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 successive 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 qualitative impression produced by the whole series. Or else I intend explicitly to count them, and then I shall have to separate them, and this separation must take place within some homogeneous medium in which the sounds, stripped of their qualities, and in a manner emptied, leave traces of their presence which are absolutely alike. The question now is, whether this medium is time or space. But a moment of time, we repeat, cannot persist in order to be added to others. If the sounds are separated, they must leave empty intervals bet ween them. If we count them, the intervals must remain though the sounds disappear: how could these intervals remain, if they were pure duration and not space? It is in space, therefore, that the operation takes place. It becomes, indeed, more and more difficult as we penetrate further into the depths of consciousness. Here we find ourselves confronted by a confused multiplicity of sensations and feelings which analysis alone can distinguish. Their number is identical with the number of the moments which we take up when we count them; but these moments, as they can be added to one another, are again points in space. Our final conclusion, therefore, is that there are two kinds of multiplicity: that of material objects, to which the conception of number is immediately applicable; and the multiplicity of states of consciousness, which cannot be regarded as numerical without the help of some symbolical representation, in which a necessary element is space.
The impenetrability of matter is not a physical but a logical necessity.
As a matter of fact, each of us makes a distinction between these two kinds of multiplicity whenever he speaks of the impenetrability of matter. We sometimes set up impenetrability as a fundamental property of bodies, known in the same way and put on the same level as e.g. weight or resistance. But a purely negative property of this kind cannot be revealed by our senses; indeed, certain experiments in mixing and combining things might lead us to call it in question if our minds were not already made up on the point. Try to picture one body penetrating another: you will at once assume that there are empty spaces in the one which will be occupied by the particles of the other; these particles in their turn cannot penetrate one another unless one of them divides in order to fill up the interstices of the other; and our thought will prolong this operation indefinitely in preference to picturing two bodies in the same place. Now, if impenetrability were really a quality of matter which was known by the senses, it is not at all clear why we should experience more difficulty in conceiving two bodies merging into one another than a surface devoid of resistance or a weightless fluid. In reality, it is not a physical but a logical necessity which attaches to the proposition: "Two bodies cannot occupy the same place at the same time." The contrary assertion involves an absurdity which no conceivable experience could succeed in dispelling.
In a word, it implies a contradiction. But does not this amount to recognizing that the very idea of the number 2, or, more generally, of any number whatever, involves the idea of juxtaposition in space? If impenetrability is generally regarded as a quality of matter, the reason is that the idea of number is thought to be independent of the idea of space. We thus believe that we are adding something to the idea of two or more objects by saying that they cannot occupy the same place: as if the idea of the number 2, even the abstract number, were not already, as we have shown, that of two different positions in space! Hence to assert the impenetrability of matter is simply to recognize the interconnexion between the notions of number and space, it is to state a property of number rather than of matter. — Yet, it will be said, do we not count feelings, sensations, ideas, all of which permeate one another, and each of which, for its part, takes up the whole of the soul? — Yes, undoubtedly; but, just because they permeate one another, we cannot count them unless we represent them by homogeneous units which occupy separate positions in space and consequently no longer permeate one another. Impenetrability thus makes its appearance at the same time as number; and when we attribute this quality to matter in order to distinguish it from everything which is not matter, we simply state under another form the distinction established above between extended objects, to which the conception of number is immediately applicable, and states of consciousness, which have first of all to be represented symbolically in space.
Homogeneous time as the medium in which conscious states form discrete series. This time is nothing but space, and pure duration is something different.
It is advisable to dwell on the last point. If, in order to count states of consciousness, we have to represent them symbolically in space, is it not likely that this symbolical representation will alter the normal conditions of inner perception? Let us recall what we said a short time ago about the intensity of certain psychic states. Representative sensation, looked at in itself, is pure quality; but, seen through the medium of extensity, this quality becomes in a certain sense quantity, and is called intensity. In the same way, our projection of our psychic states into space in order to form a discrete multiplicity is likely to influence these states themselves and to give them in reflective consciousness a new form, which immediate perception did not attribute to them. Now, let us notice that when we speak of time, we generally think of a homogeneous medium in which our conscious states are ranged alongside one another as in space, so as to form a discrete multiplicity. Would not time, thus understood, be to the multiplicity of our psychic states what intensity is to certain of them, — a sign, a symbol, absolutely distinct from true duration? Let us ask consciousness to isolate itself from the external world, and, by a vigorous effort of abstraction, to become itself again. We shall then put this question to it: does the multiplicity of our conscious states bear the slightest resemblance to the multiplicity of the units of a number? Has true duration anything to do with space? Certainly, our analysis of the idea of number could not but make us doubt this analogy, to say no more. For if time, as the reflective consciousness represents it, is a medium in which our conscious states form a discrete series so as to admit of being counted, and if on the other hand our conception of number ends in spreading out in space everything which can be directly counted, it is to be presumed that time, understood in the sense of a medium in which we make distinctions and count, is nothing but space. That which goes to confirm this opinion is that we are compelled to borrow from space the images by which we describe what the reflective consciousness feels about time and even about succession; it follows that pure duration must be something different. Such are the questions which we have been led to ask by the very analysis of the notion of discrete multiplicity. But we cannot throw any light upon them except by a direct study of the ideas of space and time in their mutual relations.
Does space exist independently of its contents, as Kant held?
We shall not lay too much stress on the question of the absolute reality of space: perhaps we might as well ask whether space is or is not in space. In short, our senses perceive the qualities of bodies and space along with them: the great difficulty seems to have been to discover whether extensity is an aspect of these physical qualities — a quality of quality — or whether these qualities are essentially unextended, space coming in as a later addition, but being self-sufficient and existing without them. On the first hypothesis, space would be reduced to an abstraction, or, speaking more correctly, an extract; it would express the common element possessed by certain sensations called representative. In the second case, space would be a reality as solid as the sensations themselves, although of a different order. We owe the exact formulation of this latter conception to Kant: the theory which he works out in the