The Analysis of Matter

compresent, and one event may be compresent with a number of others which are[Pg 373] separated by transitions; in that case, the one event is said to last for a long time. We can even obtain a continuous time in our theory, if the number of events compresent with a given event is infinite, and their beginnings and ends do not synchronize, i.e. one of them may be compresent with two others which are not compresent with each other. But I see no reason to suppose that the number of events compresent with a given event is infinite, or to desire a theory which makes time continuous; I therefore lay no stress upon this possibility. In a transaction, or during a rhythm, the causal antecedent may consist of more than one event, and so may the causal consequent; but the events which constitute the causal antecedent must all be co-punctual, and so must those which constitute the causal consequent. Any event of the antecedent group will be called a "parent" of any event of the consequent group. When two events are connected by a chain of events, each of which is a parent of the next, the one is said to be an "ancestor" of the other, and the other a "descendant" of the one. Two events may be connected by many causal chains, but all will consist of a finite number of events, and we assume that, in the case of any two given events, there is a maximum to the number of generations in the various lines of descent connecting them. This maximum number is the measure of "interval" when the interval is time-like. When the interval is space-like, the definition of interval is slightly more complicated. To define space-like intervals, we must first say a few words about light. When a luminous event travels from one body to another, I regard the whole as one static event, involving no internal change or process. Consequently, from the standpoint of the event itself, if one could imagine a being of whose biography it formed a part, there is no time between the beginning and the end. Since nothing travels faster than light, it is impossible that two parts of one luminous event should[Pg 374] be compresent with two events of which one is a causal descendant of another; therefore there is no extraneous source from which the luminous event can discover that it is lasting a long time, and there is, in fact, no meaning in saying that it is lasting a long time. But when we say that it is reflected back to its starting-point, we mean that it has undergone a transaction which has turned it into a new luminous event, and that this new event is compresent with causal descendants of events compresent with the earlier one, these compresent events being not luminous, but of the kinds associated with matter. Now, given any two events and , neither of which is an ancestor of the other, it is possible to find a luminous event compresent with and with a descendant of . We then say that the events and have a space-like separation, whose measure is that of the time-like separation between and . In the above theory, it is assumed that, in all cases where one process or piece of matter has an effect upon another, there is at least one event which is compresent with both. This is the form taken by the denial of action at a distance. If we assume, as we have been doing, that change is discontinuous, a single period of a rhythm will contain some finite number of points. Suppose, now, that there are two rhythms such that the initial event of a period in the one is always identical with the initial event of a period in the other, but the other events are diverse; and suppose that the first rhythm contains event in a period while the second contains . Then a period of the first rhythm will contain points, and one of the second will contain . We said that the "interval" between two events was to be the number of points in the longest causal route from one to the other; hence the interval between the beginning and end of a period in either rhythm is measured by the greater of the two numbers and . Suppose this is . Then we may regard the -rhythm as having a smaller "velocity" than the -rhythm, while the frequencies[Pg 375] of the two rhythms would be the same. This suggests, in a certain class of cases, a possibility of defining "velocity" otherwise than by relative motion. How far the resulting properties of "velocity" would resemble those resulting from the usual definition, I do not know. There is no difficulty in defining what is to be meant by saying that a steady event "moves." An event occupies a number of points of space-time, which can be regarded as a four-dimensional tube divisible into sections such that all the points in one section are simultaneous, and are all later or all earlier than all the points in another section. We shall then regard our event as moving along the tube, and occupying the various instantaneous sections successively. But this does not imply any process or change within ; it merely implies transitions among events compresent with but not all compresent with each other. It seems, therefore, that everything essential to theoretical physics can be stated in terms of our theory. According to the above theory, motion is discontinuous. But this hypothesis is required for one purpose only, namely for the definition of interval. It is easy to introduce such axioms as shall make our space-time continuous, and secure, as in current physics, that discontinuity shall be confined to quantum phenomena, i.e. to what we have called "transactions." But if this is done, our definition of interval must be abandoned, and interval resumes its place as something mysterious and unaccountable. There is no logical reason why it should not have such a place; the laws of transactions have such a place in our account. But it is always intellectually satisfying when we can reduce the number of inexplicabilities. So far as I can discover, there is no good ground for supposing that motion is continuous; it is therefore worth while to develop a discontinuous hypothesis if we can thereby increase the unity and diminish the arbitrariness in our account of the physical world.