Wildman, Wesley J. and Robert John Russell. “Chaos: A Mathematical Introduction with Philosophical Reflections.”
Wesley J. Wildman and Robert
John Russell’s article surveys the mathematical details of a single equation,
the logistic equation, which has become a hallmark of this field, at least
within the circles of “theology and science.” The logistic equation displays
many of the generic features of chaotic dynamical systems: the transition from regular
to apparently random behavior, the presence of period doubling bifurcation
cascades, the influence of attractors, and the underlying characteristics of a
fractal. They then raise philosophical questions based on the mathematical
analysis and conclude with possible theological implications.
The logistic equation is a
simple, quadratic equation or “map,” x_{n+1} = kx_{n}(1x_{n}),
which iteratively generates a sequences of states of the system represented by
the variable x. The tuning constant k represents the influence of the
environment on the system. One starts from an initial state x_{0} and a
specified value for the tuning constant k to generate x_{1}.
Substituting x_{1} back into the map generates x_{2}, and so
on. Although incredibly simple at face value, the logistic map actually
displays remarkably complex behavior, much of which is still the focus of
active scientific research.
The behavior of the iterated
sequence produced by the logistic map can be divided into five regimes. The
constant k determines which regime the sequence occupies as well as much of the
behavior within that regime. In Regime I, the sequence converges to 0. In
Regime II, the sequence converges on a single positive limit which depends on
k. In Regime III, bifurcations set in and increase in powers of two as k
increases. Moreover, the initial conditions have a significant permanent effect
on the system in the form of “phase shifts.” Chaos sets in in Regime IV. Here
chaotic sequences are separated by densely packed bifurcation regions and there
is maximal dependence on initial conditions. For most values of k, the
sequences seem to fluctuate at random and the periodic points found in previous
regimes appear to be absent. Nevertheless, for almost all values of k we
actually find highly intricate bifurcation structures, and the sequences fall
within broad bands, suggesting an underlying orderliness to the system. Finally
in Regime V, chaos is found on the Cantor subset of x.
There is no universally
accepted mathematical definition of chaos capturing all cases of interest. Defining chaos simply as randomness proves
too vague because this term acquires new and more precise shades of meaning in
the mathematics of chaos theory.
Defining chaos in terms of sensitive dependence on initial conditions
(the butterfly effect) results in the inclusion of many maps that otherwise
display no chaotic behavior. The
definition adopted here requires a chaotic map to meet three conditions: mixing (the effect of repeated stretching
and folding), density of periodic points (a condition suggesting orderliness),
and sensitive dependence.
Interestingly, in the case of the logistic map and many similar chaotic
maps, mixing is the fundamental condition, as it entails the other two.
The paper also addresses the
question of the predictability of chaotic systems. On the one hand, a chaotic system such as the logistic map is
predictable in principle, since the sequence of iterations is generated by a
strict governing equation. On the other
hand, chaotic systems are “eventu ally unpredictable” in practice, since most
values of the initial conditions cannot be specified precisely, and even if
they could, the information necessary to specify them cannot be stored
physically. Yet these systems are also “temporarily predictable” in practice,
since one can predict the amount of time which will elapse before mathematical
calculations will cease to match the state of the system. This leads to a definition of ‘chaotic
randomness’ as a tertium quid
between strict randomness (as in one common interpretation of quantum physics),
and the complete absence of randomness.
What implications does
mathematical chaos have for a philosophy of nature? It is superficial to say that the mathematical determinism of
chaotic equations requires metaphysical determinism in nature, because of
complexities in the experimental testing of the mathematical models used in
chaos theory. In particular, it may be very difficult to distinguish
phenomenologically between chaos, sufficiently complicated periodicity, and
strict randomness, even though these are entirely distinct mathematically.
There are additional practical limitations to the testing of chaotic models of
natural systems, including sensitivity to the effects of the environment (such
as heat noise or longrange interactions), and the fact that the development of
the physical system eventually out paces even the fastest calculations.
Two philosophical
conclusions are drawn from this. On the
one hand, the causal wholepart relations between environment and system, the
causal connnectedness implied in the butterfly effect, and the fact that much
of the apparent randomness of nature can now be brought under the umbrella of
chaos, are best seen as supporting evidence for the hypothesis of metaphysical
determinism. On the other hand,
however, there are profound epistemic and explanatory limitations on the
testing of chaos theory due to the peculiar nature of chaotic randomness. In this sense, chaos theory places a
fundamental and unexpected new limit on how well the hypothesis of metaphysical
determinism can be supported.
On the basis of these
philosophical conclusions, what relevance does chaos theory have for theology?
On the one hand, it will be “bad news” to those who simply assume that nature
is open to the free actions of God and people, and particularly bad news to
those who mistakenly appeal to chaos theory to establish this. On the other hand, chaos theory will be
irrelevant to theologians operating with a supervening solution to the problem
of divine action, such as Kant’s, that is able to affirm human freedom and
divine action even in the presence of strict metaphysical determinism. At still another level chaos theory is “good
news” to the theological project and “bad news” for “polemical determinists.” Due to the fundamental, new limitation in
the testability of chaos theory, one can never fully exclude the possibility
that classical physics as we now have it, including chaos theory, will be
replaced by a better model of the world at the classical
level which allows for divine causality in some way. This “opens a window of
hope for speaking intelligibly about special, naturallawconforming divine
acts, and it is a window that seems to be impossible in principle to close.”
The article includes an
extended bibliography of textbooks, key technical articles, experimental
applications, useful introductions and surveys, and selected works on chaos
theory and theology.
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