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Dictionary of the History of Ideas | ||

#### CHANCE

So far as we can judge, primitive man does not con-

ceptualize his world of experience in
any comprehen-

sive way. To him, some
events just happen; some he

can control himself; some he can influence by
sympa-

thetic magic; for some he can
enlist the aid of the

unseen world of spirits which surrounds him. He
knows

of no general laws; and hence he knows of no absence

of general
laws. If he ever thought about the matter

at all he might, perhaps, have
considered that many

events happen simply because they fall that way;
and

their falling so (Old French *la cheance,*
from Latin

*cadere*) was in the nature of the world, as we should

say today, “just one of those things.”

The emergence of more organized thought and lan-

guage was slow to change essential ideas about hap-

penings. As man collected his experiences, formed and

named his concepts, and began to perceive regularities

in the heavens and
on earth, he developed the idea

of cause and effect, and as time went on,
it seemed

to him that more and more events are causally linked.

But
whether every event had a cause was a question

which he was late in asking
(and for that matter, has

not yet answered). Some events were explicable in
a

straightforward way; but others were equally certainly

inexplicable,
and many more had to be explained in

terms of minor deities invented for
the purpose. In

polytheistic societies, such as the Egyptian, the
Greek,

and the Roman, it was held possible to influence events

by
enlisting the aid of some superhuman being, with

sacrifice, donation, or
even punishment (as when tribes

thrashed their idols); but these beings
themselves were

not omnipotent and it would seem—though the
records

are, not surprisingly, silent on such questions—that a

garded as proceeding blindly without direct interven-

tion of God or man, or without being subject in all

its aspects to law.

Nevertheless, nature proceeded in a manner which

man perceived more and more
to be orderly. We now

encounter one of those peculiar dichotomies of
which

history affords so many instances: the emergence of

gambling, on
the one hand, and the employment of

fortuitous events for divination, on
the other. The

gambler deliberately threw his fortunes at the mercy

of
uncontrolled events; the diviner used uncontrolled

events to control his
future.

The Germans of Tacitus' time, for example, decided

many of their tribal
procedures by a random process.

The priests would write a number of runes
on slips

of bark, offer a prayer for guidance, choose one hap-

hazardly, and follow the advice which it
gave (or, at

any rate, gave according to their interpretation). The

Jews made important choices by lot. The Romans had

their Sybilline books
and their Etruscan custom of

haruspication (divination from entrails). To
modern

eyes such procedures would look very like settling a

doubtful
issue by tossing up for it, but that was not

how it appeared to the
ancients. It was their way of

interrogating their Deity, of referring the
decision to

a Better Informed Authority.

At the same time, gambling became widespread. One

of the oldest poems on
record, in the Rig-Veda, is a

Gambler's Lament, in which the poet bewails
the loss

of all his possessions but, unfortunately for us, says

nothing about the kind of game he was playing. In

very early settlements
there occur deposits of huckle

bones (small bones in the foot of sheep or
goat) which

were assembled by man, almost certainly for playing

some
kind of game. These “astragali” have four clearly

defined surfaces and were probably the antecedents of

the ordinary
six-faced cube or die, specimens of which

are datable as far back as 3000
B.C.

The Greeks thought poorly of dice-playing. For them

it was an amusement for
children and old men. This,

among other things, may be the reason why no
Greek

writers other than Aristotle and Epicurus showed any

interest in
chance, and as far as is known, none arrived

at any idea of the statistical
regularities embodied in

series of repetitive events. The Romans were
inveterate

gamblers, especially in Imperial times; the emperor

Claudius wrote a treatise on dice, which unfortunately

has not survived.
The Germans were even worse and

an individual would on occasion gamble
himself into

slavery. We know a little about the type of dice-playing

which was indulged in. It was almost certainly the

ancestor of the medieval
game of hazard, itself the

ancestor of the American game of craps. (The word

“hazard,” from Arabic *al
zhar,* “the die,” was probably

brought
back to Europe by the Crusaders. It was the

name of a game, not a concept
of random occurrence.)

Just how much the ancients knew about calculating

chances is doubtful, but
it cannot have been very exact

knowledge, even though a gambler can hardly
fail to

have formed some notion of regular occurrence “in

the long run.” Early examples exist of loaded dice,

which
indicates that some persons at least were not

content to leave things in
the lap of the goddess For-

tuna. But anything
approaching a calculus of chances

was not even adumbrated.

The advent of Christianity, and later of Islam,

brought about a number of
important changes, both

in the philosophical concept of chance and in
moral

attitudes towards gambling. To the monotheist every

event,
however trivial, was under the direction of the

Almighty or one of his
agents. In this sense there was

no chance. Everything happened under the
divine

purpose. Hence there grew up the belief that events

which we
describe as fortuitous or random or subject

to chance are no different from
any other happenings,

except that we do not know why they happen.
Chance,

then, became a name which man gave to his own

ignorance and
not a property of events or things.

This belief has endured until the present day. Saint

Augustine, Saint Thomas
Aquinas, Spinoza all held it.

The physicists of the nineteenth century
mostly sub-

scribed to it, though not
necessarily for theological

reasons. The more Nature was discovered to be
subject

to law (or, if one prefers it, the more man shaped his

concepts into regular patterns to correspond with ob-

servation), the more it became evident that
“chance”

events appeared as such only because
something re-

mained to be discovered or
because their causality was

too complex for exact analysis. In the first
half of the

twentieth century we find a distinguished French

probabilist, Paul Lévy, remarking that chance ap-

peared to him to be a concept invented by man which

was unknown to Nature; and Einstein, notwithstanding

developments in
subatomic physics (see below), never

accepted chance as an essential
unanalyzable element

of the universe.

We return to the effect of Christianity on the concept

of chance. Augurs,
sybils, diviners, prognosticators

generally, were frowned on by the Church
from early

times. This was not merely because the new priesthood

could
tolerate no competition from the old. Under the

new religion it was impious
to interrogate God by

forcing Him, so to speak, to disclose His
intentions.

Moreover, gaming soon became associated with less

socially
tolerable activities—drinking, blasphemy,

violence—and as such was sternly discouraged. We still

possess a
sermon of Saint Cyprian of Carthage against

nardino of Siena was inveighing against gambling and

its vices to the same tune. None of this, of course,

arrested gaming for very long. The number and fre-

quency of the edicts issued against gaming are sufficient

evidence of its prevalence, on the one hand, and its

persistence, on the other. However, ineradicable as

gambling proved to be, the official attitude of the

Church was probably strong enough to prevent any

serious study of it.

Up to the middle of the fourteenth century the main

instruments of gaming
were dice. The Western world

then invented or acquired playing cards, whose
precise

origin, numerous legends notwithstanding, is still un-

known. Cards began to displace dice, but more
slowly

than might have been expected, probably on account

of their
cost. It was not until the beginning of the

eighteenth century that dice
began to lose their popu-

larity in favor of
cards. Roulette wheels and one-armed

bandits are, of course, products of
modern technology.

It might have been supposed that, after playing with

astragali, dice, and
cards for several thousand years,

man would have arrived relatively early
at some con-

cept of the laws of chance. There
is no evidence that

he did so much before the fourteenth century, and
even

then, after faint beginnings, it was three hundred years

before
the subject began to be understood. The earliest

European record of any
attempt to enumerate the

relative frequency of dice-falls occurs in a
medieval

poem *De vetula* (dated somewhere
between 1220 and

1250), one manuscript of which contains a tabulation

of the ways of throwing three dice. It is an isolated

contribution and for
the next recorded attempt at the

calculation of chances we have to notice a
treatise on

card-play by the gambling scholar, Girolamo Cardano.

This
remarkable man, part genius and part charlatan,

was an inveterate gambler
and a very competent

mathematician. His book, written perhaps in 1526
but

published only posthumously (1663), contains a clear

notion of the
definition of chances in terms of the

relative frequency of events and of
the multiplicative

law of independent probabilities. A translation
into

English and a biography of Cardano by Oystein Ore

appeared in
1953.

So far as concerns extant literature, Cardano's work

is also isolated. Some
Italian mathematicians of the

sixteenth century considered a few problems
in dice-

play, and in particular, we have a
fragment by Galileo

(about 1620), in which he correctly enumerates the
falls

of three dice. Undoubtedly there must have been much

discussion
about chances, especially in those countries

where men of science mingled
freely with men of

affairs; but little or nothing was published. The
calculus

of chances as we know it first became the subject of

general mathematical interest in France at the closing

half of
the seventeenth century, in the form of corre-

spondence between Pascal and Pierre de Fermat. The

time was ripe
for a rapid expansion of the mathe-

matical
theory of chance. The first book on the subject,

by Christiaan Huygens, was
published in 1657. In 1713

there appeared the remarkable study by James
(Jakob

or Jacques) Bernoulli called *Ars
conjectandi* in which

he derived the so-called binomial
distribution and

raised the fundamental question of the convergence

of
proportions in a series of trials to a “true” chance.

Once so launched, the mathematical theory advanced

rapidly. A little over a
hundred years later appeared

a major masterwork, Pierre Simon de Laplace's
*Théorie analytique des
probabilités* (1812). The subject was by

now not only interesting and respectable, but applica-

ble to scientific problems and, before long, to commer-

cial and industrial problems. It has been intensively

cultivated ever since.

In one respect commerce took advantage of chance

events. Some Italian shops
of the fifteenth century

would have a sack full of small presents standing
by

the counter and would invite customers to take a lucky

dip. This
*lotteria* developed into the present-day system

of raising money by selling chances on prizes. The

system spread over
Europe but lent itself so readily

to fraud that it was either forbidden or,
in most coun-

tries, conducted as a state
monopoly.

The subject which was formerly called the Doctrine

of Chances, and is now
more commonly but less accu-

rately called
the Mathematical Theory of Probability,

is mostly a deductive science.
Given a reference set

of events and their probabilities, the object is to
work

out the probabilities of some contingent event; e.g.,

given that
the chance of throwing any face of a die

is 1/6, find the probability that
all six faces will appear

in a given number of throws greater than six.
Inter-

esting as the subject is to the
mathematician and useful

as it may be to the statistician, it is not of
concern

in the history of ideas except insofar as its results are

required, as we shall see below, in scientific inference.

Once again we must go back a little in time. At the

end of the seventeenth
century the philosophical stud-

ies of cause and
chance, and the mathematics of the

Doctrine of Chances were poles apart.
They now began

to move closer together. It was not long before the

events of the dice board and the card table began to

be seen as particular
cases of fortuitous events of a

more general kind, emanating in some rather
mysteri-

ous way which conjured order out
of chaos. In short,

it began to be realized that chance, which
conceptually

was almost the negation of order, was subject to law,

although to law of a rather different kind in that it

admitted exceptions.
The English savant, Dr. John

equality of the sex ratio at birth and saw something

of Divine Providence in the phenomenon by which

the apparently random occurrence of the individual

event resulted cumulatively in a stable sex ratio. Thirty

years later, J. P. Süssmilch, an honored name in the

history of statistics, reflected the same thought in the

title of his

*magnum opus*on the divine order:

*Die*

Göttliche Ordnung(1741). In one form or another the

Göttliche Ordnung

idea has remained current ever since. There are few

people who have reflected on the curious way in which

random events have a stable pattern “in the long run”

who have not been intrigued by the way in which order

emerges from disorder in series of repeated trials. Even

events which recur relatively infrequently may have

a pattern; for example, the nineteenth-century Belgian

astronomer and statistician Adolphe Quételet, one of

the fathers of modern statistics, was struck by

*L'effrayante exactitude avec laquelle les suicides se*

reproduisent(“The frightening regularity marking the

reproduisent

recurrence of suicides”).

During the eighteenth and nineteenth centuries the

realization grew
continually stronger that aggregates

of events may obey laws even when
individuals do not.

Uncertain as is the duration of any particular
human

life, the solvency of a life insurance company is guar-

anteed; uncertain as may be the sex of an
unborn child,

the approximate equality of numbers of the two sexes

is
one of the most certain things in the world.

This development had an important impact on the

theory of chance itself.
Previously chance was a nui-

sance, at least to
those who wished to foresee and

control the future. Man now began to use it
for other

purposes, or if not to use it, to bring it under control,

to
measure its effect, and to make due allowance for

it. For example, errors
of observation were found to

follow a definite law, and it became possible
to state

limits of error in measurements in precise probabilistic

terms. In the twentieth century we have seen similar

ideas worked out to a
high degree of precision: in the

theory of sampling, where we are content
to scrutinize

only a subset of a population, relying on the laws of

chance to give us a reasonably representative subset;

or in the theory of
experimental design, in which un-

wanted
influences are distributed at random in such

a way that chance destroys (or
reduces to minimal risk)

the possibility that they may distort the
interpretation

of the experiment. Man cannot remove chance effects,

but he has learned to control them.

In practice, there is little difference of opinion

among the experts as to
what should be done in any

given set of practical circumstances affected by
random

influences. But, though they may agree on procedure

and
interpretation, there underlies the theory of chance

and probability a profound difference of opinion as to

the
basis of the inferences which derive from probabil-

istic considerations.

We must now draw a distinction between chance

and probability. To nearly all
medieval logicians prob-

ability was an
attitude of mind. It expressed the doubt

which a person entertained towards
some proposition.

It was recognized (e.g., by Aquinas) that there were

degrees of doubt, although nobody got so far as to

suggest that probability
could be measured. It was not

necessarily related to the frequency with
which an

event occurred. Saint Thomas would have considered

the word
“probability” as equally applicable to the

proposition that there was a lost continent of Atlantis

as to the
proposition that next summer will be a fine

one.

The Doctrine of Chances, on the other hand, was

related to the relative
frequency of occurrence of the

various modalities of a class of events. The
two ideas

have been confounded over the centuries, and even

today
there are strongly differing schools of thought

on the subject. One school
takes probability as a

more-or-less subjective datum, and would try to
em-

brace all doubtful propositions, whether
relating to

unique or to repetitive situations, within a probabilistic

theory of doubt and belief. The other asserts that nu-

merical probabilities can be related only to relative

frequency. Both points of view have been very ably

expounded, the main
protagonists of the subjective

viewpoint being Bruno de Finetti and L. J.
Savage and

those of the frequency viewpoint, John Venn (1866)

and R.
von Mises (1928). The two are not, perhaps,

irreconcilable, but they have
never been successfully

reconciled, at least to the satisfaction of the
partici-

pants in the argument. The
nearest approach, perhaps,

is that of Sir Harold Jeffreys (1939).

To modern eyes, the matter becomes of critical

importance when we realize
that all science proceeds

essentially from hypotheses of doubtful validity
or

generality through experiment and confirmation, to

more firmly
based hypotheses. The problem, then, is

whether we can use probability
theory, of whatever

basic character, in the scientist's approach to
forming

his picture of the universe. The first man to consider

the
problem in mathematical detail was Thomas Bayes,

a Methodist parson whose
paper was published post-

humously (1764)
and whose name is now firmly at-

tached to a
particular type of inference. Shorn of its

mathematical trappings, Bayesian
inference purports

to assign numerical probabilities to alternative hy-

potheses which can explain observation. It
can do so

only by assigning prior probabilities to the hypotheses,

prior, that is, in the sense that they are given before

the observations
are collected. Here rests the conflict

former like to express their degree of doubt about the

alternative hypotheses at the outset in terms of numer-

ical probabilities, and to modify those probabilities in

the light of further experience; the latter prefer to

reserve their initial doubts for a final synthetic judg-

ment at the conclusion of the experiment. The course

of thought during the nineteenth century was un-

doubtedly influenced by Laplace, who accepted Bayes's

treatment, although recognizing the difficulty of re-

solving many practical situations into prior alternatives

of equal probability.

The basis of the controversy may be set out in fairly

simple terms. A
naïve statement of an argument in

scientific inference would run
like this:

We experiment and find that the event is rea- sonably closely realized (or not realized);

We accept (or reject) the hypothesis, or at any rate regard it as confirmed (or not confirmed).

question is whether, if we interpret “to be expected”

in terms of probability in the sense of the Doctrine

of Chances (e.g., on the hypothesis that a penny is

unbiased the chances are that in 100 tosses it will come

down heads about half the times), we can, so to speak,

invert the situation and make numerical statements

about the

*probability of the hypothesis.*Bayes saw the

problem, but to attain practical results, he had to

assume a postulate to the effect that if a number of

different hypotheses were exhaustive and all consonant

with the observed event, and nothing is known to the

contrary, they were to be supposed to have equal prior

probabilities. This so-called Principle of Indifference

or of Nonsufficient Reason has been warmly contested

by the anti-Bayesians.

There seems to be no decisive criterion of choice

between the Bayesian and
non-Bayesian approaches.

As with attitudes towards frequency or
nonfrequency

theories of probability, a man must make up his own

mind
about the criticisms that have been made of each.

Fortunately, in practice
conclusions drawn from the

same data rarely differ—or if they do
it appears that

the inference is entangled with personal experiences,

emotions, or prejudices which are not common to both

parties to the
dispute.

Until the end of the nineteenth century, chance and

probability, however
regarded axiomatically, were still

considered by most scientists and
philosophers alike

as expressions of ignorance, not as part of the basic

structure of the world. The fall of a die might be the

most
unpredictable of events, but its unpredictability

was due to the fact that
we could not compute its

trajectory with any accuracy; given enough informa-

tion about initial conditions and
sufficient mathe-

matical skill we could
calculate exactly how it would

fall and the element of chance would vanish.
Notwith-

standing the philosophic
doubts raised by Hume and

his successors about causality, the world was
(and still

is) interpreted by most people in a causal way. The

laws of
chance were not *sui generis;* they were the

result of the convolution of a multiplicity of causes.

As A. Cournot put
it, following Aristotle, a chance

event was the result of the intersection
of many caus-

ally determined lines.

This edifice began to crack with the discovery of

radioactivity. Here were
phenomena which appeared

to generate themselves in a basically chance
manner,

uninfluenced by pressure, temperature, or any external

change
which man could induce in their environment.

It has even been suggested
that a truly random se-

quence could be
generated by noting the intervals

between impacts in a Geiger counter. It
began to look

as if chance behavior was part of the very structure

of
the atomic world, and before long (ca. 1925), P. A. M.

Dirac, Werner
Heisenberg, and others were expres-

sing
subatomic phenomena as waves of probability.

We are still fairly close to the period in which these

ideas were put
forward, and in assessing them we have

to take into account the general
cultural and psycho-

logical environment
of the times. Immediately after

the First World War there was an upsurge of
revolt

against the repressive society of the later nineteenth

century,
and any idol which could be shown to have

feet of clay was joyfully
assaulted. Scientists, whether

natural or social, are no more immune than
poets to

such movements. The warmth of the reception given

to the
theory of relativity (far more enthusiastic than

the experimental evidence
justified), to the quantum

theory, and to Freudian psychology was in part
due

to this desire to throw off the shackles of the past; and

the
elevation of chance to a fundamental rule of be-

havior may have embodied a similar iconoclastic ele-

ment. It is too soon to say; but now that the
honeymoon

period is over there are some who would revert to the

older
view and consider that perhaps it is our ignorance

again which is being
expressed in the probabilistic

element of modern physics.

There remain, then, several important questions on

which unanimity is far
from being reached: whether

a theory of probability can embrace attitudes
of doubt

of all kinds, whether chance phenomena are part of

the basic
structure of the world, what is the best

method of setting up a theory of
inference in terms

able. Perhaps these questions may not be resolved until

a great deal more knowledge is gained about how the

human mind works. In the meantime the theory of

probability continues to develop in a constructive

manner and is an important adjunct to man's efforts

to measure and control the world.

##
*BIBLIOGRAPHY*

J. Arbuthnot, “An Argument for Divine Providence,

Taken From
the Constant Regularity Observ'd in the Birth

of Both
Sexes,” *Philosophical Transactions of the
Royal Society,*
27 (1710), 186-90. T. Bayes, “An Essay
Towards

Solving a Problem in the Doctrine of Chances,”

*Philo-*

sophical Transactions of the Royal Society,53 (1763),

sophical Transactions of the Royal Society,

370-418. Jakob (Jacques) Bernoulli,

*Ars conjectandi*(Basel,

1713, posthumous; Brussels, 1968). Rudolf Carnap,

*Logical*

Foundations of Probability(Chicago, 1950; 2nd ed. 1962).

Foundations of Probability

A. A. Cournot,

*Essai sur les fondements*... (1851), trans.

M. H. Moore as

*Essay on the Foundations of our Knowledge*

(New York, 1956), Chs. IV, V, VI. F. N. David, “Studies in

the History of Probability and Statistics, I. Dicing and

Gaming,”

*Biometrika,*42 (1955), 1; idem,

*Games, Gods and*

Gambling(London, 1962). Bruno de Finetti, “La Prévision:

Gambling

ses lois logiques, ses sources subjectives,” in

*Annales de*

l'Institut Henri Poincaré,7 (1937), 1-68; trans. H. E. Kyburg,

l'Institut Henri Poincaré,

Jr., in

*Studies in Subjective Probability*(New York, 1964).

Sir Harold Jeffreys,

*Theory of Probability*(Oxford, 1939; 3rd

ed. 1961). M. G. Kendall, “On the Reconciliation of Theories

of Probability,”

*Biometrika,*36 (1949), 101; idem, “Studies

in the History of Probability and Statistics, II. The Begin-

nings of a Probability of Calculus,”

*Biometrika,*43 (1956),

1; ibid., V. “A Note on Playing Cards,”

*Biometrika,*44 (1957),

260. J. M. Keynes,

*A Treatise on Probability*(London, 1921).

Pierre Simon, Marquis de Laplace,

*Théorie analytique des*

probabilités(Paris, 1812); is found in

probabilités

*A Philosophical Essay*

on Probabilities,trans. F. L. Truscott and F. L. Emory

on Probabilities,

(London and New York, 1902; New York, 1951). Oystein

Ore,

*Cardano, The Gambling Scholar*(Princeton, 1953). L. J.

Savage,

*Foundation of Statistical Inference,*2nd ed. (New

York, 1964). J. P. Süssmilch,

*Die Göttliche Ordnung in den*

Veränderungen des menschlichen Geschlechts aus der Geburt,

dem Tode und der Fortpflanzung desselben verwiesen(Berlin,

Veränderungen des menschlichen Geschlechts aus der Geburt,

dem Tode und der Fortpflanzung desselben verwiesen

1741). Isaac Todhunter,

*A History of the Mathematical The-*

ory of Probability... (Cambridge and London, 1865). John

ory of Probability

Venn,

*The Logic of Chance*(London, 1866). R. von Mises,

*Wahrscheinlichkeit, Statistik und Wahrheit*(Vienna, 1928);

trans. as

*Probability, Statistics and Truth*(London, 1939;

New York, 1961).

MAURICE KENDALL

[See also Causation; Chance Images; Epicureanism; For-tune; Free Will and Determinism; Game Theory; Indeter-

minacy; Number; Probability.]

Dictionary of the History of Ideas | ||