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.