id="ulink_6ff4958b-57ca-50c8-a56f-1138ab47fef3">12 L'homme et l'intelligence, p. 36.
13 Ibid. p. 37.
14 Ibid. p. 43.
15 The Expression of the Emotions, 1st ed., pp. 72, 69, 70.
16 C. Féré, Sensation et Mouvement. Paris, 1887.
17 Grundzüge der Physiologischen Psychologie, 2nd ed., (1880), Vol. ii, p. 437.
18 "On the Temperature Sense," Mind, 1885.
19 Rood, Modern Chromatics,(1879), pp. 181-187.
20 Handbuch der Physiologischen Optik, 1st ed. (1867), pp. 318-319.
21 Éléments de psychophysique. Paris, 1883.
22 See the account given of these experiments in the Revue philosophique, 1887, Vol. i, p. 71, and Vol. ii, p. 180.
23 Éléments de psychophysique, pp. 61, 69.
24 In the particular case where we admit without restriction Weber's Law ΔE/E=const., integration gives S=C log. E/Q. Q being a constant. This is Fechner's "logarithmic law."
25 Latterly it has been assumed that ΔS is proportional to S.
26 Revue scientifique, March 13 and April 24, 1875.
CHAPTER II
THE MULTIPLICITY OF CONSCIOUS STATES1
THE IDEA OF DURATION
What is number?
Number maybe defined in general as a collection of units, or, speaking more exactly, as the synthesis of the one and the many. Every number is one, since it is brought before the 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 considered separately. Without attempting for the present any thorough examination of these conceptions of unity and multiplicity, let us inquire whether the idea of number does not imply the representation of something else as well.
The units which make up a number must be identical.
It is not enough to say that number is a collection of units; we must add that these units are identical with one another, or at least that they are assumed to be identical 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.
But they must also be distinct.
And yet they must be somehow distinct from one another, since otherwise they would merge into a single unit. Let us assume that 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 accompany every idea of number, even of an abstract number?
We can not form an image or idea of number without the accompanying intuition of space.
Any one can answer this question by reviewing the various forms which the idea of number has assumed for him since his childhood. It will be seen that we began by imagining e.g. a row of balls, that these balls afterwards became points, and, finally, 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 number. 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 calculation 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 certainly possible to perceive in time, and in time only, a succession which is nothing but a succession, 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 involuntarily 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 conceive 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 vanished 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