Theological Implications of Chaos Theory
Determinism, Total Predictability and the Uncertainty Principle in Chaotic Systems: Theological Implications
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
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:
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
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:
Diogenes Allen. 1985. Philosophy for Understanding Theology. Atlanta: John Knox Press.
Jack Cohen and Ian Stewart. 1994. The Collapse of Chaos: Discovering Simplicity in a Complex World. New York: Viking.
Paul Davies. 1992. The Mind of God. New York: Simon & Schuster.
James Gleick. 1987. CHAOS: Making a New Science. New York: Penguin.
----------. 1993. "Chaos and Beyond" from Chaos: The New Science, Nobel Conference XXVI. Ed. John Holte. Lanham, MD: University Press of America, Inc.
Nina Hall, Ed. 1991. Exploring Chaos: A Guide to the New Science of Disorder. New York: W.W. Norton.
David Harvey. 1990. The Condition of Postmodernity. Cambridge: Blackwell.
Stephen Hawking. 1988. A Brief History of Time. New York: Bantam.
----------. 1993. Black Holes and Baby Universes and Other Essays. New York: Bantam.
Philip Hefner. 1993. The Human Factor: Evolution, Culture, and Religion. Minneapolis: Fortress Press.
Douglas R. Hofstadter. 1981. "Chaos Theory." Scientific American. November.
Stanley L. Jaki. 1990. The Only Chaos & Other Essays. Lanham, MD: University Press of America, Inc.
Stephen H. Kellert. 1993. In the Wake of Chaos. Chicago: University of Chicago Press.
Thomas S. Kuhn. 1970 . The Structure of Scientific Revolutions. 2nd ed. Chicago: University of Chicago Press.
Edward N. Lorenz. 1993. The Essence of Chaos. Seattle: University of Washington Press.
Ivars Peterson. 1993. Newton's Clock: Chaos in the Solar System. New York: W.H. Freeman & Co.
John Polkinghorne. 1993. "Chaos and Cosmos: A Theological Approach" from Chaos: The New Science, Nobel Conference XXVI. Ed. John Holte. Lanham, MD: University Press of America, Inc.
Karl R. Popper. 1956. The Open Universe: An Argument for Indeterminism. Totowa, NJ: Rowman & Littlefield.
Ilya Prigogine. 1993. "Time, Dynamics, and Chaos: Integrating Poincar's «Non-Integrable Systems»" from Chaos: The New Science, Nobel Conference XXVI. Ed. John Holte. Lanham, MD: University Press of America, Inc.
Elliott Sober. 1994. From a Biological Point of View: Essays in Evolutionary Philosophy. Cambridge: Cambridge University Press.
E. B. Ubarov & Alan Isaacs. 1993. The Penguin Dictionary of Science. London: Penguin.
HTML compiled 19 March, 1996.