The logic of modern physics

by P. W. Bridgman

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.

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

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

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

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

THE CONCEPT OF VELOCITY

The concept of velocity, as ordinarily defined, involves the two concepts of space and time. The operations by which we measure the velocity of an object are these: we first observe the time at which the object is at one position, and then later observe the time at which it is at a second position, divide the distance between the two positions by the time interval, and if necessary, when the velocity is variable, take the limit. As long as we deal with fairly low velocities we do not have to inquire carefully as to the kind of time we use in these operations, but when the velocities become high, we do have to take care to use the local times at the two positions of the body, which means that we must have a time system spread over space, or, in other words, the "extended" time system. This velocity concept, defined in this way, may be used as a tool in describing nature, and it will be found that nature has certain properties; for example, the velocity of light is 3 x 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.

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

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