Matter will there exhibit itself to us under a thousand different

states, from those rarefied gases which seem to form the nebulæ and

which are luminous with I know not what glimmering of mysterious origin,

even to the incandescent stars and to the planets so near and yet so

different.

Perchance even, the stars will some day teach us something about life;

that seems an insensate dream and I do not at all see how it can be

realized; but, a hundred years ago, would not the chemistry of the stars

have also appeared a mad dream?

But limiting our views to horizons less distant, there still will remain

to us promises less contingent and yet sufficiently seductive. If the

past has given us much, we may rest assured that the future will give us

still more.

In sum, it is incredible how useful belief in astrology has been to

humanity. If Kepler and Tycho Brahe made a living, it was because they

sold to naïve kings predictions founded on the conjunctions of the

stars. If these princes had not been so credulous, we should perhaps

still believe that nature obeys caprice, and we should still wallow in

ignorance.

CHAPTER VII

THE HISTORY OF MATHEMATICAL PHYSICS

_The Past and the Future of Physics._--What is the present state of

mathematical physics? What are the problems it is led to set itself?

What is its future? Is its orientation about to be modified?

Ten years hence will the aim and the methods of this science appear to

our immediate successors in the same light as to ourselves; or, on the

contrary, are we about to witness a profound transformation? Such are

the questions we are forced to raise in entering to-day upon our

investigation.

If it is easy to propound them: to answer is difficult. If we felt

tempted to risk a prediction, we should easily resist this temptation,

by thinking of all the stupidities the most eminent savants of a hundred

years ago would have uttered, if some one had asked them what the

science of the nineteenth century would be. They would have thought

themselves bold in their predictions, and after the event, how very

timid we should have found them. Do not, therefore, expect of me any

prophecy.

But if, like all prudent physicians, I shun giving a prognosis, yet I

can not dispense with a little diagnostic; well, yes, there are

indications of a serious crisis, as if we might expect an approaching

transformation. Still, be not too anxious: we are sure the patient will

not die of it, and we may even hope that this crisis will be salutary,

for the history of the past seems to guarantee us this. This crisis, in

fact, is not the first, and to understand it, it is important to recall

those which have preceded. Pardon then a brief historical sketch.

_The Physics of Central Forces._--Mathematical physics, as we know, was

born of celestial mechanics, which gave birth to it at the end of the

eighteenth century, at the moment when it itself attained its complete

development. During its first years especially, the infant strikingly

resembled its mother.

The astronomic universe is formed of masses, very great, no doubt, but

separated by intervals so immense that they appear to us only as

material points. These points attract each other inversely as the square

of the distance, and this attraction is the sole force which influences

their movements. But if our senses were sufficiently keen to show us all

the details of the bodies which the physicist studies, the spectacle

thus disclosed would scarcely differ from the one the astronomer

contemplates. There also we should see material points, separated from

one another by intervals, enormous in comparison with their dimensions,

and describing orbits according to regular laws. These infinitesimal

stars are the atoms. Like the stars proper, they attract or repel each

other, and this attraction or this repulsion, following the straight

line which joins them, depends only on the distance. The law according

to which this force varies as function of the distance is perhaps not

the law of Newton, but it is an analogous law; in place of the exponent

-2, we have probably a different exponent, and it is from this change of

exponent that arises all the diversity of physical phenomena, the

variety of qualities and of sensations, all the world, colored and

sonorous, which surrounds us; in a word, all nature.

Such is the primitive conception in all its purity. It only remains to

seek in the different cases what value should be given to this exponent

in order to explain all the facts. It is on this model that Laplace, for

example, constructed his beautiful theory of capillarity; he regards it

only as a particular case of attraction, or, as he says, of universal

gravitation, and no one is astonished to find it in the middle of one of

the five volumes of the 'Mécanique céleste.' More recently Briot

believes he penetrated the final secret of optics in demonstrating that

the atoms of ether attract each other in the inverse ratio of the sixth

power of the distance; and Maxwell himself, does he not say somewhere

that the atoms of gases repel each other in the inverse ratio of the

fifth power of the distance? We have the exponent -6, or -5, in place of

the exponent -2, but it is always an exponent.

Among the theories of this epoch, one alone is an exception, that of

Fourier; in it are indeed atoms acting at a distance one upon the other;

they mutually transmit heat, but they do not attract, they never budge.

From this point of view, Fourier's theory must have appeared to the eyes

of his contemporaries, to those of Fourier himself, as imperfect and

provisional.

This conception was not without grandeur; it was seductive, and many

among us have not finally renounced it; they know that one will attain

the ultimate elements of things only by patiently disentangling the

complicated skein that our senses give us; that it is necessary to

advance step by step, neglecting no intermediary; that our fathers were

wrong in wishing to skip stations; but they believe that when one shall

have arrived at these ultimate elements, there again will be found the

majestic simplicity of celestial mechanics.

Neither has this conception been useless; it has rendered us an

inestimable service, since it has contributed to make precise the

fundamental notion of the physical law.

I will explain myself; how did the ancients understand law? It was for

them an internal harmony, static, so to say, and immutable; or else it

was like a model that nature tried to imitate. For us a law is something

quite different; it is a constant relation between the phenomenon of

to-day and that of to-morrow; in a word, it is a differential equation.

Behold the ideal form of physical law; well, it is Newton's law which

first clothed it forth. If then one has acclimated this form in physics,

it is precisely by copying as far as possible this law of Newton, that

is by imitating celestial mechanics. This is, moreover, the idea I have

tried to bring out in Chapter VI.

_The Physics of the Principles._--Nevertheless, a day arrived when the

conception of central forces no longer appeared sufficient, and this is

the first of those crises of which I just now spoke.

What was done then? The attempt to penetrate into the detail of the

structure of the universe, to isolate the pieces of this vast mechanism,

to analyze one by one the forces which put them in motion, was

abandoned, and we were content to take as guides certain general

principles the express object of which is to spare us this minute study.

How so? Suppose we have before us any machine; the initial wheel work

and the final wheel work alone are visible, but the transmission, the

intermediary machinery by which the movement is communicated from one to

the other, is hidden in the interior and escapes our view; we do not

know whether the communication is made by gearing or by belts, by

connecting-rods or by other contrivances. Do we say that it is

impossible for us to understand anything about this machine so long as

we are not permitted to take it to pieces? You know well we do not, and

that the principle of the conservation of energy suffices to determine

for us the most interesting point. We easily ascertain that the final

wheel turns ten times less quickly than the initial wheel, since these

two wheels are visible; we are able thence to conclude that a couple

applied to the one will be balanced by a couple ten times greater

applied to the other. For that there is no need to penetrate the

mechanism of this equilibrium and to know how the forces compensate each

other in the interior of the machine; it suffices to be assured that

this compensation can not fail to occur.

Well, in regard to the universe, the principle of the conservation of

energy is able to render us the same service. The universe is also a

machine, much more complicated than all those of industry, of which

almost all the parts are profoundly hidden from us; but in observing the

motion of those that we can see, we are able, by the aid of this

principle, to draw conclusions which remain true whatever may be the

details of the invisible mechanism which animates them.

The principle of the conservation of energy, or Mayer's principle, is

certainly the most important, but it is not the only one; there are

others from which we can derive the same advantage. These are:

Carnot's principle, or the principle of the degradation of energy.

Newton's principle, or the principle of the equality of action and

reaction.

The principle of relativity, according to which the laws of physical

phenomena must be the same for a stationary observer as for an observer

carried along in a uniform motion of translation; so that we have not

and can not have any means of discerning whether or not we are carried

along in such a motion.

The principle of the conservation of mass, or Lavoisier's principle.

I will add the principle of least action.

The application of these five or six general principles to the different

physical phenomena is sufficient for our learning of them all that we

could reasonably hope to know of them. The most remarkable example of

this new mathematical physics is, beyond question, Maxwell's

electromagnetic theory of light.

We know nothing as to what the ether is, how its molecules are disposed,

whether they attract or repel each other; but we know that this medium

transmits at the same time the optical perturbations and the electrical

perturbations; we know that this transmission must take place in

conformity with the general principles of mechanics, and that suffices

us for the establishment of the equations of the electromagnetic field.

These principles are results of experiments boldly generalized; but they

seem to derive from their very generality a high degree of certainty. In

fact, the more general they are, the more frequent are the opportunities

to check them, and the verifications multiplying, taking the most

varied, the most unexpected forms, end by no longer leaving place for

doubt.

_Utility of the Old Physics._--Such is the second phase of the history

of mathematical physics and we have not yet emerged from it. Shall we

say that the first has been useless? that during fifty years science

went the wrong way, and that there is nothing left but to forget so many

accumulated efforts that a vicious conception condemned in advance to

failure? Not the least in the world. Do you think the second phase could

have come into existence without the first? The hypothesis of central

forces contained all the principles; it involved them as necessary

consequences; it involved both the conservation of energy and that of

masses, and the equality of action and reaction, and the law of least

action, which appeared, it is true, not as experimental truths, but as

theorems; the enunciation of which had at the same time something more

precise and less general than under their present form.

It is the mathematical physics of our fathers which has familiarized us

little by little with these various principles; which has habituated us

to recognize them under the different vestments in which they disguise

themselves. They have been compared with the data of experience, it has

been seen how it was necessary to modify their enunciation to adapt them

to these data; thereby they have been extended and consolidated. Thus

they came to be regarded as experimental truths; the conception of

central forces became then a useless support, or rather an

embarrassment, since it made the principles partake of its hypothetical

character.

The frames then have not broken, because they are elastic; but they have

enlarged; our fathers, who established them, did not labor in vain, and

we recognize in the science of to-day the general traits of the sketch

which they traced.

CHAPTER VIII

THE PRESENT CRISIS OF MATHEMATICAL PHYSICS

_The New Crisis._--Are we now about to enter upon a third period? Are we

on the eve of a second crisis? These principles on which we have built

all, are they about to crumble away in their turn? This has been for

some time a pertinent question.

When I speak thus, you no doubt think of radium, that grand

revolutionist of the present time, and in fact I shall come back to it

presently; but there is something else. It is not alone the conservation

of energy which is in question; all the other principles are equally in

danger, as we shall see in passing them successively in review.

_Carnot's Principle._--Let us commence with the principle of Carnot.

This is the only one which does not present itself as an immediate

consequence of the hypothesis of central forces; more than that, it

seems, if not to directly contradict that hypothesis, at least not to be

reconciled with it without a certain effort. If physical phenomena were

due exclusively to the movements of atoms whose mutual attraction

depended only on the distance, it seems that all these phenomena should

be reversible; if all the initial velocities were reversed, these atoms,

always subjected to the same forces, ought to go over their trajectories

in the contrary sense, just as the earth would describe in the

retrograde sense this same elliptic orbit which it describes in the

direct sense, if the initial conditions of its motion had been reversed.

On this account, if a physical phenomenon is possible, the inverse

phenomenon should be equally so, and one should be able to reascend the

course of time. Now, it is not so in nature, and this is precisely what

the principle of Carnot teaches us; heat can pass from the warm body to

the cold body; it is impossible afterward to make it take the inverse

route and to reestablish differences of temperature which have been

effaced. Motion can be wholly dissipated and transformed into heat by

friction; the contrary transformation can never be made except

partially.

We have striven to reconcile this apparent contradiction. If the world

tends toward uniformity, this is not because its ultimate parts, at

first unlike, tend to become less and less different; it is because,

shifting at random, they end by blending. For an eye which should

distinguish all the elements, the variety would remain always as great;

each grain of this dust preserves its originality and does not model

itself on its neighbors; but as the blend becomes more and more

intimate, our gross senses perceive only the uniformity. This is why,

for example, temperatures tend to a level, without the possibility of

going backwards.

A drop of wine falls into a glass of water; whatever may be the law of

the internal motion of the liquid, we shall soon see it colored of a

uniform rosy tint, and however much from this moment one may shake it

afterwards, the wine and the water do not seem capable of again

separating. Here we have the type of the irreversible physical

phenomenon: to hide a grain of barley in a heap of wheat, this is easy;

afterwards to find it again and get it out, this is practically

impossible. All this Maxwell and Boltzmann have explained; but the one

who has seen it most clearly, in a book too little read because it is a

little difficult to read, is Gibbs, in his `Elementary Principles of

Statistical Mechanics.'

For those who take this point of view, Carnot's principle is only an

imperfect principle, a sort of concession to the infirmity of our

senses; it is because our eyes are too gross that we do not distinguish

the elements of the blend; it is because our hands are too gross that we

can not force them to separate; the imaginary demon of Maxwell, who is

able to sort the molecules one by one, could well constrain the world to

return backward. Can it return of itself? That is not impossible; that

is only infinitely improbable. The chances are that we should wait a

long time for the concourse of circumstances which would permit a

retrogradation; but sooner or later they will occur, after years whose

number it would take millions of figures to write. These reservations,

however, all remained theoretic; they were not very disquieting, and

Carnot's principle retained all its practical value. But here the scene

changes. The biologist, armed with his microscope, long ago noticed in

his preparations irregular movements of little particles in suspension;

this is the Brownian movement. He first thought this was a vital

phenomenon, but soon he saw that the inanimate bodies danced with no

less ardor than the others; then he turned the matter over to the

physicists. Unhappily, the physicists remained long uninterested in this

question; one concentrates the light to illuminate the microscopic

preparation, thought they; with light goes heat; thence inequalities of

temperature and in the liquid interior currents which produce the

movements referred to. It occurred to M. Gouy to look more closely, and

he saw, or thought he saw, that this explanation is untenable, that the

movements become brisker as the particles are smaller, but that they are

not influenced by the mode of illumination. If then these movements

never cease, or rather are reborn without cease, without borrowing

anything from an external source of energy, what ought we to believe? To

be sure, we should not on this account renounce our belief in the

conservation of energy, but we see under our eyes now motion transformed

into heat by friction, now inversely heat changed into motion, and that

without loss since the movement lasts forever. This is the contrary of

Carnot's principle. If this be so, to see the world return backward, we

no longer have need of the infinitely keen eye of Maxwell's demon; our

microscope suffices. Bodies too large, those, for example, which are a

tenth of a millimeter, are hit from all sides by moving atoms, but they

do not budge, because these shocks are very numerous and the law of

chance makes them compensate each other; but the smaller particles

receive too few shocks for this compensation to take place with

certainty and are incessantly knocked about. And behold already one of

our principles in peril.

_The Principle of Relativity._--Let us pass to the principle of

relativity; this not only is confirmed by daily experience, not only is

it a necessary consequence of the hypothesis of central forces, but it

is irresistibly imposed upon our good sense, and yet it also is

assailed. Consider two electrified bodies; though they seem to us at

rest, they are both carried along by the motion of the earth; an

electric charge in motion, Rowland has taught us, is equivalent to a

current; these two charged bodies are, therefore, equivalent to two

parallel currents of the same sense and these two currents should

attract each other. In measuring this attraction, we shall measure the

velocity of the earth; not its velocity in relation to the sun or the

fixed stars, but its absolute velocity.

I well know what will be said: It is not its absolute velocity that is

measured, it is its velocity in relation to the ether. How

unsatisfactory that is! Is it not evident that from the principle so

understood we could no longer infer anything? It could no longer tell us

anything just because it would no longer fear any contradiction. If we

succeed in measuring anything, we shall always be free to say that this

is not the absolute velocity, and if it is not the velocity in relation

to the ether, it might always be the velocity in relation to some new

unknown fluid with which we might fill space.

Indeed, experiment has taken upon itself to ruin this interpretation of

the principle of relativity; all attempts to measure the velocity of the

earth in relation to the ether have led to negative results. This time

experimental physics has been more faithful to the principle than

mathematical physics; the theorists, to put in accord their other

general views, would not have spared it; but experiment has been

stubborn in confirming it. The means have been varied; finally Michelson

pushed precision to its last limits; nothing came of it. It is precisely

to explain this obstinacy that the mathematicians are forced to-day to

employ all their ingenuity.

Their task was not easy, and if Lorentz has got through it, it is only

by accumulating hypotheses.

The most ingenious idea was that of local time. Imagine two observers

who wish to adjust their timepieces by optical signals; they exchange

signals, but as they know that the transmission of light is not

instantaneous, they are careful to cross them. When station B perceives

the signal from station A, its clock should not mark the same hour as

that of station A at the moment of sending the signal, but this hour

augmented by a constant representing the duration of the transmission.

Suppose, for example, that station A sends its signal when its clock

marks the hour _O_, and that station B perceives it when its clock marks

the hour _t_. The clocks are adjusted if the slowness equal to _t_

represents the duration of the transmission, and to verify it, station B

sends in its turn a signal when its clock marks _O_; then station A

should perceive it when its clock marks _t_. The timepieces are then

adjusted.

And in fact they mark the same hour at the same physical instant, but on

the one condition, that the two stations are fixed. Otherwise the

duration of the transmission will not be the same in the two senses,

since the station A, for example, moves forward to meet the optical

perturbation emanating from B, whereas the station B flees before the

perturbation emanating from A. The watches adjusted in that way will not

mark, therefore, the true time; they will mark what may be called the

_local time_, so that one of them will be slow of the other. It matters

little, since we have no means of perceiving it. All the phenomena which

happen at A, for example, will be late, but all will be equally so, and

the observer will not perceive it, since his watch is slow; so, as the

principle of relativity requires, he will have no means of knowing

whether he is at rest or in absolute motion.

Unhappily, that does not suffice, and complementary hypotheses are

necessary; it is necessary to admit that bodies in motion undergo a

uniform contraction in the sense of the motion. One of the diameters of

the earth, for example, is shrunk by one two-hundred-millionth in

consequence of our planet's motion, while the other diameter retains its

normal length. Thus the last little differences are compensated. And

then, there is still the hypothesis about forces. Forces, whatever be

their origin, gravity as well as elasticity, would be reduced in a

certain proportion in a world animated by a uniform translation; or,

rather, this would happen for the components perpendicular to the

translation; the components parallel would not change. Resume, then, our

example of two electrified bodies; these bodies repel each other, but at

the same time if all is carried along in a uniform translation, they are

equivalent to two parallel currents of the same sense which attract each

other. This electrodynamic attraction diminishes, therefore, the

electrostatic repulsion, and the total repulsion is feebler than if the

two bodies were at rest. But since to measure this repulsion we must

balance it by another force, and all these other forces are reduced in

the same proportion, we perceive nothing. Thus all seems arranged, but

are all the doubts dissipated? What would happen if one could

communicate by non-luminous signals whose velocity of propagation

differed from that of light? If, after having adjusted the watches by

the optical procedure, we wished to verify the adjustment by the aid of

these new signals, we should observe discrepancies which would render

evident the common translation of the two stations. And are such signals

inconceivable, if we admit with Laplace that universal gravitation is

transmitted a million times more rapidly than light?

Thus, the principle of relativity has been valiantly defended in these

latter times, but the very energy of the defense proves how serious was

the attack.

_Newton's Principle._--Let us speak now of the principle of Newton, on

the equality of action and reaction. This is intimately bound up with

the preceding, and it seems indeed that the fall of the one would

involve that of the other. Thus we must not be astonished to find here

the same difficulties.

Electrical phenomena, according to the theory of Lorentz, are due to the

displacements of little charged particles, called electrons, immersed in

the medium we call ether. The movements of these electrons produce

perturbations in the neighboring ether; these perturbations propagate

themselves in every direction with the velocity of light, and in turn

other electrons, originally at rest, are made to vibrate when the

perturbation reaches the parts of the ether which touch them. The

electrons, therefore, act on one another, but this action is not direct,

it is accomplished through the ether as intermediary. Under these

conditions can there be compensation between action and reaction, at

least for an observer who should take account only of the movements of

matter, that is, of the electrons, and who should be ignorant of those

of the ether that he could not see? Evidently not. Even if the

compensation should be exact, it could not be simultaneous. The

perturbation is propagated with a finite velocity; it, therefore,

reaches the second electron only when the first has long ago entered

upon its rest. This second electron, therefore, will undergo, after a

delay, the action of the first, but will certainly not at that moment

react upon it, since around this first electron nothing any longer

budges.

The analysis of the facts permits us to be still more precise. Imagine,

for example, a Hertzian oscillator, like those used in wireless

telegraphy; it sends out energy in every direction; but we can provide

it with a parabolic mirror, as Hertz did with his smallest oscillators,

so as to send all the energy produced in a single direction. What

happens then according to the theory? The apparatus recoils, as if it

were a cannon and the projected energy a ball; and that is contrary to

the principle of Newton, since our projectile here has no mass, it is

not matter, it is energy. The case is still the same, moreover, with a

beacon light provided with a reflector, since light is nothing but a

perturbation of the electromagnetic field. This beacon light should

recoil as if the light it sends out were a projectile. What is the force

that should produce this recoil? It is what is called the

Maxwell-Bartholi pressure. It is very minute, and it has been difficult

to put it in evidence even with the most sensitive radiometers; but it

suffices that it exists.

If all the energy issuing from our oscillator falls on a receiver, this

will act as if it had received a mechanical shock, which will represent

in a sense the compensation of the oscillator's recoil; the reaction

will be equal to the action, but it will not be simultaneous; the

receiver will move on, but not at the moment when the oscillator

recoils. If the energy propagates itself indefinitely without

encountering a receiver, the compensation will never occur.

Shall we say that the space which separates the oscillator from the

receiver and which the perturbation must pass over in going from the one

to the other is not void, that it is full not only of ether, but of air,

or even in the interplanetary spaces of some fluid subtile but still

ponderable; that this matter undergoes the shock like the receiver at

the moment when the energy reaches it, and recoils in its turn when the

perturbation quits it? That would save Newton's principle, but that is

not true. If energy in its diffusion remained always attached to some

material substratum, then matter in motion would carry along light with

it, and Fizeau has demonstrated that it does nothing of the sort, at

least for air. Michelson and Morley have since confirmed this. It might

be supposed also that the movements of matter proper are exactly

compensated by those of the ether; but that would lead us to the same

reflections as before now. The principle so understood will explain

everything, since, whatever might be the visible movements, we always

could imagine hypothetical movements which compensate them. But if it is

able to explain everything, this is because it does not enable us to

foresee anything; it does not enable us to decide between the different

possible hypotheses, since it explains everything beforehand. It

therefore becomes useless.

And then the suppositions that it would be necessary to make on the

movements of the ether are not very satisfactory. If the electric

charges double, it would be natural to imagine that the velocities of

the diverse atoms of ether double also; but, for the compensation, it

would be necessary that the mean velocity of the ether quadruple.

This is why I have long thought that these consequences of theory,

contrary to Newton's principle, would end some day by being abandoned,

and yet the recent experiments on the movements of the electrons issuing

from radium seem rather to confirm them.

_Lavoisier's Principle._--I arrive at the principle of Lavoisier on the

conservation of mass. Certainly, this is one not to be touched without

unsettling all mechanics. And now certain persons think that it seems

true to us only because in mechanics merely moderate velocities are

considered, but that it would cease to be true for bodies animated by

velocities comparable to that of light. Now these velocities are

believed at present to have been realized; the cathode rays and those of

radium may be formed of very minute particles or of electrons which are

displaced with velocities smaller no doubt than that of light, but which

might be its one tenth or one third.

These rays can be deflected, whether by an electric field, or by a

magnetic field, and we are able, by comparing these deflections, to

measure at the same time the velocity of the electrons and their mass

(or rather the relation of their mass to their charge). But when it was

seen that these velocities approached that of light, it was decided that

a correction was necessary. These molecules, being electrified, can not

be displaced without agitating the ether; to put them in motion it is

necessary to overcome a double inertia, that of the molecule itself and

that of the ether. The total or apparent mass that one measures is

composed, therefore, of two parts: the real or mechanical mass of the

molecule and the electrodynamic mass representing the inertia of the

ether.

The calculations of Abraham and the experiments of Kaufmann have then

shown that the mechanical mass, properly so called, is null, and that

the mass of the electrons, or, at least, of the negative electrons, is

of exclusively electrodynamic origin. This is what forces us to change

the definition of mass; we can not any longer distinguish mechanical

mass and electrodynamic mass, since then the first would vanish; there

is no mass other than electrodynamic inertia. But in this case the mass

can no longer be constant; it augments with the velocity, and it even

depends on the direction, and a body animated by a notable velocity will

not oppose the same inertia to the forces which tend to deflect it from

its route, as to those which tend to accelerate or to retard its

progress.

There is still a resource; the ultimate elements of bodies are

electrons, some charged negatively, the others charged positively. The

negative electrons have no mass, this is understood; but the positive

electrons, from the little we know of them, seem much greater. Perhaps

they have, besides their electrodynamic mass, a true mechanical mass.

The real mass of a body would, then, be the sum of the mechanical masses

of its positive electrons, the negative electrons not counting; mass so

defined might still be constant.

Alas! this resource also evades us. Recall what we have said of the

principle of relativity and of the efforts made to save it. And it is

not merely a principle which it is a question of saving, it is the

indubitable results of the experiments of Michelson.

Well, as was above seen, Lorentz, to account for these results, was

obliged to suppose that all forces, whatever their origin, were reduced

in the same proportion in a medium animated by a uniform translation;

this is not sufficient; it is not enough that this take place for the

real forces, it must also be the same for the forces of inertia; it is

therefore necessary, he says, that _the masses of all the particles be

influenced by a translation to the same degree as the electromagnetic

masses of the electrons_.

So the mechanical masses must vary in accordance with the same laws as

the electrodynamic masses; they can not, therefore, be constant.

Need I point out that the fall of Lavoisier's principle involves that of

Newton's? This latter signifies that the center of gravity of an

isolated system moves in a straight line; but if there is no longer a

constant mass, there is no longer a center of gravity, we no longer know

even what this is. This is why I said above that the experiments on the

cathode rays appeared to justify the doubts of Lorentz concerning

Newton's principle.

From all these results, if they were confirmed, would arise an entirely

new mechanics, which would be, above all, characterized by this fact,

that no velocity could surpass that of light,[9] any more than any

temperature can fall below absolute zero.

[9] Because bodies would oppose an increasing inertia to the causes

which would tend to accelerate their motion; and this inertia

would become infinite when one approached the velocity of light.

No more for an observer, carried along himself in a translation he does

not suspect, could any apparent velocity surpass that of light; and this

would be then a contradiction, if we did not recall that this observer

would not use the same clocks as a fixed observer, but, indeed, clocks

marking 'local time.'

Here we are then facing a question I content myself with stating. If

there is no longer any mass, what becomes of Newton's law? Mass has two

aspects: it is at the same time a coefficient of inertia and an

attracting mass entering as factor into Newtonian attraction. If the

coefficient of inertia is not constant, can the attracting mass be? That

is the question.

_Mayer's Principle._--At least, the principle of the conservation of

energy yet remained to us, and this seemed more solid. Shall I recall to

you how it was in its turn thrown into discredit? This event has made

more noise than the preceding, and it is in all the memoirs. From the

first words of Becquerel, and, above all, when the Curies had discovered

radium, it was seen that every radioactive body was an inexhaustible

source of radiation. Its activity seemed to subsist without alteration

throughout the months and the years. This was in itself a strain on the

principles; these radiations were in fact energy, and from the same

morsel of radium this issued and forever issued. But these quantities of

energy were too slight to be measured; at least that was the belief and

we were not much disquieted.

The scene changed when Curie bethought himself to put radium in a

calorimeter; it was then seen that the quantity of heat incessantly

created was very notable.

The explanations proposed were numerous; but in such case we can not

say, the more the better. In so far as no one of them has prevailed over

the others, we can not be sure there is a good one among them. Since

some time, however, one of these explanations seems to be getting the

upper hand and we may reasonably hope that we hold the key to the

mystery.

Sir W. Ramsay has striven to show that radium is in process of

transformation, that it contains a store of energy enormous but not

inexhaustible. The transformation of radium then would produce a million

times more heat than all known transformations; radium would wear itself

out in 1,250 years; this is quite short, and you see that we are at

least certain to have this point settled some hundreds of years from

now. While waiting, our doubts remain.

CHAPTER IX

THE FUTURE OF MATHEMATICAL PHYSICS

_The Principles and Experiment._--In the midst of so much ruin, what

remains standing? The principle of least action is hitherto intact, and

Larmor appears to believe that it will long survive the others; in

reality, it is still more vague and more general.

In presence of this general collapse of the principles, what attitude

will mathematical physics take? And first, before too much excitement,

it is proper to ask if all that is really true. All these derogations to

the principles are encountered only among infinitesimals; the microscope

is necessary to see the Brownian movement; electrons are very light;

radium is very rare, and one never has more than some milligrams of it

at a time. And, then, it may be asked whether, besides the infinitesimal

seen, there was not another infinitesimal unseen counterpoise to the

first.

So there is an interlocutory question, and, as it seems, only experiment

can solve it. We shall, therefore, only have to hand over the matter to

the experimenters, and, while waiting for them to finally decide the

debate, not to preoccupy ourselves with these disquieting problems, and

to tranquilly continue our work as if the principles were still

uncontested. Certes, we have much to do without leaving the domain where

they may be applied in all security; we have enough to employ our

activity during this period of doubts.

_The Rôle of the Analyst._--And as to these doubts, is it indeed true

that we can do nothing to disembarrass science of them? It must indeed

be said, it is not alone experimental physics that has given birth to

them; mathematical physics has well contributed. It is the experimenters

who have seen radium throw out energy, but it is the theorists who have

put in evidence all the difficulties raised by the propagation of light

across a medium in motion; but for these it is probable we should not

have become conscious of them. Well, then, if they have done their best

to put us into this embarrassment, it is proper also that they help us

to get out of it.

They must subject to critical examination all these new views I have

just outlined before you, and abandon the principles only after having

made a loyal effort to save them. What can they do in this sense? That

is what I will try to explain.

It is a question before all of endeavoring to obtain a more satisfactory

theory of the electrodynamics of bodies in motion. It is there

especially, as I have sufficiently shown above, that difficulties

accumulate. It is useless to heap up hypotheses, we can not satisfy all

the principles at once; so far, one has succeeded in safeguarding some

only on condition of sacrificing the others; but all hope of obtaining

better results is not yet lost. Let us take, then, the theory of

Lorentz, turn it in all senses, modify it little by little, and perhaps

everything will arrange itself.

Thus in place of supposing that bodies in motion undergo a contraction

in the sense of the motion, and that this contraction is the same

whatever be the nature of these bodies and the forces to which they are

otherwise subjected, could we not make a more simple and natural

hypothesis? We might imagine, for example, that it is the ether which is

modified when it is in relative motion in reference to the material

medium which penetrates it, that, when it is thus modified, it no longer

transmits perturbations with the same velocity in every direction. It

might transmit more rapidly those which are propagated parallel to the

motion of the medium, whether in the same sense or in the opposite

sense, and less rapidly those which are propagated perpendicularly. The

wave surfaces would no longer be spheres, but ellipsoids, and we could

dispense with that extraordinary contraction of all bodies.

I cite this only as an example, since the modifications that might be

essayed would be evidently susceptible of infinite variation.

_Aberration and Astronomy._--It is possible also that astronomy may some

day furnish us data on this point; she it was in the main who raised the

question in making us acquainted with the phenomenon of the aberration

of light. If we make crudely the theory of aberration, we reach a very

curious result. The apparent positions of the stars differ from their

real positions because of the earth's motion, and as this motion is

variable, these apparent positions vary. The real position we can not

ascertain, but we can observe the variations of the apparent position.

The observations of the aberration show us, therefore, not the earth's

motion, but the variations of this motion; they can not, therefore, give

us information about the absolute motion of the earth.

At least this is true in first approximation, but the case would be no

longer the same if we could appreciate the thousandths of a second. Then

it would be seen that the amplitude of the oscillation depends not alone

on the variation of the motion, a variation which is well known, since

it is the motion of our globe on its elliptic orbit, but on the mean

value of this motion, so that the constant of aberration would not be

quite the same for all the stars, and the differences would tell us the

absolute motion of the earth in space.

This, then, would be, under another form, the ruin of the principle of

relativity. We are far, it is true, from appreciating the thousandth of

a second, but, after all, say some, the earth's total absolute velocity

is perhaps much greater than its relative velocity with respect to the

sun. If, for example, it were 300 kilometers per second in place of 30,

this would suffice to make the phenomenon observable.

I believe that in reasoning thus one admits a too simple theory of

aberration. Michelson has shown us, I have told you, that the physical

procedures are powerless to put in evidence absolute motion; I am

persuaded that the same will be true of the astronomic procedures,

however far precision be carried.

However that may be, the data astronomy will furnish us in this regard

will some day be precious to the physicist. Meanwhile, I believe that

the theorists, recalling the experience of Michelson, may anticipate a

negative result, and that they would accomplish a useful work in

constructing a theory of aberration which would explain this in advance.

_Electrons and Spectra._--This dynamics of electrons can be approached

from many sides, but among the ways leading thither is one which has

been somewhat neglected, and yet this is one of those which promise us

the most surprises. It is movements of electrons which produce the lines

of the emission spectra; this is proved by the Zeeman effect; in an

incandescent body what vibrates is sensitive to the magnet, therefore

electrified. This is a very important first point, but no one has gone

farther. Why are the lines of the spectrum distributed in accordance

with a regular law? These laws have been studied by the experimenters in

their least details; they are very precise and comparatively simple. A

first study of these distributions recalls the harmonics encountered in

acoustics; but the difference is great. Not only are the numbers of

vibrations not the successive multiples of a single number, but we do

not even find anything analogous to the roots of those transcendental

equations to which we are led by so many problems of mathematical

physics: that of the vibrations of an elastic body of any form, that of

the Hertzian oscillations in a generator of any form, the problem of

Fourier for the cooling of a solid body.

The laws are simpler, but they are of wholly other nature, and to cite

only one of these differences, for the harmonics of high order, the

number of vibrations tends toward a finite limit, instead of increasing

indefinitely.

That has not yet been accounted for, and I believe that there we have

one of the most important secrets of nature. A Japanese physicist, M.

Nagaoka, has recently proposed an explanation; according to him, atoms

are composed of a large positive electron surrounded by a ring formed of

a great number of very small negative electrons. Such is the planet

Saturn with its rings. This is a very interesting attempt, but not yet

wholly satisfactory; this attempt should be renewed. We will penetrate,

so to speak, into the inmost recess of matter. And from the particular

point of view which we to-day occupy, when we know why the vibrations of

incandescent bodies differ thus from ordinary elastic vibrations, why

the electrons do not behave like the matter which is familiar to us, we

shall better comprehend the dynamics of electrons and it will be perhaps

more easy for us to reconcile it with the principles.

_Conventions Preceding Experiment._--Suppose, now, that all these

efforts fail, and, after all, I do not believe they will, what must be

done? Will it be necessary to seek to mend the broken principles by

giving what we French call a _coup de pouce_? That evidently is always

possible, and I retract nothing of what I have said above.

Have you not written, you might say if you wished to seek a quarrel with

me--have you not written that the principles, though of experimental

origin, are now unassailable by experiment because they have become

conventions? And now you have just told us that the most recent

conquests of experiment put these principles in danger.

Well, formerly I was right and to-day I am not wrong. Formerly I was

right, and what is now happening is a new proof of it. Take, for

example, the calorimetric experiment of Curie on radium. Is it possible

to reconcile it with the principle of the conservation of energy? This

has been attempted in many ways. But there is among them one I should

like you to notice; this is not the explanation which tends to-day to

prevail, but it is one of those which have been proposed. It has been

conjectured that radium was only an intermediary, that it only stored

radiations of unknown nature which flashed through space in every

direction, traversing all bodies, save radium, without being altered by

this passage and without exercising any action upon them. Radium alone

took from them a little of their energy and afterward gave it out to us

in various forms.

What an advantageous explanation, and how convenient! First, it is

unverifiable and thus irrefutable. Then again it will serve to account

for any derogation whatever to Mayer's principle; it answers in advance

not only the objection of Curie, but all the objections that future

experimenters might accumulate. This new and unknown energy would serve

for everything.

This is just what I said, and therewith we are shown that our principle

is unassailable by experiment.

But then, what have we gained by this stroke? The principle is intact,

but thenceforth of what use is it? It enabled us to foresee that in such

or such circumstance we could count on such a total quantity of energy;

it limited us; but now that this indefinite provision of new energy is

placed at our disposal, we are no longer limited by anything; and, as I

have written in 'Science and Hypothesis,' if a principle ceases to be

fecund, experiment without contradicting it directly will nevertheless

have condemned it.

_Future Mathematical Physics._--This, therefore, is not what would have

to be done; it would be necessary to rebuild anew. If we were reduced to

this necessity; we could moreover console ourselves. It would not be

necessary thence to conclude that science can weave only a Penelope's

web, that it can raise only ephemeral structures, which it is soon

forced to demolish from top to bottom with its own hands.

As I have said, we have already passed through a like crisis. I have

shown you that in the second mathematical physics, that of the

principles, we find traces of the first, that of central forces; it will

be just the same if we must know a third. Just so with the animal that

exuviates, that breaks its too narrow carapace and makes itself a fresh

one; under the new envelope one will recognize the essential traits of

the organism which have persisted.

We can not foresee in what way we are about to expand; perhaps it is the

kinetic theory of gases which is about to undergo development and serve

as model to the others. Then the facts which first appeared to us as

simple thereafter would be merely resultants of a very great number of

elementary facts which only the laws of chance would make cooperate for

a common end. Physical law would then assume an entirely new aspect; it

would no longer be solely a differential equation, it would take the

character of a statistical law.

Perhaps, too, we shall have to construct an entirely new mechanics that

we only succeed in catching a glimpse of, where, inertia increasing with

the velocity, the velocity of light would become an impassable limit.

The ordinary mechanics, more simple, would remain a first approximation,

since it would be true for velocities not too great, so that the old

dynamics would still be found under the new. We should not have to

regret having believed in the principles, and even, since velocities too

great for the old formulas would always be only exceptional, the surest

way in practise would be still to act as if we continued to believe in

them. They are so useful, it would be necessary to keep a place for

them. To determine to exclude them altogether would be to deprive

oneself of a precious weapon. I hasten to say in conclusion that we are

not yet there, and as yet nothing proves that the principles will not

come forth from out the fray victorious and intact.[10]

[10] These considerations on mathematical physics are borrowed from

my St. Louis address.

PART III

THE OBJECTIVE VALUE OF SCIENCE

CHAPTER X

IS SCIENCE ARTIFICIAL?

1. _The Philosophy of M. LeRoy_

There are many reasons for being sceptics; should we push this

scepticism to the very end or stop on the way? To go to the end is the

most tempting solution, the easiest and that which many have adopted,

despairing of saving anything from the shipwreck.

Among the writings inspired by this tendency it is proper to place in

the first rank those of M. LeRoy. This thinker is not only a philosopher

and a writer of the greatest merit, but he has acquired a deep knowledge

of the exact and physical sciences, and even has shown rare powers of

mathematical invention. Let us recapitulate in a few words his doctrine,

which has given rise to numerous discussions.

Science consists only of conventions, and to this circumstance solely

does it owe its apparent certitude; the facts of science and, _a

fortiori_, its laws are the artificial work of the scientist; science

therefore can teach us nothing of the truth; it can only serve us as

rule of action.

Here we recognize the philosophic theory known under the name of

nominalism; all is not false in this theory; its legitimate domain must

be left it, but out of this it should not be allowed to go.

This is not all; M. LeRoy's doctrine is not only nominalistic; it has

besides another characteristic which it doubtless owes to M. Bergson, it

is anti-intellectualistic. According to M. LeRoy, the intellect deforms

all it touches, and that is still more true of its necessary instrument

'discourse.' There is reality only in our fugitive and changing

impressions, and even this reality, when touched, vanishes.

And yet M. LeRoy is not a sceptic; if he regards the intellect as

incurably powerless, it is only to give more scope to other sources of

knowledge, to the heart, for instance, to sentiment, to instinct or to

faith.

However great my esteem for M. LeRoy's talent, whatever the ingenuity of

this thesis, I can not wholly accept it. Certes, I am in accord on many

points with M. LeRoy, and he has even cited, in support of his view,

various passages of my writings which I am by no means disposed to

reject. I think myself only the more bound to explain why I can not go

with him all the way.

M. LeRoy often complains of being accused of scepticism. He could not

help being, though this accusation is probably unjust. Are not

appearances against him? Nominalist in doctrine, but realist at heart,

he seems to escape absolute nominalism only by a desperate act of faith.

The fact is that anti-intellectualistic philosophy in rejecting analysis

and 'discourse,' just by that condemns itself to being intransmissible;

it is a philosophy essentially internal, or, at the very least, only its

negations can be transmitted; what wonder then that for an external

observer it takes the shape of scepticism?

Therein lies the weak point of this philosophy; if it strives to remain

faithful to itself, its energy is spent in a negation and a cry of

enthusiasm. Each author may repeat this negation and this cry, may vary

their form, but without adding anything.

And, yet, would it not be more logical in remaining silent? See, you

have written long articles; for that, it was necessary to use words. And

therein have you not been much more 'discursive' and consequently much

farther from life and truth than the animal who simply lives without

philosophizing? Would not this animal be the true philosopher?

However, because no painter has made a perfect portrait, should we

conclude that the best painting is not to paint? When a zoologist

dissects an animal, certainly he 'alters it.' Yes, in dissecting it, he

condemns himself to never know all of it; but in not dissecting it, he

would condemn himself to never know anything of it and consequently to

never see anything of it.

Certes, in man are other forces besides his intellect; no one has ever

been mad enough to deny that. The first comer makes these blind forces

act or lets them act; the philosopher must _speak_ of them; to speak of

them, he must know of them the little that can be known, he should

therefore _see_ them act. How? With what eyes, if not with his

intellect? Heart, instinct, may guide it, but not render it useless;

they may direct the look, but not replace the eye. It may be granted

that the heart is the workman, and the intellect only the instrument.

Yet is it an instrument not to be done without, if not for action, at

least for philosophizing? Therefore a philosopher really

anti-intellectualistic is impossible. Perhaps we shall have to declare

for the supremacy of action; always it is our intellect which will thus

conclude; in allowing precedence to action it will thus retain the

superiority of the thinking reed. This also is a supremacy not to be

disdained.

Pardon these brief reflections and pardon also their brevity, scarcely

skimming the question. The process of intellectualism is not the subject

I wish to treat: I wish to speak of science, and about it there is no

doubt; by definition, so to speak, it will be intellectualistic or it

will not be at all. Precisely the question is, whether it will be.

2. _Science, Rule of Action_

For M. LeRoy, science is only a rule of action. We are powerless to know

anything and yet we are launched, we must act, and at all hazards we

have established rules. It is the aggregate of these rules that is

called science.

It is thus that men, desirous of diversion, have instituted rules of

play, like those of tric-trac for instance, which, better than science

itself, could rely upon the proof by universal consent. It is thus

likewise that, unable to choose, but forced to choose, we toss up a

coin, head or tail to win.

The rule of tric-trac is indeed a rule of action like science, but does

any one think the comparison just and not see the difference? The rules

of the game are arbitrary conventions and the contrary convention might

have been adopted, _which would have been none the less good_. On the

contrary, science is a rule of action which is successful, generally at

least, and I add, while the contrary rule would not have succeeded.

If I say, to make hydrogen cause an acid to act on zinc, I formulate a

rule which succeeds; I could have said, make distilled water act on

gold; that also would have been a rule, only it would not have

succeeded. If, therefore, scientific 'recipes' have a value, as rule of

action, it is because we know they succeed, generally at least. But to

know this is to know something and then why tell us we can know nothing?

Science foresees, and it is because it foresees that it can be useful

and serve as rule of action. I well know that its previsions are often

contradicted by the event; that shows that science is imperfect, and if

I add that it will always remain so, I am certain that this is a

prevision which, at least, will never be contradicted. Always the

scientist is less often mistaken than a prophet who should predict at

random. Besides the progress though slow is continuous, so that

scientists, though more and more bold, are less and less misled. This is

little, but it is enough.

I well know that M. LeRoy has somewhere said that science was mistaken

oftener than one thought, that comets sometimes played tricks on

astronomers, that scientists, who apparently are men, did not willingly

speak of their failures, and that, if they should speak of them, they

would have to count more defeats than victories.

That day, M. LeRoy evidently overreached himself. If science did not

succeed, it could not serve as rule of action; whence would it get its

value? Because it is 'lived,' that is, because we love it and believe in

it? The alchemists had recipes for making gold, they loved them and had

faith in them, and yet our recipes are the good ones, although our faith

be less lively, because they succeed.

There is no escape from this dilemma; either science does not enable us

to foresee, and then it is valueless as rule of action; or else it

enables us to foresee, in a fashion more or less imperfect, and then it

is not without value as means of knowledge.

It should not even be said that action is the goal of science; should we

condemn studies of the star Sirius, under pretext that we shall probably

never exercise any influence on that star? To my eyes, on the contrary,

it is the knowledge which is the end, and the action which is the means.

If I felicitate myself on the industrial development, it is not alone

because it furnishes a facile argument to the advocates of science; it

is above all because it gives to the scientist faith in himself and also

because it offers him an immense field of experience where he clashes

against forces too colossal to be tampered with. Without this ballast,

who knows whether he would not quit solid ground, seduced by the mirage

of some scholastic novelty, or whether he would not despair, believing

he had fashioned only a dream?

3. _The Crude Fact and the Scientific Fact_

What was most paradoxical in M. LeRoy's thesis was that affirmation that

_the scientist creates the fact_; this was at the same time its

essential point and it is one of those which have been most discussed.

Perhaps, says he (I well believe that this was a concession), it is not

the scientist that creates the fact in the rough; it is at least he who

creates the scientific fact.

This distinction between the fact in the rough and the scientific fact

does not by itself appear to me illegitimate. But I complain first that

the boundary has not been traced either exactly or precisely; and then

that the author has seemed to suppose that the crude fact, not being

scientific, is outside of science.

Finally, I can not admit that the scientist creates without restraint

the scientific fact, since it is the crude fact which imposes it upon

him.

The examples given by M. LeRoy have greatly astonished me. The first is

taken from the notion of atom. The atom chosen as example of fact! I

avow that this choice has so disconcerted me that I prefer to say

nothing about it. I have evidently misunderstood the author's thought

and I could not fruitfully discuss it.

The second case taken as example is that of an eclipse where the crude

phenomenon is a play of light and shadow, but where the astronomer can

not intervene without introducing two foreign elements, to wit, a clock

and Newton's law.

Finally, M. LeRoy cites the rotation of the earth; it has been answered:

but this is not a fact, and he has replied: it was one for Galileo, who

affirmed it, as for the inquisitor, who denied it. It always remains

that this is not a fact in the same sense as those just spoken of and

that to give them the same name is to expose one's self to many

confusions.

Here then are four degrees:

1º. It grows dark, says the clown.

2º. The eclipse happened at nine o'clock, says the astronomer.

3º. The eclipse happened at the time deducible from the tables

constructed according to Newton's law, says he again.

4º. That results from the earth's turning around the sun, says Galileo

finally.

Where then is the boundary between the fact in the rough and the

scientific fact? To read M. LeRoy one would believe that it is between

the first and the second stage, but who does not see that there is a

greater distance from the second to the third, and still more from the

third to the fourth.

Allow me to cite two examples which perhaps will enlighten us a little.

I observe the deviation of a galvanometer by the aid of a movable mirror

which projects a luminous image or spot on a divided scale. The crude

fact is this: I see the spot displace itself on the scale, and the

scientific fact is this: a current passes in the circuit.

Or again: when I make an experiment I should subject the result to

certain corrections, because I know I must have made errors. These

errors are of two kinds, some are accidental and these I shall correct

by taking the mean; the others are systematic and I shall be able to

correct those only by a thorough study of their causes. The first result

obtained is then the fact in the rough, while the scientific fact is the

final result after the finished corrections.

Reflecting on this latter example, we are led to subdivide our second

stage, and in place of saying:

2. The eclipse happened at nine o'clock, we shall say:

2_a_. The eclipse happened when my clock pointed to nine, and

2_b_. My clock being ten minutes slow, the eclipse happened at ten

minutes past nine.

And this is not all: the first stage also should be subdivided, and not

between these two subdivisions will be the least distance; it is

necessary to distinguish between the impression of obscurity felt by one

witnessing an eclipse, and the affirmation: It grows dark, which this

impression extorts from him. In a sense it is the first which is the

only true fact in the rough, and the second is already a sort of

scientific fact.

Now then our scale has six stages, and even though there is no reason

for halting at this figure, there we shall stop.

What strikes me at the start is this. At the first of our six stages,

the fact, still completely in the rough, is, so to speak, individual, it

is completely distinct from all other possible facts. From the second

stage, already it is no longer the same. The enunciation of the fact

would suit an infinity of other facts. So soon as language intervenes, I

have at my command only a finite number of terms to express the shades,

in number infinite, that my impressions might cover. When I say: It

grows dark, that well expresses the impressions I feel in being present

at an eclipse; but even in obscurity a multitude of shades could be

imagined, and if, instead of that actually realized, had happened a

slightly different shade, yet I should still have enunciated this

_other_ fact by saying: It grows dark.

Second remark: even at the second stage, the enunciation of a fact can

only be _true or false_. This is not so of any proposition; if this

proposition is the enunciation of a convention, it can not be said that

this enunciation is _true_, in the proper sense of the word, since it

could not be true apart from me and is true only because I wish it to

be.

When, for instance, I say the unit for length is the meter, this is a

decree that I promulgate, it is not something ascertained which forces

itself upon me. It is the same, as I think I have elsewhere shown, when

it is a question, for example, of Euclid's postulate.

When I am asked: Is it growing dark? I always know whether I ought to

reply yes or no. Although an infinity of possible facts may be

susceptible of this same enunciation, it grows dark, I shall always

know whether the fact realized belongs or does not belong among those

which answer to this enunciation. Facts are classed in categories, and

if I am asked whether the fact that I ascertain belongs or does not

belong in such a category, I shall not hesitate.

Doubtless this classification is sufficiently arbitrary to leave a large

part to man's freedom or caprice. In a word, this classification is a

convention. _This convention being given_, if I am asked: Is such a fact

true? I shall always know what to answer, and my reply will be imposed

upon me by the witness of my senses.

If therefore, during an eclipse, it is asked: Is it growing dark? all

the world will answer yes. Doubtless those speaking a language where

bright was called dark, and dark bright, would answer no. But of what

importance is that?

In the same way, in mathematics, _when I have laid down the definitions,

and the postulates which are conventions_, a theorem henceforth can only

be true or false. But to answer the question: Is this theorem true? it

is no longer to the witness of my senses that I shall have recourse, but

to reasoning.

A statement of fact is always verifiable, and for the verification we

have recourse either to the witness of our senses, or to the memory of

this witness. This is properly what characterizes a fact. If you put the

question to me: Is such a fact true? I shall begin by asking you, if

there is occasion, to state precisely the conventions, by asking you, in

other words, what language you have spoken; then once settled on this

point, I shall interrogate my senses and shall answer yes or no. But it

will be my senses that will have made answer, it will not be _you_ when

you say to me: I have spoken to you in English or in French.

Is there something to change in all that when we pass to the following

stages? When I observe a galvanometer, as I have just said, if I ask an

ignorant visitor: Is the current passing? he looks at the wire to try to

see something pass; but if I put the same question to my assistant who

understands my language, he will know I mean: Does the spot move? and he

will look at the scale.

What difference is there then between the statement of a fact in the

rough and the statement of a scientific fact? The same difference as

between the statement of the same crude fact in French and in German.

The scientific statement is the translation of the crude statement into

a language which is distinguished above all from the common German or

French, because it is spoken by a very much smaller number of people.

Yet let us not go too fast. To measure a current I may use a very great

number of types of galvanometers or besides an electrodynamometer. And

then when I shall say there is running in this circuit a current of so

many amperes, that will mean: if I adapt to this circuit such a

galvanometer I shall see the spot come to the division _a_;

but that will mean equally: if I adapt to this circuit such an

electrodynamometer, I shall see the spot go to the division _b_. And

that will mean still many other things, because the current can manifest

itself not only by mechanical effects, but by effects chemical, thermal,

luminous, etc.

Here then is one same statement which suits a very great number of facts

absolutely different. Why? It is because I assume a law according to

which, whenever such a mechanical effect shall happen, such a chemical

effect will happen also. Previous experiments, very numerous, have never

shown this law to fail, and then I have understood that I could express

by the same statement two facts so invariably bound one to the other.

When I am asked: Is the current passing? I can understand that that

means: Will such a mechanical effect happen? But I can understand also:

Will such a chemical effect happen? I shall then verify either the

existence of the mechanical effect, or that of the chemical effect; that

will be indifferent, since in both cases the answer must be the same.

And if the law should one day be found false? If it was perceived that

the concordance of the two effects, mechanical and chemical, is not

constant? That day it would be necessary to change the scientific

language to free it from a grave ambiguity.

And after that? Is it thought that ordinary language by aid of which are

expressed the facts of daily life is exempt from ambiguity?

_Shall we thence conclude that the facts of daily life are the work of

the grammarians?_

You ask me: Is there a current? I try whether the mechanical effect

exists, I ascertain it and I answer: Yes, there is a current. You

understand at once that that means that the mechanical effect exists,

and that the chemical effect, that I have not investigated, exists

likewise. Imagine now, supposing an impossibility, the law we believe

true, not to be, and the chemical effect not to exist. Under this

hypothesis there will be two distinct facts, the one directly observed

and which is true, the other inferred and which is false. It may

strictly be said that we have created the second. So that error is the

part of man's personal collaboration in the creation of the scientific

fact.

But if we can say that the fact in question is false, is this not just

because it is not a free and arbitrary creation of our mind, a disguised

convention, in which case it would be neither true nor false. And in

fact it was verifiable; I had not made the verification, but I could

have made it. If I answered amiss, it was because I chose to reply too

quickly, without having asked nature, who alone knew the secret.

When, after an experiment, I correct the accidental and systematic

errors to bring out the scientific fact, the case is the same; the

scientific fact will never be anything but the crude fact translated

into another language. When I shall say: It is such an hour, that will

be a short way of saying: There is such a relation between the hour

indicated by my clock, and the hour it marked at the moment of the

passing of such a star and such another star across the meridian. And

this convention of language once adopted, when I shall be asked: Is it

such an hour? it will not depend upon me to answer yes or no.

Let us pass to the stage before the last: the eclipse happened at the

hour given by the tables deduced from Newton's laws. This is still a

convention of language which is perfectly clear for those who know

celestial mechanics or simply for those who have the tables calculated

by the astronomers. I am asked: Did the eclipse happen at the hour

predicted? I look in the nautical almanac, I see that the eclipse was

announced for nine o'clock and I understand that the question means: Did

the eclipse happen at nine o'clock? There still we have nothing to

change in our conclusions. _The scientific fact is only the crude fact

translated into a convenient language._

It is true that at the last stage things change. Does the earth rotate?

Is this a verifiable fact? Could Galileo and the Grand Inquisitor, to

settle the matter, appeal to the witness of their senses? On the

contrary, they were in accord about the appearances, and whatever had

been the accumulated experiences, they would have remained in accord

with regard to the appearances without ever agreeing on their

interpretation. It is just on that account that they were obliged to

have recourse to procedures of discussion so unscientific.

This is why I think they did not disagree about a _fact_: we have not

the right to give the same name to the rotation of the earth, which was

the object of their discussion, and to the facts crude or scientific we

have hitherto passed in review.

After what precedes, it seems superfluous to investigate whether the

fact in the rough is outside of science, because there can neither be

science without scientific fact, nor scientific fact without fact in the

rough, since the first is only the translation of the second.

And then, has one the right to say that the scientist creates the

scientific fact? First of all, he does not create it from nothing, since

he makes it with the fact in the rough. Consequently he does not make it

freely and _as he chooses_. However able the worker may be, his freedom

is always limited by the properties of the raw material on which he

works.

After all, what do you mean when you speak of this free creation of the

scientific fact and when you take as example the astronomer who

intervenes actively in the phenomenon of the eclipse by bringing his

clock? Do you mean: The eclipse happened at nine o'clock; but if the

astronomer had wished it to happen at ten, that depended only on him, he

had only to advance his clock an hour?

But the astronomer, in perpetrating that bad joke, would evidently have

been guilty of an equivocation. When he tells me: The eclipse happened

at nine, I understand that nine is the hour deduced from the crude

indication of the pendulum by the usual series of corrections. If he has

given me solely that crude indication, or if he has made corrections

contrary to the habitual rules, he has changed the language agreed upon

without forewarning me. If, on the contrary, he took care to forewarn

me, I have nothing to complain of, but then it is always the same fact

expressed in another language.

In sum, _all the scientist creates in a fact is the language in which he

enunciates it_. If he predicts a fact, he will employ this language, and

for all those who can speak and understand it, his prediction is free

from ambiguity. Moreover, this prediction once made, it evidently does

not depend upon him whether it is fulfilled or not.

What then remains of M. LeRoy's thesis? This remains: the scientist

intervenes actively in choosing the facts worth observing. An isolated

fact has by itself no interest; it becomes interesting if one has reason

to think that it may aid in the prediction of other facts; or better,

if, having been predicted, its verification is the confirmation of a

law. Who shall choose the facts which, corresponding to these

conditions, are worthy the freedom of the city in science? This is the

free activity of the scientist.

And that is not all. I have said that the scientific fact is the

translation of a crude fact into a certain language; I should add that

every scientific fact is formed of many crude facts. This is

sufficiently shown by the examples cited above. For instance, for the

hour of the eclipse my clock marked the hour [alpha] at the instant of

the eclipse; it marked the hour [beta] at the moment of the last transit

of the meridian of a certain star that we take as origin of right

ascensions; it marked the hour [gamma] at the moment of the preceding

transit of this same star. There are three distinct facts (still it will

be noticed that each of them results itself from two simultaneous facts

in the rough; but let us pass this over). In place of that I say: The

eclipse happened at the hour 24 ([alpha]-[beta])/([beta]-[gamma]), and

the three facts are combined in a single scientific fact. I have

concluded that the three readings, [alpha], [beta], [gamma] made on my

clock at three different moments lacked interest and that the only thing

interesting was the combination ([alpha]-[beta])/([beta]-[gamma]) of

the three. In this conclusion is found the free activity of my mind.

But I have thus used up my power; I can not make this combination

([alpha]-[beta])/([beta]-[gamma]) have such a value and not such

another, since I can not influence either the value of [alpha], or that

of [beta], or that of [gamma], which are imposed upon me as crude

facts.

In sum, facts are facts, and _if it happens that they satisfy a

prediction, this is not an effect of our free activity_. There is no

precise frontier between the fact in the rough and the scientific fact;

it can only be said that such an enunciation of fact is _more crude_ or,

on the contrary, _more scientific_ than such another.

4. _'Nominalism' and 'the Universal Invariant'_

If from facts we pass to laws, it is clear that the part of the free

activity of the scientist will become much greater. But did not M. LeRoy

make it still too great? This is what we are about to examine.

Recall first the examples he has given. When I say: Phosphorus melts at

44°, I think I am enunciating a law; in reality it is just the

definition of phosphorus; if one should discover a body which,

possessing otherwise all the properties of phosphorus, did not melt at

44°, we should give it another name, that is all, and the law would

remain true.

Just so when I say: Heavy bodies falling freely pass over spaces

proportional to the squares of the times, I only give the definition of

free fall. Whenever the condition shall not be fulfilled, I shall say

that the fall is not free, so that the law will never be wrong. It is

clear that if laws were reduced to that, they could not serve in

prediction; then they would be good for nothing, either as means of

knowledge or as principle of action.

When I say: Phosphorus melts at 44°, I mean by that: All bodies

possessing such or such a property (to wit, all the properties of

phosphorus, save fusing-point) fuse at 44°. So understood, my

proposition is indeed a law, and this law may be useful to me, because

if I meet a body possessing these properties I shall be able to predict

that it will fuse at 44°.

Doubtless the law may be found to be false. Then we shall read in the

treatises on chemistry: "There are two bodies which chemists long

confounded under the name of phosphorus; these two bodies differ only by

their points of fusion." That would evidently not be the first time for

chemists to attain to the separation of two bodies they were at first

not able to distinguish; such, for example, are neodymium and

praseodymium, long confounded under the name of didymium.

I do not think the chemists much fear that a like mischance will ever

happen to phosphorus. And if, to suppose the impossible, it should

happen, the two bodies would probably not have _identically_ the same

density, _identically_ the same specific heat, etc., so that after

having determined with care the density, for instance, one could still

foresee the fusion point.

It is, moreover, unimportant; it suffices to remark that there is a law,

and that this law, true or false, does not reduce to a tautology.

Will it be said that if we do not know on the earth a body which does

not fuse at 44° while having all the other properties of phosphorus, we

can not know whether it does not exist on other planets? Doubtless that

may be maintained, and it would then be inferred that the law in

question, which may serve as a rule of action to us who inhabit the

earth, has yet no general value from the point of view of knowledge, and

owes its interest only to the chance which has placed us on this globe.

This is possible, but, if it were so, the law would be valueless, not

because it reduced to a convention, but because it would be false.

The same is true in what concerns the fall of bodies. It would do me no

good to have given the name of free fall to falls which happen in

conformity with Galileo's law, if I did not know that elsewhere, in such

circumstances, the fall will be _probably_ free or _approximately_ free.

That then is a law which may be true or false, but which does not reduce

to a convention.

Suppose the astronomers discover that the stars do not exactly obey

Newton's law. They will have the choice between two attitudes; they may

say that gravitation does not vary exactly as the inverse of the square

of the distance, or else they may say that gravitation is not the only

force which acts on the stars and that there is in addition a different

sort of force.

In the second case, Newton's law will be considered as the definition of

gravitation. This will be the nominalist attitude. The choice between

the two attitudes is free, and is made from considerations of

convenience, though these considerations are most often so strong that

there remains practically little of this freedom.

We can break up this proposition: (1) The stars obey Newton's law, into

two others; (2) gravitation obeys Newton's law; (3) gravitation is the

only force acting on the stars. In this case proposition (2) is no

longer anything but a definition and is beyond the test of experiment;

but then it will be on proposition (3) that this check can be exercised.

This is indeed necessary, since the resulting proposition (1) predicts

verifiable facts in the rough.

It is thanks to these artifices that by an unconscious nominalism the

scientists have elevated above the laws what they call principles. When

a law has received a sufficient confirmation from experiment, we may

adopt two attitudes: either we may leave this law in the fray; it will

then remain subjected to an incessant revision, which without any doubt

will end by demonstrating that it is only approximative. Or else we may

elevate it into a _principle_ by adopting conventions such that the

proposition may be certainly true. For that the procedure is always the

same. The primitive law enunciated a relation between two facts in the

rough, _A_ and _B_; between these two crude facts is introduced an

abstract intermediary _C_, more or less fictitious (such was in the

preceding example the impalpable entity, gravitation). And then we have

a relation between _A_ and _C_ that we may suppose rigorous and which is

the _principle_; and another between _C_ and _B_ which remains a _law_

subject to revision.

The principle, henceforth crystallized, so to speak, is no longer

subject to the test of experiment. It is not true or false, it is

convenient.

Great advantages have often been found in proceeding in that way, but it

is clear that if _all_ the laws had been transformed into principles

_nothing_ would be left of science. Every law may be broken up into a

principle and a law, but thereby it is very clear that, however far this

partition be pushed, there will always remain laws.

Nominalism has therefore limits, and this is what one might fail to

recognize if one took to the very letter M. LeRoy's assertions.

A rapid review of the sciences will make us comprehend better what are

these limits. The nominalist attitude is justified only when it is

convenient; when is it so?

Experiment teaches us relations between bodies; this is the fact in the

rough; these relations are extremely complicated. Instead of envisaging

directly the relation of the body _A_ and the body _B_, we introduce

between them an intermediary, which is space, and we envisage three

distinct relations: that of the body _A_ with the figure _A'_ of space,

that of the body _B_ with the figure _B'_ of space, that of the two

figures _A'_ and _B'_ to each other. Why is this detour advantageous?

Because the relation of _A_ and _B_ was complicated, but differed little

from that of _A'_ and _B'_, which is simple; so that this complicated

relation may be replaced by the simple relation between _A'_ and _B'_

and by two other relations which tell us that the differences between

_A_ and _A'_, on the one hand, between _B_ and _B'_, on the other hand,

are _very small_. For example, if _A_ and _B_ are two natural solid

bodies which are displaced with slight deformation, we envisage two

movable _rigid_ figures _A'_ and _B'_. The laws of the relative

displacement of these figures _A'_ and _B'_ will be very simple; they

will be those of geometry. And we shall afterward add that the body _A_,

which always differs very little from _A'_, dilates from the effect of

heat and bends from the effect of elasticity. These dilatations and

flexions, just because they are very small, will be for our mind

relatively easy to study. Just imagine to what complexities of language

it would have been necessary to be resigned if we had wished to

comprehend in the same enunciation the displacement of the solid, its

dilatation and its flexure?

The relation between _A_ and _B_ was a rough law, and was broken up; we

now have two laws which express the relations of _A_ and _A'_, of _B_

and _B'_, and a principle which expresses that of _A'_ with _B'_. It is

the aggregate of these principles that is called geometry.

Two other remarks. We have a relation between two bodies _A_ and _B_,

which we have replaced by a relation between two figures _A'_ and _B'_;

but this same relation between the same two figures _A'_ and _B'_ could

just as well have replaced advantageously a relation between two other

bodies _A''_ and _B''_, entirely different from _A_ and _B_. And that in

many ways. If the principles of geometry had not been invented, after

having studied the relation of _A_ and _B_, it would be necessary to

begin again _ab ovo_ the study of the relation of _A''_ and _B''_.

That is why geometry is so precious. A geometrical relation can

advantageously replace a relation which, considered in the rough state,

should be regarded as mechanical, it can replace another which should be

regarded as optical, etc.

Yet let no one say: But that proves geometry an experimental science; in

separating its principles from laws whence they have been drawn, you

artificially separate it itself from the sciences which have given birth

to it. The other sciences have likewise principles, but that does not

preclude our having to call them experimental.

It must be recognized that it would have been difficult not to make this

separation that is pretended to be artificial. We know the rôle that the

kinematics of solid bodies has played in the genesis of geometry; should

it then be said that geometry is only a branch of experimental

kinematics? But the laws of the rectilinear propagation of light have

also contributed to the formation of its principles. Must geometry be

regarded both as a branch of kinematics and as a branch of optics? I

recall besides that our Euclidean space which is the proper object of

geometry has been chosen, for reasons of convenience, from among a

certain number of types which preexist in our mind and which are called

groups.

If we pass to mechanics, we still see great principles whose origin is

analogous, and, as their 'radius of action,' so to speak, is smaller,

there is no longer reason to separate them from mechanics proper and to

regard this science as deductive.

In physics, finally, the rôle of the principles is still more

diminished. And in fact they are only introduced when it is of

advantage. Now they are advantageous precisely because they are few,

since each of them very nearly replaces a great number of laws.

Therefore it is not of interest to multiply them. Besides an outcome is

necessary, and for that it is needful to end by leaving abstraction to

take hold of reality.

Such are the limits of nominalism, and they are narrow.

M. LeRoy has insisted, however, and he has put the question under

another form.

Since the enunciation of our laws may vary with the conventions that we

adopt, since these conventions may modify even the natural relations of

these laws, is there in the manifold of these laws something independent

of these conventions and which may, so to speak, play the rôle of

_universal invariant_? For instance, the fiction has been introduced of

beings who, having been educated in a world different from ours, would

have been led to create a non-Euclidean geometry. If these beings were

afterward suddenly transported into our world, they would observe the

same laws as we, but they would enunciate them in an entirely different

way. In truth there would still be something in common between the two

enunciations, but this is because these beings do not yet differ enough

from us. Beings still more strange may be imagined, and the part common

to the two systems of enunciations will shrink more and more. Will it

thus shrink in convergence toward zero, or will there remain an

irreducible residue which will then be the universal invariant sought?

The question calls for precise statement. Is it desired that this common

part of the enunciations be expressible in words? It is clear, then,

that there are not words common to all languages, and we can not pretend

to construct I know not what universal invariant which should be

understood both by us and by the fictitious non-Euclidean geometers of

whom I have just spoken; no more than we can construct a phrase which

can be understood both by Germans who do not understand French and by

French who do not understand German. But we have fixed rules which

permit us to translate the French enunciations into German, and

inversely. It is for that that grammars and dictionaries have been made.

There are also fixed rules for translating the Euclidean language into

the non-Euclidean language, or, if there are not, they could be made.

And even if there were neither interpreter nor dictionary, if the

Germans and the French, after having lived centuries in separate worlds,

found themselves all at once in contact, do you think there would be

nothing in common between the science of the German books and that of

the French books? The French and the Germans would certainly end by

understanding each other, as the American Indians ended by understanding

the language of their conquerors after the arrival of the Spanish.

But, it will be said, doubtless the French would be capable of

understanding the Germans even without having learned German, but this

is because there remains between the French and the Germans something in

common, since both are men. We should still attain to an understanding

with our hypothetical non-Euclideans, though they be not men, because

they would still retain something human. But in any case a minimum of

humanity is necessary.

This is possible, but I shall observe first that this little humanness

which would remain in the non-Euclideans would suffice not only to make

possible the translation of _a little_ of their language, but to make

possible the translation of _all_ their language.

Now, that there must be a minimum is what I concede; suppose there

exists I know not what fluid which penetrates between the molecules of

our matter, without having any action on it and without being subject to

any action coming from it. Suppose beings sensible to the influence of

this fluid and insensible to that of our matter. It is clear that the

science of these beings would differ absolutely from ours and that it

would be idle to seek an 'invariant' common to these two sciences. Or

again, if these beings rejected our logic and did not admit, for

instance, the principle of contradiction.

But truly I think it without interest to examine such hypotheses.

And then, if we do not push whimsicality so far, if we introduce only

fictitious beings having senses analogous to ours and sensible to the

same impressions, and moreover admitting the principles of our logic, we

shall then be able to conclude that their language, however different

from ours it may be, would always be capable of translation. Now the

possibility of translation implies the existence of an invariant. To

translate is precisely to disengage this invariant. Thus, to decipher a

cryptogram is to seek what in this document remains invariant, when the

letters are permuted.

What now is the nature of this invariant it is easy to understand, and a

word will suffice us. The invariant laws are the relations between the

crude facts, while the relations between the 'scientific facts' remain

always dependent on certain conventions.

CHAPTER XI

SCIENCE AND REALITY

5. _Contingence and Determinism_

I do not intend to treat here the question of the contingence of the

laws of nature, which is evidently insoluble, and on which so much has

already been written. I only wish to call attention to what different

meanings have been given to this word, contingence, and how advantageous

it would be to distinguish them.

If we look at any particular law, we may be certain in advance that it

can only be approximate. It is, in fact, deduced from experimental

verifications, and these verifications were and could be only

approximate. We should always expect that more precise measurements will

oblige us to add new terms to our formulas; this is what has happened,

for instance, in the case of Mariotte's law.

Moreover the statement of any law is necessarily incomplete. This

enunciation should comprise the enumeration of _all_ the antecedents in

virtue of which a given consequent can happen. I should first describe

_all_ the conditions of the experiment to be made and the law would then

be stated: If all the conditions are fulfilled, the phenomenon will

happen.

But we shall be sure of not having forgotten _any_ of these conditions

only when we shall have described the state of the entire universe at

the instant _t_; all the parts of this universe may, in fact, exercise

an influence more or less great on the phenomenon which must happen at

the instant _t_ + _dt_.

Now it is clear that such a description could not be found in the

enunciation of the law; besides, if it were made, the law would become

incapable of application; if one required so many conditions, there

would be very little chance of their ever being all realized at any

moment.

Then as one can never be certain of not having forgotten some essential

condition, it can not be said: If such and such conditions are

realized, such a phenomenon will occur; it can only be said: If such and

such conditions are realized, it is probable that such a phenomenon will

occur, very nearly.

Take the law of gravitation, which is the least imperfect of all known

laws. It enables us to foresee the motions of the planets. When I use

it, for instance, to calculate the orbit of Saturn, I neglect the action

of the stars, and in doing so I am certain of not deceiving myself,

because I know that these stars are too far away for their action to be

sensible.

I announce, then, with a quasi-certitude that the coordinates of Saturn

at such an hour will be comprised between such and such limits. Yet is

that certitude absolute? Could there not exist in the universe some

gigantic mass, much greater than that of all the known stars and whose

action could make itself felt at great distances? That mass might be

animated by a colossal velocity, and after having circulated from all

time at such distances that its influence had remained hitherto

insensible to us, it might come all at once to pass near us. Surely it

would produce in our solar system enormous perturbations that we could

not have foreseen. All that can be said is that such an event is wholly

improbable, and then, instead of saying: Saturn will be near such a

point of the heavens, we must limit ourselves to saying: Saturn will

probably be near such a point of the heavens. Although this probability

may be practically equivalent to certainty, it is only a probability.

For all these reasons, no particular law will ever be more than

approximate and probable. Scientists have never failed to recognize this

truth; only they believe, right or wrong, that every law may be replaced

by another closer and more probable, that this new law will itself be

only provisional, but that the same movement can continue indefinitely,

so that science in progressing will possess laws more and more probable,

that the approximation will end by differing as little as you choose

from exactitude and the probability from certitude.

If the scientists who think thus are right, still could it be said that

_the_ laws of nature are contingent, even though _each_ law, taken in

particular, may be qualified as contingent? Or must one require, before

concluding the contingence _of the_ natural laws, that this progress

have an end, that the scientist finish some day by being arrested in his

search for a closer and closer approximation, and that, beyond a certain

limit, he thereafter meet in nature only caprice?

In the conception of which I have just spoken (and which I shall call

the scientific conception), every law is only a statement imperfect and

provisional, but it must one day be replaced by another, a superior law,

of which it is only a crude image. No place therefore remains for the

intervention of a free will.

It seems to me that the kinetic theory of gases will furnish us a

striking example.

You know that in this theory all the properties of gases are explained

by a simple hypothesis; it is supposed that all the gaseous molecules

move in every direction with great velocities and that they follow

rectilineal paths which are disturbed only when one molecule passes very

near the sides of the vessel or another molecule. The effects our crude

senses enable us to observe are the mean effects, and in these means,

the great deviations compensate, or at least it is very improbable that

they do not compensate; so that the observable phenomena follow simple

laws such as that of Mariotte or of Gay-Lussac. But this compensation of

deviations is only probable. The molecules incessantly change place and

in these continual displacements the figures they form pass successively

through all possible combinations. Singly these combinations are very

numerous; almost all are in conformity with Mariotte's law, only a few

deviate from it. These also will happen, only it would be necessary to

wait a long time for them. If a gas were observed during a sufficiently

long time it would certainly be finally seen to deviate, for a very

short time, from Mariotte's law. How long would it be necessary to wait?

If it were desired to calculate the probable number of years, it would

be found that this number is so great that to write only the number of

places of figures employed would still require half a score places of

figures. No matter; enough that it may be done.

I do not care to discuss here the value of this theory. It is evident

that if it be adopted, Mariotte's law will thereafter appear only as

contingent, since a day will come when it will not be true. And yet,

think you the partisans of the kinetic theory are adversaries of

determinism? Far from it; they are the most ultra of mechanists. Their

molecules follow rigid paths, from which they depart only under the

influence of forces which vary with the distance, following a perfectly

determinate law. There remains in their system not the smallest place

either for freedom, or for an evolutionary factor, properly so-called,

or for anything whatever that could be called contingence. I add, to

avoid mistake, that neither is there any evolution of Mariotte's law

itself; it ceases to be true after I know not how many centuries; but at

the end of a fraction of a second it again becomes true and that for an

incalculable number of centuries.

And since I have pronounced the word evolution, let us clear away

another mistake. It is often said: Who knows whether the laws do not

evolve and whether we shall not one day discover that they were not at

the Carboniferous epoch what they are to-day? What are we to understand

by that? What we think we know about the past state of our globe, we

deduce from its present state. And how is this deduction made? It is by

means of laws supposed known. The law, being a relation between the

antecedent and the consequent, enables us equally well to deduce the

consequent from the antecedent, that is, to foresee the future, and to

deduce the antecedent from the consequent, that is, to conclude from the

present to the past. The astronomer who knows the present situation of

the stars can from it deduce their future situation by Newton's law, and

this is what he does when he constructs ephemerides; and he can equally

deduce from it their past situation. The calculations he thus can make

can not teach him that Newton's law will cease to be true in the future,

since this law is precisely his point of departure; not more can they

tell him it was not true in the past. Still, in what concerns the

future, his ephemerides can one day be tested and our descendants will

perhaps recognize that they were false. But in what concerns the past,

the geologic past which had no witnesses, the results of his

calculation, like those of all speculations where we seek to deduce the

past from the present, escape by their very nature every species of

test. So that if the laws of nature were not the same in the

Carboniferous age as at the present epoch, we shall never be able to

know it, since we can know nothing of this age, only what we deduce from

the hypothesis of the permanence of these laws.

Perhaps it will be said that this hypothesis might lead to contradictory

results and that we shall be obliged to abandon it. Thus, in what

concerns the origin of life, we may conclude that there have always been

living beings, since the present world shows us always life springing

from life; and we may also conclude that there have not always been,

since the application of the existent laws of physics to the present

state of our globe teaches us that there was a time when this globe was

so warm that life on it was impossible. But contradictions of this sort

can always be removed in two ways; it may be supposed that the actual

laws of nature are not exactly what we have assumed; or else it may be

supposed that the laws of nature actually are what we have assumed, but

that it has not always been so.

It is evident that the actual laws will never be sufficiently well known

for us not to be able to adopt the first of these two solutions and for

us to be constrained to infer the evolution of natural laws.

On the other hand, suppose such an evolution; assume, if you wish, that

humanity lasts sufficiently long for this evolution to have witnesses.

The _same_ antecedent shall produce, for instance, different consequents

at the Carboniferous epoch and at the Quaternary. That evidently means

that the antecedents are closely alike; if all the circumstances were

identical, the Carboniferous epoch would be indistinguishable from the

Quaternary. Evidently this is not what is supposed. What remains is that

such antecedent, accompanied by such accessory circumstance, produces

such consequent; and that the same antecedent, accompanied by such other

accessory circumstance, produces such other consequent. Time does not

enter into the affair.

The law, such as ill-informed science would have stated it, and which

would have affirmed that this antecedent always produces this

consequent, without taking account of the accessory circumstances, this

law, which was only approximate and probable, must be replaced by

another law more approximate and more probable, which brings in these

accessory circumstances. We always come back, therefore, to that same

process which we have analyzed above, and if humanity should discover

something of this sort, it would not say that it is the laws which have

evoluted, but the circumstances which have changed.

Here, therefore, are several different senses of the word contingence.

M. LeRoy retains them all and he does not sufficiently distinguish them,

but he introduces a new one. Experimental laws are only approximate, and

if some appear to us as exact, it is because we have artificially

transformed them into what I have above called a principle. We have made

this transformation freely, and as the caprice which has determined us

to make it is something eminently contingent, we have communicated this

contingence to the law itself. It is in this sense that we have the

right to say that determinism supposes freedom, since it is freely that

we become determinists. Perhaps it will be found that this is to give

large scope to nominalism and that the introduction of this new sense of

the word contingence will not help much to solve all those questions

which naturally arise and of which we have just been speaking.

I do not at all wish to investigate here the foundations of the

principle of induction; I know very well that I should not succeed; it

is as difficult to justify this principle as to get on without it. I

only wish to show how scientists apply it and are forced to apply it.

When the same antecedent recurs, the same consequent must likewise

recur; such is the ordinary statement. But reduced to these terms this

principle could be of no use. For one to be able to say that the same

antecedent recurred, it would be necessary for the circumstances _all_

to be reproduced, since no one is absolutely indifferent, and for them

to be _exactly_ reproduced. And, as that will never happen, the

principle can have no application.

We should therefore modify the enunciation and say: If an antecedent _A_

has once produced a consequent _B_, an antecedent _A'_, slightly

different from _A_, will produce a consequent _B'_, slightly different

from _B_. But how shall we recognize that the antecedents _A_ and _A'_

are 'slightly different'? If some one of the circumstances can be

expressed by a number, and this number has in the two cases values very

near together, the sense of the phrase 'slightly different' is

relatively clear; the principle then signifies that the consequent is a

continuous function of the antecedent. And as a practical rule, we reach

this conclusion that we have the right to interpolate. This is in fact

what scientists do every day, and without interpolation all science

would be impossible.

Yet observe one thing. The law sought may be represented by a curve.

Experiment has taught us certain points of this curve. In virtue of the

principle we have just stated, we believe these points may be connected

by a continuous graph. We trace this graph with the eye. New experiments

will furnish us new points of the curve. If these points are outside of

the graph traced in advance, we shall have to modify our curve, but not

to abandon our principle. Through any points, however numerous they may

be, a continuous curve may always be passed. Doubtless, if this curve is

too capricious, we shall be shocked (and we shall even suspect errors of

experiment), but the principle will not be directly put at fault.

Furthermore, among the circumstances of a phenomenon, there are some

that we regard as negligible, and we shall consider _A_ and _A'_ as

slightly different if they differ only by these accessory circumstances.

For instance, I have ascertained that hydrogen unites with oxygen under

the influence of the electric spark, and I am certain that these two

gases will unite anew, although the longitude of Jupiter may have

changed considerably in the interval. We assume, for instance, that the

state of distant bodies can have no sensible influence on terrestrial

phenomena, and that seems in fact requisite, but there are cases where

the choice of these practically indifferent circumstances admits of more

arbitrariness or, if you choose, requires more tact.

One more remark: The principle of induction would be inapplicable if

there did not exist in nature a great quantity of bodies like one

another, or almost alike, and if we could not infer, for instance, from

one bit of phosphorus to another bit of phosphorus.

If we reflect on these considerations, the problem of determinism and of

contingence will appear to us in a new light.

Suppose we were able to embrace the series of all phenomena of the

universe in the whole sequence of time. We could envisage what might be

called the _sequences_; I mean relations between antecedent and

consequent. I do not wish to speak of constant relations or laws, I

envisage separately (individually, so to speak) the different sequences

realized.

We should then recognize that among these sequences there are no two

altogether alike. But, if the principle of induction, as we have just

stated it, is true, there will be those almost alike and that can be

classed alongside one another. In other words, it is possible to make a

classification of sequences.

It is to the possibility and the legitimacy of such a classification

that determinism, in the end, reduces. This is all that the preceding

analysis leaves of it. Perhaps under this modest form it will seem less

appalling to the moralist.

It will doubtless be said that this is to come back by a detour to M.

LeRoy's conclusion which a moment ago we seemed to reject: we are

determinists voluntarily. And in fact all classification supposes the

active intervention of the classifier. I agree that this may be

maintained, but it seems to me that this detour will not have been

useless and will have contributed to enlighten us a little.

6. _Objectivity of Science_

I arrive at the question set by the title of this article: What is the

objective value of science? And first what should we understand by

objectivity?

What guarantees the objectivity of the world in which we live is that

this world is common to us with other thinking beings. Through the

communications that we have with other men, we receive from them

ready-made reasonings; we know that these reasonings do not come from us

and at the same time we recognize in them the work of reasonable beings

like ourselves. And as these reasonings appear to fit the world of our

sensations, we think we may infer that these reasonable beings have seen

the same thing as we; thus it is we know we have not been dreaming.

Such, therefore, is the first condition of objectivity; what is

objective must be common to many minds and consequently transmissible

from one to the other, and as this transmission can only come about by

that 'discourse' which inspires so much distrust in M. LeRoy, we are

even forced to conclude: no discourse, no objectivity.

The sensations of others will be for us a world eternally closed. We

have no means of verifying that the sensation I call red is the same as

that which my neighbor calls red.

Suppose that a cherry and a red poppy produce on me the sensation _A_

and on him the sensation _B_ and that, on the contrary, a leaf produces

on me the sensation _B_ and on him the sensation _A_. It is clear we

shall never know anything about it; since I shall call red the sensation

_A_ and green the sensation _B_, while he will call the first green and

the second red. In compensation, what we shall be able to ascertain is

that, for him as for me, the cherry and the red poppy produce the _same_

sensation, since he gives the same name to the sensations he feels and I

do the same.

Sensations are therefore intransmissible, or rather all that is pure

quality in them is intransmissible and forever impenetrable. But it is

not the same with relations between these sensations.

From this point of view, all that is objective is devoid of all quality

and is only pure relation. Certes, I shall not go so far as to say that

objectivity is only pure quantity (this would be to particularize too

far the nature of the relations in question), but we understand how some

one could have been carried away into saying that the world is only a

differential equation.

With due reserve regarding this paradoxical proposition, we must

nevertheless admit that nothing is objective which is not transmissible,

and consequently that the relations between the sensations can alone

have an objective value.

Perhaps it will be said that the esthetic emotion, which is common to

all mankind, is proof that the qualities of our sensations are also the

same for all men and hence are objective. But if we think about this, we

shall see that the proof is not complete; what is proved is that this

emotion is aroused in John as in James by the sensations to which James

and John give the same name or by the corresponding combinations of

these sensations; either because this emotion is associated in John with

the sensation _A_, which John calls red, while parallelly it is

associated in James with the sensation _B_, which James calls red; or

better because this emotion is aroused, not by the qualities themselves

of the sensations, but by the harmonious combination of their relations

of which we undergo the unconscious impression.

Such a sensation is beautiful, not because it possesses such a quality,

but because it occupies such a place in the woof of our associations of

ideas, so that it can not be excited without putting in motion the

'receiver' which is at the other end of the thread and which corresponds

to the artistic emotion.

Whether we take the moral, the esthetic or the scientific point of view,

it is always the same thing. Nothing is objective except what is

identical for all; now we can only speak of such an identity if a

comparison is possible, and can be translated into a 'money of exchange'

capable of transmission from one mind to another. Nothing, therefore,

will have objective value except what is transmissible by 'discourse,'

that is, intelligible.

But this is only one side of the question. An absolutely disordered

aggregate could not have objective value since it would be

unintelligible, but no more can a well-ordered assemblage have it, if it

does not correspond to sensations really experienced. It seems to me

superfluous to recall this condition, and I should not have dreamed of

it, if it had not lately been maintained that physics is not an

experimental science. Although this opinion has no chance of being

adopted either by physicists or by philosophers, it is well to be warned

so as not to let oneself slip over the declivity which would lead

thither. Two conditions are therefore to be fulfilled, and if the first

separates reality[11] from the dream, the second distinguishes it from

the romance.

[11] I here use the word real as a synonym of objective; I thus

conform to common usage; perhaps I am wrong, our dreams are

real, but they are not objective.

Now what is science? I have explained in the preceding article, it is

before all a classification, a manner of bringing together facts which

appearances separate, though they were bound together by some natural

and hidden kinship. Science, in other words, is a system of relations.

Now we have just said, it is in the relations alone that objectivity

must be sought; it would be vain to seek it in beings considered as

isolated from one another.

To say that science can not have objective value since it teaches us

only relations, this is to reason backward, since, precisely, it is

relations alone which can be regarded as objective.

External objects, for instance, for which the word _object_ was

invented, are really _objects_ and not fleeting and fugitive

appearances, because they are not only groups of sensations, but groups

cemented by a constant bond. It is this bond, and this bond alone, which

is the object in itself, and this bond is a relation.

Therefore, when we ask what is the objective value of science, that does

not mean: Does science teach us the true nature of things? but it means:

Does it teach us the true relations of things?

To the first question, no one would hesitate to reply, no; but I think

we may go farther; not only science can not teach us the nature of

things; but nothing is capable of teaching it to us, and if any god knew

it, he could not find words to express it. Not only can we not divine

the response, but if it were given to us we could understand nothing of

it; I ask myself even whether we really understand the question.

When, therefore, a scientific theory pretends to teach us what heat is,

or what is electricity, or life, it is condemned beforehand; all it can

give us is only a crude image. It is, therefore, provisional and

crumbling.

The first question being out of reason, the second remains. Can science

teach us the true relations of things? What it joins together should

that be put asunder, what it puts asunder should that be joined

together?

To understand the meaning of this new question, it is needful to refer

to what was said above on the conditions of objectivity. Have these

relations an objective value? That means: Are these relations the same

for all? Will they still be the same for those who shall come after us?

It is clear that they are not the same for the scientist and the

ignorant person. But that is unimportant, because if the ignorant person

does not see them all at once, the scientist may succeed in making him

see them by a series of experiments and reasonings. The thing essential

is that there are points on which all those acquainted with the

experiments made can reach accord.

The question is to know whether this accord will be durable and whether

it will persist for our successors. It may be asked whether the unions

that the science of to-day makes will be confirmed by the science of

to-morrow. To affirm that it will be so we can not invoke any _a priori_

reason; but this is a question of fact, and science has already lived

long enough for us to be able to find out by asking its history whether

the edifices it builds stand the test of time, or whether they are only

ephemeral constructions.

Now what do we see? At the first blush, it seems to us that the theories

last only a day and that ruins upon ruins accumulate. To-day the

theories are born, to-morrow they are the fashion, the day after

to-morrow they are classic, the fourth day they are superannuated, and

the fifth they are forgotten. But if we look more closely, we see that

what thus succumb are the theories properly so called, those which

pretend to teach us what things are. But there is in them something

which usually survives. If one of them taught us a true relation, this

relation is definitively acquired, and it will be found again under a

new disguise in the other theories which will successively come to reign

in place of the old.