates by computation, as humans might imagine doing. Augustine and theGita agree that God is able to guide material events effortlessly, like a practiced pianist improvising on musical themes without worrying about the detailed movements of his fingers. Our main point is that matter acting in accord with classical physical laws is indeed freely controllable, but the exertion of such control requires both omniscience and omnipotence. Long before the advent of modern physical theories, ancient traditions attributed such unlimited powers to God.
We can conclude, therefore, that the laws of classical physics are compatible with the idea that God directly controls the behavior of matter. The same compatibility can also be demonstrated for the laws of quantum mechanics, although the complexity of quantum mechanical theory prevents us from going into this topic here.
Of course, it has not been shown that all events in nature do conform to the known laws of physics, although many scientists assume that this must be so (or that it will be so once some final minor additions to the laws are made). If material phenomena wander from the laws of physics, that does, of course, leave room for action by God. But even if we suppose that all measurable phenomena do follow the known laws, it is still possible for the course of events to be under divine control from moment to moment.
Sadaputa Dasa (Richard L. Thompson) earned his Ph.D. in mathematics from Cornell University. He is the author of several books, of which the most recent is Vedic Cosmography and Astronomy (see page 20 to order). Write to him c/o BTG in San Diego.