**Determinism and Chaos**

**Richard H. Bube
** Department of Materials Science & Engineering

Stanford University

Stanford, CA 94305

From: *Perspectives on
science and Christian Faith ***41** (December 1989)

**T**he nature of the interaction between
"determinism" and "chance" has been the
subject of continued debate in one form or another from the early days of
recorded human thought. Theologically, it is well known as the "predestination vs. free will" debate, although the actual
connection between the theological debate and the scientific debate is tenuous
at best. The development of science in the last few centuries has given it a new
intensity, since scientific descriptions must fall either into the category of
deterministic or chance (probabilistic), neither of which as an isolated world
view is compatible with biblical concepts of human responsibility. A thumbnail
sketch of the question and its implications has recently been given in this
journal.^{1} I believe it is fair to say that
evangelical Christians with a commitment to both authentic science and authentic
biblical theology largely follow the lead of those like MacKay who maintain the
existence of a reality in which both determinism and chance are intricately and
sometimes even mysteriously interrelated.^{2}

A curious and fairly dramatic twist to this debate has been given in recent years by the scientific recognition of the state known as "chaos." In the popular mind, chaos is what one would expect in a completely random or chance-oriented environment; we have been delivered from chaos by the existence of order (deterministic relationships). Some of the early ideas of creation dealt with God's overcoming chaos with order, again emphasizing the common expectation that these two kinds of description are mutually exclusive.

The contention that we ought to expect complex interactions between determinism and chance, or between order and chaos, has found a rather dramatic expression in recent discovery of those specific effects that have come to be known as "chaos." It is the purpose of this communication to illustrate the type of effect observed (in one of its simplest manifestations).

A recent insert, in *Science* magazine,
entitled "A Simple Model of
Chaos,"^{3} describes a model based on
population biology that illustrates nicely how "chaos" can
proceed from a deterministically described process. The population of a
particular insect species in one year N_{t} is related
to the population in the following year N_{t+1}
approximately by the following relation:

N_{t+1} =
*alpha*N_{t}(1 -
N_{t}) (1)

Here N_{t} is expressed in appropriate
units so that its numerical values fall between 0 and 1 (a negative value for
N_{t+1} would imply extinction), and *alpha *is a constant that controls the specific form that
the results of Eq. (1) take over a number of generations. In order to express
the implications of Eq. (1) it is necessary to choose a value for *alpha* and an initial value for
N_{t}. It is the extreme sensitivity of Eq. (1) to
small variations in the initial value of N_{t} for
certain values of *alpha* that characterizes chaotic
behavior.

Figure 1 shows the variation of the
"population" with the number of generations from 1 to 100,
for values of *alpha* between 2.9 and 3.5, and an initial
value of N_{t} = 0.50. It can be seen directly from Eq.
(1) that if *alpha* = 2.0 when
N_{t} = 0.50, N_{t+1} =
N_{t}, and the population is unchanged with
successive generations. When *alpha* = 2.9, early
generations show alternating values which quickly decay down to a "population" of about 0.65 within 20 to 30 generations. When
*alpha* = 3.0, the decay of the two alternating values is
much slower and persists out to 100 generations, so that the 99th generation
shows N_{t} = 0.68 and the 100th generation shows
N_{t} = 0.64. When *alpha* = 3.3,
there are again two alternative values (0.82 and 0.47) but these are stable in
alternate years over the range from 1 to 100 generations. When *alpha* = 3.5, the number of alternative values (0.87, 0.82,
0.50 and 0.38) jumps to four, and these are unchanging from 1 to 100
generations. Although these four cases show increasing complexity, they also
give the appearance of an ordered and structured complexity.

When *alpha =* 3.57, this ordered
behavior gives way to chaos. The data points in Figure 2 show the "populations" for
*alpha* = 3.9 and for
three cases in which N_{t} = 0.49, 0.50 and 0.51, three
numbers differing by only 2% from one another. Right from the first generation
on, the points jump around in a random fashion. For the first 10 generations the
points for the three different initial values of N_{t}
are approximately the same, but after 20 or 30 generations, major differences
between the three sets of data arising from different initial values of
N_{t} are evident. Table 1 lists the specific "population" values shown in Figure 2 for a few selected
later generations, showing the very strong influence of the small difference in
the initial values of N_{t}.

These results illustrate how the condition of chaos can be generated from a deterministic relationship. Counter examples are also available that show how an orderly pattern or structure can be obtained from a large set of random events suitably limited by appropriate boundary conditions: the generation of order out of chaos. There is no necessary profound philosophical or theological implications in these results, but they do warn us to avoid simplistic dichotomies between deterministic and chance processes as we face interactions between science and theology.

©1989

** **

**NOTES**

^{1}R.H. Bube, "Penetrating the
Word Maze: Determinism/Chance," *Perspectives on Science and
Christian Faith*, ** 41**, March (1989), p. 37.

^{2}D.M. MacKay, *The Open Mind
and Other Essays: A Scientist in God's World*, edited by Melvin Tinker.
(Leicester, England: Inter-Varsity Press, 1988).

^{3}R. Pool, "A Simple Model of
Chaos," *Science* 243, 311 (1989).