Title: The logic of modern physics
Author: Bridgman, Percy Williams
Language: English
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                        THE LOGIC OF
                       MODERN PHYSICS



                             BY

                      P. W. BRIDGMAN

       HOLLIS PROFESSOR OF MATHEMATICS AND NATURAL HISTORY
                       IN HARVARD UNIVERSITY



                          NEW YORK
                   THE MACMILLAN COMPANY



                       COPYRIGHT, 1927,
                  By THE MACMILLAN COMPANY.



PREFACE


This excursion into the field of fundamental criticism by one whose
activities have hitherto been confined almost entirely to experiment is
not evidence of senile decay, as might be cynically assumed. I have
always, throughout all my experimental work, felt an imperative need of
a better understanding of the foundations of our physical thought and
have for a long time made more or less unsystematic attempts to reach
such an understanding. Only now, however, has a half sabbatical year
given me leisure to attempt a more or less orderly exposition.

In spite of previous writings on the broad fundamentals by Clifford,
Stallo, Mach, and Poincaré, to mention only a few, I believe a new
essay of this critical character needs no apology. For entirely apart
from the question of whether many of the points of view of these essays
can be maintained, the discovery of new facts in the domain of
relativity and quantum theory has shifted the center of interest and
emphasis. All the quite recent activity with the new quantum mechanics
seems to call for a new examination of fundamental matters which shall
recognize, at least by implication, the existence of the special
phenomena of the quantum domain. However, the necessity for
re-examination does not mean at all that many of the results of previous
criticism may not still be accepted; some of these results have become
so thoroughly incorporated into physical thinking that we can assume
them without mention. Thus the fundamental attitude of this essay is
empiricism, which is now justified as the attitude of the physicist in
large part by the inquiry into the physiological origin of our concepts
of space, time, and mechanics with which the previous essays were
largely concerned.

None of the previous essays have consciously or immediately affected the
details of this; in fact I have not read any of them within several
years. If passages here recall passages already written, it is because
the ideas have been assimilated and the precise origin forgotten; it is
probably worth while to let such passages stand without revision,
because such ideas gain in plausibility through having been found
acceptable to independent thought.

I am much indebted to Professor R. F. Alfred Hoernlé of the Department
of Philosophy of Johannesburg University, South Africa, for suggesting
several modifications to make the text more acceptable to a philosopher,
and slight amplifications for the benefit of readers not familiar with
all the details of recent technical developments in physics.



INTRODUCTION


One of the most noteworthy movements in recent physics is a change of
attitude toward what may be called the interpretative aspect of physics.
It is being increasingly recognized, both in the writings and the
conversation of physicists, that the world of experiment is not
understandable without some examination of the purpose of physics and of
the nature of its fundamental concepts. It is no new thing to attempt a
more critical understanding of the nature of physics, but until recently
all such attempts have been regarded with a certain suspicion or even
sometimes contempt. The average physicist is likely to deprecate his own
concern with such questions, and is inclined to dismiss the speculations
of fellow physicists with the epithet "metaphysical." This attitude has
no doubt had a certain justification in the utter unintelligibility to
the physicist of many metaphysical speculations and the sterility of
such speculations in yielding physical results. However, the growing
reaction favoring a better understanding of the interpretative
fundamentals of physics is not a pendulum swing of the fashion of
thought toward metaphysics, originating in the upheaval of moral values
produced by the great war, or anything of the sort, but is a reaction
absolutely forced upon us by a rapidly increasing array of cold
experimental facts.

This reaction, or rather new movement, was without doubt initiated by
the restricted theory of relativity of Einstein. Before Einstein, an
ever increasing number of experimental facts concerning bodies in rapid
motion required increasingly complicated modifications in our naïve
notions in order to preserve self-consistency, until Einstein showed
that everything could be restored again to a wonderful simplicity by a
slight change in some of our fundamental concepts. The concepts which
were most obviously touched by Einstein were those of space and time,
and much of the writing consciously inspired by Einstein has been
concerned with these concepts. But that experiment compels a critique of
much more than the concepts of space and time is made increasingly
evident by all the new facts being discovered in the quantum realm.

The situation presented to us by these new quantum facts is twofold. In
the first place, all these experiments are concerned with things so
small as to be forever beyond the possibility of direct experience, so
that we have the problem of translating the evidence of experiment into
other language. Thus we observe an emission line in a spectroscope and
may infer an electron jumping from one energy level to another in an
atom. In the second place, we have the problem of understanding the
translated experimental evidence. Now of course every one knows that
this problem is making us the greatest difficulty. The experimental
facts are so utterly different from those of our ordinary experience
that not only do we apparently have to give up generalizations from past
experience as broad as the field equations of electrodynamics, for
instance, but it is even being questioned whether our ordinary forms of
thought are applicable in the new domain; it is often suggested, for
example, that the concepts of space and time break down.

The situation is rapidly becoming acute. Since I began writing this
essay, there has been a striking increase in critical activity inspired
by the new quantum mechanics of 1925-26, and it is common to hear
expositions of the new ideas prefaced by analysis of what experiment
really gives to us or what our fundamental concepts really mean. The
change in ideas is now so rapid that a number of the statements of this
essay are already antiquated as expressions of the best current opinion;
however I have allowed these statements to stand, since the fundamental
arguments are in nowise affected and we have no reason to think that
present best opinions are in any way final. We have the impression of
being in an important formative period; if we are, the complexion of
physics for a long time in the future will be determined by our present
attitude toward fundamental questions of interpretation. To meet this
situation it seems to me that something more is needed than the
hand-to-mouth philosophy that is now growing up to meet special
emergencies, something approaching more closely to a systematic
philosophy of all physics which shall cover the experimental domains
already consolidated as well as those which are now making us so much
trouble. It is the attempt of this essay to give a more or less
inclusive critique of all physics. Our problem is the double one of
understanding what we are trying to do and what our ideals should be in
physics, and of understanding the nature of the structure of physics as
it now exists. These two ends are together furthered by an analysis of
the fundamental concepts of physics; an understanding of the concepts we
how have discloses the present structure of physics and a realization of
what the concepts should be involves the ideals of physics. This essay
will be largely concerned with the fundamental concepts; it will appear
that almost all the concepts can profit from re-examination.

The material of this essay is largely obtained by observation of the
actual currents of opinion in physics; much of what I have to say is
more or less common property and doubtless every reader will find
passages that he will feel have been taken out of his own mouth. On
certain broad tendencies in present day physics, however, I have put my
own interpretation, and it is more than likely that this interpretation
will be unacceptable to many. But even if not acceptable, I hope that
the stimulus of combatting the ideas offered here may be of value.

Certain limitations will have to be set to our inquiry in order to keep
it within manageable compass. It is of course the merest truism that all
our experimental knowledge and our understanding of nature is impossible
and non-existent apart from our own mental processes, so that strictly
speaking no aspect of psychology or epistemology is without pertinence.
Fortunately we shall be able to get along with a more or less naive
attitude toward many of these matters. We shall accept as significant
our common sense judgment that there is a world external to us, and
shall limit as far as possible our inquiry to the behavior and
interpretation of this "external" world. We shall rule out inquiries
into our states of consciousness as such. In spite, however, of the best
intentions, we shall not be able to eliminate completely considerations
savoring of the metaphysical, because it is evident that the nature of
our thinking mechanism essentially colors any picture that we can form
of nature, and we shall have to recognize that unavoidable
characteristics of any outlook of ours are imposed in this way.



CONTENTS

Preface

Introduction

CHAPTER

I. BROAD POINTS OF VIEW

    New Kinds of Experience Always Possible
    The Operational Character of Concepts
      Einstein's Contribution in Changing Our Attitude
          toward Concepts
      Detailed Discussion of the Concept of Length
      The Relative Character of Knowledge
      Meaningless Questions
    General Comments on the Operational Point of
          View

II. OTHER GENERAL CONSIDERATIONS
    The Approximate Character of Empirical Knowledge
    Explanations and Mechanisms
    Models and Constructs
    The Rôle of Mathematics in Physics

III. DETAILED CONSIDERATION OF VARIOUS CONCEPTS
        OF PHYSICS

    The Concept of Space
    The Concept of Time
    The Causality Concept
    The Concept of Identity
    The Concept of Velocity
    The Concepts of Force and Mass
    The Concept of Energy
    The Concepts of Thermodynamics
    Electrical Concepts
    The Nature of Light and the Concepts of Relativity
    Other Relativity Concepts
    Rotational Motion and Relativity
    Quantum Concepts

IV. SPECIAL VIEWS OF NATURE

    The Simplicity of Nature
    Determinism
    On the Possibility of Describing Nature Completely
          in Terms of Analysis
    A Glimpse Ahead

   INDEX



THE LOGIC OF MODERN PHYSICS



CHAPTER I

BROAD POINTS OF VIEW


WHATEVER may be one's opinion as to our permanent acceptance of the
analytical details of Einstein's restricted and general theories of
relativity, there can be no doubt that through these theories physics is
permanently changed. It was a great shock to discover that classical
concepts, accepted unquestioningly, were inadequate to meet the actual
situation, and the shock of this discovery has resulted in a critical
attitude toward our whole conceptual structure which must at least in
part be permanent. Reflection on the situation after the event shows
that it should not have needed the new experimental facts which led to
relativity to convince us of the inadequacy of our previous concepts,
but that a sufficiently shrewd analysis should have prepared us for at
least the possibility of what Einstein did.

Looking now to the future, our ideas of what external nature is will
always be subject to change as we gain new experimental knowledge, but
there is a part of our attitude to nature which should not be subject to
future change, namely that part which rests on the permanent basis of
the character of our minds. It is precisely here, in an improved
understanding of our mental relations to nature, that the permanent
contribution of relativity is to be found. We should now make it our
business to understand so thoroughly the character of our permanent
mental relations to nature that another change in our attitude, such as
that due to Einstein, shall be forever impossible. It was perhaps
excusable that a revolution in mental attitude should occur once,
because after all physics is a young science, and physicists have been
very busy, but it would certainly be a reproach if such a revolution
should ever prove necessary again.



NEW KINDS OF EXPERIENCE ALWAYS POSSIBLE


The first lesson of our recent experience with relativity is merely an
intensification and emphasis of the lesson which all past experience has
also taught, namely, that when experiment is pushed into new domains, we
must be prepared for new facts, of an entirely different character from
those of our former experience. This is taught not only by the discovery
of those unsuspected properties of matter moving with high velocities,
which inspired the theory of relativity, but also even more emphatically
by the new facts in the quantum domain. To a certain extent, of course,
the recognition of all this does not involve a change of former
attitude; the _fact_ has always been for the physicist the one ultimate
thing from which there is no appeal, and in the face of which the only
possible attitude is a humility almost religious. The new feature in the
present situation is an intensified conviction that in reality new
orders of experience do exist, and that we may expect to meet them
continually. We have already encountered new phenomena in going to high
velocities, and in going to small scales of magnitude: we may similarly
expect to find them, for example, in dealing with relations of cosmic
magnitudes, or in dealing with the properties of matter of enormous
densities, such as is supposed to exist in the stars.

Implied in this recognition of the possibility of new experience beyond
our present range, is the recognition that no element of a physical
situation, no matter how apparently irrelevant or trivial, may be
dismissed as without effect on the final result until proved to be
without effect by actual experiment.

The attitude of the physicist must therefore be one of pure empiricism.
He recognizes no _a priori_ principles which determine or limit the
possibilities of new experience. Experience is determined only by
experience. This practically means that we must give up the demand that
all nature be embraced in any formula, either simple or complicated. It
may perhaps turn out eventually that as a matter of fact nature can be
embraced in a formula, but we must so organize our thinking as not to
demand it as a necessity.



THE OPERATIONAL CHARACTER OF CONCEPTS

_Einstein's Contribution in Changing Our Attitude
Toward Concepts_


Recognizing the essential unpredictability of experiment beyond our
present range, the physicist, if he is to escape continually revising
his attitude, must use in describing and correlating nature concepts of
such a character that our present experience does not exact hostages of
the future. Now here it seems to me is the greatest contribution of
Einstein. Although he himself does not explicitly state or emphasize it,
I believe that a study of what he has done will show that he has
essentially modified our view of what the concepts useful in physics are
and should be. Hitherto many of the concepts of physics have been
defined in terms of their properties. An excellent example is afforded
by Newton's concept of absolute time. The following quotation from the
Scholium in Book I of the _Principia_ is illuminating:


I do not define Time, Space, Place or Motion, as being well known to
all. Only I must observe that the vulgar conceive those quantities under
no other notions but from the relation they bear to sensible objects.
And thence arise certain prejudices, for the removing of which, it will
be convenient to distinguish them into Absolute and Relative, True and
Apparent, Mathematical and Common.

(I) Absolute, True, and Mathematical Time, of itself, and from its own
nature flows equably without regard to anything external, and by another
name is called Duration.


Now there is no assurance whatever that there exists in nature anything
with properties like those assumed in the definition, and physics, when
reduced to concepts of this character, becomes as purely an abstract
science and as far removed from reality as the abstract geometry of the
mathematicians, built on postulates. It is a task for experiment to
discover whether concepts so defined correspond to anything in nature,
and we must always be prepared to find that the concepts correspond to
nothing or only partially correspond. In particular, if we examine the
definition of absolute time in the light of experiment, we find nothing
in nature with such properties.

The new attitude toward a concept is entirely different. We may
illustrate by considering the concept of length: what do we mean by the
length of an object? We evidently know what we mean by length if we can
tell what the length of any and every object is, and for the physicist
nothing more is required. To find the length of an object, we have to
perform certain physical operations. The concept of length is therefore
fixed when the operations by which length is measured are fixed: that
is, the concept of length involves as much as and nothing more than the
set of operations by which length is determined. In general, we mean by
any concept nothing more than a set of operations; _the concept is
synonymous with the corresponding set of operations_. If the concept is
physical, as of length, the operations are actual physical operations,
namely, those by which length is measured; or if the concept is mental,
as of mathematical continuity, the operations are mental operations,
namely those by which we determine whether a given aggregate of
magnitudes is continuous. It is not intended to imply that there is a
hard and fast division between physical and mental concepts, or that one
kind of concept does not always contain an element of the other; this
classification of concept is not important for our future
considerations.

We must demand that the set of operations equivalent to any concept be a
unique set, for otherwise there are possibilities of ambiguity in
practical applications which we cannot admit.

Applying this idea of "concept" to absolute time, we do not understand
the meaning of absolute time unless we can tell how to determine the
absolute time of any concrete event, _i.e._, unless we can measure
absolute time. Now we merely have to examine any of the possible
operations by which we measure time to see that all such operations are
relative operations. Therefore the previous statement that absolute time
does not exist is replaced by the statement that absolute time is
meaningless. And in making this statement we are not saying something
new about nature, but are merely bringing to light implications already
contained in the physical operations used in measuring time.

It is evident that if we adopt this point of view toward concepts,
namely that the proper definition of a concept is not in terms of its
properties but in terms of actual operations, we need run no danger of
having to revise our attitude toward nature. For if experience is always
described in terms of experience, there must always be correspondence
between experience and our description of it, and we need never be
embarrassed, as we were in attempting to find in nature the prototype of
Newton's absolute time. Furthermore, if we remember that the operations
to which a physical concept are equivalent are actual physical
operations, the concepts can be defined only in the range of actual
experiment, and are undefined and meaningless in regions as yet
untouched by experiment. It follows that strictly speaking we cannot
make statements at all about regions as yet untouched, and that when we
do make such statements, as we inevitably shall, we are making a
conventionalized extrapolation, of the looseness of which we must be
fully conscious, and the justification of which is in the experiment of
the future.

There probably is no statement either in Einstein or other writers that
the change described above in the use of "concept" has been
self-consciously made, but that such is the case is proved, I believe,
by an examination of the way concepts are now handled by Einstein and
others. For of course the true meaning of a term is to be found by
observing what a man does with it, not by what he says about it. We may
show that this is the actual sense in which concept is coming to be used
by examining in particular Einstein's treatment of simultaneity.

Before Einstein, the concept of simultaneity was defined in terms of
properties. It was a property of two events, when described with respect
to their relation in time, that one event was either before the other,
or after it, or simultaneous with it. Simultaneity was a property of the
two events alone and nothing else; either two events were simultaneous
or they were not. The justification for using this term in this way was
that it seemed to describe the behavior of actual things. But of course
experience then was restricted to a narrow range. When the range of
experience was broadened, as by going to high velocities, it was found
that the concepts no longer applied, because there was no counterpart in
experience for this absolute relation between two events. Einstein now
subjected the concept of simultaneity to a critique, which consisted
essentially in showing that the operations which enable two events to be
described as simultaneous involve measurements on the two events made by
an observer, so that "simultaneity" is, therefore, not an absolute
property of the two events and nothing else, but must also involve the
relation of the events to the observer. Until therefore we have
experimental proof to the contrary, we must be prepared to find that the
simultaneity of two events depends on their relation to the observer,
and in particular on their velocity. Einstein, in thus analyzing what is
involved in making a judgment of simultaneity, and in seizing on the act
of the observer as the essence of the situation, is actually adopting a
new point of view as to what the concepts of physics should be, namely,
the operational view.

Of course Einstein actually went much further than this, and found
precisely how the operations for judging simultaneity change when the
observer moves, and obtained quantitative expressions for the effect of
the motion of the observer on the relative time of two events. We may
notice, parenthetically, that there is much freedom of choice in
selecting the exact operations; those which Einstein chose were
determined by convenience and simplicity with relation to light beams.
Entirely apart from the precise quantitative relations of Einstein's
theory, however, the important point for us is that if we had adopted
the operational point of view, we would, before the discovery of the
actual physical facts, have seen that simultaneity is essentially a
relative concept, and would have left room in our thinking for the
discovery of such effects as were later found.



_Detailed Discussion of the Concept of Length_


We may now gain further familiarity with the operational attitude toward
a concept and some of its implications by examining from this point of
view the concept of length. Our task is to find the operations by which
we measure the length of any concrete physical object. We begin with
objects of our commonest experience, such as a house or a house lot.
What we do is sufficiently indicated by the following rough description.
We start with a measuring rod, lay it on the object so that one of its
ends coincides with one end of the object, mark on the object the
position of the other end of the rod, then move the rod along in a
straight line extension of its previous position until the first end
coincides with the previous position of the second end, repeat this
process as often as we can, and call the length the total number of
times the rod was applied. This procedure, apparently so simple, is in
practice exceedingly complicated, and doubtless a full description of
all the precautions that must be taken would fill a large treatise. We
must, for example, be sure that the temperature of the rod is the
standard temperature at which its length is defined, or else we must
make a correction for it; or we must correct for the gravitational
distortion of the rod if we measure a vertical length; or we must be
sure that the rod is not a magnet or is not subject to electrical
forces. All these precautions would occur to every physicist. But we
must also go further and specify all the details by which the rod is
moved from one position to the next on the object--its precise path
through space and its velocity and acceleration in getting from one
position to another. Practically of course, precautions such as these
are not mentioned, but the justification is in our experience that
variations of procedure of this kind are without effect on the final
result. But we always have to recognize that all our experience is
subject to error, and that at some time in the future we may have to
specify more carefully the acceleration, for example, of the rod in
moving from one position to another, if experimental accuracy should be
so increased as to show a measurable effect. In _principle_ the
operations by which length is measured should be _uniquely_ specified.
If we have more than one set of operations, we have more than one
concept, and strictly there should be a separate name to correspond to
each different set of operations.

So much for the length of a stationary object, which is complicated
enough. Now suppose we have to measure a moving street car. The
simplest, and what we may call the "naïve" procedure, is to board the
car with our meter stick and repeat the operations we would apply to a
stationary body. Notice that this procedure reduces to that already
adopted in the limiting case when the velocity of the street car
vanishes. But here there may be new questions of detail. How shall we
jump on to the car with our stick in hand? Shall we run and jump on from
behind, or shall we let it pick us up from in front? Or perhaps does now
the material of which the stick is composed make a difference, although
previously it did not? All these questions must be answered by
experiment. We believe from present evidence that it makes no difference
how we jump on to the car, or of what material the rod is made, and that
the length of the car found in this way will be the same as if it were
at rest. But the experiments are more difficult, and we are not so sure
of our conclusions as before. Now there are very obvious limitations to
the procedure just given. If the street car is going too fast, we can
not board it directly, but must use devices, such as getting on from a
moving automobile; and, more important still, there are limitations to
the velocity that can be given to street cars or to meter sticks by any
practical means in our control, so that the moving bodies which can be
measured in this way are restricted to a low range of velocity. If we
want to be able to measure the length of bodies moving with higher
velocities such as we find existing in nature (stars or cathode
particles), we must adopt another definition and other operations for
measuring length, which also reduce to the operations already adopted in
the static case. This is precisely what Einstein did. Since Einstein's
operations were different from our operations above, _his "length" does
not mean the same as our "length."_ We must accordingly be prepared to
find that the length of a moving body measured by the procedure of
Einstein is not the same as that above; this of course is the fact, and
the transformation formulas of relativity give the precise connection
between the two lengths.

Einstein's procedure for measuring the length of bodies in motion was
dictated not only by the consideration that it must be applicable to
bodies with high velocities, but also by mathematical convenience, in
that Einstein describes the world mathematically by a system of
coördinate geometry, and the "length" of an object is connected simply
with quantities in the analytic equations.

It is of interest to describe briefly Einstein's actual operations for
measuring the length of a body in motion; it will show how operations
which may be simple from a mathematical point of view may appear
complicated from a physical viewpoint. The observer who is to measure
the length of a moving object must first extend over his entire plane of
reference (for simplicity the problem is considered two-dimensional) a
system of time coordinates, _i.e._, at each point of his plane of
reference there must be a clock, and all these clocks must be
synchronized. At each clock an observer must be situated. Now to find
the length of the moving object at a specified instant of time (it is a
subject for later investigation to find whether its length is a function
of time), the two observers who happen to coincide in position with the
two ends of the object at the specified time on their clocks are
required to find the distance between their two positions by the
procedure for measuring the length of a stationary object, and this
distance is by definition the length of the moving object in the given
reference system. This procedure for measuring the length of a body in
motion hence involves the idea of simultaneity, through the simultaneous
position of the two ends of the rod, and we have seen that the
operations by which simultaneity are determined are relative, changing
when the motion of the system changes. We hence are prepared to find a
change in the length of a body when the velocity of the measuring system
changes, and this in fact is what happens. The precise numerical
dependence is worked out by Einstein, and involves other considerations,
in which we are not interested at present.

The two sorts of length, the naive one and that of Einstein, have
certain features in common. In either case in the limit, as the velocity
of the measuring system approaches zero, the operations approach those
for measuring the length of a stationary object. This, of course, is a
requirement in any good definition, imposed by considerations of
convenience, and it is too obvious a matter to need elaboration. Another
feature is that the operations equivalent to either concept both involve
the motion of the system, so that we must recognize the possibility that
the length of a moving object may be a function of its velocity. It is a
matter of experiment, unpredictable until tried, that within the limits
of present experimental error the naive length is not affected by
motion, and Einstein's length is.

So far, we have extended the concept of length in only one way beyond
the range of ordinary experience, namely to high velocities. The
extension may obviously be made in other directions. Let us inquire what
are the operations by which we measure the length of a very large
object. In practice we probably first meet the desirability of a change
of procedure in measuring large pieces of land. Here our procedure
depends on measurements with a surveyor's theodolite. This involves
extending over the surface of the land a system of coordinates, starting
from a base line measured with a tape in the conventional way, sighting
on distant points from the extremities of the line, and measuring the
angles. Now in this extension we have made one very essential change:
the angles between the lines connecting distant points are now angles
between beams of light. We assume that a beam of light travels in a
straight line. Furthermore, we assume in extending our system of
triangulation over the surface of the earth that the geometry of light
beams is Euclidean. We do the best we can to check the assumptions, but
at most can never get more than a partial check.

Thus Gauss[1] checked whether the angles of a large terrestrial triangle
add to two right angles and found agreement within experimental error.
We now know from the experiments of Michelson[2] that if his
measurements had been accurate enough he would not have got a check, but
would have had an excess or defect according to the direction in which
the beam of light travelled around the triangle with respect to the
rotation of the earth. But if the geometry of light beams is Euclidean,
then not only must the angles of a triangle add to two right angles, but
there are definite relations between the lengths of the sides and the
angles, and to check these relations the sides should be measured by the
old procedure with a meter stick. Such a check on a large scale has
never been attempted, and is not feasible. It seems, then, that our
checks on the Euclidean character of optical space are all of restricted
character. We have apparently proved that up to a certain scale of
magnitude optical space is Euclidean with respect to measures of angle,
but this may not necessarily involve that space is also Euclidean with
respect to measures of length, so that space need not be completely
Euclidean. There is a further most important restriction in that our
studies of non-Euclidean geometry have shown that the _percentage_
excess of the angles of a non-Euclidean triangle over 180° may depend
on the magnitude of the triangle, so that it may well be that we have
not detected the non-Euclidean character of space simply because our
measurements have not been on a large enough scale.

[Footnote 1: C. F. Gauss, Gesammelte Werke, especially vol. IV.]

[Footnote 2: See a discussion of the theory of this experiment by L.
Silberstein, Jour. Opt. Soc. Amer. 5, 291-307. 1921.]

We thus see that the concept of length has undergone a very essential
change of character even within the range of terrestrial measurements,
in that we have substituted for what I may call the tactual concept an
optical concept, complicated by an assumption about the nature of our
geometry. From a very direct concept we have come to a very indirect
concept with a most complicated set of operations. Strictly speaking,
length when measured in this way by light beams should be called by
another name, since the operations are different. The practical
justification for retaining the same name is that within our present
experimental limits a numerical difference between the results of the
two sorts of operations has not been detected.

We are still worse off when we make the extension to solar and stellar
distances. Here space is entirely optical in character, and we never
have an opportunity of even partially comparing tactual with optical
space. No direct measures of length have ever been made, nor can we even
measure the three angles of a triangle and so check our assumption that
the use of Euclidean geometry in extending the concept of space is
justified. We never have under observation more than two angles of a
triangle, as when we measure the distance of the moon by observation
from the two ends of the earth's diameter. To extend to still greater
distance our measures of length, we have to make still further
assumptions, such as that inferences from the Newtonian laws of
mechanics are valid. The accuracy of our inferences about lengths from
such measurements is not high. Astronomy is usually regarded as a
science of extraordinarily high accuracy, but its accuracy is very
restricted in character, namely to the measurement of angles. It is
probably safe to say that no astronomical distance, except perhaps that
of the moon, is known with an accuracy greater than 0.1%. When we push
our estimates to distances beyond the confines of the solar system in
which we are assisted by the laws of mechanics, we are reduced in the
first place to measurements of parallax, which at best have a quite
inferior accuracy, and which furthermore fail entirely outside a rather
restricted range. For greater stellar distances we are driven to other
and much rougher estimates, resting for instance on the extension to
great distances of connections found within the range of parallax
between brightness and spectral type of a star, or on such assumptions
as that, because a group of stars looks as if it were all together in
space and had a common origin, it actually is so. Thus at greater and
greater distances not only does experimental accuracy become less, but
the very nature of the operations by which length is to be determined
becomes indefinite, so that the distances of the most remote stellar
objects as estimated by different observers or by different methods may
be very divergent. A particular consequence of the inaccuracy of the
astronomical measures of great distances is that the question of whether
large scale space is Euclidean or not is merely academic.

We thus see that in the extension from terrestrial to great stellar
distances the concept of length has changed completely in character. To
say that a certain star is 10^5 light years distant is actually and
conceptually an entire different _kind_ of thing from saying that a
certain goal post is 100 meters distant. Because of our conviction that
the character of our experience may change when the range of phenomena
changes, we feel the importance of such a question as whether the space
of distances of 10^5 light years is Euclidean or not, and are
correspondingly dissatisfied that at present there seems no way of
giving meaning to it.

We encounter difficulties similar to those above, and are also compelled
to modify our procedures, when we go to small distances. Down to the
scale of microscopic dimensions a fairly straightforward extension of
the ordinary measuring procedure is sufficient, as when we measure a
length in a micrometer eyepiece of a microscope. This is of course a
combination of tactual and optical measurements, and certain
assumptions, justified as far as possible by experience, have to be made
about the behavior of light beams. These assumptions are of a quite
different character from those which give us concern on the astronomical
scale, because here we meet difficulty from interference effects due to
the finite scale of the structure of light, and are not concerned with a
possible curvature of light beams in the long reaches of space. Apart
from the matter of convenience, we might also measure small distances by
the tactual method.

As the dimensions become smaller, certain difficulties become
increasingly important that were negligible on a larger scale. In
carrying out physically the operations equivalent to our concepts, there
are a host of practical precautions to be taken which could be
explicitly enumerated with difficulty, but of which nevertheless any
practical physicist is conscious. Suppose, for example, we measure
length tactually by a combination of Johanssen gauges. In piling these
together, we must be sure that they are clean, and are thus in actual
contact. Particles of mechanical dirt first engage our attention. Then
as we go to smaller dimensions we perhaps have to pay attention to
adsorbed films of moisture, then at still smaller dimensions to adsorbed
films of gas, until finally we have to work in a vacuum, which must be
the more nearly complete the smaller the dimensions. About the time that
we discover the necessity for a complete vacuum, we discover that the
gauges themselves are atomic in structure, that they have no definite
boundaries, and therefore no definite length, but that the length is a
hazy thing, varying rapidly in time between certain limits. We treat
this situation as best we can by taking a time average of the apparent
positions of the boundaries, assuming that along with the decrease of
dimensions we have acquired a corresponding extravagant increase in
nimbleness. But as the dimensions get smaller continually, the
difficulties due to this haziness increase indefinitely in percentage
effect, and we are eventually driven to give up altogether. We have made
the discovery that there are _essential_ physical limitations to the
operations which defined the concept of length. [We perhaps do not
regard the substitution of optical for tactual space on the astronomical
scale as compelled by the same sort of physical necessity, because I
suppose the possible eventual landing of men in the moon will always be
one of the dreams of humanity.] At the same time that we have come to
the end of our rope with our Johanssen gauge procedure, our companion
with the microscope has been encountering difficulties due to the finite
wave length of light; this difficulty he has been able to minimize by
using light of progressively shorter wave lengths, but he has eventually
had to stop on reaching X-rays. Of course this optical procedure with
the microscope is more convenient, and is therefore adopted in practice.

Let us now see what is implied in our concept of length extended to
ultramicroscopic dimensions. What, for instance, is the meaning of the
statement that the distance between the planes of atoms in a certain
crystal is 3 x 10^-8 cm.? What we would like to mean is that ⅓ x 10^8
of these planes piled on top of each other give a thickness of 1 cm.;
but of course such a meaning is not the actual one. The actual meaning
may be found by examining the operations by which we arrived at the
number 3 x 10^-8. As a matter of fact, 3 x 10^-8 was the number obtained
by solving a general equation derived from the wave theory of light,
into which certain numerical data obtained by experiments with X-rays
had been substituted. Thus not only has the character of the concept of
length changed from tactual to optical, but we have gone much further in
committing ourselves to a definite optical theory. If this were the
whole story, we would be most uncomfortable with respect to this branch
of physics, because we are so uncertain of the correctness of our
optical theories, but actually a number of checks can be applied which
greatly restore our confidence. For instance, from the density of the
crystal and the grating space, the weight of the individual atoms may be
computed, and these weights may then be combined with measurements of
the dimensions of other sorts of crystal into which the same atoms enter
to give values of the densities of these crystals, which may be checked
against experiment. All such checks have succeeded within limits of
accuracy which are fairly high. It is important to notice that, in spite
of the checks, the character of the concept is changing, and begins to
involve such things as the equations of optics and the assumption of the
conservation of mass.

We are not content, however, to stop with dimensions of atomic order,
but have to push on to the electron with a diameter of the order of
10^-18 cm. What is the possible meaning of the statement that the
diameter of an electron is 10^-18 cm.? Again the only answer is found by
examining the operations by which the number 10^-18 was obtained. This
number came by solving certain equations derived from the field
equations of electrodynamics, into which certain numerical data obtained
by experiment had been substituted. The concept of length has therefore
now been so modified as to include that theory of electricity embodied
in the field equations, and, most important, assumes the correctness of
extending these equations from the dimensions in which they may be
verified experimentally into a region in which their correctness is one
of the most important and problematical of present-day questions in
physics. To find whether the field equations are correct on a small
scale, we must verify the relations demanded by the equations between
the electric and magnetic forces and the space coordinates, to determine
which involves measurement of lengths. But if these space coordinates
cannot be given an independent meaning apart from the equations, not
only is the attempted verification of the equations impossible, but the
question itself is meaningless. If we stick to the concept of length by
itself, we are landed in a vicious circle. As a matter of fact, the
concept of length disappears as an independent thing, and fuses in a
complicated way with other concepts, all of which are themselves altered
thereby, with the result that the total number of concepts used in
describing nature at this level is reduced in number. A precise analysis
of the situation is difficult, and I suppose has never been attempted,
but the general character of the situation is evident. Until at least a
partial analysis is attempted, I do not see how any meaning can be
attached to such questions as whether space is Euclidean in the small
scale.

It is interesting to observe that any increased accuracy in knowledge of
large scale phenomena must, as far as we now can see, arise from an
increase in the accuracy of measurement of small things, that is, in the
measurement of small angles or the analysis of minute differences of
wave lengths in the spectra. To know the very large takes us into the
same field of experiment as to know the very small, so that
operationally the large and the small have features in common.

This somewhat detailed analysis of the concept of length brings out
features common to all our concepts. If we deal with phenomena outside
the domain in which we originally defined our concepts, we may find
physical hindrances to performing the operations of the original
definition, so that the original operations have to be replaced by
others. These new operations are, of course, to be so chosen that they
give, within experimental error, the same numerical results in the
domain in which the two sets of operations may be both applied; but we
must recognize in principle that in changing the operations we have
really changed the concept, and that to use the same name for these
different concepts over the entire range is dictated only by
considerations of convenience, which may sometimes prove to have been
purchased at too high a price in terms of unambiguity. We must always be
prepared some day to find that an increase in experimental accuracy may
show that the two different sets of operations which give the same
results in the more ordinary part of the domain of experience, lead to
measurably different results in the more unfamiliar parts of the domain.
We must remain aware of these joints in our conceptual structure if we
hope to render unnecessary the services of the unborn Einsteins.

The second feature common to all concepts brought out by the detailed
discussion of length is that, as we approach the experimentally
attainable limit, concepts lose their individuality, fuse together, and
become fewer in number, as we have seen that at dimensions of the order
of the diameter of an electron the concepts of length and the electric
field vectors fuse into an amorphous whole. Not only does nature as
experienced by us become different in character on its horizons, but it
becomes simpler, and therefore our concepts, which are the building
stones of our descriptions, become fewer in number. This seems to be an
entirely natural state of affairs. How the number of concepts is often
kept formally the same as we approach the horizon will be discussed
later in special cases.

A precise analysis of our conceptual structure has never been attempted,
except perhaps in very restricted domains, and it seems to me that there
is room here for much important future work. Such an analysis is not to
be attempted in this essay, but only some of the more important
qualitative aspects are to be pointed out. It will never be possible to
give a clean-cut logical analysis of the conceptual situation, for the
nature of our concepts, according to our operational point of view, is
the same as the nature of experimental knowledge, which is often hazy.
Thus in the transition regions where nature is getting simpler and the
number of operationally independent concepts changes, a certain haziness
is inevitable, for the actual change in our conceptual structure in
these transition regions is continuous, corresponding to the continuity
of our experimental knowledge, whereas formally the number of concepts
should be an integer.



_The Relative Character of Knowledge_


Two other consequences of the operational point of view must now be
examined. First is the consequence that all our knowledge is relative.
This may be understood in a general or a more particular sense. The
general sense is illustrated in Haldane's book on the _Reign of
Relativity_. Relativity in the general sense is the merest truism if the
operational definition of concept is accepted, for experience is
described in terms of concepts, and since our concepts are constructed
of operations, all our knowledge must unescapably be relative to the
operations selected. But knowledge is also relative in a narrower sense,
as when we say there is no such thing as absolute rest (or motion) or
absolute size, but rest and size are relative terms. Conclusions of this
kind are involved in the specific character of the operations in terms
of which rest or size are defined. An examination of the operations by
which we determine whether a body is at rest or in motion shows that the
operations are relative operations: rest or motion is determined with
respect to some other body selected as the standard. In saying that
there is no such thing as absolute rest or motion we are not making a
statement about nature in the sense that might be supposed, but we are
merely making a statement about the character of our descriptive
processes. Similarly with regard to size: examination of the operations
of the measuring process shows that size is measured relative to the
fundamental measuring rod.

The "absolute" therefore disappears in the original meaning of the word.
But the "absolute" may usefully return with an altered meaning, and we
may say that a thing has absolute properties if the numerical magnitude
is the same when measured with the same formal procedure by all
observers. Whether a given property is absolute or not can be determined
only by experiment, landing us in the paradoxical position that the
absolute is absolute only relative to experiment. In some cases, the
most superficial observation shows that a property is not absolute, as,
for example, it is at once obvious that measured velocity changes with
the motion of the observer. But in other cases the decision is more
difficult. Thus Michelson thought he had an absolute procedure for
measuring length, by referring to the wave length of the red cadmium
line as standard;[3] it required difficult and accurate experiment to
show that this length varies with the motion of the observer. Even then,
by changing the definition of the length of a moving object, we believe
that length might be made to re-assume its desired absolute character.

[Footnote 3: A. A. Michelson, Light Waves and Their Uses, University of
Chicago Press, 1903, Chap. V.]

To stop the discussion at this point might leave the impression that
this observation of the relative character of knowledge is of only a
very tenuous and academic interest, since it appears to be concerned
mostly with the character of our descriptive processes, and to say
little about external nature. [What this means we leave to the
metaphysician to decide.] But I believe there is a deeper significance
to all this. It must be remembered that all our argument starts with the
concepts as given. Now these concepts involve physical operations; in
the discovery of what operations may be usefully employed in describing
nature is buried almost all physical experience. In erecting our
structure of physical science, we are building on the work of all the
ages. There is then this purely physical significance in the statement
that all motion is relative, namely that no operations of measuring
motion have been found to be useful in describing simply the behavior of
nature which are not operations relative to a single observer; in making
this statement we are stating something about nature. It takes an
enormous amount of real physical experience to discover relations of
this sort. The discovery that the number obtained by counting the number
of times a stick may be applied to an object can be simply used in
describing natural phenomena was one of the most important and
fundamental discoveries ever made by man.



_Meaningless Questions_


Another consequence of the operational character of our concepts, almost
a corollary of that considered above, is that it is quite possible, nay
even disquietingly easy, to invent expressions or to ask questions that
are meaningless. It constitutes a great advance in our critical attitude
toward nature to realize that a great many of the questions that we
uncritically ask are without meaning. If a specific question has
meaning, it must be possible to find operations by which an answer may
be given to it. It will be found in many cases that the operations
cannot exist, and the question therefore has no meaning. For instance,
it means nothing to ask whether a star is at rest or not. Another
example is a question proposed by Clifford, namely, whether it is not
possible that as the solar system moves from one part of space to
another the absolute scale of magnitude may be changing, but in such a
way as to affect all things equally, so that the change of scale can
never be detected. An examination of the operations by which length is
measured in terms of measuring rods shows that the operations do not
exist (because of the nature of our definition of length) for answering
the question. The question can be given meaning only from the point of
view of some imaginary superior being watching from an external point of
vantage. But the operations by which such a being measures length are
different from the operations of our definition of length, so that the
question acquires meaning only by changing the significance of our
terms--in the original sense the question means nothing.

To state that a certain question about nature is meaningless is to make
a significant statement about nature itself, because the fundamental
operations are determined by nature, and to state that nature cannot be
described in terms of certain operations is a significant statement.

It must be recognized, however, that there is a sense in which no
serious question is entirely without meaning, because doubtless the
questioner had in mind some intention in asking the question. But to
give meaning in this sense to a question, one must inquire into the
meaning of the concepts as used by the questioner, and it will often be
found that these concepts can be defined only in terms of fictitious
properties, as Newton's absolute time was defined by its properties, so
that the meaning to be ascribed to the question in this way has no
connection with reality. I believe that it will enable us to make more
significant and interesting statements, and therefore will be more
useful, to adopt exclusively the operational view, and so admit the
possibility of questions entirely without meaning.

This matter of meaningless questions is a very subtle thing which may
poison much more of our thought than that dealing with purely physical
phenomena. I believe that many of the questions asked about social and
philosophical subjects will be found to be meaningless when examined
from the point of view of operations. It would doubtless conduce greatly
to clarity of thought if the operational mode of thinking were adopted
in all fields of inquiry as well as in the physical. Just as in the
physical domain, so in other domains, one is making a significant
statement about his subject in stating that a certain question is
meaningless.

In order to emphasize this matter of meaningless questions, I give here
a list of questions, with which the reader may amuse himself by finding
whether they have meaning or not.

(1) Was there ever a time when matter did not exist?

(2) May time have a beginning or an end?

(3) Why does time flow?

(4) May space be bounded?

(5) May space or time be discontinuous?

(6) May space have a fourth dimension, not directly detectible, but
given indirectly by inference?

(7) Are there parts of nature forever beyond our detection?

(8) Is the sensation which I call blue really the _same_ as that which
my neighbor calls blue? Is it possible that a blue object may arouse in
him the same sensation that a red object does in me and _vice versa_?

(9) May there be missing integers in the series of natural numbers as we
know them?

(10) Is a universe possible in which 2 + 2 ≠ 4?

(11) Why does negative electricity attract positive?

(12) Why does nature obey laws?

(13) Is a universe possible in which the laws are different?

(14) If one part of our universe could be completely isolated from the
rest, would it continue to obey the same laws?

(15) Can we be sure that our logical processes are valid?



GENERAL COMMENTS ON THE OPERATIONAL POINT
OF VIEW


To adopt the operational point of view involves much more than a mere
restriction of the sense in which we understand "concept," but means a
far-reaching change in all our habits of thought, in that we shall no
longer permit ourselves to use as tools in our thinking concepts of
which we cannot give an adequate account in terms of operations. In some
respects thinking becomes simpler, because certain old generalizations
and idealizations become incapable of use; for instance, many of the
speculations of the early natural philosophers become simply unreadable.
In other respects, however, thinking becomes much more difficult,
because the operational implications of a concept are often very
involved. For example, it is most difficult to grasp adequately all that
is contained in the apparently simple concept of "time," and requires
the continual correction of mental tendencies which we have long
unquestioningly accepted.

Operational thinking will at first prove to be an unsocial virtue; one
will find oneself perpetually unable to understand the simplest
conversation of one's friends, and will make oneself universally
unpopular by demanding the meaning of apparently the simplest terms of
every argument. Possibly after every one has schooled himself to this
better way, there will remain a permanent unsocial tendency, because
doubtless much of our present conversation will then become unnecessary.
The socially optimistic may venture to hope, however, that the ultimate
effect will be to release one's energies for more stimulating and
interesting interchange of ideas.

Not only will operational thinking reform the social art of
conversation, but all our social relations will be liable to reform. Let
any one examine in operational terms any popular present-day discussion
of religious or moral questions to realize the magnitude of the
reformation awaiting us. Wherever we temporize or compromise in applying
our theories of conduct to practical life we may suspect a failure of
operational thinking.



CHAPTER II

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."



CHAPTER III

DETAILED CONSIDERATION OF VARIOUS
CONCEPTS OF PHYSICS


WE now begin our detailed consideration of the most important concepts
of physics. It is entirely beyond the scope of this essay to attempt
more than an indication of some of the most important matters. Neither
is it to be expected that the parts of this discussion will always have
a very close connection with each other; the purpose of the discussion
is to aid in acquiring the greatest possible self-consciousness of the
whole structure of physics.



THE CONCEPT OF SPACE


A logically satisfying definition of what we understand by the concept
of space is doubtless difficult to give, but we shall not be far from
the mark if we think of it as the aggregate of all those concepts which
have to do with position. Position means position of something. The
position of things is determined by some system of measurement; perhaps
the simplest is that implied in a Cartesian coördinate system with its
three measurements of length. Hence much of the essential discussion of
space has already been given in connection with the concept of length.
We have seen that measurements of length are made with physical
measuring rods applied to some physical object. We cannot measure the
distance between two points in empty space, because if space were empty
there would be nothing to identify the position of the ends of the
measuring rod when we move it from one position to the next. We see,
then, from the point of view of operations that the framework of
Cartesian geometry, often imagined in an ideal mathematical sense, is
really a physical framework, and that what we mean by spatial properties
is nothing but the properties of this framework. When we say that space
is Euclidean, we mean that the physical space of meter sticks is
Euclidean: it is meaningless to ask whether empty space is Euclidean.
Geometry, therefore, in so far as its results are expected to apply to
the external physical world, and in as far as it is not a logical system
built up from postulates, is an experimental science. This view is now
well understood and accepted, but there was a time when it was not
accepted, but vigorously attacked; the change of attitude toward this
question is symptomatic of a change of attitude toward many other
similar questions.

We have already emphasized that the space of astronomy is not a physical
space of meter sticks, but is a space of light waves. We may, therefore,
have different kinds of space, depending on the fundamental operations.
The space of meter sticks we have called "tactual space", and the space
of light beams "optical space". If we ask whether astronomical space is
Euclidean, we mean merely to ask whether those features of optical space
which are within the reach of astronomical measurement are Euclidean.
The only possible attitude with respect to this question, or such
related questions as whether the total volume of space is finite, or
whether space has curvature, is that it is entirely for experiment to
decide, and that we have no right to form any preconceived notion
whatever. It is therefore beyond the scope of this discussion.

It is interesting to notice that the restricted theory of relativity
virtually assumes, although often without making the explicit statement,
that tactual and optical space are the same. This equivalence results
from the properties assumed for light beams. The distance of a mirror
may be found equally well by measuring it with meter sticks, or by
determining the time required by a light signal to travel there and
back. This situation is, however, logically unsatisfying, because it
must be assumed that the operations for measuring time are independently
defined, and we shall see that they are not. It is a consequence of the
assumed equivalence of tactual and optical space that the path of a beam
of light is a straight line, as a straight line is determined by
operations with meter sticks. When we come to astronomical phenomena,
the physical operations with meter sticks can no longer be carried out,
and it is meaningless to ascribe to beams of light on an astronomical
scale the same geometrical properties that we do on a small scale.



THE CONCEPT OF TIME


According to our viewpoint, the concept of time is determined by the
operations by which it is measured. We have to distinguish two sorts of
time; the time of events taking place near each other in space, or local
time, and the time of events taking place at considerably separated
points in space, or extended time. As we now know, the concept of
extended time is inextricably mixed up with that of space. This is not
primarily a statement about nature at all, and might have been made
simply by the observation that the operations by which extended time is
measured involve those for measuring space. Of course historically the
doctrine of relativity was responsible for the critical attitude which
led to an examination of the operations of measuring time, but
relativity was not necessary for a realization of the spatial
implications of time, any more than the discovery of Planck's quantum
unit _h_ was necessary for the invention by Planck of his absolute units
of measurement, although historically he was inspired to make this
invention by discovering _h_, and in his own mind seems to have thought
of the connection as a necessary one.[7]

[Footnote 7: Max Planck, The Theory of Heat Radiation, translated by
Masius, P. Blakiston's Son & Co., 1914 edition, p. 174.]

The physical operations at the basis of the measurement of time have
never been subjected to the critical examination which seems to be
required. One method of measurement, for instance, involves the
properties of light.

A meter stick is set up with mirrors at the two ends, and a light beam
travels back and forth between the two mirrors without absorption. The
time required for a single passage back and forth is defined as the unit
of time, and time is measured simply by counting these intervals. But
such a procedure is unsatisfactory if we are to permit ourselves all
those operations which are demanded by even the simplest postulate of
relativity, for we must be able to move our clock from place to place,
transfer it from one system to another in relative motion, and with it
determine the properties of light beams in the stationary or moving
system. We recognize in principle that the length of the meter stick may
be different when it is in motion, that it may change also during the
acceleration incident to moving it from one place to another, and that
until proved to the contrary the velocity of light may be a function of
velocity or acceleration. The complicated interplay of all these
possibilities leaves us in much doubt as to the physical significance of
such postulates as, for example, that the velocity of light is the same
in the moving system and the stationary system. In order to ascribe any
simple significance to postulates about the velocity of light, it would
seem that we must have an instrument for measuring this velocity, and
therefore for measuring time, which does not itself involve the
properties of light. To do this we might seek to specify the measurement
of time in purely mechanical terms, as for instance in terms of the
vibration of a tuning fork, or the rotation of a flywheel. But here
again we encounter great difficulties, because we recognize that the
dimensions of our mechanical clock may change when it is set in motion,
and that the mass of its parts may also change. We want to use the clock
as a physical instrument in determining the laws of mechanics, which of
course are not determined until we can measure time, and we find that
the laws of mechanics enter into the operation of the clock.

The dilemma which confronts us here is not an impossible one, and is in
fact of the same nature as that which confronted the first physicist who
had to discover simultaneously the approximate laws of mechanics and
geometry with a string which stretched when he pulled it. We must first
guess at what the laws are approximately, then design an experiment so
that, in accordance with this guess, the effect of motion on some
phenomenon is much greater than the expected effect on the clock, then
from measurements with uncorrected clock time find an approximate
expression for the effect of motion on mass or length, with which we
correct the clock, and so on ad infinitum. However, so far as I know,
the possibility of such a procedure has not been analyzed, and until the
analysis is given, our complacency is troubled by a real disquietude,
the intensity of which depends on the natural skepticism of our
temperament.

In practice, the difficulties of such a logical treatment are so great
that the matter has been entirely glossed over. It is convenient to
postulate a clock, of unknown construction, but such that the velocity
of light, when measured in terms of it, has certain properties. Such,
for example, is the point of view in Birkhoff's recent book.[8]

[Footnote 8: G. D. Birkhoff. Relativity and Modern Physics, Harvard
University Press, 1923.]

The difficulty with this method is that the resulting edifice is as
divorced from physical reality as is the logical geometry of postulates.
We cannot be at all sure that the properties of light as measured with
our physical clocks are the same as the theoretical properties. The
difficulty is particularly important and fundamental in the general
theory of relativity; the basis of the whole theory is the infinitesimal
interval _ds_, which is supposed to be given. Once given, the
mathematics follows. But in a physical world, _ds_ is _not_ given, but
must be found by physical operations, and these operations involve
measurements of length and of time with clocks whose construction is not
specified. In any actual physical application the question must be
answered whether the physical instrument used in measuring the temporal
part of _ds_ is really a clock or not. There is at present no criterion
by which this question can be answered. If the vibrating atom is a
clock, then the light of the sun is shifted toward the infra-red, but
how do we know that the atom is a clock (some say yes, others no)? If we
find the displacement physically have we thereby proved that general
relativity is physically true, or have we proved that the atom is a
clock, or have we merely proved that there is a particular kind of
connection between the atom and the rest of nature, leaving the
possibility open that neither is the atom a clock nor general relativity
true? In practice, of course, we shall adopt the solution which is
simplest and most satisfying aesthetically, and doubtless shall say that
the atom is a clock and relativity true. But if we adopt this simple
view, we must also cultivate the abiding consciousness that at some time
in the future troubles may have their origin here.

It seems to me that the logical position of general relativity theory is
merely this: Given any physical system, then it is possible to assign
values to _ds_ such that relations mathematically deduced by the
principle of relativity correspond to relations between measurable
quantities in the physical system; but that the things that we
physically call _ds_ are anything more than approximately connected with
the _ds's_ required to give the mathematical relations, is at present no
more than a pious faith.

To return to the concept of time, we have already stated that there are
two main problems, that of measuring time at a single point of space,
and that of spreading a time system over all space. The second aspect of
the problem is that to which attention has been directed by relativity
theory; the following detailed examination shows how the operations of
relativity for setting and synchronizing clocks at distant places
involve the measurement of space. It is a fundamental postulate that the
adjustment of the clocks is to be accomplished by light signals. The
synchronization of the clocks is now simple enough. We merely demand
that light signals sent from the master clock at intervals of one second
arrive at any distant clock at intervals of one second as measured by
it, and we change the rate of the distant clock until it measures these
intervals as one second. After its rate has been adjusted, the distant
clock is to be so _set_ that when a light signal is despatched from the
master clock at its indicated zero of time the time of arrival recorded
at the distant clock shall be such that the distance of the clock from
the master clock divided by the time of arrival shall give the velocity
of light, assumed already known. This operation involves a measurement
of the distance of the distant clock, so that in spreading the time
coordinates over space the measurement of space is involved by
definition, and the measurement of time is, therefore, not a
self-contained thing. This is the physical basis for the treatment of
space and time as a four-dimensional manifold. Although mathematically
the numbers measuring space and time enter the formulas symmetrically,
nevertheless the physical operations by which these numbers are obtained
are entirely distinct and never fuse, and I believe it can lead only to
confusion to see in the possibility of a four dimensional treatment
anything more than a purely formal matter.

The notion of extended time, therefore, involves the measurement of
space. It is an interesting question whether the notion of local time
also involves the measurement of space. A rigorous answer to this
question involves giving the specifications for the construction of a
clock, which we have seen has not yet been done. It seems to me
probable, however, that the construction of even a single local clock
involves in some way the _measurement_ of space. If, for example, we use
a vibrating tuning fork, we must find how the time of vibration depends
on the amplitude of vibration, and this involves space measurement, or
if we use a rotating flywheel, we have to correct for the change of
moment of inertia due to the change of dimensions when it is set into
motion or brought into a gravitational field, and all this involves
space measurement. However, these considerations are not certain, and
perhaps the question is not important.

There is now the further consideration that actually in practice the
concept of local time is not entirely divorced from that of extended
time, for two bodies cannot occupy the same space at the same time, and
the time of any event is actually measured on an instrument at some
distance, communication being maintained by light or elastic signals.
But experience convinces us that in the limit, as the phenomenon to be
measured gets closer to the clock, there is no measurable difference,
whether communication with the clock is maintained by light, or
acoustical or tactual signals, so that we have come in physical practice
to accept measurement of the time of events in the immediate
neighborhood of the clock (local time) as one of the ultimately simple
things behind which we do not attempt to go.

Local time is, therefore, a concept treated by the physicist even now as
simple and unanalyzable. This concept is what most people have in mind
when they think of time. Time, according to this concept, is something
with the properties of local time; it was something of this kind that
Newton must have meant by his absolute time, and it is the tacit
retention of this sort of concept that is responsible for the difficulty
so often found in grasping the idea of the relativity of simultaneity,
which is of course entirely foreign to our experience of simultaneity in
local time. An examination of the operations involved in extending time
has shown how the concept of extended time is different from that of
simple local time; this difference leads to appreciably different
numerical relations when we are dealing with high velocities or great
distances. Local time is proved by experience not to be a satisfactory
concept for dealing with events separated by great distances in space or
with phenomena involving high velocities. For instance, we must not talk
about the age of a beam of light, although the concept of age is one of
the simplest derivatives of the concept of local time. Neither must we
allow ourselves to think of events taking place in Arcturus _now_ with
all the connotations attached to events taking place _here_ now. It is
difficult to inhibit this habit of thought, but we must learn to do it.
The naïve feeling is very strong that it does _mean_ something to talk
about the entire present state of the universe independent of the
process by which news of the condition of distant parts is determined by
us. I believe that an examination of this feeling will show that it is
psychological in character; what we mean by the totality of the present
is merely the entire present content of our consciousness. This is
apparently a simple direct thing; we do not appreciate until we make
further analysis that our present consciousness of the existence of the
moon or a star is due to light signals, and that therefore the
apparently simple immediate consciousness of events distant in space
involves complicated physical operations.

Similarly, if we continue to use local time, we get into trouble, when
we go to high velocities, with our simple concept of velocity, which may
be defined in terms of a combination of space and time concepts. The
concept of local time thus loses its value and becomes merely a blunted
tool when we try to carry it out of its original range. But the concept
of extended time, with which we have to replace local time, is a
complicated thing, to which we have not yet got ourselves accustomed; it
may perhaps prove to be so complicated as never to be a very useful
intuitive tool of thought.

All these considerations about time have been concerned only with
intervals of such an order of magnitude that they are readily
experienced by any individual. If we have to deal with intervals either
very long or very short, it is obvious that our entire procedure
changes, and consequently the concept changes. In extending the time
concept to eras remote in the past, for example, we try as always, to
choose the new operations so as to piece on continuously with those of
ordinary experience. A precise analysis of the change in the concept of
time when applied to the remote past does not seem to be of great
significance for our present physical purpose, and will not be attempted
here. It is perhaps worth while to point out, however, that all our
other concepts, as well as that of time, must be modified when applied
to the remote past; an example is the concept of truth. It is amusing to
try to discover what is the precise meaning in terms of operations of a
statement like this: "It is true that Darius the Mede arose at 6:30 on
the morning of his thirtieth birthday."

Of more concern for our physical purposes is the modification which the
time concept undergoes when applied to very short intervals. What is the
meaning, for example, in saying that an electron when colliding with a
certain atom is brought to rest in 10^-18 seconds? Here I believe the
situation is very similar to that with regard to short lengths. The
nature of the physical operations changes entirely, and as before, comes
to contain operations of an electrical and optical character. The
immediate significance of 10^-18 is that of a number, which when
substituted into the equations of optics, produces agreement with
observed facts. Thus short intervals of time acquire meaning only in
connection with the equations of electrodynamics, whose validity is
doubtful and which can be tested only in terms of the space and time
coordinates which enter them. Here is the same vicious circle that we
found before. Once again we find that concepts fuse together on the
limit of the experimentally attainable.

This discussion of the concept of time will doubtless be felt by some to
be superficial in that it makes no mention of the _properties_ of the
physical time to which the concept is designed to apply. For instance,
we do not discuss the one dimensional flow of time, or the
irrevocability of the past. Such a discussion, however, is beyond our
present purpose, and would take us deeper than I feel competent to go,
and perhaps beyond the verge of meaning itself. Our discussion here is
from the point of view of operations: we assume the operations to be
given, and do not attempt to ask why precisely these operations were
chosen, or whether others might not be more suitable. Such properties of
time as its irrevocability are implicitly contained in the operations
themselves, and the physical essence of time is buried in that long
physical experience that taught us what operations are adapted to
describing and correlating nature. We may digress, however, to consider
one question. It is quite common to talk about a reversal of the
direction of flow of time. Particularly, for example, in discussing the
equations of mechanics, it is shown that if the direction of flow of
time is reversed, the whole history of the system is retraced. The
statement is sometimes added that such a reversal is actually
impossible, because it is one of the properties of physical time to flow
always forward. If this last statement is subjected to an operational
analysis, I believe that it will be found not to be a statement about
nature at all, but merely a statement about operations. It is
_meaningless_ to talk about time moving backward: by definition,
_forward_ is the direction in which time flows.



THE CAUSALITY CONCEPT


The causality concept is unquestionably one of the most fundamental,
perhaps as fundamental as that of space and time, and therefore at least
equally entitled to a first place in the discussion. But as ordinarily
understood, there are certain spatial and temporal implications in the
causality concept, so that it can best be discussed in this order after
our examination of space and time.

There is an aspect of the causality concept that in many respects is
closely related to the question of "explanation", for to find the causes
of an event usually involves at the same time finding its explanation.
But there are nevertheless sufficient differences to warrant a separate
discussion.

It seems fairly evident that there was originally in the causality
concept an animistic element much like that in the concept of force to
be discussed later. The physical essence of the concept as we now have
it, freed as much as possible from the animistic element, seems to be
somewhat as follows. We assume in the first place an isolated system on
which we can perform unlimited identical experiments, that is, the
system may be started over again from a definite initial condition as
often as desired.[9]

[Footnote 9: We must include in general in the concept of "initial
condition" the past history of the system. In order not to make this
condition so broad as to defeat itself, we have to add the observation
that actually identity of past history is necessary over only a
comparatively short interval of time. Logical precision seems
unattainable here--the physical concepts themselves have not the
necessary precision.]

We assume further that when so started, the system always runs through
exactly the same sequence of events in all its parts. This contains the
assumption that the course of events runs independent of the absolute
time at which they occur--there is no change with time of the properties
of the universe.[10] It is a result of experience that systems with
these properties actually exist. An alternative way of stating our
fundamental hypothesis is that two or more isolated similar systems
started from the same initial condition run through the same future
course of events. Upon the system given in this way, which by itself
runs a definite course of events, we assume that we can superpose from
the outside certain changes, which have no connection with the previous
history of the system, and are completely arbitrary. Now of course in
nature, as we observe it, there is no such thing as an arbitrary change,
without connection with past history, so that strictly our assumption is
a pure fiction. It is here that the animistic element still seems to
persist, although perhaps not necessarily.

[Footnote 10: As so often in physics, we appear to be doing two things
at once here. It is doubtful whether we can give a meaning to "definite
initial condition" apart from the future behavior of the system, so that
we have no real right to infer from uniform future behavior both a
constancy of the laws of nature, independent of time, and a constancy of
initial condition. I very much question whether a thoroughgoing
operational analysis would show that there are really two independent
concepts here, and whether the use of two formally quite different
concepts is anything more than a convenience in expression. It seems to
me that it may be just as meaningless to ask whether the laws of nature
are independent of time as it was to ask with Clifford whether the
absolute scale of magnitude may not be changing as the solar system
travels through space.]

We regard our acts as not determined by the external world, so that
changes produced in the external world by acts of our wills are, to a
certain degree of approximation, arbitrary. The system, then, on which
we are experimenting, is one capable of isolation from us in that we may
regard ourselves as outside the system, and having no connection with
it. The system, furthermore, is capable of isolation from the rest of
the physical universe, in that events taking place outside the system
have no connection with those taking place inside.[11] Experience gives
the justification for assuming that physical isolation of this sort is
possible. Actually, of course, isolation is never complete, but only
partial, up to presumably any desired degree of approximation.

[Footnote 11: Here again, the concept of "isolation" or "connection" is
defined only in terms of the behavior of the system, and it is not clear
whether this is really an operationally independent concept or not.]

The statement that two exactly similar isolated systems, starting from
the same initial conditions (including past history in the general idea
of initial condition) will run through the same future course of events
involves as a corollary that if differences develop in the behavior of
two such apparently similar systems these differences are evidence of
other previous differences. The thesis that this corresponds to
experience may be called the thesis of essential connectivity and is
perhaps the broadest we have: it is the thesis that differences between
the behavior of systems do not occur isolated but are associated with
other differences. It is essentially the same thesis as that already
mentioned in connection with "explanation", namely that it is possible
to correlate any of the phenomena of nature with other phenomena.

If now the connectivity or correlation between phenomena is of a special
kind, we have a causal connection; namely, if whenever we arbitrarily
impress event A on a system we find that event B, always occurs, whereas
if we had not impressed A, B would not have occurred, then we say that A
is the cause of B, and B the effect of A. By suitably choosing the event
A, we may find the effect of any event of which the system is
susceptible.

The relation between A and B is an unsymmetrical one, by the very nature
of the definition, the cause being the arbitrary variable element, and
the effect that which accompanies it. Furthermore, A may obviously be
the cause of more than one event B, and may cause a whole train of
events.

The causal concept analyzed in this way is not simple by any means. We
do not have a simple event A causally connected with a simple event B,
but the whole background of the system in which the events occur is
included in the concept, and is a vital part of it. If the system,
including its past history, were different, the nature of the relation
between A and B might change entirely. The causality concept is
therefore a relative one, in that it involves the whole system in which
the events take place.

In practice we now take an exceedingly pregnant step and seek to extend
the concept, and rid ourselves as much as possible of its relativity. It
is a matter of experience that there are often a great number of systems
in which A is the cause of B. In many cases the causal relation persists
through such a very wide range of systems that we lose sight entirely of
the system, and come to assume that we have an _absolute_ causal
connection between A and B. For instance, when I strike a bell, and hear
the sound, the causal connection persists through such a great number of
different kinds of system that I might think that here is an absolute
causal connection. Such an absolute causal connection would mean that
always under all circumstances, the striking of the bell is accompanied
by a sound. But _all_ conditions means only _all_ those conditions
covered by experiment. Thus in the case of the bell, all our experiments
were made in the presence of the atmosphere. The causal connection
between the striking of the bell and the sound should have been always
recognized in principle as relative to the presence of the atmosphere.
Indeed, later experiments in the absence of the atmosphere show that the
atmosphere does play an essential part. Now as a matter of fact, the
atmosphere is so comparatively easy to remove that we very readily
include the atmosphere in the chain of causal connection. But if the
atmosphere had been impossible to remove, like the old ether of space,
our idea of the causal connections between the striking of the bell and
its sound might have been quite different. In actual physical
applications of the causality concept, the constant background which is
maintained during all the variations by which the causal connection is
established usually has to be inferred from the context.

It is a matter of perhaps universal experience that the event A is
accompanied by not only one event, which is the effect of A by
definition, but A entails a whole causal train of events. It seems to be
a generalization from experience that the causally connected train of
events started by A is a never ending train, provided the system is
large enough. This is perhaps not necessary in the general case, but if
the event A involves imparting external energy to the system, or the
action of external force (momentum change), there can be no question.

That there is a causal train started by A is particularly evident if A
and B are separated in space. Thus in the case of the bell, the impulse
given to the air by the vibration of the bell is propagated through the
air as an elastic wave, which thus constitutes the causal train of
events. The phenomenon of propagation is characteristic of causal
connections of a mechanical character, and is the justification for the
introduction of the time concept in connection with the causality
concept, where it now appears for the first time. It is evident that
when a disturbance is propagated to a distant point, the effect
_follows_ the cause in time, as time is usually measured.

We extend this result, and usually think that the effect _necessarily_
follows the cause. We now examine whether this is a necessary result of
the causality concept. If we are to talk about the time of events at
different places, we must have some way of setting clocks all over
space. If this is done arbitrarily, there is no necessary connection
between the local clock times of a cause and its effect, but
nevertheless the causality concept involves a certain temporal relation
even in this most general case. Suppose that event A takes place at
point 1 and its effect, event B, at point 2. We station a confederate at
2 who sends a light signal (or any other sort of signal) to 1, as soon
as the event B occurs at 2. Then it is a consequence of the nature of
the causality concept that the signal cannot arrive at 1 before event A
occurs. For if it did arrive before A, we should merely omit to perform
A, which by hypothesis is arbitrary, and entirely in our control, and
then our assumption would be violated that the system is such that the
event B occurs only when A also occurs. The same argument shows _a
fortiori_ that if the effect B occurs at the same place as its cause A,
it cannot precede it in time. I cannot see that the nature of the
causality concept imposes any further restriction on the time of B. The
restricted principle of relativity, however, in postulating that no
signal can be propagated faster than a light signal, virtually makes a
further assumption about the temporal connection of causally connected
events, namely, that the event B at 2 cannot occur before the arrival at
2 of a light signal which started from 1 at the instant that A occurred
at 1. For if B did occur earlier, we could use events A and B as a
signaling code, thus violating our hypothesis.

There is thus a closest connection in time, when time is extended over
space as the theory of relativity directs, between cause and effect,
depending on their separation in space; from this arises the relativity
concept of the causal cone, which in the four dimensional manifold of
space-time divides the aggregate of all those events which may be
causally related from the aggregate of those which are separated by such
a small interval in time and such a large interval in space that
communication by light signals and therefore a causal connection is not
possible. Given now two events A and B which are related as cause to
effect in one system of reference, then they must be causally related
also in every other system of reference. For if they were not, we could
by definition of causality suppress the event A in one of the systems in
which the causal relation does not hold, and this, because of the nature
of the concept of event, involves suppressing A in all the systems, thus
violating our hypothesis of a causal connection in the original system.
The concept of event involved in this argument will be examined later. It
appears then, that the fundamental postulate of relativity (that the
form of natural laws is the same in all reference systems) demands that
the temporal order of events causally connected be the same in all
reference systems.

The whole universe at this present moment is often supposed to be
causally connected with all succeeding states. This means that if we
could repeat experience, starting from the same initial conditions, the
future course of events would always be found to be the same. The truth
of this conviction can never be tested by direct experiment, but it is
something at which we arrive by the usual physical process of successive
approximation. It is difficult to formulate precisely what we mean by
"present" state of the universe, and there is every reason to think that
such a formulation is not unique, but the concept contains the necessary
implication that none of the events constituting the "present" can be
causally connected. The events in distant places which constitute the
present must be separated by an interval of time less than time required
by light to travel between the two places.

The conviction, arising from experience, that the future is determined
by the present and correspondingly the present by the past, is often
phrased differently by saying that the present causally determines the
future. This is in a certain sense a generalization of the causality
concept. It is one of the principal jobs of physics to analyze this
complex causal connection into components, representing as far as
possible the future state of the system as the sum of independent trains
of events started by each individual event of the present. How far such
an analysis is possible must be decided by experiment. It is certainly
possible to a very large extent in most cases, but there seems to be no
reason to expect that a complete analysis is possible. So far as the
system is describable in terms of linear differential equations, the
causal trains started by different events propagate themselves in space
and time without interference and with simple addition of effects, and
conversely the present may be analyzed back into the simple sum of
elementary events in the past, but if the equations governing the motion
of the system are not linear, effects are not additive, and such a
causal analysis into elements is not possible. No emphasis is to be laid
here on the _differential_ aspect of the equations: it is quite possible
that finite difference equations may have the same property of
additivity. Although there can be no question that linear equations
enormously preponderate, neither can there be any doubt that some
phenomena cannot be described in terms of linear equations (_e.g._,
ferro-magnetism), so that there seems no reason to think that a causal
analysis is always possible. I believe, however, that the assumption
that such an analysis into small scale elements is possible is tacitly
made in the thought of many physicists. If the analysis is not possible,
we may expect to find results following the cooperation of several
events which cannot be built up from the results of the events occurring
individually.

When a causal analysis is possible, finding the simplest events which
act as the origin of independent causal trains is equivalent to finding
the ultimate elements in a scheme of explanation, so that here we merge
with the concept of explanation, as already mentioned. As was true of
the explanatory sequence, so here there can be no _formal_ end of the
causal sequence, because we can always ask for the cause of the last
member. But it may be physically meaningless to extend the causal
sequence beyond a certain point. We have seen from the point of view of
operations that the causal concept demands the possibility of variation
in the system. It is therefore meaningless to say that A is the cause of
B unless we can experience systems in which A does not occur. Now if in
extending the causal sequence, we eventually arrive at a condition so
broad that physically no further variation can be made, our causal
sequence has to stop.

Corresponding to this property of the causality concept, the causal
sequence may be terminated either formally, by postulate, or naturally,
by the intrinsic physical nature of the elements of the sequence. Thus
if we say that light gets from point to point because it is propagated
by a medium of unalterable properties, which fills all space, which is
always present and can never be eliminated physically, we have by the
postulated properties of the medium brought the possibility of further
inquiry to a close, because to take the next step and ask the cause of
the properties of the ether, demands that we be able to perform
experiments with the ether altered or absent. Such an ending of the
sequence is evidently pure formalism, without physical significance. But
other considerations may give physical significance. Thus if there are
other sorts of experiment that can be explained by assuming a universal
medium of the same properties, the concept proves not only to be useful,
but to have a certain degree of physical significance. An example of an
inevitable termination of the causal sequence is afforded by the
possibility, already mentioned, that the value of the gravitational
constant may be determined by the total quantity of matter in the
universe. Without further qualification, this is an entirely sterile
statement, but if it can be shown that there is a simple numerical
connection, the matter takes on interest, and we may seek further for a
correlation between the numerical relation and other things.

This analysis of the causality concept does not pretend to be complete
and leaves many interesting questions untouched. Perhaps one of the most
interesting of these questions is whether we can separate into cause and
effect two phenomena which _always_ accompany each other, and whether
therefore the classification of phenomena into causally connected groups
is an exhaustive classification. But the discussion is broad enough for
our purpose here; the most important points of view to acquire are that
the causality concept is relative to the whole background of the system
which contains the causally connected events, and that we must assume
the possibility of an unlimited number of identical experiments, so that
the causality concept applies only to sub-groups of events separated out
from the aggregate of all events.



THE CONCEPT OF IDENTITY


One of the most fundamental of all the concepts with which we describe
the external world is that of identity; in fact, thinking would be
almost inconceivable without such a concept. By this concept we bridge
the passage of time; it enables us to say that a particular object in
our present experience is the same as an object of our past experience.
From the point of view of operations, the meaning of identity is
determined by the operations by which we make the judgment that this
object is the _same_ as that one of my past experience. In practice
there are many indirect ways of making this judgment, but I believe the
essence of the situation lies in the possibility of continuous
connection between the object of the present and the past by continuous
observation (either direct or indirect) through all intermediate time.
We must, for example, be able to look continuously at the object, and
state that while we look at it, it remains the same. This involves the
possession by the object of certain characteristics--it must be a
discrete thing, separated from its surroundings by physical
discontinuities which persist. The concept of identifiability applies,
therefore, only to certain classes of physical objects; no one thinks of
trying to identify the wind of to-day with the wind of yesterday. It is
somewhat easier to identify a liquid such as water in its flow in a
stream, because we can make the motion of the water visible by solid
particles suspended in it, but even here it is not easy to prove to a
captious critic that it is really the water and not the suspended
particles of solid that we are identifying. Even solids, when our
measurements are sufficiently refined, seem to lose their discontinuous
edges, as has been suggested in the discussion of the approximate
character of experimental arithmetic, and the identity concept becomes
hazy.

There can be no question that the concept of identity is a tool
perfectly well adapted to deal approximately with nature in the region
of our ordinary experience, but we have to ask a more serious question.
Does not the apparent demand of our thinking apparatus to be furnished
with discrete and identifiable things to think about impose a very
essential restriction on any picture of the physical universe which we
are able to form? We are continually surprising ourselves in the
invention of discrete structure further and further down in the scale of
things, whole sole _raison d'être_ is to be found entirely within
ourselves. Thus Osborne Reynolds[12] has speculated seriously and most
elaborately about an atomic structure in the ether, and we find
Eddington[13] hinting at the existence of structure of an order of
magnitude of 10^-40 cm. On a much larger scale of magnitude we also
think in the same terms, and conceive positive and negative elementary
charges with hard and impenetrable cores, which involves a complete
change in the law of force at points sufficiently close. What physical
assurance have we that an electron in jumping about in an atom preserves
its identifiability in anything like the way that we suppose, or that
the identity concept applies here at all?

[Footnote 12: Osborne Reynolds, The Sub-Mechanics of the Universe, 254
pp., Cambridge University Press, 1903.]

[Footnote 13: A. S. Eddington, Report on Gravitation, Lon. Phys. Soc.,
1918, p. 91.]

In fact, the identity concept seems to lose all meaning in terms of
actual operations on this level of experience.

The mind seems essentially incapable of dealing with continuity as a
property of physical things; it is not even able to talk about
continuity except in negative terms. To each attempted description of
the properties of a truly continuous substance, it can say "No, it is
not that", but cannot imagine experience which corresponds to what it
conceives a really continuous thing ought to feel like. In terms of
operations, continuity has only a sort of negative meaning. Now certain
implications of this inability of the mind can be removed by appropriate
postulates, as, for example, we can postulate the complete annihilation
of a negative by a positive charge, as is now being done in certain
speculations.[14] There is point in doing this, because the annihilation
of two charges has physical meaning. But it is a question whether all
implications of this habit of thought can be removed, and whether any
picture that we can form of nature will not be tinged—sickbed o'er
with the pale cast of thought.

[Footnote 14: For example, J. H. Jeans, Nat. 114, 828-829, 1924.]

The operational view suggests that in this last we are coming perilously
close to a meaningless question, although there is a certain sense in
which there is meaning here. It may turn out as a matter of fact that we
shall not be able to carry our delving into small-scale phenomena deeper
than a certain point, and that nature will appear to be finite downward,
so that we shall bring up against a wall of some kind. But to ask in
such a situation whether we have come to the end because nature is
_really_ finite, or whether we only appear to be at an end because of
some property of our minds, such as inability to deal with continuity,
is, I believe, a meaningless question.

In actual use the identity concept is extended, and identity is used in
other senses than the fundamental one examined above. For instance, we
speak of two observers seeing the same object, or if the object moves or
does something, we may speak of two observers perceiving the same
happening. A happening about which the judgment of sameness is possible
when perceived by different observers (or mathematically expressed when
observed in two reference frames) is what we mean by an event, which is
one of the fundamental concepts of relativity theory. What now is
involved in this concept of event, or what do we mean when we say that
two observers experience the same event? A first crude attempt might say
that the event is the same if it is described in the same way by the two
observers. But this leads us into all the complicated questions of the
meaning of language, which we would gladly avoid, and is furthermore not
true, because the whistle of a locomotive, for example, does not have
the same pitch for two observers moving with different velocities. A
satisfactory analysis of the situation is difficult to give, but I
believe the essence lies in the discrete character of the event, just as
the identity concept when applied only to objects involved discreteness.
The event is bounded on all sides by discontinuities, both in space and
time. Now it seems to be a result of experience that discontinuities
have a certain absolute significance, in that there is a one-to-one
correspondence between the discontinuities observed in any one reference
system and those observed in any other. Corresponding discontinuities in
two reference systems are by definition the same. An event is by
definition the aggregate of all phenomena bounded by certain
discontinuities, and two reference systems are by definition describing
the same event if the discontinuous boundaries of the event are the
same, irrespective of the appearance of the event in the two systems.
The emission of a light signal, for example, is an event according to
this definition, although it may appear as red light in one reference
system and green in another.

We now see that the concept of event is only an approximate concept, as
was also that of identity, and for the same reason, namely, there are no
such things in experience as sharp discontinuities, but as our
measurements become more refined, the edges of supposed discontinuities
become blurred. As we go to smaller scales of magnitude this blurring
becomes more important, until the physical possibility of performing
those operations by which the discontinuities are detected entirely
disappears, and the concept of event acquires, in terms of operations,
an entirely different meaning. We continue to think of the event in the
same way as before in terms of a mental model, but the true operational
significance now depends on the particular phenomenon under
consideration. The concept of event is really not the same sort of thing
when applied to the emission of a quantum of radiation from an atom, or
the emission of gamma radiation from a radioactive disintegration, or
the flashing of a signal from a dark lantern by opening and closing a
shutter. Here as always, when our range of experience is extended, we
must be prepared at some future time to find that, by extending the
ordinary concept of event to small-scale phenomena by the device of the
mental model, we have by implication smuggled into our picture phenomena
which do not exist, so that it will be necessary to revise our thinking,
casting it into terms corresponding to direct experience.



THE CONCEPT OF VELOCITY


The concept of velocity, as ordinarily defined, involves the two
concepts of space and time. The operations by which we measure the
velocity of an object are these: we first observe the time at which the
object is at one position, and then later observe the time at which it
is at a second position, divide the distance between the two positions
by the time interval, and if necessary, when the velocity is variable,
take the limit. As long as we deal with fairly low velocities we do not
have to inquire carefully as to the kind of time we use in these
operations, but when the velocities become high, we do have to take care
to use the local times at the two positions of the body, which means
that we must have a time system spread over space, or, in other words,
the "extended" time system. This velocity concept, defined in this way,
may be used as a tool in describing nature, and it will be found that
nature has certain properties; for example, the velocity of light is 3 x
10^10 cm./sec. Further, no material thing can be given a velocity as
high as this, but as its velocity is made to approach this value,
increments of energy increasing without limit are required.

But now it is very much a question for examination whether the velocity
concept defined in this particular way has been chosen wisely as a tool
for describing natural phenomena. It is quite possible to modify the
velocity concept, that is, to set up other operations which correspond
to our instinctive feeling of what velocity is in terms of immediate
sensation and such that all numerical measures are unmodified at low
velocities.[15] For example, a traveller in an automobile measures his
velocity by observing the clock on his instrument board and the mile
stones which he passes on the road. This operation differs from that of
the definition above in that the time is no longer extended time, but is
the local time of the moving object. The space coordinates used in this
alternative operation at first seem a hybrid sort of thing, but they are
what the observer would actually most naturally use: they are what he
would measure with a tape measure fixed to a point of the road and
allowed to unwind as he proceeds, or what is measured by a vessel at sea
with a log line let out behind.

[Footnote 15: It is an interesting question for the psychologist whether
the velocity concept is not a more primitive thing in order of
apprehension than that of time, and whether the concept of time is not
derived from observing things in motion, or whether indeed there is any
necessary connection at all between velocity and time in terms of
untutored experience.]

Or there is still another most interesting way of defining velocity, in
which the analysis into space and time is not made at all, but velocity
is directly measured by building up the given velocity by physical
addition of a unit velocity selected arbitrarily. This matter is
discussed at some length in my book "_Dimensional Analysis_",[16] but is
of sufficient pertinence here to describe briefly. We may in the first
place construct a concrete standard for velocity, as, for example, by
stretching a string between two pegs on a board with a fixed weight. If
we strike the string, a disturbance travels along the string which we
can follow with the eye, and we define unit velocity as the velocity of
this disturbance. An object has greater than unit velocity if it outruns
the disturbance, and less if it lags behind. We may now duplicate our
standard, making another board with pegs and stretched string, and check
the equality of the two velocities by observing that the two
disturbances run together. We now define two units of velocity as the
velocity of anything which runs with the disturbance of the string of
the second board, when the second board is made to move bodily with such
a velocity that it runs with the disturbance of the first string. The
process may be extended indefinitely, and any velocity measured.

[Footnote 16: Yale University Press, 1922.]

If either of these two alternative definitions of velocity were adopted,
it would be found that the velocity of light is infinite. Further, there
would be no limit to the velocity which can be imparted to material
bodies on giving them unlimited energy, which is what we are prepared to
regard from ordinary experience is natural and simple. The infinite
velocity for light, on the other hand, is most unnatural, particularly
if we favor a medium point of view. We are here faced with a
dilemma--all sorts of phenomena cannot at the same time be treated
simply. If we attach the most fundamental significance to the behavior
of material bodies, we shall do well to adopt one of the alternative
definitions of velocity. If, on the other hand, we regard the phenomena
of light as the most fundamental, we shall endeavor to form our
definition so that the properties of light are simple. This was
precisely the point of view of Einstein; it is characteristic of his
entire scheme of restricted relativity that light is the fundamental
thing, and this influenced him in adopting the first definition of
velocity. Now one can have no quarrel with this desire to make light
fundamental (the wisdom of doing this is to be justified by the
results), and if the properties of light are to be treated
mathematically, one can easily see the desirability of getting rid of
infinite attributes, and so admit the desirability of making the
velocity of light finite. But all this involves another very insidious
assumption which we ourselves have tacitly used in all our preceding
discussion, namely, that the notion of velocity properly pertains to
light at all. Einstein has very definitely adopted this point of view,
and so determined the character of the entire structure of relativity. I
believe, on the contrary, that it is very gravely to be questioned
whether the identification of light with a _thing travelling_, which is
involved in applying the velocity concept, should be made. This
discussion must be postponed, however, until we deal with the properties
of light. The important points for us to notice at present are that the
definition of velocity actually used involves the concept of extended
time, and that it would be possible to define velocity in different
ways, which would give quite a different complexion to phenomena at high
velocities, but which would leave untouched our ordinary experience.

The velocities at which the precise form of definition becomes important
are higher than can be reached in ordinary mechanical experiments. Such
velocities can be attained in terrestrial laboratories only with
electrified particles, as in experiments in high vacua or with
radioactive disintegrations. It is interesting to notice that we very
seldom attempt a direct measurement of velocity in such experiments by
following a discrete particle in its flight and finding the time
required to pass over a measured distance, but the velocities are
measured indirectly, by calculation from the equations of
electrodynamics and in terms of such immediately observed things as
curvature of path. It is true that one or two experiments have attempted
a more direct measure of velocity, but it seems there is room for more
work here.



THE CONCEPTS OF FORCE AND MASS


Another concept of great importance is that of force. Since the usual
analysis finds a connection between force and acceleration, and
acceleration involves velocity, this is a natural place for the
discussion of force. This concept has been subjected to much analysis by
various writers. In origin the concept doubtless arises from the
muscular sensations of resistance experienced from external bodies. This
crude concept may at once be put on a quantitative basis by substituting
a spring balance for our muscles, or instead of the spring balance we
may use any elastic body, and measure the force exerted by it in terms
of its deformation. Of course, the various precautions which must be
taken in carrying out this idea physically are complex; the matter of
precautions against temperature changes, for example, is one of the most
easily understood. The concept of force so defined is limited to static
systems; it is the task of statics to find the relation between the
forces in systems at rest. We next extend the force concept to systems
not in equilibrium, in which there are accelerations, and we must
conceive that at first all our experiments are made in an isolated
laboratory far out in empty space, where there is no gravitational field.
We here encounter a new concept, that of mass, which as it is originally
met is entangled with the force concept, but may later be disentangled
by a process of successive approximations. The details of the various
steps in the process of approximation are very instructive as typical of
all methods in physics, but need not be elaborated here. Suffice it to
say that we are eventually able to give to each rigid material body a
numerical tag characteristic of the body, such that the product of this
number and the acceleration it receives under the action of any given
force applied to it by a spring balance is numerically equal to the
force, the force being defined, except for a correction, in terms of the
deformation of the balance, exactly as it was in the static case. In
particularly, the relation found between mass, force, and acceleration
applies to the spring balance itself by which the force is applied, so
that a correction has to be applied for a diminution of the force
exerted by the balance arising from its own acceleration.

We now extend the scope of our measurements by bringing our laboratory
into the gravitational field of the earth, and immediately our
experience is extended, in that we continually see bodies accelerated
with no spring balance (that is, no force) acting on them. We extend the
concept of force, and say that any body accelerated is acted on by a
force, and the magnitude of this force is defined as that which would
have been necessary to produce in the same body the same acceleration
with a spring balance in empty space. There is physical justification
for this extension in that we find we can remove the acceleration which
a body acquires in a gravitational field by exerting on it with a spring
balance a force of exactly the specified amount in the opposite
direction. This extended idea of force may also be applied to systems in
which there are electrical actions.

We thus see that in extending the notion of force from bodies in rest to
bodies in motion, the character of the concept has changed, because the
operations by which force is measured change--the force acting on a body
is now measured in terms of its acceleration. But in determining the
force from the acceleration, we have to know the mass. This mass has to
be independently measured with the original concept of force; otherwise
we have no basis for such simple statements as that the force of gravity
on a body is proportional to its mass. All this applies to the ordinary
range of experiments with low velocities. If now we extend the range of
measurements, we find phenomena which we had not expected; for example,
there seem to be difficulties in the way of indefinitely increasing the
velocity of a material body, as of a charged atom. We begin to ask
searching questions: is the force of gravity independent of velocity at
high velocities, or is the mass independent of velocity under the same
conditions or independent of the gravitational field, etc.?

In attempting to answer these new questions, we find difficulty with the
concepts in terms of which they are formulated. There are no operations
by which we can find whether force is independent of velocity unless we
first know the mass, or any operations by which a mass can be measured
unless we know a force. The purely mechanical systems with the highest
velocities of which we have any experimental knowledge are the heavenly
bodies. The motion of these is, with the important exception of Mercury,
that predicted by the ordinary laws of mechanics, so that at first it
might appear that we have here confirmation of the laws of mechanics for
bodies with comparatively high velocities. But it must be remembered
that all we can observe of the heavenly bodies is their positions, and
that we cannot perform on these bodies all the operations by which we
can check the laws of mechanics for terrestrial phenomena. If, for
example, mass and the force with which gravity acts on mass were both
equally affected by velocity, the motion of the heavenly bodies would be
exactly the same as that observed now. Hence as we increase the range of
velocity, the concepts of force and mass simultaneously lose their
definiteness, and become partially fused. This is typical of what we
have now come always to expect near the limit of the experimentally
attainable; experience becomes less rich, the choice of physical
operations more restricted, concepts change and become fewer in number.
If we are to retain the same formal number of concepts, we must
introduce arbitrary conventions or definitions. These definitions are to
be determined largely by convenience. In the case of mechanical systems,
this motive of convenience is supplied by considerations from outside
the domain of mechanical phenomena. The highest velocities of practice
are not reached in mechanical, but in electrical systems, in experiments
with vacuum tubes, etc. Considerations of convenience are therefore
dictated from the electrical point of view. These considerations will be
gone into in much more detail later; the conclusion is all that we need
here, which is that it is convenient to assume for the charge of the
electron a constant number, independent of the velocity, and this
involves making its mass variable in a definite way with velocity. Now
if the principle of relativity is accepted, the mass of mechanical
objects must vary with velocity in the same way as the mass of
electrical charges. Since the variability of this latter is fixed,
mechanical mass becomes a definite function of velocity, and the force
is therefore also fixed in any specific physical case.

The fundamental definition of force given above is highly academic,
involving as it does hypothetical experiments in laboratories situated
far out in empty space. Some sort of procedure like this seems to
correspond to more or less explicit statements to be found in the
literature of mechanics. The meaning in terms of actual operations to be
given to such definitions involves complicated inferential reasoning. We
would make much closer connection with the conditions of actual
experiment if in the definition we substituted for the hypothetical
operations in empty space more or less approximately realizable
operations on bodies sliding on level table tops without friction. I
suppose our instinctive feeling for the laws of mechanics is such that
we are convinced that definitions in terms of an interstellar space
laboratory or a level table top are actually the same. But in principle
we must recognize that when the operations are different, the concepts
are different, and if we adopt something equivalent to the table top
definition, as it seems we are physically forced to do, we must leave
open in our thinking the possibility of finding in the present penumbra,
when our accuracy is sufficiently increased, such phenomena perhaps as
directional attributes of mass in a gravitational field.

We have just considered the sort of problem that we encounter on
ordinary scales of magnitude on going from low to high velocities; what
becomes of the concepts of force and mass when we go to a very small
scale? Down to the atomic scale we may at least slur over the new
physical difficulties, for although we cannot of course experiment with
actual atoms, we can nevertheless make measurements of the Brownian[17]
movement of suspensions in liquids settling in a gravitational field,
for example, and the extrapolation to the atom is not a very great one.
The mass of each individual atom is obtained by what is equivalent to a
process of counting, assuming the law of conservation of mass on an
atomic scale. This is justified by all chemical experience. The mass of
the component parts of the atoms, the electrons, may also perhaps be
given a unique significance after we have decided on the laws of the
electrical field, by experiments on acceleration in electrical fields.
The question which interests in principle here is what meaning, if any,
shall be attached to the mass of the elements of the electron.

[Footnote 17: This phenomenon is discussed at length in the book by J.
Perrin, Brownian Movement and Molecular Reality, translated by F. Soddy,
Taylor and Francis, London, 1909.]

It is evident that we here go beyond any possible experience, at least
for the present, and that experience has again become poorer and our
concepts fewer in number. All that we can now demand is that certain
combinations of numbers, some of which represent mechanical mass and
others electrical charge, have proper relations to each other when
integrated throughout the entire body of the electron. Similar questions
confront us when we ask what are the forces which the parts of the
electron exert on each other. We return to this question in considering
the nature of the electrical concepts. In any event, the concepts of
both force and mass are entirely altered in this domain.

It is interesting to note, in passing, that present electrical theory
gives no meaning to the mass of the elements of the electron, since the
total electromagnetic mass of the electron is built up from the _mutual_
terms in the action of the elements--the total mass is not a _linear_
resultant of the action of the elements.



THE CONCEPT OF ENERGY


In examining the concept of energy, we start with purely mechanical
energy. In isolated mechanical systems, in which there are only
conservative forces, the sum of kinetic and potential energy is
constant. The kinetic energy may be defined as ∑ ½ mv^2, formed for
all parts of the body. The potential energy is determined by the
position of the parts of the system, and has physical significance only
with reference to a datum position, that is, only changes of potential
energy have meaning in terms of operations. The total energy ascribed to
the system has therefore an element of arbitrariness in that the datum
position may be chosen at random, and energy acquires meaning only on
tracing the history back to the epoch of the datum position.

The concept of energy may be extended from mechanical systems to all
systems with which we are acquainted; the operations by which meaning is
given to the extended energy concept involve the generalized
conservation principle, or the first law of thermodynamics. The
extension to thermal systems is immediate; the inclusion of optical and
electrical systems in the scheme was a most important physical step,
which of course required careful experimental justification. Because of
its wide range of application, the energy concept has now come to be
regarded as one of the most important in physics; this idea was held by
Ostwald[18] twenty and more years ago, and is now much to the front
because of the connection between mass and energy indicated by the
theory of relativity, and the important rôle assigned to energy levels
in spectrum analysis.

[Footnote 18: W. Ostwald, Die Energie, Barth, Leipzig, 1908.]

What now is the precise nature and significance of the general energy
concept? In the first place the conservation property of energy is one
of the simplest and most obvious of the properties of matter, so that in
this property of energy is seen a reason for ascribing to it certain of
the properties of matter, in particular and most important, that of
localization in space. We must recognize, however, that this idea of a
location in space is injected into the situation entirely by ourselves,
and corresponds to nothing directly given by the operations of
experiment. The idea has had a most important effect, however. Witness,
for instance, the importance ascribed to the discovery by Kelvin of a
function by which the total energy of an electric field can be
represented as distributed through space;[19] this was one of the most
important props of the medium point of view.

[Footnote 19: This function is ⅛π times the scalar product of
electric force and displacement. If Maxwell's definition of displacement
is adopted, the factor ⅛π is replaced by ½, and an accurate analogy
results between the energy stored in the ether and the elastic energy
stored in a bent spring.]

A more critical examination is likely to diminish considerably our
satisfaction with this naive analogy drawn between matter and energy.
With regard to matter, we may still be tolerably satisfied with our
ascription to matter of location in space, but it is quite different
with regard to conservation of matter. In just what sense is matter
conserved? Certainly not in terms of mass, as we at one time thought.
Nevertheless we undeniably have a feeling that there is some sort of
conservation property here, and are driven to formulate it badly in
terms of a hypothetically constant number of protons and electrons. I
have long thought that Newton was groping after some very similar idea
when he so far forgot himself as to define mass as quantity of matter, a
definition perfectly meaningless to a rigorous and unsympathetic
interpretation. On the other hand, whatever meaning may reside in our
idea of conservation of matter, it certainly is not, in at least one
important respect, like the conservation of energy. For the energy of an
isolated mechanical system is a function of the frame of reference in
which it is described; merely by giving velocity to the reference frame
and altering in no way the mechanical system we may change its kinetic,
and so its total, energy by any direct amount. This does not even
remotely resemble ordinary matter. I cannot see that the operations
which are equivalent to the energy concept justify us in saying more
than that energy is a property of a material system; the operations do
not seem to give any unique meaning to a location associated with
energy.

We now ask what significance is to be ascribed to the sort of
conservation that energy does have. We restrict ourselves first to
mechanical systems. The motions of a mechanical system satisfy certain
differential equations of the second order, and the actual motion is to
be found by an integration of the equations. In the integral of a
differential equation certain constants appear which are determined by
the initial conditions, and are therefore the same during all the future
motion of the system; obviously these constants of motion correspond to
conservative properties. This reasoning can of course be at once
extended. Any system, mechanical or not, whose motion is determined by
differential equations, will have certain conservative properties. For
the systems of mechanics energy is one of the conservative functions;
others are momentum and moment of momentum. Energy is particularly
simple, in that it is connected with measurable properties of the system
by a simple formula (∑ ½ mv^2), and is furthermore scalar, which is
also a property of quantity of matter. But to go further and ascribe to
energy other properties of matter, such as localization in space, is
entirely overlooking the essential difference in the character of the
operation by which matter and quantity of energy are measured, that is,
overlooking the essential difference in their physical character.

The possible extension of the energy concept from mechanics to
thermodynamics receives a sufficient physical explanation in terms of
our views of the essentially mechanical character of thermal phenomena.
That the idea can be extended also to simple electrical or magnetic
systems, in which the effect of velocity of propagation is neglected, is
a consequence of the fact that in these systems the equations of motion
remain of the same general mechanical type, it having been shown by
Maxwell that the equations of such systems may be written in the
generalized Lagrangean form. When, however, we extend our formulas to
systems in which the velocity of propagation is important (that is, when
we consider the field equations in their general form) we find that the
Lagrangean equations no longer apply to matter taken by itself, and
energy is no longer conserved in the original sense. A new function
appears, however, which behaves mathematically in the same way that the
energy did before. The equations of motion of the system remain
Lagrangean in form if the mechanical parts of the system are
supplemented by the electric and magnetic fields in space. In this
extended form we have, therefore, a conservative function as before, and
the energy concept may be retained in this enlarged aspect. The physical
operations by which energy is determined are entirely altered, however,
and the physical character of the concept is changed. No more than
before is there justification for localizing energy in space, or
ascribing to it other properties of matter. Yet the materialization of
the energy concept, and the consequent desire that energy be localized
in space, is one of the strongest arguments in many minds for the
existence of a medium.

As far as I can see, therefore, the existence of conservative functions
is involved in the possibility of describing natural phenomena with
differential equations. That further there is a conservative function of
the precise form found in mechanics is a consequence of the particular
form of the equations and the nature of the forces. The question of the
significance of the fact that the forces of nature appear to be
conservative, with respect to this particular function of mechanics, is
of much interest, but it is not our immediate concern now. We are
interested rather to ask under what general conditions we shall have
conservative functions. Quantum theory strongly suggests that when we
pass to phenomena on a small enough scale, we may no longer be able to
employ differential equations in our descriptions, and hence the
previous reason for the existence of constants connected with the motion
disappears. Now there is one obvious remark to be made about this more
general situation. Whenever the future history of a system is so
connected with its present condition that we can retrace our way to the
present from any future configuration, we shall always have conservative
functions. For any future configuration contains certain fixed (or
conservative) features, in that we can reconstruct the unique present
from any future state. There is no reason to expect that the operations
by which we find the fixed features will always be simple, as in the
mechanical case. Now the determination of the future by the present, and
conversely the possibility of reconstructing the present from the future
(or the past from the present), is, we are convinced, a property which
is at least approximately true of phenomena down to a smaller scale of
magnitude than we have yet reached, and so we expect to find these
conservative functions in systems whose ultimate laws of motion are much
more general than any with which we are yet familiar. The particular
form of the conservative function depends on the character of the
system. That there is a scalar conservative function for ordinary
systems depends of course on particular properties of the system, but we
are at least prepared to find that a scalar conservative function does
not necessarily mean a differential equation of the second order.

The potential energy of a system has a particular significance with
respect to this point of view. In an ordinary mechanical system, the
potential energy simply measures the work done by the applied forces in
being displaced from the initial to the final positions; that is, the
potential energy is a measure of the deviation from the initial
position, and so measures a certain feature of the history of the
system. In the more general system, in which we may not have
differential equations, we may look for something analogous to the
potential energy which shall measure the displacement of the system from
its initial configuration. Such a measure is always possible as long as
the past can be reconstructed from the present (or the present from the
future). We recall a remark of Poincaré's[20] to the effect that we of
necessity must always have conservation, because if we have a system in
which conservation apparently fails, we merely have to invent a new form
of potential energy. This remark is obviously not of entire generality,
but applies only to such systems as those considered here, in which the
past may be reconstructed from the present.

[Footnote 20: Henri Poincaré, Wissenschaft und Hypothese, translated
into German by F. and L. Lindemann, Teubner, Leipzig, 1906, Chap. VIII.]

Of late there has been much discussion of the advisability, on the basis
of certain quantum phenomena, of giving up conservation as a principle
applied to the details of the emission and absorption of light,
retaining it only in a statistical sense. It seems to me that the
question here in the minds of physicists was always merely one of
convenience, and that few, if any, doubted the ultimate applicability of
the principle of Poincaré, or thought that we were here concerned with a
system of such great generality that the past could not be reconstructed
from the present. The question was merely whether those variables in
terms of which the potential energy is defined make close enough
connection with other things of immediate experimental significance, or
whether on the whole the retention of a potential energy is not more
trouble than is justified by its convenience, making a treatment from
the statistical point of view preferable. However, this is all now a
matter of more or less past history, because with the recent extension
of the experiments of Compton,[21] we seem to have experimental evidence
for the validity of the conservation law in detail for elementary
quantum processes, with a corresponding simple potential energy.

[Footnote 21: W. Bothe and H. Geiger, 2S. f. Phys. 32, 639-663, 1925. A.
H. Compton, Proc. Nat. Acad. Soc., II, 303-306, 1925.]

Going still deeper, however, there are quantum phenomena which still may
have to be treated by statistical methods, and this may mean giving up
conservation in detail. We have no experimental evidence, for example,
of what the electron is doing while jumping from one quantum orbit to
another. A situation like this merely means that those details which
determine the future in terms of the past may lie so deep in the
structure that at present we have no immediate experimental knowledge of
them, and we may for the present be compelled to give a treatment from a
statistical point of view based on considerations of probability. But I
suppose that no one, except perhaps Norman Campbell,[22] will maintain
that such a situation is more than temporary, or will cease to search
for consequences of these details of structure which may be open to
experimental verification.

[Footnote 22: Norman Campbell, Time and Chance, Phil. Mag. I, 1106-1117,
1926.]

Similarly, we cannot permanently be satisfied with a picture of
radioactive phenomena which represents radioactive disintegration as a
matter of chance.

The general conclusion to which all this discussion leads is that energy
is probably not entitled to the fundamental position that physical
thought is inclined to give it, but that it is a more or less incidental
consequence of more deep-seated properties, and that the character of
these deep-seated properties is subject to only the most general
restrictions, so that very little about the nature of the details can be
inferred from the existence of any energy function.



THE CONCEPTS OF THERMODYNAMICS


We shall not be concerned here with the many technical questions which
are the proper subject of treatises on thermodynamics, but shall attempt
an examination only of some fundamental concepts.

The most fundamental of these, which sets thermodynamics off apart from
the simpler subjects, is probably that of temperature. In origin this
concept was without question physiological, in much the same way as the
mechanical concept of force was physiological. But just as the force
concept was made more precise, so the temperature concept may be more or
less divorced from its crude significance in terms of immediate
sensation and be given a more precise meaning. This precision may be
obtained through the notion of equilibrium states. We have in the first
place the fundamental experimental fact that when a small body is placed
inside a large system, which we recognize by crude means as
comparatively invariable in temperature as time goes on, the small body
very soon acquires a steady condition, that is, it comes to equilibrium
with its surroundings. We now have the further experimental fact that if
the small body A is in equilibrium with its environment, and body B is
also in equilibrium with the same environment, there will be no change
of condition of A and B when they are brought into contact with each
other--that is, A and B are each in equilibrium with the other and also
with the environment and therefore, by definition, at the same
temperature as the environment. The temperature of the environment is
now measured in terms of some of the properties of A and B which crude
experience has shown change with the physiological temperature of A and
B. The physiological notion of temperature is thus made more precise by
being connected with the physical phenomenon of equilibrium.

Now it is at once evident that stated in this way without qualification
we have said things that are not true. It is not true in general that,
when A is in equilibrium with an environment and B is in equilibrium
with the same environment, A will be in equilibrium with B. Suppose, for
example, that the environment is a stream of water and A is a tiny water
wheel moving freely in its bearings, and that B is a similar wheel with
much friction. Then we know that B will become warm, and will not be in
equilibrium with A when brought into contact with it. Or we may choose
for A a mercury thermometer with bulb covered with putty, and for B a
similar thermometer with bulb sheathed in platinum, and we know that the
two thermometers will not register the same temperature in the water
stream. Or still more simply, we may try to read the temperature of the
air in our garden on any bright day with a silvered and with a blackened
bulb thermometer; we know that the two thermometers will read different
temperatures. It is evident, therefore, that we shall have to specify
much more carefully the conditions under which equilibrium holds if we
are to give precise significance to the temperature concept.

It seems fairly evident in the first place that we shall have to rule
out systems in which there is large scale mechanical motion; the simple
notion of temperature does not apply to a system moving with respect to
us. Only when the two thermometers A and B move with the same velocity
as the stream do we have three-fold equilibrium between the stream, A
and B. We may state this in another way by saying that the temperature
of a moving body must be measured on a thermometer stationary with
respect to the body, but this is only a matter of words, and properly
speaking the temperature concept applies only to a certain aspect of the
relation between two bodies mutually at rest. We here entirely neglect
relativity questions such, for example, as the proper way of correcting
for the change of dimensions of moving thermometers.

If now the body whose temperature we are measuring does not move with
the same velocity in all its parts, we may still give a meaning to local
temperature by dividing the body into parts so small that the velocity
of each part is essentially uniform, and measuring the temperature of
each part with a thermometer stationary with respect to it. We are now
confronted with the question of how far to carry the process of
subdivision. Suppose we have a fluid whose motion is completely
turbulent when measured with instruments of the ordinary scale of
magnitude. For such a fluid the fundamental equilibrium proportions hold
between two measuring bodies A and B and the fluid, provided that the
bodies A and B are so large that the motion is completely turbulent on
their scale of magnitude. We may then define the temperature of the
turbulent fluid from the standpoint of these large scale bodies. But we
may also define the temperature from the small scale point of view as
the average of the temperatures recorded by sufficiently small
thermometers, each moving with the velocity of a local bit of the fluid.
These two temperatures will in general be different, and we must more or
less arbitrarily select one which we define as the true temperature. It
would seem that the small scale temperature is the better one to choose,
because there is a certain degree of arbitrariness in specifying the
scale from which the motion shall be judged completely turbulent But on
the other hand, there are difficulties in the small scale definition,
because the turbulence may become more and more fine grained, until we
end with the motion of the molecules themselves, when the operations
certainly fail which give meaning to the temperature concept. In this
case of molecular turbulence, we are driven back to the large scale
definition, which obviously corresponds to ordinary physical practice.

It appears then that the temperature concept is not a clean cut thing,
which can be made to apply to all experience, but that it is more or
less arbitrary, involving the scale of our measuring instruments. In any
special case, the meaning of the temperature concept must be set by
special convention. In practice this does not often make difficulty,
because in the majority of cases there is no large scale motion with
respect to the thermometers.

Consider now the other aspect of the equilibrium problem suggested by
the thermometers with blackened and silvered bulbs in the sunshine. Our
common experience tells us how to deal with this situation effectively
enough for ordinary purposes. We recognize that the possibility of
temperature equilibrium is disturbed by the radiation, and we protect
the bulbs of the thermometers from the sun's radiation by appropriate
shields. But this only minimizes the difficulty. For the shield is
warmed by the sun, and in turn warms to a less degree by its radiation
the bulb of the thermometer within. We must recognize that every body,
no matter what its temperature, is always emitting radiation, so that
the bulb of our thermometer is always in a radiation field. At first
this puts us in a serious quandary as to the whole question of
equilibrium and the meaning of temperature. The situation is saved by
the experimental observation that there is a particular radiation field
which affects all thermometers equally, namely, the field inside an
infinite body all at the same temperature. Logically this looks like the
vicious circle again, for we have not yet defined what we mean by the
same temperature. But actually we avoid the circle here, as in so many
other physical cases, by a process of asymptotic approximation. The
procedure is perhaps something like this: we find that if we experiment
with larger and larger bodies, isolated and at great distances from
other bodies at approximately the same temperature as judged by crude
physiological sensations, two thermometers, identical except that the
bulb of one is blackened and that of the other is silvered, record more
nearly the same temperature as time goes on and as the thermometers are
sunk to greater depths in the body. In actual practice, of course, the
radiational opacity of most bodies is so high that these precautions
against the effects of external radiation can usually be entirely
ignored. At high temperatures, on the other hand, radiation has to be
explicitly dealt with.

The conclusion for us from these considerations is that operationally
the concept of temperature is tied up with that of radiation--the
equilibrium concept of temperature is strictly never exactly applicable;
it is only a limiting sort of concept applicable when the radiation
field is of a special sort, namely, that of a black body.

In spite of the explicit recognition which we have to give radiation in
defining temperature, we usually entirely lose sight of it in thinking
about the mechanism of ordinary physical processes, as for instance when
we picture the temperature of a gas as determined by the kinetic energy
of its molecules. Now I have no doubt that negligence of this sort can
be justified, but the necessary logical analysis is apparently
complicated, and involves a great many different sorts of experiment by
methods of asymptotic approximation, by which we establish the existence
of various sorts of physical constants, such as constants of emission
and absorption and reflection and scattering and fluorescence and
thermal conductivity. We do not need to make the analysis here, but I
believe that some time it would be worth while to attempt it. Such an
analysis will justify the principle so often used: that if a body is in
thermal equilibrium the various processes involved, such as radiation or
thermal conductivity, must when taken separately also be in equilibrium.
Doubtless, if our experience had been confined to higher temperatures,
like that of the sun, this notion of different mechanisms acting
independently would have been more difficult to acquire.

We next consider another fundamental concept of thermodynamics, that of
quantity of heat. We are at first perhaps inclined to think of this as a
comparatively straightforward concept, given immediately in terms of
experience, but an analysis of the operations by which we measure
quantity of heat will show that the situation is really most
complicated. Consider, for example, Joule's experiment in which the
mechanical equivalent of heat was measured by determining the rise of
temperature of the water in a container when stirred by paddles driven
by a falling weight. We do not question that the rise of temperature of
the water has its origin in the mechanical work done on it by the
paddles. But what about the rise of temperature of the container? We
shall doubtless say that part of this rise comes from heat communicated
to it by the warmer water in contact with it, and part from mechanical
work done on it by turbulent impact of the water. But by what operations
shall we measure what part of the energy communicated to the container
is heat and what part mechanical work? We try to give an idealized
answer to this question in terms of Maxwell demons stationed at all
parts of the boundary of the containing vessel with small scale
measuring instruments. To measure the heat entering at any point I can
see nothing else for the Lilliputian observers to do but to determine
the temperature gradient at every point of the boundary from temperature
observations at two different levels, and calculate the heat inflow from
the gradient and the thermal conductivity of the material of the
walls--there seems no way of measuring a flow of heat as such. The
inflow of mechanical energy must be calculated from a detailed knowledge
of the elastic waves and other large scale deformations of the walls.
Here again there is an arbitrary element in our procedure; if our
mechanical measuring instruments are on too gross a scale, we may miss
mechanical energy which we would catch with finer instruments.

This situation which we have just submitted to detailed analysis is, I
believe, typical of the general situation; it is not possible in the
general case to find anything which we can call heat as such. Without
further explicit examination, we can unambiguously speak of a body
losing or gaining heat only when there has been no energy interchange of
any other sort with other bodies. In such a case the heat is measured in
terms of the temperature change of the body. The heat concept is in the
general case a sort of wastebasket concept, defined negatively in terms
of the energy left over when all other forms of energy have been allowed
for.

The essential fact that a quantity of heat can by itself be defined only
in terms of a drop of temperature is somewhat obscured by the usual
method of thermodynamic analysis. In describing a Carnot engine, for
example, it is specified that the engine shall work between a source and
a sink so large that their temperature is not changed by the heat given
out or absorbed by them, so that the impression is natural that heat may
in some way be measured apart from temperature changes. This of course
is not the case; we merely require that the source and sink be so large
that their temperature changes are of a different order of magnitude
from those in the working substance itself, so that with respect to the
working substance, source and sink may be considered to be at constant
temperature.

Assuming now that we are able to measure quantity of heat in those cases
in which the concept has meaning, let us examine the first law of
thermodynamics, which we write in the form:

_dQ_ + _dW_ = _dE_

Here _dQ_ is the heat imparted to a given body by other bodies, _dW_ is
the work of all kinds done on it from outside, and _dE_ is the increase
of internal energy. Now if this equation says what appears at a naïve
first glance, it should say that we find experimentally that the
relation written always holds between the measured quantities _dQ_,
_dW_, and _dE_. We have seen that in the general case it is not possible
to assign a unique operational significance to _dQ_ and _dW_, and
presumably not to their sum. We ignore for the present difficulties of
this kind and confine attention on _dE_; how shall we measure it? I
believe it does not take much examination to convince us that there are
no physical operations for measuring _dE_ as such, and that therefore
the equation expressing the first law must have a different significance
from that which appears on the surface. This is often recognized in the
statement that the essence of the first law is that _dE_ is an exact
differential determined only by the variables which fix the internal
condition of the body, and not a function of the path by which the body
is carried from one condition to another. But what shall we mean by
internal condition, and how shall we be sure that we have found all the
variables required to specify it completely? Internal condition may be a
most complicated thing and require many variables, as shown by a piece
of iron with a complicated magnetic history or by a piece of aluminum
about to undergo recrystallization after overstrain. Here again I
believe there is no physical procedure by which general meaning can be
given to this concept of internal condition. In specific cases we can
state what the variables are which determine internal condition, and the
criterion that we have found the correct internal variables is that _dE_
shall be a complete differential in terms of them. The first law of
thermodynamics properly understood is not at all a statement that energy
is conserved, for the energy concept without conservation is
meaningless. The essence of the first law is contained in the statement
that the energy concept exists (or has meaning in terms of operations).

The first law is often thought to be one of the most general of physics,
but in a paradoxical sense it is the most special of all laws, because
no general meaning can be given to the energy concept, but only specific
meaning in special cases. The first law owes its complete generality to
the fact that no specific case has yet been found of so broad a
character that it cannot be included under one or another special case.

Examination will at once justify this view. Thus we find a great many
systems which are adequately described in terms of two variables,
pressure and temperature, in that a function of _p_ and _t_ can be found
such that its differential equals _dQ_ + _dW_. There are other systems
in which the six components of stress and _t_ completely fix the
internal condition in the sense that they determine a _dE_. In other
systems the specification of a magnetic field may be necessary, or an
electric, or a gravitational field. No case is known which cannot be
handled in terms of the action of external forces of the proper kind,
but there is no general procedure, and the first law owes its generality
to the exhaustive cataloging of special cases.

We may now return to the question left in abeyance above of the
ambiguity in _dQ_ + _dW_. In all the cases in which the specific
variables can be found which define _dE_, _dQ_ and _dW_ also have
meaning. Consider, for example, a gas, the internal condition of which
may be characterized in terms of _t_ and _p_. The mere fact that the
internal condition can be specified in terms of two variables, one a
mechanical variable, shows that the substance is mechanically
homogeneous. Being mechanically homogeneous, we do not have the
possibility of ambiguous values of _dW_ varying with the scale of the
measuring instruments, and in fact we know that dW = p dv. Similarly the
gas being homogeneous and at rest as a whole allows unique values for
_dQ_. Of course this cannot obscure the physical fact that even in such
a gas, when we go to a small enough scale, we find inhomogeneities
arising from the Brownian movement, etc. Practically our statement means
that the inhomogeneities are so fine grained that over a very wide range
of scale of the measuring instruments we find the same definite results.
The same sort of considerations apply to more complicated systems. If
_dE_ is a complete differential in terms of _t_ and six stress
components, this means again that the body is homogeneous, its condition
is determined by temperature and stress, which are the same throughout
the body, and again there is no possible ambiguity from the scale of the
instruments which measure _dW_ and _dQ_. It seems in general, then, that
if the body allows operations by which _dE_ acquires meaning, at the
same time _dQ_ and _dW_ are provided for. In working out this idea in
full detail, some care must be given to the question of order of
differentials. _dQ_, for unit time and unit volume, is strictly equal to
k∇^2 t, where _k_ is thermal conductivity, so that in determining _dQ_
the second derivatives of temperature are involved.

If the body is obviously not homogeneous, it is still a matter of
experience that it can be divided into small pieces, each of which are
by themselves sufficiently homogeneous, and the first law in its usual
form may be applied to each of the pieces.

Finally, we emphasize a fact already implicitly mentioned, namely, that
no physical significance can be directly given to flow of heat, and
there are no operations for measuring it. All we can measure are
temperature distributions and rates of rise of temperature. As at
present defined, a heat current is a pure invention, without physical
reality, for any determined heat flow may always be modified by the
addition of a solenoidal vector, with change in no measurable quantity.
If someone states that throughout all space there is a uniform heat
current of 10^6 cal./cm.^2 sec. in the direction of Polaris, no disproof
can be given, for such a stream is solenoidal, and as much heat flows
out of every closed surface in unit time as flows in. Such a solenoidal
flow is meaningless in terms of operations; we could give meaning to
such a flow only in terms of some slight modification of the solenoidal
condition introduced by the measuring instrument. In all ordinary
conditions the flow of heat given by the simple relation q = k Grad t
corresponds exactly to what our atomic pictures lead to expect in those
cases where the details of the picture can be worked out. But there may
be cases where it is advantageous to supplement the ordinary heat flow
(= k Grad t) by the pure fiction of a solenoidal flow, because in this
way it may be possible to account for new phenomena which appear when
the solenoidal conditions are slightly departed from. Thus if in a
conductor at uniform temperature carrying a steady electric current we
say that a heat current is also flowing proportional to the electric
current and therefore solenoidal, we may provide the possibility for a
simple correlation of phenomena found under those more complicated
conditions when an electric current flows in a conductor of non-uniform
temperature in a magnetic field. If it should turn out that the heat
current is _uniquely_ determined by considerations of this character,
then we have taken the first step away from the pure formalism which
this sort of thing otherwise is in the direction of giving physical
reality to the invention of "heat current."

There are other interesting questions of a fundamental thermodynamic
character, such for example, as whether the entropy concept has any
general significance apart from the scale of our measuring instruments,
and what is the operational significance of applying thermodynamic
concepts to radiation, but we shall not consider these questions here.



ELECTRICAL CONCEPTS


We now set ourselves the problem of finding the meaning of the various
concepts in terms of which we describe the behavior of electrical
systems, assuming that we understand what we mean by "electrical." We
start with the simplest electrical systems, namely, those in which we
deal with static phenomena on a large scale. In such systems there are
independent physical operations by which we may find the magnitude of
any charge, provided that it is effectively concentrated in a
geometrical point. The measurements involved in these operations are
measurements of ordinary mechanical forces; we assume that our knowledge
of mechanics has already taught us how to make such measurements. An
electrically charged body experiences forces, which may be measured by
tying a string to it and pulling on the string with a spring balance
hard enough to keep the body in equilibrium. Three charges are
numerically equal if when each is placed at unit distance from another,
in the absence of the third (or other charge), the forces are always the
same. If furthermore the forces are of unit magnitude, the charges are
defined as unit charge. Having obtained unit charge, we define the
magnitude of any other charge as equal to the force which it experiences
when placed at unit distance from unit charge. This of course is all
very trite; the important thing for us is merely that magnitude of
charge, or quantity of electricity, is an independent physical concept,
and that unique operations exist for determining it. These operations
presuppose the ability to perform certain operations of mechanics.
Having now learned how to measure electrical quantities, we discover
experimentally the inverse square law of force, and later arrive at the
concept of the electric field. As we have seen, the field is an
invention; here we shall use this concept only for the purpose for which
the invention was made, and shall not involve ourselves in any of the
implications of ascribing physical reality to the field. Notice that as
long as we deal only with point charges we do not have to define field
strength in terms of the limiting procedure of making the exploring
charge smaller, for the limiting small charge is necessary only to avoid
the reaction of the exploring charge on the positions of the charges
which generate the field. All this again is trite; the important point
is that the operations by which the inverse square law and the concept
of the field are established presuppose that the charge is given as an
independent concept, since the operations involve a knowledge of
charges. The operations also involve the measurement of forces by the
ordinary _static_ procedure of mechanics with spring balances. With the
means now at our command we establish one very important property of
electric charges, namely that the total amount of charge on an isolated
body of finite size is conserved, no matter how the charge is forced to
rearrange itself by the motion of charges on adjacent bodies.

By procedures exactly like those outlined above, we may treat all the
corresponding magnetic quantities; there is formal parallelism between
the two sets of phenomena, but there is the physical difference that we
have to realize a single magnetic pole by the device of using a very
long slender magnet.

We now give our electrical system more freedom, in that we allow the
charges to be in motion with respect to each other. Perhaps the most
immediate question which we now have to ask is whether charge continues
to be conserved when set in motion, or whether the total charge on an
isolated body is a function of its velocity? To answer this question we
must generalize the procedure by which we assigned a numerical value to
a stationary charge. Perhaps the simplest way is to allow two unit
charges each to move with constant velocity, remaining at unit distance
apart, and measure with a spring balance the force required to keep them
at constant distance apart. Now we immediately find that the force is
altered under these conditions, so that our first impulse is to say that
the charge is a function of the velocity. But as we experiment further,
we find that the state of affairs is very complicated; the force between
the two charges at any moment of their motion depends not only on the
charges, their distance apart, and their velocities, but also on the
angle between the line joining them and the direction of motion in the
lines. Further experiment of other kinds yields other information; it
requires a force to maintain a charge in uniform motion in a magnetic
field, or to maintain a magnetic pole in motion in an electric field. A
moving electric charge exerts a force on a stationary magnetic pole, so
that by definition the moving charge is surrounded by a magnetic field,
and similarly a moving magnetic pole is surrounded by an electric field.
Returning to our two moving electric charges, we are impelled to ask
whether, if all these complications are possible, the numerical constant
(unity for static charges) in the inverse square law of force is a
function of velocity as well as the magnitude of the charges themselves?
If we broaden the question in this way, as we apparently must, our
problem becomes indeterminate, for we are trying to answer two different
questions with a single kind of measurement, namely of the force between
moving charges. I have had no better luck on trying other methods of
measurement. Apparently the operations do not exist by which unique
meaning can be given to the question of whether the magnitude of a
charge is a function of its velocity. On realizing this situation, we
are at first embarrassed to know how to proceed, but we reflect that the
embarrassment is not of our own making, but corresponds to a physical
fact. The concept of charge as a unique and independent thing
essentially pertains only to static systems. We may extend the concept
to moving systems if we wish, as a matter of convenience to ourselves,
but must recognize that such an extension is an invention of ours and
not a reality of nature. Now we do make such an extension, and we make
it in the simplest possible way, that is, we define the charge on an
isolated body in motion as that which we should find on it if we reduced
it to rest and made measurements according to the regular static
procedure. That this is a convenient thing to do depends on the
experimental result that the charge so found is independent of the way
in which velocity is imparted to or removed from the body; in other
words, whenever the body is reduced to rest, the same charge is always
found on it.

Although this is pure definition on our part, it turns out to have a
most simple and convenient connection with experimental facts which were
discovered after the decision to treat a moving charge in this way was
made; the discovery is of the atomic structure of electricity. If then
we agree to call each elementary charge a constant independent of the
velocity, the total charge on a body becomes merely proportional to the
count of the total number of atomic charges on the body, which is
certainly highly convenient and suggestive.

Having now fixed what we mean by the magnitude of a moving charge, we
are ready to turn to the general problem of the behavior of any system
of charged bodies in motion. For the present we consider only phenomena
of the scale of everyday experience. The most general problem that has
meaning here is to determine all measurable properties of the system in
terms of those data which experiment shows can be arbitrarily specified.
Now we have already emphasized that the electromagnetic field itself is
an invention, and is never subject to direct observation. What we
observe are material bodies, with or without charges (including
eventually in this category electrons), their positions, motions, and
the forces to which they are subject. The forces are to be measured
according to definition in mechanical terms, either by the strains in
members of a framework if the system is in equilibrium, or in terms of
accelerations and masses if it is not in equilibrium. The
electromagnetic field as such is not the final object of our
calculations, but the calculation of it is only an intermediate
auxiliary step, convenient to make because our mathematical formulation
gives so simple a connection between electromagnetic field, charges, and
mechanical action that the latter can be calculated at once in terms of
the former. In fact the connection is so simple that in many cases we
have come to regard our problem as solved if we can compute the
electromagnetic field, overlooking the fact that the field has no
immediate meaning in terms of experience.

Electromagnetic theory now presents us with a solution of the general
problem; this solution is contained in the four-field equations of
Maxwell, the constitutive equations, and those additional equations
(quite often lost sight of) which give the forces exerted by the field
on electric charges, or currents, or dielectrics. Let us inquire how we
may set about testing the physical correctness of these equations. We
may begin with one of the simplest possible tests, and inquire whether
the equations are correct in stating that the force acting on a charge
moving in an electric field is simply the product of the charge and the
field strength. This, on the face of it, is a surprising statement. The
field itself is affected by the motion of the charges which generate it,
and it is natural to expect a converse effect. If, furthermore, we have
sympathy with the medium point of view, it is easy to think that
whatever it is in the medium that gets hold of a charge and exerts a
force on it will find it harder to take hold when the charge is in
motion.

In attempting to check our statement experimentally, the only additional
complication, as compared with the static case which we have already
checked, is afforded by the motion of the charge, for we have defined
the magnitude of a charge in motion, so there is no difficulty here, and
we may furthermore suppose that the field is generated by stationary
charges, so that we need not trouble to inquire whether the procedure by
which the field was originally defined is here applicable. The task of
checking the equation then reduces to the simple physical task of
measuring the force on the moving charge. How shall we do this? If the
velocity is low, we may tie a string to the charge and measure the force
with a spring balance (or its equivalent). But now an examination of the
equations shows that in more complicated phenomena perceptible
deviations from the static behavior are to be expected only at much
higher velocities than can be attained by towing charges with a string
and a spring balance, so that it is evidently necessary to check the
simple equation for the force on a moving charge also at high velocity.
Since at high velocity the spring balance method for measuring forces
fails, we are driven to the only procedure that we have, namely a
measurement in terms of the resultant acceleration, calculating the
force by Newton's first law of mechanics. But this involves a knowledge
of the mass of the moving body, which we recognize in general may be a
function of the velocity. Now we have already seen, in discussing the
concepts of mechanics, that the operations by which mechanical mass is
defined cannot be carried out at high velocities, so that either the
concept of mechanical mass becomes meaningless at high velocities, or we
must adopt another definition. In attempting to give this new definition
of mass at high velocities, we are driven to a result of special
relativity theory, namely that all mass, mechanical or electrical, must
be the same function of velocity. If now electrical mass can be found in
terms of velocity, our immediate problem is solved and we shall be in a
position to complete the experimental check of the equation. But as a
matter of fact, in order to determine electrical mass, we have to use
that equation which we are now engaged in trying to establish. Logically
we have again the vicious circle, the physical significance of which is
that independent operations do not exist for giving unique meaning to
the concept of force on a charge at high velocity.

We seemed so close to our goal a minute ago; that we may allow ourselves
to jump the logical chasm, and assume that the equation is correct.
Electrical mass now becomes a definite function of velocity, mechanical
mass the same function, and we are in a position to compare the actual
acceleration received by a charge in a field with that calculated by the
equation. Our conviction, on the basis of all experience up to the
present, is that the two accelerations will be found to agree.

The equation then does somehow make correct connection with experience
in that a consequence of the equation can be verified experimentally, in
spite of the fact that as the equation stands it is meaningless, because
the operations do not exist by which meaning can be given to the
individual terms. At low velocities the equation really says what it
seems to say, because the individual terms have meaning in terms of
operations; and, what is more, what the equation says agrees with
experiment. At high velocities the equation does not mean at all what
appears on the surface; by itself it has no meaning; it has meaning only
when considered as a member of a system of equations, and only in so far
as the system of equations makes by implication statements about nature
that have meaning in terms of operations that can be carried out
physically. The individual terms of the equation of the system do not
have meaning at high velocities, and in fact there are more terms than
there are independent physical operations.

An exact analysis from the operational point of view of the significance
of the equations at high velocities has perhaps never been made, and is
not necessary for our immediate purpose. The discussion has brought out,
however, that the number of physically independent concepts has been cut
down by two at least, in that we have made purely formal definitions of
the meaning of quantity of electricity, and of the force exerted by a
field on a charge at high velocity. There is no reason to think that
there is anything unique about this analysis, or that formal definitions
might not have been given to other concepts than charge and force. We
can only state that as far as physical content goes the equations have
at least two degrees of freedom. It should then be possible to find
quite different sorts of equations which agree equally well with
experience. In particular, since we have seen that the force on a moving
charge has no meaning in terms of independent operations, it should be
possible by arbitrary definition to make this force any function of
velocity that we please (of course reducing to the proper value at low
velocities), and then to determine the other equations so that the
entire group of equations is consistent with experiment. So far as I
know, no one has tried to give such a modified set of equations, and
indeed there is no particular reason why anyone should bother to do
this, because the present equations are simple enough, and the modified
equations, although perhaps differing greatly in appearance from the
present ones, would have no advantage in any greater or different
physical content.

But there is no reason to think that the present state of affairs will
always continue. We have seen that the decrease in the number of
concepts corresponds to our inability to measure as many sorts of
physical things at high velocities as at low. Now it is the task of the
future experimenter so to refine the possibilities of measurement at
high velocities as to restore these two degrees of freedom. In
particular, mass should be made measurable in mechanical terms at high
velocities. When this restoration has been made, and all the quantities
in our equations receive independent physical meaning, the significance
of the equations in terms of operations will be quite altered, although
the formal appearance will be unchanged. We must then be prepared to
find, as always when we change the range of phenomena, that the
equations in their present form do not correspond to the facts at all,
and that one of the alternative forms allowed by our present two degrees
of freedom is the correct form. But until the new experimental facts
have been obtained, it seems hardly worth while to attempt to specify
the doubly infinite variety of forms which the equations might have
consistently with present experiment.[23]

[Footnote 23: Since this was written, a paper has appeared by V. Bush,
Jour. of Math. and Phys., vol. V., No. 3, 1926, in which it is shown
that there are advantages in supposing the charge of an electron to
change when it is set in motion.]

So far we have discussed the extension of ordinary electric phenomena in
only one direction, to high velocities. There is another extension which
is much more important physically, namely to very small scales of
magnitude. This extension is necessary to an understanding of the
properties of matter in bulk, the electrical nature of the atom having
been once established. Our problem is to show how the statistical
average of the behavior of a large number of electrons gives the large
scale effects which are within the reach of observation, and which are
described by the equations we have just discussed. To get this
statistical average we must be able to calculate at least certain
features of the behavior of the individual electrons, which means that
we must know the form of the equations down to dimensions of the order
of those of an electron, or smaller. Now if one contrasts the scale of
the supposed dimensions of the electron with the smallest dimensions on
which we can make independent experimental verification of those
equations, he must admit that there is an enormous chance for change in
the type of equation beyond the limit that we can reach by direct
experiment, and the chances of guessing the correct extension of the
equation to small dimensions are correspondingly almost vanishingly
small. (We may perhaps say that experiments on the Brownian motion on a
scale a good many atoms in diameter bring us the closest possible
directly, which means that we are 10^6 or 10^7 fold away from electronic
dimensions.) In spite, however, of the apparently enormous chances
against it, this program of extending the field equations to small
dimensions and following out the consequences was exactly the program
which Lorentz set himself.[24] That Lorentz saw that such a program
might be carried through must be recognized as a vision of extraordinary
genius, and that he was willing to devote to it the years of arduous and
detailed calculation that he did is evidence of a pertinacity of purpose
of the highest moral order.

[Footnote 24: See for example, H. A. Lorentz, The Theory of Electrons,
B. G. Teubner, 1916.]

We now have to examine critically this program and to inquire what is
the significance of the measure of success that Lorentz attained. The
precise extension of the equations that he made was very simple, for the
large scale equations of Maxwell were taken over with as little change
as possible. The equations are so familiar that it is not necessary for
us to write them in detail; they express relations between the electric
and magnetic force vectors (force and induction now becoming the same
thing, the difference between them in ponderable bodies being one of the
things that is to be explained in terms of the electrons), the space
density of electric charge, its velocity, and the force acting on
elementary charge. We have to notice that although formally the
equations have changed little in appearance, nevertheless the physical
content, as judged by the operations, has changed a great deal.
Consider, for instance, the meaning of charge density. In the Maxwell
equations, _ρ_ was merely the number of discrete elementary charges per
unit volume, the distances between these charges being supposed so small
compared with the scale of the phenomena involved that their average
effect could be fairly represented in terms of their numbers. In the
Lorentz equations, on the other hand, _ρ_ has a value different from 0
only inside the electron; everywhere else _ρ_ = 0. Now an examination
of the previous discussion, in which we questioned whether the magnitude
of the charge might be a function of its velocity, will show that there
are no physical operations whatever by which meaning can be given to
_ρ_ at individual points inside an electron. There is a single
condition on this _ρ_, namely, that its integral throughout the total
volume assigned to the electron shall equal the total static charge of
the electron. Obviously a single scalar condition is a pretty blunt tool
with which to attempt to determine a point function throughout a volume.
Again, the equations talk about the velocity of the charge at interior
points of the electron; what possible physical operations are there by
which meaning can be assigned to the velocity of an amorphous
structureless substance in regions inaccessible to experiment? Here
again, the concept as a detailed description of the behavior at a point
has become meaningless, and again there is a single integral condition,
namely, that the _v_ associated with every _ρ_ must be such that when
integrated over the volume of the electron it will give a total
transport of charge equal to that carried by the electron in its motion.
This again is a single condition on a function distributed through
space. Still again, the equations contain the electric and magnetic
vectors at points inside the electron. What is the possible meaning of
these field vectors in terms of operations? Our procedure for finding
the field at a point involves by definition finding the force on an
electric charge placed at that point. But there is no charge smaller
than an electron, and the procedure degenerates into a fiction. Again
there is a single integral condition on the field vectors; the integral
of the force on the assumed charge density when taken over the total
volume of the electron must give a value corresponding to experiment.
Except for this single condition, the concept of the field at points
inside the electron is an invention without physical reality. Not only
is the field concept meaningless at points inside the electron, but it
is meaningless at points outside within a certain distance, because the
exploring charge can never be made smaller than the electron itself, and
so can never come closer than a certain distance.

The actual state of affairs is much worse than has already appeared. It
was shown in the discussion of space and time that no independent
physical meaning can be attached to lengths and times as small as have
to be assumed in describing the behavior of the individual electrons.
The operations Div, Curl, d/dt which enter the field equations are,
therefore, physically meaningless as they stand; they have only a
mathematical meaning which begins to acquire physical complexion in a
most complicated way when the equations are integrated over large enough
volumes.

It is evident, therefore, that the concepts which enter the field
equations have entirely lost their large scale significance; they have
become blurred, fused together, and fewer in number. A precise analysis
of this situation has probably never been attempted and would obviously
be difficult: it would be interesting to know at least how many really
independent concepts there are at this order of phenomena. An attempt at
an analysis would probably be worth while from a physical point of view
in suggesting possible experiments by which the number of physically
independent concepts could be extended.

Since the quantities in the field equations are meaningless in the naked
form in which they enter the equations, it is meaningless to inquire
whether the equations as they stand are true or not. In our present
state of experimental knowledge it is also meaningless to ask whether,
for example, the inverse square law between electric charges continues
to hold, or whether an accelerated charge radiates. These questions have
meaning only when applied to phenomena on a scale large enough to
correspond to possible experiment.

There is a rather interesting obverse to the statement that it is
meaningless to ask whether the field equations are true, namely, that it
may not be meaningless to state that they are false. A statement is not
true unless it is true in every particular, but it is false if it is
false in a single particular. If we can show that a single consequence
of the field equations of Lorentz, when integrated or averaged in such a
way as to correspond to experimental possibilities, is false, then the
equations must be false. It seems that, regarded as a complete
description of physical behavior on a small scale, the equations must be
judged false, because they contain no suggestion of quantum phenomena.

Even if we have to recognize that the equations are false, there can be
no question that they correspond to an important part of reality, and
that they have been of the greatest service to physics. What is the
significance of the success that they have attained? It is to be noticed
that all the phenomena to which the Lorentz equations have been
successfully applied, although not large scale phenomena in the ordinary
sense of the word, are nevertheless phenomena involving the coöperation
of a number of atoms, and that the equations unquestionably fail when
applied to phenomena involving single electrons. It appears from our
best present evidence that on a small scale the behavior of nature is
governed by quantum principles and is therefore quite different from
large scale behavior, Which we have seen is governed by the Maxwell
equations. There must of course be a transition zone in which the
character of phenomena changes from quantum to Maxwell. Now any program
like that of Lorentz is almost inevitably bound to begin to give correct
results when we get up as far as the transition zone, for the simple
reason that the relations of Maxwell have been put into the equations
and are always there ready to appear as soon as the quantum relations
begin to give way. The physical significance of the success of the
Lorentz program seems to be that the transition from Maxwell to quantum
takes place at a stage pretty far down toward the individual atoms. To
find the precise details of the transition from Maxwell to quantum
phenomena constitutes a large part of the program of the immediate
future.

All this skepticism about the classical work of Lorentz is likely to be
rather irritating or depressing, particularly if one attempts to imagine
what other course could have been adopted. Indeed it does seem that we
find ourselves in a real quandary; Lorentz was practically forced,
because of the character of the mathematical tools at his command, to
take the course that he did, in spite of any recognition of the physical
meaninglessness of the mathematical operations. We have already seen
that conventional mathematics does not correspond to the physical
reality; it cannot easily make a qualified statement subject to
limitations, and it recognizes no difference between the physically big
and the physically little and the corresponding change in the
operational meaning of its symbols. It begins by being a most useful
servant when dealing with phenomena of the ordinary scale of magnitude,
but ends by dragging us by the scruff of the neck willy nilly into the
inside of the electron where it forces us to repeat meaningless
gibberish. Larmor recognized this, and in his electron theory, developed
practically contemporaneously with that of Lorentz, endeavored to treat
electrons as wholes, and not to make meaningless statements about their
insides.[25] But he was much less successful than Lorentz in making his
analysis give physical results, and one may suspect that it was at least
in part due to difficulty with his tools.

[Footnote 25: Joseph Larmor, Æther and Matter, Cambridge University,
Press, 1900. In this book the electron is treated as a point singularity
in the ether.]

What we should like to be able to do is easy to see. The things that go
into our equations must have independent physical meaning, and the
character of our mathematical formulation should change to keep pace
with the change in the physical operations which give meaning to the
terms. For example, electrical density has a meaning for large scale
phenomena, but means nothing on a small scale. Our ultimate electric
unit is the electron; when we get down to this scale of magnitude, our
mathematics ought to be making statements about the relative behavior of
discrete electrons, and not mention so much as by implication the
density at points inside an electron. But this sort of thing we
apparently cannot yet do; the proper mathematical language has not been
developed. Such a language, when developed, must not only be able to
resist the temptation to burrow inside the electron, but must also try
to get along without the field concept, which we have seen is liable to
so much physical abuse, and must reduce effects in complicated
electrical systems to the ultimate elements that have physical meaning,
namely, a dual action between pairs of electrical charges, with no
implications about physical action where the charges are not.



THE NATURE OF LIGHT AND THE CONCEPTS OF
RELATIVITY


We have already discussed several aspects of the theory of relativity in
connection with the relation to it of some of our fundamental concepts.
There are still other topics connected with relativity which demand
attention; most of these involve the properties of light. It will now be
convenient to discuss together the properties of light and these
concepts of relativity. We restrict our discussion of light to those
simple properties which bear on the theory of relativity.

Practically all our thinking about optical phenomena is done in terms of
an invention, by means of which these phenomena are assimilated to
those of ordinary mechanical experience, and so made easier to think
about. To realize that invention has been active here, we must think
ourselves back into that naive frame of mind in which experience is
given directly in terms of sensation. The most elementary examination of
what light means in terms of direct experience shows that we never
experience light itself, but our experience deals only with things
lighted. This fundamental fact is never modified by the most complicated
or refined physical experiments that have ever been devised; from the
point of view of operations, light means nothing more than things
_lighted_. Now experience shows that these things lighted may stand to
each other in varied relations; in attempting to reduce these relations
to order and understandability we make a certain invention. This is
prompted by several cardinal experimental facts: in the first place,
things lighted have a simple geometrical relation to each other, in that
screens placed on straight lines between the lighted objects may
suppress the illumination of one or the other and themselves become
illuminated. This leads to the concept of rectilinear beams of light,
which is no more than a description of the geometrical relation between
lighted objects. Then we have the experimental fact of the asymmetrical
relation of the lighted objects, described in terms of sources and
sinks. Finally, we have the discovery made at a much later stage, and
not possible until physical measurements had reached a high refinement,
that light has properties analogous to the velocity of material things.
This was first discovered in connection with astronomical phenomena in
the shift of the time of eclipse of Jupiter's satellites and in
aberration, but was later found to hold for purely terrestrial
phenomena, in that a beam of light reflected from a distant mirror does
not return to the source until after the lapse of a time interval that
can be measured with means sufficiently refined. This property of return
after the lapse of time is precisely like that of material things, such
as a messenger despatched for an answer, or a ball or a water wave
bouncing from a wall. These various properties of light lead quite
naturally and almost inevitably to the invention of light as a thing
that _travels_, "thing" not necessarily connoting material thing.

The question now for us is whether we shall regard this as a mere
invention, made for convenience in thinking, or shall go further and
ascribe physical reality to it, that is, shall we think of light as
capable of independent physical existence in the space between the
matter that constitutes the source and the mirror? Now in spite of the
resemblances pointed out above, there is at least one universal and
fundamental difference between a thing that travels and light. We have
independent physical evidence of the continued existence of the ball,
for example, at all intermediate points of space; we can see it, or hear
it, or feel the wind in the air as it passes, or even touch it. All
these phenomena are independent of the initial and terminal phenomena,
and hence by our criterion for the physical reality of an invention, we
are justified in ascribing physical reality to the ball in transit. But
with the beam of light it is entirely different; the only way by which
we can obtain physical evidence of the intermediate existence of the
beam is by interposing some sort of a screen, and this act destroys just
that part of the beam whose existence we have thereby detected. There is
no physical phenomenon, whatever by which light may be detected apart
from the phenomena of the source and the sink (understanding that a
mirror is included in the idea of, sink); that is, no phenomenon exists
independent of the phenomenon which led us to the invention of a thing
travelling. Hence from the point of view of operations it is meaningless
or trivial to ascribe physical reality to light in intermediate space,
and light as a thing travelling must be recognized to be a pure
invention.

The status of light is exactly the same as that of an electric field;
there is not the slightest warrant for ascribing physical reality to
either at points of empty space--light and field-at-a-point have no
meaning until we go there and make experiments with some material thing.
Of course the electromagnetic theory of light makes this resemblance
inevitable, provided the theory and our views of the nature of light and
the field are correct.

It cannot be denied that there are some phenomena which uncritically
considered appear to justify thinking of light as a thing that travels;
these will now be discussed. Probably the argument to which most
significance is usually ascribed is derived from the phenomena of
energy. The passing of light from source to sink is accompanied by the
transfer of energy. But energy is conserved, so that we have to ask
where the energy is in the time interval between the emission of light
from the source and its absorption by the sink. There is an obvious
answer: the energy is in transit, of course, somewhere in the
intermediate space between source and sink. If we think of light as
propagated through a medium, then the medium is such that energy may
reside in it, as in the electromagnetic theory of light, or if light is
more material and ballistic in character, the thing that travels has
itself energy. We notice in the first place that the conservation
principle involves the time concept, because what we mean by
conservation is that the total energy of the universe, at a fixed
instant of time, is constant. That is, we have to integrate over all
space the local energy at a definite instant of time, and this involves
spreading the time concept over all space. It is further evident that
unless we spread the time concept over space in the right way we shall
not get conservation. The proof that it is possible to spread the time
concept over space in such a way as to give conservation involves a
knowledge of the properties of light. It would seem, then, that we ought
not to assume conservation in deducing the properties of light, when a
knowledge of the properties of light is necessary to establish
conservation. These considerations cannot be accepted as final, however,
until a detailed analysis has been made, and this would be most
complicated. But there is a more important consideration derived from
our previous critique of the energy concept, namely, that there is no
basis for asserting that energy is localized in space at all; energy is
not a physical thing, but rather what we would call a property of a
system as a whole. If this view of energy be granted, the whole energy
argument for light as a thing travelling, and also for the existence of
a medium, falls. I believe that similar considerations apply to any
arguments from the conservation of momentum.

The possibility of detecting light in apparently empty space by a screen
constitutes perhaps the most immediate reason for considering light as a
thing that travels. This point of view I believe is characteristic of
the entire attitude of Einstein in deducing the theorems of the special
theory of relativity. Einstein's light signal is for the purposes of the
deduction thought of as a simple spherical wave spreading from the
source and capable of being watched as it spreads by an observer outside
the system, in much the same way that a water wave can be watched. Of
course the light signal cannot actually be watched in transit, but we
can come fairly close to this ideal by placing screens at any point we
please to make the wave visible. It is true that the mere act of showing
the existence of the light destroys that part of the beam whose
existence is detected, but the screen needs only an infinitesimal amount
of light to make it visible, and so by the usual physical argument we
may suppose that the detecting screen produces only an infinitesimal
modification of the total original light.

Our satisfaction with this picture evaporates if our present quantum
views of the nature of light are correct. We can no longer think of the
spherical light pulse as of irreducible simplicity, but it is an
exceedingly complicated thing, perhaps more complicated than a gas from
the point of view of kinetic theory, and simulates simplicity by some
sort of averaging of the effects of the elementary quantum processes of
which it is composed. If the principles of relativity are to continue to
be regarded as fundamental, or even if they are to remain intelligible,
we must apply our reasoning, not to spherical wavelets, but to the
elementary process of which these wavelets are composed. Now the
elementary quantum act is essentially a twofold thing: there is a
discrete act of emission at some discrete material particle, and the act
is consummated by another discrete act (absorption or scattering) at
some other discrete particle. We cannot yet fully characterize the
details of this twofold process, but have to connect the place at which
absorption takes place with the place of emission by statistical
considerations. It is evident, however, that to think of emission as
starting some process like a spherical wavelet travelling like a thing
through space presents an entirely incorrect view, because in the wave
there is no hint of the discrete place which is to terminate it. We may
say crudely that there is no way by which the wave can know what
discrete material particle is to complete the emission process. We may
perhaps try to save the situation by remembering that a spherical wave
is polarized and so has a unique direction associated with it; but
further examination shows that this does not help, because the unique
direction is that of no energy flow, and absorption can take place in
any direction _except_ this. It appears then that instead of being a
help, the thing travelling point of view is a positive embarrassment
when we try to picture by means of it the essentially twofold nature of
the elementary quantum act.

Another plausible argument for light as a thing travelling may be
deduced from our principle of connectivity. Imagine a dark lantern with
a shutter that can be opened or closed so as to emit a momentary flash
of light, a distant mirror, and a receiving instrument near the source.
One of the properties of light that we always assume is that no
permanent trace of the act of emission is left behind in the source. The
most minute examination of all the details of the lantern and its
surroundings at some time after the emission of the flash has not yet
shown any phenomenon that betrays a remembrance of the emission of the
flash, unless perhaps we measure the total energy or momentum and have
some way of knowing what the energy or momentum would have been if the
flash had not been emitted, and in any event we cannot specify the
moment in past time when the signal was emitted. In the same way we
cannot tell from an examination of a mirror whether it has at any time
in the past reflected a beam of light. Consider now two systems, each
consisting of a source and a mirror distant 3 x 10^10 cm., identical in
all respects, except that in one a light signal was flashed from the
source 1.5 seconds ago, and in the other only 0.5 seconds ago. According
to our hypothesis, the most complete examination of source and mirror in
either system fails to show the slightest difference, but nevertheless
there is something essentially different about the two systems, for in
one a light signal arrives at the screen in 0.5 seconds, and in the
other not until 1.5 seconds. This violates what we have suggested might
be regarded as the cardinal and most general principle of all physics,
the principle of essential connectivity, which states that differences
between two systems must be associated with other differences. A most
obvious and simple way of maintaining our principle is merely to point
out that the system really included more than we investigated: the
system properly consists of source, mirror, screen, and all intermediate
space, so that if we had examined intermediate space we should have
found light there in transit in different positions in the two systems,
thus correlating with the differences in subsequent history. This
argument appeals to me as perhaps the strongest that can be advanced for
the view of light as a thing travelling. But it seems in no way
conclusive. Our principle of essential connectivity made no mention of
the time concept, but we have somehow smuggled it in making the
application above. We sought to give a complete description of our
system at some one instant of time, and this involved spreading the time
concept over space. This itself is a questionable operation and may be
done in different ways. But, more important, what is the justification
for supposing that the system can be completely described by giving a
complete description of all the measurable parts of it at some one
epoch? We have seen that in the most general case the principle of
essential connectivity must recognize that the concept of "initial
condition" of a system involves all the past history, and we may have
here a case in point. The answer can be given only by experiment. In
dealing with ordinary experience, when we do not have to distinguish
between local and extended time, and are not dealing with optical
phenomena, there can be little question that experience at least
approximately justifies the expectation that future behavior is
determined in terms of the present condition and that present condition
may be specified in terms of the results of present operations performed
in the system. But before extending this principle to phenomena in which
we have to distinguish between local and extended time, we have to
answer just that question which we are now considering; namely, whether
there are physical phenomena taking place in apparently empty space, and
whether therefore empty space has to be included in the system. We find
ourselves again treading the vicious circle. Perhaps experience will
show that the extension of the principle of connectivity to optical
phenomena involves something like this: namely, the future at any point
in a material system is determined by a complete description of the
_present_ state of the system in the immediate vicinity, and by a
_history_ of the behavior at more distant points, this history extending
over longer and longer intervals of time as the point becomes more
remote.

I believe, however, that these possibilities will not seem very
satisfactory, and that most physicists will discover in themselves a
very strong disposition to feel that the future is determined in terms
of a complete description of some sort of instantaneous configuration,
time being extended in some suitable way over space. This instinctive
demand that the future be determined in terms of the present may easily
be consistent with the optical phenomena in our two systems consisting
of source, mirror, and screen, without involving the material existence
of light in empty space, provided that our assumption that the emission
of light leaves behind it no permanent record in the source was
incorrect. It may be that detailed examination of a source after
emission will disclose permanent traces, from which the instant of
emission may be found by extrapolation. If the conviction of determinism
of the future is strong, the physicist may well be impelled to search
here for new phenomena indicating such a memory of emission.

Let us now inquire how our physical structure might be affected if we
should give up the identification of light with a thing travelling. One
consequence is that light need no longer be thought of as having the
property of velocity, since velocity, in terms of immediate experience,
is a property of things moving from place to place. Giving up the
concept of light as a thing travelling would enable us, then, to adopt
an alternative method of describing nature with a different concept of
velocity; we have seen that it is possible to define velocity in terms
of operations different from the usual ones, in such a way as to give
the usual numerical results at small velocities, but different results
at high velocities, and in particular to give an infinite velocity for
light.[26]

[Footnote 26: No difficulty arises from the asymmetric character of
light in assigning an infinite velocity to light because those physical
operations by which we discover which is the source and which the sink
are entirely distinct from the operations by which a velocity is
measured; or in other words, even in the limit, it still has meaning to
say that an infinite velocity has a direction associated with it.]

There is now no objection to an infinite number associated with light,
if we no longer think of light as having physical velocity. We may, if
we like, continue for the sake of convenience to talk of the velocity of
light, clearly understanding that the infinite value which must be
ascribed to this velocity corresponds to the fact that the physical
concept of velocity does not apply in all respects to light. We should
now have to revise our process for extending the time concept over
space, because this was formerly so done as to give light a finite
velocity. We are now to make this velocity infinite, which is obviously
to be done simply by setting a distant clock on zero at the instant it
receives a light signal flashed from our clock at its zero. The behavior
of material things now takes on a simple aspect--there is no longer a
finite upper limit to the velocity that can be given a material thing,
and light has no longer the paradoxical property of bearing the same
finite relation to each of two material systems which differ from each
other by a finite amount (that is, the first postulate of relativity
that the velocity of light is 3 x 10^10 in all reference systems). Light
instead now bears the relation of infinitude to each of two systems
which differ from each other by only a finite amount, and this is
natural from the mathematical point of view.

However, all is not simplified by this change in the method of setting
clocks, but a price has to be paid. The price is that we have to give up
the simple connection between the velocity of a thing and its "go and
come" time. Our changes have not affected local time; the time of
passage of light to a distant mirror and return to the source is not
changed, and is therefore still finite, although we describe the
velocity of light as infinite. Now examination shows at once that there
is no immediate connection between the concepts of "velocity" and of "go
and come" time, because the operations involved are different. A
measurement of a linear velocity according to our definition involves
two clocks at two different places or else a clock travelling with the
object, while "go and come" time demands only a single clock at a single
place, and also involves necessarily a reversal of direction of motion
in the object under measurement. We see then that, according to the
definition adopted for velocity, we have the choice either of doing as
Einstein did in the restricted theory of relativity and making "go and
come" time very simply related to velocity; or we may say that refined
physical measurements show that something of significance happens when
the direction of motion is reversed, and that phenomena are not
symmetrical with respect to a reversal of direction. The asymmetry which
results from reversing the direction of motion we may visualize as a
sort of curvature in space and time, as of a small piece of an arc of a
circle bent back on itself, with the two ends diverging. This
alternative way of treating velocity would mean that velocity can be
measured simply only by a specially situated observer; this need not be
considered disturbing, because in fact the operations have been defined
only with respect to such an observer.

Which of these two possible treatments of velocity shall be adopted is
to a certain extent a matter of convenience, determined by the sort of
phenomena in which we are most interested and wish most to simplify.
Einstein's chief concern was with optical phenomena, so that the motive
for his choice is evident. In this choice of Einstein it is not very
evident that the desire to make "go and come" time simply connected with
velocity played a very prominent part, but it seems rather that the
desire to think of light as a thing travelling, with a finite velocity,
was much more influential. This way of thinking of light is fundamental
to all the treatment of restricted relativity; without this sort of
picture all the mathematical deductions would lose their simplicity and
convincingness, for in all the deductions we inevitably think of
ourselves as an observer from outside, watching a thing that we call
light travelling back and forth like any physical thing.

Now there can be no doubt that, when choice is possible, convenience and
simplicity are important considerations; but I believe that there is
another much more important consideration, namely, the most perfect
reproduction possible of the physical situation. It seems to me that it
is very questionable whether Einstein, and all the rest of modern
physics, for that matter, have not paid too high a price for simplicity
and mathematical tractability in choosing to treat light as a thing that
travels. Physically it is the essence of light that it is not a thing
that travels, and in choosing to treat it as a thing that does, I do not
see how we can expect to avoid the most serious difficulties. Of course
the whole problem of the nature of light is now giving the most acute
difficulty. The thing-travelling point of view, even as treated by
Einstein, does not land us in a situation which is at all satisfying
logically. We are familiar with only two kinds of thing travelling, a
disturbance in a medium, and a ballistic thing like a projectile. But
light is not like a disturbance in a medium, for otherwise we should
find a different velocity when we move with respect to the medium, and
no such phenomenon exists; neither is light like a projectile, because
the velocity of light with respect to the observer is independent of the
velocity of the source. On the other hand, in aberration we have a
phenomenon similar to that shown by projectiles. The properties of light
are more like those of a projectile than is perhaps commonly realized,
as is shown in the papers of La Rosa[27] on the ballistic theory of
light. The properties of light remain incongruous and inconsistent when
we try to think of them in terms of material things.

[Footnote 27: M. La Rosa, Scientia, July-August, 1924.]

Einstein's restricted relativity has made a great contribution in so
grouping and coordinating the phenomena that they can all be embraced in
a simple mathematical formula, but he does not seem to have presented
them in such a light that they are simple or easy to grasp physically.
The explanatory aspect is completely absent from Einstein's work.

In view of all our present difficulties it would seem that we ought at
least to try to start over again from the beginning and devise concepts
for the treatment of all optical phenomena which come closer to physical
reality. No one realizes more vividly than I that this is a most
difficult thing to do. If we are ever successful in carrying through
such a modified treatment, it is evident that not only will the
structure of most of our physics be altered, but in particular the
formal approach to those phenomena now treated by relativity theory must
be changed, and therefore the appearance of the entire theory altered. I
believe that it is a very serious question whether we shall not
ultimately see such a change, and whether Einstein's whole formal
structure is not a more or less temporary affair.

Although it is exceedingly difficult to forsee what the treatment of the
future will be like, it is easy to surmise certain of its features. In
essence the elementary process of all radiation perceived as radiation
is twofold. There is some process at the source and some accompanying
process at the sink, and nothing else, as far as we have any physical
evidence; furthermore, the elementary act is unsymmetrical, in that the
source and the sink are physically differentiated from each other. This
is the most complete expression of the physical facts; there is nowhere
any physical evidence for the inclusion of a third element (the ether).
Therefore all the phenomena apprehended by an observer (and this
embraces all physical phenomena) can be determined only by the source
and the sink and the relation to each other of source and sink, for
there is nothing else that has physical meaning in terms of operations.
This formula covers not only the possibility of such first order
phenomena as aberration and the Döppler effect, but also shows that
such second order effects as that looked for by Michelson and Morley
must be non-existent. It will thus be seen that some of the consequences
of relativity theory are implicitly contained in certain very broad
points of view. One interesting question that must be answered before we
can get very far with a new treatment is whether the elementary optical
process is of _necessity_ twofold, or whether we may have emission
without absorption, that is, radiation into empty space. Lewis seems to
imply in recent papers that this is not possible.[28] The astronomers
have already pointed out difficulties in explaining phenomena like the
temperature equilibrium of the planets if we suppose this is the case.

[Footnote 28: For example, in the book: G. N. Lewis, The Anatomy of
Science, Yale University Press, 1926, p. 129.]



OTHER RELATIVITY CONCEPTS


We now turn to some of the other concepts of relativity. One of the most
important of these is the "event"; in fact this concept is made
fundamental by Whitehead.[29] We have already discussed the concept of
"event" under the "identity" concept with which it is closely involved.
The event is usually thought of by Einstein as merely an aggregate of
four coordinates, three of space and one of time. The principle of
general relativity, namely, that the laws of nature shall be of
invariant form, when formulated mathematically, involves the assumption
that nature may be analyzed into events, and is expressed by the
requirement that the mathematical relations between the coordinates of a
chain of events shall be invariant. The same idea is also expressed by
Einstein in another form, namely, that nature may be completely
characterized in terms of space-time coincidences. In elaborating this
idea, Einstein assumes that the results of all measurement may be given
in terms of such coincidences.

Now it appears to me most questionable whether the analysis of nature
into events is possible or sufficient. With regard to the coincidence
point of view, it seems perfectly obvious that the world of our
immediate _sensation_ cannot be described in terms of coincidences; how,
for example, shall we describe in terms of space-time coincidence the
photometric comparison of the intensity of two sources of illumination,
or the comparison of the pitch of two sounds, or the location of a sound
by the binaural effect?

[Footnote 29: A. N. Whitehead, An Enquiry Concerning the Principles of
Natural Knowledge, Cambridge University Press, 1919, Chap. V.]

To justify the coincidence point of view we apparently have to analyze
down to the colorless elements beyond our sense perception. It does not
seem unreasonable, perhaps, to expect that the universe is completely
determined in terms of the positions as a function of time of all the
positive and negative electrons; but to introduce such a thesis now
certainly goes beyond present experimental warrant, and is contrary to
the general spirit of relativity, which nowhere else involves any
reference to the small scale structure of things. Even if we were
willing to overlook all these objections, we would still have the fact
that the difference between a positive and negative electron is not
contained in any specification of the mere coordinates.

A further very important doubt in principle as to the possibility of the
analysis of nature into events is afforded by the character of the
concept of event itself. We have seen that the idea of event involves
the existence of discontinuities, and that this can correspond only
approximately to the physical fact, because discontinuities apparently
lose their abruptness as we make our measurements more refined. The
thesis that nature can be described in terms of discontinuities of a
very small scale seems much too special to be made a fundamental part of
a theory of the general pretensions of that of relativity. In fact this,
as well as a consideration to be mentioned later, suggests that the
argument and result of general relativity may be intrinsically
restricted to large scale phenomena.

We now pass from these somewhat special questions to ask why it is that
Einstein was able in the general theory of relativity to obtain new and
physically correct results from general reasoning of an apparently
purely mathematical character. We are convinced that purely mathematical
reasoning never can yield physical results--that if anything physical
comes out of mathematics it must have been put in another form. Our
problem is to find where the physics got into the general theory.

There are two questions to be disentangled here: we have to consider in
the first place the significance of the fact that Einstein has been able
to describe relations in nature in mathematical form, and in the second
place of the fact that he was able to arrive at the mathematical
formulation of these physical relations by reasoning of apparently a
purely mathematical character, from postulates of merely formal
mathematical content (invariance of natural laws in generalized
coordinates). Now the theory of relativity does not seem to differ in
the first respect from any other branch of mathematical physics, such as
the classical mathematical theory of electricity and magnetism, for
instance, and this matter has already been touched in an earlier
chapter. It is a fact that the behavior of nature can in many cases be
expressed to a high degree of precision in mathematical language, and
relativity is not unique in this respect. In any event, we must not
allow this possibility of mathematical formulation to obscure the
essential fact that all physical knowledge is by its nature only
approximate, so that we may expect at any time to find, when we have
carried our measurements to a higher degree of precision, that our
mathematical expression of the laws was not quite exact, as seems now to
be the case with Newton's law of gravitation, for example. I do not
suppose that Einstein would claim that the statements of relativity
differ in this respect from any of our other statements about nature,
although apparently some of his followers see something more here. (From
the operational viewpoint the meaning to be attached to "something more"
is somewhat obscure.)

With respect to the second question, we may stop to notice that the
special theory stands in quite a different position from the general
theory. The special theory is much more physical throughout; its
postulates are physical in character, and it is obvious that the physics
got into the results through the postulates. It seems to me without
question that Einstein showed the intuitive insight of a great genius in
recognizing that there are mutual relations between physical phenomena
which can be described in very much simplified language in terms of
concepts slightly modified from those already in common use. In view of
the remarks made on the nature of light, it is legitimate to wonder,
however, whether the formulations of even the special theory will always
stand. It seems to be true that _all_ the facts of nature, even in the
absence of a gravitational field, cannot be connected by the simple
formulations of the special theory; that the physical relations are
simple only in a sub-group; and that if we wish to deal with _all_
optical phenomena, we have carried our simplifications too far, for the
emission of a light signal is not a simple event, and light is not in
nature like a thing travelling. Just the sorts of physical thing which
are ignored in treating light as the special theory does are coming to
be more and more important in the minds of physicists, and this is a
reason for wondering whether ultimately Einstein's special theory may
not be regarded merely as a very convenient way of tying together a
large group of important physical phenomena, but not as being by any
means a full or complete statement of natural relations.

With respect to the general theory, however, I believe the situation is
quite different. The fundamental postulate that the laws of nature are
of invariant form in all coördinate systems is highly mathematical, and
of an entirely man-made character. Of what concern of nature's is it how
man may choose to describe her phenomena, and how can we expect the
limitations of our descriptive process to limit the thing described?
Furthermore, Einstein's method of connecting his mathematical
formulation and nature by way of coincidences of 4-events (three space,
one time coordinates) seems to be very far removed from reality, since
it entirely leaves out the descriptive background in terms only of which
the 4-event takes on physical significance. Nevertheless, three definite
conclusions about the physical universe have been taken out of the hat
by the conjuror Einstein (shift of the perihelion of Mercury,
displacement of apparent position of stars at the edge of the sun's
disk, and the shift toward the infra-red of spectrum lines from a source
in a gravitational field), and the problem for us as physicists is to
discover by what process these results were obtained.

An examination of what Einstein actually did in deriving his results
will show, I believe, that the situation is really different from that
suggested above. In the first place, the requirement that the laws of
nature be of invariant form actually places no restriction, as any one
can see by setting himself the task of expressing, for example, an
inverse cube law for gravitation in terms of generalized coordinates.
The work of expressing such a law can be attacked in a perfectly routine
way. (The essential difference between the invariability requirements of
the special and general theories is to be noted; the special theory
requires that the velocity of light, for instance, have the same
_numerical_ value in all allowed systems: the general theory merely that
all laws have the same _literal_ form, but with variable numerical
coefficients.) But, as Einstein says, if any one actually attempts to
carry through the work of expressing an inverse cube law in generalized
coordinates, he will find the task prohibitively complicated, and will
seek for some simpler formulation. What Einstein actually did,
therefore, was to require that the laws of nature be _simple_ in
generalized form. Now we know that the law of gravitation as formerly
expressed in ordinary coordinates as an inverse square law was
approximately exact, and was also simple. Any deviations from this law
are small, and all experience leads us to expect that to the first order
of small quantities the deviations can be taken care of mathematically
in the form of small correction terms to this law. This by itself gives
nothing, however, because a small correction term can be added to our
equations in an infinite number of ways. If, however, we know that the
equation must be of a certain type after the correction terms have been
added, the possibilities are so much restricted that the form of the
correction term may be determined. In arguing as to the probable type of
the equation, Einstein advanced the considerations by which physics gets
into the situation.

In the first place, the special theory had prepared us for the
possibility of finding that our measuring instruments might be modified
in a gravitational field, analogously to the shortening of a meter stick
when set into motion. In fact, special theory had indicated that in an
accelerated system the modifications might be too complicated to be
treated by that theory. In the absence, then, of specific information we
must be prepared for the most general possible alteration in space-time
in a gravitational field. In describing space-time we must therefore use
coördinates adapted to handling the most general possible relations,
and these are the generalized coördinates of Riemann, which had been
already discussed by mathematicians. Going back now to Einstein's
criterion that the equations are to be simple, we have the demand that
the equations be simple in generalized coördinates, and of course they
must also reduce to the ordinary equations (that is, the equations of
special relativity) in space where there is no gravitational field. In
deciding the further question as to what the type of equation probably
is, we are influenced by considerations of convenience as well as by
physical considerations. Practically the only type of equation that can
be handled mathematically is linear, so that we shall certainly try
first whether this type of equation may not continue to hold. Now the
Newtonian law of the inverse square may be expressed in terms of a
linear differential equation of the second order in the old Cartesian
coördinates (Poisson's equation), so that our most immediate suggestion
is that the equations remain linear and of the second order in
generalized coördinates. As a matter of fact, this requirement turns
out to be sufficient to determine the small correction term by which the
ordinary equations can be generalized; Einstein's papers must be studied
to see how this works out in detail.

All this looks pretty mathematical, but as a matter of fact there is
much physical content, because systems which can be described by linear
equations of the second order have definite physical properties. The
requirement that the equations be linear corresponds to one of the most
fundamental properties of our universe--the causality concept would not
be possible or would be much modified in a universe governed by
non-linear equations, for the joint effect of two causes acting together
would not be the sum of their effects acting separately, so that the
analysis of a situation into simple elements would be impossible and the
causality concept probably would not have arisen. Furthermore, an
equation of the type of Poisson of the second order means that there are
propagation phenomena, and equations of mechanics of the second order
involve the existence of a scalar energy function. If, then, the
behavior of the universe can be described by differential equations at
all, these equations must be linear of the second order if the universe
is to have the broadest physical characteristics of our own universe.
What Einstein really did, therefore, was to demand that even when
space-time is warped by the presence of a gravitational field, those
physical phenomena which can be described in terms of differential
equations continue to be described by linear differential equations of
the second order; that is, that nature continues to be describable in
terms of a causality concept, with propagation phenomena, and a simple
energy function. The consequences of a guess like this about the
properties of nature appeal to our physical intuition as being worth
following out, and of course we know the experimental justification.

Several general comments may now be made on the structure reached in
this way. In the first place, the whole structure is only descriptive in
character; we find certain correlations in nature which we describe with
considerable completeness in mathematical equations, without introducing
any new element of explanation or of mechanism. We have seen that as we
increase our range from the realm of ordinary phenomena to phenomena of
different character we arrive at a stage where for a time the process of
explanation apparently halts, and we have to be satisfied with a
statement of mere correlation between elements; later, however, these
elements may be accepted as the ultimates in a broadened scheme of
explanation, and the explanatory process resumed. Are we at such a stage
now with the general relativity theory, and may later a new scheme of
explanation be established based on the correlations of Einstein? This
is of course a matter of individual judgment; I personally question
whether the elements of Einstein's formulation, such as curvature of
space-time, are closely enough connected with immediate physical
experience ever to be accepted as an ultimate in a scheme of
explanation, and I very much feel the need for a formulation in more
intimate physical terms.

In the second place, we must repeat the comment already made in
discussing time, namely, that there is still a very wide gap between the
theory and its physical application, in that we have no way of
identifying our physical clocks and our physical measures of time with
the thing called time in the formulas. This gap must be filled by a
specification of the physical structure of a clock.

It has always been very puzzling to understand why Einstein has so
strenuously insisted that the shift toward the infra-red is an integral
part of the general theory, and that if the shift is not found, the
theory must fall. In other words, Einstein insists that the assumption
that an atom is a clock is an integral part of his theory. I believe
that this attitude may be due to a realization by Einstein of that very
flaw in the logical structure which we are now emphasizing. In the
absence of any method of specifying the details of construction of at
least one clock, relativity becomes a purely academic affair, unless
there exist in nature concrete things which may serve as clocks.
Einstein _must_ either be able to tell how to construct a clock, or else
be able to point to a specific example of a clock. He chose the atom as
the specific thing. Doubtless the reason was the apparent simplicity of
the vibrating mechanism of an atom, as shown by the precise equality of
the frequencies emitted by all atoms of the same element. If the atom is
not a clock, where in nature can one be found? But in the last few years
we have come to appreciate the exceedingly complicated quantum structure
of an atom, and Einstein's thesis loses much of its instinctive appeal.

Since Einstein created the theory of relativity, it is perhaps
ungracious to question his right to stipulate that the assumption that
the atom is a clock is an integral part of the theory. This, however,
degenerates to a mere matter of language, and does not touch the
arbitrary nature of the procedure. It does not prevent us from having a
second brand of relativity theory, that of X instead of Einstein,
exactly like that of Einstein except that perhaps now the "clock" is
constructed in terms of the life period of a radioactively
disintegrating element. The only way to eliminate the arbitrariness
seems to be to postulate that _all_ natural processes, which run
naturally of themselves independently of what we may do, may equally
well serve as clocks and give the same results. But in answering the
question of the operational meaning of "independently of what we may do"
we shall effectively have to answer the question of what is a clock.
This point of view may possibly, however, get us a little nearer to our
goal of finding how to specify the structure of a clock.

Finally, the general theory is not completely general, but applies only
to a certain range of phenomena, just as we saw that the special theory
does not embrace all optical phenomena. The general theory applies only
to those phenomena which can be described in terms of differential
equations, that is, par excellence, to large scale phenomena. If quantum
phenomena cannot be described by differential equations,[30] as
apparently now they cannot, general relativity cannot by its very nature
be applicable. General relativity does not give us a comprehensive
formulation of the behavior of all nature, and as far as we can see, we
are still as far as ever from such a general formulation.

[Footnote 30: This statement now takes on a very questionable aspect in
view of the new quantum wave mechanics (March, 1927).]



ROTATIONAL MOTION AND RELATIVITY


Physically there is a great difference between the behavior of systems
in uniform relative rectilinear motion and those in uniform relative
rotation. The special theory of relativity states that there is a triply
infinite number of systems with all possible uniform rectilinear
velocities with respect to each other, in all of which physical
phenomena have exactly the same mutual relations, that is, natural laws
are the same. Now the mere formulation of the principle suggests the
sense in which "system" is here used. It is obvious that "system" refers
only to a part of the universe; we are not making a hopelessly academic
statement about what would happen if we had an infinite number of
universes to experiment with, but are talking about operations that may
be approximately realized in our own universe. The "system" of the
formulation we may think of as a completely equipped laboratory, out in
empty space, so far from the heavenly bodies that they can have no
effect. The different systems of the formulation are different
laboratories, all built to exactly the same architectural blue prints.
The phenomenon to which the postulates of relativity apply are phenomena
which pertain entirely only to one or another of these laboratories. The
meaning of this restriction is not completely definite and has, in any
special case, to be judged partly by the context. Obviously, to see from
the window of a laboratory another laboratory passing with a certain
velocity cannot be counted as one of the allowed phenomena. Still less
is it one of the allowed phenomena to observe that the center of gravity
of the entire stellar universe has a certain velocity of translation
with respect to the laboratory. The special principle of relativity
contains by implication therefore the statement that certain very large
and important classes of physical phenomena may be isolated and treated
as taking place unaffected by the rest of the universe. Granted now the
possibility of isolation, we have a second statement, which is usually
treated as if it were the entire statement of the restricted principle,
namely, that there is a triply infinite set of systems in which these
phenomena run in the same way independent of the relative motion of the
systems with respect to each other. When once the significance of the
observation is grasped that absolute motion has no meaning in terms of
operations, we see that this last statement takes immediately a most
simple and satisfying aspect, in fact, so simple and inevitable that we
are inclined to see in this the complete essence of the situation and
regard the meaninglessness of absolute motion as affording peremptory
proof of the restricted principle.

With this bias we now turn to examine the facts of rotary motion, and
are disconcerted to find them quite different. No meaning in terms of
measuring operations can be given to absolute rotary motion any more
than to absolute translation, but nevertheless phenomena are obviously
entirely different in different systems in relative rotary motion
(phenomena of rupture, for example), so that apparently there are
physical phenomena by which the concept of absolute rotary motion might
be given a certain physical significance. Given two worlds like our own
in empty space, but surrounded by impenetrable clouds, and each provided
with a Foucault pendulum, then we believe that it is physically possible
that we may find on one of these worlds the plane of rotation of the
pendulum gradually changing in direction, while on the other it remains
stationary. This difference we regard as possible without other
accompanying physical phenomena which are causally related to the
rotation of the pendulum (of course we have to make the two worlds of
infinitely rigid material and eliminate other phenomena which we regard
as purely incidental), so that we apparently have here a contradiction
of our cardinal physical principle of essential connectivity. We are
certainly not inclined to give up our principle, and we believe that as
a physical fact, if the clouds could be evaporated, an observer in one
world would find that he was rotating with respect to the system of the
fixed stars, whereas the corresponding observer on the other world would
find that he was stationary. Our principle of essential connectivity is
therefore maintained, in that the rotation of the plane of the pendulum
is connected with a rotation with respect to the rest of the universe of
the entire world in which the pendulum is mounted. As far as I am aware,
no other way of maintaining our principle has ever been suggested. But
this demands that we give up our physical hypothesis of the possibility
of isolating a system. There is here no question of limiting behavior;
we believe that no matter how far our rotating world gets from the rest
of the universe the Foucault pendulum would always behave in the same
way; the system can never be isolated, but such local phenomena as the
invariance of the plane of the pendulum are always essentially
determined by the rest of the universe.

If now our system cannot be isolated, we must return to the phenomena of
translational motion. In principle the act of isolation cannot be
performed, the rest of the universe cannot be disregarded, and we should
expect that different states of translational motion as well as
different states of rotational motion with respect to the rest of the
universe would have an effect on phenomena. We set ourselves the problem
of understanding this apparent enormous difference between phenomena of
translation and rotation. We remark that what apparently is a difference
in principle may, in virtue of the approximate character of all
measurement, be only a difference in magnitude, and that translational
effects may exist too small to detect. A physical basis for such a
difference may be found in the enormously different numerical values of
translational and rotational velocities with respect to the rest of the
universe attainable in practice. In describing phenomena of cosmic
magnitude, we may plausibly measure the phenomena in units commensurable
with the scale of the phenomena. Thus in measuring linear distances, we
may perhaps choose as the unit of length the diameter of the stellar
universe, and in measuring rotation, a complete reversal of direction
with respect to the entire universe. This last means a change of angular
orientation of 2 π, the first means a length of the order of 10^6 light
years. Measured in such cosmic units the angular velocities attainable
in practice are incomparably greater than linear velocities. We now see
that it is possible that the real state of affairs is as follows:
namely, phenomena in any system are affected by motion with respect to
the entire universe, whether that motion is of translation or of
rotation, and the magnitude of the effect is connected with the velocity
of the motion by a factor which is of the general order of unity when
velocity is measured in cosmic units. This last is merely an application
of the argument so often made in physics as to the order of magnitude of
unknown numerical factors, and will be found expanded on page 88 of my
book on _Dimensional Analysis_. The linear velocities attainable in
practice are now so exceedingly low that their effect has not yet been
detected experimentally, but angular velocities are high, and the effect
is easily demonstrable. In this light the special principle of
relativity is no different in character from any other physical law; it
is only approximate, and some day our measurements may become refined
enough to detect its limitations.

We have made a hypothesis here, which we may call the hypothesis of the
immanence of the entire universe, namely, that isolation is impossible,
or that the rest of the universe, no matter how distant, always has a
local effect on at least some phenomena. This is essentially the
hypothesis of Mach,[31] and leads to a situation which can, I think, be
contemplated with logical equanimity, although it has always seemed to
many physicists most highly antiphysical in character.

[Footnote 31: E. Mach, The Science of Mechanics, translated by
McCormack, The Open Court Publishing Co., Chicago, 1893. See especially
p. 235.]

It must certainly be admitted that most physical experience justifies us
in thinking that effects may be made as small as we please by getting
far enough away from the cause of the effect. But if we accept the
considerations of the preceding pages, we must be prepared to admit that
as phenomena change in range their character may change, and that in
these new realms we must, at first at least, be satisfied with a mere
statement of correlations. Certainly we have very strong physical
evidence of a formal correlation between the Foucault pendulum and the
rest of the universe. But a correlation of this sort may be without
significance because of its very breadth; we never can prove the
significance of the correlation by performing an experiment with the
rest of the universe absent. Have we really done anything more than
merely get things into such a formal situation that they cannot be
assailed, a possibility which the mere laws of our thinking seem always
to leave open, as has been suggested, or is there any physical content
to what we have done? We have seen that if our correlation is also
suggested by other phenomena, then we may accept it as having physical
content. Now there is just a glimmer of a suggestion that our hypothesis
of the immanence of the universe may be needed in other ways. The
gravitational constant and the velocity of light are always treated as
arbitrary magnitudes thrust on the universe from outside with no
connection with other phenomena. Nevertheless, I suppose that no one
regards this situation as ultimately satisfactory and does not entertain
the hope that some day we may be able to give some sort of account of
the numerical magnitude of these constants. We have not hitherto
succeeded in finding any connection between these constants and small
scale phenomena such as the charge on the electron, its mass, etc., so
that there is some plausibility in expecting that a connection may be
sometime found with cosmic things; indeed general relativity theory
already prepares us for exactly this possibility. Now the velocity of
light and the gravitational constant control small scale experiments,
for of course these two constants can be measured by local experiments,
so that if the cosmic connection is found, we should have a control of
local behavior by cosmic things, and therefore another example of the
immanence of the entire universe. There is no need for me to waste time
in apologizing for the highly speculative character of all this. It is
worth while to emphasize, however, that our general considerations on
the meaning of "explanation" have prepared us to admit as reasonable
just the sort of explanation contained in the hypothesis of the
immanence of the universe, and therefore to reserve a place in our
physical thinking for possibilities of this sort, in spite of the fact
that such considerations are not usually entertained, and may seem to
many opposed to the spirit of physics.



QUANTUM CONCEPTS[32]


The history of quantum theory up to the present is a repetition in many
respects of that of the early theories of electricity, in that all our
thinking has been in mechanical terms. As far as we now know, quantum
phenomena are always associated with atoms. We make for the atom a
mental model with all the properties of the mechanisms of the ordinary
scale of magnitude and with a few impressed properties in addition which
represent the new quantum relations. As we now think of it, the atom has
a massive core about which electrons revolve under an inverse square
law, the connection between the mass of the electron, its acceleration,
and the force acting on it being that usual in Newtonian mechanics.

[Footnote 32: This section was written early in 1926 without access to
recent literature. Our attitude toward quantum phenomena has been so
much changed since then by the "new" quantum mechanics, that a number of
the following statements are superseded as a statement of present
opinion. However it has seemed worth while to let the section stand as
written, because many of the developments actually taken in the new
mechanics follow the lines that it is here urged they ought to take, and
in so far afford interesting confirmation of the point of view of this
essay.]

The space in which the electron circulates is thought of as Euclidean,
and the motion is described in time, which may be measured with clocks
in the usual way. The general equations of electrodynamics do not apply;
there are no propagation effects inside the atom, the motion of the
electrons does not produce a magnetic field, and there is no radiation
when the electron is in one of its possible stable states, in spite of
the acceleration. We may, if we please, in working out the character of
the motion, entirely neglect the electrical origin of the inverse square
law, and treat this merely as an impressed force without further
implications. Superposed on the ordinary spatial, temporal, and
mechanical characteristics of the model are additional quantum
properties, one which determines the particular orbit in which the
electron moves [∫ pdq = nh], and another which determines the
frequency of the radiation emitted when the electron passes from one
allowed orbit to another. No mechanism is suggested to account for these
quantum conditions, although the conditions are formulated in mechanical
terms.

We now have to ask what is the meaning in terms of operations of our
usual concepts of space-time and mechanics when applied to phenomena of
this order. It is of course evident, as has already been emphasized,
that the concepts have entirely changed in character, because we do not
measure an electron orbit, for example, by stepping off the diameter
with meter sticks, or by measuring the time required for light to travel
across the diameter. The particular feature of immediate interest in
this changed situation is the change in number of our concepts on the
atomic level. I shall not attempt to find by an exact analysis the
number of independent concepts at this level; probably such an analysis
is not possible. We may, however, make an approximate suggestion.
Apparently the most important concept in describing relations inside a
quantum system corresponds to that of energy on the ordinary scale.
Changes of energy determine the frequency of emitted radiation, as well
as the relations during collisions of atoms and electrons; these
collisional relations make direct connection with experiment through the
voltages applied to electrons in collision experiments. The analogue of
the momentum concept also seems to have independent significance, as
shown by the Compton effect. The frequency of emitted radiation is also
something with independent experimental significance. I believe that
these three things are all that have direct significance for quantum
experiments made up to the present time. In any event, it is perfectly
evident that on the quantum level the concepts which at present have
operational significance are considerably fewer than on the level of
ordinary experience.

Apart from the question of convenience, there may be justification in
continuing to use our old mechanical forms of thought if new
experimental relations are thereby suggested. That a very large number
of such as yet undiscovered relations may be suggested in some such way
is at once evident. Thus we have no present knowledge of any phenomenon
associated with what the electron does when passing from one energy
level to another. How long does it take to make the passage? What is its
path during passage? Is it subject to the ordinary laws of
electrodynamics during passage? When and where is the radiation emitted
that corresponds to passage? When the electron leaves one stable orbit
is the orbit on which it will eventually land already determined? Does
the radiation train emitted during a change from one energy level to
another have a definite length in space, or may it have a variable
length and correspondingly something that corresponds to variable
amplitude? What happens to the radiation when the electron passages are
interfered with before the emission of a quantum has been completed?
What is the mechanism by which the quantum conditions are imposed? Is it
not possible that part of the clew to the riddle of the manner of
transition from purely quantum behavior to the behavior of classical
mechanics may be found in the behavior of the electron during passage
from one energy level to another? Certainly we have a tendency to the
classical behavior under those conditions, such as at high temperature
or in strongly condensed systems, in which the time occupied in passage
might be expected to become a more important part of the total time.

Corresponding to these questions there should be many as yet
undiscovered phenomena, and the mechanical point of view therefore has
its value in suggesting experiments to detect such effects. It is of
course too early to see what the final result will be here; we cannot
tell whether eventually enough new experimental kinds of behavior will
be found to restore the number of independent concepts to that of the
level of ordinary experience or not, or whether indeed it will turn out
that a greater number of concepts is required. It is contrary to our
instincts to expect a greater number, and a smaller number now seems to
us not unnatural, but the considerations of this essay should prepare us
for either possibility.

It is often said that quantum phenomena are inconsistent with ordinary
mechanics, and proofs of this assertion are often offered. I believe
that no such proof, in the spirit in which the attempt is usually made,
can be correct, for it seems to me that the remark of Poincaré applies,
namely, that any sort of behavior can be imitated by a mechanical
system, provided it is only complicated enough. A peremptory proof of
this can be given to any one who is not a believer in vitalism. If a
sentient being can be regarded as a mechanical system, we merely have to
station inside each atom a Maxwell demon, with instructions to make the
atom react according to quantum rules. Opposed to the spirit of this
sort of reduction of quantum phenomena to mechanical terms, we have to
remember that it makes sense to talk about the character of our
conceptual structure only when the number of concepts is reduced to the
number that have independent operational significance, that is, to the
minimum number.

In the meantime let us examine what may be the significance in the light
of present experiment of statements like those ascribed to Bohr that our
usual concepts of space and time may be inapplicable in dealing with
quantum phenomena. This idea is often given the more explicit form that
space and time may be essentially discontinuous at the quantum level.
From the operational point of view, it is most difficult to see exactly
what this more explicit statement means, at least in terms of those
operations by which length and time were originally defined. Thus if
space were discontinuous, it might mean that a point exists which may be
reached by laying off a meter stick fourteen times, for example, and
another point by laying off sixteen times, but that no point can be
found with fifteen applications. Such a state of affairs seems to be
inconsistent with our definition of the counting operation and to have
no concern with any properties of space; for what shall we mean by
laying off a meter stick sixteen times if it cannot be laid off fifteen
times? It is conceivable that space might end, in the sense that beyond
a certain limit there might be some irremovable physical hindrance to
the continued laying off of distances with a meter stick (although I
think that we should be inclined to describe such a state of affairs in
terms of matter enclosing empty space rather than as the end of space),
but to say that space may be discontinuous seems to be meaningless. In
the same way, I believe it meaningless to speak of discontinuous time.
We may have phenomena discontinuous in space and time, but not
discontinuous space or time.

It seems then that we must give up the idea that in the quantum domain
the usual concepts of space and time may fail, in the specific sense
that they may become discontinuous. What may we understand by the
failure of these concepts in a more general sense? No one of course
would expect that even eventually the concepts will have the same
operational significance for the inside of an atom that they have on the
ordinary scale; it must be a modified sort of concept with which we are
concerned, such as we have already seen is given by the field equations
of electrodynamics. If now the number of operationally independent
concepts on the quantum level turns out to be the same as on the level
of ordinary experience, and if there is also the possibility of
continuous transition from the operations of the quantum domain to those
of ordinary experience, then it seems to me that we should say that our
usual concepts of space and time still apply in the quantum domain. But
if the number of operationally independent concepts is either greater or
less than on the ordinary level, then I believe we must say that the
ordinary concepts of space and time cannot apply. One might still look
for the possibility of separating out from the complex of concepts on
the quantum level a group which might change continuously to those of
space and time on the ordinary level, but I think that such a
possibility is very remote when one considers that the total number of
concepts changes, and that in the zone where the number changes the
definitions are not unique by which one extrapolates a concept from one
domain to another.

If Bohr's idea is true that space and time cannot be used in describing
ultimate quantum phenomena, one of the most immediate implications in
terms of experiment might be that phenomena corresponding to
intermediate positions of the electron between stable orbits do not
exist.

Finally, we must comment on the general tactics of the quantum
situation. It would seem that there have already been a sufficient
number of unsuccessful attempts to formulate quantum behavior in terms
of ordinary mechanics to justify the expectation that ultimately
something quite different must evolve. The difficulties of an unmodified
carrying over of ordinary mechanical notions to quantum phenomena may be
illustrated by a simple example. Consider a particle of mass _m_
rotating in a frictionless circular track of radius _r_. Then according
to quantum conditions it can move stably on this track only with certain
definite velocities, such that ∫ pdq = mv 2πr = nh. Suppose now the
particle rotating with one of the allowed velocities, and a tangential
force applied. If the usual mechanical notions of force are still valid,
the particle must respond by moving in its track with continually
increasing velocity. After the velocity has been increased by a small
amount, we remove the force. The motion is now no longer one of the
allowed ones, and the particle must in some way change its velocity; it
must either slow down or speed up. In the first case it must either
radiate energy, which a system of the simple mechanical properties we
have supposed is not capable of doing, or else the law of conservation
of energy fails, and also Newton's first law of motion during the
process of acquiring the steady condition. If, on the other hand, the
particle speeds up, it must increase its energy from nowhere, and again
ordinary mechanics does not apply.

It seems then a mistake to attempt to formulate the quantum conditions
in terms of the notions of ordinary mechanics (momentum, and position
coördinates in either the ordinary or the generalized Lagrangean
sense). It would seem, on the other hand, plausible to expect that
mechanics is not a fundamental thing, but is in some way an effect
produced by the aggregate action of a great many elementary quantum
processes. Amplitude of radiational vibration, for example, may be such
a statistical aspect of a great many processes, in some such way as on
the ordinary level of experience temperature is a statistical aspect of
the average kinetic energy of the atoms. One possibility of this kind
has already been more explicitly indicated; in the elementary process of
emission of radiation, frequency and energy are not two independently
assignable variables, but are connected [E = hν]. That is, on the
quantum level radiation has only a single property, which is properly
neither energy or frequency. [We are now neglecting the polarization
aspect of radiation.] On a higher level, that of ordinary radiation, the
single elementary property has expanded itself into two (energy and
frequency) through the additional variable of the number of elementary
quantum processes in the complex radiation.

The program of the immediate future should be an extension of something
of this sort, namely, to invent new concepts corresponding to the
experimentally independent things on the quantum level (such perhaps as
the resultant of the fusion of the energy and frequency concepts for
radiation), and then to show how the ordinary concepts of mechanics (and
very likely those also of space and time) are generated by statistical
effects in aggregates of great numbers. Perhaps it is yet too early for
an attempt of this sort, because it may seem that there are still too
many possibilities of new experimental discoveries which might upset the
results of elaborate theoretical speculation. If this should really be
felt to be the case, I believe that physics ought for the present to
hold in partial abeyance its theoretical activities in this field, and
devote itself to acquiring as rapidly as possible the necessary
experimental facts. We may emphasize again that the possibility of
carrying out this plausible program can be proved only by experiment; it
may be that more concepts will be required on the quantum level than for
ordinary experience.

The invention of new concepts is certainly not an easy thing, and is
something which physics has always deliberately, and perhaps
justifiably, shirked, as shown by the persistent attempts to carry the
notions of mechanics down into the finest structure of things. This
shirking has not had bad results, but on the contrary good results, as
long as physics has been primarily concerned with phenomena near the
range of ordinary experience, but I believe that as we get farther and
farther away from ordinary experience, the invention of new concepts
will become an increasing necessity.



CHAPTER IV

SPECIAL VIEWS OF NATURE


IN this last chapter we propose to discuss certain special hypotheses
about the structure of nature, and certain other matters that could best
be left until we had examined our fundamental concepts.

We have seen that in setting up the general rules which are to guide us
in describing and correlating nature, we have to take extreme care to
allow no special hypotheses to creep in, as otherwise we might be
restraining possible future experience. Even here there is no hard and
fast line of separation of the general from the special, and one might
entangle himself in inextricable difficulties if his ideals were too
meticulous. How for example is the critic to be answered who says: "Your
very endeavor to formulate principles so broad as not to restrict future
experience means, when examined in the light of operations, that you are
seeking for principles which _past_ experience suggests will not limit
the future. It is in the very nature of things impossible to escape all
the implications of past experience and therefore to find any completely
general principle." I believe that we must admit the critic is right,
and that rigorously our goal is impossible of attainment. We may say in
partial self-defense that all the discussion of this essay has been
subject to one explicit assumption, namely, that the working of our
minds is understood, which of course involves the assumption that our
minds continue to function in the future in the same way as in the past.
Even with this proviso we can not rigorously avoid the implications of
the past, but there can be no practical question that we recognize
certain assumptions about the behavior of nature to be so special as to
limit seriously the physical possibilities, and other assumptions to be
less restricting. In the previous discussion we had to make assumptions,
but I hope these assumptions will be recognized by all with physical
experience to be so broad as not to restrict us seriously. More special
assumptions or hypotheses have their very great use, however, when we
attempt to push forward the domains of experimental knowledge, because
they may suggest new experiments or aid in correlating information
already obtained. These special hypotheses may cover a very wide range
of generality; some of them are general enough in character to be
discussed here.

Among these special hypotheses there is a group which play an important
part in the speculation of most physicists, and which have features in
common. These are: the hypotheses of the simplicity of nature, of the
finiteness of nature in the direction of the very small, and of the
determinateness of the future in terms of the present. That these views
have points of similarity is obvious if we consider a hypothetical
special case. Suppose that no physical structure beyond the electrons
and protons can be discovered, or is even suggested by any known
phenomenon, so that the entire future behavior of a system can be
determined by a specification of the present relations of all its
protons and electrons; in this case nature would be both simple and
finite and the future determined by the present.



THE SIMPLICITY OF NATURE


Of these hypotheses, perhaps the most important is that of the
simplicity of nature, because of its wide spread diffusion and the
effect it has had on physical thought. The hypothesis of simplicity
assumes several forms; some physicists are convinced that the laws which
govern nature are simple, others that the ultimate stuff of which nature
is composed is simple (perhaps protons and electrons and energy), or
there may be a combination of both views into the belief that ultimately
we shall find simple ultimate elements behaving according to simple
laws. In one respect it is obvious that nature is not simple, namely
numerically--try counting the electrons or atoms or stars!

Consider now the first of these aspects of the thesis of simplicity,
which may be expressed as the conviction that the behavior of the entire
universe can be comprehended in a few principles of great breadth and
simplicity, such as the inverse square law of force, or the second law
of thermodynamics, or perhaps still better the equality of the
elementary positive and negative charges, which apparently holds to an
enormous degree of precision. In explanation of a view like this there
is in the first place the mental urge, because we can take a
satisfaction almost æsthetic in contemplating such a universe, and
there is in the second place a strong suggestion from experience.
Practically all the history of physics is a history of the reduction of
the complicated to the simpler. For example, the behavior of a large
part of the world of immediate experience can be reduced to the simple
laws of mechanics. The behavior of another very large group of natural
phenomena can be reduced to thermodynamics. The behavior of the heavenly
bodies, which at first was described in a rather complicated way in the
Ptolemaic system of astronomy, can be reduced to those same laws of
mechanics which we find in our immediate neighborhood, with the one
addition of the universal law of gravitation, which later refined
experiment discloses is really active in our immediate surroundings.
Similarly the laws of thermodynamics (except that part dealing with
radiation) are reduced to the ordinary laws of mechanics through the
additional assumption of the atomic structure of matter. Truly a
stupendous accomplishment that may well color our whole future outlook.
One may find great justification here for the belief that all nature
will ultimately be reduced to a similar simplicity, and, in particular,
justification for the attempt to find the explanation of all nature in
the action of mechanical laws. Now, of course, as a matter of physical
and historical fact, this program could not be carried through, but
obdurate physical phenomena were discovered. Electric phenomena, which
at first seemed so promising, refused to fit into the scheme, and the
converse attempt, to explain mechanical effects in terms of electrical
effects, also failed. We still carry our ordinary mechanical notions
down into the realm of small electric effects, and still talk, for
instance, about non-electrical forces which hold an electron together.
Nor are there experiments affording sufficient basis for believing that
all the mass of a positive nucleus is electrical in character. We also
think of electrical charges as having the property of identifiability,
which involves the possession of sharp edges and a change in the law of
force at small distances, and this is certainly a property carried over
from our large scale experience.

It seems fairly evident then that the laws of nature cannot be reduced
to either those of mechanics or of electricity, nor probably, as is
suggested by quantum phenomena, to a combination of both. This of course
does not preclude the possibility that the laws still may be simple when
expressed in other forms. An example of such a broad general law that
goes deeper than mechanics or electrodynamics is probably afforded by
the second law of thermodynamics when extended to include radiation
phenomena.

Examples of attempts to find other such simple laws are Tolman's 
Principle of Similitude,[33] and Lewis's theory of Ultimate Rational 
Units,[34] and his recently enunciated principle of Complete 
Reversibility.[35] The first two of these attempts I do not believe are 
successful, for reasons I have stated elsewhere,[36] the third also 
seems somewhat doubtful. 

[Footnote 33: R. C. Tolman, Phys. Rev. 3, 244-255, 1914; 6, 219-233,
1915; 15, 521, 1920; Jour. Amer. Chem. Soc., 43, 866-875, 1921.]

[Footnote 34: G. N. Lewis. Vol. 15, 1921 of the Contributions from the
Jefferson Physical Laboratory, dedicated to Professor Hall, Cambridge,
Mass.; Phil. Mag., 49, 739-750, 1925.]

[Footnote 35: G. N. Lewis, Proc. Nat. Acad. Sci., 11, 179-183, 422-428,
1925.]

[Footnote 36: P. W. Bridgman, Phys. Rev. 8, 423-431, 1916; "Dimensional
Analysis," p. 105.]

With regard to the general question of simple laws, there are at least
two attitudes; one is that there are probably simple general laws still
undiscovered, the other is that nature has a predilection for simple
laws. I do not see how there can be any quarrel with the first of these
attitudes. Let us examine the second. We have in the first place to
notice that "simple" means simple to us, when stated in terms of our
concepts. This is in itself sufficient to raise a presumption against
this general attitude. It is evident that our thinking must follow those
lines imposed by the nature of our thinking mechanism: does it seem
likely that all nature accepts these same limitations? If this were the
case, our conceptions ought to stand in certain simple and definite
relations to nature. Now if our discussion has brought out any one
thing, it is that our concepts are not well defined things, but they are
hazy and do not fit nature exactly, and many of them fit even
approximately only within restricted range. The task of finding concepts
which shall adequately describe nature and at the same time be easily
handled by us, that is, be simple, is the most important and difficult
of physics, and we never achieve more than approximate and temporary
success. Consider the example of time. The original concept of local
time, which for long seemed satisfactory, turns out to be inadequate,
and has to be replaced by extended time, which is so complicated that it
is questionable whether we shall ever be able to grasp it with the
confidence that we must demand in a useful concept (by "grasp" I mean
intuitive command of all the implications of the operations which are
involved). The concept has not yet been found which describes simply the
temporal relations of the universe.

Not only are concepts hazy around the edges and so incapable of fitting
nature exactly, but there is always the chance that there are concepts
other than those which we have adopted which would fit our present
phenomena. Finding concepts to fit nature is much like solving a
cross-word puzzle. In the puzzle there may be some parts of the pattern
which we fill completely and easily, but sometimes we find parts in
which we can fill in everything except one or two obstinate definitions,
so that we are sure we are on the right track, and rack our brains for
the missing words, when with a flash of inspiration we see that the
obstinate words can be fitted in by a complete change in those which we
had already accepted. It may be that we are soon to witness a similar
change in our concept of the nature of light. An important difference
between the cross-word puzzle and nature is that we can never tell when
we have filled in all the squares in any of the parts of nature's
puzzle; there is always the possibility of new phenomena which our
present scheme does not touch.

Considering, then, the nature of our conceptual material, it seems to me
that the overwhelming presumption is against the laws of nature having
any predisposition to simplicity as formulated in terms of our concepts
(which is of course all that simplicity means), and the wonder is that
there are apparently so many simple laws. There is this observation to
be made about all the simple laws of nature that have hitherto been
formulated; they apply only over a certain range. We have not extended
the laws of gravitation to small bodies, nor have we found that our
electrical laws will work on a cosmic scale. It does not seem so very
surprising that over a limited domain, in which the most important
phenomena are of a restricted type, the conduct of nature should follow
comparatively simple rules.

A tempting question is whether there may not be some laws of nature that
are _really_ simple, without relation to our mode of formulation, such
as the law of the inverse square. I leave it to the reader to decide
whether this question has meaning. In this connection it is possibly
significant that the average physicist is strangely reluctant to tamper
with the inverse square law. I find in myself a lack of sympathy, which
I cannot justify by any of the considerations of this essay, with
attempts like the recent one of Swann,[37] for example, to explain a
wide variety of hitherto obstinate effects by the assumption of slightly
unequal departures from the inverse square law by the electrons and
protons. Of course I hope that this feeling will turn out not to be
prejudice, but will perhaps be justified by some such general
observation as that a departure from the inverse square law so slight as
by definition to be forever beyond detection by direct experiment is
meaningless; but of this I am not at all sure.

[Footnote 37: W. F. G. Swann, Phys. Rev., 25, 253, 1925.]

We are now ready to consider the second respect in which nature may be
simple, namely, because the material of which it is built may reduce to
a few sorts of elements. In this discussion it will be convenient to
consider also at the same time the more inclusive simplicity arising
from simple laws acting on simple elements. The immediate question for
us here is one of fact: does nature seem to be getting intrinsically
simpler as we get toward small scale phenomena? There is much room for
difference of opinion here; personally I feel that this expected
simplicity is not in evidence, at least to the extent that we could
desire. For instance, the fact that the electrons must have both
electrical and mechanical properties is a straw in this direction.

It must also be remembered that a certain simulation of simplicity is
inevitable as we approach the limits of experimental knowledge, whatever
the actual structure of nature, for the mere reason that near the limit
our possible experimental operations become fewer in number, and our
concepts fewer also. The question which we are trying to answer has,
therefore, its real meaning only in terms of the possible future. Do we
believe that if we drive in our stakes at a certain point on our present
frontiers, this point will gradually, as physics advances, become
possessed of a continually richer experience, so that nature at this
point will appear increasingly complicated? Or do we expect a
termination of this process of expansion fairly soon? It seems to me
that as a matter of experimental fact there is no doubt that the
universe at any definite level is on the average becoming increasingly
complicated, and that the region of apparent simplicity continually
recedes. This, however, is not the opinion of all observers. Thus
Bertrand Russell, in "What I Believe", page 10, writes, "Physical
Science is then approaching the stage where it will be complete, and
therefore uninteresting."

This is perhaps a particularly favorable epoch in the history of physics
to urge the essential complexity of nature, because all our new quantum
phenomena indicate a vast wealth of hitherto unsuspected relations on
the very edge of the attainable. There is one aspect of quantum
relations, as also of our ideas of the nature of the structure of the
nucleus of the atom, which is particularly significant in this respect,
namely, that we have to describe phenomena by statistical methods. Now a
statistical method is used either to conceal a vast amount of actual
ignorance, or else to smooth out the details of a vast amount of actual
physical complication, most of which is unessential for our purposes.
There can be no doubt of the amount of ignorance that the statistical
method conceals when applied to these phenomena, but there are also
strong indications, particularly when applied to the nucleus, that it
covers a vast amount of actual physical complications. The nucleus of a
radium atom becomes unstable on the average every 10^4 years, which may
be plausibly taken to indicate that every 10^4 years the radium nucleus
gets itself into some particular configuration. Considering the time
scale on which we suppose events in the atom to take place, and also
considering the fact that radioactive disintegration seems unaffected by
outside agencies, this would indicate a perfectly appalling amount of
structure. We are similarly driven to statistical methods in quantum
theory, as for example, in Einstein's analysis of the details of
equilibrium between emitting and absorbing atoms and radiation.

In general, we cannot admit for a minute that a statistical method,
unless used to smooth out irrelevant details, can ever mark more than a
temporary stage in our progress, because the assumption of events taking
place according to pure chance constitutes the complete negation of our
fundamental assumption of connectivity; such statistical methods always
indicate the presence of physical complications which it must be our aim
to disentangle eventually.

It appears then that present experimental evidence makes very probable
structures beyond the electron and the quantum; we may go even further
and say that there is no experimental evidence that the sequence of
phenomena in nature as we go to ever smaller scales is a terminated
sequence, or that a drop of water is not in itself essentially infinite.
(This statement contains by implications the meaning that we attach to
infinite.) All the more, then, there is no evidence that nature reduces
to _simplicity_ as we burrow down into the small scale.

Whatever may be one's opinion as to the simplicity of either the laws or
the material structure of nature, there can be no question that the
possessors of some such conviction have a real advantage in the race for
physical discovery. Doubtless there are many simple connections still to
be discovered, and he who has a strong conviction of the existence of
these connections is much more likely to find them than he who is not at
all sure they are there, and is merely hunting for anything that may
turn up. It is largely a matter of psychology. Everyone knows that the
mere suggestion that a problem has a solution, or the knowledge that
someone has already solved it, is often sufficient to suggest a relation
that otherwise might not have been noticed. The chances are, therefore,
that the relations between phenomena will be found by those who are
previously convinced that the relations exist. The observation that most
of the discoveries are made by men with particular sorts of conviction
naturally strengthens the belief that their convictions are true. But
this picture has an obverse side. The man who is convinced that there is
a relation where none exists may waste all his time in vain seeking for
it. Granted that nature has no particular predisposition to simple
relations, the conviction that there are such relations is, from the
point of view of any one individual, as likely to be a hindrance as a
help. From the point of view of physical society, on the other hand, it
is desirable that there be such convictions, for in such a society there
will be more discoveries than in a society without such convictions. We
have here again the old conflict between the individual and society. As
in all other similar conflicts, society will not be able to demand
permanently from the individual the acceptance of any conviction or
creed which is not true, no matter what the gain in other ways to
society. If nature is not simple, physicists will not continue to
believe that it is, even if such a conviction does increase the total
number of discoveries. It is an impossible attitude to expect that one
can maintain. Does this then mean that physics is to face a drab future,
becoming continually more prosaic, with new discoveries ever rarer, made
by a continually decreasing number of misguided but fortunate
enthusiasts? There may be such a danger, but the greatest part of the
danger is avoided if its nature is clearly recognized. One of the
problems of the future is the self-conscious development of a more
powerful technique for the discovery of new relations without the
necessity for preconceived opinions on the part of the observer.

There is an aspect here of our physical research that is often lost
sight of, namely, the small proportion of successful discoveries
compared with the number of investigators. Certainly the number of
unsuccessful attempts, even in the case of those fortunate individuals
who make the great discoveries, is very much greater than the number of
their successful attempts. (Faraday's reputed satisfaction with a ⅒%
return comes to mind.) This must always be taken into account in
estimating the probable chances of correctness of any new theory. With
so many physicists working to devise new theories, the chances are high
that many false theories will be found, in which a number of phenomena
may apparently fit together into a new relation, but which eventually
prove to be inconsistent with other phenomena, so that the proposed
theory has to be abandoned. As physics advances and the number of
investigators and the amount of physical material increases, one has to
be more and more exacting in one's requirements of a new theory. One
must be particularly on guard against numerical coincidences. An
interesting chapter might be written on numerical relations which have
been hopefully published, but later had to be abandoned as without
significance.



DETERMINISM


If we are right in supposing that physical evidence gives no warrant for
the idea that nature is finite downward, we have not only repudiated the
thesis of simplicity but we have also made a very important observation
on the other general thesis mentioned at the beginning of this chapter,
namely, the thesis of physical determinism. By determinism we understand
the belief that the future of the whole universe, or of an isolated part
of it, is determined in terms of a complete description of its present
condition. [What we mean by present condition will be discussed later.]
It is popularly assumed that every physicist subscribes to some such
thesis as this. But now if there is infinite structure even in a small
isolated part of the universe, a complete description of it is
impossible, and the doctrine as stated must be abandoned. It seems to me
that all present physical evidence prepares us to admit this
possibility. I suppose, however, that most physicists would subscribe to
some modification of the original thesis, perhaps along the following
lines. Given a description of an isolated part of the physical universe
in the most complete terms that have physical meaning, that is, down to
the smallest elements of which our physical operations give us
cognizance, then the future history of the system is determined within a
certain penumbra of uncertainty, this penumbra growing broader as we
penetrate to finer details of the structure of the system or as times
goes on, until eventually all but certain very general properties of the
original system, such as its total energy, are forever lost in the haze,
and we have a system which was unpredictable. I suppose that it is a
further conviction of at least many physicists that by sufficiently
refining our measurements, the amount of haze at any fixed point in the
future may be made indefinitely small, and many might even go further
and hope by studying the haze (perhaps statistically) to obtain some
inferential evidence of structure beyond that yet experienced. In fact
it may be that this last contains the germs of the ultimate method of
investigation, if we ever reach a stage when we can no longer refine our
methods of measurement.

Determinism to the physicist is simply a way of stating certain
implications of his conviction of the connectivity of nature. We have
seen that the broadest possible statement of the thesis of connectivity
is: Given two isolated systems with identical past histories up to a
certain epoch, then the future histories will also be identical. The
thesis of the determinism of the future by the present constitutes a
specialization of this general thesis in that we suppose that identity
of _all_ past history is not necessary for identity of future behavior,
but only identity of present condition. The general and the special
thesis are not equivalent by any means: if past histories are identical
then present conditions are also identical, but the converse does not
necessarily hold at all.

Now I believe that the general thesis (which I suppose all physicists
will admit, but whose truth is nevertheless subject to the verification
of experience) gets turned into the special thesis by a feeling of
somewhat metaphysical content, which we may perhaps state by saying that
we can see no way by which the past can affect the future except through
the present. We do not like to think of the effect of a cause distant in
the past jumping over the present and affecting the future without
touching the present at all. It is the analogue of that attitude of mind
to which action at a distance in space is inconceivable; just as it is
difficult to conceive of a body here affecting a body there without in
some way an action propagated through intermediate space, so we do not
like to think of a past cause jumping over time and producing a future
effect without some sort of continuity in the causal chain through all
intermediate time.

So far our discussion has been purposely loose: it is evident that what
we mean by "present state" is crying for definition. What is meant by
this may depend somewhat on the specific hypothesis that one adopts
about the structure of nature. Historically the conviction of future
determinism has been most intimately associated with a mechanical
picture of the structure of the universe, so that it may be well to
begin from this point of view. Suppose the simplest possible system
composed of point masses without structure, as in the kinetic theory of
gases. What sort of specifications do we believe necessary to fix the
present state of such a system? The mechanical view of nature gives a
definite answer. By present state we mean the positions and velocities
of all the masses. This is sufficient for the complete determination of
any purely mechanical system, in which the forces between the elements
are known functions of only their relative positions. By a sort of
extension of these ideas valid for mechanical systems, it seems to be
often thought that the present state of _any_ system is determined by a
complete specification of the positions and velocities of all the
ultimate elements of the system (provided always of course that this
number is finite). This principle, however, does not appear to bear the
check of experiment when applied to electrical systems with radiation.
The theorems of the retarded potential show that such systems are
determined by the present position and velocities of the charges in the
immediate vicinity, and by the corresponding data at remote points given
for proper epochs in the past; in this case, therefore, past and present
history are necessary to determine the future. But if we consider the
electrical field as part of the system, we may fix the future in terms
of the present positions of the charges, their velocities, and the
values of the field vectors all over space, thus returning to a certain
formal resemblance to mechanical systems, and suggesting a reason for
ascribing physical reality to the electric field. This analogy with a
mechanical system is, however, loose; complete analogy would allow the
instantaneous values of the time derivatives of the field to be given
also, and this is not possible.

How is it that velocity can strictly be regarded as characteristic of
the _present_ state of the system? Certainly the usual operations for
measuring velocity demand that we know the configuration of the system
at two different times, and calculate the velocity from certain
differences of the system at these two times. The velocity is defined as
a limiting result, but even in the limit the essential physical fact
does not disappear that we must know the positions of the system at two
times. We may now go further; if the velocity is properly included in
the present attributes of the system, we can see no reason for not
including a specification of all the higher time derivatives also. In
the case of the simple gaseous system under present consideration we can
answer this question by examining the operations by which we actually go
to work to determine the future of such a system. The problem of
determining the future condition of such a system reduces to the problem
of writing the differential equations of motion of all its parts. If the
system is a mechanical system, as in this case, these equations are of
the second order in the time derivatives of the position coordinates,
and also involve the forces, which we suppose are known in terms of the
relative positions of the parts of the system. Given, then, the
positions and the way in which the forces depend on the relative
positions of the parts, the equations of motion can be written down for
any configuration of the system, and these equations may be integrated
(at least approximately) in terms of the proper initial conditions. Now
the only boundary conditions on a second order equation are the initial
positions and velocities. This is the reason that velocities have to be
specified in giving the present condition of the system, and that it is
not necessary to give the higher derivatives. Apparently the reason why
we instinctively include velocity among the present properties of the
system is not because velocity is by its nature strictly a present
property of the elements of the system, but rather because our wide
experience with mechanical systems has shown that as a matter of fact
velocity is necessary in such systems to determine future motion.

But now if the equations of motion of the parts of the system are not
those of mechanics, they will in general be much more complicated in
appearance and will involve higher derivatives of the time than the
second. Suppose for the moment that the equations contain only
derivatives and the mutual positions of the parts of the system. Then to
integrate the equations and determine the motion we have to know the
initial positions and the initial values of all the derivatives up to an
order one lower than the highest which occurs in the equations. The
equations of motion of an electron are even more complicated than this,
in that the positions of distant parts of the system have to be given
throughout an interval of time instead of merely an instant. It would
seem that the feeling that the present state of a system may be
determined in terms of positions and velocities does not as a matter of
fact apply to all the systems of our experience.

The discussion up to this point has been subject to the fundamental
assumption that the behavior of the system is entirely determined if we
can give the position of each part as a function of time. This
assumption is implicitly contained in Einstein's formulation of the
general principle of relativity, namely, that there is nothing more to a
physical system than a set of space-time coincidences, and that the
system is fixed in terms of the space time coördinates of all its
parts. Already in discussing the assumption of relativity we have
indicated reasons for dissatisfaction with this as a means of
reproducing all experience, because in giving only the space-time
coordinates of events we have entirely omitted the descriptive
background of the equations, which gives physical color to the system in
question. This discussion also assumes that a specification of the
positions, velocities, and higher derivatives (if necessary) of the
elements of the system is possible, which amounts essentially to the
assumption that the system contains only a finite number of elements.
Now in view of the experimental fact that there is no reason for
supposing that the structure of the universe is finite, this conclusion
must be modified, but I do not believe that the necessary modification
affects the essential argument. In view of the possible infinite
structure it would seem that we cannot expect more than that the future
is determined by the present within a certain penumbra of uncertainty,
and this penumbra may be made less important by digging down deeper into
the structure when specifying the present condition.

We have also slurred over the ambiguities in "present" condition when
the system is spread over space. Probably a unique ascription of meaning
to "present" is not possible for an extended system, but at least one
possibility is indicated by relativity theory. Imagine a staff of
assistants distributed throughout space, each equipped with clocks
synchronized and set with the master clock by light signals in the
conventional manner, and each fully equipped with the necessary
measuring instruments. Then what we mean at this point of the argument
by "present" state of the system is the aggregate of all the information
about the _positions_ and _velocities_ of the ultimate elements which I
determine in my immediate vicinity at my origin of time plus the reports
of similar observations made by all the assistants, each local
observation being made at the time origin of each local clock.

Going back now to the main argument, we have shown that the feeling that
the present condition of the universe may be specified in terms of
positions and velocities arose from experience with purely mechanical
systems, and that the more general formulation, in which we add to the
velocities the higher time derivatives, applies only to systems in which
the ultimate elements move according to differential equations of higher
order than the second. Furthermore, our analysis seems to have shown
that systems in which there is radiation do not allow a determination of
the future in terms of a present condition specified in terms such as
these. It seems, however, that the general principle of the determinism
of the future by the present may be saved by a change in the definition
of what we mean by the present condition of the system, ridding it of
its mechanical and other special implications, and making more immediate
connection with direct experiment. Let us understand by present
condition of a system the aggregate of all information that can be
obtained by any physical means whatever, with any sort of physical
instrument, not attempting to get out of this analysis information about
hypothetical ultimate physical elements, with the proviso that the
measurements are to be made now, extending the concept of "now" to
points distant in space in the way intimated above. With such a general
definition of the meaning of "present" we can now deal with systems in
which there is radiation, noticing that our assistant observers must be
stationed throughout apparently empty space as well as in the
neighborhood of matter. That this does adequately cover the case of
radiation is suggested by considering again the two systems of dark
lanterns with screens and distant mirrors which we have previously
considered, in one system a light signal having been despatched 0.5
second ago and in the other 1.5 seconds ago. Our thesis demands that
there be some present difference in these two systems, because their
future history is different, in one of them a light signal arriving
after the lapse of 1.5 seconds, and in the other after only 0.5 second.
Now there is a present difference as reported by our assistants, for the
assistant stationed half way between lantern and mirror reports in one
system a flash of light on the side of a screen which is turned toward
the lantern, and in the other system on the side of the screen turned
toward the mirror.

This more general point of view answers the question whether velocity
may be regarded as a present attribute of the system, for the parts of a
system which are in motion have momentum, and momentum may be detected
by placing against such parts comparatively rigid members which will
receive a minute deformation, so that velocity has a meaning in terms of
physical measurements made at a single instant of time.

There is a subtle and difficult question here, namely, whether in
talking about operations of measurement we can ever get rid of temporal
implications, and therefore, whether a condition of the system in which
temporal implications remain can properly be described as "present." I
shall not attempt to answer this question: there must be some
practically satisfying answer, involving perhaps the physical analogue
of differentials of different orders in mathematics, short of carrying
the analysis to such a degree of refinement that the concept of present
becomes meaningless, as we can see might easily happen.

With this enlarged understanding of what we mean by present state of the
system, it seems to me that physical evidence is now rather favorable to
the view that the present determines the future, subject to
qualification about the penumbra, at least as far as large scale
phenomena are concerned. It appears much more doubtful when we come to
small scale phenomena, and in particular it is doubtful whether the
principle can be applied to the details of the quantum process, and in
fact it is not certain that it has meaning. It is certain that if it is
true an enormous amount of structure beyond any that has yet been
detected is implied.



ON THE POSSIBILITY OF DESCRIBING NATURE
COMPLETELY IN TERMS OF ANALYSIS


There is a certain thesis that is loosely related to the view that
nature is finite downward, namely, that an explanation of the universe
is possible in which we start with small scale things, and explain large
scale phenomena in terms of their small scale constituents, the thesis,
in other words, that all the properties of the large are contained in
the properties of the small and that the large may be constructed out of
the small. Some such thesis as this seems implied in the general
attitude of many physicists. Let us examine the physical basis for this.
To maintain this thesis would demand that aggregates of things never
acquire properties in virtue of their numbers which they do not already
possess as individuals. Is this true? Consider, for example, the
two-dimensional geometry on the surface of a sphere. This is
non-Euclidean. Is the geometry of the individual elements of the surface
of the sphere non-Euclidean, or do they acquire this property in
changing scale? Is the kinetic energy of a number of electrons all
moving together in such a way as to constitute an electric current the
sum of the kinetic energies of the individual electrons, or is there an
additional term? Is the mass of an electron the sum of the masses of its
elements?

A mathematical consideration is suggestive here. Those properties of a
system which can be described in terms of linear differential equations
have the property of additivity; the effect of a number of elements is
the sum of the effects separately, and no new properties appear in the
aggregate which were not present in the individual elements. But if
there are combination terms (as in the electrical energy, which contains
the square of the field), then the sum is more than (or different from)
its parts, and new effects may appear in the aggregate. Now of course
the linear equation is of enormous importance in describing nature, but
many examples of systems with other types of equation can be found, as
that above for electromagnetic mass. In expecting to find in nature such
non-additive effects, we need not commit ourselves at all to the view
that nature is governed by differential equations, but by analogy may
expect similar effects if difference equations, for instance, should
prove to be fundamental, or even something beyond present mathematical
formulation.

It is certainly very much easier to handle a system physically if the
total action can be built up from that of its parts, because the
analysis which establishes the connection between the elements is easier
to perform. It is obviously easier to show that an explanation in such
terms is correct, because we have seen that explanation involves making
experiments with representative elements absent or altered, and it is
easier to vary the small things than the large things. Those
explanations which involve working from the small up will therefore be
made first, and will appear to be of disproportionate importance. Places
where I look for an explanation from the large to the small are perhaps
in accounting for the values of the gravitational constant and the
velocity of light and in those phenomena which general relativity theory
indicates may depend on all the matter in the universe, as the Foucault
pendulum experiment. We must, of course, also be prepared for such
non-linear effects in the domain of unexplored quantum phenomena.



A GLIMPSE AHEAD


Some of the general considerations of this essay may, with considerable
plausibility, be expected to play a part in the future of both
speculative and experimental physics. The most important effect may be
expected from the clearer recognition of the operational character of
our physical concepts. Indeed during the writing of this essay there has
been a very marked increase in emphasis on the necessity of
understanding in terms of physical operations such fundamental concepts
as that of the electron, by the new quantum mechanics [the mechanics of
Heisenberg-Born and Schrödinger of 1925-26].

We are to expect then in the first place a more self-conscious and
detailed analysis of the operational structure of all our physical
concepts. [It has been beyond the scope of this essay even to begin to
attempt a systematic and thoroughgoing analysis of this character.] This
future analysis will show precisely how, as we extend the range of
experience, the physical character of the operations changes by which we
define our concepts, as, for example, in mechanics the notion of force
disappears at high velocity and is replaced perhaps by the notion of
momentum. In the region of change in the nature of our concepts, special
study will be made of the accuracy of our physical measurements, and new
experiments devised of greater accuracy, in order that we may know
precisely to what extent the new concepts are equivalent to the old.
Past experience suggests that we may perhaps expect to find new
phenomena especially in those regions where the difficulty of carrying
out the usual operation forces us to change the operational character of
our concepts. There will be questions of a more or less formal nature to
answer, as for example the _best_ way of extending concepts when there
are several possible courses open to us.

We may expect more interesting results, however, when we get so far
beyond ordinary experience that the character of the possible physical
operations has become so restricted as to result in an apparent decrease
in the number of independent concepts. It seems plausible to expect that
the structure of nature is more fundamentally connected with the number
of independent concepts necessary for a complete description than with
the precise details of the structure of the individual concepts, such,
for example, as whether space is measured optically or tactually. In
those regions where the number of concepts decreases, we must make the
most thoroughgoing experimental examination to discover if possible new
sorts of operations by which the number of concepts may be brought back
to normal. In searching for such new experimental operations it seems to
me that by far the greatest promise for the immediate future is offered
by improvements in our powers of dealing with individual atomic and
electronic processes, such as we now have to a limited extent in the
various spinthariscope methods of counting radioactive disintegrations,
or Wilson's β-track experiments.[38] In this self-conscious search for
phenomena which increase the number of operationally independent
concepts, we may expect to find a powerful systematic method directing
the discovery of new and essentially important physical facts.

[Footnote 38: C. T. R. Wilson, Proc, Roy. Soc., 87, 277, 1912.]

We can only conjecture whether the number of fundamental concepts will
prove impossible of further increase or not, but present experience
seems to give greater probability to the view that as we penetrate
deeper the number of fundamental concepts will always tend to become
fewer. We have already certainly one example in that the temperature
concept disappears when we get to the atomic scale of magnitude, and
possibly a second example in the building up of separate concepts for
energy and frequency by the combination of great numbers of that one
operationally simple thing which characterizes the elementary quantum
process in ordinary radiation.

Different sorts of relations between concepts are conceivable in the
transition zone where the number changes. We may find that other
examples are like that of temperature, which is simply a statistical
effect of a great many phenomena which may be described individually in
terms of the ordinary concepts of mechanics, so that in this case the
number of concepts changes merely by temperature dropping out, leaving
the others more or less unaffected. Or all the concepts may be more
closely interwoven, so that when the total number of concepts changes it
may not be possible to separate out a group of concepts whose defining
operations are unchanged. In such a case we must say that the original
concepts are not applicable on the new level. The most immediate
application of this idea has been already mentioned, namely, to the
concepts of space and time. If the operations by which space and time
are measured on the ordinary scale of magnitude cannot be carried down
as a whole into the region of quantum phenomena, then we must say that
the ordinary concepts of space and time are not applicable to these
phenomena.

Closely connected with the sharper analysis of the operational structure
of our concepts, we may expect in the future also a closer analysis of
our inventions. This will take the form of a search for new physical
facts which shall give to our inventions the character of physical
reality. In case prolonged search fails to disclose such phenomena (as
is probably now the case with the field concept of electrodynamics), we
must then find some way of embodying explicitly in our thinking the fact
that we are dealing with pure inventions and not realities.



INDEX

  Absolute, 26
  Absolute time, 4
  Action at a distance, 46
  Analysis of large into small, 51,
    220
  Arithmetic, 35
  Atom, 59

  Bell, 84
  Birkhoff, 72
  Black body, 112
  Bohr, 190, 192
  Born, 222
  Boscovitch, 46
  Bothe and Geiger, 116
  Bridgman, 201
  Brownian movement, 107, 129,
    143
  Bush, 142

  Caloric fluid, 59
  Campbell, 117
  Carnot engine, 125
  Causality, 80 ff
  Causal train of events, 85
  Clifford, 28
  Clock, 70 ff, 176
  Compton, 116, 188
  Continuity, 94
  Conservation of charge, 135,
    136
  Conservative functions, 113
  Constructs, 53
  Correlation, 37
  Cosmic units, 182
  Cross word puzzle, 202

  Descriptive background, 64
  Determinism, 114, 209 ff
  Discontinuous space, 191
  Döppler effect, 166
  ds, 72

  Eddington, 93
  Einstein, vii, 1, 2, 3, 4, 7, 8,
    9, 12, 13, 14, 64, 100, 155,
    162, 163, 164, 167, 169, 170,
    171, 172, 173, 175, 176, 177,
    206
  Electrical concepts, 131 ff
  Electrical explanation of
    universe, 50
  Electrical mass, 139
  Electric field, 56, 133
  Empiricism, 3
  Energy, 108 ff, 126 ff
  Euclidean space, 14, 15, 16, 18,
    23, 52, 61, 67
  Event, 95, 167
  Explanation, 37
  Explanatory crisis, 41
  Extended time, 77

  Faraday, 44, 57, 58, 209
  Final explanations, 48
  First law of thermodynamics,
    126 ff
  Force, 102 ff
  Foucault pendulum, 180, 184
  Fourth dimension, 74
  Future, 222

  Gauss, 15, 34
  "Go and come" time, 112
  Gravitational constant, 91

  Haldane, 25
  Heat flow, 130
  Heisenberg, 222
  Hertz, 44
  Hoernlé, vi

  Identity, 91 ff
  Isolation, 82

  Joule, 124

  Kelvin, 45, 110
  Kinetic theory of gases, 40

  Lagrangean equations, 112
  Larmor, 149
  La Rosa, 164
  Length, 9 ff
  Lewis, 166, 201
  Light, 150 ff
  Local time, 75
  Lorentz, 143, 147, 148, 149

  Mach, 183
  Mass, 102 ff
  Mathematics, 60 ff
  Maxwell, 44, 58, 112, 137, 148
  Meaningless questions, 28 ff
  Measurement approximate, 33
  Mechanism, 45
  Mercury, 105
  Michelson, 15, 26
  Michelson and Morley, 66
  Models, 45, 52

  Newton, 4, 110

  Operational character of
    concepts, 5
  Operational thinking, 32
  Optical space, 67
  Ostwald, 109

  Penumbra, 34
  Perrin, 107
  Physical reality, 59
  Planck, 69
  Poincaré, 48, 115, 116, 190
  Pythagoras, 61

  Quantum act, 156
  Quantum theory, 40, 47, 186

  Radiation and temperature, 123
  Relative character of
    knowledge, 25
  Relativity, 150 ff
  Reynolds, 93
  Rotational motion, 178
  Russell, 205

  Schrödinger, 222
  Space, 66
  Silberstein, 11
  Simplicity of nature, 198
  Simultaneity, 7, 8
  Spring balance, 103
  Statistical methods, 115, 117
  Stress, 54
  Swann, 204

  Table top, 106
  Tactual space, 67
  Temperature, 118 ff
  Thermodynamics, 117
  Thing traveling, 101, 152, 157,
    164
  Time, 69
  Tolman, 201
  Truth, 78
  Turbulent motion, 120, 124

  Velocity, 97 ff, 213 ff
  Velocity of light, 100

  Whitehead, 167
  Wilson, 224



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