Fallacy
of the Assumption of Statistical Independence Between Successive Indeterminant
Events
by Thomas V. Gillman ’49

As
a preface I would like to mention a concept that bears on the relation
between physics as encompassed by science and metaphysics the branch of
philosophy that treats of the ultimate nature of existence, reality, and
experience.

Recent research in cosmology indicates that there exists a universal wave
function that determines everything in the universe. This unique field
gives being to the recognized physical fields-gravitational and electromagnetic
forces, the strong force that among other things is responsible for the
sun shining, and the weak force instrumental in radioactive decay. The
further possibility exists that life is a direct expression of the effects
of such a "force" field and that the evolutionary properties
of living matter provide evidence of the indeterminate nature of the universal
wave function.
There is a simple experiment that I believe demonstrates the existence
and the workings of such a universal wave function. The results, at the
very least, represent an instance of the indeterminate nature of the probability
aspect of wave mechanics at the macro level.

A Heuristic Experiment
The procedure is simply a matter of flipping a coin and recording the
result of each toss, head or tail, as well as the sequence of the results
for an arbitrarily large number of tosses- something in the order of 100
tosses. No attempt is made to maintain a uniform time interval between
tosses, since time apparently does not enter in.

The expectation is that in the long term the number of resulting heads
and tails will be approximately equal, since the likelihood of a head
or tail is 1/2. According to probability theory each toss of the coin
is an independent event, therefore, there is not supposed to be any relation
between successive tosses of the coin. The probability of any particular
sequence of heads or tails is therefore the product of their "independent"
probabilities. For example, the probability of tossing seven heads in
succession would be (112)7 or 1/128-a fairly unlikely sequence of events
but well within the range of expectation. On the other hand, it is common
knowledge that in many games of chance players often experience "runs
of luck" in which the outcome temporarily favors them. How are such
unlikely courses of events to be explained?

In conducting this experiment it is common that over the course of 100
or so tosses a sequence of at least six or more heads or tails will occur.
Even longer sequences are interrupted by only one or two inverse events,
thus establishing a trend or a "run" as it is often described.

Graphic Results
If one plots the sequence of tosses on the horizontal axis and the algebraic
results of the coin tossing on the vertical axis, with the simple assumption
that each toss represents a unit gain or loss of some sort of "potential"
from one toss to the next, some fascinating patterns emerge. These correspond
to the so-called "runs" of good or bad luck that gamblers experience.
A more interesting finding is that these deviations tend to propagate
or persist. That is, the number of heads or tails sometimes does not even
out for long sequences. [Note that these sequences are time independent
and therefore do not represent periods, but they do seem to indicate a
"progression".]

A question that comes to mind is whether or not the cumulative "potential"
indicated on the graphs provides evidence for the existence of a deterministic
element that enters into these results. The gambler will tell you that
when he is on a "roll" he is able to "influence" the
course of events. Who is to say? What is obvious is that if one bets in
concert with one of these "potential" swings-these apparent
"drifts" of the probability function-one is going to be ahead
of the odds for an indeterminate but substantial number of events!

Whereas in the previous plot, the number of opposing events (heads and
tails) is not far from the expected proportion, viz., 50:50, this is not
always the case, as shown in the following graph.

Hypothesis
While there is little basis for conclusion at this stage, the results
do lead to speculation. The partially determinate nature of the outcomes
of events that have traditionally been treated as indeterminate, random,
and independent may point to the possible fluctuations of something of
the nature of a general field which "determines" the course
of events. Events at the macro level are usually analyzed in terms of
cause and effect, but there are many events where the outcomes cannot
be predicted but can only be described statistically.

Another instance is radioactive decay at the subatomic level. Anyone who
has listened to a Geiger counter knows that these events occur randomly
but in time-wise bursts that exhibit no regularity. And there is no way
of predicting the decay of a particular radioactive atom in terms of time
or location. The best we can do is to establish a so-called half-life
— a period during which half of the atoms originally present will
have decayed. We have no way of knowing what conditions, if any, "cause"
the occurrence of the decay event.

The most that we know, at present, is something about the elementary particles
of matter that are involved in the process. It has been discovered that,
in the vernacular of elementary particle physics, the '"weak"
interaction causes radioactive decay of nuclear constituents and unstable
leptons, and is mediated by the massive W and Z bosons. Whatever precipitates
(causes) these weak interactions within a space-time frame is unknown,
at least as far as I am aware.

Further speculation
Now wouldn't it be interesting if it were found that what we call probability
is nothing more than the way in which the occurrence of events is modulated
by something in the nature of a general wave? Further, wouldn't it be
a kick if the effects of a general field are reflected in the activities
of living matter, the primary characteristic of which is purposiveness
or goal-oriented behavior?

Suppose that living matter has the power to causally influence
the outcome of events. This would help to explain the apparent evolutionary
discontinuities that are reflected in the geologic record. This leads
to the further possibility that evolution occurs, not as a result of environmental
change, but reflects the implicit capability of living matter to affect
change as a way of adapting to changing environmental requirements or
opportunities? This is in direct contrast to Darwin's theory of the survival
of the fittest, or the occurrence of natural selection among the chance
variants or "sports" that are speculated to arise spontaneously.

Running parallel to such speculation is the heuristic work of the engineer
and behaviorist William Powers. (William T. Powers Living Control Systems:
Selected Papers (Gravel Switch, KY: The Control Systems Group, Inc., 1989).)
He shows that control in living systems is neither subject to chance nor
to the causal control of outside agencies. Behavior is not a direct response
to external stimuli but is under the direct and nonprobabilistic control
of feedback mechanisms built into the organism. We find that this cybernetic
mechanism is typical of living organisms and, therefore, is a major design
aspect implicit in the life functions.

Returning to a consideration of the results of the coin-toss experiment,
the necessary next experiment would be to look to the identification of
something in the nature of a bifurcation that will predict the onset of
another probability swing or trend in the course of action. That appears
to be the nature of evolutionary change, indeed of all change. If such
change can be controlled, as we attempt to do through planning, then we
have evidence for the intervention of a life force (willpower?) in the
determination of the outcome of events.
The Statistical Postulate of Quantum Mechanics
In a discussion of quantum mechanics, physicist Victor J. Stenger (ViCtor
J. Stenger, The Unconscious Quantum (Amherst, NY: Prometheus Books, 1995),
pp 56-60) indicates:

"In 1926 Max Born proposed what was to become a primary postulate
of quantum mechanics in the von Neumann scheme. According to this postulate,
the wave function is used to compute the probability P for a particle
to be found in a particular state. This probability was to be proportional
to II^{2},
the square of the magnitude of the wave function ....
This postulate was extended by Wolfgang Pauli to include the probability
for finding a particle at a particular position.

"Pauli proposed that the probability P for finding
a particle in an infinitesimal volume element AV located in a specific
region of space is equal to the square of the magnitude of the wave function
computed at
that point multiplied by V:
P = II^{2}
AV. Since we can measure volume in any units we wish, no loss of generality
occurs if we assume a unit volume, V
= 1, and simply write P = II^{2}
and understand it to mean probability per unit volume, that is, probability
density.

Paraphrasing Pauli's postulate in terms of my conjecture about a universal
wave: The probability of an event, that is, conversion of the energy of
the universal wave into a material state at a particular place (the result
of the toss of a coin is thought of as equivalent to the conversion of
energy into a particle) is equal to the square of the wave function,.
Mathematically, squaring of the quantum "state" converts it
from imaginary and complex to real and rational, and this would be analogous
to the occurrence of a real event. A positive value may correspond to
a constructive (energy-binding) event typical of the action of living
systems, while a negative value would represent a destructive (entropic)
event.

Stenger mentions that the role of statistics in quantum mechanics was
supported by Einstein's calculation of the probabilities for atomic transitions;
however, the uncertain nature of its predictions was one of the aspects
that Einstein found unsatisfying about quantum mechanics. Einstein is
well known for having said, 'God does not play dice,' What he was really
objecting to was the notion that statistics was the final word. He found
it hard to accept that no underlying causal laws determined the behavior
of individual quantum particles at the most fundamental level. As Stenger
explains,

"Actually, statistics enters quantum mechanics only in an indirect
way. The time-dependent Schrödinger equation predicts the exact value
of the wave function
at future times given its value at some initial time. Probability enters
with the Bom postulate when the time comes to make a prediction on the
expected value of some measurement."

When such a measurement is attempted, however, we are attempting to calculate
at some instant in time when in fact the application of the Schr6dinger
equation varies constantly over time. So we are left with a probability
distribution rather than a specific value at t_{x}.

Thereby, quantum mechanics is often said to be "deterministic"
in that its basic equation, the Schrödinger equation, precisely determines
the time evolution of the wave function. However, it is indeterministic
in the sense that knowledge of the wave function is not always
sufficient to predict the outcome of a measurement — or of an event.

By the probability postulate, the wave function allows for the prediction
of the average motion of a system [the probable outcome of a coin
toss] but not the outcome in any particular instance, which is what the
above experiment demonstrates. I believe that we are approaching the time
when some deterministic theory will evolve that goes beyond quantum mechanics
and which applies to individual quantum systems as well as to causal events
at the macro level.

According to the recent theoretical development of Dr. Frank Tipler, we
have an all-pervasive physical field which gives being to all being —
which gives life to all living things-and which itself is generated by
the ultimate life which it defines. Through this "physical"
field we humans are apparently capable of superimposing our own wills
on the ordinarily indeterminate laws of probability and the chaotic physical
laws that prevail throughout the universe. This we do when we exercise
our intellect and creativity.