Let us now pass to the problem of congruence for time. As is the case with space, mathematical time is an amorphous continuum presenting no definite metrics. The concept of congruence or of the equality of successive durations has no precise meaning; and a definition of time-congruence can be obtained only after we have imposed some conventional measuring standard on an indifferent duration. We might define as equal the durations separating the rising and the setting of the sun, or, again, the durations separating successive rainstorms, etc. Theoretically, all such definitions would be legitimate, since in the final analysis duration, just like space, presents no intrinsic metrics.
In common practice, however, we find that just as in the case of space, a definite species of congruence appears to be imposed on human beings and animals alike. For instance, so long as we are in a conscious state, we all agree that the successive tickings of the clock correspond to what our psychological sense of duration would qualify as congruent intervals. In other words, we appear to be gifted with a species of psychological time-keeping device which allows us to differentiate between durations which we have come to call “seconds” and those which we have come to call “hours.” What is commonly called the sense of rhythm, which manifests itself in the primitive music of drum beats and in the dances of savages, is nothing but a consequence of this intuitive understanding of time-congruence.
A superficial analysis of the situation might lead us to assume that, contrary to our previous assertions, a definite species of congruence was intrinsic to duration. This opinion, however, is quite untenable unless supplemented by additional explanations. The problem with which we are confronted is in all respects similar to the one we mentioned when discussing space. Of itself, space, or at least mathematical space, is amorphous. In spite of this fact, we have no difficulty in agreeing that a coin is round while an egg is oval. But we saw elsewhere that our unreasoned predilection for a definite type of congruence in space was imposed by the behaviour of material bodies when displaced. Visually these bodies could be made to present us with the same appearances, whether situated here or there, and for that reason they imposed themselves as defining congruent space intervals. Furthermore, our own human limbs, the steps we take, the ground on which we walk, all suggest the same definition of congruence.
Likewise in the case of time-congruence. The human organism is replete with rhythmical motions such as the beating of the heart, the periodic contractions of the arteries, or the respiratory motions of the lungs. Again, an infant feels the pangs of hunger at periodic intervals. Quite independently of these internal processes and changes, certain external mechanical motions affecting our bodies must be considered. Thus, when we walk with our normal gait, our successive steps follow one another in time in a certain definite way, and the same applies to the swinging of our arms. By an effort of our will we might vary the rhythm of our steps, alternating short steps with long ones, now tarrying, now hastening. But we should soon be aware that something was wrong, for we should find that our irregular motions demanded additional effort, so that on relaxing our will we should relapse automatically into our normal gait. It may be that in addition to these periodic motions we have mentioned, cyclic processes occur in our brains, conferring on our consciousness direct norms of time-congruence. This, however, is a point for physiologists to decide. At any rate, when it is realised that these various methods, whether they appeal to our respiratory motions or to our successive steps when walking normally, all issue in the same definition for time-congruence, there is no cause to be surprised at a definite sense of time-congruence imposing itself naturally upon us, and this in spite of the fact that pure duration of itself possesses no metrics.
Thus far we have been considering time-congruence from the standpoint of the savage and the animal. For the most rudimentary purposes, a definition of equal durations obtained in this crude way would suffice; yet it is obvious that no great precision could thus be arrived at. Our crude psychological clock is subject to variations according to our mental and physical condition; furthermore, though it can differentiate between what we have come to call “one second” and “one minute,” it would be incapable of differentiating between fifty-nine minutes and one hour. In the case of animals this imprecision in their sense of duration is manifested when shepherd dogs lead the flocks back during an eclipse or hens lay eggs under the influence of artificial light, believing that day has come. However, as these last illustrations permit of other interpretations, we need not dwell on them and may confine ourselves to the well-known fact that our human sense of duration varies between one individual and another, and between one psychological state and another.
In addition to these difficulties, our understanding of equal durations is entirely subjective and can give rise to no common knowledge. We must needs objectivise it, hence appeal to some objective phenomenon which all men can observe. Let us assume, for the sake of argument, that the monotonous dripping of water drops falling on a pan may have served as a primitive clock. Still, something would be lacking. We should not be completely satisfied, for we might sense now and then an irregularity in the successive thuds of the drops falling on the pan. Furthermore, if we were to establish two such water clocks we should observe that the drops sometimes fell in unison and sometimes in succession. We should wonder then which of the clocks was most reliable and be in a quandary to know to which to appeal. Just as in the case of space, where we substituted wooden rods for the lengths of our steps, then metallic rods maintained at constant pressure and temperature, so now, in the case of time-keeping devices, we must find some way of selecting clocks which will satisfy all our requirements.
We thus arrive at more scientific methods of time-determination. The psychological clock, water clock, sand clock and burning of candles, which had served men in the earlier days of civilisation, have now been discarded in favour of more precise mechanisms. These, as we shall see, will not differ perceptibly in their indications from our psychological clock, but their measurements will be given with increased refinement. We are thus led to examine what are the conditions that must be satisfied for a time-keeping device to be considered perfect.
In a general way we obtain a definition of time-congruence by appealing to the principle of causality. The following passage, quoted from Weyl, expresses this fact:
“If an absolutely isolated physical system (i.e., one not subject to external influences) reverts once again to exactly the same state as that in which it was at some earlier instant, then the same succession of states will be repeated in time and the whole series of events will constitute a cycle. In general, such a system is called a clock. Each period of the cycle lasts equally long.”
We shall obtain a more profound understanding of the nature of the problem by considering the various methods of time-determination that have been contemplated by physicists.