SPECIAL VIEWS OF NATURE
IN this last chapter we propose to discuss certain special hypotheses
about the structure of nature, and certain other matters that could best
be left until we had examined our fundamental concepts.
We have seen that in setting up the general rules which are to guide us
in describing and correlating nature, we have to take extreme care to
allow no special hypotheses to creep in, as otherwise we might be
restraining possible future experience. Even here there is no hard and
fast line of separation of the general from the special, and one might
entangle himself in inextricable difficulties if his ideals were too
meticulous. How for example is the critic to be answered who says: "Your
very endeavor to formulate principles so broad as not to restrict future
experience means, when examined in the light of operations, that you are
seeking for principles which _past_ experience suggests will not limit
the future. It is in the very nature of things impossible to escape all
the implications of past experience and therefore to find any completely
general principle." I believe that we must admit the critic is right,
and that rigorously our goal is impossible of attainment. We may say in
partial self-defense that all the discussion of this essay has been
subject to one explicit assumption, namely, that the working of our
minds is understood, which of course involves the assumption that our
minds continue to function in the future in the same way as in the past.
Even with this proviso we can not rigorously avoid the implications of
the past, but there can be no practical question that we recognize
certain assumptions about the behavior of nature to be so special as to
limit seriously the physical possibilities, and other assumptions to be
less restricting. In the previous discussion we had to make assumptions,
but I hope these assumptions will be recognized by all with physical
experience to be so broad as not to restrict us seriously. More special
assumptions or hypotheses have their very great use, however, when we
attempt to push forward the domains of experimental knowledge, because
they may suggest new experiments or aid in correlating information
already obtained. These special hypotheses may cover a very wide range
of generality; some of them are general enough in character to be
discussed here.
Among these special hypotheses there is a group which play an important
part in the speculation of most physicists, and which have features in
common. These are: the hypotheses of the simplicity of nature, of the
finiteness of nature in the direction of the very small, and of the
determinateness of the future in terms of the present. That these views
have points of similarity is obvious if we consider a hypothetical
special case. Suppose that no physical structure beyond the electrons
and protons can be discovered, or is even suggested by any known
phenomenon, so that the entire future behavior of a system can be
determined by a specification of the present relations of all its
protons and electrons; in this case nature would be both simple and
finite and the future determined by the present.
THE SIMPLICITY OF NATURE
Of these hypotheses, perhaps the most important is that of the
simplicity of nature, because of its wide spread diffusion and the
effect it has had on physical thought. The hypothesis of simplicity
assumes several forms; some physicists are convinced that the laws which
govern nature are simple, others that the ultimate stuff of which nature
is composed is simple (perhaps protons and electrons and energy), or
there may be a combination of both views into the belief that ultimately
we shall find simple ultimate elements behaving according to simple
laws. In one respect it is obvious that nature is not simple, namely
numerically--try counting the electrons or atoms or stars!
Consider now the first of these aspects of the thesis of simplicity,
which may be expressed as the conviction that the behavior of the entire
universe can be comprehended in a few principles of great breadth and
simplicity, such as the inverse square law of force, or the second law
of thermodynamics, or perhaps still better the equality of the
elementary positive and negative charges, which apparently holds to an
enormous degree of precision. In explanation of a view like this there
is in the first place the mental urge, because we can take a
satisfaction almost æsthetic in contemplating such a universe, and
there is in the second place a strong suggestion from experience.
Practically all the history of physics is a history of the reduction of
the complicated to the simpler. For example, the behavior of a large
part of the world of immediate experience can be reduced to the simple
laws of mechanics. The behavior of another very large group of natural
phenomena can be reduced to thermodynamics. The behavior of the heavenly
bodies, which at first was described in a rather complicated way in the
Ptolemaic system of astronomy, can be reduced to those same laws of
mechanics which we find in our immediate neighborhood, with the one
addition of the universal law of gravitation, which later refined
experiment discloses is really active in our immediate surroundings.
Similarly the laws of thermodynamics (except that part dealing with
radiation) are reduced to the ordinary laws of mechanics through the
additional assumption of the atomic structure of matter. Truly a
stupendous accomplishment that may well color our whole future outlook.
One may find great justification here for the belief that all nature
will ultimately be reduced to a similar simplicity, and, in particular,
justification for the attempt to find the explanation of all nature in
the action of mechanical laws. Now, of course, as a matter of physical
and historical fact, this program could not be carried through, but
obdurate physical phenomena were discovered. Electric phenomena, which
at first seemed so promising, refused to fit into the scheme, and the
converse attempt, to explain mechanical effects in terms of electrical
effects, also failed. We still carry our ordinary mechanical notions
down into the realm of small electric effects, and still talk, for
instance, about non-electrical forces which hold an electron together.
Nor are there experiments affording sufficient basis for believing that
all the mass of a positive nucleus is electrical in character. We also
think of electrical charges as having the property of identifiability,
which involves the possession of sharp edges and a change in the law of
force at small distances, and this is certainly a property carried over
from our large scale experience.
It seems fairly evident then that the laws of nature cannot be reduced
to either those of mechanics or of electricity, nor probably, as is
suggested by quantum phenomena, to a combination of both. This of course
does not preclude the possibility that the laws still may be simple when
expressed in other forms. An example of such a broad general law that
goes deeper than mechanics or electrodynamics is probably afforded by
the second law of thermodynamics when extended to include radiation
phenomena.
Examples of attempts to find other such simple laws are Tolman's
Principle of Similitude,[33] and Lewis's theory of Ultimate Rational
Units,[34] and his recently enunciated principle of Complete
Reversibility.[35] The first two of these attempts I do not believe are
successful, for reasons I have stated elsewhere,[36] the third also
seems somewhat doubtful.
[Footnote 33: R. C. Tolman, Phys. Rev. 3, 244-255, 1914; 6, 219-233,
1915; 15, 521, 1920; Jour. Amer. Chem. Soc., 43, 866-875, 1921.]
[Footnote 34: G. N. Lewis. Vol. 15, 1921 of the Contributions from the
Jefferson Physical Laboratory, dedicated to Professor Hall, Cambridge,
Mass.; Phil. Mag., 49, 739-750, 1925.]
[Footnote 35: G. N. Lewis, Proc. Nat. Acad. Sci., 11, 179-183, 422-428,
1925.]
[Footnote 36: P. W. Bridgman, Phys. Rev. 8, 423-431, 1916; "Dimensional
Analysis," p. 105.]
With regard to the general question of simple laws, there are at least
two attitudes; one is that there are probably simple general laws still
undiscovered, the other is that nature has a predilection for simple
laws. I do not see how there can be any quarrel with the first of these
attitudes. Let us examine the second. We have in the first place to
notice that "simple" means simple to us, when stated in terms of our
concepts. This is in itself sufficient to raise a presumption against
this general attitude. It is evident that our thinking must follow those
lines imposed by the nature of our thinking mechanism: does it seem
likely that all nature accepts these same limitations? If this were the
case, our conceptions ought to stand in certain simple and definite
relations to nature. Now if our discussion has brought out any one
thing, it is that our concepts are not well defined things, but they are
hazy and do not fit nature exactly, and many of them fit even
approximately only within restricted range. The task of finding concepts
which shall adequately describe nature and at the same time be easily
handled by us, that is, be simple, is the most important and difficult
of physics, and we never achieve more than approximate and temporary
success. Consider the example of time. The original concept of local
time, which for long seemed satisfactory, turns out to be inadequate,
and has to be replaced by extended time, which is so complicated that it
is questionable whether we shall ever be able to grasp it with the
confidence that we must demand in a useful concept (by "grasp" I mean
intuitive command of all the implications of the operations which are
involved). The concept has not yet been found which describes simply the
temporal relations of the universe.
Not only are concepts hazy around the edges and so incapable of fitting
nature exactly, but there is always the chance that there are concepts
other than those which we have adopted which would fit our present
phenomena. Finding concepts to fit nature is much like solving a
cross-word puzzle. In the puzzle there may be some parts of the pattern
which we fill completely and easily, but sometimes we find parts in
which we can fill in everything except one or two obstinate definitions,
so that we are sure we are on the right track, and rack our brains for
the missing words, when with a flash of inspiration we see that the
obstinate words can be fitted in by a complete change in those which we
had already accepted. It may be that we are soon to witness a similar
change in our concept of the nature of light. An important difference
between the cross-word puzzle and nature is that we can never tell when
we have filled in all the squares in any of the parts of nature's
puzzle; there is always the possibility of new phenomena which our
present scheme does not touch.
Considering, then, the nature of our conceptual material, it seems to me
that the overwhelming presumption is against the laws of nature having
any predisposition to simplicity as formulated in terms of our concepts
(which is of course all that simplicity means), and the wonder is that
there are apparently so many simple laws. There is this observation to
be made about all the simple laws of nature that have hitherto been
formulated; they apply only over a certain range. We have not extended
the laws of gravitation to small bodies, nor have we found that our
electrical laws will work on a cosmic scale. It does not seem so very
surprising that over a limited domain, in which the most important
phenomena are of a restricted type, the conduct of nature should follow
comparatively simple rules.
A tempting question is whether there may not be some laws of nature that
are _really_ simple, without relation to our mode of formulation, such
as the law of the inverse square. I leave it to the reader to decide
whether this question has meaning. In this connection it is possibly
significant that the average physicist is strangely reluctant to tamper
with the inverse square law. I find in myself a lack of sympathy, which
I cannot justify by any of the considerations of this essay, with
attempts like the recent one of Swann,[37] for example, to explain a
wide variety of hitherto obstinate effects by the assumption of slightly
unequal departures from the inverse square law by the electrons and
protons. Of course I hope that this feeling will turn out not to be
prejudice, but will perhaps be justified by some such general
observation as that a departure from the inverse square law so slight as
by definition to be forever beyond detection by direct experiment is
meaningless; but of this I am not at all sure.
[Footnote 37: W. F. G. Swann, Phys. Rev., 25, 253, 1925.]
We are now ready to consider the second respect in which nature may be
simple, namely, because the material of which it is built may reduce to
a few sorts of elements. In this discussion it will be convenient to
consider also at the same time the more inclusive simplicity arising
from simple laws acting on simple elements. The immediate question for
us here is one of fact: does nature seem to be getting intrinsically
simpler as we get toward small scale phenomena? There is much room for
difference of opinion here; personally I feel that this expected
simplicity is not in evidence, at least to the extent that we could
desire. For instance, the fact that the electrons must have both
electrical and mechanical properties is a straw in this direction.
It must also be remembered that a certain simulation of simplicity is
inevitable as we approach the limits of experimental knowledge, whatever
the actual structure of nature, for the mere reason that near the limit
our possible experimental operations become fewer in number, and our
concepts fewer also. The question which we are trying to answer has,
therefore, its real meaning only in terms of the possible future. Do we
believe that if we drive in our stakes at a certain point on our present
frontiers, this point will gradually, as physics advances, become
possessed of a continually richer experience, so that nature at this
point will appear increasingly complicated? Or do we expect a
termination of this process of expansion fairly soon? It seems to me
that as a matter of experimental fact there is no doubt that the
universe at any definite level is on the average becoming increasingly
complicated, and that the region of apparent simplicity continually
recedes. This, however, is not the opinion of all observers. Thus
Bertrand Russell, in "What I Believe", page 10, writes, "Physical
Science is then approaching the stage where it will be complete, and
therefore uninteresting."
This is perhaps a particularly favorable epoch in the history of physics
to urge the essential complexity of nature, because all our new quantum
phenomena indicate a vast wealth of hitherto unsuspected relations on
the very edge of the attainable. There is one aspect of quantum
relations, as also of our ideas of the nature of the structure of the
nucleus of the atom, which is particularly significant in this respect,
namely, that we have to describe phenomena by statistical methods. Now a
statistical method is used either to conceal a vast amount of actual
ignorance, or else to smooth out the details of a vast amount of actual
physical complication, most of which is unessential for our purposes.
There can be no doubt of the amount of ignorance that the statistical
method conceals when applied to these phenomena, but there are also
strong indications, particularly when applied to the nucleus, that it
covers a vast amount of actual physical complications. The nucleus of a
radium atom becomes unstable on the average every 10^4 years, which may
be plausibly taken to indicate that every 10^4 years the radium nucleus
gets itself into some particular configuration. Considering the time
scale on which we suppose events in the atom to take place, and also
considering the fact that radioactive disintegration seems unaffected by
outside agencies, this would indicate a perfectly appalling amount of
structure. We are similarly driven to statistical methods in quantum
theory, as for example, in Einstein's analysis of the details of
equilibrium between emitting and absorbing atoms and radiation.
In general, we cannot admit for a minute that a statistical method,
unless used to smooth out irrelevant details, can ever mark more than a
temporary stage in our progress, because the assumption of events taking
place according to pure chance constitutes the complete negation of our
fundamental assumption of connectivity; such statistical methods always
indicate the presence of physical complications which it must be our aim
to disentangle eventually.
It appears then that present experimental evidence makes very probable
structures beyond the electron and the quantum; we may go even further
and say that there is no experimental evidence that the sequence of
phenomena in nature as we go to ever smaller scales is a terminated
sequence, or that a drop of water is not in itself essentially infinite.
(This statement contains by implications the meaning that we attach to
infinite.) All the more, then, there is no evidence that nature reduces
to _simplicity_ as we burrow down into the small scale.
Whatever may be one's opinion as to the simplicity of either the laws or
the material structure of nature, there can be no question that the
possessors of some such conviction have a real advantage in the race for
physical discovery. Doubtless there are many simple connections still to
be discovered, and he who has a strong conviction of the existence of
these connections is much more likely to find them than he who is not at
all sure they are there, and is merely hunting for anything that may
turn up. It is largely a matter of psychology. Everyone knows that the
mere suggestion that a problem has a solution, or the knowledge that
someone has already solved it, is often sufficient to suggest a relation
that otherwise might not have been noticed. The chances are, therefore,
that the relations between phenomena will be found by those who are
previously convinced that the relations exist. The observation that most
of the discoveries are made by men with particular sorts of conviction
naturally strengthens the belief that their convictions are true. But
this picture has an obverse side. The man who is convinced that there is
a relation where none exists may waste all his time in vain seeking for
it. Granted that nature has no particular predisposition to simple
relations, the conviction that there are such relations is, from the
point of view of any one individual, as likely to be a hindrance as a
help. From the point of view of physical society, on the other hand, it
is desirable that there be such convictions, for in such a society there
will be more discoveries than in a society without such convictions. We
have here again the old conflict between the individual and society. As
in all other similar conflicts, society will not be able to demand
permanently from the individual the acceptance of any conviction or
creed which is not true, no matter what the gain in other ways to
society. If nature is not simple, physicists will not continue to
believe that it is, even if such a conviction does increase the total
number of discoveries. It is an impossible attitude to expect that one
can maintain. Does this then mean that physics is to face a drab future,
becoming continually more prosaic, with new discoveries ever rarer, made
by a continually decreasing number of misguided but fortunate
enthusiasts? There may be such a danger, but the greatest part of the
danger is avoided if its nature is clearly recognized. One of the
problems of the future is the self-conscious development of a more
powerful technique for the discovery of new relations without the
necessity for preconceived opinions on the part of the observer.
There is an aspect here of our physical research that is often lost
sight of, namely, the small proportion of successful discoveries
compared with the number of investigators. Certainly the number of
unsuccessful attempts, even in the case of those fortunate individuals
who make the great discoveries, is very much greater than the number of
their successful attempts. (Faraday's reputed satisfaction with a ⅒%
return comes to mind.) This must always be taken into account in
estimating the probable chances of correctness of any new theory. With
so many physicists working to devise new theories, the chances are high
that many false theories will be found, in which a number of phenomena
may apparently fit together into a new relation, but which eventually
prove to be inconsistent with other phenomena, so that the proposed
theory has to be abandoned. As physics advances and the number of
investigators and the amount of physical material increases, one has to
be more and more exacting in one's requirements of a new theory. One
must be particularly on guard against numerical coincidences. An
interesting chapter might be written on numerical relations which have
been hopefully published, but later had to be abandoned as without
significance.
DETERMINISM
If we are right in supposing that physical evidence gives no warrant for
the idea that nature is finite downward, we have not only repudiated the
thesis of simplicity but we have also made a very important observation
on the other general thesis mentioned at the beginning of this chapter,
namely, the thesis of physical determinism. By determinism we understand
the belief that the future of the whole universe, or of an isolated part
of it, is determined in terms of a complete description of its present
condition. [What we mean by present condition will be discussed later.]
It is popularly assumed that every physicist subscribes to some such
thesis as this. But now if there is infinite structure even in a small
isolated part of the universe, a complete description of it is
impossible, and the doctrine as stated must be abandoned. It seems to me
that all present physical evidence prepares us to admit this
possibility. I suppose, however, that most physicists would subscribe to
some modification of the original thesis, perhaps along the following
lines. Given a description of an isolated part of the physical universe
in the most complete terms that have physical meaning, that is, down to
the smallest elements of which our physical operations give us
cognizance, then the future history of the system is determined within a
certain penumbra of uncertainty, this penumbra growing broader as we
penetrate to finer details of the structure of the system or as times
goes on, until eventually all but certain very general properties of the
original system, such as its total energy, are forever lost in the haze,
and we have a system which was unpredictable. I suppose that it is a
further conviction of at least many physicists that by sufficiently
refining our measurements, the amount of haze at any fixed point in the
future may be made indefinitely small, and many might even go further
and hope by studying the haze (perhaps statistically) to obtain some
inferential evidence of structure beyond that yet experienced. In fact
it may be that this last contains the germs of the ultimate method of
investigation, if we ever reach a stage when we can no longer refine our
methods of measurement.
Determinism to the physicist is simply a way of stating certain
implications of his conviction of the connectivity of nature. We have
seen that the broadest possible statement of the thesis of connectivity
is: Given two isolated systems with identical past histories up to a
certain epoch, then the future histories will also be identical. The
thesis of the determinism of the future by the present constitutes a
specialization of this general thesis in that we suppose that identity
of _all_ past history is not necessary for identity of future behavior,
but only identity of present condition. The general and the special
thesis are not equivalent by any means: if past histories are identical
then present conditions are also identical, but the converse does not
necessarily hold at all.
Now I believe that the general thesis (which I suppose all physicists
will admit, but whose truth is nevertheless subject to the verification
of experience) gets turned into the special thesis by a feeling of
somewhat metaphysical content, which we may perhaps state by saying that
we can see no way by which the past can affect the future except through
the present. We do not like to think of the effect of a cause distant in
the past jumping over the present and affecting the future without
touching the present at all. It is the analogue of that attitude of mind
to which action at a distance in space is inconceivable; just as it is
difficult to conceive of a body here affecting a body there without in
some way an action propagated through intermediate space, so we do not
like to think of a past cause jumping over time and producing a future
effect without some sort of continuity in the causal chain through all
intermediate time.
So far our discussion has been purposely loose: it is evident that what
we mean by "present state" is crying for definition. What is meant by
this may depend somewhat on the specific hypothesis that one adopts
about the structure of nature. Historically the conviction of future
determinism has been most intimately associated with a mechanical
picture of the structure of the universe, so that it may be well to
begin from this point of view. Suppose the simplest possible system
composed of point masses without structure, as in the kinetic theory of
gases. What sort of specifications do we believe necessary to fix the
present state of such a system? The mechanical view of nature gives a
definite answer. By present state we mean the positions and velocities
of all the masses. This is sufficient for the complete determination of
any purely mechanical system, in which the forces between the elements
are known functions of only their relative positions. By a sort of
extension of these ideas valid for mechanical systems, it seems to be
often thought that the present state of _any_ system is determined by a
complete specification of the positions and velocities of all the
ultimate elements of the system (provided always of course that this
number is finite). This principle, however, does not appear to bear the
check of experiment when applied to electrical systems with radiation.
The theorems of the retarded potential show that such systems are
determined by the present position and velocities of the charges in the
immediate vicinity, and by the corresponding data at remote points given
for proper epochs in the past; in this case, therefore, past and present
history are necessary to determine the future. But if we consider the
electrical field as part of the system, we may fix the future in terms
of the present positions of the charges, their velocities, and the
values of the field vectors all over space, thus returning to a certain
formal resemblance to mechanical systems, and suggesting a reason for
ascribing physical reality to the electric field. This analogy with a
mechanical system is, however, loose; complete analogy would allow the
instantaneous values of the time derivatives of the field to be given
also, and this is not possible.
How is it that velocity can strictly be regarded as characteristic of
the _present_ state of the system? Certainly the usual operations for
measuring velocity demand that we know the configuration of the system
at two different times, and calculate the velocity from certain
differences of the system at these two times. The velocity is defined as
a limiting result, but even in the limit the essential physical fact
does not disappear that we must know the positions of the system at two
times. We may now go further; if the velocity is properly included in
the present attributes of the system, we can see no reason for not
including a specification of all the higher time derivatives also. In
the case of the simple gaseous system under present consideration we can
answer this question by examining the operations by which we actually go
to work to determine the future of such a system. The problem of
determining the future condition of such a system reduces to the problem
of writing the differential equations of motion of all its parts. If the
system is a mechanical system, as in this case, these equations are of
the second order in the time derivatives of the position coordinates,
and also involve the forces, which we suppose are known in terms of the
relative positions of the parts of the system. Given, then, the
positions and the way in which the forces depend on the relative
positions of the parts, the equations of motion can be written down for
any configuration of the system, and these equations may be integrated
(at least approximately) in terms of the proper initial conditions. Now
the only boundary conditions on a second order equation are the initial
positions and velocities. This is the reason that velocities have to be
specified in giving the present condition of the system, and that it is
not necessary to give the higher derivatives. Apparently the reason why
we instinctively include velocity among the present properties of the
system is not because velocity is by its nature strictly a present
property of the elements of the system, but rather because our wide
experience with mechanical systems has shown that as a matter of fact
velocity is necessary in such systems to determine future motion.
But now if the equations of motion of the parts of the system are not
those of mechanics, they will in general be much more complicated in
appearance and will involve higher derivatives of the time than the
second. Suppose for the moment that the equations contain only
derivatives and the mutual positions of the parts of the system. Then to
integrate the equations and determine the motion we have to know the
initial positions and the initial values of all the derivatives up to an
order one lower than the highest which occurs in the equations. The
equations of motion of an electron are even more complicated than this,
in that the positions of distant parts of the system have to be given
throughout an interval of time instead of merely an instant. It would
seem that the feeling that the present state of a system may be
determined in terms of positions and velocities does not as a matter of
fact apply to all the systems of our experience.
The discussion up to this point has been subject to the fundamental
assumption that the behavior of the system is entirely determined if we
can give the position of each part as a function of time. This
assumption is implicitly contained in Einstein's formulation of the
general principle of relativity, namely, that there is nothing more to a
physical system than a set of space-time coincidences, and that the
system is fixed in terms of the space time coördinates of all its
parts. Already in discussing the assumption of relativity we have
indicated reasons for dissatisfaction with this as a means of
reproducing all experience, because in giving only the space-time
coordinates of events we have entirely omitted the descriptive
background of the equations, which gives physical color to the system in
question. This discussion also assumes that a specification of the
positions, velocities, and higher derivatives (if necessary) of the
elements of the system is possible, which amounts essentially to the
assumption that the system contains only a finite number of elements.
Now in view of the experimental fact that there is no reason for
supposing that the structure of the universe is finite, this conclusion
must be modified, but I do not believe that the necessary modification
affects the essential argument. In view of the possible infinite
structure it would seem that we cannot expect more than that the future
is determined by the present within a certain penumbra of uncertainty,
and this penumbra may be made less important by digging down deeper into
the structure when specifying the present condition.
We have also slurred over the ambiguities in "present" condition when
the system is spread over space. Probably a unique ascription of meaning
to "present" is not possible for an extended system, but at least one
possibility is indicated by relativity theory. Imagine a staff of
assistants distributed throughout space, each equipped with clocks
synchronized and set with the master clock by light signals in the
conventional manner, and each fully equipped with the necessary
measuring instruments. Then what we mean at this point of the argument
by "present" state of the system is the aggregate of all the information
about the _positions_ and _velocities_ of the ultimate elements which I
determine in my immediate vicinity at my origin of time plus the reports
of similar observations made by all the assistants, each local
observation being made at the time origin of each local clock.
Going back now to the main argument, we have shown that the feeling that
the present condition of the universe may be specified in terms of
positions and velocities arose from experience with purely mechanical
systems, and that the more general formulation, in which we add to the
velocities the higher time derivatives, applies only to systems in which
the ultimate elements move according to differential equations of higher
order than the second. Furthermore, our analysis seems to have shown
that systems in which there is radiation do not allow a determination of
the future in terms of a present condition specified in terms such as
these. It seems, however, that the general principle of the determinism
of the future by the present may be saved by a change in the definition
of what we mean by the present condition of the system, ridding it of
its mechanical and other special implications, and making more immediate
connection with direct experiment. Let us understand by present
condition of a system the aggregate of all information that can be
obtained by any physical means whatever, with any sort of physical
instrument, not attempting to get out of this analysis information about
hypothetical ultimate physical elements, with the proviso that the
measurements are to be made now, extending the concept of "now" to
points distant in space in the way intimated above. With such a general
definition of the meaning of "present" we can now deal with systems in
which there is radiation, noticing that our assistant observers must be
stationed throughout apparently empty space as well as in the
neighborhood of matter. That this does adequately cover the case of
radiation is suggested by considering again the two systems of dark
lanterns with screens and distant mirrors which we have previously
considered, in one system a light signal having been despatched 0.5
second ago and in the other 1.5 seconds ago. Our thesis demands that
there be some present difference in these two systems, because their
future history is different, in one of them a light signal arriving
after the lapse of 1.5 seconds, and in the other after only 0.5 second.
Now there is a present difference as reported by our assistants, for the
assistant stationed half way between lantern and mirror reports in one
system a flash of light on the side of a screen which is turned toward
the lantern, and in the other system on the side of the screen turned
toward the mirror.
This more general point of view answers the question whether velocity
may be regarded as a present attribute of the system, for the parts of a
system which are in motion have momentum, and momentum may be detected
by placing against such parts comparatively rigid members which will
receive a minute deformation, so that velocity has a meaning in terms of
physical measurements made at a single instant of time.
There is a subtle and difficult question here, namely, whether in
talking about operations of measurement we can ever get rid of temporal
implications, and therefore, whether a condition of the system in which
temporal implications remain can properly be described as "present." I
shall not attempt to answer this question: there must be some
practically satisfying answer, involving perhaps the physical analogue
of differentials of different orders in mathematics, short of carrying
the analysis to such a degree of refinement that the concept of present
becomes meaningless, as we can see might easily happen.
With this enlarged understanding of what we mean by present state of the
system, it seems to me that physical evidence is now rather favorable to
the view that the present determines the future, subject to
qualification about the penumbra, at least as far as large scale
phenomena are concerned. It appears much more doubtful when we come to
small scale phenomena, and in particular it is doubtful whether the
principle can be applied to the details of the quantum process, and in
fact it is not certain that it has meaning. It is certain that if it is
true an enormous amount of structure beyond any that has yet been
detected is implied.
ON THE POSSIBILITY OF DESCRIBING NATURE
COMPLETELY IN TERMS OF ANALYSIS
There is a certain thesis that is loosely related to the view that
nature is finite downward, namely, that an explanation of the universe
is possible in which we start with small scale things, and explain large
scale phenomena in terms of their small scale constituents, the thesis,
in other words, that all the properties of the large are contained in
the properties of the small and that the large may be constructed out of
the small. Some such thesis as this seems implied in the general
attitude of many physicists. Let us examine the physical basis for this.
To maintain this thesis would demand that aggregates of things never
acquire properties in virtue of their numbers which they do not already
possess as individuals. Is this true? Consider, for example, the
two-dimensional geometry on the surface of a sphere. This is
non-Euclidean. Is the geometry of the individual elements of the surface
of the sphere non-Euclidean, or do they acquire this property in
changing scale? Is the kinetic energy of a number of electrons all
moving together in such a way as to constitute an electric current the
sum of the kinetic energies of the individual electrons, or is there an
additional term? Is the mass of an electron the sum of the masses of its
elements?
A mathematical consideration is suggestive here. Those properties of a
system which can be described in terms of linear differential equations
have the property of additivity; the effect of a number of elements is
the sum of the effects separately, and no new properties appear in the
aggregate which were not present in the individual elements. But if
there are combination terms (as in the electrical energy, which contains
the square of the field), then the sum is more than (or different from)
its parts, and new effects may appear in the aggregate. Now of course
the linear equation is of enormous importance in describing nature, but
many examples of systems with other types of equation can be found, as
that above for electromagnetic mass. In expecting to find in nature such
non-additive effects, we need not commit ourselves at all to the view
that nature is governed by differential equations, but by analogy may
expect similar effects if difference equations, for instance, should
prove to be fundamental, or even something beyond present mathematical
formulation.
It is certainly very much easier to handle a system physically if the
total action can be built up from that of its parts, because the
analysis which establishes the connection between the elements is easier
to perform. It is obviously easier to show that an explanation in such
terms is correct, because we have seen that explanation involves making
experiments with representative elements absent or altered, and it is
easier to vary the small things than the large things. Those
explanations which involve working from the small up will therefore be
made first, and will appear to be of disproportionate importance. Places
where I look for an explanation from the large to the small are perhaps
in accounting for the values of the gravitational constant and the
velocity of light and in those phenomena which general relativity theory
indicates may depend on all the matter in the universe, as the Foucault
pendulum experiment. We must, of course, also be prepared for such
non-linear effects in the domain of unexplored quantum phenomena.
A GLIMPSE AHEAD
Some of the general considerations of this essay may, with considerable
plausibility, be expected to play a part in the future of both
speculative and experimental physics. The most important effect may be
expected from the clearer recognition of the operational character of
our physical concepts. Indeed during the writing of this essay there has
been a very marked increase in emphasis on the necessity of
understanding in terms of physical operations such fundamental concepts
as that of the electron, by the new quantum mechanics [the mechanics of
Heisenberg-Born and Schrödinger of 1925-26].
We are to expect then in the first place a more self-conscious and
detailed analysis of the operational structure of all our physical
concepts. [It has been beyond the scope of this essay even to begin to
attempt a systematic and thoroughgoing analysis of this character.] This
future analysis will show precisely how, as we extend the range of
experience, the physical character of the operations changes by which we
define our concepts, as, for example, in mechanics the notion of force
disappears at high velocity and is replaced perhaps by the notion of
momentum. In the region of change in the nature of our concepts, special
study will be made of the accuracy of our physical measurements, and new
experiments devised of greater accuracy, in order that we may know
precisely to what extent the new concepts are equivalent to the old.
Past experience suggests that we may perhaps expect to find new
phenomena especially in those regions where the difficulty of carrying
out the usual operation forces us to change the operational character of
our concepts. There will be questions of a more or less formal nature to
answer, as for example the _best_ way of extending concepts when there
are several possible courses open to us.
We may expect more interesting results, however, when we get so far
beyond ordinary experience that the character of the possible physical
operations has become so restricted as to result in an apparent decrease
in the number of independent concepts. It seems plausible to expect that
the structure of nature is more fundamentally connected with the number
of independent concepts necessary for a complete description than with
the precise details of the structure of the individual concepts, such,
for example, as whether space is measured optically or tactually. In
those regions where the number of concepts decreases, we must make the
most thoroughgoing experimental examination to discover if possible new
sorts of operations by which the number of concepts may be brought back
to normal. In searching for such new experimental operations it seems to
me that by far the greatest promise for the immediate future is offered
by improvements in our powers of dealing with individual atomic and
electronic processes, such as we now have to a limited extent in the
various spinthariscope methods of counting radioactive disintegrations,
or Wilson's β-track experiments.[38] In this self-conscious search for
phenomena which increase the number of operationally independent
concepts, we may expect to find a powerful systematic method directing
the discovery of new and essentially important physical facts.
[Footnote 38: C. T. R. Wilson, Proc, Roy. Soc., 87, 277, 1912.]
We can only conjecture whether the number of fundamental concepts will
prove impossible of further increase or not, but present experience
seems to give greater probability to the view that as we penetrate
deeper the number of fundamental concepts will always tend to become
fewer. We have already certainly one example in that the temperature
concept disappears when we get to the atomic scale of magnitude, and
possibly a second example in the building up of separate concepts for
energy and frequency by the combination of great numbers of that one
operationally simple thing which characterizes the elementary quantum
process in ordinary radiation.
Different sorts of relations between concepts are conceivable in the
transition zone where the number changes. We may find that other
examples are like that of temperature, which is simply a statistical
effect of a great many phenomena which may be described individually in
terms of the ordinary concepts of mechanics, so that in this case the
number of concepts changes merely by temperature dropping out, leaving
the others more or less unaffected. Or all the concepts may be more
closely interwoven, so that when the total number of concepts changes it
may not be possible to separate out a group of concepts whose defining
operations are unchanged. In such a case we must say that the original
concepts are not applicable on the new level. The most immediate
application of this idea has been already mentioned, namely, to the
concepts of space and time. If the operations by which space and time
are measured on the ordinary scale of magnitude cannot be carried down
as a whole into the region of quantum phenomena, then we must say that
the ordinary concepts of space and time are not applicable to these
phenomena.
Closely connected with the sharper analysis of the operational structure
of our concepts, we may expect in the future also a closer analysis of
our inventions. This will take the form of a search for new physical
facts which shall give to our inventions the character of physical
reality. In case prolonged search fails to disclose such phenomena (as
is probably now the case with the field concept of electrodynamics), we
must then find some way of embodying explicitly in our thinking the fact
that we are dealing with pure inventions and not realities.
INDEX
Absolute, 26
Absolute time, 4
Action at a distance, 46
Analysis of large into small, 51,
220
Arithmetic, 35
Atom, 59
Bell, 84
Birkhoff, 72
Black body, 112
Bohr, 190, 192
Born, 222
Boscovitch, 46
Bothe and Geiger, 116
Bridgman, 201
Brownian movement, 107, 129,
143
Bush, 142
Caloric fluid, 59
Campbell, 117
Carnot engine, 125
Causality, 80 ff
Causal train of events, 85
Clifford, 28
Clock, 70 ff, 176
Compton, 116, 188
Continuity, 94
Conservation of charge, 135,
136
Conservative functions, 113
Constructs, 53
Correlation, 37
Cosmic units, 182
Cross word puzzle, 202
Descriptive background, 64
Determinism, 114, 209 ff
Discontinuous space, 191
Döppler effect, 166
ds, 72
Eddington, 93
Einstein, vii, 1, 2, 3, 4, 7, 8,
9, 12, 13, 14, 64, 100, 155,
162, 163, 164, 167, 169, 170,
171, 172, 173, 175, 176, 177,
206
Electrical concepts, 131 ff
Electrical explanation of
universe, 50
Electrical mass, 139
Electric field, 56, 133
Empiricism, 3
Energy, 108 ff, 126 ff
Euclidean space, 14, 15, 16, 18,
23, 52, 61, 67
Event, 95, 167
Explanation, 37
Explanatory crisis, 41
Extended time, 77
Faraday, 44, 57, 58, 209
Final explanations, 48
First law of thermodynamics,
126 ff
Force, 102 ff
Foucault pendulum, 180, 184
Fourth dimension, 74
Future, 222
Gauss, 15, 34
"Go and come" time, 112
Gravitational constant, 91
Haldane, 25
Heat flow, 130
Heisenberg, 222
Hertz, 44
Hoernlé, vi
Identity, 91 ff
Isolation, 82
Joule, 124
Kelvin, 45, 110
Kinetic theory of gases, 40
Lagrangean equations, 112
Larmor, 149
La Rosa, 164
Length, 9 ff
Lewis, 166, 201
Light, 150 ff
Local time, 75
Lorentz, 143, 147, 148, 149
Mach, 183
Mass, 102 ff
Mathematics, 60 ff
Maxwell, 44, 58, 112, 137, 148
Meaningless questions, 28 ff
Measurement approximate, 33
Mechanism, 45
Mercury, 105
Michelson, 15, 26
Michelson and Morley, 66
Models, 45, 52
Newton, 4, 110
Operational character of
concepts, 5
Operational thinking, 32
Optical space, 67
Ostwald, 109
Penumbra, 34
Perrin, 107
Physical reality, 59
Planck, 69
Poincaré, 48, 115, 116, 190
Pythagoras, 61
Quantum act, 156
Quantum theory, 40, 47, 186
Radiation and temperature, 123
Relative character of
knowledge, 25
Relativity, 150 ff
Reynolds, 93
Rotational motion, 178
Russell, 205
Schrödinger, 222
Space, 66
Silberstein, 11
Simplicity of nature, 198
Simultaneity, 7, 8
Spring balance, 103
Statistical methods, 115, 117
Stress, 54
Swann, 204
Table top, 106
Tactual space, 67
Temperature, 118 ff
Thermodynamics, 117
Thing traveling, 101, 152, 157,
164
Time, 69
Tolman, 201
Truth, 78
Turbulent motion, 120, 124
Velocity, 97 ff, 213 ff
Velocity of light, 100
Whitehead, 167
Wilson, 224