Take only a single example: The theory of the undulations of the ether

taught us that light is a motion; to-day fashion favors the

electromagnetic theory which teaches us that light is a current. We do

not consider whether we could reconcile them and say that light is a

current, and that this current is a motion. As it is probable in any

case that this motion would not be identical with that which the

partisans of the old theory presume, we might think ourselves justified

in saying that this old theory is dethroned. And yet something of it

remains, since between the hypothetical currents which Maxwell supposes

there are the same relations as between the hypothetical motions that

Fresnel supposed. There is, therefore, something which remains over and

this something is the essential. This it is which explains how we see

the present physicists pass without any embarrassment from the language

of Fresnel to that of Maxwell. Doubtless many connections that were

believed well established have been abandoned, but the greatest number

remain and it would seem must remain.

And for these, then, what is the measure of their objectivity? Well, it

is precisely the same as for our belief in external objects. These

latter are real in this, that the sensations they make us feel appear to

us as united to each other by I know not what indestructible cement and

not by the hazard of a day. In the same way science reveals to us

between phenomena other bonds finer but not less solid; these are

threads so slender that they long remained unperceived, but once noticed

there remains no way of not seeing them; they are therefore not less

real than those which give their reality to external objects; small

matter that they are more recently known, since neither can perish

before the other.

It may be said, for instance, that the ether is no less real than any

external body; to say this body exists is to say there is between the

color of this body, its taste, its smell, an intimate bond, solid and

persistent; to say the ether exists is to say there is a natural kinship

between all the optical phenomena, and neither of the two propositions

has less value than the other.

And the scientific syntheses have in a sense even more reality than

those of the ordinary senses, since they embrace more terms and tend to

absorb in them the partial syntheses.

It will be said that science is only a classification and that a

classification can not be true, but convenient. But it is true that it

is convenient, it is true that it is so not only for me, but for all

men; it is true that it will remain convenient for our descendants; it

is true finally that this can not be by chance.

In sum, the sole objective reality consists in the relations of things

whence results the universal harmony. Doubtless these relations, this

harmony, could not be conceived outside of a mind which conceives them.

But they are nevertheless objective because they are, will become, or

will remain, common to all thinking beings.

This will permit us to revert to the question of the rotation of the

earth which will give us at the same time a chance to make clear what

precedes by an example.

7. _The Rotation of the Earth_

"... Therefore," have I said in _Science and Hypothesis_, "this

affirmation, the earth turns round, has no meaning ... or rather these

two propositions, the earth turns round, and, it is more convenient to

suppose that the earth turns round, have one and the same meaning."

These words have given rise to the strangest interpretations. Some have

thought they saw in them the rehabilitation of Ptolemy's system, and

perhaps the justification of Galileo's condemnation.

Those who had read attentively the whole volume could not, however,

delude themselves. This truth, the earth turns round, was put on the

same footing as Euclid's postulate, for example. Was that to reject it?

But better; in the same language it may very well be said: These two

propositions, the external world exists, or, it is more convenient to

suppose that it exists, have one and the same meaning. So the hypothesis

of the rotation of the earth would have the same degree of certitude as

the very existence of external objects.

But after what we have just explained in the fourth part, we may go

farther. A physical theory, we have said, is by so much the more true as

it puts in evidence more true relations. In the light of this new

principle, let us examine the question which occupies us.

No, there is no absolute space; these two contradictory propositions:

'The earth turns round' and 'The earth does not turn round' are,

therefore, neither of them more true than the other. To affirm one while

denying the other, _in the kinematic sense_, would be to admit the

existence of absolute space.

But if the one reveals true relations that the other hides from us, we

can nevertheless regard it as physically more true than the other, since

it has a richer content. Now in this regard no doubt is possible.

Behold the apparent diurnal motion of the stars, and the diurnal motion

of the other heavenly bodies, and besides, the flattening of the earth,

the rotation of Foucault's pendulum, the gyration of cyclones, the

trade-winds, what not else? For the Ptolemaist all these phenomena have

no bond between them; for the Copernican they are produced by the one

same cause. In saying, the earth turns round, I affirm that all these

phenomena have an intimate relation, and _that is true_, and that

remains true, although there is not and can not be absolute space.

So much for the rotation of the earth upon itself; what shall we say of

its revolution around the sun? Here again, we have three phenomena which

for the Ptolemaist are absolutely independent and which for the

Copernican are referred back to the same origin; they are the apparent

displacements of the planets on the celestial sphere, the aberration of

the fixed stars, the parallax of these same stars. Is it by chance that

all the planets admit an inequality whose period is a year, and that

this period is precisely equal to that of aberration, precisely equal

besides to that of parallax? To adopt Ptolemy's system is to answer,

yes; to adopt that of Copernicus is to answer, no; this is to affirm

that there is a bond between the three phenomena, and that also is true,

although there is no absolute space.

In Ptolemy's system, the motions of the heavenly bodies can not be

explained by the action of central forces, celestial mechanics is

impossible. The intimate relations that celestial mechanics reveals to

us between all the celestial phenomena are true relations; to affirm the

immobility of the earth would be to deny these relations, that would be

to fool ourselves.

The truth for which Galileo suffered remains, therefore, the truth,

although it has not altogether the same meaning as for the vulgar, and

its true meaning is much more subtle, more profound and more rich.

8. _Science for Its Own Sake_

Not against M. LeRoy do I wish to defend science for its own sake; maybe

this is what he condemns, but this is what he cultivates, since he loves

and seeks truth and could not live without it. But I have some thoughts

to express.

We can not know all facts and it is necessary to choose those which are

worthy of being known. According to Tolstoi, scientists make this choice

at random, instead of making it, which would be reasonable, with a view

to practical applications. On the contrary, scientists think that

certain facts are more interesting than others, because they complete an

unfinished harmony, or because they make one foresee a great number of

other facts. If they are wrong, if this hierarchy of facts that they

implicitly postulate is only an idle illusion, there could be no science

for its own sake, and consequently there could be no science. As for me,

I believe they are right, and, for example, I have shown above what is

the high value of astronomical facts, not because they are capable of

practical applications, but because they are the most instructive of

all.

It is only through science and art that civilization is of value. Some

have wondered at the formula: science for its own sake; and yet it is as

good as life for its own sake, if life is only misery; and even as

happiness for its own sake, if we do not believe that all pleasures are

of the same quality, if we do not wish to admit that the goal of

civilization is to furnish alcohol to people who love to drink.

Every act should have an aim. We must suffer, we must work, we must pay

for our place at the game, but this is for seeing's sake; or at the very

least that others may one day see.

All that is not thought is pure nothingness; since we can think only

thoughts and all the words we use to speak of things can express only

thoughts, to say there is something other than thought, is therefore an

affirmation which can have no meaning.

And yet--strange contradiction for those who believe in time--geologic

history shows us that life is only a short episode between two

eternities of death, and that, even in this episode, conscious thought

has lasted and will last only a moment. Thought is only a gleam in the

midst of a long night.

But it is this gleam which is everything.

* * * * *

SCIENCE AND METHOD

* * * * *

INTRODUCTION

I bring together here different studies relating more or less directly

to questions of scientific methodology. The scientific method consists

in observing and experimenting; if the scientist had at his disposal

infinite time, it would only be necessary to say to him: 'Look and

notice well'; but, as there is not time to see everything, and as it is

better not to see than to see wrongly, it is necessary for him to make

choice. The first question, therefore, is how he should make this

choice. This question presents itself as well to the physicist as to the

historian; it presents itself equally to the mathematician, and the

principles which should guide each are not without analogy. The

scientist conforms to them instinctively, and one can, reflecting on

these principles, foretell the future of mathematics.

We shall understand them better yet if we observe the scientist at work,

and first of all it is necessary to know the psychologic mechanism of

invention and, in particular, that of mathematical creation. Observation

of the processes of the work of the mathematician is particularly

instructive for the psychologist.

In all the sciences of observation account must be taken of the errors

due to the imperfections of our senses and our instruments. Luckily, we

may assume that, under certain conditions, these errors are in part

self-compensating, so as to disappear in the average; this compensation

is due to chance. But what is chance? This idea is difficult to justify

or even to define; and yet what I have just said about the errors of

observation, shows that the scientist can not neglect it. It therefore

is necessary to give a definition as precise as possible of this

concept, so indispensable yet so illusive.

These are generalities applicable in sum to all the sciences; and for

example the mechanism of mathematical invention does not differ sensibly

from the mechanism of invention in general. Later I attack questions

relating more particularly to certain special sciences and first to pure

mathematics.

In the chapters devoted to these, I have to treat subjects a little more

abstract. I have first to speak of the notion of space; every one knows

space is relative, or rather every one says so, but many think still as

if they believed it absolute; it suffices to reflect a little however to

perceive to what contradictions they are exposed.

The questions of teaching have their importance, first in themselves,

then because reflecting on the best way to make new ideas penetrate

virgin minds is at the same time reflecting on how these notions were

acquired by our ancestors, and consequently on their true origin, that

is to say, in reality on their true nature. Why do children usually

understand nothing of the definitions which satisfy scientists? Why is

it necessary to give them others? This is the question I set myself in

the succeeding chapter and whose solution should, I think, suggest

useful reflections to the philosophers occupied with the logic of the

sciences.

On the other hand, many geometers believe we can reduce mathematics to

the rules of formal logic. Unheard-of efforts have been made to do this;

to accomplish it, some have not hesitated, for example, to reverse the

historic order of the genesis of our conceptions and to try to explain

the finite by the infinite. I believe I have succeeded in showing, for

all those who attack the problem unprejudiced, that here there is a

fallacious illusion. I hope the reader will understand the importance of

the question and pardon me the aridity of the pages devoted to it.

The concluding chapters relative to mechanics and astronomy will be

easier to read.

Mechanics seems on the point of undergoing a complete revolution. Ideas

which appeared best established are assailed by bold innovators.

Certainly it would be premature to decide in their favor at once simply

because they are innovators.

But it is of interest to make known their doctrines, and this is what I

have tried to do. As far as possible I have followed the historic order;

for the new ideas would seem too astonishing unless we saw how they

arose.

Astronomy offers us majestic spectacles and raises gigantic problems. We

can not dream of applying to them directly the experimental method; our

laboratories are too small. But analogy with phenomena these

laboratories permit us to attain may nevertheless guide the astronomer.

The Milky Way, for example, is an assemblage of suns whose movements

seem at first capricious. But may not this assemblage be compared to

that of the molecules of a gas, whose properties the kinetic theory of

gases has made known to us? It is thus by a roundabout way that the

method of the physicist may come to the aid of the astronomer.

Finally I have endeavored to give in a few lines the history of the

development of French geodesy; I have shown through what persevering

efforts, and often what dangers, the geodesists have procured for us the

knowledge we have of the figure of the earth. Is this then a question of

method? Yes, without doubt, this history teaches us in fact by what

precautions it is necessary to surround a serious scientific operation

and how much time and pains it costs to conquer one new decimal.

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?

CHAPTER II

THE FUTURE OF MATHEMATICS

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 [Theta] 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.

ARITHMETIC

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.

ALGEBRA

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.

GEOMETRY

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.

CANTORISM

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.

THE INVESTIGATION OF THE POSTULATES

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.

CHAPTER III

MATHEMATICAL CREATION

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.

CHAPTER IV

CHANCE

I

"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.

II

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.

III

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.

IV

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

_A_^{_n_}. 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 _n_th place the card

occupying the [phi](_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.

V

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.

VI

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 [alpha] and [alpha] +

[epsilon] will then be sensibly equal to the probability of its being

comprised between [alpha] + [epsilon] and [alpha] + 2[epsilon],

_provided_ [epsilon] _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 [alpha] to [alpha]+[epsilon] it is red, from

[alpha]+[epsilon] to [alpha]+2[epsilon] 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:

p_1, p_2, p_3, p_4, p_5, p_6.

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 p_4 for example represents always the

probability that the three cards which occupied after the _n_th 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

p_1 ... p_6 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 p_1 and p_2 = 1 - p_1.

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 (p_1 - p_2)^{_n_}.

The difference p_1 - p_2 is certainly less than 1; so that if _n_ is

very great my expectation will be zero; we need not learn p_1 and p_2

to be aware that the game is equitable.

There would always be an exception if one of the numbers p_1 and p_2

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.

VII

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.

VIII

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.