22 THE SYNTHETIC CHARACTER OF INTEGRATION
Contrary to the formation of an arithmetical sum, which, as we have just said, is strictly analytic in character, integration must be regarded as an essentially synthetic operation in that it simultaneously embraces each element of the sum to be calculated, preserving the 'indistinction' appropriate to the parts of a continuum, since, by the very nature of continuity, these parts cannot be fixed and determined things. Moreover, whenever one wishes to calculate the sum of the discontinuous elements of an indefinite sequence, this 'indistinction' must likewise be maintained, although for a slightly different reason, for even if the magnitude of each may be conceived of as determined, the total number of elements may not, and we can even say more exactly that their multitude surpasses all number; nevertheless, there are some cases in which the sum of the elements of such a sequence tends toward a certain definite limit, even when their multitude increases indefinitely. Although such a manner of speaking might at first seem a little strange, one could also say that such a discontinuous sequence is indefinite by 'extrapolation', while a continuous set is so by 'interpolation'; what is meant by this is that if one takes a given portion of a discontinuous sequence, bounded by any two of its terms, such a portion will in no way be indefinite, as it is determined both as a whole and with respect to its elements; the indefinitude of the sequence lies in the fact that it extends beyond this portion, without ever arriving at a final term; on the contrary, the indefinitude of a continuous set,
determined as such, is to be found precisely in its interior, since its elements are not determined, and since it has no final terms, the continuous being always divisible; in this respect each case is thus as it were the inverse of the other. The summation of an indefinite numerical sequence will never be completed if each term must be taken one by one, since there is no final term whereby the sequence could come to an end; such a summation is possible only in the case where a synthetic procedure lets us seize in a single stroke, as it were, the indefinitude considered in its entirety, without this at all presupposing the distinct consideration of its elements, which, moreover, is impossible, by the very fact that they constitute an indefinite multitude. And similarly, when an indefinite sequence is given to us implicitly by its law of formation, as in the case of the sequence of whole numbers, we can say that it is thus given to us completely in a synthetic manner, and that it cannot be given otherwise; indeed, to do so analytically would be to lay out each term distinctly, which is an impossibility.
Therefore, whenever we have a given example of indefinitude to consider, whether it be a continuous set or a discontinuous sequence, it will be necessary in every case to have recourse to a synthetic operation in order to reach its limits; progression by degrees would be useless here and could never bring us to our goal, for such a progression can arrive at a final term only on the twofold condition that both this term and the number of degrees to be covered in order to reach it, be determined. That is why we did not say that the limits of the indefinite could not be reached at all, which would be unjustifiable when its limits do exist, but only that they cannot be reached analytically: the indefinite cannot be exhausted by degrees, but it can be embraced in its totality by certain transcendent operations, of which integration is the classic example in the mathematical order. One could point out that progression by degrees here corresponds precisely to the variation of quantity, directly in the case of discontinuous sequences and, in cases of continuous variation, following therefrom, so to speak, to the extent permitted by the discontinuous nature of number; on the other hand, synthetic operations immediately place one outside of and beyond the domain of variation, as must necessarily be the case according with
what we said above, in order for a 'passage to the limit' actually to be realized; in other words, analysis pertains only to variables, taken in the very course of their variation, while synthesis alone attains their limits, which is the only definitive and really valuable result, since, to be able to speak of results, one must clearly arrive at something relating exclusively to fixed and determined quantities.
Furthermore, one can of course find analogous synthetic operations in domains apart from quantity, for the idea of an indefinite development of possibilities is clearly applicable to other things than quantity, as, for example, to a given state of manifested existence and the conditions, whatever they might be, to which the state is subject, whether considered with respect to the whole of the cosmos, or to one being in particular; that is, one can take either a 'macrocosmic' or a 'microcosmic' point of view. [1] One could say that in this case 'passage to the limit' corresponds to the definitive fixation of the results of manifestation in the principial order; indeed, by this alone does the being finally escape from the change and 'becoming' that is necessarily inherent to all manifestation as such; and one can thus see that this fixation is in no way a 'final term' of the development of manifestation, but rather that it is essentially situated outside of and beyond that development, since it belongs to another order of reality, transcendent in relation to manifestation and 'becoming'; in this regard, the distinction between the manifested order and the principial order thus corresponds analogically to that which we established between the domains of variable and fixed quantities. What is more, when it is a question of fixed quantities, it is obvious that no modification can be introduced by any operation whatsoever, and that, consequently, 'passage to the limit' cannot produce anything in this domain, but can only give us knowledge of it; likewise, the principial order being immutable, arriving at it is not a question of 'effectuating' something that did not exist before, but rather of effectively taking cognizance, in a permanent and absolute manner, of that which is. Given the subject of this study, we must naturally consider more particularly and above
all, what properly concerns the quantitative domain, in which, as we have seen, the idea of the development of possibilities is translated by the notion of variation, whether in the direction of indefinite increase or of indefinite decrease; but these few will suffice to show that by an appropriate analogical transposition all of this is capable of receiving an incomparably greater significance than that which it appears to have in and of itself, since integration and other operations of the same kind will thereby veritably appear as symbols of metaphysical 'realization' itself.
By this one sees the extent of the difference between traditional science, which allows such considerations, and the profane science of the moderns; and, in this connection, we shall add yet another remark directly relating to the distinction between analytic and synthetic knowledge. Profane science, indeed, is essentially and exclusively analytical; it never considers principles, losing itself instead in the details of phenomena, of which the indefinite and indefinitely changing multiplicity are for it truly inexhaustible, such that it can never arrive at any real or definitive result as far as knowledge is concerned; it keeps solely to phenomena themselves, that is, to exterior appearances, and is incapable of reaching the heart of things, for which Leibnitz had already reproached Cartesian mechanism. This is moreover one of the reasons by which modern 'agnosticism' is explained, for, since there are things that can be known only synthetically, whoever proceeds by analysis alone is thereby led to declare such things 'unknowable', since in this respect they really are so, just as those who keep to the analytic view of the indefinite believe its indefinitude to be absolutely inexhaustible, whereas in reality it is so only analytically. It is true that synthetic knowledge is essentially what one might call 'global' knowledge, as is the knowledge of a continuous set or an indefinite sequence the elements of which are not and cannot be set out distinctly; but, apart from the fact that this knowledge is ultimately all that really matters, one can always-since everything is contained in it in principle-descend from it to the consideration of such particular things as one might wish, just as, if an indefinite sequence, for example, is given synthetically through the knowledge of its law of formation, one can as occasion arises always calculate any of its particular terms, while on
the contrary when one takes as one's starting-point these same particular things considered in and of themselves, and in all their indefinite detail, one can never rise to the level of principles; and, as we said at the beginning, it is in this regard that the method and point of view of traditional science is as it were inverse to that of profane science, as synthesis itself is to analysis. Moreover, we have here only an application of the obvious truth that, although the 'lesser' can be drawn from the 'greater', one can never cause the 'greater' to come from the 'lesser'; nevertheless, this is precisely what modern science claims to do, with its mechanistic and materialistic conceptions and its exclusively quantitative point of view; but it is precisely because this is impossible that such science is, in reality, incapable of giving the true explanation of anything whatever. [2]