Bohm, David, Causality and Chance in Modern Physics,
London: Routledge and Kegan Paul, 1957, pp. 20-23/55-7
Summary
8. CONTINGENCY, CHANCE, AND
STATISTICAL LAW
Now contingencies are, as we have pointed
out in Section 1, possibilities existing outside the context under discussion.
The essential characteristic of contingencies is that their nature cannot
be defined or inferred solely in terms of the properties of things within
the context in question. In other words, they have a certain relative
independence of what is inside this context. However, as we have
seen, our general experience shows that all things are interconnected in
some way and to some degree. Hence we never expect to find complete
independence. But to the extent that the interconnection is negligible,
we may abstract out from the real process and its interconnections the
notion of chance contingencies, which are idealized, as completely independent
of the context under discussion. Thus, like the notion of necessary
causal connections, the notion of chance contingencies is seen to be an
approximation, which gives a partial treatment of certain aspects of the
real process, but which eventually has to be corrected and completed by
a consideration of the causal interconnections that always exist between
the processes taking place in different contexts. In order to bring
out in more detail what is meant by chance, we may consider a typical chance
event; namely, an automobile accident. Now it is evident that
just where, when, and how a particular accident takes place depends on
an enormous number of factors, a slight change of any one of which could
greatly change the character of the accident or even avoid it altogether.
For example, in a collision of two cars, if one of the motorists had started
out ten seconds earlier or ten seconds later, or if he had stopped to buy
cigarettes, or slowed down to avoid a cat that happened to cross the road,
or for any one of an unlimited number of similar reasons, this particular
accident would not even have happened; while even a slightly different
turn of the steering wheel might either have.prevented the accident altogether
or might have changed its character completely, either for the better or
for the worse. We see, then, that relative to a context in which
we consider,for example, the actions and precautions that can be taken
by a particular motorist, each accident has an aspect that is fortuitous.
By this we mean that what happens is contingent on what are, to a high
degree of approximation, independent factors, existing outside the context
in question, which have no essential relationship to the characteristic
traits that define just what sort of a person this motorist is and how
he will behave in a given situation. For this reason, we say that
relative to such a context a particular collision is not a necessary or
inevitable development, but rather that it is an accident and comes about
by chance, from which it also follows that within this context, the question
of just where, when, and how such a collision will take place, as well
as that of whether it will take place or not, is unpredictable.
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So much for an individual accident.
Let us now consider a series of similar accidents. First of all,
we note that there is an irregular and unpredictable variation or
fluctuation in the precise details of the various accidents (e. g. precisely
when and where they take place, precisely what is destroyed, etc.).
The origin of this variation is easily understood since a great many of
the independent factors on which the details of the accidents depend fluctuate
in a way having no systematic relationship to what a particular motorist
may be doing.
As the number of accidents under consideration becomes
larger and larger, however, new properties begin to appear; for one finds
that individual variations tend to cancel out, and statistical regularities
begin to show themselves. Thus, the total number of accidents in
a particular region generally does not change very much from ear to year,
and the changes that do take place often show a regular trend. Moreover,
this trend can be altered in a systematic way by the alteration of specified
factors on which accidents depend. Thus, when laws are passed punishing
careless driving and requiring regular inspection of mechanicalparts, tyres,
etc., the mean rate of accidents in any given region has been almost always
found to undergo a definite trend downward. In the case of an individual
motorist taking a particular trip, no very definite predictions can in
general be made concerning the effects of such measures, since there are
still an enormous number of sources of accidents that have not yet been
eliminated; yet statistically, as we have seen, variations in a particular
cause produce a regular and predictable trend in the effect.
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The behaviour described above is found in
a very wide range of fields, including social, economic, medical, and scientific
statistics and many other applications. In all these fields, there
is a characteristic irregular fluctuation or variation in the behaviour
of individual objects, events, and phenomena, the details of which are
not predictable within the context under discussion. This is combined
with regular trends in the behaviour of a long series or large aggregate
of such objects, events, or phenomena. These regular trends lead
to what we maycall statistical laws, which permit the approximate prediction
of the properties of the "long run" or average behaviour of a long series
or large aggregate ofindividuals, without the need to go to a broader context
in which we would take into account additional causal factors that contribute
to governing the details of the fluctuations of the individual members
of such series of aggregates.
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The tendency for contingencies lying outside
a given context to fluctuate approximately independently of happenings
inside that context has demonstrated itself to be so widespread that one
may enunciate it as a principle; namely, the principle of randomness.
By randomness we mean just that this independence leads to fluctuation
of these contingencies in a very complicated way over a wide range of possibilities,
but in such a manner that statistical averages have aregular and approximately
predictable behaviour.
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It is clear, then, that when we know that
a certain fluctuation is due to chance contingencies lying outside the
context of the causal laws under discussion, we know more than the mere
fact that the causal laws in question do not give perfectly accurate predictions;
we know also that the contingencies will produce complicated fluctuations
having regular statistical trends. Consider, for example, the problem
of error in measurement discussed in the previous section. Such errors
are generally divided into two classes, systematic and random. Systematic
errors arise, but they are just due to external causes, and not to real
chance contingencies that fluctuate independently of the context in question.
To reduce systematic errors, we must obtain an improved understanding and
control of the factors that are responsible for the error. The random
part of the error can, however, be reduced simply by taking the average
of more and more measurements. For, according to a well-known theorem,
the effects of chance fluctuations tend to cancel out in such a way that
this part of the error is inversely proportional to the square root of
the number of measurements. This shows how the fact that a certain
effect comes from chance contingencies implies more than the fact that
the causes lie outside the context under discussion. It implies,
in addition, a certain objective characteristic of randomness in the factors
in which the effect originates. We see, then, that it is appropriate
to speak about objectively valid laws of chance, which tell us about a
side of nature that is not treated completely by the causal laws alone.
Indeed, the laws of chance are just as necessary as the causal laws themselves.
For example, the random character of chance fluctuations is, in a wide
variety of situations, made inevitable by the extremely complex and manifold
character of the external contingencies on which the fluctuations depend.
(Thus random errors in measurement arise, as we have seen, in a practically
unlimited number of different kinds of factors that are essentially independent
of the quantity that is being measured.) Moreover, this random character
of the fluctuations is quite often an inherent and indispensable part of
the normal functioning of many kinds of things, and of their modes of being.
Thus, it would be impossible for a modern city to continue to exist in
its normal condition unless there were a tendency towards the cancellation
of chance fluctuations in traffic, in the demand for various kinds of food,clothing,
etc., in the times at which various individuals get sick or die, etc.
In all kinds of fields we find a similar dependence on the characteristic
effects of chance. Thus, when sand and cement are mixed, one does
not carefully distribute each individual grain of sand and cement so as
to obtain a uniform mixture, but rather one stirs the sand and cement together
and depends on chance to produce a uniform mixture....
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A systematic application of the theory
of probability to atomic physics is carried out in the study of statistical
mechanics, which makes possible fairly precise calculation of a great many
macroscopic properties of the system (e.g. entropy, heat capacity, equation
of state, etc.) on the basis of the microscopic laws, and which also provides
a model permitting a quantitative treatment of the way in which the macroscopic
laws of thermodynamics arise out of the microscopic motions. It has
also been applied in the study of Brownian motion, and in the study of
the fluctuations of the macroscopic properties of matter near the criticalpoints
of liquids. Thus, the theory of probability has made an important contribution
to ourunderstanding of the relationship between microscopic and macroscopic
levels by permitting us to take into account chance phenomena originating
in the microscopic level without the need for either a precise and a detailed
calculation of the motions of all the individual molecules in a large aggregate
or a precise knowledge of the laws of the microscopic level.
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13. THE ENRICHMENTS OF THE CONCEPTUAL
STRUCTURE OF
CLASSICAL PHYSICS AND THE PHILOSOPHY OF MECHANISM
We have seen in the preceding sections
that, even apart from the development of the notion of fields, there occurred,
during the eighteenth and nineteenth centuries a number of very important
additional steps that considerably enriched the conceptual structure of
physics. These included the introduction of the concept of levels,
that of quantitative changes that lead to qualitative changes, and that
of chance fluctuations that tend towards approximately determinate laws
for the mean behaviour of large aggregates. While, as we have already
pointed out in Section 4, none of these concepts is in direct contradiction
to a mechanistic philosophy, every one of them constitutes, in its general
trend and spirit at least, a step away from the idea that there is an absolute
and final fundamental law, which is purely quantitative in form, and which
would by itself permit, in principle at least, the complete and perfect
calculation of every feature of everything in the whole universe.
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To see why these new concepts tend to lead
away from mechanism we first recall that in the original form of the mechanistic
philosophy, both the notion of qualitative changes and that of chance were
regarded as nothing more than subjective aids to our thinking about the
properties of matter en masse, so that they did not represent anything
that was actually supposed to exist objectively in material systems.
We have already seen in Section II, however, that at least within the macroscopic
domain, in a qualitative transformation, new qualities satisfying new laws
become the significant and dominant causes in the domain in question.
Moreover, the objective reality of the breaks in quantitative macroscopic
properties, as well as that of the insensitivity of the qualitative change
to quantitative details also cannot be denied. Similarly, it is evident
(on the basis of the discussions given in Chapter 1, Sections 8 and 9)
that chance fluctuations exist objectively within specified contexts, and
that the theory of probability provides a relatively precise mathematical
expression of objective properties of these fluctuations, including the
statistical regularities which arise (on the basis of the cancellation
of large numbers of chance fluctuations).
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It is a most important characteristic of the mechanistic
philosophy, however, that it permits one to make a limitless number of
adjustments in his detailed point of view, without giving up what is essential
to the mechanistic position. Thus, with the notion of qualitative
changes, a great many physicists have effectively accepted the idea that
these may well be objective (or at least as objective as anything else
is). Nevertheless, they assert that such changes are not of fundamental
significance, because they must, in principle at least, follow completely
and perfectly in all of their details, in every respect, and without any
approximation, from the quantitative laws of motion of the fundamental
elements that make up the system, whatever these elements may be.
Hence, it is maintained that qualitative changes are like passing shadows
that have absolutely no independent existence of their own, but which depend
for all their attributes on the quantitative laws governing the basic elements
entering into the theory. It is evident that such an attitude implies
also that the notion of a series of levels of law is likewise nothing more
than a set of approximations to the absolute and final fundamental law,
approximations in which the laws of the various levels depend completely
for all their characteristics on the fundamental law, while the fundamental
law has no dependence whatever on the laws of the various levels. Similarly,
it is consistent with this point of view to suppose that chance and statistical
laws arise out of nothing more than the sheer complexity and multiplicity
of the motions of the basic entities entering into the fundamental causal
law and that into the formulation of this latter law no element of chance
whatever will appear.
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We recall that historically
the mechanistic philosophy was expressed in terms of the assumption that
the basic units out of which the universe was supposed to be built are
indivisible atoms. The purely quantitative laws governing the motions
of these atoms, were then regarded as the laws from which everything else
followed. It was discovered later,however, that the atoms are not
really the fundamental units, because ... [and so the story goes on
...]
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