2 THE CONTRADICTION OF 'INFINITE NUMBER'

As we will see yet more clearly in the following, there are some cases in which it suffices to replace the idea of the so-called infinite with that of the indefinite in order to dispel all difficulties immediately; but there are others in which even this is not possible, because it is a question of something clearly determined-'fixed', so to speak, by hypothesis-which, as such, cannot be called indefinite, according to our last remarks above. Thus, for example, one can say that the sequence of numbers is indefinite, but not that a certain number, however great one supposes it to be and whatever position it occupies in the sequence, is indefinite. The idea of an 'infinite number', understood as 'the greatest of all numbers', or 'the number of all numbers', or, again, 'the number of all units', is in itself a truly contradictory idea, the impossibility of which would remain even were one to renounce the unjustifiable use of the word 'infinite'. There cannot be a number greater than all others, for however great a number might be, one can always form a greater one from it by adding a unit, in accordance with the law of formation which we set forth above. This amounts to saying that the sequence of numbers cannot have a final term, and it is precisely because it does not 'terminate' that it is truly indefinite; as the number of all the terms of the sequence could itself only be the last of them, it can be said that the sequence is not 'numerable', and this is an idea we shall have to return to more fully in what follows. The impossibility of an 'infinite number' can be established further by various arguments. Leibnitz, who at least recognized this quite clearly, [1] used one that consisted in comparing the sequence of even numbers to that of whole numbers: to every number there corresponds another number equal to its double, such that one can make the two sequences correspond term by term, with the result that the number of terms must be the same in both; but there are obviously twice as many whole numbers as there are even, since the even numbers alternate by twos in the sequence of whole numbers; one thus ends up with a manifest contradiction. One can generalize this argument by taking, instead of the sequence of even numbers, that is, multiples of two, that of multiples of any number whatsoever, and the reasoning will be identical; or again, in the same way one could take the sequence of the squares of whole numbers, [2] or more generally that of their powers of any exponent. Whatever the case, the conclusion will always be the same: a sequence containing only a part of the whole numbers will have the same number of terms as another containing all of them, which would amount to saying that the whole is not greater than its part, and, as soon as one allows that there is a number of all numbers, this contradiction will be inescapable. Nevertheless, some have thought to avoid it by supposing at the same time that there are numbers for which multiplication by a certain number or elevation to a certain power is not possible, precisely because such operations would yield a result exceeding the so-called 'infinite number'; there are even those who have indeed been led to envisage numbers said to be 'greater than infinite', whence such theories as that of Cantor's 'transfinite', which may be quite ingenious, but are no longer logically valid: [3] is it even conceivable that one could dream of calling a number 'infinite' when it is on the contrary so 'finite' that it is not even the greatest of all numbers? Moreover, with such theories there would be numbers to which none of the rules of ordinary calculation would apply any longer, or, in short, numbers that would no longer truly be numbers but merely called such by convention. [4] This inevitably occurs when, seeking to conceive of an 'infinite number' otherwise than as the greatest of all numbers, one envisages various 'infinite numbers', supposedly unequal to each other, to which we attribute properties that no longer have anything in common with those of ordinary numbers; thus one escapes one contradiction only to fall into others, and all this is at bottom only the product of the most meaningless 'conventionalism' imaginable. Thus, the idea of a so-called 'infinite number', whatever manner in which it is presented and whatever name by which one wishes to designate it, always comprises contradictory elements; moreover, one has no need of such an absurd supposition from the moment one forms a proper conception of what the indefinitude of number really is, and when one further recognizes that number, despite its indefinitude, is by no means applicable to all that exists. We need not dwell upon this last point here, as we have already sufficiently explained it elsewhere. Number is only a mode of quantity, and quantity itself only a category or special mode of being, not coextensive with it, or, more precisely still, quantity is only a condition proper to one certain state of existence in the totality of universal existence; but this is precisely the point that most moderns have difficulty understanding, habituated as they are to wanting to reduce everything to quantity and even to evaluating everything numerically. [5] However, in the very domain of quantity there are things that escape number, as we shall see when we come to the subject of continuity; and even without departing from the sole consideration of discontinuous quantity, one is already forced to admit, at least implicitly, that number is not applicable to everything, when one recognizes that the multitude of all numbers cannot constitute a number, which, moreover, is finally only an application of the incontestable truth that what limits a certain order of possibilities must necessarily be beyond and outside that which it limits. [6] Only it must be understood that such a multitude, be it discontinuous, as in the case of the sequence of numbers, or continuous-a subject we shall have to return to shortly-can in no wise be called infinite, and in such cases there can never be anything but the indefinite; and it is this notion of multitude that we are now going to examine more closely.