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?
[12]Osborne Reynolds, The Sub-Mechanics of the Universe, 254 pp., Cambridge University Press, 1903.
[13]A. S. Eddington, Report on Gravitation, Lon. Phys. Soc., 1918, p. 91.
In fact, the identity concept seems to lose all meaning in terms of actual operations on this level of experience.