'VANISHING QUANTITIES'
For Leibnitz, the justification for 'passage to the limit' ultimately consists in the fact that the particular case of the 'vanishing quantities', as he says, must in a certain sense be included within the general rule by virtue of continuity; moreover, these vanishing quantities cannot be regarded as 'absolute nothings', or as pure zeros, for by reason of the same continuity they maintain among themselves determined ratios-and generally differ from unity-in the very instant in which they vanish, which implies that they are still real quantities, although 'unassignable' with respect to ordinary quantities. [1] However, if these vanishing quantities-or the infinitesimal quantities, which amounts to the same thing-are not 'absolute nothings', even when it is a question of differentials of orders higher than the first, they must still be considered 'relative nothings', which is to say that, while retaining the character of real quantities, they can and must be negligible with regard to ordinary quantities, with which they are 'incomparable'; [2] but multiplied by 'infinite' quantities, or quantities incomparably greater than ordinary ones, they again produce these ordinary quantities, which could not be so
if they were absolutely nothing. In light of the definitions we presented earlier, one can see that the consideration of the ratios of vanishing but still determined quantities refers to the differential calculus, while the consideration of the multiplication of these quantities by 'infinite' quantities, yielding ordinary quantities, refers to the integral calculus. The difficulty in all this is to admit that quantities that are not absolutely null must nonetheless be treated in the calculus as if they were, which risks giving the impression that it is merely a question of simple approximation; again, in this regard Leibnitz sometimes seems to invoke the 'law of continuity', by which the 'borderline case' finds itself included within the general rule, as if this were the only postulate his method required; this argument is quite unclear, however, and one should rather return to the notion of 'incomparables', as he himself often does, moreover, in order to justify the elimination of infinitesimal quantities from the results of the calculus.
Indeed, Leibnitz considers as equal not only those quantities of which the difference is null, but even those of which the difference is incomparable with respect to the quantities themselves; this notion of 'incomparables' is, for him, the foundation not only for the elimination of infinitesimal quantities, which thus disappear in the face of ordinary quantities, but also for the distinction between different orders of infinitesimal or differential quantities, the quantities of each order being incomparable with respect to those of the preceding, as those of the first order are with respect to ordinary quantities, but without ever arriving at 'absolute nothings'. 'I call two magnitudes incomparable,' says Leibnitz, 'when one, despite multiplication by any finite number whatsoever, can nonetheless not exceed the other, in the same way that Euclid treated it in the fifth definition of his fifth book. [3] However, there is nothing there to indicate whether this definition should be understood of fixed and determined, or of variable, quantities; but one can admit that in all its generality it must apply without distinction to both cases; the entire question would then be one of knowing whether two fixed quantities, however different they might be within the scale of magnitudes,
could ever be regarded as truly 'incomparable', or whether they would only be so relative to the means of measurement at our disposal. But we shall not dwell further on this point, since Leibnitz himself declared elsewhere that this is not the case with differentials, [4] from which it is necessary to conclude, not only that the comparison of the grain of sand is in itself manifestly faulty, but also that it fundamentally does not answer, even in his own thought, to the true notion of 'incomparables', at least insofar as this notion must be applied to the infinitesimal quantities.
Some people, however, have believed that the infinitesimal calculus can be rendered perfectly rigorous only on the condition that the infinitesimal quantities be regarded as null, and at the same time they have wrongly thought that one can suppose an error to be null as long as one can also suppose it to be as small as one likes; wrongly, we say, for that would be the same as to admit that a variable, as such, could reach its limit. Here is what Carnot has to say on the subject:
There are those who believe they have sufficiently established the principle of infinitesimal analysis with the following reasoning: it is obvious, they say, and universally acknowledged, that the errors to which the procedure of infinitesimal analysis would give riseif there were any-could always be supposed as small as one might wish; it is also obvious that any error one is free to suppose as small as one likes is null, for since one can suppose it to be as small as one wishes, one can suppose it to be zero; therefore, the results of the infinitesimal analysis are rigorously exact. This argument, plausible at first sight, is nevertheless anything but valid, for it is false to say that because one is free to render an error as small as one likes one can thus render it absolutely null.... One is faced with the necessary alternative either of committing an error, however slight one might suppose it to be, or of falling back on a formula that says nothing, and such is precisely the crux of the difficulty with the infinitesimal analysis. [5]
It is certain that any formula in which a ratio appears in the form 0 / 0 'says nothing', and one could even say that it has no meaning in and of itself; it is only in virtue of a convention-justified, moreover-that one can give any sense to the expression 0 / 0, regarding it as a symbol of indeterminacy; [6] but this very indeterminacy then means that the ratio in this form can be equal to anything, whereas on the contrary it must maintain a determined value in every particular case; it is the existence of this determined value that Leibnitz puts forward, [7] and in itself this argument is completely unassailable. [8] However, it is quite necessary to recognize that the notion of 'vanishing quantities' has 'the tremendous drawback of considering quantities in that state in which they so to speak cease to be quantities', to use Lagrange's expression; but contrary to what Leibnitz thought, there is no need to consider them precisely in the instant in which they vanish, nor even to suppose that they really could vanish, for in that case they would indeed cease to be quantities. Moreover, this essentially supposes that strictly speaking there is no 'infinitely small' quantity, for this 'infinitely small' quantity-or at least what would be called such in Leibnitz's language-could only be zero, just as an 'infinitely great' quantity, taken in the same sense, could only be an 'infinite number'; but in reality zero is not a number, and 'null quantities' have no more existence than do 'infinite quantities'. The mathematical zero, in its rigorous and strict sense, is but a negation, at least as far as its quantitative aspect is concerned, and one cannot say that the absence of quantity itself constitutes a quantity; we shall return to this point shortly, in order to develop more completely the consequences that result from it.
In sum, the expression 'vanishing quantities' has above all the drawback of producing an equivocation, and of leading to the belief that infinitesimal quantities can be considered as quantities that are effectively annulled, for without altering the meaning of these words, it is difficult to understand how, when it is a question of quantities, 'to vanish' could mean anything other than to be annulled. In reality, these infinitesimal quantities, understood as indefinitely decreasing quantities, which is their true significance, can never be called 'vanishing' in the proper sense of the word. It would most certainly have been preferable had the notion never been introduced, as it is fundamentally bound up with Leibnitz's conception of continuity, and, as such, inevitably contains the same element of contradiction inherent in the illogicality of this latter. Now if an error, despite being able to be rendered as small as one likes, can never become absolutely null, how can the infinitesimal calculus be truly rigorous, and if the error is in fact only practically negligible, would it not be necessary to conclude that the calculus is thus reduced to a simple method of approximation, or at least, as Carnot says, of 'compensation'? This is a question that we must resolve in what follows; but as we have here been brought to speak of zero and of the so-called 'null quantity', it will be worthwhile to deal with this other subject first, the importance of which, as we shall see, is far from negligible.