HYPOTHESES IN PHYSICS.
The Role of Experiment and Generalization. ―Experiment
is the sole source of truth. It alone can teach us something new; it alone can
give us certainty. These are two points that cannot be questioned. But then, if
experiment is everything, what place is left for mathematical physics? What can
experimental physics do with such an auxiliary ― an
auxiliary, moreover, which seems useless, and even may be dangerous?
However, mathematical physics exists. It
has rendered undeniable service, and that is a fact which has to be explained.
It is not sufficient merely to observe ; we must use our observations, and for
that purpose we must generalize. This is what has always been done, only as the
recollection of past errors has made man more and more circumspect, he has
observed more and more and generalized less and less. Every age has scoffed at
its predecessor, accusing it of having generalized too boldly and too naively.
Descartes used to commiserate the Ionians. Descartes in his turn makes us
smile, and no doubt some day our children will laugh at us. Is there no way of
getting at once to the gist of the matter, and thereby escaping the raillery
which we foresee? Cannot we be content with experiment alone? No, that is
impossible; that would be a complete misunderstanding of the true character of
science. The man of science must work with method. Science is built up of
facts, as a house is built of stones; but an accumulation of facts is no more a
science than a heap of stones is a house. Most important of all, the man of
science must exhibit foresight. Carlyle has written somewhere something after
this fashion. " 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."1 Carlyle was a
compatriot of Bacon, and, like him, he wished to proclaim his worship of the God of Things as they are.
[1 V. Past and Present, end of Chapter I., Book II.-[TR.]]
But Bacon would not have said that. That
is the language of the historian. The physicist would most likely have said :
"John Lackland passed by here. It is all the same to me, for he will not
pass this way again."
We all know that there are good and bad
experiments. The latter accumulate in vain. Whether there are a hundred or a
thousand, one single piece of work by a real master―by a Pasteur, for example―will be sufficient to sweep them into oblivion. Bacon
would have thoroughly understood that, for he invented the phrase experimentum crucis; but Carlyle would
not have understood it. A fact is a fact. A student has read such and such a
number on his thermometer. He has taken no precautions. It does not matter; he
has read it, and if it is only the fact which counts, this is a reality that is
as much entitled to be called a reality as the peregrinations of King John
Lackland. What, then, is a good experiment? It is that which teaches us
something more than an isolated fact. It is that which enables us to predict,
and to generalize. Without generalization, prediction is impossible. The
circumstances under which one has operated will never again be reproduced
simultaneously. The fact observed will never be repeated. All that can be
affirmed is that under analogous circumstances an analogous fact will be
produced. To predict it, we must therefore invoke the aid of analogy―that is to say, even at this stage, we must generalize.
However timid we may be, there must be interpolation. Experiment only gives us
a certain number of isolated points. They must be connected by a continuous
line, and this is a true generalization. But more is done. The curve thus
traced will pass between and near the points observed ; it will not pass
through the points themselves. Thus we are not restricted to generalizing our experiment,
we correct it ; and the physicist who would abstain from these corrections, and
really content himself with experiment pure and simple, would be compelled to
enunciate very extraordinary laws indeed. Detached facts cannot therefore
satisfy us, and that is why our science must be ordered, or, better still,
generalized.
It is often said that experiments should
be made without preconceived ideas. That is impossible. Not only would it make
every experiment fruitless, but even if we wished to do so, it could not be
done. Every man has his own conception of the world, and this he cannot so
easily lay aside. We must, for example, use language, and our language is
necessarily steeped in preconceived ideas. Only they are unconscious
preconceived ideas, which are a thousand times the most dangerous of all. Shall
we say, that if we cause others to intervene of which we are fully conscious,
that we shall only aggravate the evil? I do not think so. I am inclined to
think that they will serve as ample counterpoises―I
was almost going to say antidotes. They will generally disagree, they will
enter into conflict one with another, and ipso
facto, they will force us to look at things under different aspects. This
is enough to free us. He is no longer a slave who can choose his master.
Thus, by generalization, every fact
observed enables us to predict a large number of others ; only, we ought not to
forget that the first alone is certain, and that all the others are merely probable.
However solidly founded a prediction may appear to us, we are never absolutely sure that experiment will not
prove it to be baseless if we set to work to verify it. But the probability of
its accuracy is often so great that practically we may be content with it. It
is far better to predict without certainty, than never to have predicted at
all. We should never, therefore, disdain to verify when the opportunity
presents itself. But every experiment is long and difficult, and the laborers
are few, and the number of facts which we require to predict is enormous ; and
besides this mass, the number of direct verifications that we can make will
never be more than a negligible quantity. Of this little that we can directly
attain we must choose the best. Every experiment must enable us to make a maximum
number of predictions having the highest possible degree of probability. The
problem is, so to speak, to increase the output of the scientific machine. I
may be permitted to compare science to a library which must go on increasing
indefinitely; the librarian has limited funds for his purchases, and he must,
therefore, strain every nerve not to waste them. Experimental physics has to
make the purchases, and experimental physics alone can enrich the library. As
for mathematical physics, her duty is to draw up the catalogue. If the
catalogue is well done the library is none the richer for it; but the reader
will be enabled to utilize its riches; and also by showing the librarian the
gaps in his collection, it will help him to make a judicious use of his funds,
which is all the more important, inasmuch as those funds are entirely
inadequate. That is the role of mathematical physics. It must direct generalization,
so as to increase what I called just now the output of science. By what means
it does this, and how it may do it without danger, is what we have now to
examine.
The
Unity of Nature. ―Let us first of all observe that every
generalization supposes in a certain measure a belief in the unity and
simplicity of Nature. As far as the unity is concerned, there can be no
difficulty. If the different parts of the universe were not as the organs of
the same body, they would not re-act one upon the other; they would mutually
ignore each other, and we in particular should only know one part. We need not,
therefore, ask if Nature is one, but how she is one.
As for the second point, that is not so clear. It is not certain that
Nature is simple. Can we without danger act as if she were ?
There was a time when the simplicity of
Mariotte's law was an argument in favor of its accuracy: when Fresnel himself,
after having said in a conversation with Laplace that Nature cares naught for
analytical difficulties, was compelled to explain his words so as not to give
offence to current opinion. Nowadays, ideas have changed considerably; but
those who do not believe that natural laws must be simple, are still often
obliged to act as if they did believe it. They cannot entirely dispense with
this necessity without making all generalization, and therefore all science,
impossible. It is clear that any fact can be generalized in an infinite number
of ways, and it is a question of choice. The choice can only be guided by
considerations of simplicity. Let us take the most ordinary case, that of
interpolation. We draw a continuous line as regularly as possible between the
points given by observation. Why do we avoid angular points and inflexions that
are too sharp? Why do we not make our curve describe the most capricious
zigzags? It is because we know beforehand, or think we know, that the law we
have to express cannot be so complicated as all that. The mass of Jupiter may
be deduced either from the movements of his satellites, or from the
perturbations of the major planets, or from those of the minor planets. If we
take the mean of the determinations obtained by these three methods, we find
three numbers very close together, but not quite identical. This result might
be interpreted by supposing that the gravitation constant is not the same in
the three cases; the observations would be certainly much better represented.
Why do we reject this interpretation? Not because it is absurd, but because it
is uselessly complicated. We shall only accept it when we are forced to, and it
is not imposed upon us yet. To sum up, in most cases every law is held to be
simple until the contrary is proved.
This custom is imposed upon physicists by
the reasons that I have indicated, but how can it be justified in the presence
of discoveries which daily show us fresh details, richer and more complex? How
can we even reconcile it with the unity of nature? For if all things are
interdependent, the relations in which so many different objects intervene can
no longer be simple.
If we study the history of science we see
produced two phenomena which are, so to speak, each the inverse of the other.
Sometimes it is simplicity which is hidden under what is apparently complex;
sometimes, on the contrary, it is simplicity which is apparent, and which
conceals extremely complex realities. What is there more complicated than the
disturbed motions of the planets, and what more simple than Newton's law?
There, as Fresnel said, Nature playing with analytical difficulties, only uses
simple means, and creates by their combination I know not what tangled skein.
Here it is the hidden simplicity which must be disentangled. Examples to the
contrary abound. In the kinetic theory of gases, molecules of tremendous
velocity are discussed, whose paths, deformed by incessant impacts, have the
most capricious shapes, and plough their way through space in every direction.
The result observable is Mariotte's simple law. Each individual fact was
complicated. The law of great numbers has re-established simplicity in the
mean. Here the simplicity is only apparent, and the coarseness of our senses
alone prevents us from seeing the complexity.
Many phenomena obey a law of proportionality.
But why? Because in these phenomena, there is something which is very small.
The simple law observed is only the translation of the general analytical rule
by which 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 only very small, the law of proportionality is only approximate, and
simplicity is only apparent. What I have just said applies to the law of the
superposition of small movements, which is so fruitful in its applications and
which is the foundation of optics.
And Newton's law itself? Its
simplicity, so long undetected, is perhaps only apparent. Who knows if it be
not due to some complicated mechanism, to the impact of some subtle matter
animated by irregular movements, and if it has not become simple merely through
the play of averages and large numbers? In any case, it is difficult not to
suppose that the true law contains complementary terms which may become
sensible at small distances. If in astronomy they are negligible, and if the
law thus regains its simplicity, it is solely on account of the enormous
distances of the celestial bodies. No doubt, if our means of investigation
became more and more penetrating, we should discover the simple beneath the
complex, and then the complex from the simple, and then again the simple
beneath the complex, and so on, without ever being able to predict what the
last term will be. We must stop somewhere, and for science to be possible we
must stop where we have found simplicity. That is the only ground on which we
can erect the edifice of our generalizations. But, this simplicity being only
apparent, will the ground be solid enough ? That is what we have now to
discover.
For this purpose let us see what part is played in our generalizations
by the belief in simplicity. We have verified a simple law in a considerable
number of particular cases. We refuse to admit that this coincidence, so often
repeated, is a result of mere chance, and we conclude that the law must be true
in the general case.
Kepler remarks that the positions of a
planet observed by Tycho are all on the same ellipse. Not for one moment does
he think that, by a singular freak of chance, Tycho had never looked at the
heavens except at the very moment when the path of the planet happened to cut
that ellipse. What does it matter then if the simplicity be real or if it hide
a complex truth? Whether it be due to the influence of great numbers which
reduces individual differences to a level, or to the greatness or the smallness
of certain quantities which allow of certain terms to be neglected-in no case
is it due to chance. This simplicity, real or apparent, has always a cause. We
shall therefore always be able to reason in the same fashion, and if a simple
law has been observed in several particular cases, we may legitimately suppose
that it still will be true in analogous cases. To refuse to admit this would be
to attribute an in-admissible role to chance. However, there is a difference.
If the simplicity were real and profound it would bear the test of the
increasing precision of our methods of measurement. If, then, we believe Nature
to be profoundly simple, we must conclude that it is an approximate and not a
rigorous simplicity. This is what was formerly done, but it is what we have no
longer the right to do. The simplicity of Kepler's laws, for instance, is only
apparent ; but that does not prevent them from being applied to almost all
systems analogous to the solar system, though that prevents them from being
rigorously exact.
Role
of Hypothesis. ―Every generalization is a hypothesis. Hypothesis
therefore plays a necessary role, which no one has ever contested. Only, it
should always be as soon as possible submitted to verification. It goes without
saying that, if it cannot stand this test, it must be abandoned without any
hesitation. This is, indeed, what is generally done; but sometimes with a
certain impatience. Ah well! this impatience is not justified. The physicist
who has just given up one of his hypotheses should, on the contrary, rejoice, for
he found an unexpected opportunity of discovery. His hypothesis, I imagine, had
not been lightly adopted. It took into account all the known factors which seem
capable of intervention in the phenomenon. If it is not verified, it is because
there is something unexpected and extraordinary about it, because we are on the
point of finding something unknown and new. Has the hypothesis thus rejected
been sterile ? Far from it. It may be even said that it has rendered more
service than a true hypothesis. Not only has it been the occasion of a decisive
experiment, but if this experiment had been made by chance, without the
hypothesis, no conclusion could have been drawn; nothing extraordinary would
have been seen; and only one fact the more would have been catalogued, without
deducing from it the remotest consequence.
Now, under what conditions is the use of
hypothesis without danger? The proposal to submit all to experiment is not
sufficient. Some hypotheses are dangerous, ―first
and foremost those which are tacit and unconscious. And since we make them
without knowing them, we cannot get rid of them. Here again, there is a service
that mathematical physics may render us. By the precision which is its
characteristic, we are compelled to formulate all the hypotheses that we would
unhesitatingly make without its aid. Let us also notice that it is important
not to multiply hypotheses indefinitely. If we construct a theory based upon
multiple hypotheses, and if experiment condemns it, which of the premises must
be changed? It is impossible to tell. Conversely, if the experiment succeeds,
must we suppose that it has verified all these hypotheses at once? Can several
unknowns be determined from a single equation?
We must also take care to distinguish
between the different kinds of hypotheses. First of all, there are those which
are quite natural and necessary. It is difficult not to suppose that the
influence of very distant bodies is quite negligible, that small movements obey
a linear law, and that effect is a continuous function of its cause. I will say
as much for the conditions imposed by symmetry. All these hypotheses affirm, so
to speak, the common basis of all the theories of mathematical physics. They
are the last that should be abandoned. There is a second category of hypotheses
which I shall qualify as indifferent. In most questions the analyst assumes, at
the beginning of his calculations, either that matter is continuous, or the
reverse, that it is formed of atoms. In either case, his results would have been
the same. On the atomic supposition he has a little more difficulty in
obtaining them―that is all. If, then, experiment confirms
his conclusions, will he suppose that he has proved, for example, the real
existence of atoms?
In optical theories two vectors are
introduced, one of which we consider as a velocity and the other as a vortex.
This again is an indifferent hypothesis, since we should have arrived at the same
conclusions by assuming the former to be a vortex and the latter to be a
velocity. The success of the experiment cannot prove, therefore, that the first
vector is really a velocity. It only proves one thing―namely, that it is a vector; and that is the only
hypothesis that has really been introduced into the premises. To give it the
concrete appearance that the fallibility of our minds demands, it was necessary
to consider it either as a velocity or as a vortex. In the same way, it was
necessary to represent it by an x or a y, but the result will not prove that we
were right or wrong in regarding it as a velocity; nor will it prove we are
right or wrong in calling it x and not y.
These indifferent hypotheses are never dangerous provided their
characters are not misunderstood. They may be useful, either as artifices for
calculation, or to assist our understanding by concrete images, to fix the
ideas, as we say. They need not therefore be rejected. The hypotheses of the
third category are real generalizations. They must be confirmed or invalidated
by experiment. Whether verified or condemned, they will always be fruitful;
but, for the reasons I have given, they will only be so if they are not too
numerous.
Origin
of Mathematical Physics. ―Let us go further and study more closely
the conditions which have assisted the development of mathematical physics. We
recognize at the outset that the efforts of men of science have always tended
to resolve the complex phenomenon given directly by experiment into a very
large number of elementary phenomena, and that in three different ways.
First, with respect to time. Instead of
embracing in its entirety the progressive development of a phenomenon, we
simply try to connect each moment with the one immediately preceding. We admit
that the present state of the world only depends on the immediate past, without
being directly influenced, so to speak, by the recollection of a more distant
past. Thanks to this postulate, instead of studying directly the whole
succession of phenomena, we may confine ourselves to writing down its
differential equation; for the laws of Kepler we substitute the law of Newton.
Next, we try to decompose the phenomena
in space. What experiment gives us is a confused aggregate of facts spread over
a scene of considerable extent. We must try to deduce the elementary phenomenon,
which will still be localized in a very small region of space.
A few examples perhaps will make my
meaning clearer. If we wished to study in all its complexity the distribution
of temperature in a cooling solid, we could never do so. This is simply
because, if we only reflect that a point in the solid can directly impart some
of its heat to a neighboring point, it will immediately impart that heat only
to the nearest points, and it is but gradually that the flow of heat will reach
other portions of the solid. The elementary phenomenon is the interchange of
heat between two contiguous points. It is strictly localized and relatively
simple if, as is natural, we admit that it is not influenced by the temperature
of the molecules whose distance apart is small.
I bend a rod: it takes a very
complicated form, the direct investigation of which would be impossible. But I
can attack the problem, however, if I notice that its flexure is only the
resultant of the deformations of the very small elements of the rod, and that
the deformation of each of these elements only depends on the forces which are
directly applied to it, and not in the least on those which may be acting on
the other elements.
In all these examples, which may be
increased without difficulty, it is admitted that there is no action at a
distance or at great distances. That is a hypothesis. It is not always true, as
the law of gravitation proves. It must therefore be verified. If it is
confirmed, even approximately, it is valuable, for it helps us to use
mathematical physics, at any rate by successive approximations. If it does not
stand the test, we must seek something else that is analogous, for there are
other means of arriving at the elementary phenomenon. If several bodies act
simultaneously, it may happen that their actions are independent, and may be
added one to the other. either as vectors or as scalar quantities. The
elementary phenomenon is then the action of an isolated body. Or suppose,
again, it is a question of small movements, or more generally of small
variations which obey the well-known law of mutual or relative independence.
The movement observed will then be decomposed into simple movements―for example, sound into its harmonics, and white light
into its monochromatic components. When we have discovered in which direction
to seek for the elementary phenomena, by what means may we reach it? First, it
will often happen that in order to predict it, or rather in order to predict
what is useful to us, it will not be necessary to know its mechanism. The law
of great numbers will suffice. Take for example the propagation of heat. Each
molecule radiates towards its neighbor―we
need not inquire according to what law; and if we make any supposition in this
respect, it will be an indifferent hypothesis, and therefore useless and
unverifiable. In fact, by the action of averages and thanks to the symmetry of
the medium, all differences are leveled, and, whatever the hypothesis may be,
the result is always the same.
The same feature is presented in the
theory of elasticity, and in that of capillarity. The neighboring molecules
attract and repel each other, we need not inquire by what law. It is enough for
us that this attraction is sensible at small distances only, and that the
molecules are very numerous, that the medium is symmetrical, and we have only
to let the law of great numbers come into play.
Here again the simplicity of the elementary phenomenon is hidden beneath
the complexity of the observable resultant phenomenon; but in its turn this
simplicity was only apparent and disguised a very complex mechanism. Evidently
the best means of reaching the elementary phenomenon would be experiment. It
would be necessary by experimental artifices to dissociate the complex system
which nature offers for our investigations and carefully to study the elements
as dissociated as possible; for example, natural white light would be
decomposed into monochromatic lights by the aid of the prism, and into polarized
lights by the aid of the polarizer. Unfortunately, that is neither always
possible nor always sufficient, and sometimes the mind must run ahead of
experiment. I shall only give one example which has always struck me rather
forcibly. If I decompose white light, I shall be able to isolate a portion of
the spectrum, but however small it may be, it will always be a certain width.
In the same way the natural lights which are called monochromatic give us a very fine array, but a y which is not,
however, infinitely fine. It might be supposed that in the experimental study
of the properties of these natural lights, by operating with finer and finer
rays, and passing on at last to the limit, so to speak, we should eventually
obtain the properties of a rigorously monochromatic light. That would not be
accurate. I assume that two rays emanate from the same source, that they are
first polarized in planes at right angles, that they are then brought back again
to the same plane of polarization, and that we try to obtain interference. If
the light were rigorously monochromatic,
there would be interference; but with our nearly monochromatic lights, there
will be no interference, and that, however narrow the ray may be. For it to be
otherwise, the ray would have to be several million times finer than the finest
known rays.
Here then we should be led astray by proceeding to the limit. The mind
has to run ahead of 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 state the problem in the form of
an equation. It only remains to deduce from it by combination the observable
and verifiable complex fact. That is what we call integration, and it is the province of the mathematician. It might
be asked, why in physical science generalization so readily takes the
mathematical form. The reason is now easy to see. It is not only because we
have to express numerical laws; it is because the observable phenomenon is due
to the superposition of a large number of elementary phenomena which are all similar to each other; and in this
way differential equations are quite naturally introduced. It is not enough
that each elementary phenomenon should obey simple laws: all those that we have
to combine must obey the same law; then only is the intervention of mathematics
of any use. Mathematics teaches us, in fact, to combine like with like. Its
object is to divine the result of a combination without having to reconstruct
that combination element by element. If we have to repeat the same operation
several times, mathematics enables us to avoid this repetition by telling the
result beforehand by a kind of induction. This I have explained before in the
chapter on mathematical reasoning. But for that purpose all these operations
must be similar; in the contrary case we must evidently make up our minds to
working them out in full one after the other, and mathematics will be useless.
It is therefore, thanks to the approximate homogeneity of the matter studied by
physicists, that mathematical physics came into existence. In the natural
sciences the following conditions are no longer to be found:-homogeneity,
relative independence of remote parts, simplicity of the elementary fact; and
that is why the student of natural science is compelled to have recourse to
other modes of generalization.
(From H. Poincare,
Science and Hypothesis, Dover
Publications, Inc. Library of Congress Catalog Card Number; 53-13673. PART IV. NATURE. CHAPTER IX.)