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

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

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

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

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

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

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

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

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

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

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

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

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

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

THE CAUSALITY CONCEPT

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

THE CONCEPT OF IDENTITY

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

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

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

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

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

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

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

THE CONCEPT OF VELOCITY

The concept of velocity, as ordinarily defined, involves the two concepts of space and time. The operations by which we measure the velocity of an object are these: we first observe the time at which the object is at one position, and then later observe the time at which it is at a second position, divide the distance between the two positions by the time interval, and if necessary, when the velocity is variable, take the limit. As long as we deal with fairly low velocities we do not have to inquire carefully as to the kind of time we use in these operations, but when the velocities become high, we do have to take care to use the local times at the two positions of the body, which means that we must have a time system spread over space, or, in other words, the "extended" time system. This velocity concept, defined in this way, may be used as a tool in describing nature, and it will be found that nature has certain properties; for example, the velocity of light is 3 x 10¹⁰ cm./sec. Further, no material thing can be given a velocity as high as this, but as its velocity is made to approach this value, increments of energy increasing without limit are required.

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