phers would have said, an acquired perception. In the second case, we give the name of intensity to the larger or smaller number of simple psychic phenomena which we conjecture to be involved in the fundamental state: it is no longer an acquired perception, but a confused perception. In fact, these two meanings of the word usually intermingle, because the simpler phenomena in volved in an emotion or an effort are generally representative, and because the majority of re presentative states, being at the same time affect ive, themselves include a multiplicity of element ary psychic phenomena. The idea of intensity is thus situated at the junction of two streams, one of which brings us the idea of extensive mag nitude from without, while the other brings us from within, in fact from the very depths of consciousness, the image of an inner multiplicity. Now, the point is to determine in what the latter image consists, whether it is the same as that of number, or whether it is quite different from it. In the following chapter we shall no longer con sider states of consciousness in isolation from one another, but in their concrete multiplicity, in so far as they unfold themselves in pure duration. And, in the same way as we have asked what would be the intensity of a representative sen sation if we did not introduce into it the idea of its cause, we shall now have to inquire what the multiplicity of our inner states becomes, w/hat form duration assumes, when the space in which 74 TIME AND FREE WILL CHAP, i it unfolds is eliminated. This second question is even more important than the first. For, if the confusion of quality with quantity were confined to each of the phenomena of conscious ness taken separately, it would give rise to obscuri ties, as we have just seen, rather than to problems. But by invading the series of our psychic states, by introducing space into our perception of dura tion, it corrupts at its very source our feeling of outer and inner change, of movement, and of freedom. Hence the paradoxes of the Eleatics, hence the problem of free will. We shall insist rather on the second point; but instead of seeking to solve the question, we shall show the mistake of those who ask it.
CHAPTER II THE MULTIPLICITY OF CONSCIOUS STATES * THE IDEA OF DURATION NUMBER may be defined in general as a collection of units, or, speaking more exactly, as the synthesis what is num- °f the one and the many. Every num- ber{> her is one, since it is brought before the 1 I had already completed the present work when I read in the Critique philosophique (for 1883 and 1884) F. Pillon's very remarkable refutation of an interesting article by G. Noel on the interconnexion of the notions of number and space. But I have not found it necessary to make any alterations in the following pages, seeing that Pillon does not distinguish between time as quality and time as quantity, between the mul tiplicity of juxtaposition and that of interpenetration. With out this vital distinction, which it is the chief aim of the present chapter to establish, it would be possible to maintain, with Pillon, that number may be built up from the relation of co-existence. But what is here meant by co-existence? If the co-existing terms form an organic whole, they will never lead us to the notion of number; if they remain distinct, they are in juxtaposition and we are dealing with space. It is no use to quote the example of simultaneous impressions received by several senses. We either leave these sensations their specific differences, which amounts to saying that we do not count them; or else we eliminate their differences, and then how are we to distinguish them if not by their position or that of their symbols? We shall see that the verb '' to dis tinguish " has two meanings, the one qualitative, the other 76 TIME AND FREE WILL CHAP, n mind by a simple intuition and is given a name; but the unity which attaches to it is that of a sum, it covers a multiplicity of parts which can be con sidered separately. Without attempting for the present any thorough examination of these con ceptions of unity and multiplicity, let us inquire whether the idea of number does not imply the representation of something else as well.
It is not enough to say that number is a collec tion of units; we must add that these units are identical with one another, or at least whiciTmake that they are assumed to be identical must b^iden- when they are counted. No doubt we can count the sheep in a flock and say that there are fifty, although they are all different from one another and are easily recognized by the shepherd: but the reason is that we agree in that case to neglect their individual differences and to take into account only what they have in common. On the other hand, as soon as we fix our attention on the particular features of objects or individuals, we can of course make an enumeration of them, but not a total. We place ourselves at these two very different points of view when we count the soldiers in a battalion and when we call the roll. Hence we may conclude that the idea of number implies the simple intuition of a multiplicity of parts or units, which are absolutely alike.
quantitative: these two meanings have been confused, in my opinion, by the philosophers who have dealt with the relations between number and space.
CHAP, n NUMERICAL MULTIPLICITY AND SPACE 77 And yet they must be somehow distinct from one another, since otherwise they would merge into a single unit. Let us assume that also be dis- all the sheep in the flock are identical; they differ at least by the position which they occupy in space, otherwise they would not form a flock. But now let us even set aside the fifty sheep themselves and retain only the idea of them. Either we include them all in the same image, and it follows as a necessary consequence that we place them side by side in an ideal space, or else we repeat fifty times in succession the image of a single one, and in that case it does seem, indeed, that the series lies in duration rather than in space. But we shall soon find out that it cannot be so. For if we picture to ourselves each of the sheep in the flock in succession and separately, we shall never have to do with more than a single sheep. In order that the number should go on increasing in proportion as we advance, we must retain the successive images and set them alongside each of the new units which we picture to ourselves: now, it is in space that such a juxtaposition takes place and not in pure duration. In fact, it will be easily granted that counting material objects means thinking all these objects together, thereby leaving them in space. But does this intuition of space accom pany every idea of number, even of an abstract number?
Any one can answer this question by reviewing CHAP, n 78 TIME AND FREE WILL the various forms which the idea of number has we cannot assumed for him since his childhood.
orTde8?^*86 Jt wi11 be seen that we beSan bv imagin- numberwith- mg e>gr a row of balls, that these balls out the ac- o o SfiaSfS afterwards became points, and, finally, ipac«. this image itself disappeared, leaving behind it, as we say, nothing but abstract number. But at this very moment we ceased to have an image or even an idea of it; we kept only the symbol which is necessary for reckoning and which is the conventional way of expressing num ber. For we can confidently assert that 12 is half of 24 without thinking either the number 12 or the number 24: indeed, as far as quick calcu lation is concerned, we have everything to gain by not doing so. But as soon as we wish to picture number to ourselves, and not merely figures or words, we are compelled to have recourse to an extended image. What leads to misunderstanding on this point seems to be the habit we have fallen into of counting in time rather than in space. In order to imagine the number 50, for example, we repeat all the numbers starting from unity, and when we have arrived at the fiftieth, we believe we have built up the number in duration and in duration only. And there is no doubt that in this way we have counted moments of duration rather than points in space; but the question is whether we have not counted the moments of duration by means of points in space. It is cer tainly possible to perceive in time, and in time CHAP, n NUMERICAL MULTIPLICITY AND SPACE 79 only, a succession which is nothing but a succes sion, but not an addition, i.e. a succession which culminates in a sum. For though we reach a sum by taking into account a succession of different terms, yet it is necessary that each of these terms should remain when we pass to the following, and should wait, so to speak, to be added to the others: how could it wait, if it were nothing but an instant of duration? And where could it wait if we did not localize it in space? We involun tarily fix at a point in space each of the moments which we count, and it is only on this condition that the abstract units come to form a sum. No doubt it is possible, as we shall show later, to con ceive the successive moments of time independently of space; but when we add to the present moment those which have preceded it, as is the case when we are adding up units, we are not dealing with these moments themselves, since they have van ished for ever, but with the lasting traces which they seem to have left in space on their passage through it. It is true that we generally dispense with this mental image, and that, after having used it for the first two or three numbers, it is enough to know that it would serve just as well for the mental picturing of the others, if we needed it. But every clear idea of number implies a visual image in space; and the direct study of the units which go to form a discrete multiplicity will lead us to the same conclusion on this point as the examination of number itself.
g0 TIME AND FREE WILL CHAP. 11 Every number is a collection of units, as we have said, and on the other hand every number is itself a unit, in so far as it is a synthesis of oi a the units which compose it. But is the k4 Ol word unit taken in the same sense in tfecauie both cases? When we assert that num- " ber is a unit, we understand by this that we master the whole of it by a simple and indivisible intuition of the mind; this unity thus includes a multiplicity, since it is the unity of a whole. But when we speak of the units which go to form number, we no longer think of these units as sums, but as pure, simple, irreducible units, intended to yield the natural series of num bers by an indefinitely continued process of ac cumulation. It seems, then, that there are two kinds of units, the one ultimate, out of which a number is formed by a process of addition, and the other provisional, the number so formed, which is multiple in itself, and owes its unity to the simplicity of the act by which the mind per ceives it. And there is no doubt that, when we picture the units which make up number, we be lieve that we are thinking of indivisible com ponents: this belief has a great deal to do with the idea that it is possible to conceive number independently of space. Nevertheless, by looking more closely into the matter, we shall see that all unity is the unity of a simple act of the mind, and that, as this is an act of unification, there must be some multiplicity for it to unify. No doubt, at CHAP, n NUMERICAL MULTIPLICITY AND SPACE 8l the moment at which I think each of these units separately, I look upon it as indivisible, since I am determined to think of its unity alone. But as soon as I put it aside in order to pass to the next, I objectify it, and by that very deed I make it a thing, that is to say, a multiplicity. To con vince oneself of this, it is enough to notice that the units by means of which arithmetic forms numbers are provisional units, which can be sub divided without limit, and that each of them is the sum of fractional quantities as small and as numerous as we like to imagine. How could we divide the unit, if it were here that ultimate unity which characterizes a simple act of the mind? How could we split it up into fractions whilst affirming its unity, if we did not regard it implicitly as an extended object, one in intuition but multiple in space? You will never get out of an idea which you have formed anything which you have not put into it; and if the unity by means of which you make up your number is the unity of an act and not of an object, no effort of analysis will bring out of it anything but unity pure and simple. No doubt, when you equate the number 3 to the sum of i + i + i, nothing prevents you from regarding the units which compose it as indivisible: but the reason is that you do not choose to make use of the multiplicity which is enclosed within each of these units. Indeed, it is probable that the number 3 first assumes to our mind this simpler shape, because we think g2 TIME AND FREE WILL CHAP, n rather of the way in which we have obtained it than of the use which we might make of it. But we soon perceive that, while all multiplication implies the possibility of treating any number whatever as a provisional unit which can be added to itself, inversely the units in their turn are true numbers which are as big as we like, but are regarded as provisionally indivisible for the purpose of com pounding them with one another. Now, the very admission that it is possible to divide the unit into as many parts as we like, shows that we regard it as extended.
For we must understand what is meant by the discontinuity of number. It cannot be denied that the formation or construction of procew of tor- a number implies discontinuity. In mation If dis-., 111 continuous, other words, as we remarked above, formed, "« in- each of the units with which we form the continuity the number 3 seems to be indivisible while we are dealing with it, and we pass abruptly from one to the other. Again, if we form the same number with halves, with quarters, with any units whatever, these units, in so far as they serve to form the said number, will still constitute elements which are provision ally indivisible, and it is always by jerks, by sudden jumps, so to speak, that we advance from one to the other. And the reason is that, in order to get a number, we are compelled to fix our attention successively on each of the units of which it is com pounded. The indivisibility of the act by which CHAP, n NUMERICAL MULTIPLICITY AND SPACE 83 we conceive any one of them is then represented under the form of a mathematical point which is separated from the following point by an interval of space. But, while a series of mathematical points arranged in empty space expresses fairly well the process by which we form the idea of number, these mathematical points have a tendency to develop into lines in proportion as our attention is diverted from them, as if they were trying to reunite with one another. And when we look at number in its finished state, this union is an accom plished fact: the points have become lines, the divisions have been blotted out, the whole displays all the characteristics of continuity. This is why number, although we have formed it according to a definite law, can be split up on any system we please. In a word, we must distinguish be tween the unity which we think of and the unity which we set up as an object after having thought of it, as also between number in process of forma tion and number once formed. The unit is irre ducible while we are thinking it and number is discontinuous while we are building it up: but, as soon as we consider number in its finished state, we objectify it, and it then appears to be divisible to an unlimited extent. In fact, we apply the term subjective to what seems to be completely and adequately known, and the term objective to what is known in such a way that a constantly increasing number of new impressions could be substituted for the idea which we actually have g4 TIME AND FREE WILL CHAP, n of it. Thus, a complex feeling will contain a fairly large number of simple elements; but, as long as these elements do not stand out with per fect clearness, we cannot say that they were com pletely realized, and, as soon as consciousness has a distinct perception of them, the psychic state which results from their synthesis will have changed for this very reason. But there is no change in the general appearance of a body, however it is analysed by thought, because these different analyses, and an infinity of others, are already visible in the mental image which we form of the body, though they are not realized: this actual and not merely virtual perception of subdivisions in what is undivided is just what we call objectivity. It then becomes easy to determine the exact part played by the subjective and the objective in the idea of number. What properly belongs to the mind is the indivisible process by which it con centrates attention successively on the different parts of a given space; but the parts which have thus been isolated remain in order to join with the others, and, once the addition is made, they may be broken up in any way whatever. They are therefore parts of space, and space is, accordingly, the material with which the mind builds up number, the medium in which the mind places it.
Properly speaking, it is arithmetic which teaches us to split up without limit the units of which number consists. Common sense is very much inclined to build up number with indivisibles.
CHAP, ii NUMERICAL MULTIPLICITY AND SPACE 85 And this is easily understood, since the pro- it follows visional simplicity of the component units what they owe to the mmd> and SSJo5ti5natne latter Pays more attention to its in space. own ac^-s than to the material on which it works. Science confines itself, here, to drawing our attention to this material: if we did not already localize number in space, science would certainly not succeed in making us transfer it thither. From the beginning, therefore, we must have thought of number as of a juxtaposition in space. This is the conclusion which we reached at first, basing ourselves on the fact that all addi tion implies a multiplicity of parts simultaneously perceived.
Now, if this conception of number is granted, it will be seen that everything is not counted in the same way, and that there are two Two kinds of...r...multiplicity: very different kinds of multiplicity.
objects, When we speak of material objects, we space; (2) refer to the possibility of seeing and conscious IT xr states, not touching them; we localize them in countable un- T..
less symbolic- space. In that case, no effort ol the sented in inventive faculty or of symbolical represpace...sentation is necessary in order to count them; we have only to think them, at first separ ately, and then simultaneously, within the very medium in which they come under our observation. The case is no longer the same when we consider purely affective psychic states, or even mental 86 TIME AND FREE WILL CHAP, n images other than those built up by means of sight and touch. Here, the terms being no longer given in space, it seems, a priori, that we can hardly count them except by some process of symbolical representation. In fact, we are well aware of a representation of this kind when we are dealing with sensations the cause of which is obviously situated in space. Thus, when we hear a noise of steps in the street, we have a confused vision of somebody walking along: each of the successive sounds is then localized at a point in space where the passer-by might tread: we count our sensations in the very space in which their tangible causes are ranged. Perhaps some people count the successive strokes of a distant bell in a similar way, their imagination pictures the bell coming and going; this spatial sort of image is sufficient for the first two units, and the others follow naturally. But most people's minds do not proceed in this way. They range the suc cessive sounds in an ideal space and then fancy that they are counting them in pure duration. Yet we must be clear on this point. The sounds of the bell certainly reach me one after the other; but one of two alternatives must be true. Either I retain each of these successive sensations in order to combine it with the others and form a group which reminds me of an air or rhythm which I know: in that case I do not count the sounds, I limit myself to gathering, so to speak, the qualita tive impression produced by the whole series. Or