23 THE ARGUMENTS OF ZENO OF ELEA

The preceding considerations implicitly contain the solution to all problems of the sort raised by Zeno of Elea in his famous arguments against the possibility of motion, or at least in what appear to be such when one takes the arguments only as they are usually presented; in fact, one might well doubt whether this was really their true significance. Indeed, it is rather unlikely that Zeno really intended to deny motion; what is more probable is that he merely wished to prove the incompatibility of the latter with the supposition, accepted notably by the atomists, of a real, irreducible multiplicity existing in the nature of things. It was therefore originally against this very multiplicity so conceived that these arguments originally must have been directed; we do not say against all multiplicity, for it goes without saying that multiplicity also exists within its order, as does motion, which, moreover, like every kind of change, necessarily supposes multiplicity. But just as motion, by reason of its character of transitory and momentary modification, is not self-sufficient and would be purely illusory were it not linked to a higher principle transcendent with respect to it, such as the 'unmoved mover' of Aristotle, so multiplicity would truly be nonexistent were it to be reduced to itself alone, and did it not proceed from unity, as is reflected mathematically in the formation of the sequence of numbers, as we have seen. What is more, the supposition of an irreducible multiplicity inevitably excludes all real connections between the elements of things, and consequently all continuity as well, for the latter is only a particular case or special form of such connections. As we have already said above, atomism necessarily implies the discontinuity of all things; ultimately, motion really is incompatible with this discontinuity, and we shall see that this is indeed what the arguments of Zeno show. Take, for example, the following argument: an object in motion can never pass from one position to another, since between the two there is always an infinity of other positions, however close, that must be successively traversed in the course of the motion, and, however much time is employed to traverse them, this infinity can never be exhausted. Assuredly, this is not a question of an infinity, as is usually said, for such would have no real meaning; but it is no less the case that in every interval one may take into account an indefinite number of positions for the moving object, and these cannot be exhausted in analytic fashion, which would involve each position being occupied one by one, as the terms of a discontinuous sequence are taken one by one. But it is this very conception of motion that is in error, for it amounts in short to regarding the continuous as if it were composed of points, or of final, indivisible elements, like the notion according to which bodies are composed of atoms; and this would amount to saying that in reality there is no continuity, for whether it is a question of points or atoms, these final elements can only be discontinuous; furthermore, it is true that without continuity there would be no possible motion, and this is all that the argument actually proves. The same goes for the argument of the arrow that flies and is nonetheless immobile, since at each instant one sees only a single position, which amounts to supposing that each position can in itself be regarded as fixed and determined, and that the successive positions thus form a sort of discontinuous series. It is further necessary to observe that it is not in fact true that a moving object is ever viewed as if it occupied a fixed position, and that quite to the contrary, when the motion is fast enough, one will no longer see the moving object distinctly, but only the path of its continuous displacement; thus for example, if a flaming ember is whirled about rapidly, one will no longer see the form of the ember, but only a circle of fire; moreover, whether one explains this by the persistence of retinal impressions, as physiologists do, or in any other way, it matters little, for it is no less obvious that in such cases one grasps the continuity of motion directly, as it were, and in a perceptible manner. What is more, when one uses the expression 'at each instant' in formulating such arguments, one is implying that time is formed from a sequence of indivisible instants, to each of which there corresponds a determined position of the object; but in reality, temporal continuity is no more composed of instants than spatial continuity is of points, and as we have already pointed out, the possibility of motion presupposes the union, or rather the combination, of both temporal and spatial continuity. It is also argued that in order to traverse a given distance, it is first necessary to traverse half this distance, then half of the remaining half, then half of the rest, and so on indefinitely, ^[1] such that one would always be faced with an indefinitude that, envisaged in this way, is indeed inexhaustible. Another almost equivalent argument is as follows: if one supposes two moving objects to be separated by a certain distance, then one of them, even if traveling faster than the other, will never be able to overtake the other, for, when it arrives at the point where it would have met the one in the lead, the latter will be in a second position, separated from the first by a smaller distance than the initial one; when it arrives at this new position, the other will be in yet a third position, separated from the second by a still smaller distance, and so on indefinitely, in such a way that, despite the fact that the distance between the two objects is always decreasing, it will never disappear altogether. The essential problem with these two arguments, as well as with the preceding, consists in the fact that they all suppose that in order to reach a certain endpoint, all the intermediate degrees must be traversed distinctly and successively. Now, we are led to one of two conclusions: either the motion in question is indeed continuous, and therefore cannot be broken down in this way, since the continuous has no irreducible elements; or the motion is composed, or at least may be considered to be composed, of a discontinuous succession of intervals, each with a determined magnitude, as with the steps taken by a man walking, ^[2] in which case the consideration of these intervals would obviously rule out that of all the various intermediate positions possible, which would not actually have to be traversed as so many distinct steps. Besides, in the first case, which is really that of a continuous variation, the end-point, assumed by definition to be fixed, cannot be reached within the variation itself, and the fact that it actually is reached demands the introduction of a qualitative heterogeneity, which this time does constitute a true discontinuity, and which is represented here by the passage from the state of motion to that of rest; this brings us to the question of 'passage to the limit', the true meaning of which still remains to be explained.