THE 'LAW OF CONTINUITY'
Whenever there exists a continuum, we can say with Leibnitz that there is something of the continuous in its nature, or, if one prefer, that there must be a certain 'law of continuity' applying to all that presents the characteristics of the continuous; this is obvious enough, but it by no means follows that such a law must then be applicable to absolutely everything, as he claims, for, if the continuous exists, so does the discontinuous, even in the domain of quantity; [1] number, indeed, is essentially discontinuous, and it is this very discontinuous quantity, and not continuous quantity, that is really the first and fundamental mode of quantity, what one might properly call pure quantity, as we have said elsewhere. [2] Moreover, nothing allows us to suppose a priori that, outside of pure quantity, a continuity of some kind exists everywhere, and, to tell the truth, it would be quite astonishing if, among all possible things, number alone had the property of being essentially discontinuous; but our
intention is not to determine the bounds within which a 'law of continuity' truly is applicable, or what restrictions should be brought to bear on all that goes beyond the domain of quantity understood in its most general sense. We shall limit ourselves to giving one very simple example of discontinuity, taken from the realm of natural phenomena: if it takes a certain amount of force to break a rope, and one applies to the rope a slightly lesser force, what will result is not a partial rupture, that is, the rupture of some part of the strands making up the rope, but merely tension, which is something completely different; if one augments the force in a continuous way, the tension will also increase continuously, but there will come a moment when the rupture will occur, and then, suddenly and as it were instantaneously, there will be an effect of quite another nature than the preceding, which manifestly implies a discontinuity; and thus it is not true to say, in completely general terms and without any sort of restriction, that natura non facit saltus [nature does not make leaps].
However that may be, it is at any rate sufficient that geometric magnitudes should be continuous, as indeed they are, in order that one always be able to take from them elements as small as one likes, hence elements that are capable of becoming smaller than any assignable magnitude; and as Leibnitz said, 'a rigorous demonstration of the infinitesimal calculus no doubt consists in this,' which applies precisely to these geometric magnitudes. The 'law of continuity' can thus serve as the fundamentum in re of these fictions that are the infinitesimal quantities, and, moreover, as well as the other fictions of imaginary roots (since Leibnitz linked the two in this respect), but for all that without it being necessary to see in it 'the touchstone of all truth', as he would perhaps have wished. Furthermore, even if one does admit a 'law of continuity', though of course still maintaining certain restrictions as to its range, and even if one recognizes that this law can serve to justify the foundation of the infinitesimal calculus, modo sano sensu intelligantur, it by no means follows that one must conceive of it exactly as Leibnitz did, or that one must accept all the consequences he attempted to draw from it; it is this conception and these consequences that we must now examine a little more closely.
In its most general form, this law finally amounts to the following, which Leibnitz stated on many occasions in different terms, but always with fundamentally the same meaning: whenever there is a certain order to principles understood here in the relative sense of whatever is taken as starting-point, there must always be a corresponding order to the consequences drawn from them. As we have already pointed out, this is then only a particular case of the 'law of justice', or of order, which postulates 'universal intelligibility'. For Leibnitz it is therefore fundamentally a consequence or application of the 'principle of sufficient reason', if not this principle itself insofar as it applies more particularly to combinations and variations of quantity. As he says, 'continuity is an ideal thing [which is moreover far from as clear a statement as one might desire], but the real is nevertheless governed by the ideal or abstract . . . because all is governed by reason. [3] There is assuredly a certain order in things, which is not in question, but this order can be conceived of quite differently from the manner of Leibnitz, whose ideas in this regard were always influenced more or less directly by his so-called 'principle of the best', which loses all meaning as soon as one has understood the metaphysical identity of the possible with the real; [4] what is more, although he was a declared adversary of narrow Cartesian rationalism, when it comes to his conception of 'universal intelligibility', one could reproach him for having too readily confused 'intelligible' with 'rational'; but we shall not dwell further on these considerations of a general order, for they would lead us far afield from our subject. In this connection we will only add that one might well be astonished that, after having affirmed that 'mathematical analysis need not depend on metaphysical controversies'-which is quite contestable, moreover, since it amounts to making of mathematics a science entirely ignorant of its own principles, in accordance with the purely profane point of view; besides, incomprehension alone can give birth to controversies in the metaphysical domain-after such an assertion Leibnitz himself, in support of his 'law of causality', to which he links this mathematical analysis, finally comes to
invoke an argument no longer metaphysical indeed, but definitely theological, which could in turn lead to many other controversies. 'It is because all is governed by reason,' he says, 'and because otherwise there would be neither science nor rules, which would not conform to the nature of the sovereign principle, [5] to which one could respond that in reality reason is only a purely human faculty, of an individual order, and that, even without having to go back to the 'sovereign principle', intelligence understood in its universal sense, that is, as the pure and transcendent intellect, is something completely different from reason, and cannot be likened to it in any way, such that if it is true that nothing is 'irrational', there are nevertheless many things that are 'supra-rational', but which on that account are no less 'intelligible'.
Let us now move on to a more precise statement of the 'law of continuity', a statement that relates more directly to the principles of the infinitesimal calculus than the preceding: 'If with respect to its data one case approaches another in a continuous fashion and finally disappears into it, it necessarily follows that the results of the cases equally approach in a continuous fashion their sought-out solutions, and that they must finally terminate in one another reciprocally. [6] There are two things here, which it is important to distinguish: first, if the difference between the two cases diminishes to the point of becoming less than any assignable magnitude in datis [in the given], the same must hold in quaesitis [in what is sought];
this, in short, is only an application of the more general statement, and this part of the law raises no objections as soon as it is admitted that continuous variations exist and that the infinitesimal calculus is properly linked precisely to the domain in which such variations are effected, namely the geometric domain, but must it be further admitted that casus in casum tandem evanescat [one case finally disappears into the other], and that consequently eventus casuum tandem in se invicem desinant [the outcomes of the cases finally end in each other]? In other words, will the difference between the two cases ever become rigorously null, in consequence of their continuous and indefinite decrease, or again, if one prefer, will their decrease, though indefinite, ever come to an end? This is fundamentally the question of knowing whether, within a continuous variation, the limit can be reached, and on this point we will first of all make this remark: as the indefinite always includes in a certain sense something of the inexhaustible, insofar as it is implied by the continuous, and as Leibnitz moreover did not suppose that the division of the continuous could ever arrive at a final term, nor even that this term could really exist, is it completely logical and coherent on his part to maintain at the same time that a continuous variation, which is effected per infinitos gradus intermedios [by infinite intermediary steps], [7] could reach its limit? This is certainly not to say that such a limit can in no way be reached, which would reduce the infinitesimal calculus to no more than a simple method of approximation; but if it is effectively reached, this must not be within the continuous variation itself, nor as a final term in the indefinite sequence of gradus mutationis [degrees of change]. Nevertheless, it is by this 'law of continuity' that Leibnitz claims to justify the 'passage to the limit', which is not the least of the difficulties to which his method gives rise from the logical point of view, and it is precisely here that his conclusions become completely unacceptable; but to make this aspect of the question entirely understandable, we must begin by clarifying the mathematical notion of the limit itself.