OTHER GENERAL CONSIDERATIONS
THE APPROXIMATE CHARACTER OF EMPIRICAL
KNOWLEDGE
ALTHOUGH many aspects of the processes by which we obtain knowledge of
the external physical world are much beyond the scope of our present
inquiry, one matter must be mentioned in detail because it tacitly
underlies all our discussion, the fact, namely, that all results of
measurement are only approximate. That such is true is evident after the
most superficial examination of any measuring process; any statement
about numerical relations between measured quantities must always be
subject to the qualification that the relation is valid only within
limits. Furthermore, all experience seems to be of this character; we
never have perfectly clean-cut knowledge of anything, but all our
experience is surrounded by a twilight zone, a penumbra of uncertainty,
into which we have not yet penetrated. This penumbra is as truly an
unexplored region as any other region beyond experiment, such as the
region of high velocities, for example, and we must hold no preconceived
notions as to what will be found within the region. The penumbra is to
be penetrated by improving the accuracy of measurement. Within what was
at one time penumbra has been found the displacement of angular position
of the stars near the edge of the solar disc, and within the penumbra as
yet unpenetrated we look for such effects as the equivalence of mass and
energy. Many of the great discoveries of the future will probably be
made within the penumbra: we have already mentioned that increased
knowledge of phenomena of a cosmic scale is to be obtained by increasing
the accuracy of measurement of the very small.
It is a general consequence of the approximate character of all
measurement that no empirical science can ever make exact statements.
This was fairly obvious in the case of mechanics, but it required a
Gauss[4] to convince us that the geometry in which we are interested as
physicists is an empirical subject, and that one cannot say that actual
space is Euclidean, but only that actual space approaches to ideal
Euclidean space within a certain degree of approximation. I believe that
we are compelled to go still further, and recognize that arithmetic, so
far as it purports to deal with actual physical objects, is also
affected with the same penumbra of uncertainty as all other empirical
science. A typical statement of empirical arithmetic is that 2 objects
plus 2 objects makes 4 objects. This statement acquires physical meaning
only in terms of certain physical operations, and these operations must
be performed in time.
[Footnote 4: C. F. Gauss, Gesammelte Werke, especially vols. IV and
VIII.]
Now the penumbra gets into this situation through the concept of object.
If the statement of arithmetic is to be an exact statement in the
mathematical sense the "object" must be a definite clean-cut thing,
which preserves its identity in time with no penumbra. But this sort of
thing is never experienced, and as far as we know does not correspond
exactly to anything in experience. It is of course true that in most
experience the penumbra is so very thin and snug-fitting that it
requires special effort to recognize its presence at all; but scrutiny,
I believe, shows that it is always there. If our experience had been
restricted to phenomena in a vacuum, and the objects we were trying to
count had been spheres of a gas which expand and interpenetrate, it is
obvious that the concept of "object" as a thing with identity would have
been much more difficult to form. Or, if our objects are tumblers of
water, we discover when our observation reaches a certain stage of
refinement that the amount of water is continually changing by
evaporation and condensation, and we are bothered by the question
whether the object is still the same after it has waxed and waned.
Coming to solids, we eventually discover that even solids evaporate, or
condense gases on them, and we see that an object with identity is an
abstraction corresponding exactly to nothing in nature. Of course the
penumbra of uncertainty which surrounds our arithmetical statements
because of this property of physical objects is so exceedingly tenuous
that practically we are not aware of its existence, and expect never to
find undiscovered phenomena within the penumbra. But in principle we
must recognize its presence, and must further recognize that _all_
empirical science must be of this character.
In most empirical sciences, the penumbra is at first prominent, and
becomes less important and thinner as the accuracy of physical
measurement is increased. In mechanics, for example, the penumbra is at
first like a thick obscuring veil at the stage where we measure forces
only by our muscular sensations, and gradually is attenuated, as the
precision of measurements increases. But with the arithmetical concept
of an individual identifiable object it is the exact reverse; a crude
point of view does not suspect the existence of the penumbra at all, and
we discover it only by highly refining our methods. Doubtless arithmetic
owes its early development to this property.
We may now go still further. Operations themselves are, of course,
derived from experience, and would be expected also to have a nebulous
edge of uncertainty. We have to ask such questions as whether the
_operations_ of arithmetic are clean-cut things. Is the operation of
multiplying 2 objects by 2 a definite operation, with no enveloping
haze? All our physical experience convinces us that if there is a
penumbra about the concept of operations of this sort it is so tenuous
as to be negligible, at least for the present; but the question affords
an interesting topic for speculation. We also have to ask whether mental
operations may similarly be enveloped in a haze.
EXPLANATIONS AND MECHANISMS
Perhaps the climax of our task of interpreting and correlating nature is
reached when we are able to find an explanation of phenomena; with the
finding of the explanation we are inclined to feel that our
understanding of the situation is complete. We now have to ask what is
the nature of the explanation which we set as the goal of our efforts.
The answer is not easy to give, and there may be difference of opinion
about it. We shall get the best answer to this, as to so many other
questions, by adopting the operational point of view, and examining what
we do in giving an explanation. I believe that examination will show
that the essence of an explanation consists in reducing a situation to
elements with which we are so familiar that we accept them as a matter
of course, so that our curiosity rests.[5] "Reducing a situation to
elements" means, from the operational point of view, discovering
familiar correlations between the phenomena of which the situation is
composed.
[Footnote 5: The ultimate elements of explanation are analogous to the
axioms of formal mathematics.]
There is involved here the thesis that it is possible to analyze nature
into correlations, without, however, any assumption whatever as to the
character of these correlations. It seems to me that such a thesis is
the most general that can be made if nature is to be intelligible at
all. This thesis underlies all the considerations of this essay, and we
shall not try to find anything more general. We shall, however,
recognize that any assumption as to the character of the correlations
constitutes a special hypothesis which may restrict the future, and that
therefore these special hypotheses are to be subjected to special
examination. We return to this matter in more detail in discussing the
causality concept, which is closely related to the concept of
explanation.
In this view of explanation there is no implication that the "element"
is either a smaller or a larger scale thing than the phenomenon being
explained; thus we may explain the properties of a gas in terms of its
constituent molecules, or perhaps some day we shall become so familiar
with the idea of a non-Euclidean space that we shall _explain_ (instead
of describe) the gravitational attraction of a stone by the earth in
terms of a space-time curvature imposed by all the rest of the matter in
the universe.
If this is accepted as the true nature of explanation, we see that an
explanation is not an absolute sort of thing, but what is satisfactory
for one man will not be for another. The savage is satisfied by
explaining the thunderstorm as the capricious act of an angry god. The
physicist demands more, and requires that the familiar elements to which
we reduce a situation be such that we can intuitively predict their
behavior. Thus even if the physicist believed in the existence of the
angry god, he would not be satisfied with this explanation of the
thunderstorm because he is not so well acquainted with angry gods as to
be able to predict when anger is followed by a storm. He would have to
know why the god had become angry, and why making a thunderstorm eased
his ire. But even with this additional qualification, scientific
explanation is obviously still a relative affair--relative to the
elements or axioms to which we make reduction and which we accept as
ultimate. These elements depend to a certain extent on the purpose in
view, and also on the range of our previous physical experience. If we
are explaining the action of a machine, we are satisfied to reduce the
action to the push and pull of the various members of the machine, it
being accepted as an ultimate that these members transmit pushes or
pulls. But the physicist who has extended his experimental knowledge
further, may want to explain how the members transmit pushes or pulls in
terms of the action on each other of the electrons in their orbits in
the atoms. The character of our explanatory structure will depend on the
character of our experimental knowledge, and will change as this
changes.
Formally, there is no limit to the process of explanation, because we
can always ask what is the explanation of the elements in terms of which
we have given the last explanation. But the point of view of operations
shows that this is mere formalism which ends only in meaningless jargon,
for we soon arrive at the limit of our experimental knowledge, and
beyond this the operations involved in the concepts of our explanations
become impossible and the concepts become meaningless.
As we extend experimental knowledge and push our explanations further
and further, we see that the explanatory sequence may be terminated in
several possible ways. In the first place, we may never push our
experiments beyond a stage into which the elements with which we are
already familiar do not enter. In this case explanation is very simple:
it involves nothing essentially new, but merely the disentanglement of
complexities. The kinetic theory of gases, in explaining the thermal
properties of a gas in terms of ordinary mechanical properties of the
molecules, suggests such a situation. Or, secondly, our experiments may
bring us into contact with situations novel to us, in which we can
recognize no familiar elements, or at least must recognize that there is
something in addition to the familiar elements. Such a situation
constitutes an explanatory crisis and explanation has to stop by
definition. Or thirdly, we may try to force our explanations into a
predetermined mold, by formally erecting or inventing beyond the range
of present experiment ultimates more or less like elements already
familiar to us, and seek to explain all present experience in terms of
these chosen ultimates.
Leaving for the present the third possibility, which is within our
control to accept or reject, and is a formal matter, it is merely a
question of experimental fact which of the first two possibilities
corresponds to the actual state of affairs. The most perfunctory
examination of the present state of physics shows that we are now facing
the second of these possibilities, and that in the new experimental
facts of relativity, and in all quantum phenomena, we are confronted
with an explanatory crisis. It has often been emphasized that Einstein's
theory of gravitation does not seek at all to give an explanation of
gravitational phenomena, but merely describes and correlates these
phenomena in comparatively simple mathematical language. No more attempt
is made to reduce the gravitational attraction between the earth and the
sun to simple terms than was made by Newton. In the realm of quantum
phenomena it is of course the merest commonplace that our old ideas of
mechanics and electrodynamics have failed, so that it is a matter of the
greatest concern to find how many, or indeed whether any, elements of
the old situations can be carried over into the new.
An examination of many of the so-called "explanations" of quantum theory
constitutes at once a justification of the definition of explanation
given above, and of the statement that in quantum phenomena we are at an
explanatory crisis. For the endeavor of all these quantum explanations
is to find in every new or more complicated situation the same elements
which have already been met in simpler situations, and which are
therefore relatively more familiar. For example, many quantum phenomena
are made to involve the emission of energy when an electron jumps from
one orbit to another. But always the elements to which reduction is made
are themselves quantum phenomena, and these are still so new and
unfamiliar that we feel an instinctive need for explanation in other
terms. We seek to understand why the electron emits when it jumps.
The explanatory crisis which now confronts us in relativity and quantum
phenomena is but a repetition of what has occurred many time in the
past. A similar crisis confronted Prometheus when he discovered fire,
and the first man who observed a straw sticking to a piece of rubbed
amber, or a suspended lodestone seeking the north star. Every kitten is
confronted with such a crisis at the end of nine days. Whenever
experience takes us into new and unfamiliar realms, we are to be at
least prepared for a new crisis.
Now what are we to do in such a crisis? It seems to me that the only
sensible course is to do exactly what the kitten does, namely, to wait
until we have amassed so much experience of the new kind that it is
perfectly familiar to us, and then to resume the process of explanation
with elements from our new experience included in our list of axioms.
Not only will observation show that this is what is now actually being
done with respect to quantum and gravitational phenomena, but it is in
harmony with the entire spirit of our outlook on nature. All our
knowledge is in terms of experience; we should not expect or desire to
erect an explanatory structure different in character from that of
experience. Our experience is finite; on the confines of the
experimentally attainable it becomes hazy, and the concepts in terms of
which we describe it fuse together and lose independent meaning.
Furthermore, at every extension of our experimental range we must be
prepared to find, and as a matter of fact we have often found, that we
encounter phenomena of an entirely novel character for which previous
experience has given us no preparation. The explanatory structure
proposed above has all these properties; it is finite, being terminated
by the edge of experiment, the final stages of our explanations are hazy
in that it becomes more and more difficult to distinguish elements of
familiar experience, and every now and then we must admit new elements
into our explanations.
The first step in resuming our explanatory progress, after we have been
confronted with such a crisis, is to seek for various sorts of
correlation between the elements of our new experience, in the confident
expectation that these elements will eventually become so familiar to us
that they may be used as the ultimates of a new explanation. This is
exactly what is now happening in quantum theory.
Diametrically opposed to the views above, there is another ideal of the
explanatory process which is held by many physicists, and which has been
mentioned above as the third possible way in which the explanatory
sequence may be terminated, namely, the endeavor to devise beyond the
limit of present experiment a structure built of elements like some of
those of our present experience, in the action of which we endeavor to
find the explanation of phenomena in the present range. Now a program
such as this, as a serious program for the final correlation of nature,
is entirely opposed to the spirit of the considerations expounded here.
There is no warrant whatever in experience for the conviction that as we
penetrate deeper and deeper we shall find the elements of previous
experience repeated, although sometimes we do find such repetitions, as
in the behavior of gases. Yet this has been the attitude of many eminent
physicists, for example, Faraday and Maxwell, in seeking to explain
distant electrical action by the propagation through a medium of a
mechanical push or pull, or by Hertz, who sought in all phenomena the
effect of concealed masses with ordinary mechanical inertia. Although as
a general principle this program seems to be absolutely without
justification, nevertheless it may be justified if the specific
character of the physical facts seems to indicate a repetition at lower
levels of elements familiar higher up. Hertz undoubtedly had this
justification, as did also Maxwell to a certain extent, in the discovery
that the general equations of electrodynamics are of the same form as
the generalized Lagrangean equations of mechanics. For Faraday, however,
there seems no such justification; the urge to this sort of thing in
Faraday came from an uncritical acceptance of his own temperamental
reactions.
From a less serious point of view it may, however, be quite justified to
make such a working hypothesis as that in the action of electrical
forces may be discovered the same elements with which we are familiar in
the everyday experiences of mechanics. For such a hypothesis often
enables us to make partial correlations which suggest new experimental
tests, and thus gives the stimulus to an extension of our experimental
horizon. Many physicists recognize the tentative character of such
attempted explanations, but others apparently take them more seriously,
as for example Lord Kelvin in his continuous life-long attempts to find
a mechanical explanation of all physical phenomena. This quotation from
Kelvin is illuminating. "I never satisfy myself until I can make a
mechanical model of a thing. If I can make a mechanical model, I can
understand it. As long as I cannot make a mechanical model all the way
through, I cannot understand it.... But I want to understand light as
well as I can without introducing things that we understand even less
of."
So much for general considerations on the nature of explanation. Coming
now to greater detail, many explanations involve what may be described
as a mechanism. It is difficult to characterize exactly what we mean by
mechanism, but it seems to be associated with an attitude of mind that
strives to realize the third of the possibilities mentioned above. As a
matter of fact, the mechanism sought for is usually of a particular
type, in that the ultimate elements selected are mechanical elements.
This point of view is particularly characteristic of the English school
of physicists. Although "mechanism" usually implies mechanical elements,
we may show by specific examples that we do actually use the word in a
broader significance. If, for example, we could devise within the core
of an atom a revolving system of electrical charges, acting on each
other with the ordinary inverse square forces of electrostatics, such
that every now and then the system becomes unstable and breaks up, we
should doubtless say that we had found a mechanism for explaining
radioactive disintegration.
However, the formulation of a precise definition of mechanism is of
secondary concern to us; we are primarily interested in understanding
the attitude of mind that feels a mechanism is necessary. A typical
example of such an urge to a mechanism is afforded by the gravitational
action between distant bodies. To many minds the concept of action at a
distance is absolutely abhorrent, not to be tolerated for an instant.
Such an intolerable situation is avoided by the invention of a medium
filling all space, which transmits a force from one body to the other
through the successive action on each other of its contiguous parts. Or
the dilemma of action at a distance may be avoided in other ways, as by
Boscovitch in the eighteenth century, who, in order to explain
gravitation, filled space with a triply infinite horde of infinitesimal
projectiles. Now of course it is a matter for experiment to decide
whether any physical reality can be ascribed to a medium which makes
gravitation possible by the action of its adjacent parts, but I can see
no justification whatever for the attitude which refuses on purely _a
priori_ grounds to accept action at a distance as a possible axiom or
ultimate of explanation. It is difficult to conceive anything more
scientifically bigoted than to postulate that all possible experience
conforms to the same type as that with which we are already familiar,
and therefore to demand that explanation use only elements familiar in
everyday experience. Such an attitude bespeaks an unimaginativeness, a
mental obtuseness and obstinacy, which might be expected to have
exhausted their pragmatic justification at a lower plane of mental
activity.
Although it will probably be fairly easy to give intellectual assent to
the strictures of the last paragraph, I believe many will discover in
themselves a longing for mechanical explanation which has all the
tenacity of original sin. The discovery of such a desire need not
occasion any particular alarm, because it is easy to see how the demand
for this sort of explanation has had its origin in the enormous
preponderance of the mechanical in our physical experience. But
nevertheless, just as the old monks struggled to subdue the flesh, so
must the physicist struggle to subdue this sometimes nearly
irresistible, but perfectly unjustifiable desire. One of the large
purposes of this exposition will be attained if it carries the
conviction that this longing is unjustifiable, and is worth making the
effort to subdue.
The situation with respect to action at a distance is typical of the
general situation. I believe the essence of the explanatory process is
such that we must be prepared to accept as an ultimate for our
explanations the mere statement of a correlation between phenomena or
situations with which we are sufficiently familiar. Thus, in quantum
theory, there is no reason why we should not be willing to accept as an
ultimate the fundamental fact that when an electron jumps radiation is
emitted, provided always that we can give independent meaning in terms
of operations to the jumping of an electron. If there is no experiment
suggesting other and intermediate phenomena, we ought to be able to rest
intellectually satisfied with this. Of course it is quite a different
matter, and entirely justified, to imagine what the assumption of finer
details in the process would involve experimentally, and then to seek
for these possible new experimental facts.
It is a consequence of this view that any correlation is adapted to be
an absolutely final element of explanation, and can never be superseded
by the discovery of new experimental facts, if the correlation is by
_definition_ beyond the reach of further experiment. Such a possibility,
for example, is contained in a correlation between the numerical
magnitude of the gravitational constant and the total mass of the
universe. Something of this sort may be well attempted by those who
desire their explanations to take a formally final shape. We shall
return to this subject later.
The instinctive demand for a mechanism is fortified by observation of
the many important cases in which mechanisms have been discovered or
invented. However, the significance of such successful attempts must be
subject to most careful scrutiny. The matter has been discussed by
Poincaré,[6] who showed that not only is it always possible to find a
mechanistic explanation of any phenomenon (Hertz's program was a
perfectly possible one), but there are always an infinite number of such
explanations. This is very unsatisfactory.
[Footnote 6: Henri Poincaré. Wissenschaft and Hypothese, Translated
into German by F. and L. Lindemann, Teubner, Leipzig, 1906. See
especially p. 217.]
We want to be able to find the _real_ mechanism. Now an examination of
specific proposed mechanisms will show that most mechanisms are more
complicated than the simple physical phenomenon which they are invented
to explain, in that they have more independently variable attributes
than the phenomenon has been yet proved to have. An example is afforded
by the mechanical models invented to facilitate the study of the
properties of simple inductive electrical circuits. The great number of
such models which have been proposed is sufficient indication of their
possible infinite number. But if the mechanism has more independently
variable attributes than the original phenomenon, it is obvious that the
question is without meaning whether the mechanism is the real one or
not, for in the mechanism there must be simple motions or combinations
of motions which have no counterpart in features of the original
phenomenon as yet discovered. Obviously, then, the operations do not
exist by which we may set up a one to one correspondence between the
properties of the mechanism and the natural phenomenon, and the question
of reality has no meaning. If, then, a mechanism is to be taken
seriously as actually corresponding to reality, we must demand that it
have no more degrees of freedom than the original phenomenon, and we
must also be sure that the phenomenon has no undiscovered features.
Physical experience shows that such conditions are most difficult to
meet, and indeed the probability is that they are impossible.
A mechanism with more independently variable attributes than the
phenomenon may prove to be a very useful tool of thought, and therefore
worth inventing and studying, but it is to be regarded no more seriously
than is a mnemonic device, or any other artifice by which a man forces
his mind to give him better service.
There is another possible program of explanation, the converse of that
considered above, namely, to explain all familiar facts of ordinary
experience in terms of less familiar facts found at a deeper level. The
most striking example of this is the recent attempt to give a complete
electrical explanation of the universe. The original attempt was to
explain electrical effects in mechanical terms; this attempt failed. At
about the same time the existence of the electron was experimentally
established, so that it was evident that electricity is a very
fundamental constituent of matter. The program of explanation was
reversed, and an electrical explanation sought for all mechanical
phenomena, including in particular mechanical mass. But this attempt has
also failed; we recognize that part of mass may be non-electrical in
character, we postulate non-electrical forces inside the electron, and
further, we usually postulate for electrons and protons the property of
impenetrability, a property derived from experience on a higher scale of
magnitude.
A program of this general sort is likely to be regarded with
considerable sympathy, and indeed the chances of success seem much
greater than do those of the converse program, for in our experience
large scale phenomena are more often built up from and analyzed into
small scale phenomena than the converse. But as a matter of principle we
must again recognize that the only appeal is to experiment, and that we
have to ask just one question: "Is it true, as a matter of fact, that
all large scale phenomena can be built up of elements of small scale
phenomena?" It seems to me that the experimental warrant for this
conviction has not yet been given. The failure of the attempted
electrical explanation of the universe is a case in point. However, the
failure to prove a proposition is no guarantee that some time it may not
be proved, and many physicists are convinced of the ultimate feasibility
of this program. Personally I feel that the large may not always be
analyzed into the smaller; the subject will be discussed again.
A conviction of the significance of microscopic analysis has many
features in common with the usual conviction of the ultimate simplicity
of nature. The thesis of simplicity involves in addition the assumption
that the kinds of small scale elements are few in number, but actually
this involves no important difference between the two convictions,
because we have seen that the elements of which we build our structure
become fewer in number as we approach the limit of the experimentally
attainable. We may properly grant to convictions of this sort pragmatic
value in suggesting new correlations and experiments, but a recognition
of the empirical basis of all physics will not allow us to go further.
MODELS AND CONSTRUCTS
In discussing the concept of length, we could find no meaning in
questions such as: "Is space on a scale of 10^-8 cm. Euclidean?"
Nevertheless it will seem to many that they do attach a perfectly
definite meaning to a question of this kind. Of course it must be agreed
that magnitudes of 10^-8 cm. cannot be thought of in terms of immediate
sensation. When one thinks of an atom as a thing with any geometrical
properties at all, I believe he will find that what he essentially does
is to imagine a model, multiplying all the hypothetical dimensions by a
factor large enough to bring it to a magnitude of ordinary experience.
This large scale model is given properties corresponding to those of the
physical thing. For example, the model of the atom which was accepted in
the fall of 1925 contains electrons rotating in orbits, and every now
and then an electron jumps from one orbit to another, and simultaneously
energy is radiated from the atom. Such a model is satisfactory if it
offers the counterpart of all the phenomena of the original atom. Now I
believe the only meaning that any one can find in his statement that the
space of the atom is Euclidean is that he believes that he can construct
in Euclidean space a model with all the observed properties of the atom.
This possibility may or may not be sufficient to give real physical
significance to the statement that the space of the atom is Euclidean.
The situation here is very much the same as it was with respect to
mechanisms. The model may have many more properties than correspond to
measurable properties of the atom, and in particular, the operations by
which the space of the model is tested for its Euclidean character may
[and as a matter of fact I believe _do_] not have any counterpart in
operations which can be carried out on the atom. Further, we cannot
attach any _real_ significance to the statement that the space of the
atom is Euclidean unless we can show that no model constructed in
non-Euclidean space can reproduce the measurable properties of the atom.
In spite of all this, I believe that the model is a useful and indeed
unescapable tool of thought, in that it enables us to think about the
unfamiliar in terms of the familiar. There are, however, dangers in its
use: it is the function of criticism to disclose these dangers, so that
the tool may be used with confidence.
Closely related to the mental model are mental constructs, of which
physics is full. There are many sorts of constructs: those in which we
are interested are made by us to enable us to deal with physical
situations which we cannot directly experience through our senses, but
with which we have contact indirectly and by inference. Such constructs
usually involve the element of invention to a greater or less degree. A
construct containing very little of invention is that of the inside of
an opaque solid body. We can never experience directly through our
senses the inside of such a solid body, because the instant we directly
experience it, it ceases by definition to be the inside. We have here a
construct, but so natural a one as to be practically unavoidable. An
example of a construct involving a greater amount of invention is the
stress in an elastic body. A stress is by definition a property of the
interior points of a body which is connected mathematically in a simple
way with the forces acting across the free surface of the body. A stress
is then, by its very nature, forever beyond the reach of direct
experience, and it is therefore a construct. The entire structure of a
stress corresponds to nothing in direct experience; it is related to
force, but is itself a six-fold magnitude, whereas a force is only
three-fold.
We have next to ask whether the stress, which we have invented to meet
the situation in a body exposed to forces, is a good construct. In the
first place, a stress has the same number of degrees of freedom as the
observable phenomenon, for it is one of the propositions of the
mathematical theory of elasticity that the boundary conditions, which
are the experimental variables, uniquely determine the stress in a
_given_ body [_i.e._ a body of given elastic constants]; and of course
it is at once obvious, by an inspection of the equations, that
conversely a possible stress system uniquely determines the boundary
conditions to the significant amount. There is, therefore, a unique
one-to-one correspondence between a stress and the physical situation it
was made to meet, and so far a stress is a good construct. Up to this
point a stress, from the point of view of the operations in terms of
which it is defined, is a purely mathematical invention, which is
justified because it is convenient in describing the behavior of bodies
subjected to the action of force. But we wish now to go farther and
ascribe physical reality to a stress, meaning by this that a stress in a
solid body shall correspond to some real physical state of the interior
points. Let us examine, from the point of view of operations, what the
meaning of a statement like this may be. Since we now wish to ascribe an
additional physical meaning to a stress beyond that of the mathematical
operations in terms of which the stress was determined, there must exist
additional physical operations corresponding to this meaning, or else
our statement is meaningless. Now of course it is a matter of the most
elementary experience that physical phenomena do exist which allow these
other independent operations. A body under stress is also in a state of
strain, which may be determined from the external deformations, or the
strain at internal points may be made more vividly real by those optical
effects of double refraction in transparent bodies which are now so
extensively used in model experiments, or if the stress is pushed beyond
a certain point, we have such new phenomena as permanent set or finally,
rupture.
We have, then, every reason to be satisfied with our construct of
stress. In the first place, from the formal point of view, it is a good
construct because there is a unique correspondence between it and the
physical data in terms of which it is defined; and in the second place
we have a right to ascribe physical reality to it because it is uniquely
connected with other physical phenomena, _independent of those which
entered its definition_. This last requirement, in fact, from the
operational point of view, amounts to nothing more than a definition of
what we mean by the reality of things not given directly by experience.
Since now in addition to satisfying the formal requirements, experience
shows that a stress is most useful in correlating phenomena, we are
justified in giving to this construct of stress a prominent place among
our concepts.
Consider now another construct, one of the most important of physics,
that of the electric field. In the first place, an examination of the
operations by which we determine the electric field at any point will
show that it is a construct, in that it is not a direct datum of
experience. To determine the electric field at a point, we place an
exploring charge at the point, measure the force on it, and then
calculate the ratio of the force to the charge. We then allow the
exploring charge to become smaller and smaller, repeating our
measurement of force on each smaller charge, and define the limit of the
ratio of the force to the charge as the electric field intensity at the
point in question, and the limiting direction of the force on a small
charge as the direction of the field. We may extend this process to
every point of space, and so obtain the concept of a field of force, by
which every point of the space surrounding electric charges is tagged
with the appropriate number and direction, the exploring charge having
completely disappeared. The field is, then, clearly a construct. Next,
from the formal point of view of mathematics, it is a good construct,
because there is a one to one correspondence between the electric field
and the electric charges in terms of which it is defined, the field
being uniquely determined by the charges, and conversely there being
only one possible set of charges corresponding to a given field. Now
nearly every physicist takes the next step, and ascribes physical
reality to the electric field, in that he thinks that at every point of
the field there is some real physical phenomenon taking place which is
connected in a way not yet precisely determined with the number and
direction which tag the point. At first this view most naturally
involved as a corollary the existence of a medium, but lately it has
become the fashion to say that the medium does not exist, and that only
the field is real. The reality of the field is self-consciously
inculcated in our elementary teaching, often with considerable
difficulty for the student. This view is usually credited to Faraday,
and is considered the most fundamental concept of all modern electrical
theory. Yet in spite of this, I believe that a critical examination will
show that the ascription of physical reality to the electric field is
entirely without justification. I cannot find a single physical
phenomenon or a single physical operation by which evidence of the
existence of the field may be obtained independent of the operations
which entered the definition. The only physical evidence we ever have of
the existence of a field is obtained by going there with an electric
charge and observing the action on the charge [when the charges are
inside atoms we may have optical phenomena], which is precisely the
operation of the definition. It is then either meaningless to say that a
field has physical reality, or we are guilty of adopting a definition of
reality which is the crassest tautology.
There can be no question whatever of the tremendous importance of the
concept of the electric field as a tool in thinking about, describing,
correlating, and predicting the properties of electrical systems;
electrical science is inconceivable without this or something
equivalent. But in addition to this aspect of the field concept, the
further tacit implication of physical reality is almost always present,
and has had the greatest influence on the character of physical thought
and experiment. Yet I do not believe that the additional implication of
physical reality has justified itself by bringing to light a single
positive result, or can offer more than the pragmatic plea of having
stimulated a large number of experiments, all with persistently negative
results. It is sufficient to mention the fate of the attempt of Faraday
and Maxwell to ascribe a stress like that of ordinary matter to the
ether, which failed because, among other reasons, nothing can exist in
the ether analogous to the _strain_ of ordinary matter, to indicate the
unfruitfulness of the idea of physical reality. It seems to me that any
pragmatic justification in postulating reality for the electric field
has now been exhausted, and that we have reached a stage where we should
attempt to get closer to the actual facts by ridding the field concept
of the implications of reality.
Another indispensable and most interesting construct is that of the
atom. This is evidently a construct, because no one ever directly
experienced an atom, and its existence is entirely inferential. The atom
was invented to explain constant combining weights in chemistry. For a
long time there was no other experimental evidence of its existence, and
it remained a pure invention, without physical reality, useful in
discussing a certain group of phenomena. It is one of the most
fascinating things in physics to trace the accumulation of independent
new physical information all pointing to the atom, until now we are as
convinced of its physical reality as of our hands and feet.
A construct which had to be abandoned because it did not turn out to
have physical reality, and which furthermore was not sufficiently useful
in the light of newly discovered phenomena, was that of a caloric fluid.
The notion of "physical reality" is not of prime importance to this
discussion of the character of our constructs; our definition of the
meaning of physical reality may not appeal to everyone. The essential
point is that our constructs fall into two classes: those to which no
physical operations correspond other than those which enter the
definition of the construct, and those which admit of other operations,
or which could be defined in several alternative ways in terms of
physically distinct operations. This difference in the character of
constructs may be expected to correspond to essential physical
differences, and these physical differences are much too likely to be
overlooked in the thinking of physicists. We must always be on our guard
not to forget the physical differences between a thing like a stress in
an elastic body and an electromagnetic field.
The moral of all this is that constructs are most useful and even
unavoidable things, but that they may have great dangers, and that a
careful critique may be necessary to avoid reading into them
implications for which there is no warrant in experience, and which may
most profoundly affect our physical outlook and course of action.
THE RÔLE OF MATHEMATICS IN PHYSICS
Practically all the formulations of theoretical physics are made in
mathematical terms; in fact to obtain such formulations is generally
felt to be the goal of theoretical physics. It is then evidently
pertinent to consider what the nature of the mathematics is to which we
assign so prominent a rôle.
We have in the first place to understand why it is possible to express
physical relations in mathematical language at all. I am not sure that
there is much meaning in this question. It is the merest truism, evident
at once to unsophisticated observation, that mathematics is a human
invention. Furthermore, the mathematics in which the physicist is
interested was developed for the explicit purpose of describing the
behavior of the external world, so that it is certainly no accident that
there is a correspondence between mathematics and nature. The
correspondence is not by any means perfect, however, but there is always
in mathematics a precise quality to which none of our information about
nature ever attains. The theorems of Euclid's geometry illustrate this
in a preeminent degree. The statement that there is just one straight
line between two points and that this is the shortest possible path
between the points is entirely different in character from any
information ever given by physical measurement, for all our measurements
are subject to error. It is possible, nevertheless, to give a certain
real physical meaning to the ideally precise statements of geometry,
because it is a result of everyday experience that as we refine the
accuracy of our physical measurements the quantitative statements of
geometry are verified within an ever decreasing margin of error. From
this arises that view of the nature of mathematics which apparently is
most commonly held; namely that if we could eliminate the imperfections
of our measurements, the relations of mathematics would be exactly
verified. Abstract mathematical principles are supposed to be active in
nature, controlling natural phenomena, as Pythagoras long ago tried to
express with his harmony of the spheres and the mystic relations of
numbers.
This idealized view of the connection of mathematics with nature could
be maintained only during that historical period when the accuracy of
physical measurement was low, and must now be abandoned. For it is no
longer true that the precise relations of Euclid's geometry may be
indefinitely approximated to by increasing the refinements of the
measuring process, but there are essential physical limitations to the
very concepts of length, etc., which enter the geometrical formulations,
set by the discrete structure of matter and of radiation. This is no
academic matter, but touches the essence of the situation. There is no
longer any basis for the idealization of mathematics, and for the view
that our imperfect knowledge of nature is responsible for failure to
find in nature the precise relations of mathematics. It is the
mathematics made by us which is imperfect and not our knowledge of
nature. [From the operational point of view it is meaningless to attempt
to separate "nature" from "knowledge of nature".] The concepts of
mathematics are inventions made by us in the attempt to describe nature.
Now we shall repeatedly see that it is the most difficult thing in the
world to invent concepts which exactly correspond to what we know about
nature, and we apparently never achieve success. Mathematics is no
exception; we doubtless come closer to the ideal here than anywhere
else, but we have seen that even arithmetic does not completely
reproduce the physical situation.
Mathematics appears to fail to correspond exactly to the physical
situation in at least two respects. In the first place, there is the
matter of errors of measurement in the range of ordinary experience. Now
mathematics can deal with this situation, although somewhat clumsily,
and only approximately, by specifically supplementing its equations by
statements about the limit of error, or replacing equations by
inequalities--in short, the sort of thing done in every discussion of
the propagation of error of measurement. In the second place, and much
more important, mathematics does not recognize that as the physical
range increases, the fundamental concepts become hazy, and eventually
cease entirely to have physical meaning, and therefore must be replaced
by other concepts which are operationally quite different. For instance,
the equations of motion make no distinction between the motion of a star
into our galaxy from external space, and the motion of an electron about
the nucleus, although physically the meaning in terms of operations of
the quantities in the equations is entirely different in the two cases.
The structure of our mathematics is such that we are almost forced,
whether we want to or not, to talk about the inside of an electron,
although physically we cannot assign any meaning to such statements. As
at present constructed, mathematics reminds one of the loquacious and
not always coherent orator, who was said to be able to set his mouth
going and go off and leave it. What we would like is some development of
mathematics by which the equations could be made to cease to have
meaning outside the range of numerical magnitude in which the physical
concepts themselves have meaning. In other words, the problem is to make
our equations correspond more closely to the physical experience back of
them; it evidently needs some sort of new invention to accomplish this.
We return later, in discussing Lorentz's equations of electrodynamics,
to the disadvantages arising from the present undiscriminating character
of mathematics. In the meantime, we must recognize that there are very
important advantages here, as well as disadvantages. All experience
justifies the expectation that the laws of nature with which we are
already familiar hold at least approximately and without violent change
in the unexplored regions immediately beyond our present reach. By
assuming an unlimited validity for the laws as we now know them,
mathematics enables us to penetrate the twilight zone, and make
predictions which may be later verified. It is only when we are carried
too far afield that we must deprecate this characteristic of our
mathematics.
There is another aspect of the use of mathematics in describing nature
that is often lost sight of; namely, that any system of equations can
contain only a very small part of the actual physical situation; there
is behind the equations an enormous descriptive background through which
the equations make connection with nature. This background includes a
description of all the physical operations by which the data are
obtained which enter the equations. For instance, when Einstein
formulates the behavior of the universe in terms of the world lines of
events, the events as they enter the equations are entirely colorless
things, merely 3 space and 1 time coördinate. To make connection with
experience there must be a descriptive background giving the physical
contents of the events; for example, there may be the statement that
some of the events are light signals. This descriptive background is
supposed to remain fixed, unaffected by any operations to which the
equations themselves are subject. If, for example, the frame of
reference of the equations is altered by changing its velocity, the
physical significance of the descriptive background is supposed to
remain unaltered, or rather no mention is usually made of this question
at all. It would seem, however, that this matter needs some discussion.
The descriptive background gets its meaning only in terms of certain
physical operations. If the descriptive background remains unaltered
when the uniform velocity of the frame of reference is changed, for
instance, this means that the motion of the frame of reference does not
at all affect the possibility of carrying out certain operations. This
is pretty close to the restricted principle of relativity itself, which
states that the form of natural laws is not affected by uniform
velocity. Until a more careful analysis of the situation is made it
would seem therefore that there is some ground for the suspicion that
the principle of relativity is involved in the possibility of giving to
physical phenomena a _complete_ mathematical formulation, understanding
"complete" to mean "including the descriptive background."