The Analysis of Matter

temperatures, etc., have all been[Pg 346] regarded as caused by various kinds of motions. There was no objection to this so far as it succeeded, but, if and where it proves insufficient, there can also be no objection to re-introducing qualitative differences into the physical world. There is, however, one essential limitation. We may find reasons for supposing qualitative differences, in order to be able to build up the kind of structure which we have inferred; but we cannot have any means of knowing what are the qualities which differ. This point was discussed in Part II., and need not now detain us. The apparatus so far assumed, apart from qualities, has been: co-punctuality, cause-and-effect, and the quantum laws. I say "cause-and-effect" because it is necessary to be able to distinguish the earlier from the later event in a transaction, and this is a smaller assumption than that of a general time-order among events in one causal series. The above apparatus sufficed except for one purpose: that of defining "repetition." The possibility of repetition is at the bottom of the common-sense distinction between space and time; the substitution of space-time should, one might suppose, make repetition impossible, and yet the whole of what is distinctive in quantum physics, and the theories of light and sound, not to mention other matters, depend upon periodicity, which involves repetition. So long as we had billiard-balls moving in an unchanging space, we could be content with repetition of configuration. But now spatial distance, which is essential to configuration, has to be analyzed into an elaborate indirect relation depending upon the existence of common causal ancestors or descendants. We must, therefore, be able to distinguish among events by means additional to their space-time relations. There is, however, a considerable difficulty in finding laws governing what we are calling "qualities." In a world of continuous processes, one would say that qualities must[Pg 347] change gradually. But in a quantum process they apparently change suddenly. Perhaps, however, this suddenness does not exist in a steady rhythmic process; or perhaps, even if it does, it may involve small changes producing a serial character in the successive qualities. Take, for example, the revolution of an electron about a nucleus. In the newer quantum theory this does not really occur, but we may consider how it could be interpreted if it were necessary to assume it. Let us make a fantastic hypothesis, purely for illustrative purposes: let us suppose that the electron and the nucleus can see each other, and that neither rotates on its own axis. Then they will get pictures of each other which change during each revolution, and repeat the cycle of changes each time. Now let us turn this hypothesis round, and begin by assuming the recurrent series of pictures. From this we can infer the revolution of the electron, provided we are free to construct space as we like, subject to certain formal laws. Now in fact we have this freedom: the "space" in which the electron revolves need only have certain abstract mathematical properties, and, so long as it has them, it may be constructed out of any material available. So long as the electron continues in one orbit, we may conceive, at any rate as a schematic simplification, that there is a persistent event which may be taken as representative of it, and in like manner that there is a persistent event P representative of the proton. Now let us suppose that, compresent with E but not with each other, there are successive events , , , ... which may be regarded as "aspects" of the proton, and are related to each other more or less in the way in which the appearances of the proton from different places would be related if the electron could see. Similarly let us assume a series of events , , , ... compresent with but not with each other, analogous to what would be appearances of the electron to the proton if the proton could see. And let us further suppose that, after a certain set[Pg 348] of such events, an exactly similar set recurs, or a very approximately similar set. This supposition provides us with the material required for a periodic relative motion. We shall say, therefore, not that perspectives differ because spatial relations change, but that change in spatial relations consists of systematic alteration in perspectives. Such a view is feasible, but it makes similarity and difference of quality essential. It ceases to be fantastic if we drop the analogy with vision except as regards purely formal characteristics. Let us now set forth the analysis of a periodic process suggested by the above, bringing it into relation with the construction of points in Chapter XXVIII. Let us assume, to begin with, that the process is discrete; this hypothesis can be dropped later, but simplifies the initial statement. Suppose, for the sake of illustration, that there are ten qualities, , , ... , and that there exist events which are subject to the following conditions: (1) , a, , ... have the quality : , , ... have the quality etc. (2) Each of the 's is compresent with its immediate neighbour to left and right, but with none of the other 's; (3) If , any point of space-time of which but not is a member has a time-like interval from any point of which but not is a member. In that case, the series of 's constitutes a periodic process, having ten 's in each period. The last digit in the suffix of an indicates the quality of the —i.e. if the last digit is , the quality is —while the remaining digits indicate the number of the period. If all the 's are events in the history of one piece of matter, that piece of matter is undergoing the periodic process. If there is a correlative series of 's in another piece of matter,[Pg 349] the two periodic processes together make up one relative motion of a periodic character, such as the revolution of an electron about a proton. Generalizing the above, while still assuming that the process is discrete, suppose we have qualities , ... , and a set of events where, as before, the last suffix indicates the quality, i.e. has the quality (). Suppose, also, that each is compresent with of its predecessors and of its successors, where ; but that no is compresent with any except these. The remaining assumptions are to be as before. Then again we obtain a rhythm which may be regarded as an analysis of periodic processes in physics. If we suppose that the 's are not compresent with any events except the other specified 's, then the group of 's with which a given is compresent constitutes a point, which may be taken as the middle point in the duration of the in question. We can take this point as representative of the in question, since their relation is one-one. Thus the in question is associated with a point, in spite of the fact that it lasts for a finite time, i.e. is compresent with events not compresent with each other. It is to be observed that, according to the theory of space-time in Chapters XXVIII. and XXIX., it is quite possible for some parts of space-time to be continuous and others discrete. I am supposing, at the moment, that we are considering a periodic process in a discrete part of space-time; this does not involve the hypothesis that all space-time is discrete. If the 's in one periodic process, as we supposed a moment ago, are not compresent with any events except certain neighbouring 's (which must be fewer than the whole of one period), then the number of points in a period is the same as[Pg 350] the number of 's, and either affords a measure of the duration of the period, measured by its proper time. It is obvious that, in a discrete part of space-time, the natural measure of distance will be number of intermediate points. We see also how the proper time of one process can differ from that of another. Let us suppose that our 's form an "isolated" process (i.e. are not compresent with anything except other 's), except at the beginning and end; the first and last 's are to be compresent with the first and last terms of another periodic process composed of 's, which also is to be isolated except at its ends. Then the proper time of the -process is measured by the number of 's between the two ends, which need not have any relation to the number of 's. This illustrates, what of course follows from relativity, that periodicity must be measured by standards intrinsic to the process concerned, not by standards appropriate to other periodic processes. Such remarks would hardly be necessary but for the fact that relativity and quantum theory at present stand apart from each other, and have not yet been brought into one whole by the physicists. The above can be stated in the language of mathematical logic, thereby making the character of the assumptions clearer and the generalization to continuous processes easier. Let be the series of qualities, the series of events in the rhythmic process. Let us imagine the events arranged in rows and columns, so that each row consists of one period and each column consists of all the events having a given quality. We assume a one-many relation , whose domain is the field of and whose converse domain is the field of . When has the relation to , we say " has the quality ." If is any term in the field of , let be the term which has the relation to ; then the next term below a in the same column (i.e. the corresponding in the next period) is the first term in the series which is after and to which has the relation .[Pg 351] The "row of " consists of all 's earlier than and not earlier than . The "column of " consists of all 's to which has the relation . We assume that with its converse domain limited to one row is one-one, so that each row (i.e. each period) is a series which is similar (in the technical sense) to the series . There is no difficulty in adapting the above analysis of periodicity to continuous processes. Instead of an enumerated set of qualities , , ..., we shall have to take some continuous series of qualities, such as the colours of the rainbow, or the notes produced on a violin by running one's finger up and down the string. The number of events compresent with a given event must now be infinite, but must still be less than the whole of one period (ignoring events outside the process concerned). The number of points in one period, or in any finite portion of it, is now infinite, and cannot therefore be used as a measure of distance. Thus in regard to metrical properties there are important differences between continuous and discrete processes. However, I shall not enlarge upon these, as I propose to consider the analysis of "interval" in a later chapter. Hitherto I have been considering processes which may be regarded as taking place in matter, or which, at any rate, do not move with the velocity of light. But light, also, is commonly regarded as consisting of a periodic process. Accepting the wave-theory of light, let us proceed to analyze its periodic character. We shall find that it differs in important respects from that of periodic processes in matter. [Pg 352] The periodic character of a light-wave cannot exist from its own point of view, but only from that of the matter which it encounters or from which it radiates. We may suppose that when light radiates from an atom at the time of a quantum change, there is, from the point of view of the atom, a temporal series of what we may call "luminous events," and that this series is periodic in the sense which we have been considering. One period of such luminous events constitutes the