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Paul H. Carr ( led the Component Technology Branch of
the Air Force Research Laboratory, Hanscom AFB, MA 01731, where he is emeritus; email
by Paul H. Carr
Abstract. Albert Einstein and Huston Smith reflect the old metaphor
that chaos and randomness are bad. Scientists recently have
discovered that many phenomena, from the fluctuations of the stock
market to variations in our weather, have the same underlying order.
Natural beauty from plants to snowflakes is described by fractal geometry;
tree branching from trunks to twigs has the same fractal scaling
as our lungs, from trachea to bronchi. Algorithms for drawing
fractals have both randomness and global determinism. Fractal statistics
is like picking a card from a stacked deck rather than from one
that is shuffled to be truly random. The polarity of randomness (or
freedom) and law characterizes the self-creating natural world. Polarity
is in consonance with Taoism and contemporary theologians
such as Paul Tillich, Alfred North Whitehead, Gordon Kaufman,
Philip Hefner, and Pierre Teilhard de Chardin. Joseph Ford뭩 new
metaphor is replacing the old: 밎od plays dice with the universe, but
they뭨e loaded dice.
Keywords: chaos and complexity; contemporary theologians; evolution;
fractal geometry; fractals; genetic algorithms; loaded dice;
polarity; randomness and law; science and religion.
Albert Einstein once said, 밒 am convinced the Old One [God] does not
play dice (Jammer 1999, 222). Huston Smith, whose book The World
Religions (1991) has sold over a million copies, stated, 밒 do not believe
that God could have created us in his image by the mutations of the genes
(Smith 2000). New findings about the fractal geometry of nature, chaos,
and complexity challenge these negative statements about the statistical
nature of the physical world (Gleick 1987). Einstein and Smith reflect the
[Zygon, vol. 39, no. 4 (December 2004).]
2004 by the Joint Publication Board of Zygon. ISSN 0591-2385
934 Zygon
old metaphor: chaos and randomness are bad. Scientists have recently
discovered that many phenomena, from the fluctuations of the stock market
to variations in our weather, have the same underlying order. Natural
beauty in mountains, plants, and snowflakes reveals a new fractal geometry
characterized by the complex interplay between randomness (symbolized
by dice) and global determinism (which loads the dice) (Mandelbrot
1983). Darwin뭩 theory of evolution is similar to fractal analysis: the randomness
of mutations and global natural selection. We shall see how old
metaphors are being replaced by the new, such as one by chaos theorist
Joseph Ford: 밎od plays dice with the universe, but they뭨e loaded dice
(in Gleick 1987, 314).
밡oise is good, says Robert Hilborn (2000; 2001). Random noise
added to a signal can increase its detectability in a system having a threshold.
A familiar example is a hearing test in which one is asked to press a
button as soon as the sound is loud enough to be heard. The noisy hiss
added to the coherent sound signal enables it to trigger the hearing threshold.
Thus, a lower-level signal with noise can be detected better than one
without noise. Of course, the noisy hiss can not be so large as to drown
out the signal. 밃 little noise is good is a more precise statement.
What might appear as random noise has in some cases been discovered to
have an underlying order. For example, the fluctuations of the stock market
obey fractal statistics (Peters 1994). A number of physicists are using
this analysis as an investment strategy to make money on the market (Bass
1999). Nature offers many examples of fractal statistics: branching in our
lungs and in plants; variations in the flooding of the Nile river, of rainfall,
and of tree-rings (Peters 1994).
Fractals have the property of self-similarity in that the parts are in some
way related to the whole. In the fractal, or Sierpinski, triangle (Figure 1),
the one central triangle has sides that are 1/2 that of the large one in which
it is enclosed. The three second-generation triangles have sides that are 1/
4 of the large one, the nine third-generation triangles have sides 1/8 of the
large triangle, and so on. The scaling factor between generations is 1/2.
We will now play the Chaos Game to generate the fractal triangle. First,
we randomly draw a point, as shown in the left triangle in Figure 2. Next,
we roll a die to find the direction of the next point. In the triangle on the
right, the roll of a die gave a 5 or a 6, which corresponds to vortex C. Then
we apply the global rule and draw a point halfway to C (5,6). This algorithm
is repeated again at each new point.
A computer was programmed to apply this rule 10,000 times. The first
50 points were discarded as 뱓ransients, and the remaining 9,950 points
formed the black granular background inside the orderly fractal (Sierpinski)
Paul H. Carr 935
Fig. 1. A fractal (Sierpinski) triangle.
Fig. 2. Illustration of the Chaos Game.
triangle. The Chaos Game shows that local randomness and global determination
can coexist to create an orderly, self-similar structure called a
fractal (Peters 1994, 1017).
The branching of the tubes in our lungs as well as that of plants is described
by fractal scaling. For example, the diameter D(G) of our bronchial
tubes is related to the diameter of our main trachea D(0) by
D(G) = D(0) 2 exp (-G/3)
where exp means that 2 is raised to the fractional power -G/3. G represents
the generation number 1, 2, 3, . . . . Each generation or set of smaller
tubes is scaled down by this factor (Peters 1994, 13). Of course, not all of
the smaller tubes are the same size, but they have a statistical distribution
about the mean value given by the global equation above. Fractals have
global determinism (average tube size) and local randomness (diameter of
individual branches). The fractal structure is more stable and error-tolerant
than the more deterministic, Euclidean geometry. This is why fractals
are so common in nature from tree trunks and branches to the intricate
vein structure of leaves.
The variations in natural phenomena such as the flooding of the Nile
River, rainfall, tree rings, and noise-caused errors in electronic transmission
lines also can be characterized by fractal statistics. H. E. Hurst (1900
936 Zygon
1978) worked on the Nile River Dam Project. Most hydrologists had
assumed that water inflow was a random process. Hurst, however, studied
the 847-year record that the Egyptians had kept of the Nile River뭩 overflows,
from 622 to 1469 A.D. The record did not appear random to Hurst.
Larger-than-average overflows were likely to be followed by more large
overflows. Suddenly, the process would change to a lower-than-average
overflow (the Joseph Effect: seven years of great plenty in the land of Egypt
were followed by seven years of famine). Overall there appeared to be
cycles, but their length was nonperiodic. Hurst뭩 mathematical analysis
revealed that the Nile river뭩 overflows were described by fractal statistics,
which is more like picking a card from a stacked deck than from one that
has been shuffled to be truly random. The stacking algorithm is the global
rule that characterizes fractal statistics.
Charles Darwin뭩 theory of evolution is similar to fractal analysis in that it
includes the individual randomness of mutations and the global determinism
of natural selection. After reading Darwin뭩 Origin of Species in 1859,
Henry D. Thoreau wrote, 밫he development theory [evolution] implies a
greater vital force in nature, because it is more flexible and accommodating,
and equivalent to a sort of constant new creation (1993, 102; emphasis
Darwinian evolution does not have design built in as a premise, but the
design emerges as variations occur and some organisms get naturally selected
over others (Ruse 2003). It is analogous to the algorithm for generating
the fractal triangle, whose intricate design containing smaller triangles
emerges as a result of applying randomness with the global law.
The engineering community has discovered that computer simulations
of Darwin뭩 evolution, called genetic algorithms, are very effective for optimizing
physical devices. Edward Altshuler of the Air Force Research Laboratory
discovered a genetic algorithm that enabled him to design antennas
having much better performance than he would have been able to conceive
of based on his many years of conventional design experience (Altshuler
and Linden 1999). The genetic algorithm starts with millions of randomized
antenna dimensions. Each antenna뭩 performance is then calculated,
and the best are selected as being closest to that desired. The next generation
of antennas is made by 뱒exually mating the best antennas: half the
dimensions of each new generation are chosen from the old generation.
These theoretical design predictions are in close agreement with experimental
performance. Darwin뭩 theory of evolution, discovered from nature,
is very effective at optimizing the man-made world.
The development of a baby뭩 neural system is a good example of natural
selection. A baby has a number of neurons in parallel. The nerve that is
Paul H. Carr 937
dominant and used is the one that survives. The other nerves atrophy and
die뾞 good example of 밬se it or lose it.
Biochemist Arthur Peacocke, winner of the 2001 Templeton Prize for
Progress in Religion, states, 밒nstead of being daunted by the role of chance
in genetic mutations as being the manifestation of irrationality in the universe,
it would be more consistent with the observations to assert that the
full gamut of the potentialities of living matter could be explored only
through the agency of rapid and frequent randomization. This is possible
at the level of DNA (1998; see also Peacocke 1995). Chance operating
within a lawlike framework is the basis of the inherent creativity of the
natural order in its ability to generate new forms of matter and life. As in
many games, the consequences of the fall of the dice depend on the rules
of the game.
Biologist Stuart Kauffman of the Santa Fe Institute writes, 밪elf-organization
mingles with natural selection in barely understood ways to yield
the magnificence of our teeming biosphere (2000, 2).
Steven Wolfram뭩 best-selling book A New Kind of Science (2002) shows
that randomness can evolve into order and vice versa. Wolfram uses a rule
or recursion relation called 밹ellular automaton to show that there are
conditions under which a random set of cells can evolve into an ordered
set. Starting with a row of randomly ordered black and white cells, he
applies the simple cellular automaton rule that a lower cell becomes black
if either of its upper neighbors is black. The row of random cells then
evolves into an orderly pattern. Conversely, there are other local recursion
rules, or cellular automatons, that cause an ordered set to evolve into a
random pattern. Cellular automata can be used to generate the hexagonal
patterns seen in snowflakes (Wolfram 2002, chap. 8) as well as pigmentation
patterns in mollusk shells. The latter, like a one-dimensional cellar
automaton, grow one line at a time, with new shell material being produced
by a lip of soft tissue at the edge of the mollusk. The simple local
rule of a cellular automaton can lead to the large-scale complexity observed
in nature.
Einstein, having published a paper on random Brownian motion, would,
I think, be fascinated by the discovery of fractal statistics if he were alive
today. And Huston Smith might want to reconsider his doubt that God
creates through the random mutations of evolution. He overlooked the
global determinism of natural selection.
Does God play dice? Yes and no. Yes, if one considers the random
nature of evolution and fractal statistics. No, if one considers their globally
deterministic laws and rules. 밎od plays dice with the universe, but
they뭨e loaded dice (Gleick 1987, 314). Like Hindu뭩 Shiva, God plays.
938 Zygon
The yes/no answer of this metaphor represents the polarity of randomness
and law that characterizes the self-creating natural world. This polarity is
in consonance with Taoism and contemporary theology. Theologian Paul
Tillich believes that we have freedom only in polar interdependence with
destiny and nature has spontaneity in polar interdependence with natural
law. This interdependence is seen by theologian Gordon Kaufman (1993)
as the serendipitous creativity of God. Continuing creativity is a common
theme for multipolar process theology and such theologians as Philip Hefner
and Pierre Teilhard de Chardin. Hefner believes that we are 밹reated cocreators
(1993), and Teilhard holds that evolution is converging toward
an Omega Point (1961, 287). Polarity is an intrinsic part of Taoism: good/
evil, male/female, light/dark, and so on. Thus, Taoism would accept randomness
and deterministic rules as complementing and balancing each
other in creative tension.
Tillich believed that religious truth is expressed by symbol and metaphor,
which should not be interpreted literally. Thus, 밎od plays dice, but
the dice are loaded should be understood as metaphor. Tillich emphasized
that God is not a being, who would be finite and limited, but the
ground of all being. Being is all-encompassing and includes both deterministic
(law) as well as statistical (dice) reality. Evolution is a manifestation
of New Being (Haught 2002). Tillich believed that God뭩 creativity
works though spontaneity of creatures and human freedom. This should
not be understood as God뭩 miraculous interference (Tillich 1963). Humans
have freedom in polar interdependence with destiny, which are analogues
of randomness (spontaneity) and law.
The polar interdependence between chance and law is seen by Kaufman
as serendipitous (fortunate) creativity, which is manifest throughout the
universe from the Big Bang on in trajectories. These directional movements
emerge in the evolutionary development of the cosmos and of life
(including human life) on planet Earth. Serendipitous creativity is a manifestation
of God.
Mathematician Alfred North Whitehead뭩 Process Philosophy and Reality
(1929) is multipolar in that events occur as the joint product of the entity뭩
past, including its genetic inheritance (law); its own action, self-creativity,
and freedom (chance); and divine purpose. Creativity, the principle of
novelty, is a universal of process theology. Our freedom eliminates a preordained
determinism. God does not coerce but lures and guides the universe
in the continuing process of evolution. God does not intervene in
discrete events but is present in all events in a role different from natural
causes. God 밶cts with and through other entities rather than acting alone
(Barbour 1997, 296). The Creator has a vision for the future rather than a
deterministic plan (Haught 2000). God has both a primordial and a conPaul
H. Carr 939
sequent nature. In the latter, the creative process affects God. Divine
purpose is a part of the evolutionary process.
We are created co-creators, according to Hefner (1993). God through
the process of evolution created us with the freedom and responsibility to
contribute to the ongoing process of creation. We are to 밷irth a future
that is most wholesome for nature and the human community that birthed
us (1993, 27).
Global laws and randomness have analogues in Christianity. Global
laws are similar to liturgy, such as the tradition of the Lord뭩 Supper. The
whisper of God뭩 grace is beyond our ability to predict and thus has an
element of surprise akin to serendipitous randomness. Grace is different
from Hinduism뭩 karma or cause and effect. 밎race is when God gives you
something you do not deserve, in contrast to: Mercy in which God keeps
you from getting what you deserve (Hedrick 1986).1
Teilhard sees evolution occurring in the spiritual as well as the biological
realm and converging toward the Omega Point, the final culmination of
the continuing creation in God. The evolution of the earth from geosphere
to hydrosphere, atmosphere, and biosphere is emerging toward a
world-encircling 뱊oosphere, created by human hearts and minds:
The end of the world: the wholesale internal introversion upon itself of the
noosphere, which has simultaneously reached the uttermost limit of its complexity
and centrality. The end of the world: the overthrow of equilibrium (Heat Death),
detaching the mind, fulfilled at last from its material matrix, so that it will henceforth
rest with all its weight on God-Omega. (Teilhard 1961, 287)
Does God play dice? The yes/no insight from fractal geometry is symbolized
in the self-similar patterns of the Sierpinski triangle, which can be
generated with an algorithm that has both a random and a lawful element.
Nature뭩 spontaneity and our freedom result in a universe that has beauty
and harmony. This self-creating universe with both randomness and law is
for me a manifestation of divine creativity. Process theology reminds us
that the universe is not static but an evolving and continuing creation,
whose intricacies result in continuing scientific discoveries. I pray that we
will have the wisdom to use the power of scientific knowledge as responsible
created co-creators and not as destroyers of our earth through the
unintended consequences of our technology. As created and creating creatures,
we can profit from religious wisdom. In it, I find hope that the
continuing creation is converging toward its ultimate consummation.
940 Zygon
A version of this article was presented at the CTNS Science and Religion Course Advanced
Workshop, 밡euroscience, Religious Experience, and the Self, Montreal, Quebec, Canada, 2
June 2001.
1. Samuel Hedrick discovered this saying at Christ Church, Oxford, U.K., where John Wesley
was educated.
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