BOOK I
SCIENCE AND THE SCIENTIST
CHAPTER I
The Choice of Facts
Tolstoi somewhere explains why 'science for its own sake' is
in his eyes an absurd conception. We can not know all facts,
since their number is practically infinite. It is necessary to
choose; then we may let this choice depend on the pure caprice
of our curiosity; would it not be better to let ourselves be guided
by utility, by our practical and above all by our moral needs;
have we nothing better to do than to count the number of lady-bugs
on our planet?
It is clear the word utility has not for him the sense men of
affairs give it, and following them most of our contemporaries.
Little cares he for industrial applications, for the marvels of
electricity or of automobilism, which he regards rather as obstacles
to moral progress; utility for him is solely what can make
man better.
For my part, it need scarce be said, I could never be content
with either the one or the other ideal; I want neither that plutocracy
grasping and mean, nor that democracy goody and mediocre,
occupied solely in turning the other cheek, where would dwell
sages without curiosity, who, shunning excess, would not die of
disease, but would surely die of ennui. But that is a matter of
taste and is not what I wish to discuss.
The question nevertheless remains and should fix our attention;
if our choice can only be determined by caprice or by immediate
utility, there can be no science for its own sake, and consequently
no science. But is that true? That a choice must be made is
incontestable; whatever be our activity, facts go quicker than we,
and we can not catch them; while the scientist discovers one fact,
there happen milliards of milliards in a cubic millimeter of his
body. To wish to comprise nature in science would be to want
to put the whole into the part.
But scientists believe there is a hierarchy of facts and that
among them may be made a judicious choice. They are right,
since otherwise there would be no science, yet science exists. One
need only open the eyes to see that the conquests of industry which
have enriched so many practical men would never have seen the
light, if these practical men alone had existed and if they had not
been preceded by unselfish devotees who died poor, who never
thought of utility, and yet had a guide far other than caprice.
As Mach says, these devotees have spared their successors the
trouble of thinking. Those who might have worked solely in
view of an immediate application would have left nothing behind
them, and, in face of a new need, all must have been begun over
again. Now most men do not love to think, and this is perhaps
fortunate when instinct guides them, for most often, when they
pursue an aim which is immediate and ever the same, instinct
guides them better than reason would guide a pure intelligence.
But instinct is routine, and if thought did not fecundate it, it
would no more progress in man than in the bee or ant. It is
needful then to think for those who love not thinking, and, as
they are numerous, it is needful that each of our thoughts be as
often useful as possible, and this is why a law will be the more
precious the more general it is.
This shows us how we should choose: the most interesting facts
are those which may serve many times; these are the facts which
have a chance of coming up again. We have been so fortunate as
to be born in a world where there are such. Suppose that instead
of 60 chemical elements there were 60 milliards of them,
that they were not some common, the others rare, but that they
were uniformly distributed. Then, every time we picked up a
new pebble there would be great probability of its being formed
of some unknown substance; all that we knew of other pebbles
would be worthless for it; before each new object we should be
as the new-born babe; like it we could only obey our caprices or
our needs. Biologists would be just as much at a loss if there
were only individuals and no species and if heredity did not
make sons like their fathers.
In such a world there would be no science; perhaps thought
and even life would be impossible, since evolution could not there
develop the preservational instincts. Happily it is not so; like
all good fortune to which we are accustomed, this is not appreciated
at its true worth.
Which then are the facts likely to reappear? They are first
the simple facts. It is clear that in a complex fact a thousand
circumstances are united by chance, and that only a chance still
much less probable could reunite them anew. But are there any
simple facts? And if there are, how recognize them? What
assurance is there that a thing we think simple does not hide a
dreadful complexity? All we can say is that we ought to prefer
the facts which seem simple to those where our crude eye discerns
unlike elements. And then one of two things: either this simplicity
is real, or else the elements are so intimately mingled as not
to be distinguishable. In the first case there is chance of our
meeting anew this same simple fact, either in all its purity or
entering itself as element in a complex manifold. In the second
case this intimate mixture has likewise more chances of recurring
than a heterogeneous assemblage; chance knows how to mix, it
knows not how to disentangle, and to make with multiple elements
a well-ordered edifice in which something is distinguishable, it
must be made expressly. The facts which appear simple, even
if they are not so, will therefore be more easily revived by chance.
This it is which justifies the method instinctively adopted by the
scientist, and what justifies it still better, perhaps, is that oft-recurring
facts appear to us simple, precisely because we are
used to them.
But where is the simple fact? Scientists have been seeking
it in the two extremes, in the infinitely great and in the infinitely
small. The astronomer has found it because the distances of
the stars are immense, so great that each of them appears but
as a point, so great that the qualitative differences are effaced,
and because a point is simpler than a body which has form and
qualities. The physicist on the other hand has sought the elementary
phenomenon in fictively cutting up bodies into infinitesimal
cubes, because the conditions of the problem, which undergo
slow and continuous variation in passing from one point of the
body to another, may be regarded as constant in the interior of
each of these little cubes. In the same way the biologist has
been instinctively led to regard the cell as more interesting than
the whole animal, and the outcome has shown his wisdom, since
cells belonging to organisms the most different are more alike,
for the one who can recognize their resemblances, than are these
organisms themselves. The sociologist is more embarrassed; the
elements, which for him are men, are too unlike, too variable, too
capricious, in a word, too complex; besides, history never begins
over again. How then choose the interesting fact, which is that
which begins again? Method is precisely the choice of facts; it
is needful then to be occupied first with creating a method, and
many have been imagined, since none imposes itself, so that sociology
is the science which has the most methods and the fewest
results.
Therefore it is by the regular facts that it is proper to begin;
but after the rule is well established, after it is beyond all doubt,
the facts in full conformity with it are erelong without interest
since they no longer teach us anything new. It is then the exception
which becomes important. We cease to seek resemblances;
we devote ourselves above all to the differences, and
among the differences are chosen first the most accentuated, not
only because they are the most striking, but because they will
be the most instructive. A simple example will make my thought
plainer: Suppose one wishes to determine a curve by observing
some of its points. The practician who concerns himself only
with immediate utility would observe only the points he might
need for some special object. These points would be badly distributed
on the curve; they would be crowded in certain regions,
rare in others, so that it would be impossible to join them by a
continuous line, and they would be unavailable for other applications.
The scientist will proceed differently; as he wishes to
study the curve for itself, he will distribute regularly the points
to be observed, and when enough are known he will join them
by a regular line and then he will have the entire curve. But
for that how does he proceed? If he has determined an extreme
point of the curve, he does not stay near this extremity, but goes
first to the other end; after the two extremities the most instructive
point will be the mid-point, and so on.
So when a rule is established we should first seek the cases
where this rule has the greatest chance of failing. Thence,
among other reasons, come the interest of astronomic facts, and
the interest of the geologic past; by going very far away in space
or very far away in time, we may find our usual rules entirely
overturned, and these grand overturnings aid us the better to see
or the better to understand the little changes which may happen
nearer to us, in the little corner of the world where we are called
to live and act. We shall better know this corner for having
traveled in distant countries with which we have nothing to do.
But what we ought to aim at is less the ascertainment of resemblances
and differences than the recognition of likenesses hidden
under apparent divergences. Particular rules seem at first discordant,
but looking more closely we see in general that they
resemble each other; different as to matter, they are alike as to
form, as to the order of their parts. When we look at them with
this bias, we shall see them enlarge and tend to embrace everything.
And this it is which makes the value of certain facts
which come to complete an assemblage and to show that it is the
faithful image of other known assemblages.
I will not further insist, but these few words suffice to show
that the scientist does not choose at random the facts he observes.
He does not, as Tolstoi says, count the lady-bugs, because, however
interesting lady-bugs may be, their number is subject to
capricious variations. He seeks to condense much experience
and much thought into a slender volume; and that is why a little
book on physics contains so many past experiences and a thousand
times as many possible experiences whose result is known
beforehand.
But we have as yet looked at only one side of the question.
The scientist does not study nature because it is useful; he studies
it because he delights in it, and he delights in it because it is
beautiful. If nature were not beautiful, it would not be worth
knowing, and if nature were not worth knowing, life would not
be worth living. Of course I do not here speak of that beauty
which strikes the senses, the beauty of qualities and of appearances;
not that I undervalue such beauty, far from it, but it has
nothing to do with science; I mean that profounder beauty which
comes from the harmonious order of the parts and which a pure
intelligence can grasp. This it is which gives body, a structure
so to speak, to the iridescent appearances which flatter our senses,
and without this support the beauty of these fugitive dreams
would be only imperfect, because it would be vague and always
fleeting. On the contrary, intellectual beauty is sufficient unto
itself, and it is for its sake, more perhaps than for the future
good of humanity, that the scientist devotes himself to long and
difficult labors.
It is, therefore, the quest of this especial beauty, the sense of
the harmony of the cosmos, which makes us choose the facts
most fitting to contribute to this harmony, just as the artist
chooses from among the features of his model those which perfect
the picture and give it character and life. And we need not
fear that this instinctive and unavowed prepossession will turn
the scientist aside from the search for the true. One may dream
a harmonious world, but how far the real world will leave it
behind! The greatest artists that ever lived, the Greeks, made
their heavens; how shabby it is beside the true heavens, ours!
And it is because simplicity, because grandeur, is beautiful,
that we preferably seek simple facts, sublime facts, that we delight
now to follow the majestic course of the stars, now to examine
with the microscope that prodigious littleness which is
also a grandeur, now to seek in geologic time the traces of a past
which attracts because it is far away.
We see too that the longing for the beautiful leads us to the
same choice as the longing for the useful. And so it is that this
economy of thought, this economy of effort, which is, according
to Mach, the constant tendency of science, is at the same time
a source of beauty and a practical advantage. The edifices that
we admire are those where the architect has known how to proportion
the means to the end, where the columns seem to carry
gaily, without effort, the weight placed upon them, like the
gracious caryatids of the Erechtheum.
Whence comes this concordance? Is it simply that the things
which seem to us beautiful are those which best adapt themselves
to our intelligence, and that consequently they are at the same
time the implement this intelligence knows best how to use?
Or is there here a play of evolution and natural selection? Have
the peoples whose ideal most conformed to their highest interest
exterminated the others and taken their place? All pursued
their ideals without reference to consequences, but while this
quest led some to destruction, to others it gave empire. One is
tempted to believe it. If the Greeks triumphed over the barbarians
and if Europe, heir of Greek thought, dominates the
world, it is because the savages loved loud colors and the clamorous
tones of the drum which occupied only their senses, while the
Greeks loved the intellectual beauty which hides beneath sensuous
beauty, and this intellectual beauty it is which makes intelligence
sure and strong.
Doubtless such a triumph would horrify Tolstoi, and he would
not like to acknowledge that it might be truly useful. But this
disinterested quest of the true for its own beauty is sane also and
able to make man better. I well know that there are mistakes,
that the thinker does not always draw thence the serenity he
should find therein, and even that there are scientists of bad
character. Must we, therefore, abandon science and study only
morals? What! Do you think the moralists themselves are irreproachable
when they come down from their pedestal?
To foresee the future of mathematics, the true method is to
study its history and its present state.
Is this not for us mathematicians in a way a professional procedure?
We are accustomed to extrapolate, which is a means
of deducing the future from the past and present, and as we well
know what this amounts to, we run no risk of deceiving ourselves
about the range of the results it gives us.
We have had hitherto prophets of evil. They blithely reiterate
that all problems capable of solution have already been solved,
and that nothing is left but gleaning. Happily the case of the
past reassures us. Often it was thought all problems were solved
or at least an inventory was made of all admitting solution.
And then the sense of the word solution enlarged, the insoluble
problems became the most interesting of all, and others unforeseen
presented themselves. For the Greeks a good solution was
one employing only ruler and compasses; then it became one
obtained by the extraction of roots, then one using only algebraic
or logarithmic functions. The pessimists thus found themselves
always outflanked, always forced to retreat, so that at present I
think there are no more.
My intention, therefore, is not to combat them, as they are
dead; we well know that mathematics will continue to develop,
but the question is how, in what direction? You will answer,
'in every direction,' and that is partly true; but if it were
wholly true it would be a little appalling. Our riches would
soon become encumbering and their accumulation would produce
a medley as impenetrable as the unknown true was for the
ignorant.
The historian, the physicist, even, must make a choice among
facts; the head of the scientist, which is only a corner of the
universe, could never contain the universe entire; so that among
the innumerable facts nature offers, some will be passed by,
others retained.
Just so, a fortiori, in mathematics; no more can the geometer
hold fast pell-mell all the facts presenting themselves to him;
all the more because he it is, almost I had said his caprice, that
creates these facts. He constructs a wholly new combination by
putting together its elements; nature does not in general give it
to him ready made.
Doubtless it sometimes happens that the mathematician undertakes
a problem to satisfy a need in physics; that the physicist
or engineer asks him to calculate a number for a certain application.
Shall it be said that we geometers should limit ourselves
to awaiting orders, and, in place of cultivating our science for
our own delectation, try only to accommodate ourselves to the
wants of our patrons? If mathematics has no other object besides
aiding those who study nature, it is from these we should
await orders. Is this way of looking at it legitimate? Certainly
not; if we had not cultivated the exact sciences for themselves,
we should not have created mathematics the instrument, and the
day the call came from the physicist we should have been
helpless.
Nor do the physicists wait to study a phenomenon until some
urgent need of material life has made it a necessity for them;
and they are right. If the scientists of the eighteenth century
had neglected electricity as being in their eyes only a curiosity
without practical interest, we should have had in the twentieth
century neither telegraphy, nor electro-chemistry, nor electro-technics.
The physicists, compelled to choose, are therefore not
guided in their choice solely by utility. How then do they choose
between the facts of nature? We have explained it in the preceding
chapter: the facts which interest them are those capable
of leading to the discovery of a law, and so they are analogous
to many other facts which do not seem to us isolated, but closely
grouped with others. The isolated fact attracts all eyes, those of
the layman as well as of the scientist. But what the genuine
physicist alone knows how to see, is the bond which unites many
facts whose analogy is profound but hidden. The story of Newton's
apple is probably not true, but it is symbolic; let us speak
of it then as if it were true. Well then, we must believe that
before Newton plenty of men had seen apples fall; not one knew
how to conclude anything therefrom. Facts would be sterile
were there not minds capable of choosing among them, discerning
those behind which something was hidden, and of recognizing
what is hiding, minds which under the crude fact perceive the
soul of the fact.
We find just the same thing in mathematics. From the varied
elements at our disposal we can get millions of different combinations;
but one of these combinations, in so far as it is isolated,
is absolutely void of value. Often we have taken great pains to
construct it, but it serves no purpose, if not perhaps to furnish a
task in secondary education. Quite otherwise will it be when
this combination shall find place in a class of analogous combinations
and we shall have noticed this analogy. We are no longer
in the presence of a fact, but of a law. And upon that day the
real discoverer will not be the workman who shall have patiently
built up certain of these combinations; it will be he who brings
to light their kinship. The first will have seen merely the crude
fact, only the other will have perceived the soul of the fact.
Often to fix this kinship it suffices him to make a new word, and
this word is creative. The history of science furnishes us a
crowd of examples familiar to all.
The celebrated Vienna philosopher Mach has said that the rôle
of science is to produce economy of thought, just as machines
produce economy of effort. And that is very true. The savage
reckons on his fingers or by heaping pebbles. In teaching children
the multiplication table we spare them later innumerable
pebble bunchings. Some one has already found out, with pebbles
or otherwise, that 6 times 7 is 42 and has had the idea of noting
the result, and so we need not do it over again. He did not
waste his time even if he reckoned for pleasure: his operation
took him only two minutes; it would have taken in all two milliards
if a milliard men had had to do it over after him.
The importance of a fact then is measured by its yield, that is
to say, by the amount of thought it permits us to spare.
In physics the facts of great yield are those entering into a
very general law, since from it they enable us to foresee a great
number of others, and just so it is in mathematics. Suppose I
have undertaken a complicated calculation and laboriously
reached a result: I shall not be compensated for my trouble if
thereby I have not become capable of foreseeing the results of
other analogous calculations and guiding them with a certainty
that avoids the gropings to which one must be resigned in a
first attempt. On the other hand, I shall not have wasted my
time if these gropings themselves have ended by revealing to me
the profound analogy of the problem just treated with a much
more extended class of other problems; if they have shown me
at once the resemblances and differences of these, if in a word
they have made me perceive the possibility of a generalization.
Then it is not a new result I have won, it is a new power.
The simple example that comes first to mind is that of an algebraic
formula which gives us the solution of a type of numeric
problems when finally we replace the letters by numbers. Thanks
to it, a single algebraic calculation saves us the pains of ceaselessly
beginning over again new numeric calculations. But this
is only a crude example; we all know there are analogies inexpressible
by a formula and all the more precious.
A new result is of value, if at all, when in unifying elements
long known but hitherto separate and seeming strangers one to
another it suddenly introduces order where apparently disorder
reigned. It then permits us to see at a glance each of these
elements and its place in the assemblage. This new fact is not
merely precious by itself, but it alone gives value to all the old
facts it combines. Our mind is weak as are the senses; it would
lose itself in the world's complexity were this complexity not harmonious;
like a near-sighted person, it would see only the details
and would be forced to forget each of these details before examining
the following, since it would be incapable of embracing all.
The only facts worthy our attention are those which introduce
order into this complexity and so make it accessible.
Mathematicians attach great importance to the elegance of
their methods and their results. This is not pure dilettantism.
What is it indeed that gives us the feeling of elegance in a solution,
in a demonstration? It is the harmony of the diverse parts,
their symmetry, their happy balance; in a word it is all that
introduces order, all that gives unity, that permits us to see
clearly and to comprehend at once both the ensemble and the
details. But this is exactly what yields great results; in fact the
more we see this aggregate clearly and at a single glance, the
better we perceive its analogies with other neighboring objects,
consequently the more chances we have of divining the possible
generalizations. Elegance may produce the feeling of the unforeseen
by the unexpected meeting of objects we are not accustomed
to bring together; there again it is fruitful, since it thus unveils
for us kinships before unrecognized. It is fruitful even when it
results only from the contrast between the simplicity of the
means and the complexity of the problem set; it makes us then
think of the reason for this contrast and very often makes us
see that chance is not the reason; that it is to be found in some
unexpected law. In a word, the feeling of mathematical elegance
is only the satisfaction due to any adaptation of the solution
to the needs of our mind, and it is because of this very
adaptation that this solution can be for us an instrument. Consequently
this esthetic satisfaction is bound up with the economy
of thought. Again the comparison of the Erechtheum
comes to my mind, but I must not use it too often.
It is for the same reason that, when a rather long calculation
has led to some simple and striking result, we are not satisfied
until we have shown that we should have been able to foresee,
if not this entire result, at least its most characteristic traits.
Why? What prevents our being content with a calculation
which has told us, it seems, all we wished to know? It is because,
in analogous cases, the long calculation might not again
avail, and that this is not so about the reasoning often half intuitive
which would have enabled us to foresee. This reasoning
being short, we see at a single glance all its parts, so that we immediately
perceive what must be changed to adapt it to all the
problems of the same nature which can occur. And then it
enables us to foresee if the solution of these problems will be
simple, it shows us at least if the calculation is worth undertaking.
What we have just said suffices to show how vain it would be
to seek to replace by any mechanical procedure the free initiative
of the mathematician. To obtain a result of real value, it is not
enough to grind out calculations, or to have a machine to put
things in order; it is not order alone, it is unexpected order,
which is worth while. The machine may gnaw on the crude fact,
the soul of the fact will always escape it.
Since the middle of the last century, mathematicians are more
and more desirous of attaining absolute rigor; they are right,
and this tendency will be more and more accentuated. In mathematics
rigor is not everything, but without it there is nothing.
A demonstration which is not rigorous is nothingness. I think
no one will contest this truth. But if it were taken too literally,
we should be led to conclude that before 1820, for example, there
was no mathematics; this would be manifestly excessive; the
geometers of that time understood voluntarily what we explain
by prolix discourse. This does not mean that they did not see it
at all; but they passed over it too rapidly, and to see it well
would have necessitated taking the pains to say it.
But is it always needful to say it so many times? Those who
were the first to emphasize exactness before all else have given
us arguments that we may try to imitate; but if the demonstrations
of the future are to be built on this model, mathematical
treatises will be very long; and if I fear the lengthenings, it is
not solely because I deprecate encumbering libraries, but because
I fear that in being lengthened out, our demonstrations may lose
that appearance of harmony whose usefulness I have just
explained.
The economy of thought is what we should aim at, so it is not
enough to supply models for imitation. It is needful for those
after us to be able to dispense with these models and, in place of
repeating an argument already made, summarize it in a few
words. And this has already been attained at times. For instance,
there was a type of reasoning found everywhere, and
everywhere alike. They were perfectly exact but long. Then
all at once the phrase 'uniformity of convergence' was hit upon
and this phrase made those arguments needless; we were no
longer called upon to repeat them, since they could be understood.
Those who conquer difficulties then do us a double service:
first they teach us to do as they at need, but above all they
enable us as often as possible to avoid doing as they, yet without
sacrifice of exactness.
We have just seen by one example the importance of words in
mathematics, but many others could be cited. It is hard to believe
how much a well-chosen word can economize thought, as
Mach says. Perhaps I have already said somewhere that mathematics
is the art of giving the same name to different things. It
is proper that these things, differing in matter, be alike in
form, that they may, so to speak, run in the same mold. When
the language has been well chosen, we are astonished to see that
all the proofs made for a certain object apply immediately to
many new objects; there is nothing to change, not even the words,
since the names have become the same.
A well-chosen word usually suffices to do away with the exceptions
from which the rules stated in the old way suffer; this
is why we have created negative quantities, imaginaries, points
at infinity, and what not. And exceptions, we must not forget,
are pernicious because they hide the laws.
Well, this is one of the characteristics by which we recognize
the facts which yield great results. They are those which allow
of these happy innovations of language. The crude fact then
is often of no great interest; we may point it out many times
without having rendered great service to science. It takes value
only when a wiser thinker perceives the relation for which it
stands, and symbolizes it by a word.
Moreover the physicists do just the same. They have invented
the word 'energy,' and this word has been prodigiously
fruitful, because it also made the law by eliminating the exceptions,
since it gave the same name to things differing in matter
and like in form.
Among words that have had the most fortunate influence I
would select 'group' and 'invariant.' They have made us see
the essence of many mathematical reasonings; they have shown
us in how many cases the old mathematicians considered groups
without knowing it, and how, believing themselves far from one
another, they suddenly found themselves near without knowing
why.
To-day we should say that they had dealt with isomorphic
groups. We now know that in a group the matter is of little
interest, the form alone counts, and that when we know a group
we thus know all the isomorphic groups; and thanks to these
words 'group' and 'isomorphism,' which condense in a few syllables
this subtile rule and quickly make it familiar to all minds,
the transition is immediate and can be done with every economy
of thought effort. The idea of group besides attaches to that
of transformation. Why do we put such a value on the invention
of a new transformation? Because from a single theorem
it enables us to get ten or twenty; it has the same value as
a zero adjoined to the right of a whole number.
This then it is which has hitherto determined the direction of
mathematical advance, and just as certainly will determine it in
the future. But to this end the nature of the problems which
come up contributes equally. We can not forget what must be
our aim. In my opinion this aim is double. Our science borders
upon both philosophy and physics, and we work for our two
neighbors; so we have always seen and shall still see mathematicians
advancing in two opposite directions.
On the one hand, mathematical science must reflect upon itself,
and that is useful since reflecting on itself is reflecting on the
human mind which has created it, all the more because it is the
very one of its creations for which it has borrowed least from
without. This is why certain mathematical speculations are
useful, such as those devoted to the study of the postulates, of
unusual geometries, of peculiar functions. The more these speculations
diverge from ordinary conceptions, and consequently
from nature and applications, the better they show us what the
human mind can create when it frees itself more and more from
the tyranny of the external world, the better therefore they let
us know it in itself.
But it is toward the other side, the side of nature, that we must
direct the bulk of our army. There we meet the physicist or
the engineer, who says to us: "Please integrate this differential
equation for me; I might need it in a week in view of a construction
which should be finished by that time." "This equation,"
we answer, "does not come under one of the integrable types;
you know there are not many." "Yes, I know; but then what
good are you?" Usually to understand each other is enough;
the engineer in reality does not need the integral in finite terms;
he needs to know the general look of the integral function, or he
simply wants a certain number which could readily be deduced
from this integral if it were known. Usually it is not known,
but the number can be calculated without it if we know exactly
what number the engineer needs and with what approximation.
Formerly an equation was considered solved only when its
solution had been expressed by aid of a finite number of known
functions; but that is possible scarcely once in a hundred times.
What we always can do, or rather what we should always seek
to do, is to solve the problem qualitatively so to speak; that is to
say, seek to know the general form of the curve which represents
the unknown function.
It remains to find the quantitative solution of the problem;
but if the unknown can not be determined by a finite calculation,
it may always be represented by a convergent infinite series
which enables us to calculate it. Can that be regarded as a true
solution? We are told that Newton sent Leibnitz an anagram
almost like this: aaaaabbbeeeeij, etc. Leibnitz naturally understood
nothing at all of it; but we, who have the key, know that
this anagram meant, translated into modern terms: "I can integrate
all differential equations"; and we are tempted to say that
Newton had either great luck or strange delusions. He merely
wished to say he could form (by the method of indeterminate
coefficients) a series of powers formally satisfying the proposed
equation.
Such a solution would not satisfy us to-day, and for two
reasons: because the convergence is too slow and because the
terms follow each other without obeying any law. On the contrary,
the series Θ seems to us to leave nothing to be desired, first
because it converges very quickly (this is for the practical man
who wishes to get at a number as quickly as possible) and next
because we see at a glance the law of the terms (this is to satisfy
the esthetic need of the theorist).
But then there are no longer solved problems and others
which are not; there are only problems more or less solved,
according as they are solved by a series converging more or less
rapidly, or ruled by a law more or less harmonious. It often
happens however that an imperfect solution guides us toward a
better one. Sometimes the series converges so slowly that the
computation is impracticable and we have only succeeded in
proving the possibility of the problem.
And then the engineer finds this a mockery, and justly, since
it will not aid him to complete his construction by the date fixed.
He little cares to know if it will benefit engineers of the twenty-second
century. But as for us, we think differently and we are
sometimes happier to have spared our grandchildren a day's
work than to have saved our contemporaries an hour.
Sometimes by groping, empirically, so to speak, we reach a
formula sufficiently convergent. "What more do you want?"
says the engineer. And yet, in spite of all, we are not satisfied;
we should have liked to foresee that convergence. Why? Because
if we had known how to foresee it once, we would know how
to foresee it another time. We have succeeded; that is a small
matter in our eyes if we can not validly expect to do so again.
In proportion as science develops, its total comprehension
becomes more difficult; then we seek to cut it in pieces and to
be satisfied with one of these pieces: in a word, to specialize.
If we went on in this way, it would be a grievous obstacle to the
progress of science. As we have said, it is by unexpected union
between its diverse parts that it progresses. To specialize too
much would be to forbid these drawings together. It is to be
hoped that congresses like those of Heidelberg and Rome, by
putting us in touch with one another, will open for us vistas over
neighboring domains and oblige us to compare them with our
own, to range somewhat abroad from our own little village; thus
they will be the best remedy for the danger just mentioned.
But I have lingered too long over generalities; it is time to
enter into detail.
Let us pass in review the various special sciences which combined
make mathematics; let us see what each has accomplished,
whither it tends and what we may hope from it. If the preceding
views are correct, we should see that the greatest advances
in the past have happened when two of these sciences have united,
when we have become conscious of the similarity of their form,
despite the difference of their matter, when they have so modeled
themselves upon each other that each could profit by the other's
conquests. We should at the same time foresee in combinations
of the same sort the progress of the future.
Progress in arithmetic has been much slower than in algebra
and analysis, and it is easy to see why. The feeling of continuity
is a precious guide which the arithmetician lacks; each whole
number is separated from the others—it has, so to speak, its own
individuality. Each of them is a sort of exception and this is
why general theorems are rarer in the theory of numbers; this
is also why those which exist are more hidden and longer elude
the searchers.
If arithmetic is behind algebra and analysis, the best thing for
it to do is to seek to model itself upon these sciences so as to
profit by their advance. The arithmetician ought therefore to
take as guide the analogies with algebra. These analogies are
numerous and if, in many cases, they have not yet been studied
sufficiently closely to become utilizable, they at least have long
been foreseen, and even the language of the two sciences shows
they have been recognized. Thus we speak of transcendent
numbers and thus we account for the future classification of
these numbers already having as model the classification of transcendent
functions, and still we do not as yet very well see how
to pass from one classification to the other; but had it been seen,
it would already have been accomplished and would no longer
be the work of the future.
The first example that comes to my mind is the theory of congruences,
where is found a perfect parallelism to the theory of
algebraic equations. Surely we shall succeed in completing this
parallelism, which must hold for instance between the theory of
algebraic curves and that of congruences with two variables.
And when the problems relative to congruences with several
variables shall be solved, this will be a first step toward the solution
of many questions of indeterminate analysis.
The theory of algebraic equations will still long hold the attention
of geometers; numerous and very different are the sides
whence it may be attacked.
We need not think algebra is ended because it gives us rules
to form all possible combinations; it remains to find the interesting
combinations, those which satisfy such and such a condition.
Thus will be formed a sort of indeterminate analysis where the
unknowns will no longer be whole numbers, but polynomials.
This time it is algebra which will model itself upon arithmetic,
following the analogy of the whole number to the integral polynomial
with any coefficients or to the integral polynomial with
integral coefficients.
It looks as if geometry could contain nothing which is not
already included in algebra or analysis; that geometric facts are
only algebraic or analytic facts expressed in another language.
It might then be thought that after our review there would
remain nothing more for us to say relating specially to geometry.
This would be to fail to recognize the importance of well-constructed
language, not to comprehend what is added to the things
themselves by the method of expressing these things and consequently
of grouping them.
First the geometric considerations lead us to set ourselves new
problems; these may be, if you choose, analytic problems, but
such as we never would have set ourselves in connection with
analysis. Analysis profits by them however, as it profits by those
it has to solve to satisfy the needs of physics.
A great advantage of geometry lies in the fact that in it the
senses can come to the aid of thought, and help find the path to
follow, and many minds prefer to put the problems of analysis
into geometric form. Unhappily our senses can not carry us very
far, and they desert us when we wish to soar beyond the classical
three dimensions. Does this mean that, beyond the restricted
domain wherein they seem to wish to imprison us, we should
rely only on pure analysis and that all geometry of more than
three dimensions is vain and objectless? The greatest masters
of a preceding generation would have answered 'yes'; to-day we
are so familiarized with this notion that we can speak of it, even
in a university course, without arousing too much astonishment.
But what good is it? That is easy to see: First it gives us a
very convenient terminology, which expresses concisely what the
ordinary analytic language would say in prolix phrases. Moreover,
this language makes us call like things by the same name
and emphasize analogies it will never again let us forget. It
enables us therefore still to find our way in this space which is
too big for us and which we can not see, always recalling visible
space, which is only an imperfect image of it doubtless, but which
is nevertheless an image. Here again, as in all the preceding
examples, it is analogy with the simple which enables us to comprehend
the complex.
This geometry of more than three dimensions is not a simple
analytic geometry; it is not purely quantitative, but qualitative
also, and it is in this respect above all that it becomes interesting.
There is a science called analysis situs and which has for its
object the study of the positional relations of the different elements
of a figure, apart from their sizes. This geometry is purely
qualitative; its theorems would remain true if the figures, instead
of being exact, were roughly imitated by a child. We may also
make an analysis situs of more than three dimensions. The
importance of analysis situs is enormous and can not be too much
emphasized; the advantage obtained from it by Riemann, one of
its chief creators, would suffice to prove this. We must achieve
its complete construction in the higher spaces; then we shall have
an instrument which will enable us really to see in hyperspace
and supplement our senses.
The problems of analysis situs would perhaps not have suggested
themselves if the analytic language alone had been spoken;
or rather, I am mistaken, they would have occurred surely, since
their solution is essential to a crowd of questions in analysis, but
they would have come singly, one after another, and without our
being able to perceive their common bond.
I have spoken above of our need to go back continually to the
first principles of our science, and of the advantage of this for
the study of the human mind. This need has inspired two endeavors
which have taken a very prominent place in the most
recent annals of mathematics. The first is Cantorism, which has
rendered our science such conspicuous service. Cantor introduced
into science a new way of considering mathematical infinity.
One of the characteristic traits of Cantorism is that in
place of going up to the general by building up constructions
more and more complicated and defining by construction, it starts
from the genus supremum and defines only, as the scholastics
would have said, per genus proximum et differentiam specificam.
Thence comes the horror it has sometimes inspired in certain
minds, for instance in Hermite, whose favorite idea was to compare
the mathematical to the natural sciences. With most of
us these prejudices have been dissipated, but it has come to
pass that we have encountered certain paradoxes, certain apparent
contradictions that would have delighted Zeno, the Eleatic
and the school of Megara. And then each must seek the remedy.
For my part, I think, and I am not the only one, that the important
thing is never to introduce entities not completely definable
in a finite number of words. Whatever be the cure adopted, we
may promise ourselves the joy of the doctor called in to follow
a beautiful pathologic case.
On the other hand, efforts have been made to enumerate the
axioms and postulates, more or less hidden, which serve as foundation
to the different theories of mathematics. Professor Hilbert
has obtained the most brilliant results. It seems at first that this
domain would be very restricted and there would be nothing
more to do when the inventory should be ended, which could not
take long. But when we shall have enumerated all, there will be
many ways of classifying all; a good librarian always finds something
to do, and each new classification will be instructive for
the philosopher.
Here I end this review which I could not dream of making
complete. I think these examples will suffice to show by what
mechanism the mathematical sciences have made their progress
in the past and in what direction they must advance in the future.
The genesis of mathematical creation is a problem which
should intensely interest the psychologist. It is the activity in
which the human mind seems to take least from the outside
world, in which it acts or seems to act only of itself and on itself,
so that in studying the procedure of geometric thought we may
hope to reach what is most essential in man's mind.
This has long been appreciated, and some time back the journal
called L'enseignement mathématique, edited by Laisant and
Fehr, began an investigation of the mental habits and methods
of work of different mathematicians. I had finished the main
outlines of this article when the results of that inquiry were
published, so I have hardly been able to utilize them and shall
confine myself to saying that the majority of witnesses confirm
my conclusions; I do not say all, for when the appeal is to universal
suffrage unanimity is not to be hoped.
A first fact should surprise us, or rather would surprise us if
we were not so used to it. How does it happen there are people
who do not understand mathematics? If mathematics invokes
only the rules of logic, such as are accepted by all normal minds;
if its evidence is based on principles common to all men, and that
none could deny without being mad, how does it come about that
so many persons are here refractory?
That not every one can invent is nowise mysterious. That
not every one can retain a demonstration once learned may also
pass. But that not every one can understand mathematical
reasoning when explained appears very surprising when we think
of it. And yet those who can follow this reasoning only with
difficulty are in the majority: that is undeniable, and will surely
not be gainsaid by the experience of secondary-school teachers.
And further: how is error possible in mathematics? A sane
mind should not be guilty of a logical fallacy, and yet there are
very fine minds who do not trip in brief reasoning such as occurs
in the ordinary doings of life, and who are incapable of following
or repeating without error the mathematical demonstrations
which are longer, but which after all are only an accumulation
of brief reasonings wholly analogous to those they make so easily.
Need we add that mathematicians themselves are not infallible?
The answer seems to me evident. Imagine a long series of
syllogisms, and that the conclusions of the first serve as premises
of the following: we shall be able to catch each of these syllogisms,
and it is not in passing from premises to conclusion that
we are in danger of deceiving ourselves. But between the
moment in which we first meet a proposition as conclusion of one
syllogism, and that in which we reencounter it as premise of
another syllogism occasionally some time will elapse, several links
of the chain will have unrolled; so it may happen that we have
forgotten it, or worse, that we have forgotten its meaning. So
it may happen that we replace it by a slightly different proposition,
or that, while retaining the same enunciation, we attribute
to it a slightly different meaning, and thus it is that we are
exposed to error.
Often the mathematician uses a rule. Naturally he begins by
demonstrating this rule; and at the time when this proof is fresh
in his memory he understands perfectly its meaning and its bearing,
and he is in no danger of changing it. But subsequently he
trusts his memory and afterward only applies it in a mechanical
way; and then if his memory fails him, he may apply it all
wrong. Thus it is, to take a simple example, that we sometimes
make slips in calculation because we have forgotten our multiplication
table.
According to this, the special aptitude for mathematics would
be due only to a very sure memory or to a prodigious force of
attention. It would be a power like that of the whist-player who
remembers the cards played; or, to go up a step, like that of the
chess-player who can visualize a great number of combinations
and hold them in his memory. Every good mathematician ought
to be a good chess-player, and inversely; likewise he should be a
good computer. Of course that sometimes happens; thus Gauss
was at the same time a geometer of genius and a very precocious
and accurate computer.
But there are exceptions; or rather I err; I can not call them
exceptions without the exceptions being more than the rule.
Gauss it is, on the contrary, who was an exception. As for myself,
I must confess, I am absolutely incapable even of adding
without mistakes. In the same way I should be but a poor chess-player;
I would perceive that by a certain play I should expose
myself to a certain danger; I would pass in review several other
plays, rejecting them for other reasons, and then finally I should
make the move first examined, having meantime forgotten the
danger I had foreseen.
In a word, my memory is not bad, but it would be insufficient
to make me a good chess-player. Why then does it not fail me in
a difficult piece of mathematical reasoning where most chess-players
would lose themselves? Evidently because it is guided
by the general march of the reasoning. A mathematical demonstration
is not a simple juxtaposition of syllogisms, it is syllogisms
placed in a certain order, and the order in which these
elements are placed is much more important than the elements
themselves. If I have the feeling, the intuition, so to speak, of
this order, so as to perceive at a glance the reasoning as a whole,
I need no longer fear lest I forget one of the elements, for each
of them will take its allotted place in the array, and that without
any effort of memory on my part.
It seems to me then, in repeating a reasoning learned, that I
could have invented it. This is often only an illusion; but even
then, even if I am not so gifted as to create it by myself, I myself
re-invent it in so far as I repeat it.
We know that this feeling, this intuition of mathematical
order, that makes us divine hidden harmonies and relations, can
not be possessed by every one. Some will not have either this
delicate feeling so difficult to define, or a strength of memory
and attention beyond the ordinary, and then they will be absolutely
incapable of understanding higher mathematics. Such are
the majority. Others will have this feeling only in a slight
degree, but they will be gifted with an uncommon memory and
a great power of attention. They will learn by heart the details
one after another; they can understand mathematics and sometimes
make applications, but they cannot create. Others, finally,
will possess in a less or greater degree the special intuition
referred to, and then not only can they understand mathematics
even if their memory is nothing extraordinary, but they may
become creators and try to invent with more or less success
according as this intuition is more or less developed in them.
In fact, what is mathematical creation? It does not consist
in making new combinations with mathematical entities already
known. Any one could do that, but the combinations so made
would be infinite in number and most of them absolutely without
interest. To create consists precisely in not making useless
combinations and in making those which are useful and which
are only a small minority. Invention is discernment, choice.
How to make this choice I have before explained; the mathematical
facts worthy of being studied are those which, by their
analogy with other facts, are capable of leading us to the knowledge
of a mathematical law just as experimental facts lead us to
the knowledge of a physical law. They are those which reveal
to us unsuspected kinship between other facts, long known, but
wrongly believed to be strangers to one another.
Among chosen combinations the most fertile will often be those
formed of elements drawn from domains which are far apart.
Not that I mean as sufficing for invention the bringing together
of objects as disparate as possible; most combinations so formed
would be entirely sterile. But certain among them, very rare,
are the most fruitful of all.
To invent, I have said, is to choose; but the word is perhaps
not wholly exact. It makes one think of a purchaser before whom
are displayed a large number of samples, and who examines
them, one after the other, to make a choice. Here the samples
would be so numerous that a whole lifetime would not suffice to
examine them. This is not the actual state of things. The sterile
combinations do not even present themselves to the mind of the
inventor. Never in the field of his consciousness do combinations
appear that are not really useful, except some that he rejects
but which have to some extent the characteristics of useful combinations.
All goes on as if the inventor were an examiner for
the second degree who would only have to question the candidates
who had passed a previous examination.
But what I have hitherto said is what may be observed or
inferred in reading the writings of the geometers, reading
reflectively.
It is time to penetrate deeper and to see what goes on in the
very soul of the mathematician. For this, I believe, I can do best
by recalling memories of my own. But I shall limit myself to
telling how I wrote my first memoir on Fuchsian functions. I
beg the reader's pardon; I am about to use some technical expressions,
but they need not frighten him, for he is not obliged to
understand them. I shall say, for example, that I have found
the demonstration of such a theorem under such circumstances.
This theorem will have a barbarous name, unfamiliar to many,
but that is unimportant; what is of interest for the psychologist
is not the theorem but the circumstances.
For fifteen days I strove to prove that there could not be any
functions like those I have since called Fuchsian functions. I
was then very ignorant; every day I seated myself at my work
table, stayed an hour or two, tried a great number of combinations
and reached no results. One evening, contrary to my
custom, I drank black coffee and could not sleep. Ideas rose in
crowds; I felt them collide until pairs interlocked, so to speak,
making a stable combination. By the next morning I had established
the existence of a class of Fuchsian functions, those which
come from the hypergeometric series; I had only to write out
the results, which took but a few hours.
Then I wanted to represent these functions by the quotient of
two series; this idea was perfectly conscious and deliberate, the
analogy with elliptic functions guided me. I asked myself what
properties these series must have if they existed, and I succeeded
without difficulty in forming the series I have called theta-Fuchsian.
Just at this time I left Caen, where I was then living, to go on
a geologic excursion under the auspices of the school of mines.
The changes of travel made me forget my mathematical work.
Having reached Coutances, we entered an omnibus to go some
place or other. At the moment when I put my foot on the step
the idea came to me, without anything in my former thoughts
seeming to have paved the way for it, that the transformations
I had used to define the Fuchsian functions were identical with
those of non-Euclidean geometry. I did not verify the idea; I
should not have had time, as, upon taking my seat in the omnibus,
I went on with a conversation already commenced, but I
felt a perfect certainty. On my return to Caen, for conscience'
sake I verified the result at my leisure.
Then I turned my attention to the study of some arithmetical
questions apparently without much success and without a suspicion
of any connection with my preceding researches. Disgusted
with my failure, I went to spend a few days at the seaside,
and thought of something else. One morning, walking on
the bluff, the idea came to me, with just the same characteristics
of brevity, suddenness and immediate certainty, that the arithmetic
transformations of indeterminate ternary quadratic forms
were identical with those of non-Euclidean geometry.
Returned to Caen, I meditated on this result and deduced the
consequences. The example of quadratic forms showed me that
there were Fuchsian groups other than those corresponding to
the hypergeometric series; I saw that I could apply to them the
theory of theta-Fuchsian series and that consequently there
existed Fuchsian functions other than those from the hypergeometric
series, the ones I then knew. Naturally I set myself
to form all these functions. I made a systematic attack upon
them and carried all the outworks, one after another. There was
one however that still held out, whose fall would involve that of
the whole place. But all my efforts only served at first the better
to show me the difficulty, which indeed was something. All this
work was perfectly conscious.
Thereupon I left for Mont-Valérien, where I was to go through
my military service; so I was very differently occupied. One
day, going along the street, the solution of the difficulty which
had stopped me suddenly appeared to me. I did not try to go
deep into it immediately, and only after my service did I again
take up the question. I had all the elements and had only to
arrange them and put them together. So I wrote out my final
memoir at a single stroke and without difficulty.
I shall limit myself to this single example; it is useless to
multiply them. In regard to my other researches I would have
to say analogous things, and the observations of other mathematicians
given in L'enseignement mathématique would only
confirm them.
Most striking at first is this appearance of sudden illumination,
a manifest sign of long, unconscious prior work. The rôle
of this unconscious work in mathematical invention appears to
me incontestable, and traces of it would be found in other cases
where it is less evident. Often when one works at a hard question,
nothing good is accomplished at the first attack. Then
one takes a rest, longer or shorter, and sits down anew to the
work. During the first half-hour, as before, nothing is found,
and then all of a sudden the decisive idea presents itself to the
mind. It might be said that the conscious work has been more
fruitful because it has been interrupted and the rest has given
back to the mind its force and freshness. But it is more probable
that this rest has been filled out with unconscious work and
that the result of this work has afterward revealed itself to the
geometer just as in the cases I have cited; only the revelation,
instead of coming during a walk or a journey, has happened
during a period of conscious work, but independently of this
work which plays at most a rôle of excitant, as if it were the goad
stimulating the results already reached during rest, but remaining
unconscious, to assume the conscious form.
There is another remark to be made about the conditions of
this unconscious work: it is possible, and of a certainty it is only
fruitful, if it is on the one hand preceded and on the other hand
followed by a period of conscious work. These sudden inspirations
(and the examples already cited sufficiently prove this)
never happen except after some days of voluntary effort which
has appeared absolutely fruitless and whence nothing good seems
to have come, where the way taken seems totally astray. These
efforts then have not been as sterile as one thinks; they have set
agoing the unconscious machine and without them it would not
have moved and would have produced nothing.
The need for the second period of conscious work, after the
inspiration, is still easier to understand. It is necessary to put
in shape the results of this inspiration, to deduce from them the
immediate consequences, to arrange them, to word the demonstrations,
but above all is verification necessary. I have spoken of
the feeling of absolute certitude accompanying the inspiration;
in the cases cited this feeling was no deceiver, nor is it usually.
But do not think this a rule without exception; often this feeling
deceives us without being any the less vivid, and we only find it
out when we seek to put on foot the demonstration. I have
especially noticed this fact in regard to ideas coming to me in the
morning or evening in bed while in a semi-hypnagogic state.
Such are the realities; now for the thoughts they force upon
us. The unconscious, or, as we say, the subliminal self plays an
important rôle in mathematical creation; this follows from what
we have said. But usually the subliminal self is considered as
purely automatic. Now we have seen that mathematical work is
not simply mechanical, that it could not be done by a machine,
however perfect. It is not merely a question of applying rules,
of making the most combinations possible according to certain
fixed laws. The combinations so obtained would be exceedingly
numerous, useless and cumbersome. The true work of the inventor
consists in choosing among these combinations so as to
eliminate the useless ones or rather to avoid the trouble of making
them, and the rules which must guide this choice are extremely
fine and delicate. It is almost impossible to state them precisely;
they are felt rather than formulated. Under these conditions,
how imagine a sieve capable of applying them mechanically?
A first hypothesis now presents itself: the subliminal self is in
no way inferior to the conscious self; it is not purely automatic;
it is capable of discernment; it has tact, delicacy; it knows how
to choose, to divine. What do I say? It knows better how to
divine than the conscious self, since it succeeds where that has
failed. In a word, is not the subliminal self superior to the
conscious self? You recognize the full importance of this question.
Boutroux in a recent lecture has shown how it came up
on a very different occasion, and what consequences would follow
an affirmative answer. (See also, by the same author, Science
et Religion, pp. 313 ff.)
Is this affirmative answer forced upon us by the facts I have
just given? I confess that, for my part, I should hate to accept
it. Reexamine the facts then and see if they are not compatible
with another explanation.
It is certain that the combinations which present themselves to
the mind in a sort of sudden illumination, after an unconscious
working somewhat prolonged, are generally useful and fertile
combinations, which seem the result of a first impression. Does
it follow that the subliminal self, having divined by a delicate
intuition that these combinations would be useful, has formed
only these, or has it rather formed many others which were
lacking in interest and have remained unconscious?
In this second way of looking at it, all the combinations would
be formed in consequence of the automatism of the subliminal
self, but only the interesting ones would break into the domain
of consciousness. And this is still very mysterious. What is the
cause that, among the thousand products of our unconscious
activity, some are called to pass the threshold, while others remain
below? Is it a simple chance which confers this privilege? Evidently
not; among all the stimuli of our senses, for example, only
the most intense fix our attention, unless it has been drawn to
them by other causes. More generally the privileged unconscious
phenomena, those susceptible of becoming conscious, are
those which, directly or indirectly, affect most profoundly our
emotional sensibility.
It may be surprising to see emotional sensibility invoked
à propos of mathematical demonstrations which, it would seem,
can interest only the intellect. This would be to forget the feeling
of mathematical beauty, of the harmony of numbers and
forms, of geometric elegance. This is a true esthetic feeling that
all real mathematicians know, and surely it belongs to emotional
sensibility.
Now, what are the mathematic entities to which we attribute
this character of beauty and elegance, and which are capable of
developing in us a sort of esthetic emotion? They are those
whose elements are harmoniously disposed so that the mind without
effort can embrace their totality while realizing the details.
This harmony is at once a satisfaction of our esthetic needs and
an aid to the mind, sustaining and guiding; And at the same
time, in putting under our eyes a well-ordered whole, it makes
us foresee a mathematical law. Now, as we have said above, the
only mathematical facts worthy of fixing our attention and
capable of being useful are those which can teach us a mathematical
law. So that we reach the following conclusion: The
useful combinations are precisely the most beautiful, I mean
those best able to charm this special sensibility that all mathematicians
know, but of which the profane are so ignorant as
often to be tempted to smile at it.
What happens then? Among the great numbers of combinations
blindly formed by the subliminal self, almost all are without
interest and without utility; but just for that reason they are
also without effect upon the esthetic sensibility. Consciousness
will never know them; only certain ones are harmonious, and,
consequently, at once useful and beautiful. They will be capable
of touching this special sensibility of the geometer of which I
have just spoken, and which, once aroused, will call our attention
to them, and thus give them occasion to become conscious.
This is only a hypothesis, and yet here is an observation which
may confirm it: when a sudden illumination seizes upon the
mind of the mathematician, it usually happens that it does not
deceive him, but it also sometimes happens, as I have said, that
it does not stand the test of verification; well, we almost always
notice that this false idea, had it been true, would have gratified
our natural feeling for mathematical elegance.
Thus it is this special esthetic sensibility which plays the rôle
of the delicate sieve of which I spoke, and that sufficiently explains
why the one lacking it will never be a real creator.
Yet all the difficulties have not disappeared. The conscious
self is narrowly limited, and as for the subliminal self we know
not its limitations, and this is why we are not too reluctant in
supposing that it has been able in a short time to make more
different combinations than the whole life of a conscious being
could encompass. Yet these limitations exist. Is it likely that
it is able to form all the possible combinations, whose number
would frighten the imagination? Nevertheless that would seem
necessary, because if it produces only a small part of these combinations,
and if it makes them at random, there would be small
chance that the good, the one we should choose, would be found
among them.
Perhaps we ought to seek the explanation in that preliminary
period of conscious work which always precedes all fruitful
unconscious labor. Permit me a rough comparison. Figure
the future elements of our combinations as something like the
hooked atoms of Epicurus. During the complete repose of the
mind, these atoms are motionless, they are, so to speak, hooked
to the wall; so this complete rest may be indefinitely prolonged
without the atoms meeting, and consequently without any combination
between them.
On the other hand, during a period of apparent rest and
unconscious work, certain of them are detached from the wall and
put in motion. They flash in every direction through the space
(I was about to say the room) where they are enclosed, as would,
for example, a swarm of gnats or, if you prefer a more learned
comparison, like the molecules of gas in the kinematic theory of
gases. Then their mutual impacts may produce new combinations.
What is the rôle of the preliminary conscious work? It is
evidently to mobilize certain of these atoms, to unhook them from
the wall and put them in swing. We think we have done no
good, because we have moved these elements a thousand different
ways in seeking to assemble them, and have found no satisfactory
aggregate. But, after this shaking up imposed upon them by our
will, these atoms do not return to their primitive rest. They
freely continue their dance.
Now, our will did not choose them at random; it pursued a
perfectly determined aim. The mobilized atoms are therefore
not any atoms whatsoever; they are those from which we might
reasonably expect the desired solution. Then the mobilized atoms
undergo impacts which make them enter into combinations among
themselves or with other atoms at rest which they struck against
in their course. Again I beg pardon, my comparison is very
rough, but I scarcely know how otherwise to make my thought
understood.
However it may be, the only combinations that have a chance
of forming are those where at least one of the elements is one
of those atoms freely chosen by our will. Now, it is evidently
among these that is found what I called the good combination.
Perhaps this is a way of lessening the paradoxical in the original
hypothesis.
Another observation. It never happens that the unconscious
work gives us the result of a somewhat long calculation all made,
where we have only to apply fixed rules. We might think the
wholly automatic subliminal self particularly apt for this sort of
work, which is in a way exclusively mechanical. It seems that
thinking in the evening upon the factors of a multiplication we
might hope to find the product ready made upon our awakening,
or again that an algebraic calculation, for example a verification,
would be made unconsciously. Nothing of the sort, as
observation proves. All one may hope from these inspirations,
fruits of unconscious work, is a point of departure for such calculations.
As for the calculations themselves, they must be made
in the second period of conscious work, that which follows the
inspiration, that in which one verifies the results of this inspiration
and deduces their consequences. The rules of these calculations
are strict and complicated. They require discipline, attention,
will, and therefore consciousness. In the subliminal self,
on the contrary, reigns what I should call liberty, if we might
give this name to the simple absence of discipline and to the
disorder born of chance. Only, this disorder itself permits unexpected
combinations.
I shall make a last remark: when above I made certain personal
observations, I spoke of a night of excitement when I worked in
spite of myself. Such cases are frequent, and it is not necessary
that the abnormal cerebral activity be caused by a physical excitant
as in that I mentioned. It seems, in such cases, that one is
present at his own unconscious work, made partially perceptible
to the over-excited consciousness, yet without having changed its
nature. Then we vaguely comprehend what distinguishes the
two mechanisms or, if you wish, the working methods of the two
egos. And the psychologic observations I have been able thus
to make seem to me to confirm in their general outlines the views
I have given.
Surely they have need of it, for they are and remain in spite
of all very hypothetical: the interest of the questions is so great
that I do not repent of having submitted them to the reader.
"How dare we speak of the laws of chance? Is not chance
the antithesis of all law?" So says Bertrand at the beginning of
his Calcul des probabiltités. Probability is opposed to certitude;
so it is what we do not know and consequently it seems what we
could not calculate. Here is at least apparently a contradiction,
and about it much has already been written.
And first, what is chance? The ancients distinguished between
phenomena seemingly obeying harmonious laws, established once
for all, and those which they attributed to chance; these were
the ones unpredictable because rebellious to all law. In each
domain the precise laws did not decide everything, they only
drew limits between which chance might act. In this conception
the word chance had a precise and objective meaning; what was
chance for one was also chance for another and even for the gods.
But this conception is not ours to-day. We have become absolute
determinists, and even those who want to reserve the rights
of human free will let determinism reign undividedly in the inorganic
world at least. Every phenomenon, however minute, has
a cause; and a mind infinitely powerful, infinitely well-informed
about the laws of nature, could have foreseen it from the beginning
of the centuries. If such a mind existed, we could not play
with it at any game of chance; we should always lose.
In fact for it the word chance would not have any meaning,
or rather there would be no chance. It is because of our weakness
and our ignorance that the word has a meaning for us. And,
even without going beyond our feeble humanity, what is chance
for the ignorant is not chance for the scientist. Chance is only
the measure of our ignorance. Fortuitous phenomena are, by
definition, those whose laws we do not know.
But is this definition altogether satisfactory? When the first
Chaldean shepherds followed with their eyes the movements of
the stars, they knew not as yet the laws of astronomy; would they
have dreamed of saying that the stars move at random? If a
modern physicist studies a new phenomenon, and if he discovers
its law Tuesday, would he have said Monday that this phenomenon
was fortuitous? Moreover, do we not often invoke what
Bertrand calls the laws of chance, to predict a phenomenon?
For example, in the kinetic theory of gases we obtain the known
laws of Mariotte and of Gay-Lussac by means of the hypothesis
that the velocities of the molecules of gas vary irregularly, that
is to say at random. All physicists will agree that the observable
laws would be much less simple if the velocities were ruled by
any simple elementary law whatsoever, if the molecules were,
as we say, organized, if they were subject to some discipline. It
is due to chance, that is to say, to our ignorance, that we can draw
our conclusions; and then if the word chance is simply synonymous
with ignorance what does that mean? Must we therefore
translate as follows?
"You ask me to predict for you the phenomena about to
happen. If, unluckily, I knew the laws of these phenomena I
could make the prediction only by inextricable calculations and
would have to renounce attempting to answer you; but as I have
the good fortune not to know them, I will answer you at once.
And what is most surprising, my answer will be right."
So it must well be that chance is something other than the
name we give our ignorance, that among phenomena whose
causes are unknown to us we must distinguish fortuitous phenomena
about which the calculus of probabilities will provisionally
give information, from those which are not fortuitous and of
which we can say nothing so long as we shall not have determined
the laws governing them. For the fortuitous phenomena themselves,
it is clear that the information given us by the calculus
of probabilities will not cease to be true upon the day when these
phenomena shall be better known.
The director of a life insurance company does not know when
each of the insured will die, but he relies upon the calculus of
probabilities and on the law of great numbers, and he is not
deceived, since he distributes dividends to his stockholders. These
dividends would not vanish if a very penetrating and very indiscreet
physician should, after the policies were signed, reveal to
the director the life chances of the insured. This doctor would
dissipate the ignorance of the director, but he would have no
influence on the dividends, which evidently are not an outcome
of this ignorance.
To find a better definition of chance we must examine some of
the facts which we agree to regard as fortuitous, and to which
the calculus of probabilities seems to apply; we then shall investigate
what are their common characteristics.
The first example we select is that of unstable equilibrium; if
a cone rests upon its apex, we know well that it will fall, but we
do not know toward what side; it seems to us chance alone will
decide. If the cone were perfectly symmetric, if its axis were
perfectly vertical, if it were acted upon by no force other than
gravity, it would not fall at all. But the least defect in symmetry
will make it lean slightly toward one side or the other, and if it
leans, however little, it will fall altogether toward that side.
Even if the symmetry were perfect, a very slight tremor, a breath
of air could make it incline some seconds of arc; this will be
enough to determine its fall and even the sense of its fall which
will be that of the initial inclination.
A very slight cause, which escapes us, determines a considerable
effect which we can not help seeing, and then we say this
effect is due to chance. If we could know exactly the laws of
nature and the situation of the universe at the initial instant,
we should be able to predict exactly the situation of this same
universe at a subsequent instant. But even when the natural
laws should have no further secret for us, we could know the
initial situation only approximately. If that permits us to foresee
the subsequent situation with the same degree of approximation,
this is all we require, we say the phenomenon has been
predicted, that it is ruled by laws. But this is not always the
case; it may happen that slight differences in the initial conditions
produce very great differences in the final phenomena; a
slight error in the former would make an enormous error in the
latter. Prediction becomes impossible and we have the fortuitous
phenomenon.
Our second example will be very analogous to the first and we
shall take it from meteorology. Why have the meteorologists
such difficulty in predicting the weather with any certainty?
Why do the rains, the tempests themselves seem to us to come by
chance, so that many persons find it quite natural to pray for
rain or shine, when they would think it ridiculous to pray for
an eclipse? We see that great perturbations generally happen in
regions where the atmosphere is in unstable equilibrium. The
meteorologists are aware that this equilibrium is unstable, that a
cyclone is arising somewhere; but where they can not tell; one-tenth
of a degree more or less at any point, and the cyclone
bursts here and not there, and spreads its ravages over countries
it would have spared. This we could have foreseen if we had
known that tenth of a degree, but the observations were neither
sufficiently close nor sufficiently precise, and for this reason all
seems due to the agency of chance. Here again we find the same
contrast between a very slight cause, unappreciable to the observer,
and important effects, which are sometimes tremendous
disasters.
Let us pass to another example, the distribution of the minor
planets on the zodiac. Their initial longitudes may have been
any longitudes whatever; but their mean motions were different
and they have revolved for so long a time that we may say they
are now distributed at random along the zodiac. Very slight
initial differences between their distances from the sun, or, what
comes to the same thing, between their mean motions, have
ended by giving enormous differences between their present
longitudes. An excess of the thousandth of a second in the daily
mean motion will give in fact a second in three years, a degree
in ten thousand years, an entire circumference in three or four
million years, and what is that to the time which has passed since
the minor planets detached themselves from the nebula of
Laplace? Again therefore we see a slight cause and a great
effect; or better, slight differences in the cause and great differences
in the effect.
The game of roulette does not take us as far as might seem
from the preceding example. Assume a needle to be turned on a
pivot over a dial divided into a hundred sectors alternately red
and black. If it stops on a red sector I win; if not, I lose. Evidently
all depends upon the initial impulse I give the needle.
The needle will make, suppose, ten or twenty turns, but it will
stop sooner or not so soon, according as I shall have pushed it
more or less strongly. It suffices that the impulse vary only by
a thousandth or a two thousandth to make the needle stop over a
black sector or over the following red one. These are differences
the muscular sense can not distinguish and which elude even the
most delicate instruments. So it is impossible for me to foresee
what the needle I have started will do, and this is why my heart
throbs and I hope everything from luck. The difference in the
cause is imperceptible, and the difference in the effect is for me
of the highest importance, since it means my whole stake.
Permit me, in this connection, a thought somewhat foreign to
my subject. Some years ago a philosopher said that the future
is determined by the past, but not the past by the future; or, in
other words, from knowledge of the present we could deduce the
future, but not the past; because, said he, a cause can have only
one effect, while the same effect might be produced by several
different causes. It is clear no scientist can subscribe to this
conclusion. The laws of nature bind the antecedent to the consequent
in such a way that the antecedent is as well determined by
the consequent as the consequent by the antecedent. But whence
came the error of this philosopher? We know that in virtue of
Carnot's principle physical phenomena are irreversible and the
world tends toward uniformity. When two bodies of different
temperature come in contact, the warmer gives up heat to the
colder; so we may foresee that the temperature will equalize.
But once equal, if asked about the anterior state, what can we
answer? We might say that one was warm and the other cold,
but not be able to divine which formerly was the warmer.
And yet in reality the temperatures will never reach perfect
equality. The difference of the temperatures only tends asymptotically
toward zero. There comes a moment when our
thermometers are powerless to make it known. But if we had thermometers
a thousand times, a hundred thousand times as sensitive,
we should recognize that there still is a slight difference, and
that one of the bodies remains a little warmer than the other, and
so we could say this it is which formerly was much the warmer.
So then there are, contrary to what we found in the former
examples, great differences in cause and slight differences in
effect. Flammarion once imagined an observer going away from
the earth with a velocity greater than that of light; for him time
would have changed sign. History would be turned about, and
Waterloo would precede Austerlitz. Well, for this observer,
effects and causes would be inverted; unstable equilibrium would
no longer be the exception. Because of the universal irreversibility,
all would seem to him to come out of a sort of chaos in
unstable equilibrium. All nature would appear to him delivered
over to chance.
Now for other examples where we shall see somewhat different
characteristics. Take first the kinetic theory of gases. How
should we picture a receptacle filled with gas? Innumerable
molecules, moving at high speeds, flash through this receptacle
in every direction. At every instant they strike against its walls
or each other, and these collisions happen under the most diverse
conditions. What above all impresses us here is not the littleness
of the causes, but their complexity, and yet the former element
is still found here and plays an important rôle. If a molecule
deviated right or left from its trajectory, by a very small
quantity, comparable to the radius of action of the gaseous molecules,
it would avoid a collision or sustain it under different conditions,
and that would vary the direction of its velocity after
the impact, perhaps by ninety degrees or by a hundred and
eighty degrees.
And this is not all; we have just seen that it is necessary to
deflect the molecule before the clash by only an infinitesimal, to
produce its deviation after the collision by a finite quantity. If
then the molecule undergoes two successive shocks, it will suffice
to deflect it before the first by an infinitesimal of the second
order, for it to deviate after the first encounter by an infinitesimal
of the first order, and after the second hit, by a finite quantity.
And the molecule will not undergo merely two shocks; it
will undergo a very great number per second. So that if the
first shock has multiplied the deviation by a very large number
A, after n shocks it will be multiplied by An. It will therefore
become very great not merely because A is large, that is to say
because little causes produce big effects, but because the exponent
n is large, that is to say because the shocks are very numerous
and the causes very complex.
Take a second example. Why do the drops of rain in a
shower seem to be distributed at random? This is again because
of the complexity of the causes which determine their formation.
Ions are distributed in the atmosphere. For a long while they
have been subjected to air-currents constantly changing, they
have been caught in very small whirlwinds, so that their final
distribution has no longer any relation to their initial distribution.
Suddenly the temperature falls, vapor condenses, and each
of these ions becomes the center of a drop of rain. To know
what will be the distribution of these drops and how many will
fall on each paving-stone, it would not be sufficient to know the
initial situation of the ions, it would be necessary to compute
the effect of a thousand little capricious air-currents.
And again it is the same if we put grains of powder in suspension
in water. The vase is ploughed by currents whose law
we know not, we only know it is very complicated. At the
end of a certain time the grains will be distributed at random,
that is to say uniformly, in the vase; and this is due precisely to
the complexity of these currents. If they obeyed some simple
law, if for example the vase revolved and the currents circulated
around the axis of the vase, describing circles, it would no
longer be the same, since each grain would retain its initial altitude
and its initial distance from the axis.
We should reach the same result in considering the mixing of
two liquids or of two fine-grained powders. And to take a
grosser example, this is also what happens when we shuffle playing-cards.
At each stroke the cards undergo a permutation
(analogous to that studied in the theory of substitutions). What
will happen? The probability of a particular permutation (for
example, that bringing to the nth place the card occupying the
ϕ(n)th place before the permutation) depends upon the player's
habits. But if this player shuffles the cards long enough, there
will be a great number of successive permutations, and the resulting
final order will no longer be governed by aught but
chance; I mean to say that all possible orders will be equally
probable. It is to the great number of successive permutations,
that is to say to the complexity of the phenomenon, that this
result is due.
A final word about the theory of errors. Here it is that the
causes are complex and multiple. To how many snares is not
the observer exposed, even with the best instrument! He should
apply himself to finding out the largest and avoiding them.
These are the ones giving birth to systematic errors. But when
he has eliminated those, admitting that he succeeds, there remain
many small ones which, their effects accumulating, may become
dangerous. Thence come the accidental errors; and we attribute
them to chance because their causes are too complicated
and too numerous. Here again we have only little causes, but
each of them would produce only a slight effect; it is by their
union and their number that their effects become formidable.
We may take still a third point of view, less important than
the first two and upon which I shall lay less stress. When we
seek to foresee an event and examine its antecedents, we strive
to search into the anterior situation. This could not be done for
all parts of the universe and we are content to know what is
passing in the neighborhood of the point where the event should
occur, or what would appear to have some relation to it. An
examination can not be complete and we must know how to
choose. But it may happen that we have passed by circumstances
which at first sight seemed completely foreign to the
foreseen happening, to which one would never have dreamed of
attributing any influence and which nevertheless, contrary to all
anticipation, come to play an important rôle.
A man passes in the street going to his business; some one
knowing the business could have told why he started at such a
time and went by such a street. On the roof works a tiler.
The contractor employing him could in a certain measure foresee
what he would do. But the passer-by scarcely thinks of the
tiler, nor the tiler of him; they seem to belong to two worlds
completely foreign to one another. And yet the tiler drops a
tile which kills the man, and we do not hesitate to say this is
chance.
Our weakness forbids our considering the entire universe
and makes us cut it up into slices. We try to do this as little
artificially as possible. And yet it happens from time to time
that two of these slices react upon each other. The effects
of this mutual action then seem to us to be due to chance.
Is this a third way of conceiving chance? Not always; in
fact most often we are carried back to the first or the second.
Whenever two worlds usually foreign to one another come thus
to react upon each other, the laws of this reaction must be very
complex. On the other hand, a very slight change in the initial
conditions of these two worlds would have been sufficient for the
reaction not to have happened. How little was needed for the
man to pass a second later or the tiler to drop his tile a second
sooner.
All we have said still does not explain why chance obeys laws.
Does the fact that the causes are slight or complex suffice for
our foreseeing, if not their effects in each case, at least what their
effects will be, on the average? To answer this question we had
better take up again some of the examples already cited.
I shall begin with that of the roulette. I have said that the
point where the needle will stop depends upon the initial push
given it. What is the probability of this push having this or
that value? I know nothing about it, but it is difficult for me
not to suppose that this probability is represented by a continuous
analytic function. The probability that the push is comprised
between α and α + ε will then be sensibly equal to the probability
of its being comprised between α + ε and α + 2ε, provided ε be
very small. This is a property common to all analytic functions.
Minute variations of the function are proportional to minute
variations of the variable.
But we have assumed that an exceedingly slight variation of
the push suffices to change the color of the sector over which the
needle finally stops. From α to α + ε it is red, from α + ε to
α + 2ε it is black; the probability of each red sector is therefore
the same as of the following black, and consequently the total
probability of red equals the total probability of black.
The datum of the question is the analytic function representing
the probability of a particular initial push. But the theorem
remains true whatever be this datum, since it depends upon a
property common to all analytic functions. From this it follows
finally that we no longer need the datum.
What we have just said for the case of the roulette applies
also to the example of the minor planets. The zodiac may be
regarded as an immense roulette on which have been tossed many
little balls with different initial impulses varying according to
some law. Their present distribution is uniform and independent
of this law, for the same reason as in the preceding case.
Thus we see why phenomena obey the laws of chance when
slight differences in the causes suffice to bring on great differences
in the effects. The probabilities of these slight differences may
then be regarded as proportional to these differences themselves,
just because these differences are minute, and the infinitesimal
increments of a continuous function are proportional to those of
the variable.
Take an entirely different example, where intervenes especially
the complexity of the causes. Suppose a player shuffles a pack
of cards. At each shuffle he changes the order of the cards, and
he may change them in many ways. To simplify the exposition,
consider only three cards. The cards which before the shuffle
occupied respectively the places 123, may after the shuffle occupy
the places
123, 231, 312, 321, 132, 213.
Each of these six hypotheses is possible and they have respectively
for probabilities:
p1, p2, p3, p4, p5, p6.
The sum of these six numbers equals 1; but this is all we know
of them; these six probabilities depend naturally upon the habits
of the player which we do not know.
At the second shuffle and the following, this will recommence,
and under the same conditions; I mean that p4 for example represents
always the probability that the three cards which occupied
after the nth shuffle and before the n + 1th the places 123,
occupy the places 321 after the n + 1th shuffle. And this remains
true whatever be the number n, since the habits of the
player and his way of shuffling remain the same.
But if the number of shuffles is very great, the cards which
before the first shuffle occupied the places 123 may, after the
last shuffle, occupy the places
123, 231, 312, 321, 132, 213
and the probability of these six hypotheses will be sensibly the
same and equal to 1/6; and this will be true whatever be the
numbers p1 ... p6 which we do not know. The great number
of shuffles, that is to say the complexity of the causes, has
produced uniformity.
This would apply without change if there were more than
three cards, but even with three cards the demonstration would
be complicated; let it suffice to give it for only two cards. Then
we have only two possibilities 12, 21 with the probabilities p1 and
p2 = 1 − p1.
Suppose n shuffles and suppose I win one franc if the cards
are finally in the initial order and lose one if they are finally
inverted. Then, my mathematical expectation will be (p1 − p2)n.
The difference p1 − p2 is certainly less than 1; so that if n
is very great my expectation will be zero; we need not learn p1
and p2 to be aware that the game is equitable.
There would always be an exception if one of the numbers
p1 and p2 was equal to 1 and the other naught. Then it would
not apply because our initial hypotheses would be too simple.
What we have just seen applies not only to the mixing of
cards, but to all mixings, to those of powders and of liquids;
and even to those of the molecules of gases in the kinetic theory
of gases.
To return to this theory, suppose for a moment a gas whose
molecules can not mutually clash, but may be deviated by hitting
the insides of the vase wherein the gas is confined. If the form
of the vase is sufficiently complex the distribution of the molecules
and that of the velocities will not be long in becoming uniform.
But this will not be so if the vase is spherical or if it
has the shape of a cuboid. Why? Because in the first case the
distance from the center to any trajectory will remain constant;
in the second case this will be the absolute value of the angle of
each trajectory with the faces of the cuboid.
So we see what should be understood by conditions too simple;
they are those which conserve something, which leave an invariant
remaining. Are the differential equations of the problem too
simple for us to apply the laws of chance? This question would
seem at first view to lack precise meaning; now we know what it
means. They are too simple if they conserve something, if they
admit a uniform integral. If something in the initial conditions
remains unchanged, it is clear the final situation can no longer
be independent of the initial situation.
We come finally to the theory of errors. We know not to
what are due the accidental errors, and precisely because we do
not know, we are aware they obey the law of Gauss. Such is the
paradox. The explanation is nearly the same as in the preceding
cases. We need know only one thing: that the errors are very
numerous, that they are very slight, that each may be as well
negative as positive. What is the curve of probability of each
of them? We do not know; we only suppose it is symmetric.
We prove then that the resultant error will follow Gauss's law,
and this resulting law is independent of the particular laws
which we do not know. Here again the simplicity of the result
is born of the very complexity of the data.
But we are not through with paradoxes. I have just recalled
the figment of Flammarion, that of the man going quicker than
light, for whom time changes sign. I said that for him all phenomena
would seem due to chance. That is true from a certain
point of view, and yet all these phenomena at a given moment
would not be distributed in conformity with the laws of chance,
since the distribution would be the same as for us, who, seeing
them unfold harmoniously and without coming out of a primal
chaos, do not regard them as ruled by chance.
What does that mean? For Lumen, Flammarion's man, slight
causes seem to produce great effects; why do not things go on as
for us when we think we see grand effects due to little causes?
Would not the same reasoning be applicable in his case?
Let us return to the argument. When slight differences in the
causes produce vast differences in the effects, why are these effects
distributed according to the laws of chance? Suppose a difference
of a millimeter in the cause produces a difference of a kilometer
in the effect. If I win in case the effect corresponds to a
kilometer bearing an even number, my probability of winning
will be 1/2. Why? Because to make that, the cause must correspond
to a millimeter with an even number. Now, according to
all appearance, the probability of the cause varying between
certain limits will be proportional to the distance apart of these
limits, provided this distance be very small. If this hypothesis
were not admitted there would no longer be any way of representing
the probability by a continuous function.
What now will happen when great causes produce small
effects? This is the case where we should not attribute the phenomenon
to chance and where on the contrary Lumen would
attribute it to chance. To a difference of a kilometer in the
cause would correspond a difference of a millimeter in the effect.
Would the probability of the cause being comprised between two
limits n kilometers apart still be proportional to n? We have
no reason to suppose so, since this distance, n kilometers, is
great. But the probability that the effect lies between two
limits n millimeters apart will be precisely the same, so it will not
be proportional to n, even though this distance, n millimeters,
be small. There is no way therefore of representing the law of
probability of effects by a continuous curve. This curve, understand,
may remain continuous in the analytic sense of the
word; to infinitesimal variations of the abscissa will correspond
infinitesimal variations of the ordinate. But practically it will
not be continuous, since very small variations of the ordinate
would not correspond to very small variations of the abscissa. It
would become impossible to trace the curve with an ordinary
pencil; that is what I mean.
So what must we conclude? Lumen has no right to say that
the probability of the cause (his cause, our effect) should be
represented necessarily by a continuous function. But then why
have we this right? It is because this state of unstable equilibrium
which we have been calling initial is itself only the final
outcome of a long previous history. In the course of this history
complex causes have worked a great while: they have contributed
to produce the mixture of elements and they have tended to make
everything uniform at least within a small region; they have
rounded off the corners, smoothed down the hills and filled up
the valleys. However capricious and irregular may have been the
primitive curve given over to them, they have worked so much
toward making it regular that finally they deliver over to us a
continuous curve. And this is why we may in all confidence
assume its continuity.
Lumen would not have the same reasons for such a conclusion.
For him complex causes would not seem agents of equalization
and regularity, but on the contrary would create only inequality
and differentiation. He would see a world more and more varied
come forth from a sort of primitive chaos. The changes he
could observe would be for him unforeseen and impossible to
foresee. They would seem to him due to some caprice or another;
but this caprice would be quite different from our chance, since
it would be opposed to all law, while our chance still has its laws.
All these points call for lengthy explications, which perhaps
would aid in the better comprehension of the irreversibility of
the universe.
We have sought to define chance, and now it is proper to put a
question. Has chance thus defined, in so far as this is possible,
objectivity?
It may be questioned. I have spoken of very slight or very
complex causes. But what is very little for one may be very
big for another, and what seems very complex to one may seem
simple to another. In part I have already answered by saying
precisely in what cases differential equations become too simple
for the laws of chance to remain applicable. But it is fitting to
examine the matter a little more closely, because we may take
still other points of view.
What means the phrase 'very slight'? To understand it we
need only go back to what has already been said. A difference
is very slight, an interval is very small, when within the limits
of this interval the probability remains sensibly constant. And
why may this probability be regarded as constant within a
small interval? It is because we assume that the law of probability
is represented by a continuous curve, continuous not only
in the analytic sense, but practically continuous, as already explained.
This means that it not only presents no absolute hiatus,
but that it has neither salients nor reentrants too acute or too
accentuated.
And what gives us the right to make this hypothesis? We
have already said it is because, since the beginning of the ages,
there have always been complex causes ceaselessly acting in the
same way and making the world tend toward uniformity without
ever being able to turn back. These are the causes which little
by little have flattened the salients and filled up the reentrants,
and this is why our probability curves now show only gentle undulations.
In milliards of milliards of ages another step will
have been made toward uniformity, and these undulations will be
ten times as gentle; the radius of mean curvature of our curve
will have become ten times as great. And then such a length as
seems to us to-day not very small, since on our curve an arc of
this length can not be regarded as rectilineal, should on the contrary
at that epoch be called very little, since the curvature will
have become ten times less and an arc of this length may be
sensibly identified with a sect.
Thus the phrase 'very slight' remains relative; but it is not
relative to such or such a man, it is relative to the actual state of
the world. It will change its meaning when the world shall have
become more uniform, when all things shall have blended still
more. But then doubtless men can no longer live and must give
place to other beings—should I say far smaller or far larger?
So that our criterion, remaining true for all men, retains an
objective sense.
And on the other hand what means the phrase 'very complex'?
I have already given one solution, but there are others. Complex
causes we have said produce a blend more and more intimate,
but after how long a time will this blend satisfy us? When
will it have accumulated sufficient complexity? When shall we
have sufficiently shuffled the cards? If we mix two powders, one
blue, the other white, there comes a moment when the tint of the
mixture seems to us uniform because of the feebleness of our
senses; it will be uniform for the presbyte, forced to gaze from
afar, before it will be so for the myope. And when it has become
uniform for all eyes, we still could push back the limit by the use
of instruments. There is no chance for any man ever to discern
the infinite variety which, if the kinetic theory is true, hides
under the uniform appearance of a gas. And yet if we accept
Gouy's ideas on the Brownian movement, does not the microscope
seem on the point of showing us something analogous?
This new criterion is therefore relative like the first; and if it
retains an objective character, it is because all men have approximately
the same senses, the power of their instruments is
limited, and besides they use them only exceptionally.
It is just the same in the moral sciences and particularly in
history. The historian is obliged to make a choice among the
events of the epoch he studies; he recounts only those which
seem to him the most important. He therefore contents himself
with relating the most momentous events of the sixteenth century,
for example, as likewise the most remarkable facts of the
seventeenth century. If the first suffice to explain the second,
we say these conform to the laws of history. But if a great event
of the seventeenth century should have for cause a small fact of
the sixteenth century which no history reports, which all the
world has neglected, then we say this event is due to chance.
This word has therefore the same sense as in the physical sciences;
it means that slight causes have produced great effects.
The greatest bit of chance is the birth of a great man. It is
only by chance that meeting of two germinal cells, of different
sex, containing precisely, each on its side, the mysterious elements
whose mutual reaction must produce the genius. One will
agree that these elements must be rare and that their meeting is
still more rare. How slight a thing it would have required to
deflect from its route the carrying spermatozoon. It would have
sufficed to deflect it a tenth of a millimeter and Napoleon would
not have been born and the destinies of a continent would have
been changed. No example can better make us understand the
veritable characteristics of chance.
One more word about the paradoxes brought out by the application
of the calculus of probabilities to the moral sciences. It
has been proven that no Chamber of Deputies will ever fail to
contain a member of the opposition, or at least such an event
would be so improbable that we might without fear wager the
contrary, and bet a million against a sou.
Condorcet has striven to calculate how many jurors it would
require to make a judicial error practically impossible. If we
had used the results of this calculation, we should certainly have
been exposed to the same disappointments as in betting, on the
faith of the calculus, that the opposition would never be without
a representative.
The laws of chance do not apply to these questions. If justice
be not always meted out to accord with the best reasons, it uses
less than we think the method of Bridoye. This is perhaps to
be regretted, for then the system of Condorcet would shield us
from judicial errors.
What is the meaning of this? We are tempted to attribute
facts of this nature to chance because their causes are obscure;
but this is not true chance. The causes are unknown to us, it is
true, and they are even complex; but they are not sufficiently so,
since they conserve something. We have seen that this it is which
distinguishes causes 'too simple.' When men are brought together
they no longer decide at random and independently one
of another; they influence one another. Multiplex causes come
into action. They worry men, dragging them to right or left,
but one thing there is they can not destroy, this is their Panurge
flock-of-sheep habits. And this is an invariant.
Difficulties are indeed involved in the application of the
calculus of probabilities to the exact sciences. Why are the
decimals of a table of logarithms, why are those of the number
π distributed in accordance with the laws of chance? Elsewhere
I have already studied the question in so far as it concerns
logarithms, and there it is easy. It is clear that a slight difference
of argument will give a slight difference of logarithm, but a great
difference in the sixth decimal of the logarithm. Always we find
again the same criterion.
But as for the number π, that presents more difficulties, and I
have at the moment nothing worth while to say.
There would be many other questions to resolve, had I wished
to attack them before solving that which I more specially set
myself. When we reach a simple result, when we find for example
a round number, we say that such a result can not be due
to chance, and we seek, for its explanation, a non-fortuitous
cause. And in fact there is only a very slight probability that
among 10,000 numbers chance will give a round number; for
example, the number 10,000. This has only one chance in 10,000.
But there is only one chance in 10,000 for the occurrence of any
other one number; and yet this result will not astonish us, nor
will it be hard for us to attribute it to chance; and that simply
because it will be less striking.
Is this a simple illusion of ours, or are there cases where this
way of thinking is legitimate? We must hope so, else were all
science impossible. When we wish to check a hypothesis, what
do we do? We can not verify all its consequences, since they
would be infinite in number; we content ourselves with verifying
certain ones and if we succeed we declare the hypothesis confirmed,
because so much success could not be due to chance.
And this is always at bottom the same reasoning.
I can not completely justify it here, since it would take too
much time; but I may at least say that we find ourselves confronted
by two hypotheses, either a simple cause or that aggregate
of complex causes we call chance. We find it natural to
suppose that the first should produce a simple result, and then,
if we find that simple result, the round number for example, it
seems more likely to us to be attributable to the simple cause
which must give it almost certainly, than to chance which could
only give it once in 10,000 times. It will not be the same if we
find a result which is not simple; chance, it is true, will not give
this more than once in 10,000 times; but neither has the simple
cause any more chance of producing it.
