Theological Implications of Chaos Theory

Determinism, Total Predictability and the Uncertainty Principle in Chaotic Systems: Theological Implications

(a naive beginning)

by Neal Magee * TH417 Theology & Evolution * Dr. J. Wentzel van Huyssteen * Fall 1994

Eighteenth century French mathematician Pierre Simon de Laplace's aspiration was to know the world in a way so that every process and every change could be understood, predicted, and anticipated. The idea of a determined universe was not a recent development in science, philosophy, or theology, and has continued to be the center of attention for many years; if we can more effectively forecast events, we can (collectively) benefit greatly from that knowledge.

1 Over the last three hundred years we have discovered the scientific laws that govern matter in all normal situations. We understand and make use of fields like thermodynamics, electromagnetics, gravity, motion, momentum, and chemistry to enhance our collective 'big picture' of the world and for our own personal benefit through technology.

2 However, that dream of total predictability never came true. Why is that? Why are we unable to predict the weather more than four or five days in advance? Why do these laws always apply but not allow us to see far into the future? This is because the solutions to the equations of physics may exhibit a property known as chaos. In this paper I will discuss the basic properties of chaos and chaotic behavior (sometimes refer red to as nonlinear systems, nonlinear dynamics, or complex systems), and its implications to the theological and scientific discussion of total predictability and determinism, paying special regard to quantum physics' special proposition through the Heisenberg Uncertainty Principle. These issues play important foundational roles in the faith-science dialogue, our idea of free will, and our views of God. In addition, this indirectly has a great impact on our understanding of the evolutionary processes at work in the universe, by asking if our world is uniquely designed, arbitrarily predetermined, or specially predestined.

Chaos is all around us in the structure of a tree, our capillaries, or river systems. It is in the turbulent flow of a fluid, cardiac arrhythmias, and protein formation. Chaotic tendencies are also in the fluctuations of business cycles, the arms race, and wildlife populations. In other words, chaos is in many of the everyday processes we encounter in the real world. Still, does that mean that the world is really without order or structure? Are events and occurrences random and without explanation, or is there good reason they act that way?

A truly random sequence of events is one in which anything that can ever happen can happen next. Usually it is also understood that the probability that a given event will happen next is the same as the probability that a like event will happen at any later time. I flip a coin, and, no matter how many heads I have flipped in a row, the possibility of another heads (or tails) is equal and not determined by any previous flips. If we already know the probability, knowing in addition the outcome of the last toss cannot improve our chances of guessing the outcome of the next one exactly.

3 There is also a difference between occurrences appearing random and truly random ones, as chaos or chance can seem the likely explanation, and be misappropriated as labels. A deterministic sequence is one in which only one thing can happen next; that is, its evolution is governed by physical laws. Randomness in the broader sense is therefore identical with the absence of determinism.

4 If the world is strictly deterministic, then all events are locked in a matrix of cause and effect. The past and the future are contained in the present, in the sense that the information needed to construct both states of the world are folded into its present state.

5 In The Open Universe, Karl Popper explains this collapsing of past, present, and future into one co-present continuum:

The intuitive idea of determinism may be summed up by saying that the world is like a motion-picture film: the picture or still which is just being projected is the present. Those parts of the film which have already been shown constitute the past. And t hose which have not yet been shown constitute the future.

In the film, the future co-exists with the past; and the future is fixed, in exactly the same sense as the past. Though the spectator may not know the future, every future event, without exception, might in principle be known with certainty, exactly like the past, since it exists in the same sense in which the past exists. In fact, the future will be known to the producer of the film to the Creator of the world.

6 Total predictability is an important concept because of its close relationship to determinism. Predictable is not the same as deterministic for the same reason that epistemology is not metaphysics: just because no one could ever use the equations to find that single trajectory the world/system follows does not mean it is not there.

7 If a model is totally predictable, it must therefore be deterministic. Unfortunately, the contrary is not necessarily true; a determined system can behave in ways that do no t conform to our predictions.

To begin to understand several senses of determinism in complex systems, we will look at four levels of inquiry, moving from simpler to more complex, and can then test for one (or more) in each system:8

  1. Differential Dynamics: Are differential equations sufficient for description of the system? In this system the future depends on the present in a mathematically specificative way. This description cannot rely on any chance, probabilities or ave rages within the subsystems--it must be calculable precisely on the microlevel. The proposal is: use mathematical expressions to model the changes of physical systems; seek to understand or predict the future by relating it to the past with mathematical rules. Perhaps these rules will provide strictly unique implications and perhaps not, but keep trying to explain or predict until you cannot any more. This is based on a cosmological belief in the notion that the next moment in the evolution of the universe flows out of this one according to intelligible rules.

  2. Unique Evolution: Is the evolution of the system uniquely fixed once we specify the state of the system at any one moment? If two worlds agree on everything at one time, they must agree on everything at all other times. This layer of determinism holds that the complete instantaneous description of a deterministic system fixes the past and future with no alternatives. On the level of a metaphysical presupposition, this simply means that the universe as a whole is a fully deterministic system.
  3. Value Determinateness: Do all properties of the system have well-defined real values? After all, if there are properties whose values are spread out or somehow indistinct, the system would seem insufficiently set, fixed, specified, and determined by the laws of physics.

  4. Total Predictability: Is the system predictable by a superior intelligence? This would mean predictability to an all-powerful intelligence or computational scheme, given complete information of instantaneous conditions and the complete set of physical laws. Additionally, there is Popper's notion of metaphysical determinism,which says that all events in this world are fixed, or unalterable, or predetermined. It does not assert that they are known to anybody, or predictable by scientific mean s. But it asserts that the future is as little changeable as the past
    9 The future is fixed now and for all time, even though the above four layers of determinism could fail to hold. This notion actually seems more concerned with the present truth values of future-tensed statements, but it does illustrate that predetermined is not the same as predestined.
    10Determinism can therefore be seen in varying potencies of metaphysics, each with a different focus or line of attack. We will now look at the properties of chaotic systems to see why we would ask these questions at all.

I will borrow a working definition for chaos theory from Dr. Stephen Kellert: the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems.11 I should briefly dissect some of these terms to better describe what is and what is not chaotic in nature:

  1. Chaos is qualitative in that it seeks to know the general character of a system's long-term behavior, rather than seeking numerical predictions about a future state. What characteristics will all solutions of a system exhibit? How does this system change from exhibiting one behavior to another?

  2. Chaotic systems are unstable since they tend not to resist any outside disturbances but instead react in significant ways. In other words, they do not shrug off external influences but are partly navigated by them.
  3. The variables describing the state of a system do not demonstrate a regular repetition of values and are therefore aperiodic. This unstable aperiodic behavior is highly complex since it never repeats and continues to show the effects of the disturbance(s).

  4. These systems are deterministic because they are made up of few, simple differential equations, and make no references to implicit chance mechanisms. This is not to be completely equated with the metaphysical or philosophical idea of determinism (that human choices could be predetermined as well).

  5. Finally, a dynamic system is a simplified model for the time-varying behavior of an actual system. These systems are described using differential equations specifying the rates of change for each variable.

Edward Lorenz would stretch the definition of chaos to include phenomena that are slightly random, provided that their much greater apparent randomness is not a by-product of their slight true randomness. In other words, real-world processes that appear to be behaving randomly,perhaps the falling leaf or the flapping flagÑshould be allowed to qualify as chaos, as long as they would continue to appear random even if any true randomness could somehow be eliminated.12

What this means is when we make slight changes to a system at one time, and the later behavior of the system may soon become completely different. In Lorenz' meteorological computer modeling, he discovered the foundation of mainstream chaos: that simply- formulated systems with few variables could display highly complex behavior that was unpredictable and unforseeable. He saw that slight differences in one variable had profound effects on the outcome of the whole system. In Chaos parlance, this is referred to as sensitive dependence on initial conditions. In real weather situations, this could mean the development of a front or pressure-system where there never would have been one in previous models. In differential plotting this took on a new form called a strange attractor (see figure 1). Initial conditions need not be the ones that existed when a system was created, but may be the ones at the beginning of any stretch of time that interests an investigator.13

The Butterfly Effect explains this sensitive dependence by hyperbole in a (now) famous paper: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?
14 Lorenz was obviously not attributing a large-scale event solely to one butterfly, but any attempt to predict the weather with long-term precision would fail utterly unless it took into account all data, including all butterflies, with complete accuracy.
15 He also postulated the contrary; that the absence of the butterfly could also prevent the tornado. A curious literary foreshadowing of this premise is found in Ray Bradbury's "A Sound of Thunder", where the death of a prehistoric butterfly, and its consequent failure to reproduce, change the outcome of a present-day presidential election.16

It should be remembered then, that nonlinear systems are not "breaking the rules" in any way, but actually play by them in the strictest sense. Chaos is an understanding of 'absolute causality' that tries to take into account all variables as important t o the process and the final outcome.

The Heisenberg Uncertainty Principle
Theoretical Physics has made a contribution to the Determinism / Predictability discussion, by suggesting that the best descriptions of the macro come from close analysis of the micro. In the 1920s, Werner Heisenberg's Principle of Quantum Uncertainty appeared and seemed a barrier for any future attempts to describe the natural world as totally predictable or deterministic. The atomic-scale phenomenon has indeterminism built into it at a fundamental level.
17 Sixty years later Stephen Hawking stands by th is notion: The Uncertainty Principle signaled an end to Laplace's dream of a theory of science, a model of the universe that would be completely deterministic. One certainly cannot predict future events exactly if one cannot even measure the present stat e of the universe precisely! Quantum mechanics, therefore, introduces an unavoidable element of unpredictability and randomness into science.
18 This would imply that the final limit on predictability is limited to the laws of elementary particles as rule d by the uncertainty principle. The fact that both the position and momentum (velocity and direction) of a subatomic particle cannot be known simultaneously should indeed tell us something of the nature and behavior of larger structures and systems.

The implications of the photoelectric effect were not realized until 1926, when Heisenberg pointed out that it made it impossible to measure the position of a particle exactly. To find a particle and size it up, you must shine light on it, and Einstein h ad shown that you couldn't use a very small amount of light; you had to use at least one packet, or quantum. This light would disturb the particle and cause it to move at a new speed in some direction. The more accurately you try to measure the position o f the particle, the greater the energy of the packet you would have to use and thus the more it would disturb the particle. However we try to measure the particle, the uncertainty in its position, times the uncertainty in its speed, would always be greater than a certain minimum amount (called Planck's constant).
19 It should be noticed that the means of testing, in this case, is what actually contributes the inaccuracy in measurement.

The practicalities of universal and completely accurate measurements aside, the uncertainty principle would seem to indicate that, at the simplest level of physical construction, we lack any ability to describe and quantify any given state of a system.
20 Because of sensitive dependence on initial conditions, and the accuracy necessary to make reliable, long-term predictions, nonlinear systems can never be truly predictable.21

Kellert's argument against determinism also rests upon uncertainty:

Quantum mechanics says a one-particle system cannot be said to have a point-like state in state space: the totality of physical information about it suffices only to identify it as a patch of finite area with a lower bound on its size.

Chaos theory says that two otherwise identical chaotic systems with slightly different initial conditions will eventually diverge greatly, no matter how small the initial difference.

Two physically identical chaotic systems with identical boundary conditions and laws and with their one particle in the same physical state at t0 can be in different states at t > t0. That is, determinism as uniqueness of evolution fails to hold.

Physicist John Earman argues that this does not necessarily defeat determinism. The uncertainty relations tell us that attempting to specify the state suffices only to associate the particle with a patch, not a point. That is, the universe may well evolve along not a one-point-thick trajectory but a slightly blurry trail with some nonzero thickness (see figure 2). He relies on the principle of unique evolution as the essence of determinism to rescue the concept itself.22

However, Chaos theorists want to shrug off quantum uncertainty as irrelevant. James Gleick insists that when we look for fundamental laws with the greatest generality, the most profound laws, the laws with the greatest explanatory power  we must look outside of quantum physics.

23 The laws of quarks and gluons, or quantum electrodynamics do not explain fluid turbulence, the formation of snowflakes, rivers, the balance of nature, or the Great Red Spot of Jupiter. He contends that if you could imagine a universe with no Heisenberg uncertainty principle, you would have a universe in which it would be precisely as difficult as it is in our universe to predict next Sunday's weather; or to predict what will happen to the price of oil next month; or to predict just about anything about the behavior of any macroscopic complex system.

24 This is because Chaos is antireductionist, and, because of its existence in the world outside of particle accelerators, will not simply behave as the sum of its parts. The sensitive dependence on initial conditions and the characteristics of strange attractors take the systems outside of periodicity, predictability, and stability.

John Polkinghorne, a particle physicist and Anglican theologian, also denies that quantum theory solves the question of determinacy and would rather rely upon the exquisite sensitivity of systems. Everyday openness should not have to depend on goings-on in the microworld.

25 He, and many other chaos theorists, thinks of cells and human beings as being as fundamental as quarks and gluons, suggesting an ontological egalitarianism which does not assign a uniquely fundamental role to elementary particle physics. He therefore hopes for an emergence of understanding up and down the ladder of complexity.
26 Ilya Prigogine goes on to demand science to describe a world of which we can conceive ourselves as inhabitants. He gives primacy to behavior over equations, of interpreting deterministic chaos as pointing to an actual physical world of subtle and supple character whose process is open to the future.
27 Edward Lorenz suspects that the general behavior of the swinging pendulum, the rolling rock, the breaking wave, and most other macroscopic phenomena would not be noticeably altered even if quantum events occurred at regular predictable instants, or at chaotically determined instants, instead of randomly.
28 A part of the mentality that is shared by these thinkers is that chaos is anti-reductionistic, and that to understand its concepts and categories we must not necessarily look to physics of the components alone.29

John Polkinghorne adds some important philosophical and theological points to this issue. The first is that chaotic systems are intrinsically unpredictable. Because unpredictability is an epistemological statement about what we can know, Polkinghorne wants to suggest that the physical world is an open process, not just spelling out what was implicit from the past, but genuinely novel, genuinely becoming the history of the universe.
30 As a critical realist, he holds that we possess maps of the physical world sufficiently accurate for many, but not every, circumstance. A critical realist also believes that what we know and what is the case are closely connected. The mainstream understanding of quantum theory sees the uncertainty principle as expressing a genuine ontological indeterminacy, rather than a merely epistemological ignorance. In an exactly similar way, it seems natural to me to interpret the undoubted unpredictability exhibited by chaotic systems as pointing to a genuine openness in the process of the physical world.
31 Furthermore, Polkinghorne supports openness to provide some sense of reconciliation of physics with our basic experience as human beings of responsibility and agency, as we help to bring about the state of our universe.

Assisted by Polkinghorne, I would like to suggest four theological points to be gleaned:32

  1. In the act of creation, freedom is given to the whole cosmos to be and make itself,much as Philip Hefner's idea of 'created co-creators',taking into account physical evil (disease and disaster), and moral evil (the bad choices of human-kind). God does not will the act of a murderer nor the incidence of cancer, but he allows both to happen in a world which he has endowed with the ability to be itself. This then brings to life the problem of to what extent God has reserved the right to intervene in this world.

  2. God's action within the cloudiness of unpredictable open process will always be hidden; it cannot be demonstrated by experiment, though it may be discerned by faith. Its nature also limits what is sensible to pray for. Origen recognized that one should not pray for the cool of spring in the heat of summer.

  3. God is not to be overly joined to physical process, thereby treating him as a cause among other causes, an invisible agent or force in cosmic process. Nor should God be seen as a "god of the gaps", since God is not progressively eliminated as we come to new knowledge and insights through science or religion.

  4. The metaphysical exploration of chaos theory is one of a world of true becoming. A deterministic world is one of being and not becoming, in which God would therefore see time all at once. Chaos theory's intimation of openness changes all that. Polkinghorne goes so far as to suggest the eternal viewpoint is no longer a coherent possibility, in the same way that classical relativity saw space-time as frozen chunks of history. The future is not there waiting for us to arrive; we make it as we go along.

There are then two major consequences for our view of God: first, God will have an intimate connection with the reality of time. This actually corresponds with the God familiar to Abraham, Isaac, and Jacob, who was deeply involved in the history of his people. There is also an eternal aspect to God, which disallows God to be in the "flux of becoming", but intimately and interactively relating with the world. Second, God does not know the future. He stresses that this is no imperfection in the divine nature, for it is not extant to know yet anyway. God is, however, ready for it,not caught unprepared,but even he does not know beforehand what the outcome of a free process or a free action will be.

This has serious repercussions to the questions of sin, evil, and theodicy. This conclusion may also come (if not already?) from a different philosophy or science, but needs support since it sounds logically appealing and (perhaps) spiritually disheartening. Is God, in this view, constrained by time in some aspect, even if by choice? The Old Testament often points to a God who changes God's mind, and Jurgen Moltmann writes about a suffering God who is affected by our decisions. Is there Biblical evidence that this is the case? Perhaps a larger question is: Can the discoveries answers to deeper philosophical problems be considered a form of revelation? Is chaos being revealed to us as a component of design in our world, or is that concept defeating t he idea of indeterminism itself? Regardless, the hypothesis that God is not aware of an uncreated future may tend to incite negative responses from more than conservative minds. With all our efforts to believe and convince and support the thought that we are truly free and privileged as responsible creatures, chaos may have more in store for us than we asked for.

PERSONAL COMMENTS: Because of pages limits and the mountains of background necessary to get into larger issues, I kept with your suggestion to focus carefully and not let the paper grow out of control. With that as a goal in my further reading and writing, I chose the area of predictability, paying attention to quantum mechanics' uncertainty principle and in relation to chaotic systems.

I have seen the attractiveness and relative popularity of chaos theory through my research as well as in the media over the last few years. While Michael Crichton and Steven Spielberg have both profited from simpler versions of "Pop Chaos", the real chaos is so much more profound and challenging to me. I have found the entire field to be very complicated in itself, as interpreted by various disciplines in such different ways. There is no single formula for chaos, nor is there an adequate one-sentence definition. There are so many important ideas left out of this paper,fractals, bifurcations, Mandelbrot series, three-variable models and Hamiltonian systems,that would all take another semester alone to figure out. As is usually the case in my research paper s, I came to the point late in my studies where I felt like I didn't know anything about this on a larger scale, and quite incompetent to try and put it all together. This is usually a good sign that I am on the right track in research, but no guarantee t hat I will pull it off! In encountering the whole issue of determinism / indeterminism, I continually dug up more and more articles, yet not many dealt directly with the concept of chaos. Stanley Jaki proved to be hostile towards any concept of chaos and regards it mainly as a contradictory enterprise. I am sorry I couldn't present his arguments here.

The entire issue around the uncertainty principle was difficult. Hawking insists that philosophers and thinkers have still not taken it seriously enough or to its logical implications. Then Gleick and others treated it as trivial on a macroscopic level anyway, arriving at unpredictability by another rationale. There is obviously constant disagreement about where science should be headed, but their comments raise good questions.

There are other areas to explore in this new field, and branches into other disciplines that may be productive for future papers or courses:


  1. Lorenz 1993, 159. As a meteorologist he anticipated sidestepping disasters and dangerous conditions.
  2. Hawking 1993, 143.
  3. Lorenz, 6-7
  4. Ibid., 7.
  5. Davies 1992, 29.
  6. Popper 1956, 5.
  7. Kellert 1993, 65.
  8. These four are from Kellert, 49-62.
  9. Popper 1956, 8.
  10. Kellert, 61.
  11. These descriptions are from Stephen Kellert's thorough definition, chapter 1. Kellert 1993, 1-7.
  12. Lorenz, 5.
  13. Lorenz, 9. Therefore, one person's initial conditions may be another's midstream or final conditions.
  14. This paper was the first to use the phrase, and was delivered to the 139th meeting of the American Association for the Advancement of Science on December 29, 1972. It can be found in Appendix 1 of Lorenz 1993.
  15. Kellert 1993, 12-13.
  16. Cited by Lorenz (1993, 15), this short story was written in the 1970s, and appears in The Stories of Ray Bradbury, 1980. New York: Alfred A. Knopf, pp. 231-241.
  17. Davies 1992, 30.
  18. Hawking 1988, 56.
  19. Hawking 1993, 77.
  20. Hawking 1988, 54-56.
  21. Lorenz, 8-9.
  22. quoted in Kellert, 69. See John Earman. 1986. A Primer on Determinism. Dordrecht: D. Reidel.
  23. Gleick 1993, 125. Kuhn points out that science is not necessarily in need of leading theories or goal-theorems, but should instead be free to work towards true and useful findings in research (Structure, chap. 12). He may then discount Physics' current fascination with a TOE (Theory of Everything).
  24. Ibid.
  25. Polkinghorne 1993, 110.
  26. Ibid.
  27. Holte, 55-84.
  28. Lorenz, 159.
  29. Polkinghorne, 110.
  30. Ibid., 108.
  31. Ibid.
  32. Ibid., 114-117. These are the ideas of his I felt are more fruitful to this discussion.


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HTML compiled 19 March, 1996.