indivisible sensation of motion or mobility. A rapid gesture, made with one's eyes shut, will assume for consciousness the form of a purely qualitative sensation as long as there is no thought of the space traversed. In a word, there are two elements to be distinguished in motion, the space traversed and the act by which we traverse it, the successive positions and the synthesis of these positions. The first of these elements is a homogeneous quantity: the second has no reality except in a consciousness: it is a quality or an intensity, whichever you prefer. But here again we meet with a case of endosmosis, an inter mingling of the purely intensive sensation of mobility with the extensive representation of the space traversed. On the one hand we attribute to the motion the divisibility of the space which it traverses, forgetting that it is quite possible to divide an object, but not an act: and on the other hand we accustom ourselves to projecting this act itself into space, to applying it to the whole of the line which the moving body traverses, in a word, to solidifying it: as if this localizing of a progress in space did not amount to asserting that, even outside consciousness, the past co-exists along with the present!
It is to this confusion between motion and the space traversed that the paradoxes of the Eleatics are due; for the interval which separates two points is infinitely divisible, and if motion con sisted of parts like those of the interval itself, CHAP, n THE ELEATIC PARADOX 113 the interval would never be crossed. But the The common truth is that each of Achilles' steps is tlon a simple indivisible act, and that, after a &ven number of these acts, Achilles paradoxes *rf ^^ have passed the tortoise. The mis- the Eieatics. take of the Eleatics arises from their identification of this series of acts, each of which is of a definite kind and indivisible, with the homo geneous space which underlies them. As this space can be divided and put together again accord ing to any law whatever, they think they are justified in reconstructing Achilles' whole move ment, not with Achilles' kind of step, but with the tortoise's kind: in place of Achilles pursuing the tortoise they really put two tortoises, regulated by each other, two tortoises which agree to make the same kind of steps or simultaneous acts, so as never to catch one another. Why does Achilles outstrip the tortoise? Because each of Achilles' steps and each of the tortoise's steps are indivisible acts in so far as they are movements, and are different magnitudes in so far as they are space: so that addition will soon give a greater length for the space traversed by Achilles than is obtained by adding together the space traversed by the tortoise and the handicap with which it started. This is what Zeno leaves out of account when he reconstructs the movement of Achilles according to the same law as the movement of the tortoise, forgetting that space alone can be divided and put together again in any way we like, and thus H4 TIME AND FREE WILL CHAP, n confusing space with motion. Hence we do not think it necessary to admit, even after the acute and profound analysis of a contemporary thinker,1 that the meeting of the two moving bodies implies a discrepancy between real and imaginary .motion, between space in itself and indefinitely divisible space, between concrete time and abstract time. Why resort to a metaphysical hypothesis, however ingenious, about the nature of space, time, and motion, when immediate intuition shows us motion within duration, and duration outside space? There is no need to assume a limit to the divisibility of concrete space; we can admit that it is infinitely divisible, provided that we make a distinction between the simultaneous positions of the two moving bodies, which are in fact in space, and their movements, which cannot occupy space, being duration rather than extent, quality and not quantity. To measure the velo city of a movement, as we shall see, is simply to ascertain a simultaneity; to introduce this velo city into calculations is simply to use a convenient means of anticipating a simultaneity. Thus mathe matics confines itself to its own province as long as it is occupied with determining the simul taneous positions of Achilles and the tortoise at a given moment, or when it admits d priori that the two moving bodies meet at a point X — a meeting which is itself a simultaneity. But it goes 1 Evellin, Infini et quantite. Paris, 1881.
CHAP.II DURATION AND SIMULTANEITY 115 beyond its province when it claims to reconstruct what takes place in the interval between two simultaneities; or rather it is inevitably led, even then, to consider simultaneities once more, fresh simultaneities, the indefinitely increasing number of which ought to be a warning that we cannot make movement out of immobilities, nor time out of space. In short, just as nothing will be found homogeneous in duration except a sym bolical medium with no duration at all, namely space, in which simultaneities are set out in line, in the same way no homogeneous element will be found in motion except that which least belongs to it, the traversed space, which is motionless.
Now, just for this reason, science cannot deal with time and motion except on condition of first science has to eliminating the essential and qualita- r~ tive element— of time, duration, and of " motion, mobility. We may easily con- before vjnce ourselves of this by examining the with them. par{. piaye(i m astronomy and mechanics by considerations of time, motion, and velocity.
Treatises on mechanics are careful to announce that they do not intend to define duration itself but only the equality of two durations. " Two intervals of time are equal when two identical bodies, in identical conditions at the beginning of each of these intervals and subject to the same actions and influences of every kind, have traversed the same space at the end of these intervals." In other words, we are to note the exact moment at Il6 TIME AND FREE WILL CHAP, n which the motion begins, i.e. the coincidence of an external change with one of our psychic states; we are to note the moment at which the motion ends, that is to say, another simultaneity; finally we are to measure the space traversed, the only thing, in fact, which is really measurable. Hence there is no question here of duration, but only of space and simultaneities. To announce that some thing will take place at the end of a time t is to declare that consciousness will note between now and then a number t of simultaneities of a certain kind. And we must not be led astray by the words " between now and then," for the interval of duration exists only for us and on account of the interpenetration of our conscious states. Outside ourselves we should find only space, and consequently nothing but simultaneities, of which we could not even say that they are objectively successive, since succession can only be thought through comparing the present with the past. — That the interval of duration itself cannot be taken into account by science is proved by the fact that, if all the motions of the universe took place twice or thrice as quickly, there would be nothing to alter either in our formulae or in the figures which are to be found in them. Consciousness would have an indefinable and as it were qualitative impression of the change, but the change would not make itself felt outside consciousness, since the same number of simultaneities would go on taking place in space. We shall see, later on, that when the CHAP, ii VELOCITY AND SIMULTANEITY 117 astronomer predicts, e.g., an eclipse, he does some thing of this kind: he shortens infinitely the inter vals of duration, as these do not count for science, and thus perceives in a very short time — a few seconds at the most — a succession of simultaneities which may take up several centuries for the con crete consciousness, compelled to live through the intervals instead of merely counting their extrem ities.
A direct analysis of the notion of velocity will bring us to the same conclusion. Mechanics gets this notion through a series of ideas, the This is seen in...the definition connexion of which it is easy enough to of velocity.
trace. It first builds up the idea of uniform motion by picturing, on the one hand, the path AB of a certain moving body, and, on the other, a physical phenomenon which is re peated indefinitely under the same conditions, e.g., a stone always falling from the same height on to the same spot. If we mark on the path AB the points M, N, P...reached by the moving body at each of the moments when the stone touches the ground, and if the intervals AM, MN and NP are found to be equal to one another, the motion will be said to be uniform: and any one of these intervals will be called the velocity of the moving body, provided that it is agreed to adopt as unit of duration the physical phenomenon which has been chosen as the term of comparison. Thus, the velocity of a uniform motion is defined by mechanics without appealing to any other notions Il8 TIME AND FREE WILL CHAP, ti than those of space and simultaneity. Now let us turn to the case of a variable motion, that is, to the case when the elements AM, MN, NP...are found to be unequal. In order to define the velocity of the moving body A at the point M, we shall only have to imagine an unlimited number of moving bodies At, A2, A3...all moving uni formly with velocities vlt vt, v3...which are arranged, e.g., in an ascending scale and which correspond to all possible magnitudes. Let us then consider on the path of the moving body A two points M' and M", situated on either side of the point M but very near it. At the same time as this moving body reaches the points M', M, M", the other moving bodies reach points M\ Mt M'^, M'2 M2 M"2...on their respective paths; and there must be two moving bodies A* and A^ such that we have on the one hand M' M = M'* MA and on the other hand MM" = M, M'V We shall then agree to say that the velocity of the moving body A at the point M lies between vk and vp. But nothing prevents our assuming that the points M' and M" are still nearer the point M, and it will then be necessary to replace vh and vp by two fresh velocities v, and vn, the one greater than vh and the other less than vp. And in proportion as we reduce the two intervals M'M and MM", we shall lessen the difference between the velocities of the uniform corresponding movements. Now, the two intervals being capable of decreasing right down to zero, there evidently exists between vt CHAP, ii VELOCITY AND SIMULTANEITY and vn a certain velocity vm, such that the differ ence between this velocity and vh, ^...on the one hand, and vp, v...on the other, can be come smaller than any given quantity. It is this common limit vm which we shall call the velocity of the moving body A at the point M. — Now, in this analysis of variable motion, as in that of uniform motion, it is a question only of spaces once traversed and of simultaneous positions once reached. We were thus justified in saying that, while all that mechanics retains of time is simul taneity, all that it retains of motion itself— restricted, as it is, to a measurement of motion — is immobility.
This result might have been foreseen by noticing that mechanics necessarily deals with equations, Mechanics an^ that an algebraic equation always expresses something already done. Now, ^ *s o* tne very essence of duration and processes? not m°tion, as they appear to our conscious- Son1 iXddm£ ness> t° be something that is unceasingly being done; thus algebra can represent the results gained at a certain moment of duration and the positions occupied by a certain moving body in space, but not duration and motion them selves. Mathematics may, indeed, increase the number of simultaneities and positions which it takes into consideration by making the intervals very small: it may even, by using the differential instead of the difference, show that it is possible to increase without limit the number of these I2O TIME AND FREE WILL CHAP, n intervals of duration. Nevertheless, however small the interval is supposed to be, it is the extremity of the interval at which mathematics always places itself. As for the interval itself, as for the duration and the motion, they are neces sarily left out of the equation. The reason is that duration and motion are mental syntheses, and not objects; that, although the moving body occupies, one after the other, points on a line, motion itself has nothing to do with a line; and finally that, although the positions occupied by the moving body vary with the different moments of duration, though it even creates distinct mo ments by the mere fact of occupying different positions, duration properly so called has no moments which are identical or external to one another, being essentially heterogeneous, continu ous, and with no analogy to number.
It follows from this analysis that space alone is homogeneous, that objects in space form a discrete multiplicity, and that every discrete Conclusion:...space alone is multiplicity is got bv a process of unhomogene-.
ous: dura- folding in space. It also follows that turn and sue-.. *" cession belong there is neither duration nor even sue- not to the ex ternal worm, cession in space, if we give to these words but to the,.. ° conscious the meaning in which consciousness mind..
takes them: each of the so-called suc cessive states of the external world exists alone; their multiplicity is real only for a consciousness that can first retain them and then set them side by side by externalizing them in relation CHAP, ii TWO KINDS OF MULTIPLICITY 121 to one another. If it retains them, it is because these distinct states of the external world give rise to states of consciousness which permeate one another, imperceptibly organize themselves into a whole, and bind the past to the present by this very process of connexion. If it externalizes them in relation to one another, the reason is that, thinking of their radical distinctness (the one having ceased to be when the other appears on the scene), it perceives them under the form of a discrete multiplicity, which amounts to settingthem out in line, in the space in which each of them existed separately. The space employed for this purpose is just that which is called homogeneous time.
But another conclusion results from this analysis, namely, that the multiplicity of conscious states, regarded in its original purity, is not at multiplicity: all like the discrete multiplicity which two senses of. _ the word "dis- goes to form a number. In such a case tinguish," the".., r,,., one quaiita- there is, as we said, a qualitative multive and the...-,,.
other quanti- tiplicitv. In short, we must admit two tative..
kinds of multiplicity, two possible senses of the word " distinguish," two conceptions, the one qualitative and the other quantitative, of the difference between same and other. Sometimes this multiplicity, this distinctness, this hetero geneity contains number only potentially, as Aristotle would have said. Consciousness, then, makes a qualitative discrimination without any further thought of counting the qualities or even of distinguishing them as several. In such 122 TIME AND FREE WILL CHAP, n a case we have multiplicity without quantity. Sometimes, on the other hand, it is a question of a multiplicity of terms which are counted or which are conceived as capable of being counted; but we think then of the possibility of externalizing them in relation to one another, we set them out in space. Unfortunately, we are so accustomed to illustrate one of these two meanings of the same word by the other, and even to perceive the one in the other, that we find it extraordinarily difficult to distinguish between them or at least to express this distinction in words. Thus I said that several conscious states are organized into a whole, per meate one another, gradually gain a richer con tent, and might thus give any one ignorant of space the feeling of pure duration; but the very use of the word " several " shows that I had already isolated these states, externalized them in relation to one another, and, in a word, set them side by side; thus, by the very language which I was compelled to use, I betrayed the deeply ingrained habit of setting out time in space. From this spatial setting out, already accomplished, we are compelled to borrow the terms which we use to describe the state of a mind which has not yet accomplished it: these terms are thus misleading from the very beginning, and the idea of a mul tiplicity without relation to number or space, although clear for pure reflective thought, cannot be translated into the language of common sense. And yet we cannot even form the idea of discrete CHAP, n TWO KINDS OF MULTIPLICITY 123 multiplicity without considering at the same time a qualitative multiplicity. When we explicitly count units by stringing them along a spatial line, is it not the case that, alongside this addition of identical terms standing out from a homogene ous background, an organization of these units is going on in the depths of the soul, a wholly dynamic process, not unlike the purely qualitative way in which an anvil, if it could feel, would realize a series of blows from a hammer? In this sense we might almost say that the numbers in daily use have each their emotional equivalent. Tradesmen are well aware of it, and instead of indicating the price of an object by a round number of shillings, they will mark the next smaller number, leaving themselves to insert afterwards a sufficient number of pence and farthings. In a word, the process by which we count units and make them into a discrete multiplicity has two sides; on the one hand we assume that they are identical, which is conceivable only on con dition that these units are ranged alongside each other in a homogeneous medium; but on the other hand the third unit, for example, when added to the other two, alters the nature, the appearance and, as it were, the rhythm of the whole; without this interpenetration and this, so to speak, qualitative progress, no addition would be possible. Hence it is through the quality of quantity that we form the idea of quantity without quality.
TIME AND FREE WILL CHAP, n It is therefore obvious that, if it did not betake itself to a symbolical substitute, our consciousness oar successive would never regard time as a homogeneare ous medium, in which the terms of a regarded as mutually ex- succession remain outside one another.
ternal. like we naturally reach this symbolical °s chiS representation by the mere fact that, "fc- in a series of identical terms, each term assumes a double aspect for our consciousness: one aspect which is the same for all of them, since we are thinking then of the sameness of the external object, and another aspect which is characteristic of each of them, because the super vening of each term brings about a new organiz ation of the whole. Hence the possibility of setting out in space, under the form of numerical multiplicity, what we have called a qualitative multiplicity, and of regarding the one as the equivalent of the other. Now, this twofold pro cess is nowhere accomplished so easily as in the perception of the external phenomenon which takes for us the form of motion. Here we cer tainly have a series of identical terms, since it is always the same moving body; but, on the other hand, the synthesis carried out by our consciousness between the actual position and what our memory calls the former positions, causes these images to permeate, complete, and, so to speak, continue one another. Hence, it is principally by the help of motion that duration assumes the form of a homogeneous medium, and that time is projected REAL DURATION 125 into space. But, even if we leave out motion, any repetition of a well-marked external pheno menon would suggest to consciousness the same mode of representation. Thus, when we hear a series of blows of a hammer, the sounds form an indivisible melody in so far as they are pure sensations, and, here again, give rise to a dynamic progress; but, knowing that the same objective cause is at work, we cut up this progress into phases which we then regard as identical; and this multiplicity of elements no longer being con ceivable except by being set out in space, since they have now become identical, we are necessarily led to the idea of a homogeneous time, the sym bolical image of real duration. In a word, our ego comes in contact with the external world at its surface; our successive sensations, although dissolving into one another, retain something of the mutual externality which belongs to their objective causes; and thus our superficial psychic life comes to be pictured without any great effort as set out in a homogeneous medium. But the symbolical character of such a picture becomes more striking as we advance further into the depths of consciousness: the deep-seated self which ponders and decides, which heats and blazes up, is a self whose states and changes permeate one another and undergo a deep alteration as soon as we separate them from one another in order to set them out in space. But as this deeper self forms one and the same person with the superficial ego,