BOOK II. § 5. Number. can make this distinction only by taking each as embodied in, or represented by, the particular number to which it gives rise, and which is its inseparable result. Each number or modification of number will then represent a particular act of counting in relation to a series of others, on which it immediately depends; and its place in that series or system of numbers to which it belongs is the only means we have for recording and distinguishing the act of counting which gives rise to it, from the endless series of acts of counting, of which, being otherwise undistinguishable, memory refuses to retain a separate trace. That is to say, any act of counting, when taken as an unit counted, is ipso facto identified with that particular portion of the continuous time-stream which it serves to count, and from the place of which in the time-stream it derives its value as a quantity. We come back, then, in the last resort to Numbers, and in the first instance to the series of whole numbers of Integers, as the basis of the whole science of Calculation, and through calculation of Measurement, since there can be no measurement of one thing by another, without first distinguishing two things from one thing, that is, without counting. But, as we have seen, all Numbers are continua; that is, cannot be distinguished one from another save by taking them as continuous portions of one and the same continuum, only rendered discrete one from another. by the abstract act of counting, that is, of ideally dividing (without occupying) that continuum. Discrete quantity is continuous quantity broken up, or considered as broken up, into smaller Book II. § 5. continua, a process to which there is no assignable limit. Number itself is discrete quantity in this sense. I think there is no avoiding this conclusion, Number. unless we assume, that an Absolute Logos of some sort or other creates itself and the universe, by means of some immanent pseudo-action and re-action between the logical principles of Identity and Contradiction, an idea which would be strange if true, besides being unintelligible whether true or not. At the same time, several things must be remembered. First, in forming any series or system of Numbers, the particular nature of the continuum, of which they form part, is abstracted from. We have seen that, as a fact, Time is the one continuum which is indispensable to the process of counting. But a knowledge of this fact is not included in the nature of Number, considered either as the means, or as the object of Calculation. Time is not the object measured by a simple succession of acts of counting, the intervals between which are wholly arbitrary, so far as their length is concerned. Similarly, the psychological act of purposive attention to a content of consciousness is indispensable to counting, and therefore to Number. But this act in its in its psychological character lies wholly outside the process-content of calculation as such. Its duration as a psychological act is not in question at all. If either time or the act of attention is made an object of measurement or calculation, it must be by way of first objectifying it as a particular object among others. Number, in short, though arising solely from the ideal division of a continuum, by means of a CH. I. § 5. Number. BOOK II. psychological act which which has duration, is no measurement either of the continuum or of the act. Yet there is an object which it measures, and therefore a sense, and that the most essential to it, in which it is measurement; the object which it creates is the object which it measures, namely, number itself, by means of the first result of its fundamental and perpetually repeated act, the act of counting, that first result being Unity, or the number One. Number (as a general term) means a number of Units. In other words, the standard of measurement in all calculation is Unity, being that determination in which the act of counting and its result coincide. This circumstance it is, which gives Calculation its specific character among all other modes or sciences of measurement. Let us see more particularly how this can be. In ideally dividing the time-continuum by the first act of counting, we look back at a portion of that continuum which is undetermined as to its beginning, and forward to another portion of it which is undetermined as to its end. In the second act of counting we determine the end of this latter portion, look back upon it in retrospect as a portion whose beginning is already determined by the first act of counting, and forward to another portion whose end is as yet undetermined. In the third act of counting the same process is repeated, and so on for as long as we can continue to count. Thus, as we advance, we continue to lay behind us in memory a series of acts of counting, each of which determines the end of one portion of the time-continuum and the beginning THE BIRTH-PLACE OF SCIENCE.NIVERSITY OF CALIFORNIA 41 BOOK II. § 5. of another, the continuum itself being otherwise undivided, that is, undetermined as to the length of any portion of it, save by the successive acts of Number. counting, which may themselves take place at quite arbitrary and variable intervals. Yet at the same time, the perception of the time-continuum itself cannot be avoided or dispensed with. For, if there were no interval perceived between the successive acts of counting, they could not be perceived as several or successive; there would be no possibility of remembering or recording a first act when performing a second, or a second when performing a third, and so on. Time-intervals are therefore necessary to a succession of acts of counting, that is, to Number, and yet there is no measure of the length of those intervals, save the remembered recorded number of times for which the successive acts of counting have been performed. Consequently the interval or difference between acts of counting, that is, between successive numbers, 1, 2, 3, &c. (as well as every increase in the number of the acts themselves), is measured by 1. Or in other words, numerical Unity, a pure Number, is the measure of the interval or difference between 2 and 1, between 3 and 2, between 4 and 3, and so on. or When, therefore, taking Number at its origin, or in its lowest and simplest terms, we objectify it as the result of repeated acts of counting, we must consider it, like Time itself, as a continuously growing quantity, the successive increments of which are noted and recorded only by figures or symbols expressing the number of single acts of CH. I. § 5. BOOK II. counting which have gone to their discrimination, in which every single increment necessarily correNumber. sponds to a single act of counting, and is therefore necessarily equal to every other. For two ways, equally legitimate and equally necessary, are then open to us in which to objectify it. If in the first instance we objectify the several acts of counting per se, we get the series, 1. 2. 3. 4. &c., while, if we objectify this same series of numbers together with the continuum which they divide, what we get is the series of intervals, 0234 &c., in which the same figures or symbols represent intervals between single acts of counting, and in which we supply, in thought, the starting point 0, the distance or difference of which from the first act of counting is determined by unity, that is, the same distance or difference which obtains between all the several subsequent acts of counting. Each interval is itself a number and nothing else, namely, the number one. And the result is plainly to transform, in thought, the original time-continuum discriminated by acts of ideal division into a purely numerical continuum, that is, a continuum in which there is no interval (but only an ideal division) interposed between the several discrete parts, called Numbers, of which it is composed. And Numbers thenceforward, for purposes of calculation, replace, and are the substitute for, that time-continuum and its ideal division by acts of purposive attention, which are the matrix out of which they originally spring. |