_CHANCE AS OPPOSED TO CAUSATION AND DESIGN._
1. The remarks in the previous chapter will have served
to clear the way for an enquiry which probably excites more
popular interest than any other within the range of our subject,
viz. the determination whether such and such events are to
be attributed to Chance on the one hand, or to Causation or
Design on the other. As the principal difficulty seems to
arise from the ambiguity with which the problem is generally
conceived and stated, owing to the extreme generality of the
conceptions involved, it becomes necessary to distinguish
clearly between the several distinct issues which are apt to
be involved.
I. There is, to begin with, a very old objection, founded
on the assumption which our science is supposed to make of
the existence of _Chance_. The objection against chance is
of course many centuries older than the Theory of Probability;
and as it seems a nearly obsolete objection at the
present day we need not pause long for its consideration.
If we spelt the word with a capital C, and maintained that
it was representative of some distinct creative or administrative
agency, we should presumably be guilty of some form
of Manicheism. But the only rational meaning of the objection
would appear to be that the principles of the science
compel us to assume that events (some events, only, that is)
happen without causes, and are thereby removed from the
customary control of the Deity. As repeatedly pointed out
already this is altogether a mistake. The science of Probability
makes no assumption whatever about the way in
which events are brought about, whether by causation or
without it. All that we undertake to do is to establish and
explain a body of rules which are applicable to classes of
cases in which we do not or cannot make inferences about
the individuals. The objection therefore must be somewhat
differently stated, and appears finally to reduce itself
to this:--that the assumptions upon which the science of
Probability rests, are not inconsistent with a disbelief in
causation within certain limits; causation being of course
understood simply in the sense of regular sequence. So
stated the objection seems perfectly valid, or rather the facts
on which it is based must be admitted; though what connection
there would be between such lack of causation and
absence of Divine superintendence I quite fail to see.
As this Theological objection died away the men of
physical science, and those who sympathized with them,
began to enforce the same protest; and similar cautions are
still to be found from time to time in modern treatises.
Hume, for instance, in his short essay on _Probability_, commences
with the remark, "though there be no such thing as
chance in the world, our ignorance of the real cause of any
event has the same influence on the understanding, &c."
De Morgan indeed goes so far as to declare that "the
foundations of the theory of Probability have ceased to exist
in the mind that has formed the conception," "that anything
ever did happen or will happen without some particular
reason why it should have been precisely what it was and
not anything else."[1] Similar remarks might be quoted from
Laplace and others.
2. In the particular form of the controversy above
referred to, and which is mostly found in the region of the
natural and physical sciences, the contention that chance
and causation are irreconcileable occupies rather a defensive
position; the main fact insisted on being that, whenever in
these subjects we may happen to be ignorant of the details
we have no warrant for assuming as a consequence that the
details are uncaused. But this supposed irreconcileability
is sometimes urged in a much more aggressive spirit in reference
to social enquiries. Here the attempt is often made to
prove causation in the details, from the known and admitted
regularity in the averages. A considerable amount of controversy
was excited some years ago upon this topic, in great
part originated by the vigorous and outspoken support of the
necessitarian side by Buckle in his _History of Civilization_.
It should be remarked that in these cases the attempt is
sometimes made as it were to startle the reader into acquiescence
by the singularity of the examples chosen. Instances
are selected which, though they possess no greater logical
value, are, if one may so express it, emotionally more effective.
Every reader of Buckle's History, for instance, will remember
the stress which he laid upon the observed fact, that the number
of suicides in London remains about the same, year by year;
and he may remember also the sort of panic with which the
promulgation of this fact was accompanied in many quarters.
So too the way in which Laplace notices that the number of
undirected letters annually sent to the Post Office remains
about the same, and the comments of Dugald Stewart upon
this particular uniformity, seem to imply that they regarded
this instance as more remarkable than many analogous ones
taken from other quarters.
That there is a certain foundation of truth in the reasonings
in support of which the above examples are advanced,
cannot be denied, but their authors appear to me very much
to overrate the sort of opposition that exists between the
theory of Chances and the doctrine of Causation. As regards
first that wider conception of order or regularity which we
have termed uniformity, anything which might be called
objective chance would certainly be at variance with this in
one respect. In Probability ultimate regularity is always
postulated; in tossing a die, if not merely the individual
throws were uncertain in their results, but even the average
also, owing to the nature of the die, or the number of the
marks upon it, being arbitrarily interfered with, of course no
kind of science would attempt to take any account of it.
3. So much must undoubtedly be granted; but must
the same admission be made as regards the succession of the
individual events? Can causation, in the sense of invariable
succession (for we are here shifting on to this narrower
ground), be denied, not indeed without suspicion of scientific
heterodoxy, but at any rate without throwing uncertainty
upon the foundations of Probability? De Morgan, as we
have seen, strongly maintains that this cannot be so. I find
myself unable to agree with him here, but this disagreement
springs not so much from differences of detail, as from those
of the point of view in which we regard the science. He
always appears to incline to the opinion that the individual
judgment in probability is to admit of justification;
that when we say, for instance, that the odds in favour of
some event are three to two, that we can explain and justify
our statement without any necessary reference to a series or
class of such events. It is not easy to see how this can be
done in any case, but the obstacles would doubtless be
greater even than they are, if knowledge of the individual
event were not merely unattained, but, owing to the absence
of any causal connection, essentially unattainable. On the
theory adopted in this work we simply postulate ignorance
of the details, but it is not regarded as of any importance
on what sort of grounds this ignorance is based. It may
be that knowledge is out of the question from the nature
of the case, the causative link, so to say, being missing. It
may be that such links are known to exist, but that either
we cannot ascertain them, or should find it troublesome to
do so. It is the fact of this ignorance that makes us appeal
to the theory of Probability, the grounds of it are of no
importance.
4. On the view here adopted we are concerned only
with averages, or with the single event as deduced from an
average and conceived to form one of a series. We start
with the assumption, grounded on experience, that there is
uniformity in this average, and, so long as this is secured to
us, we can afford to be perfectly indifferent to the fate, as
regards causation, of the individuals which compose the
average. The question then assumes the following form:--Is
this assumption, of average regularity in the aggregate,
inconsistent with the admission of what may be termed
causeless irregularity in the details? It does not seem to me
that it would be at all easy to prove that this is so. As
a matter of fact the two beliefs have constantly co-existed in
the same minds. This may not count for much, but it suggests
that if there be a contradiction between them it is by
no means palpable and obvious. Millions, for instance, have
believed in the general uniformity of the seasons taken one
with another, who certainly did not believe in, and would
very likely have been ready distinctly to deny, the existence
of necessary sequences in the various phenomena which compose
what we call a season. So with cards and dice; almost
every gambler must have recognized that judgment and
foresight are of use in the long run, but writers on chance
seem to think that gamblers need a good deal of reasoning to
convince them that each separate throw is in its nature essentially
predictable.
5. In its application to moral and social subjects,
what gives this controversy its main interest is its real or
supposed bearing upon the vexed question of the freedom
of the will; for in this region Causation, and Fatalism or
Necessitarianism, are regarded as one and the same thing.
Here, as in the last case, that wide and somewhat vague
kind of regularity that we have called Uniformity, must be
admitted as a notorious fact. Statistics have put it out of
the power of any reasonably informed person to feel any
hesitation upon this point. Some idea has already been
gained, in the earlier chapters, of the nature and amount
of the evidence which might be furnished of this fact, and
any quantity more might be supplied from the works of
professed writers upon the subject. If, therefore, Free-will be
so interpreted as to imply such essential irregularity as defies
prediction both in the average, and also in the single case,
then the negation of free-will follows, not as a remote logical
consequence, but as an obvious inference from indisputable
facts of experience.
Few persons, however, would go so far as to interpret it
in this sense. All that troubles them is the fear that somehow
this general regularity may be found to carry with it
causation, certainly in the sense of regular invariable sequence,
and probably also with the further association of
compulsion. Rejecting the latter association as utterly
unphilosophical, I cannot even see that the former consequence
can be admitted as really proved, though it doubtless
gains some confirmation from this source.
6. The nature of the argument against free-will,
drawn from statistics, at least in the form in which it is very
commonly expressed, seems to me exceedingly defective.
The antecedents and consequents, in the case of our volitions,
must clearly be supposed to be very nearly _immediately_
in succession, if anything approaching to causation is to be
established: whereas in statistical enquiries the data are
often widely separate, if indeed they do not apply merely
to single groups of actions or results. For instance, in the
case of the misdirected letters, what it is attempted to prove
is that each writer was so much the 'victim of circumstances'
(to use a common but misleading expression) that he could
not have done otherwise than he did under his circumstances.
But really no accumulation of figures to prove that the
number of such letters remains the same year by year, can
have much bearing upon this doctrine, even though they
were accompanied by corresponding figures which should
connect the forgetfulness thus indicated with some other
characteristics in the writers. So with the number of
suicides. If 250 people do, or lately did, annually put an
end to themselves in London, the fact, as it thus stands by
itself, may be one of importance to the philanthropist and
statesman, but it needs bringing into much closer relation
with psychological elements if it is to convince us that the
actions of men are always instances of inflexible order. In
fact, instead of having secured our A and B here in closest
intimacy of succession to one another,--to employ the symbolic
notation commonly used in works on Inductive Logic
to illustrate the causal connection,--we find them separated
by a considerable interval; often indeed we merely
have an A or a B by itself.
7. Again, another deficiency in such reasoning seems
to be the laying undue weight upon the mere regularity or
persistency of the statistics. These may lead to very important
results, but they are not exactly what is wanted
for the purpose of proving anything against the freedom
of the will; it is not indeed easy to see what connection
this has with such facts as that the annual number of thefts
or of suicides remains at pretty nearly the same figure.
Statistical uniformity seems to me to establish nothing else,
at least directly, in the case of human actions, than it does
in that of physical characteristics. Take but one instance,
that of the misdirected letters. We were already aware
that the height, weight, chest measurement, and so on, of
a large number of persons preserved a tolerably regular
average amidst innumerable deflections, and we were prepared
by analogy to anticipate the same regularity in their
mental characteristics. All that we gain, by counting the
numbers of letters which are posted without addresses, is
a certain amount of direct evidence that this is the case.
Just as observations of the former kind had already shown
that statistics of the strength and stature of the human
body grouped themselves about a mean, so do those of the
latter that a similar state of things prevails in respect of the
readiness and general trustworthiness of the memory. The
evidence is not so direct and conclusive in the latter case,
for the memory is not singled out and subjected to measurement
by itself, but is taken in combination with innumerable
other influencing circumstances. Still there can be little
doubt that the statistics tell on the whole in this direction,
and that by duly varying and extending them they may
obtain considerable probative force.
The fact is that Probability has nothing more to do with
Natural Theology, either in its favour or against it, than the
general principles of Logic or Induction have. It is simply a
body of rules for drawing inferences about classes of events
which are distinguished by a certain quality. The believer
in a Deity will, by the study of nature, be led to form an
opinion about His works, and so to a certain extent about
His attributes. But it is surely unreasonable to propose
that he should abandon his belief because the sequence of
events,--not, observe, their general tendency towards happiness
or misery, good or evil,--is brought about in a way
different from what he had expected; whether it be by displaying
order where he had expected irregularity, or by
involving the machinery of secondary causes where he had
expected immediate agency.
8. It is both amusing and instructive to consider
what very different feelings might have been excited in our
minds by this co-existence of, what may be called, ignorance
of individuals and knowledge of aggregates, if they had presented
themselves to our observation in a reverse order.
Being utterly unable to make assured predictions about a
single life, or the conduct of individuals, people are sometimes
startled, and occasionally even dismayed, at the unexpected
discovery that such predictions can be confidently
made when we are speaking of large numbers. And so
some are prompted to exclaim, This is denying Providence!
it is utter Fatalism! But let us assume, for a moment, that
our familiarity with the subject had been experienced, in the
first instance, in reference to the aggregates instead of the
individual lives. It is difficult, perhaps, to carry out such a
supposition completely; though we may readily conceive
something approaching to it in the case of an ignorant clerk
in a Life Assurance Office, who had never thought of life, except
as having such a 'value' at such an age, and who had
hardly estimated it except in the form of averages. Might
we not suppose him, in some moment of reflectiveness, being
astonished and dismayed at the sudden realization of the
utter uncertainty in which the single life is involved? And
might not his exclamation in turn be, Why this is denying
Providence! It is utter chaos and chance! A belief in a
Creator and Administrator of the world is not confined to
any particular assumption about the nature of the immediate
sequence of events, but those who have been accustomed
hitherto to regard the events under one of the aspects above
referred to, will often for a time feel at a loss how to connect
them with the other.
9. So far we have been touching on a very general
question; viz. the relation of the fundamental postulates of
Probability to the conception of Order or Uniformity in the
world, physical or moral. The difficulties which thence arise
are mainly theological, metaphysical or psychological. What
we must now consider are problems of a more detailed or
logical character. They are prominently these two; (1) the
distinction between chance arrangement and _causal_ arrangement
in physical phenomena; and (2) the distinction between
chance arrangement and _designed_ arrangement where
we are supposed to be contemplating rational agency as
acting on one side at least.
II. The first of these questions raises the antithesis
between chance and causation, not as a general characteristic
pervading all phenomena, but in reference to some specified
occurrence:--Is this a case of chance or not? The most
strenuous supporters of the universal prevalence of causation
and order admit that the question is a relevant one, and
they must therefore be supposed to have some rule for
testing the answers to it.
Suppose, for instance, a man is seized with a fit in a
house where he has gone to dine, and dies there; and some
one remarks that that was the very house in which he was
born. We begin to wonder if this was an odd coincidence
and nothing more. But if our informant goes on to tell us
that the house was an old family one, and was occupied by
the brother of the deceased, we should feel at once that
these facts put the matter in a rather different light. Or
again, as Cournot suggests, if we hear that two brothers
have been killed in battle on the same day, it makes a great
difference in our estimation of the case whether they were
killed fighting in the same engagement or whether one fell
in the north of France and the other in the south. The
latter we should at once class with mere coincidences, whereas
the former might admit of explanation.
10. The problem, as thus conceived, seems to be one
rather of Inductive Logic than of Probability, because there
is not the slightest attempt to calculate chances. But it
deserves some notice here. Of course no accurate thinker
who was under the sway of modern physical notions would
for a moment doubt that each of the two elements in question
had its own 'cause' behind it, from which (assuming perfect
knowledge) it might have been confidently inferred. No
more would he doubt, I apprehend, that if we could take a
sufficiently minute and comprehensive view, and penetrate
sufficiently far back into the past, we should reach a stage at
which (again assuming perfect knowledge) the co-existence of
the two events could equally have been foreseen. The
employment of the word _casual_ therefore does not imply any
rejection of a cause; but it does nevertheless correspond to a
distinction of some practical importance. We call a coincidence
casual, I apprehend, when we mean to imply that no
knowledge of one of the two elements, which we can suppose
to be practically attainable, would enable us to expect the
other. We know of no generalization which covers them
both, except of course such as are taken for granted to be
inoperative. In such an application it seems that the word
'casual' is not used in antithesis to 'causal' or to 'designed',
but rather to that broader conception of order or regularity
to which I should apply the term Uniformity. The casual
coincidence is one which we cannot bring under any special
generalization; certain, probable, or even plausible.
A slightly different way of expressing this distinction is
to regard these 'mere coincidences' as being simply cases in
point of _independent_ events, in the sense in which independence
was described in a former chapter. We saw
that any two events, A and B, were so described when each
happens with precisely the same relative statistical frequency
whether the other happens or not. This state of things
seems to hold good of the successions of heads and tails in
tossing coins, as in that of male and female births in a
town, or that of the digits in many mathematical tables.
Thus we suppose that when men are picked up in the street
and taken into a house to die, there will not be in the long
run any preferential selection for or against the house in
which they were born. And all that we necessarily mean to
claim when we deny of such an occurrence, in any particular
case, that it is a mere coincidence, is that that particular
case must be taken out of the common list and transferred
to one in which there _is_ some such preferential selection.
11. III. The next problem is a somewhat more intricate
one, and will therefore require rather careful subdivision.
It involves the antithesis between Chance and
Design. That is, we are not now (as in the preceding case)
considering objects in their physical aspect alone, and taking
account only of the relative frequency of their co-existence or
sequence; but we are considering the agency by which they
are produced, and we are enquiring whether that agency
trusted to what we call chance, or whether it employed what
we call design.
The reader must clearly understand that we are not now
discussing the mere question of fact whether a certain
assigned arrangement _is_ what we call a chance one. This,
as was fully pointed out in the fourth chapter, can be settled
by mere inspection, provided the materials are extensive
enough. What we are now proposing to do is to carry on
the enquiry from the point at which we then had to leave it
off, by solving the question, Given a certain arrangement, is
it more likely that this was _produced_ by design, or by some
of the methods commonly called chance methods? The distinction
will be obvious if we revert to the succession of
figures which constitute the ratio π. As I have said, this
arrangement, regarded as a mere succession of digits, appears
to fulfil perfectly the characteristics of a chance arrangement.
If we were to omit the first four or five digits,
which are familiar to most of us, we might safely defy any
one to whom it was shown to say that it was not got at by
simply drawing figures from a bag. He might look at it for
his whole life without detecting that it was anything but the
result of such a chance selection. And rightly so, because
regarded as a mere arrangement it _is_ a chance one: it fulfils
all the requirements of such an arrangement.[2] The question
we are now proceeding to discuss is this: Given any such
arrangement how are we to determine the process by which
it was arrived at?
We are supposed to have some event before us which
might have been produced in either of two alternative
ways, i.e. by chance or by some kind of deliberate design;
and we are asked to determine the odds in favour of one
or other of these alternatives. It is therefore a problem in
Inverse Probability and is liable to all the difficulties to
which problems of this class are apt to be exposed.
12. For the theoretic solution of such a question we
require the two following data:--
(1) The relative frequency of the two classes of agencies,
viz. that which is to act in a chance way and that which
is to act designedly.
(2) The probability that each of these agencies, if it
were the really operative one, would produce the event in
question.
The latter of these data can generally be secured without
any difficulty. The determination of the various contingencies
on the chance hypothesis ought not, if the example
were a suitable one, to offer any other than arithmetical
difficulties. And as regards the design alternative, it is
generally taken for granted that if this had been operative
it would certainly have produced the result aimed at. For
instance, if ten pence are found on a table, all with head
uppermost, and it be asked whether chance or design had
been at work here; we feel no difficulty up to a certain
point. Had the pence been tossed we should have got ten
heads only once in 1024 throws; but had they been placed
designedly the result would have been achieved with certainty.
But the other postulate, viz. that of the relative prevalence
of these two classes of agencies, opens up a far more
serious class of difficulties. Cases can be found no doubt,
though they are not very frequent, in which this question
can be answered approximately, and then there is no further
trouble. For instance, if in a school class-list I were to see
the four names Brown, Jones, Robinson, Smith, standing
in this order, it might occur to me to enquire whether this
arrangement were alphabetical or one of merit. In our
enlarged sense of the terms this is equivalent to chance
and design as the alternatives; for, since the initial letter of
a boy's name has no known connection with his attainments,
the successive arrangement of these letters on any other
than the alphabetical plan will display the random features,
just as we found to be the case with the digits of an incommensurable
magnitude. The odds are 23 to 1 against 4 names
coming undesignedly in alphabetical order; they are
equivalent to certainty in favour of their doing so if this
order had been designed. As regards the relative frequency
of the two kinds of orders in school examinations I do not
know that statistics are at hand, though they could easily
be procured if necessary, but it is pretty certain that the
majority adopt the order of merit. Put for hypothesis the
proportion as high as 9 to 1, and it would still be found more
likely than not that in the case in question the order was
really an alphabetical one.
13. But in the vast majority of cases we have no
such statistics at hand, and then we find ourselves exposed
to very serious ambiguities. These may be divided into
two distinct classes, the nature of which will best be seen
by the discussion of examples.
In the first place we are especially liable to the drawback
already described in a former chapter as rendering
mere statistics so untrustworthy, which consists in the fact
that the proportions are so apt to be disturbed almost from
moment to moment by the possession of fresh hints or information.
We saw for instance why it was that statistics of
mortality were so very unserviceable in the midst of a
disease or in the crisis of a battle. Suppose now that on
coming into a room I see on the table ten coins lying face
uppermost, and am asked what was the likelihood that the
arrangement was brought about by design. Everything
turns upon special knowledge of the circumstances of the
case. Who had been in the room? Were they children,
or coin-collectors, or persons who might have been supposed
to have indulged in tossing for sport or for gambling purposes?
Were the coins new or old ones? a distinction of
this kind would be very pertinent when we were considering
the existence of any motive for arranging them the same
way uppermost. And so on; we feel that our statistics are
at the mercy of any momentary fragment of information.
14. But there is another consideration besides this.
Not only should we be thus influenced by what may be
called external circumstances of a general kind, such as the
character and position of the agents, we should also be influenced
by what we supposed to be the conventional[3] estimate
with which this or that particular chance arrangement was
then regarded. Thus from time to time as new games of
cards become popular new combinations acquire significance;
and therefore when the question of design takes the form of
possible cheating a knowledge of the current estimate of
such combinations becomes exceedingly important.
15. The full significance of these difficulties will best
be apprehended by the discussion of a case which is not
fictitious or invented for the purpose, but which has actually
given rise to serious dispute. Some years ago Prof. Piazzi
Smyth published a work[4] upon the great pyramid of Ghizeh,
the general object of which was to show that that building
contained, in its magnitude, proportions and contents, a
number of almost imperishable natural standards of length,
volume, &c. Amongst other things it was determined that
the value of π was accurately (the degree of accuracy is not,
I think, assigned) indicated by the ratio of the sides to the
height. The contention was that this result could not be
accidental but must have been designed.
As regards the estimation of the value of the chance
hypothesis the calculation is not quite so clear as in the
case of dice or cards. We cannot indeed suppose that,
for a given length of base, _any_ height can be equally possible.
We must limit ourselves to a certain range here; for if too
high the building would be insecure, and if too low it would
be ridiculous. Again, we must decide to how close an
approximation the measurements are made. If they are
guaranteed to the hundredth of an inch the coincidence
would be of a quite different order from one where the
guarantee extended only to an inch. Suppose that this
has been decided, and that we have ascertained that out
of 10,000 possible heights for a pyramid of given base just
that one has been selected which would most nearly yield
the ratio of the radius to the circumference of a circle.
The remaining consideration would be the relative frequency
of the 'design' alternative,--what is called its _à priori_
probability,--that is, the relative frequency with
which such builders can be supposed to have aimed at that
ratio; with the obvious implied assumption that if they did
aim at it they would certainly secure it. Considering our
extreme ignorance of the attainments of the builders it is
obvious that no attempt at numerical appreciation is here
possible. If indeed the 'design' was interpreted to mean
conscious resolve to produce that ratio, instead of mere resolve
to employ some method which happened to produce
it, few persons would feel much hesitation. Not only do
we feel tolerably certain that the builders did not know the
value of π, except in the rude way in which all artificers
must know it; but we can see no rational motive, if they
did know it, which should induce them to perpetuate it
in their building. If, however, to adopt an ingenious suggestion,[5]
we suppose that the builder may have proceeded
in the following fashion, the matter assumes a different
aspect. Suppose that having decided on the height of his
pyramid he drew a circle with that as radius: that, laying
down a cord along the line of this circle, he drew this cord
out into a square, which square marked the base of the
building. Hardly any simpler means could be devised in
a comparatively rude age; and it is obvious that the circumference
of the base, being equal to the length of the
cord, would bear exactly the admitted ratio to the height.
In other words, the exact attainment of a geometric value
does not imply a knowledge of that ratio, but merely of some
method which involves and displays it. A teredo can bore,
as well as any of us, a hole which displays the geometric properties
of a circle, but we do not credit it with corresponding
knowledge.
As before said, all numerical appreciation of the likelihood
of the design alternative is out of the question. But,
_if_ the precision is equal to what Mr Smyth claimed, I suppose
that most persons (with the above suggestion before
them) will think it somewhat more likely that the coincidence
was not a chance one.
16. There still remains a serious, and highly interesting
speculative consideration. In the above argument we took
it for granted, in calculating the chance alternative, that
only _one_ of the 10,000 possible values was favourable; that
is, we took it for granted that the ratio π was the only one
whose claims, so to say, were before the court. But it is
clear that if we had obtained just double this ratio the result
would have been of similar significance, for it would have
been simply the ratio of the circumference to the diameter.
In fact, Mr Smyth's selected ratio,--the height to twice the
breadth of the base as compared with the diameter to the
circumference,--is obviously only one of a plurality of ratios.
Again; if the measured results had shown that the ratio of
the height to one side of the base was 1 : sqrt{2} (i.e. that of a
side to a diagonal of a square) or 1 : sqrt{3} (i.e. that of a side
to a diagonal of a cube) would not such results equally show
evidence of design? Proceeding in this way, we might
suggest one known mathematical ratio after another until
most of the 10,000 supposed possible values had been taken
into account. We might then argue thus: since almost
every possible height of the pyramid would correspond to
_some_ mathematical ratio, a builder, ignorant of them all
alike, would be not at all unlikely to stumble upon one or
other of them: why then attribute design to him in one case
rather than another?
17. The answer to this objection has been already
hinted at. Everything turns upon the _conventional_ estimate
of one result as compared with another. Revert, for simplicity
to the coins. Ten heads is just as likely as alternate
heads and tails, or five heads followed by five tails; or, in
fact, as any one of the remaining 1023 possible cases. But
universal convention has picked out a run of ten as being
remarkable. Here, of course, the convention seems a very
natural and indeed inevitable one, but in other cases it is
wholly arbitrary. For instance, in cards, "queen of spades
and knave of diamonds" is exactly as uncommon as any
other such pair: moreover, till _bezique_ was introduced it
offered presumably no superior interest over any other
specified pair. But during the time when that game was
very popular this combination was brought into the category
of coincidences in which interest was felt; and, given dishonesty
amongst the players, its chance of being designed
stood at once on a much better footing.[6]
Returning then to the pyramid, we see that in balancing
the claims of chance and design we must, in fairness to the
latter, reckon to its account several other values as well as
that of π, e.g. sqrt{2} and sqrt{3}, and a few more such simple and
familiar ratios, as well as some of their multiples. But
though the number of such values which _might_ be reckoned,
on the ground that they are actually known to us, is infinite,
yet the number that _ought_ to be reckoned, on the ground
that they could have been familiar to the builders of a
pyramid, are very few. The order of probability for or against
the existence of design will not therefore be seriously altered
here by such considerations.[7]
18. Up to this point it will be observed that what we
have been balancing against each other are two forms of
agency,--of human agency, that is,--one acting through
chance, and the other by direct design. In this case we
know where we are, for we can thoroughly understand agency
of this kind. The problem is indeed but seldom numerically
soluble, and in most cases not soluble at all, but it is at any
rate capable of being clearly stated. We know the kind of
answer to be expected and the reasons which would serve to
determine it, if they were attainable.
The next stage in the enquiry would be that of balancing
ordinary human chance agency against,--I will not call it
direct spiritualist agency, for that would be narrowing the
hypothesis unnecessarily,--but against all other possible
causes. Some of the investigations of the Society for
Psychical Research will furnish an admirable illustration of
what is intended by this statement. There is a full discussion
of these applications in a recent essay by Mr F. Y.
Edgeworth;[8] but as his account of the matter is connected
with other calculations and diagrams I can only quote it in
part. But I am in substantial agreement with him.
"It is recorded that 1833 guesses were made by a 'percipient'
as to the suit of cards which the 'agent' had fixed
upon. The number of successful guesses was 510, considerably
above 458, the number which, as being the quarter of 1833,
would, on the supposition of pure chance, be more
likely than any other number. Now, by the Law of Error,
we are able approximately to determine the probability of
such an excess occurring by chance. It is equal to the extremity
of the tail of a probability-curve such as [those we
have already had occasion to examine].... The proportion
of this extremity of the tail to the whole body is 0.003 to 1.
That fraction, then, is the probability of a chance shot
striking that extremity of the tail; the probability that, if
the guessing were governed by pure chance, a number of
successful guesses equal or greater than 510 would occur":
odds, that is, of about 332 to 1 against such occurrence.
19. Mr Edgeworth holds, as strongly as I do, that for purposes
of calculation, in any strict sense of the word, we ought
to have some determination of the data on the non-chance side
of the hypothesis. We ought to know its relative frequency
of occurrence, and the relative frequency with which it
attains its aims. I am also in agreement with him that
"what that other cause may be,--whether some trick, or
unconscious illusion, or _thought-transference_ of the sort which
is vindicated by the investigators--it is for common-sense
and ordinary Logic to consider."
I am in agreement therefore with those who think that
though we cannot form a quantitative opinion we can in
certain cases form a tolerably decisive one. Of course if we
allow the last word to the supporters of the chance hypothesis
we can never reach proof, for it will always be open to
them to revise and re-fix the antecedent probability of the
counter hypothesis. What we may fairly require is that
those who deny the chance explanation should assign some
sort of minimum value to the probability of occurrence on
the other supposition, and we can then try to surmount this
by increasing the rarity of the actually produced phenomenon
on the chance hypothesis. If, for instance, they declare that
in their estimation the odds against any other than the chance
agency being at work are greater than 332 to 1, we must
try to secure a yet uncommoner occurrence than that in
question. If the supporters of thought-transference have
the courage of their convictions,--as they most assuredly
have,--they would not shrink from accepting this test. I
am inclined to think that even at present, on such evidence
as that above, the probability that the results were got at by
ordinary guessing is very small.
20. The problems discussed in the preceding sections
are at least intelligible even if they are not always resolvable.
But before finishing this chapter we must take notice
of some speculations upon this part of the subject which do not
seem to keep quite within the limits of what is intelligible.
Take for instance the question discussed by Arbuthnott (in
a paper in the _Phil. Transactions_, Vol. XXVII.) under the
title "An Argument for Divine Providence, taken from the
constant Regularity observed in the birth of both sexes." Had
his argument been of the ordinary teleological kind; that
is, had he simply maintained that the existent ratio of approximate
equality, with a six per cent. surplusage of males,
was a beneficent one, there would have been nothing here to
object against. But what he contemplated was just such a
balance of alternate hypotheses between chance and design
as we are here considering. His conclusion in his own words
is, "it is art, not chance, that governs."
It is difficult to render such an argument precise without
rendering it simply ridiculous. Strictly understood it can
surely bear only one of two interpretations. On the one
hand we may be personifying Chance: regarding it as an
agent which must be reckoned with as being quite capable
of having produced man, or at any rate having arranged the
proportion of the sexes. And then the decision must be
drawn, as between this agent and the Creator, which of the
two produced the existent arrangement. If so, and Chance
be defined as any agent which produces a chance or random
arrangement, I am afraid there can be little doubt that it
was this agent that was at work in the case in question.
The arrangement of male and female births presents, so far
as we can see, one of the most perfect examples of chance:
there is ultimate uniformity emerging out of individual
irregularity: all the 'runs' or successions of each alternative
are duly represented: the fact of, say, five sons having been
already born in a family does not seem to have any certain
effect in diminishing the likelihood of the next being a son,
and so on. Such a nearly perfect instance of 'independent
events' is comparatively very rare in physical phenomena.
It is all that we can claim from a chance arrangement.[9] The
only other interpretation I can see is to suggest that there
was but one agent who might, like any one of us, have either
tossed up or designed, and we have to ascertain which course
he probably adopted in the case in question. Here too, if
we are to judge of his mode of action by the tests we should
apply to any work of our own, it would certainly look very
much as if he had adopted some scheme of tossing.
21. The simple fact is that any rational attempt to
decide between chance and design as agencies must be confined
to the case of finite intelligences. One of the important
determining elements here, as we have seen, is the
state of knowledge of the agent, and the conventional estimate
entertained about this or that particular arrangement;
and these can be appreciated only when we are dealing with
beings like ourselves.
For instance, to return to that much debated question
about the arrangement of the stars, there can hardly be any
doubt that what Mitchell,--who started the discussion,--had
in view was the decision between Chance and Design. He
says (_Trans. Roy. Soc._ 1767) "The argument I intend to
make use of... is of that kind which infers either design or
some general law from a general analogy and from the greatness
of the odds against things having been in the present
situation if it was not owing to some such cause." And he
concludes that had the stars "been scattered by mere chance
as it might happen" there would be "odds of near 500,000
to 1 that no six stars out of that number [1500], scattered at
random in the whole heavens, would be within so small a
distance from each other as the Pleiades are." Under any
such interpretation the controversy seems to me to be idle.
I do not for a moment dispute that there is some force in
the ordinary teleological argument which seeks to trace signs
of goodness and wisdom in the general tendency of things.
But what do we possibly understand about the nature of
creation, or the designs of the Creator, which should enable
us to decide about the likelihood of his putting the stars in
one shape rather than in another, or which should allow any
significance to "mere chance" as contrasted with his supposed
all-pervading agency?
22. Reduced to intelligible terms the two following
questions seem to me to emerge from the controversy:--
(I.) The stars being distributed through space, some of
them would of course be nearly in a straight line behind
others when looked at from our planet. Supposing that
they were tolerably uniformly distributed, we could calculate
about how many of them would thus be seen in apparent
close proximity to one another. The question is then put,
Are there more of them near to each other, two and two,
than such calculation would account for? The answer is that
there are many more. So far as I can see the only direct
inference that can be drawn from this is that they are _not_
uniformly distributed, but have a tendency to go in pairs.
This, however, is a perfectly sound and reasonable application
of the theory. Any further conclusions, such as that
these pairs of stars will form systems, as it were, to themselves,
revolving about one another, and for all practical purposes
unaffected by the rest of the sidereal system, are of
course derived from astronomical considerations.[10] Probability
confines itself to the simple answer that the distribution is
not uniform; it cannot pretend to say whether, and by what
physical process, these binary systems of stars have been
'caused'.[11]
23. (II.) The second question is this, Does the distribution
of the stars, after allowing for the case of the binary
stars just mentioned, resemble that which would be produced
by human agency sprinkling things 'at random'?
(We are speaking, of course, of their distribution as it appears
to us, on the visible heavens, for this is nearly all that
we can observe; but if they extend beyond the telescopic
range in every direction, this would lead to practically much
the same discussion as if we considered their actual arrangement
in space.) We have fully discussed, in a former chapter,
the meaning of 'randomness.' Applying it to the case
before us, the question becomes this, Is the distribution
tolerably uniform on the whole, but with innumerable individual
deflections? That is, when we compare large areas,
are the ratios of the number of stars in each equal area
approximately equal, whilst, as we compare smaller and
smaller areas, do the relative numbers become more and
more irregular? With certain exceptions, such as that of
the Milky Way and other nebular clusters, this seems to be
pretty much the case, at any rate as regards the bulk of the
stars.[12]
All further questions: the decision, for instance, for or
against any form of the Nebular Hypothesis: or, admitting
this, the decision whether such and such parts of the visible
heavens have sprung from the same nebula, must be left to
Astronomy to adjudicate.
NOTE ON THE PROPORTIONS OF THE SEXES.
The following remarks were rather too long for convenient insertion on
p. 259, and are therefore appended here.
The 'random' character of male and female births has generally been
rested almost entirely on statistics of place and time. But what is
more wanted, surely, is the proportion displayed when we compare a
number of _families_. This seems so obvious that I cannot but suppose
that the investigation must have been already made somewhere, though I
have not found any trace of it in the most likely quarters. Thus
Prof. Lexis (_Massenerscheinungen_) when supporting his view that the
proportion between the sexes at birth is almost the only instance
known to him, in natural phenomena, of true normal dispersion about a
mean, rests his conclusions on the ordinary statistics of the
registers of different countries.
It certainly needs proof that the same characteristics will hold good
when the family is taken as the unit, especially as some theories (e.g. that
of Sadler) would imply that 'runs' of boys or girls would be proportionally
commoner than pure chance would assign. Lexis has shown that this is
most markedly the case with _twins_: i.e., to use an obviously intelligible
notation, (M for male, F for female), that M.M. and F.F. are very much
commoner in proportion than M.F.
I have collected statistics including over 13,000 male and female
births, arranged in families of four and upwards. They were taken from
the pedigrees in the Herald's Visitations, and therefore represent as
a rule a somewhat select class, viz. the families of the eldest sons
of English country gentlemen in the sixteenth century. They are not
sufficiently extensive yet for publication, but I give a summary of
the results to indicate their tendency so far. The upper line of
figures in each case gives the _observed_ results: i.e. in the case
of a family of four, the numbers which had four male, three male and
one female, two male and two female, and so on. The lower line gives
the _calculated_ results; i.e. the corresponding numbers which would
have been obtained had batches of M.s and F.s been drawn from a bag in
which they were mixed in the ratio assigned by the total observed
numbers for those families.
512 families of 4; | m^4 m^3 f m^2 f^2 mf^3 f^4
yielding | 81 + 148 + 161 + 98 + 24 (observed.)
1188 M. : 860 F. | 57 + 168 + 184 + 88 + 15 (calculated.)
512 families of 5; | m^5 m^4 f m^3 f^2 m^2 f^3 mf^4 f^5
yielding | 50 + 82 + 161 + 143 + 61 + 15 (obs.)
1402 M. : 1158 F. | 25 + 103 + 172 + 143 + 59 + 10 (calc.)
512 families of 6; | m^6 m^5 f m^4 f^2 m^3 f^3 m^2 f^4 mf^5 f^6
yielding | 30 + 48 + 115 + 146 + 126 + 40 + 7 (obs.)
1612 M. : 1460 F. | 10 + 56 + 133 + 159 + 108 + 41 + 5 (calc.)
The numbers for the larger families are as yet too small to be worth
giving, but they show the same tendency. It will be seen that in every case
the observed central values are less than the calculated; and that the
observed extreme values are much greater than the calculated. The results
seem to suggest (so far) that a family cannot be likened to a chance drawing
of the requisite number from _one_ bag. A better analogy would be to suppose
two bags, one with M.s in excess and the other with F.s in less excess, and
that some persons draw from one and some from the other. But fuller
statistics are needed.
It will be observed that the total excess of male births is large. This
_may_ arise from undue omission of females; but I have carefully confined
myself to the two or three last generations, in each pedigree, for greater
security.
1. _Essay on Probabilities_, p. 114.
2. Doubts have been expressed about the truly random character of the
digits in this case (v. De Morgan, _Budget of Paradoxes_, p. 291),
and Jevons has gone so far as to ask (_Principles of Science_,
p. 529), "Why should the value of π, when expressed to a great
number of figures, contain the digit 7 much less frequently than any
other digit!" I do not quite understand what this means. If such a
question were asked in relation to any unusual divergence from the
_à priori_ chance in a case of throwing dice, say, we should
probably substitute for it the following, as being more appropriate
to our science:--Assign the degree of improbability of the event in
question; i.e. its statistical rarity. And we should then proceed
to judge, in the way indicated in the text, whether this
improbability gave rise to any grounds of suspicion.
The calculation is simple. The actual number of 7's, in the 708
digits, is 53: whilst the fair average would be 71. The question is,
What is the chance of such a departure from the average in 708
turns? By the usual methods of calculation (v. Galloway on
_Probability_) the chances against an excess or defect of 18 are
about 44 : 1, in respect of any specified digit. But of course what
we want to decide are the chances against _some one of the ten_
showing this divergence. This I estimate as being approximately
determined by the fraction (44/45)^{10}, viz. 0.8. This represents
odds of only about 4 : 1 against such an occurrence, which is
nothing remarkable. As a matter of fact several digits in the two
other magnitudes which Mr Shanks had calculated to the same length,
viz. Tan^{-1} 1/5 and Tan^{-1} 1/239, show the same divergencies
(v. _Proc. Roy. Soc._ xxi. 319).
I may call attention here to a point which should have been noticed
in the chapter on Randomness. We must be cautious when we decide
upon the random character by mere inspection. It is very instructive
here to compare the digits in π with those within the 'period' of a
circulating decimal of very long period. That of 1 ÷ 7699, which
yields the full period of 7698 figures, was calculated some years
ago by two Cambridge graduates (Mr Lunn and Mr Suffield), and
privately printed. If we confine our examination to a portion of the
succession the random character seems plausible; i.e. the digits,
and their various combinations, come out in nearly, but not exactly,
equal numbers. So if we take batches of 10; the averages hover
nicely about 45. But if we took the whole period which
'circulates,' we should find these characteristics overdone, and the
random character would disappear. That is, instead of a merely
ultimate approximation to equality we should have (as far as this is
possible) an absolute attainment of it.
3. Of course this conventional estimate is nothing different in kind
from that which may attach to _any_ order or succession. Ten heads
in succession is intrinsically or objectively indistinguishable in
character from alternate heads and tails, or seven heads and three
tails, &c. Its distinction only consists in its almost universal
acceptance as remarkable.
4. _Our Inheritance in the Great Pyramid_, Ed. III. 1877.
5. Made in _Nature_ (Jan. 24, 1878) by Mr J. G. Jackson. It must be
remarked that Mr Smyth's alternative statement of his case leads up
to that explanation:--"The vertical height of the great pyramid is
the radius of a theoretical circle the length of whose curved
circumference is exactly equal to the sum of the lengths of the four
straight sides of the actual and practical square base." As regards
the alternatives of chance and design, here, it must be remembered
in justice to Mr Smyth's argument that the antithesis he admits to
chance is not human, but divine design.
6. See Cournot, _Essai sur les fondements de nos connaissances_.
Vol. I. p. 71.
7. It deserves notice that considerations of this kind have found
their way into the Law Courts though of course without any attempt
at numerical valuation. Thus, in the celebrated De Ros trial, in so
far as the evidence was indirect, one main ground of suspicion seems
to have been that Lord De Ros, when dealing at whist, obtained far
more court cards than chance could be expected to assign him; and
that in consequence his average gains for several years in
succession were unusually large. The counsel for the defence urged
that still larger gains had been secured by other players without
suspicion of unfairness,--(I cannot find that it was explained over
how large an area of experience these instances had been sought; nor
how far the magnitude of the stakes, as distinguished from the
number of successes, accounted for that of the actual gains),--and
that large allowance must be made for skill where the actual gains
were computed. (See the _Times_' report, Feb. 11, 1837.)
8. _Metretike_. At the end of this volume will be found a useful list
of a number of other publications by the same author on allied
topics.
9. That is, if we look simply to statistical results, as Arbuthnott
did, and as we should do if we were examining the tosses of a
penny. If the remarkable theory of Dr Düsing (_Die Regulierung des
Geschlechts-verhältnisses..._ Jena, 1884) be confirmed, the matter
would assume a somewhat different aspect. He attempts to show, both
on physiological grounds, and by analysis of statistics referring to
men and animals, that there is a decidedly _compensatory_ process at
work. That is, if for any cause either sex attains a preponderance,
agencies are at once set in motion which tend to redress the
balance. This is a modification and improvement of the older theory,
that the relative age of the parents has something to do with the
sex of the offspring.
Quetelet (_Letters_, p. 61) has attempted to _prove_ a proposition
about the succession of male and female births by certain
experiments supposed to be tried upon an urn with black and white
balls in it. But this is going too far. (See the note at the end of
this chapter.)
10. It is precisely analogous to the conclusion that the _flowers_ of
the daisies (as distinguished from the _plants_, v. p. 109) are not
distributed at random, but have a tendency to go in groups of two or
more. Mere observation shows this: and then, from our knowledge of
the growth of plants we may infer that these little groups spring
from the same root.
11. In this discussion, writers often speak of the probability of a
"_physical connection_" between these double stars. The phrase seems
misleading, for on the usual hypothesis of universal gravitation
_all_ stars are physically connected, by gravitation. It is
therefore better, as above, to make it simply a question of relative
proximity, and to leave it to astronomy to infer what follows from
unusual proximity.
12. Professor Forbes in the paper in the Philosophical Magazine
already referred to (Ch. VII. §18) gave several diagrams to show
what were the actual arrangements of a random distribution. He
scattered peas over a chess-board, and then counted the number which
rested on each square. His figures seem to show that the general
appearance of the stars is much the same as that produced by such a
plan of scattering.
Some recent investigations by Mr R. A. Proctor seem to show,
however, that there are at least two exceptions to this tolerably
uniform distribution. (1) He has ascertained that the stars are
decidedly more thickly aggregated in the Milky Way than
elsewhere. So far as this is to be relied on the argument is the
same as in the case of the double stars; it tends to prove that the
proximity of the stars in the Milky Way is not merely apparent, but
actual. (2) He has ascertained that there are two large areas, in
the North and South hemispheres, in which the stars are much more
thickly aggregated than elsewhere. Here, it seems to me,
Probability proves nothing: we are simply denying that the
distribution is uniform. What may follow in the way of inferences
as to the physical process of causation by which the stars have been
disposed is a question for the Astronomer. See Mr Proctor's _Essays
on Astronomy_, p. 297. Also a series of Essays in _The Universe and
the coming Transits_.