When Achilles pursues the tortoise, each of his steps must be treated as indivisible, and so must each step of the tortoise. After a certain number of steps, Achilles will have overtaken the tortoise. There is nothing more simple. If you insist on dividing the two motions further, distinguish both on the one side and on the other, in the course of Achilles and in that of the tortoise, the sub- mulHj)les of the steps of each of them; but respect the natural articulations of the two courses. As long as you respect them, no difficulty will arise, because you will follow the indications of experience. But Zeno's device is to reconstruct the movement of Achilles according to a law arbitrarily chosen. Achilles with a first step is sup- posed to arrive at the point where the tortoise was, with a second step at the point which it has moved to while he was making the first, and so on. In this case, AchiUea would always have a new step to take. But obviously, to overtake the tortoise, he goes about it in quite another way. The movement considered by Zeno would only be the equivalent of the movement of Achilles if we could treat the movement as we treat the interval passed through, decomposable and recomposable at will. Once you sub- scribe to this first absurdity, all the others follow.'
1 That is, we do not cooaider the sophism of Zcno refuted bj the fact th»t the geometrical progresBion 0(1 +^ + ^ + ia+ ■•• etc.)— in which a deaignatea the initial distance between Achillea and the tortoise, &nd n the reUtion of their reapeotive velocities— has a finite aum if n ia greater than 1. On thia point we may refer to the arguments of P. Evellin. which we regard aa conclusive (see Eveltin, Infini et qiianli/i, Paris, 1880, pp. 63-97; ct. Revue phSoeophiqwt, vol. xi., 1881, pp. 564-
Nothing would be easier, now, than to extend Zeno's argument to quaUtative becoming and to evolutionary becoming. We should find the same contradictions in these. That the child can become a youth, ripen to maturity and decline to old age, we understand when we consider that vital evolution is here the reality itself. Infancy, adolescence, maturity, old age, are mere views of the mind, possible stops imagined by us, from without, along the continuity of a progress. On the contrary, let childhood, adolescence, maturity and old age be given as integral parts of the evolution, they become reoi stops, and we can no longer conceive how evolution is possible, for rests placed beside rests will never be equivalent to a movement. How, with what is made, can we reconsti- tute what is being made? How, for instance, from child- hood once posited as a thing, shall we pass to adolescence, when, by the hypothesis, childhood only ia given? If we look at it closely, we shall see that our habitual manner of speaking, which is fashioned after our habitual manner of thinking, leads us to actual logical deadlocks — dead- locks to which we allow ourselves to be led without anxiety, because we feel confasedly that we can always get out of them if we like: all that we have to do, in fact, is to give up the cinematographical habits of our intellect. \\'heii we say "The child becomes a man," let us take care not to fathom too deeply the literal meaning of the expression, or we shall find that, when we posit the subject "child," the attribute "man" does not yet apply to it, and that,
568). The truth is that mstbomatica, &b we have tried to show in a former work, deals and can deal only with length*. It baa therefor* had to seek devices, trat. U> transrer to the movement, which it not a length, the divisibility of the line pasted over, and then to reooneila with experience the idea (contrary to experience and full of abiunlitiM) of movemeat that is a length, that ia, of a movement pkutd upeii its tnjectoi7 and arbitrarily deoompoaable like it.
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when we express the attribute "man," it applies no more to the subject " child." The reahty, which is the transition from childhood to manhood, has slipped between our fingers. We have only the imaginary stops "child" and "man," and we are very near to saying that one of these stops is the other, just as the arrow of Zeno is, according to that philosopher, at all the points of the course. The truth is that if language here were molded on reahty, we should not say "The child becomes the man," but "There is becoming frona the child to the man." In the first proposition, " becomes" is a verb of indeterminate meaning, intended to mask the absurdity into which we fall when we attribute the state "man" to the subject "child." It behaves in much the same way as the movement, always the same, of the cinematographical film, a movement bidden m the apparatus and whose function it is to super- pose the successive pictures on one another in order to imitate the movement of the real object. In the second proposition, "becoming" is a subject. It comes to the front. It is the reahty itself; childhood and manhood are then only possible stops, mere views of the mind; we now have to do with the objective movement itself, and no longer with its cinematographical imitation. But the first manner of expression is alone conformable to our habits of language. We must, in order to adopt the second, escape from the cinematographical mechanism of thought.
We must make complete abstraction of this mechan- ism, if we wish to get rid at one stroke of the theoretical absurdities that the question of movement raises. All is obscure, all is contradictory when we try, with states, to build up a transition. The obscurity is cleared up, the contradiction vanishes, as soon as we place ourselves along the transition, in order to distinguish states in it
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by maldiig cross cuts therein in thought. The reason is that there is more in the transition than the series of states, that is to say, the possible cuts — more in the move- ment than the scries of positions, that is to say, the possible stops. Only, the first way of looking at things is con- formable to the processes of the human mind ; the second requires, on the contrary, that we reverse the bent of our intellectual habits. No wonder, then, if philasophy at first recoiled before such an effort. The Greeks trusted to nature, trusted the natural propensity of the mind, trusted language above all, in so far as it naturally externalizes thought. Rather than lay blame on the attitude of thou^t and language toward the course of things, they preferred to pronounce the course of things itself to be wrong.
Such, indeed, was the sentence passed by the philoso- phers of the Eleatic school. And they passed it with- out any reservation whatever. As becoming shocks the habits of thought and fits ill into the molds of language, they declared it unreal. In spatial movement and in change in general they saw only pure illusion. This con- clusion could be softened down without changing the premisses, by saying that the reality changes, but that it ought not to change. Experience confronts us with be- coming: that is sensible reality. But the intelliffible reality, that which ought to be, is more real still, and that reality does not change. Beneath the qualitative becoming, beneath the evolutionary becoming, beneath the extensive becoming, the mind must seek that which defies change, the definable quality, the form or essence, the end. Such was the fundamental principle of the philosophy which developed throughout the classic age, the philosophy of Forms, or, to use a term more akin to the Greek, the philoso- phy of Ideas.
The wordei^oa, which we translate here by "Idea," has,
in fact, this threefold meaning. It denotes (1) the quality, (2) the form or essence, (3) the end or design (in the sense of intention) of the act being performed, that is to say, at bottom, the design (in the sense of dramng) of the act sup- posed accomplished. These three aspects are those of the adjective, substantive and verb, and correspond to lite three essential categories of language. After the explanations we have given above, we might, and perhaps we ought to, translate ci3oa by "view" or rather by "moment." For tidot is the stable view taken of the instability of things: the quality, which is a moment of becoming; the/arm, which is B moment of evolution; the essence, which is the mean form above and below which the other forms are arranged as alterations of the mean; finally, the intention or mental design which presides over the action being accomplished, and which is nothing else, we said, than the nmlerial design, traced out and contemplated beforehand, of the action accomplished. To reduce things to Ideas is therefore to resolve becoming into its principal moments, each of these being, moreover, by the hypothesis, screened from the laws of time and, as it were, plucked out of eternity. That is to say that we end in the philosophy of Ideas when we apply the cinematographical mechanism of the intellect to the analysis of the real.
But, when we put immutable Ideas at the base of the moving reality, a whole physics, a whole cosmology, a whole theology follows necessarily. We must insist on the point. Not that we mean to summarize in a few pages a philosophy BO complex and so comprehensive as that of the Greeks, But, since we have described the cinematographical mech- anism of the intellect, it is important that we ^ould show to what idea of reality the play of this mechanism leads. It is the very idea, we believe, that we find in the ancient philosophy. The main lines of the doctrine that was
developed from Plato to Plotinus, passing through Aristotla (and even, in a certain measure, through the Stoics), have nothing accidental, nothing contingent, nothing that must be regarded as a philosopher's fancy. They indicate the vision that a systematic intellect obtains of the universal becoming when regarding it by means of snapshots, taken at intervals, of its flowing. So that, even to-day, we shall philosophize in the manner of the Greeks, we shall re- discover, without needing to know them, such and such of their general conclusions, in the exact proportion that we trust in the cinematographical instioct of our thought.
We said there is more in a movement than in the suc- cessive positions attributed to the moving object, more in a becoming than in the forms passed through in turn, more in the evolution of form than the forms assumed one after another. Philosophy can therefore derive terms of the second kind from those of the first, but not the first from the second: from the first terms speculation must take its start. But the intellect reverses the order of the two groups; and, on this point, ancient philosophy pro- ceeds as the intellect does. It installs itself in the im- mutable, it posits only Ideas. Yet becoming exists: it is a fact. How, then, having posited immutability alone, shall we make change come forth from it? Not by the addition of anything, for, by the hypothesis, there exists nothing positive outside Ideas. It must therefore be by a diminution. So at the base of ancient philosophy lies necessarily this postulate : that there is more in the motion- less than in the moving, and that we pass from immuta- bility to becoming by way of diminution or attenuation.
It is therefore something negative, or zero at most, that must be added to Ideas to obtain change. In that consists the Platonic "non-being," the Aristotelian "matter" — a
metaphysical zero which, joined to the Idea, Hke the arith- metical zero to unity, multiplies it in space and time. By it the motionless and simple Idea is refracted into a move- ment spread out indefinitely. In right, there ought to be nothing but immutable Ideas, immutably fitted to each other. In fact, matter comes to add to them its void, and thereby lets loose the universal becoming. It is an elusive nothing, that creeps between the Ideas and creates endless agitation, eternal disquiet, like a suspicion insinuated be- tween two loving hearts. Degrade the immutable Ideas: you obtain, by that alone, the perpetual flux of things. The Ideas or Forms are the whole of intelligible reality, that IB to say, of truth, in that they represent, all together, the theoretical equilibrium of Being. As to sensible reality, it is a perpetual oscillation from one aide to the other of this point of equilibrium.