'DEGREES OF INFINITY'
We have not yet had occasion in the preceding pages to see all the confusions that are inevitably introduced when the idea of the infinite is taken otherwise than in its one true and properly metaphysical sense; more than one example could be found, notably, in Leibnitz's long discussion with Jean Bernoulli on the reality of infinite and infinitely small quantities, which moreover never came to any definitive conclusion; nor, indeed, could it have done so, given the continual confusion on both sides, and the lack of principles from which this confusion proceeded; moreover, whatever the order of ideas in question, ultimately it is always the lack of principles which alone renders questions insoluble. One might well be astonished to learn, among other things, that Leibnitz distinguished between 'infinite' and 'interminable', and that he had thus not absolutely rejected the idea-nonetheless manifestly contra-dictory-of a 'terminating infinite', and went so far as to ask himself 'whether it might be possible for there to exist, for example, an infinite straight line that might nevertheless terminate at both ends. [1] No doubt he is reluctant to admit this possibility, 'all the more so since it seems to me,' he says elsewhere, 'that the infinite, taken rigorously, must have its source in the interminable, without which I see no means of finding a proper foundation for distinguishing it from the finite. [2] But even if one puts it more affirmatively (which he did not do) and says that 'the infinite has its source in the interminable,' one still does not take them to be absolutely identical, but rather as distinguished from one another to a certain degree; and as long as that is
so, one risks finding oneself checked by a crowd of strange and contradictory ideas. It is true that Leibnitz declares that he would not willingly admit these ideas without first being 'forced by indubitable demonstrations', but it is already serious enough to attribute a certain degree of importance to them, and even to be able to envisage them other than as pure impossibilities; as for the idea of a sort of 'terminating eternity', to take one example from those he sets forth in this connection, we can see in it only the product of a confusion between the notions of eternity and duration, which is absolutely unjustifiable with respect to metaphysics. We readily grant that the time in which we pass our corporeal lives really is indefinite, which is in no way incompatible with its 'terminating at both ends', which is to say, in conformity with the traditional cyclic conception, that it has both a beginning and an end; we also grant that there exist other modes of duration, such as that which the Scholastics call aevum, the indefinitude of which is, if one may so express it, indefinitely greater than that of this time; but all these modes, in all their possible extension, are nonetheless only indefinite, since it is always a question of particular conditions of existence proper to this or that state; and, precisely insofar as each is a kind of durationwhich implies succession-not one can be identified with or assimilated to eternity, with which it has no more connection than does the finite, whatever its mode, nor again with the true Infinite, for the notion of a relative eternity has no more meaning than that of a relative infinite. In all of this we have only various orders of indefinitude, as will be seen more clearly later on, but Leibnitz, for want of having made the necessary and essential distinctions, and above all for not having laid down before all else the principle that alone would have prevented him from going astray, found himself very much at a loss to refute Bernoulli's opinions; indeed, so equivocal and hesitant were Leibnitz's responses that Bernoulli even took him to be much closer than was really the case to his own ideas about the 'infinity of worlds' and the different 'degrees of infinity'.
This notion of the so-called 'degrees of infinity' amounts in short to supposing that there can exist worlds incomparably greater and incomparably smaller than our own, the corresponding parts of each being in equal proportion to one another, such that the
inhabitants of any one of these worlds would have just as much reason to call theirs infinite as we would with respect to ours; for our part we would rather say they would have just as little reason. Such a manner of envisaging things would not appear absurd a priori without the introduction of the idea of the infinite, which is certainly nothing to the purpose, for however great one imagines them to be, each of these worlds is nonetheless limited; how then can they be called infinite? The truth is that none of them can really be so, if only because they are conceived as multiple, for here we return to the contradiction of a plurality of infinites; and besides, even if it happens that some or even many consider our world to be infinite, this assertion nonetheless can offer no acceptable meaning. Moreover, one might wonder if they really are different worlds, or if, quite simply, they are not rather more or less extended parts of the same world, since by hypothesis they must all be subject to the same conditions of existence-notably to spatiality-and simply developed on an enlarged or diminished scale. It is in a completely different sense that one can truly speak, not of an infinity, but of an indefinitude of worlds, since apart from the conditions of existence such as space and time, which are proper to our world considered in all the extension of which it is susceptible, there is an indefinitude of others, equally possible; a world, or, in short, a state of existence, is thus defined by the totality of the conditions to which it is subject; but, by the very fact that it will always be conditioned, that is, determined and limited, and hence unable to contain all possibilities, it can never be regarded as infinite, but only indefinite. [3]
Fundamentally, the consideration of 'worlds' in the sense understood by Bernoulli, incomparably larger or smaller in relation to one another, is not very different from what Leibnitz resorted to when he envisaged 'the firmament with respect to the earth, and the earth with respect to a grain of sand,' and the latter with respect to 'a particle of magnetic material passing through a lens.' Only here Leibnitz does not claim to speak of gradus infinitatis [grade of infinity] in the strict sense; on the contrary, he even means to show that 'one need not take the infinite rigorously,' and he is content to
envisage 'incomparables', to which no logical objection can be raised. The shortcoming of his comparison is of quite another order, and as we have already said, lies in the fact that it is only capable of giving an inexact, or even completely false, idea of the infinitesimal quantities as they figure in the calculus. In what follows we shall have occasion to substitute for this consideration that of the true multiple degrees of indefinitude, taken in increasing as well as decreasing order; for the moment, therefore, we shall not dwell further on it.
In short, the difference between Bernoulli and Leibnitz is that for the first, even though he presents them only as a probable conjecture, it is truly a question of 'degrees of infinity', while the second, doubting their probability and even their possibility, limits himself to replacing them with what could be called 'degrees of incomparability'. Aside from this difference, which is moreover assuredly extremely important, they share in common the notion of a series of worlds that are similar, but on different scales. This notion is not without a certain incidental connection with discoveries made in the same period with the microscope, and with certain views that arose as a consequence-although later observations were in no way to justify them-such as the theory of the 'encasement of embryos'; now it is not true of embryos that every part of the living being must be actually and physically 'preformed', and the organization of a cell bears no resemblance to that of the entire body of which it is an element. There seems to be no doubt that this was in fact the origin of Bernoulli's notion, at any rate; indeed, among other things highly significant in this regard, he says that the particles of a body coexist in the whole 'in the same way that, in accordance with Harvey and others, though not with Leeuwenhoeck, there exist within an animal innumerable ovules, within each ovule one or several animalcules, within each animalcule again innumerable ovules, and so on to infinity. [4] As for Leibnitz, his was likely a completely different point of departure; thus, the idea that all the stars that we can see can only be components of the body of an incomparably greater being, recalls the Kabbalistic conception of the 'Great Man', but singularly
materialized and 'spatialized' through a sort of ignorance of the true analogical value of traditional symbolism; likewise, the idea of the 'animal', that is, the living being, subsisting corporeally after death, but 'in miniature', is obviously inspired by the traditional Judaic concept of the luz or 'kernel of immortality',[5] which Leibnitz equally distorted by connecting it with the notion of worlds incomparably smaller than our own, saying, 'nothing prevents animals from being transferred to such worlds after death; indeed, I think that death is no more than a contraction of the animal, just as generation is no more than an evolution, [6] this last word being taken here simply in its etymological sense of 'development'. All this is fundamentally only an example of the dangers that exist when one wishes to make traditional notions agree with the views of profane science, which can only be done to the detriment of the former; these notions are most clearly independent of the theories brought about by microscopic observations, and in comparing and muddling them, Leibnitz was already acting as would the occultists later on, for they particularly delighted in these sorts of unjustified comparisons. Moreover, the superposition of 'incomparables' of different orders seemed to him in conformity with his notion of the 'best of worlds', furnishing a means of investing it with 'as much being or reality as possible', to quote from his definition; and as we have already pointed out elsewhere, [7] this idea of the 'best of worlds' is also derived from yet another ill-applied traditional doctrine, this one borrowed from the symbolic geometry of the Pythagoreans. According to this geometry, of all lines of equal length, the circumference of a circle is that which encloses the maximum surface area, and of all bodies of equal surface area, the sphere is likewise that which contains the maximum volume, and this is one of the reasons why these figures were regarded as the most perfect. But if in this respect there is a maximum, there is nonetheless no minimum, that
is, there exist no figures enclosing a surface area or a volume less than all others, and this is why Leibnitz was led to think that, although there is a 'best of worlds', there is no 'worst of worlds', that is, a world containing less being than any other possible world. Moreover, we know that this notion of the 'best of worlds', like that of 'incomparables', is linked to the well-known comparisons involving the 'garden full of plants' and the 'pond filled with fish', where 'each twig of the plant, each member of the animal, each drop of its humors, is again such a garden or such a pond'; [8] and this naturally brings us to another, related question, that of the 'infinite division of matter'.