3 THE INNUMERABLE MULTITUDE
As we have seen, Leibnitz by no means admits 'infinite number', since on the contrary he expressly declares that this would imply contradiction in whatever sense one took it; on the other hand, he does admit what he calls an 'infinite multitude', though without making it clear-as the Scholastics would at least have done-that in any case it can only be an infinitum secundum quid, the sequence of numbers being, for him, an example of such a multitude. From another point of view, however, in the quantitative domain, and even in that of continuous magnitude, the idea of the infinite always appears to him as suspect of at least possible contradiction, for, far from being an adequate idea, it inevitably entails a certain amount of confusion, and we cannot be certain that an idea implies no contradiction unless we distinctly conceive all of its elements; [1] this hardly allows according this idea a 'symbolic' - we would rather say 'representative'-character, and as we shall see later, this is why he
never dared to give a clear verdict on the reality of the 'infinitely small'; but this very perplexity, this doubtful attitude, brings out even better the lack of principle that led him to admit that one could speak of an 'infinite multitude'. From this one might also wonder if he did not think that in order to be 'infinite', as he calls it, such a multitude must not only be 'numerable', which is obvious, but that it must not even be quantitative at all, taking quantity in all its extension and in all its modes; this would be true in certain cases, but not in all; however it may be, it remains a point on which he never clearly explained himself.
The idea of a multitude that surpasses all number, and that consequently is not a number, seems to have astonished most of those who have discussed the conceptions of Leibnitz, be they 'finitists' or 'infinitists'; it is nevertheless far from unique to Leibnitz, as they have generally seemed to believe, and, on the contrary, was quite common among the Scholastics. [2] This idea was applied specifically to everything that is neither a number nor 'numerable', that is, all that does not relate to the domain of discontinuous quantity, whether it be a question of things belonging to other modes of quantity, or of what is entirely outside of the quantitative domain, for it concerned an idea belonging to the order of 'transcendentals', or general modes of being, which, contrary to its special modes like quantity, are coextensive with it. [3] This also allows one to speak of the multitude of divine attributes for example, or again of the multitude of angels, that is, of beings belonging to states that are not
subject to quantity, where, consequently, there can be no question of number; it is also this that allows one to speak of the states of being or degrees of existence as multiple or as constituting an indefinite multitude, even though quantity is only one special condition of a single one of them. On the other hand, since the idea of multitude, contrary to that of number, is applicable to all that exists, there must necessarily be multitudes of a quantitative order, notably in the domain of continuous quantity, and this is why we said just now that it would not be correct to consider every case of the socalled 'infinite multitude', that is, that which surpasses all number, as entirely escaping the domain of quantity. Furthermore, number itself can also be regarded as a species of multitude, but on the added condition that it be a 'multitude measured by the unit', according to the expression of Saint Thomas Aquinas; all other sorts of multitude, being 'innumerable', are 'non-measured', which is not to say they are infinite, but merely that they are indefinite.
While on the subject, it is appropriate to note a rather singular fact: for Leibnitz, this multitude, which does not constitute a number, is nonetheless a 'result of units. [4] How should we understand this, and indeed, what are the units in question? The word unit can be taken in two completely different senses: [5] on the one hand, there is the arithmetical or quantitative unit, which is the first element of number, its point of departure, and, on the other hand, there is what is analogously designated as metaphysical Unity, which is identified with pure Being itself; we see no other possible meaning outside of these; but furthermore, whenever one speaks of 'units' in the plural, this can obviously only be understood in the quantitative sense. If this is so, however, then the sum of these units cannot be anything other than a number, and can in no way transcend number; it is true that Leibnitz said 'result' and not 'sum', but this distinction, even if it is intentional, nonetheless remains an unfortunate obscurity. Besides, he declares elsewhere that multitude, without being a number, is nevertheless conceived by analogy
with number: 'When there are more things,' he says, 'than can be comprehended by any number, we yet attribute to them analogically a number that we call infinite,' although this would only be a 'manner of speaking', a modus loquendi, [6] and even, in this form, a most incorrect manner of speaking, since in reality the thing in question is not a number at all; but whatever the imperfections of expression and the confusions to which they might give rise, we must in any case admit that an identification of multitude with number was assuredly not at the root of his thought.
Another point to which Leibnitz seems to attach great importance is that the 'infinite', such as he conceives of it, does not constitute a whole; [7] this is a condition he regards as necessary if the idea is to escape contradiction, but here we have another rather obscure point. One might well wonder what sort of 'whole' is in question here, and it is first of all necessary to put aside entirely the idea of the universal All, which is on the contrary, as we have said from the beginning, the metaphysical Infinite itself, the only true Infinite, which could by no means be in question here; indeed, whether it is a question of continuous or discontinuous, the 'indefinite multitude' that Leibnitz envisages in any case only makes sense in a restricted and contingent domain of a cosmological and not metaphysical order. It is obviously a question, moreover, of a whole conceived as composed of parts, whereas, as we have explained elsewhere, [8] the universal All is properly 'without parts', by very reason of its infinity, since these parts are necessarily relative and finite and thus could
not have any real connection with it, which amounts to saying that for it they do not exist. So, as regards the question posed, we must confine ourselves to the consideration of a particular whole; but here again, and precisely in what concerns the mode of composition of such a whole and its relation with its parts, there are two cases to consider, corresponding to two very different senses of the same word 'whole'. First, there is the whole that is nothing more or other than the simple sum of its parts, of which it is composed in the manner of an arithmetical sum, which Leibnitz says is obviously fundamental, for this mode of formation is precisely that which is proper to number, and he does not allow us to go beyond number; but in fact this notion, far from representing the only way in which a whole can be conceived, is not even that of a true whole in the most rigorous sense of the term. Indeed, a whole that is thus only the sum or result of its parts and which consequently is logically posterior to them, is, as such, nothing other than an ens rationis [a being of reason or of the mind], for it is 'one' and 'whole' only in the measure that we conceive it as such; in itself it is strictly speaking only a 'collection', and it is we who, by the manner in which we envisage it, confer upon it in a certain relative sense the character of unity and totality. On the contrary, a true whole possessing this character by its very nature, must be logically anterior to its parts and independent of them: such is the case with a continuous set, which we can divide into parts arbitrarily, that is, into parts of any size, without in the least presupposing the actual existence of these parts; here, it is we who give a reality to the parts as such, by an ideal or effective division, and this case is thus the exact inverse of the preceding.
Now, the whole question comes back in short to knowing whether, when Leibnitz says that 'the infinite is not a whole,' he excludes this second sense as well as the first; it seems that he does, and this is probable since it is the only case in which a whole would truly be 'one', and since the infinite, according to him, is nec unum, nec totum [neither one nor a whole]. What further confirms this is that this latter, and not the former, is what applies to a living being or an organism when it is considered from the point of view of totality; now Leibnitz says: 'Even the Universe is not a whole, and it must not be conceived of as an animal with God for its soul, as the
ancients thought. [9] However, if this is so, one does not really see how the ideas of the infinite and the continuous can be connected, as he most often takes them to be, since the idea of the continuous is, at least in a certain sense, linked precisely to this second conception of totality; but this is a point that will be better understood in the light of what is to follow. In any case, what is certain is that if Leibnitz had conceived of a third sense of the word 'whole', a purely metaphysical sense superior to the other two, namely the idea of the universal All as we set it forth at the very beginning, he would not have been able to say that the idea of the infinite excludes totality, for he declares moreover: 'The real infinite is perhaps the absolute itself, which is not composed of parts, but having parts, comprehends them by eminent reason, as to the degree of its perfection. [10] Here, one could say, there is at the very least a 'glimmer', for this time, almost by exception, he takes the word 'infinite' in its true sense, although it would be erroneous to say that this infinite 'has parts', however one wishes to understand this; but it is then strange that he again expresses his thought only in a doubtful and perplexing form, as if he were not exactly settled as to the significance of the idea; and indeed perhaps he never was, for otherwise one could not explain why he so often turned away from its proper meaning, and why, when he speaks of the infinite, it is sometimes so difficult to know whether his intention was to take this term rigorously, albeit wrongly, or whether he had in view only a simple 'manner of speaking'.