A Pluralistic Universe

if the tortoise can be overtaken at all, he can be overtaken in (say) twenty seconds, but he would still have insisted that he can’t be overtaken at all. Leave Achilles and the tortoise out of the account altogether, he would have said⁠—they complicate the case unnecessarily. Take any single process of change whatever, take the twenty seconds themselves elapsing. If time be infinitely divisible, and it must be so on intellectualist principles, they simply cannot elapse, their end cannot be reached; for no matter how much of them has already elapsed, before the remainder, however minute, can have wholly elapsed, the earlier half of it must first have elapsed. And this ever rearising need of making the earlier half elapse first leaves time with always something to do before the last thing is done, so that the last thing never gets done. Expressed in bare numbers, it is like the convergent series ½ plus ¼ plus ⅛⁠ ⁠… , of which the limit is one. But this limit, simply because it is a limit, stands outside the series, the value of which approaches it indefinitely but never touches it. If in the natural world there were no other way of getting things save by such successive addition of their logically involved fractions, no complete units or whole things would ever come into being, for the fractions’ sum would always leave a remainder. But in point of fact nature doesn’t make eggs by making first half an egg, then a quarter, then an eighth, etc., and adding them together. She either makes a whole egg at once or none at all, and so of all her other units. It is only in the sphere of change, then, where one phase of a thing must needs come into being before another phase can come that Zeno’s paradox gives trouble.

And it gives trouble then only if the succession of steps of change be infinitely divisible. If a bottle had to be emptied by an infinite number of successive decrements, it is mathematically impossible that the emptying should ever positively terminate. In point of fact, however, bottles and coffeepots empty themselves by a finite number of decrements, each of definite amount. Either a whole drop emerges or nothing emerges from the spout. If all change went thus drop-wise, so to speak, if real time sprouted or grew by units of duration of determinate amount, just as our perceptions of it grow by pulses, there would be no Zenonian paradoxes or Kantian antinomies to trouble us. All our sensible experiences, as we get them immediately, do thus change by discrete pulses of perception, each of which keeps us saying “more, more, more,” or “less, less, less,” as the definite increments or diminutions make themselves felt. The discreteness is still more obvious when, instead of old things changing, they cease, or when altogether new things come. Fechner’s term of the “threshold,” which has played such a part in the psychology of perception, is only one way of naming the quantitative discreteness in the change of all our sensible experiences. They come to us in drops. Time itself comes in drops.

Our ideal decomposition of the drops which are all that we feel into still finer fractions is but an incident in that great transformation of the perceptual order into a conceptual order of which I spoke in my last lecture. It is made in the interest of our rationalizing intellect solely. The times directly felt in the experiences of living subjects have originally no common measure. Let a lump of sugar melt in a glass, to use one of M. Bergson’s instances. We feel the time to be long while waiting for the process to end, but who knows how long or how short it feels to the sugar? All felt times coexist and overlap or compenetrate each other thus vaguely, but the artifice of plotting them on a common scale helps us to reduce their aboriginal confusion, and it helps us still more to plot, against the same scale, the successive possible steps into which nature’s various changes may be resolved, either sensibly or conceivably. We thus straighten out the aboriginal privacy and vagueness, and can date things publicly, as it were, and by each other. The notion of one objective and “evenly flowing” time, cut into numbered instants, applies itself as a common measure to all the steps and phases, no matter how many, into which we cut the processes of nature. They are now definitely contemporary, or later or earlier one than another, and we can handle them mathematically, as we say, and far better, practically as well as theoretically, for having thus correlated them one to one with each other on the common schematic or conceptual time-scale.

Motion, to take a good example, is originally a turbid sensation, of which the native shape is perhaps best preserved in the phenomenon of vertigo. In vertigo we feel that movement is, and is more or less violent or rapid, more or less in this direction or that, more or less alarming or sickening. But a man subject to vertigo may gradually learn to coordinate his felt motion with his real position and that of other things, and intellectualize it enough to succeed at last in walking without staggering. The mathematical mind similarly organizes motion in its way, putting it into a logical definition: motion is now conceived as “the occupancy of serially successive points of space at serially successive instants of time.” With such a definition we escape wholly from the turbid privacy of sense. But do we not also escape from sense-reality altogether? Whatever motion really may be, it surely is not static; but the definition we have gained is of the absolutely static. It gives a set of one-to-one relations between space-points and time-points, which relations themselves are as fixed as the points are. It gives positions assignable ad infinitum, but how the body gets from one position to another it omits