HYPOTHESES IN PHYSICS
THE RÔLE OF EXPERIMENT AND GENERALIZATION.--Experiment is the sole
source of truth. It alone can teach us anything new; it alone can give
us certainty. These are two points that can not be questioned.
But then, if experiment is everything, what place will remain for
mathematical physics? What has experimental physics to do with such an
aid, one which seems useless and perhaps even dangerous?
And yet mathematical physics exists, and has done unquestionable
service. We have here a fact that must be explained.
The explanation is that merely to observe is not enough. We must use our
observations, and to do that we must generalize. This is what men always
have done; only as the memory of past errors has made them more and more
careful, they have observed more and more, and generalized less and
less.
Every age has ridiculed the one before it, and accused it of having
generalized too quickly and too naïvely. Descartes pitied the Ionians;
Descartes, in his turn, makes us smile. No doubt our children will some
day laugh at us.
But can we not then pass over immediately to the goal? Is not this the
means of escaping the ridicule that we foresee? Can we not be content
with just the bare experiment?
No, that is impossible; it would be to mistake utterly the true nature
of science. The scientist must set in order. Science is built up with
facts, as a house is with stones. But a collection of facts is no more a
science than a heap of stones is a house.
And above all the scientist must foresee. Carlyle has somewhere said
something like this: "Nothing but facts are of importance. John Lackland
passed by here. Here is something that is admirable. Here is a reality
for which I would give all the theories in the world." Carlyle was a
fellow countryman of Bacon; but Bacon would not have said that. That is
the language of the historian. The physicist would say rather: "John
Lackland passed by here; that makes no difference to me, for he never
will pass this way again."
We all know that there are good experiments and poor ones. The latter
will accumulate in vain; though one may have made a hundred or a
thousand, a single piece of work by a true master, by a Pasteur, for
example, will suffice to tumble them into oblivion. Bacon would have
well understood this; it is he who invented the phrase _Experimentum
crucis_. But Carlyle would not have understood it. A fact is a fact. A
pupil has read a certain number on his thermometer; he has taken no
precaution; no matter, he has read it, and if it is only the fact that
counts, here is a reality of the same rank as the peregrinations of King
John Lackland. Why is the fact that this pupil has made this reading of
no interest, while the fact that a skilled physicist had made another
reading might be on the contrary very important? It is because from the
first reading we could not infer anything. What then is a good
experiment? It is that which informs us of something besides an isolated
fact; it is that which enables us to foresee, that is, that which
enables us to generalize.
For without generalization foreknowledge is impossible. The
circumstances under which one has worked will never reproduce themselves
all at once. The observed action then will never recur; the only thing
that can be affirmed is that under analogous circumstances an analogous
action will be produced. In order to foresee, then, it is necessary to
invoke at least analogy, that is to say, already then to generalize.
No matter how timid one may be, still it is necessary to interpolate.
Experiment gives us only a certain number of isolated points. We must
unite these by a continuous line. This is a veritable generalization.
But we do more; the curve that we shall trace will pass between the
observed points and near these points; it will not pass through these
points themselves. Thus one does not restrict himself to generalizing
the experiments, but corrects them; and the physicist who should try to
abstain from these corrections and really be content with the bare
experiment, would be forced to enunciate some very strange laws.
The bare facts, then, would not be enough for us; and that is why we
must have science ordered, or rather organized.
It is often said experiments must be made without a preconceived idea.
That is impossible. Not only would it make all experiment barren, but
that would be attempted which could not be done. Every one carries in
his mind his own conception of the world, of which he can not so easily
rid himself. We must, for instance, use language; and our language is
made up only of preconceived ideas and can not be otherwise. Only these
are unconscious preconceived ideas, a thousand times more dangerous than
the others.
Shall we say that if we introduce others, of which we are fully
conscious, we shall only aggravate the evil? I think not. I believe
rather that they will serve as counterbalances to each other--I was
going to say as antidotes; they will in general accord ill with one
another--they will come into conflict with one another, and thereby
force us to regard things under different aspects. This is enough to
emancipate us. He is no longer a slave who can choose his master.
Thus, thanks to generalization, each fact observed enables us to foresee
a great many others; only we must not forget that the first alone is
certain, that all others are merely probable. No matter how solidly
founded a prediction may appear to us, we are never _absolutely_ sure
that experiment will not contradict it, if we undertake to verify it.
The probability, however, is often so great that practically we may be
content with it. It is far better to foresee even without certainty than
not to foresee at all.
One must, then, never disdain to make a verification when opportunity
offers. But all experiment is long and difficult; the workers are few;
and the number of facts that we need to foresee is immense. Compared
with this mass the number of direct verifications that we can make will
never be anything but a negligible quantity.
Of this few that we can directly attain, we must make the best use; it
is very necessary to get from every experiment the greatest possible
number of predictions, and with the highest possible degree of
probability. The problem is, so to speak, to increase the yield of the
scientific machine.
Let us compare science to a library that ought to grow continually. The
librarian has at his disposal for his purchases only insufficient funds.
He ought to make an effort not to waste them.
It is experimental physics that is entrusted with the purchases. It
alone, then, can enrich the library.
As for mathematical physics, its task will be to make out the catalogue.
If the catalogue is well made, the library will not be any richer, but
the reader will be helped to use its riches.
And even by showing the librarian the gaps in his collections, it will
enable him to make a judicious use of his funds; which is all the more
important because these funds are entirely inadequate.
Such, then, is the rôle of mathematical physics. It must direct
generalization in such a manner as to increase what I just now called
the yield of science. By what means it can arrive at this, and how it
can do it without danger, is what remains for us to investigate.
THE UNITY OF NATURE.--Let us notice, first of all, that every
generalization implies in some measure the belief in the unity and
simplicity of nature. As to the unity there can be no difficulty. If the
different parts of the universe were not like the members of one body,
they would not act on one another, they would know nothing of one
another; and we in particular would know only one of these parts. We do
not ask, then, if nature is one, but how it is one.
As for the second point, that is not such an easy matter. It is not
certain that nature is simple. Can we without danger act as if it were?
There was a time when the simplicity of Mariotte's law was an argument
invoked in favor of its accuracy; when Fresnel himself, after having
said in a conversation with Laplace that nature was not concerned about
analytical difficulties, felt himself obliged to make explanations, in
order not to strike too hard at prevailing opinion.
To-day ideas have greatly changed; and yet, those who do not believe
that natural laws have to be simple, are still often obliged to act as
if they did. They could not entirely avoid this necessity without making
impossible all generalization, and consequently all science.
It is clear that any fact can be generalized in an infinity of ways, and
it is a question of choice. The choice can be guided only by
considerations of simplicity. Let us take the most commonplace case,
that of interpolation. We pass a continuous line, as regular as
possible, between the points given by observation. Why do we avoid
points making angles and too abrupt turns? Why do we not make our curve
describe the most capricious zig-zags? It is because we know beforehand,
or believe we know, that the law to be expressed can not be so
complicated as all that.
We may calculate the mass of Jupiter from either the movements of its
satellites, or the perturbations of the major planets, or those of the
minor planets. If we take the averages of the determinations obtained by
these three methods, we find three numbers very close together, but
different. We might interpret this result by supposing that the
coefficient of gravitation is not the same in the three cases. The
observations would certainly be much better represented. Why do we
reject this interpretation? Not because it is absurd, but because it is
needlessly complicated. We shall only accept it when we are forced to,
and that is not yet.
To sum up, ordinarily every law is held to be simple till the contrary
is proved.
This custom is imposed upon physicists by the causes that I have just
explained. But how shall we justify it in the presence of discoveries
that show us every day new details that are richer and more complex? How
shall we even reconcile it with the belief in the unity of nature? For
if everything depends on everything, relationships where so many diverse
factors enter can no longer be simple.
If we study the history of science, we see happen two inverse phenomena,
so to speak. Sometimes simplicity hides under complex appearances;
sometimes it is the simplicity which is apparent, and which disguises
extremely complicated realities.
What is more complicated than the confused movements of the planets?
What simpler than Newton's law? Here nature, making sport, as Fresnel
said, of analytical difficulties, employs only simple means, and by
combining them produces I know not what inextricable tangle. Here it is
the hidden simplicity which must be discovered.
Examples of the opposite abound. In the kinetic theory of gases, one
deals with molecules moving with great velocities, whose paths, altered
by incessant collisions, have the most capricious forms and traverse
space in every direction. The observable result is Mariotte's simple
law. Every individual fact was complicated. The law of great numbers has
reestablished simplicity in the average. Here the simplicity is merely
apparent, and only the coarseness of our senses prevents our perceiving
the complexity.
Many phenomena obey a law of proportionality. But why? Because in these
phenomena there is something very small. The simple law observed, then,
is only a result of the general analytical rule that the infinitely
small increment of a function is proportional to the increment of the
variable. As in reality our increments are not infinitely small, but
very small, the law of proportionality is only approximate, and the
simplicity is only apparent. What I have just said applies to the rule
of the superposition of small motions, the use of which is so fruitful,
and which is the basis of optics.
And Newton's law itself? Its simplicity, so long undetected, is perhaps
only apparent. Who knows whether it is not due to some complicated
mechanism, to the impact of some subtile matter animated by irregular
movements, and whether it has not become simple only through the action
of averages and of great numbers? In any case, it is difficult not to
suppose that the true law contains complementary terms, which would
become sensible at small distances. If in astronomy they are negligible
as modifying Newton's law, and if the law thus regains its simplicity,
it would be only because of the immensity of celestial distances.
No doubt, if our means of investigation should become more and more
penetrating, we should discover the simple under the complex, then the
complex under the simple, then again the simple under the complex, and
so on, without our being able to foresee what will be the last term.
We must stop somewhere, and that science may be possible we must stop
when we have found simplicity. This is the only ground on which we can
rear the edifice of our generalizations. But this simplicity being only
apparent, will the ground be firm enough? This is what must be
investigated.
For that purpose, let us see what part is played in our generalizations
by the belief in simplicity. We have verified a simple law in a good
many particular cases; we refuse to admit that this agreement, so often
repeated, is simply the result of chance, and conclude that the law must
be true in the general case.
Kepler notices that a planet's positions, as observed by Tycho, are all
on one ellipse. Never for a moment does he have the thought that by a
strange play of chance Tycho never observed the heavens except at a
moment when the real orbit of the planet happened to cut this ellipse.
What does it matter then whether the simplicity be real, or whether it
covers a complex reality? Whether it is due to the influence of great
numbers, which levels down individual differences, or to the greatness
or smallness of certain quantities, which allows us to neglect certain
terms, in no case is it due to chance. This simplicity, real or
apparent, always has a cause. We can always follow, then, the same
course of reasoning, and if a simple law has been observed in several
particular cases, we can legitimately suppose that it will still be true
in analogous cases. To refuse to do this would be to attribute to chance
an inadmissible rôle.
There is, however, a difference. If the simplicity were real and
essential, it would resist the increasing precision of our means of
measure. If then we believe nature to be essentially simple, we must,
from a simplicity that is approximate, infer a simplicity that is
rigorous. This is what was done formerly; and this is what we no longer
have a right to do.
The simplicity of Kepler's laws, for example, is only apparent. That
does not prevent their being applicable, very nearly, to all systems
analogous to the solar system; but it does prevent their being
rigorously exact.
THE RÔLE OF HYPOTHESIS.--All generalization is a hypothesis. Hypothesis,
then, has a necessary rôle that no one has ever contested. Only, it
ought always, as soon as possible and as often as possible, to be
subjected to verification. And, of course, if it does not stand this
test, it ought to be abandoned without reserve. This is what we
generally do, but sometimes with rather an ill humor.
Well, even this ill humor is not justified. The physicist who has just
renounced one of his hypotheses ought, on the contrary, to be full of
joy; for he has found an unexpected opportunity for discovery. His
hypothesis, I imagine, had not been adopted without consideration; it
took account of all the known factors that it seemed could enter into
the phenomenon. If the test does not support it, it is because there is
something unexpected and extraordinary; and because there is going to be
something found that is unknown and new.
Has the discarded hypothesis, then, been barren? Far from that, it may
be said it has rendered more service than a true hypothesis. Not only
has it been the occasion of the decisive experiment, but, without having
made the hypothesis, the experiment would have been made by chance, so
that nothing would have been derived from it. One would have seen
nothing extraordinary; only one fact the more would have been catalogued
without deducing from it the least consequence.
Now on what condition is the use of hypothesis without danger?
The firm determination to submit to experiment is not enough; there are
still dangerous hypotheses; first, and above all, those which are tacit
and unconscious. Since we make them without knowing it, we are powerless
to abandon them. Here again, then, is a service that mathematical
physics can render us. By the precision that is characteristic of it, it
compels us to formulate all the hypotheses that we should make without
it, but unconsciously.
Let us notice besides that it is important not to multiply hypotheses
beyond measure, and to make them only one after the other. If we
construct a theory based on a number of hypotheses, and if experiment
condemns it, which of our premises is it necessary to change? It will be
impossible to know. And inversely, if the experiment succeeds, shall we
believe that we have demonstrated all the hypotheses at once? Shall we
believe that with one single equation we have determined several
unknowns?
We must equally take care to distinguish between the different kinds of
hypotheses. There are first those which are perfectly natural and from
which one can scarcely escape. It is difficult not to suppose that the
influence of bodies very remote is quite negligible, that small
movements follow a linear law, that the effect is a continuous function
of its cause. I will say as much of the conditions imposed by symmetry.
All these hypotheses form, as it were, the common basis of all the
theories of mathematical physics. They are the last that ought to be
abandoned.
There is a second class of hypotheses, that I shall term neutral. In
most questions the analyst assumes at the beginning of his calculations
either that matter is continuous or, on the contrary, that it is formed
of atoms. He might have made the opposite assumption without changing
his results. He would only have had more trouble to obtain them; that is
all. If, then, experiment confirms his conclusions, will he think that
he has demonstrated, for instance, the real existence of atoms?
In optical theories two vectors are introduced, of which one is regarded
as a velocity, the other as a vortex. Here again is a neutral
hypothesis, since the same conclusions would have been reached by taking
precisely the opposite. The success of the experiment, then, can not
prove that the first vector is indeed a velocity; it can only prove one
thing, that it is a vector. This is the only hypothesis that has really
been introduced in the premises. In order to give it that concrete
appearance which the weakness of our minds requires, it has been
necessary to consider it either as a velocity or as a vortex, in the
same way that it has been necessary to represent it by a letter, either
_x_ or _y_. The result, however, whatever it may be, will not prove that
it was right or wrong to regard it as a velocity any more than it will
prove that it was right or wrong to call it _x_ and not _y_.
These neutral hypotheses are never dangerous, if only their character is
not misunderstood. They may be useful, either as devices for
computation, or to aid our understanding by concrete images, to fix our
ideas as the saying is. There is, then, no occasion to exclude them.
The hypotheses of the third class are the real generalizations. They are
the ones that experiment must confirm or invalidate. Whether verified or
condemned, they will always be fruitful. But for the reasons that I have
set forth, they will only be fruitful if they are not too numerous.
ORIGIN OF MATHEMATICAL PHYSICS.--Let us penetrate further, and study
more closely the conditions that have permitted the development of
mathematical physics. We observe at once that the efforts of scientists
have always aimed to resolve the complex phenomenon directly given by
experiment into a very large number of elementary phenomena.
This is done in three different ways: first, in time. Instead of
embracing in its entirety the progressive development of a phenomenon,
the aim is simply to connect each instant with the instant immediately
preceding it. It is admitted that the actual state of the world depends
only on the immediate past, without being directly influenced, so to
speak, by the memory of a distant past. Thanks to this postulate,
instead of studying directly the whole succession of phenomena, it is
possible to confine ourselves to writing its 'differential equation.'
For Kepler's laws we substitute Newton's law.
Next we try to analyze the phenomenon in space. What experiment gives us
is a confused mass of facts presented on a stage of considerable extent.
We must try to discover the elementary phenomenon, which will be, on the
contrary, localized in a very small region of space.
Some examples will perhaps make my thought better understood. If we
wished to study in all its complexity the distribution of temperature in
a cooling solid, we should never succeed. Everything becomes simple if
we reflect that one point of the solid can not give up its heat directly
to a distant point; it will give up its heat only to the points in the
immediate neighborhood, and it is by degrees that the flow of heat can
reach other parts of the solid. The elementary phenomenon is the
exchange of heat between two contiguous points. It is strictly
localized, and is relatively simple, if we admit, as is natural, that it
is not influenced by the temperature of molecules whose distance is
sensible.
I bend a rod. It is going to take a very complicated form, the direct
study of which would be impossible. But I shall be able, however, to
attack it, if I observe that its flexure is a result only of the
deformation of the very small elements of the rod, and that the
deformation of each of these elements depends only on the forces that
are directly applied to it, and not at all on those which may act on the
other elements.
In all these examples, which I might easily multiply, we admit that
there is no action at a distance, or at least at a great distance. This
is a hypothesis. It is not always true, as the law of gravitation shows
us. It must, then, be submitted to verification. If it is confirmed,
even approximately, it is precious, for it will enable us to make
mathematical physics, at least by successive approximations.
If it does not stand the test, we must look for something else
analogous; for there are still other means of arriving at the elementary
phenomenon. If several bodies act simultaneously, it may happen that
their actions are independent and are simply added to one another,
either as vectors or as scalars. The elementary phenomenon is then the
action of an isolated body. Or again, we have to deal with small
movements, or more generally with small variations, which obey the
well-known law of superposition. The observed movement will then be
decomposed into simple movements, for example, sound into its harmonics,
white light into its monochromatic components.
When we have discovered in what direction it is advisable to look for
the elementary phenomenon, by what means can we reach it?
First of all, it will often happen that in order to detect it, or rather
to detect the part of it useful to us, it will not be necessary to
penetrate the mechanism; the law of great numbers will suffice.
Let us take again the instance of the propagation of heat. Every
molecule emits rays toward every neighboring molecule. According to what
law, we do not need to know. If we should make any supposition in regard
to this, it would be a neutral hypothesis and consequently useless and
incapable of verification. And, in fact, by the action of averages and
thanks to the symmetry of the medium, all the differences are leveled
down, and whatever hypothesis may be made, the result is always the
same.
The same circumstance is presented in the theory of electricity and in
that of capillarity. The neighboring molecules attract and repel one
another. We do not need to know according to what law; it is enough for
us that this attraction is sensible only at small distances, and that
the molecules are very numerous, that the medium is symmetrical, and we
shall only have to let the law of great numbers act.
Here again the simplicity of the elementary phenomenon was hidden under
the complexity of the resultant observable phenomenon; but, in its turn,
this simplicity was only apparent, and concealed a very complex
mechanism.
The best means of arriving at the elementary phenomenon would evidently
be experiment. We ought by experimental contrivance to dissociate the
complex sheaf that nature offers to our researches, and to study with
care the elements as much isolated as possible. For example, natural
white light would be decomposed into monochromatic lights by the aid of
the prism, and into polarized light by the aid of the polarizer.
Unfortunately that is neither always possible nor always sufficient, and
sometimes the mind must outstrip experiment. I shall cite only one
example, which has always struck me forcibly.
If I decompose white light, I shall be able to isolate a small part of
the spectrum, but however small it may be, it will retain a certain
breadth. Likewise the natural lights, called _monochromatic_, give us a
very narrow line, but not, however, infinitely narrow. It might be
supposed that by studying experimentally the properties of these natural
lights, by working with finer and finer lines of the spectrum, and by
passing at last to the limit, so to speak, we should succeed in learning
the properties of a light strictly monochromatic.
That would not be accurate. Suppose that two rays emanate from the same
source, that we polarize them first in two perpendicular planes, then
bring them back to the same plane of polarization, and try to make them
interfere. If the light were _strictly_ monochromatic, they would
interfere. With our lights, which are nearly monochromatic, there will
be no interference, and that no matter how narrow the line. In order to
be otherwise it would have to be several million times as narrow as the
finest known lines.
Here, then, the passage to the limit would have deceived us. The mind
must outstrip the experiment, and if it has done so with success, it is
because it has allowed itself to be guided by the instinct of
simplicity.
The knowledge of the elementary fact enables us to put the problem in an
equation. Nothing remains but to deduce from this by combination the
complex fact that can be observed and verified. This is what is called
_integration_, and is the business of the mathematician.
It may be asked why, in physical sciences, generalization so readily
takes the mathematical form. The reason is now easy to see. It is not
only because we have numerical laws to express; it is because the
observable phenomenon is due to the superposition of a great number of
elementary phenomena _all alike_. Thus quite naturally are introduced
differential equations.
It is not enough that each elementary phenomenon obeys simple laws; all
those to be combined must obey the same law. Then only can the
intervention of mathematics be of use; mathematics teaches us in fact to
combine like with like. Its aim is to learn the result of a combination
without needing to go over the combination piece by piece. If we have to
repeat several times the same operation, it enables us to avoid this
repetition by telling us in advance the result of it by a sort of
induction. I have explained this above, in the chapter on mathematical
reasoning.
But for this, all the operations must be alike. In the opposite case, it
would evidently be necessary to resign ourselves to doing them in
reality one after another, and mathematics would become useless.
It is then thanks to the approximate homogeneity of the matter studied
by physicists that mathematical physics could be born.
In the natural sciences, we no longer find these conditions:
homogeneity, relative independence of remote parts, simplicity of the
elementary fact; and this is why naturalists are obliged to resort to
other methods of generalization.