9 INDEFINITELY INCREASING; INDEFINITELY DECREASING

BEFORE CONTINUING the examination of questions properly relating to the continuous, we must return to what was said above about the non-existence of a _fractio omnium infima_, which will allow us to see how the correlation or symmetry that exists in certain respects between indefinitely increasing and indefinitely decreasing quantities can be represented numerically. We have seen that in the domain of discontinuous quantity, as long as it is only the sequence of whole numbers that needs to be considered, these numbers must be regarded as increasing indefinitely starting from the unit, but that there can obviously be no question of an indefinite decrease since the unit is essentially indivisible; were the numbers to be taken in the decreasing direction, one would necessarily find oneself stopped at the unit itself, so that the representation of the indefinite by whole numbers is limited to a single direction, that of indefinite increase. On the other hand, when it is a question of continuous quantity, one can envisage indefinitely decreasing quantities as well as indefinitely increasing ones; and the same occurs in discontinuous quantity itself as soon as, in order to express this possibility, the consideration of fractional numbers is introduced. Indeed, one can envisage a sequence of fractions continuing to decrease indefinitely; that is, however small a fraction might be, a smaller one could always be formed, and this decrease can no more arrive at a _fractio minima_ [smallest fraction] than can the increase of whole numbers at a _numerus maximus_ [greatest number].

If we wish to use a numerical representation in order to make evident the correlation between the indefinitely increasing and the indefinitely decreasing, it suffices to consider the sequence of whole numbers together with that of their inverses; a number is said to be the inverse of another when the product of the two is equal to the unit, and for this reason the inverse of the number n is represented by the notation 1/n. Whereas the sequence of whole numbers goes on increasing indefinitely starting from the unit, the sequence of their inverses decreases indefinitely, starting from the same unit, which is its own inverse, and which is therefore the common point of departure for the two sequences; to each number in one sequence there thus corresponds a number in the other, and inversely, such that the two sequences are equally indefinite, and in exactly the same way, though in contrary directions. The inverse of a number is obviously as small as the number itself is great, since their product always remains constant; however great a number n might be, the number n + 1 will be greater still by virtue of the very law of formation for the indefinite sequence of whole numbers, and similarly, as small as a number 1/n might be, the number 1/(n+1) will be smaller still; and this clearly proves the impossibility of any ‘smallest of numbers’, which notion is no less contradictory than is that of a ‘greatest of numbers’, for, if it is impossible to stop at a determined number in the increasing direction, it will be no more possible to stop in the decreasing direction. Moreover, since this correlation which is found in numerical discontinuity occurs first of all as a consequence of the application of this discontinuity to the continuous, as we said concerning fractional numbers, the introduction of which it naturally supposes, it can only express the correlation that exists within the continuous itself between the indefinitely increasing and the indefinitely decreasing in its own way, which is necessarily conditioned by the nature of number. Therefore, whenever continuous quantities are considered capable of becoming as great or as small as one likes, that is, greater or smaller than any determined quantity, one can always observe a symmetry and, in a manner of speaking, a parallelism presented by these two inverse kinds of variability. This remark will subsequently help us to understand better the possibility of different orders of infinitesimal quantities.

It would be good to point out that although the symbol 1/n evokes the idea of fractional numbers, and although it is in fact incontestably derived from them, the inverses of the whole numbers need not be defined here as such, and this in order to avoid the difficulty presented by the ordinary notion of fractional numbers from the strictly arithmetical point of view, that is, the conception of fractions as ‘parts of the unit’. Indeed, it suffices to consider the two sequences to be constituted by numbers respectively greater and smaller than the unit, that is, as two orders of magnitude that have their common limit in the latter, and that at the same time both can be regarded as issuing from this unit, which is truly the primary source of all numbers; what is more, if one wished to consider the two indefinite sets as forming a single sequence, one could say that the unit occupies the exact mid-point within this sequence, since, as we have seen, there are exactly as many numbers in the one set as in the other. Moreover, if, to generalize further, instead of considering only the sequence of whole numbers and their inverses, one wished to introduce fractional numbers properly speaking, nothing would be changed as far as the symmetry of increasing and decreasing quantities is concerned: on one side one would have all the numbers greater than the unit, and on the other all those smaller than the unit; here, again, for any number a/b > 1, there will be a corresponding number b/a < 1 in the other group, and reciprocally, such that (a/b) (b/a) = 1, just as earlier we had (n) (1/n) = 1, and there will thus be exactly the same number of terms in each of these two indefinite groups separated by the unit; it must moreover be understood that when we say ‘the same number of terms’, we simply mean that the two multitudes correspond term by term, and not that they can themselves on that account be considered ‘numerable’. Any two inverse numbers multiplied together always produce again the unit from which they proceeded; one can say further that the unit, occupying the mid-point between the two groups, and being the only number that can be regarded as belonging to both at once^[1]— although in reality it would be more correct to say that it unites rather than separates them—corresponds to the state of perfect equilibrium, and contains in itself all numbers which issue from it in pairs of inverse or complementary numbers, each pair by virtue of this complementarity constituting a relative unity in its indivisible duality;^[2] but we shall return a little later to this last consideration and to the consequences it implies. Instead of saying that the series of whole numbers is indefinitely increasing and that of their inverses indefinitely decreasing, one could also say, in the same sense, that the numbers thus tend on the one hand toward the indefinitely great and on the other toward the indefinitely small, on condition that we understand by this the actual limits of the domain in which these numbers are considered, for a variable quantity can only tend toward a limit. The domain in question is, in short, that of numerical quantity, taken in every possible extension;^[3] this again amounts to saying that its limits are not determined by such and such a particular number, however great or small it might be supposed, but by the very nature of number as such. By the very fact that number, like everything else of a determined nature, excludes all that it is not, there can be no question of the infinite; moreover, we have just said that the indefinitely great must inevitably be conceived of as a limit, although it is in no way a terminus ultimus [ultimate limit] of the series of numbers, and in this connection one can point out that the expression ‘tend toward infinity’, frequently employed by mathematicians in the sense of ‘increase indefinitely’, is again an absurdity, since the infinite obviously implies the absence of any limit, and that consequently there is nothing toward which it is possible to tend. What is also rather remarkable is that certain mathematicians, while recognizing the inaccuracy and improper character of the expression ‘tend toward infinity’, on the other hand feel no scruple at all about taking the expression ‘tend toward zero’ in the sense of ‘decrease indefinitely’; zero, however, or the ‘null quantity’, is, with respect to decreasing quantities, exactly the same as the so-called ‘quantitative infinite’ is with respect to increasing quantities; but we shall have to return to these questions later, particularly when we come to the subject of zero and its different meanings. Since the sequence of numbers in its entirety is not ‘terminated’ by a given number, it follows that there is no number however great that could be identified with the indefinitely great in the sense just understood; and, naturally, the same is true for the indefinitely small. One can only regard a number as practically indefinite, if one may so express it, when it can no longer be expressed by language or represented by writing, which in fact inevitably occurs the moment one considers numbers that go on increasing or decreasing; here we have a simple matter of ‘perspective’, if one wishes, but all in all even this is in keeping with the character of the indefinite, insofar as the latter is ultimately nothing other than that of which the limits can be, not done away with, since this would be contrary to the very nature of things, but simply pushed back to the point of being entirely lost from view. In this connection some rather curious questions should be considered; thus, one could ask why the Chinese language symbolically represents the indefinite by the number ten thousand; the expression ‘the ten thousand beings’, for example, means all beings, which really make up an indefinite or ‘innumerable’ multitude. What is quite remarkable is that it is precisely the same in Greek, where a single word likewise serves to express both ideas at once, with a simple difference in accentuation, obviously only a quite secondary detail, and doubtless only due to the need to distinguish the two meanings in usage: μύριοι, ‘ten thousand’; μυρίοι, ‘an indefinitude’. The true reason for this is the following: the number ten thousand is the fourth power of ten; now, according to the formulation of the Tao Te Ching, ‘one produced two, two produced three, three produced all numbers,’ which implies that four, produced immediately after three, is in a way equivalent to the whole set of numbers, and this because, when one has the quaternary by adding the first four numbers, one also has the denary, which represents a complete numerical cycle: 1 + 2 + 3 + 4 = 10, which, as we have already said on other occasions, is the numerical formula of the Pythagorean Tetraktys. One can further add that this representation of numerical indefinitude has its correspondence in the spatial order: it is common knowledge that raising a number from one degree to the next highest power represents in this order the addition of a dimension; now, our space having only three dimensions, its limits are transcended when one goes beyond the third power, which, in other words, amounts to saying that elevation to the fourth power marks the very term of its indefinitude, since, as soon as it is effected, one has thereby departed from space and passed on to another order of possibilities.