ZERO IS NOT A NUMBER
The indefinite decrease of numbers can no more end in a 'null number' than their indefinite increase can in an 'infinite number', and for the same reason, since each of these numbers must be the inverse of the other; indeed, in accordance with what was said earlier on the subject of inverse numbers, as each of the two sets-the one increasing, the other decreasing-is equally distant from the unit, the common point of departure for both, and as there must further necessarily be as many terms in the one as in the other, their final terms-namely, the 'infinite number' and the 'null number'if they existed, would themselves have to be equally distant from the unit, and thus the inverses of one another. [1] Under these conditions, if the sign ∞ is in reality only a symbol for indefinitely increasing quantities, then logically the sign o should likewise be able to be taken as a symbol for indefinitely decreasing quantities, in order to express in notation the symmetry that, as we have said, exists between the two; but unfortunately this sign o already has quite another significance, for it originally served to designate the complete absence of quantity, whereas the sign ∞ has no real sense that
would correspond to the former. Here, as with the 'vanishing quantities', we have yet another source of confusion, and in order to avoid this it would be necessary to create another symbol, apart from zero, for indefinitely decreasing quantities, since these quantities are characterized precisely by the fact that they can never be annulled, despite any variation they might undergo; at any rate, with the notation currently employed by mathematicians, it seems almost impossible to prevent confusions from arising.
If we emphasize the fact that zero, insofar as it represents the complete absence of quantity, is not a number and cannot be considered as such-even though this might appear obvious enough to those who have never had occasion to take cognizance of certain disputes-this is because, as soon as one admits the existence of a 'null number', which would have to be the 'smallest of numbers', one is inevitably led by way of correlation to suppose as its inverse an 'infinite number', in the sense of the 'greatest of numbers'. If, therefore, one accepts the postulate that zero is a number, the arguments in favor of an 'infinite number' follow in a perfectly logical manner; [2] but it is precisely this postulate that we must reject, for if the consequences deduced from it are contradictory-and we have seen that the existence of an 'infinite number' is indeed so-then the postulate in itself must already imply contradiction. Indeed, the negation of quantity can in no way be assimilated to a particular quantity; the negation of number or of magnitude can in no sense and to no degree constitute a species of number or magnitude; to claim the contrary would be to maintain that a thing could be 'equivalent to a species of its contradictory', to use Leibnitz's expression, and would be as much as to say immediately that the negation of logic is itself logic.
It is therefore contradictory to speak of zero as a number, or to suppose that a 'zero in magnitude' is still a magnitude, from which would inevitably result the consideration of as many distinct zeros as there are different kinds of magnitude; in reality, there can only be zero pure and simple, which is none other than the negation of
quantity, whatever the mode envisaged. [3] When such is accepted as the true sense of the arithmetical zero, taken 'rigorously', it becomes obvious that this sense has nothing in common with the notion of indefinitely decreasing quantities, which are always quantities; they are never an absence of quantity, nor again are they anything that is as it were intermediate between zero and quantity, which would be yet another completely unintelligible conception, and which in its own order would recall that of Leibnitzian 'virtuality', which we had occasion to mention earlier.
We can now return to the other meaning that zero actually has in common notation, in order to see how the confusions we spoke of were introduced. We said earlier that in a way a number can be regarded as practically indefinite when it is no longer possible for us to express or represent it distinctly in any way; such a number, whatever it might be, can only be symbolized in the increasing order by the sign ∞, insofar as this represents the indefinitely great; it is therefore not a question of a determined number, but rather of an entire domain, and this is necessary moreover if it is to be possible to envisage inequalities and even different orders of magnitude within the indefinite. Mathematical notation lacks a symbol for the corresponding domain in the decreasing order, what might be called the domain of the indefinitely small; but since a number belonging to this domain is, in fact, negligible in calculations, it is in practice habitually considered to be null, even though this is only a simple approximation resulting from the inevitable imperfection of our means of expression and measurement, and it is doubtless for this reason that it came to be represented by the same symbol o that also represents the rigorous absence of quantity. It is only in this
sense that the sign o becomes in a way symmetrical to the sign ∞ and that the two can be placed respectively at the two extremities of the sequence of numbers as we envisaged it earlier, with the whole numbers and their inverses extending indefinitely in the two opposite directions of increase and decrease. This sequence then presents itself in the following form: $0 \ldots 1 / 4,1 / 3,1 / 2,1,2,3,4 \ldots ∞; but we must take care to recall that $o$ and ∞ represent not two determined numbers terminating the series in either direction, but two indefinite domains, in which on the contrary there can be no final terms, precisely by reason of their indefinitude; moreover, it is obvious that here zero can be neither a 'null number', which would be a final term in the decreasing direction, nor again a negation or absence of quantity, which would have no place in this sequence of numerical quantities.
As we explained previously, any two numbers in the sequence that are equidistant from the central unit are inverses or complementaries of one another, thus producing the unit when multiplied together: $(1 / n)(n)=1$, such that for the two extremities of the sequence, one would be led to write $(0)(\infty)=1$ as well; but, since the signs o and ∞, the two factors of this product, do not represent determined numbers, it follows that the expression (o)( ∞ ) itself constitutes a symbol of indeterminacy, or what one would call an 'indeterminate form', and one must therefore write $(0)(\infty)=n$, where $n$ could be any number; [4] it is no less true that in any case one will thus be brought to ordinary finitude, the two opposed indefinites so to speak neutralizing one another. Here, once again, one can clearly see that the symbol ∞ most emphatically does not represent the Infinite, for the Infinite, in its true sense, can have neither opposite nor complementarity, nor can it enter into correlation with anything at all, no more with zero, in whatever sense it might be understood, than with the unit, or with any number, or again with any particular thing of any order whatsoever, quantitative or not; being the absolute and universal All, it contains Non-Being as well as Being, such that zero itself, whenever it is not regarded as purely nothing, must also necessarily be considered to be contained within the Infinite.
In alluding here to Non-Being, we touch on another meaning of zero quite different from those we have just considered, the most important from the point of view of metaphysical symbolism; but in this regard, in order to avoid all confusion between the symbol and that which it represents, it is necessary to make it quite clear that the metaphysical Zero, which is Non-Being, is no more the zero of quantity than the metaphysical Unit, which is Being, is the arithmetical unit. What is thus designated by these terms is so only by analogical transposition, since as soon as one places oneself within the Universal one is obviously beyond every special domain such as that of quantity. Furthermore, it is not insofar as it represents the indefinitely small that zero by such a transposition can be taken as a symbol of Non-Being, but, following its most rigorous mathematical usage, rather insofar as it represents the absence of quantity, which in its order indeed symbolizes the possibility of non-manifestation, just as the unit, since it is the point of departure for the indefinite multiplicity of numbers, symbolizes the possibility of manifestation as Being is the principle of all manifestation. [5]
This again leads us to note that zero, however it may be envisaged, can in no case be taken for pure nothingness, which corresponds metaphysically only to impossibility, and which in any case cannot logically be represented by anything. This is all too obvious when it is a question of the indefinitely small; it is true that this is only a derivative sense, so to speak, due, as we were just saying, to a sort of approximate assimilation of quantities negligible for us, to the total absence of quantity; but insofar as it is a question of this very absence of quantity, what is null in this connection certainly cannot be so in other respects, as is apparent in an example such as the point, which, being indivisible, is by that very fact without extension, that is, spatially null, [6] but which, as we have explained elsewhere, is nonetheless the very principle of all extension. [7] It is quite strange, moreover, that mathematicians are generally inclined to envisage zero as a pure nothingness, when it is nevertheless
impossible for them not to regard it at the same time as endowed with an indefinite potentiality, since, placed to the right of another digit termed 'significant', it contributes to forming the representation of a number that, by the repetition of this same zero, can increase indefinitely, as is the case with the number ten and its successive powers for example. If zero were really only pure nothingness, this could not be so; and indeed, in that case it would only be a useless sign, entirely deprived of effective value; here we have yet another inconsistency to add to the list of those that we have already had occasion to point out in the conceptions of modern mathematicians.