21 THE INDEFINITE IS ANALYTICALLY INEXHAUSTIBLE
In the two cases just considered, that of the indefinitely increasing and that of the indefinitely decreasing, a quantity of a given order can be regarded as the sum of an indefinitude of elements, each of which is an infinitesimal quantity with respect to the entire sum. In order to be able to speak of infinitesimal quantities, it is moreover necessary that it be a question of elements that are not determined with respect to their sum, and this is indeed the case whenever the sum is indefinite with respect to the elements in question; this follows immediately from the essential character of indefinitude itself, inasmuch as the latter obviously implies the idea of 'becoming', as we have said before, and consequently a certain* indeterminacy. It is of course understood that this indeterminacy can only be relative, and exists only from a certain point of view or with respect to a certain thing: such is the case, for example, with a sum that is an ordinary quantity, and hence not indefinite in and of itself, but only with respect to its infinitesimal elements; at any rate, if it were otherwise, and if this notion of indeterminacy were not introduced, one would be reduced to the mere conception of 'incomparables', interpreted in the crude sense of the grain of sand in comparison to the earth, and the earth in comparison to the heavens.
The sum in question can by no means be effected in the manner of an arithmetical sum, since for that it would be necessary for an
indefinite series of successive additions to be achieved, which is contradictory; in the case in which the sum is an ordinary and determined quantity as such, it is obviously necessary, as we already said when we set forth the definition of the integral calculus, that the number, or rather the multitude, of elements increase indefinitely while at the same time the magnitude of each decreases indefinitely, and in this sense the indefinitude of its elements is truly inexhaustible. But if the sum cannot be effected in this way, as the final result of a multitude of distinct and successive operations, it can on the other hand be comprehended at one stroke, by a single operation, namely, integration; [1] here we have the inverse operation of differentiation, since it reconstitutes the sum starting from its infinitesimal elements, while differentiation on the contrary moves from the sum to the elements, furnishing the means of formulating the law for the instantaneous variations of the quantity of which the expression is given.
Thus, whenever it is a question of indefinitude, the notion of an arithmetical sum is no longer applicable, and one must resort to the notion of integration in order to compensate for the impossibility of 'numbering' the infinitesimal elements, an impossibility which, of course, results from the very nature of these elements, and not from any imperfection on our part. In passing we may observe that as regards the application of this to geometric magnitudes (which, moreover, is ultimately the true raison d'être of the infinitesimal calculus), this is a method of measurement completely different from the usual method founded on the division of a magnitude into definite portions, of which we spoke previously in connection with 'units of measurement'. The latter always amounts in short to a substitution of the discontinuous for the continuous by 'cutting up' the sum into various portions equal to a magnitude of the same species
taken as the unit, [2] in order to be able to apply the resulting number directly to the measurement of continuous magnitudes, which cannot actually be done except by altering the nature of the magnitudes in order to make it assimilable, so to speak, to that of number. The other method, on the contrary, respects the true character of continuity as much as possible, regarding it as a sum of elements that are fixed and determined, but that are essentially variable and by virtue of their variability capable of becoming smaller than any assignable magnitude; this method thereby allows the spatial quantity between the limits of these elements to be reduced as much as one likes, and it is therefore the least imperfect representation of continuous variation one can give, in that it takes account of the nature of number, which in spite of everything cannot be changed.
These observations will allow us to understand more precisely in what sense one can say, as we did at the beginning, that the limits of the indefinite can never be reached through any analytical procedure, or, in other words, that the indefinite, while not absolutely and in every way inexhaustible, is at least analytically inexhaustible. In this regard, we must naturally consider those procedures analytical which, in order to reconstitute a whole, consist in taking its elements distinctly and successively; such is the procedure for the formation of an arithmetical sum, and it is precisely in this regard that it differs essentially from integration. This is particularly interesting from our point of view, for one can see in it, as a very clear example, the true relationship between analysis and synthesis: contrary to current opinion, according to which analysis is as it were a preparation for synthesis, or again something leading to it, so much so that one must always begin with analysis, even when one does not intend to stop there, the truth is that one can never actually arrive at synthesis through analysis. All synthesis, in the true sense
of the word, is something immediate, so to speak, something that is not preceded by any analysis and is entirely independent of it, just as integration is an operation carried out in a single stroke, by no means presupposing the consideration of elements comparable to those of an arithmetical sum; and as this arithmetical sum can yield no means of attaining and exhausting the indefinite, this latter must, in every domain, be one of those things that by their very nature resist analysis and can be known only through synthesis. [3]