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 recognised,
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 "physical."
This attitude has no doubt had a certain
justification in the utter unintelligibility
to the physicist of many physical speculations
and the sterility of such speculations in
yielding physical results. However, the growing
reaction favouring a better understanding
of the interpretative fundamentals of physics
is not a pendulum swing of the fashion of
thought toward physics, 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 naive 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 two-fold. 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
generalisations 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 now have discloses the
present structure of physics and a realisation
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 combating 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 behaviour 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
savouring of the physical, because it is
evident that the nature of our thinking mechanism
essentially colours any picture that we can
form of nature, and we shall have to recognise
that unavoidable characteristics of any outlook
of ours are imposed in this way.
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 fast 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 recognises
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 f act nature can be embraced
in a formula, but we must so organise our
thinking as not to demand it as a necessity.
THE OPERATIONAL CHARACTER OF CONCEPTS
Einstein's Contribution in Changing Our Attitude
Toward Concepts
Recognising 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 emphasise 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.
(1) 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 a 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 conventionalised
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 behaviour 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 analysing
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
recognise 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 "naive"
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 coördinates,
i. e., at each point of his plane of reference
there must be a clock, and all these clocks
must be synchronised. 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 recognise
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
coördinates, 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 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 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.
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 0.19. 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 105 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 behaviour 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 minimise 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 10-1
cm.? What we would like to mean is that 1/3
x 108 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 10S-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 g 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 -12 cm. What is the possible
meaning of the statement that the diameter
of an electron is 10 -13 cm.? Again the only
answer is found by examining the operations
by which the number 10 -13 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 coördinates,
to determine which involves measurement of
lengths. But if these space coördinates 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 recognise 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 Einstein's.
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, 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 reassume its desired absolute
character.
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 physician
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 behaviour 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 recognised, 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 emphasise 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 neighbour 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 generalisations and idealisations 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 temporise or compromise in applying our
theories of conduct to practical life we
may suspect a failure of operational thinking.
...
The Logic of Modern Physics (1927), publ.
MacMillan (New York) Edition, 1927.
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Evans Experientialism 
