6 'WELL-FOUNDED FICTIONS'

The thought most characteristic of Leibnitz, although he does not always affirm it with the same force, and on which he sometimes even seems, albeit exceptionally, not to wish to deliver a categorical verdict, is that basically infinite and infinitely small quantities are only fictions; but, he adds, they are 'well-founded fictions', and by this he does not simply mean that they are useful for calculation, [1] or even for 'finding real truths', although sometimes he does also insist on this usefulness; but he constantly repeats that these fictions are 'founded in reality', that they are fundamentum in re, which obviously implies something of a more than purely utilitarian value; and for him this value itself must after all be explained by the basis these fictions have in reality. In any case, he believes that for the method to be reliable, it suffices to envisage, not infinite and infinitely small quantities in the rigorous sense of these expressions, since this would have no corresponding reality, but simply quantities as great or as small as one likes, or as is necessary in order for the error to be rendered less than any given quantity. It is still necessary to examine whether it is true that, as he declares, this error is thereby null, that is, whether this manner of envisaging the infinitesimal calculus gives him a perfectly rigorous foundation, but we shall have to return to this question later. However it might be with respect to this last point, for him statements concerned with the infinite and infinitely small quantities fall under the category of assertions that according to him are only toleranter verae [reasonably true], or 'tolerable', and must be 'redressed' by an explanation, as when one regards negative quantities as 'less than zero', or as in a number of other cases in which the language of geometry implies 'a certain figurative and cryptic manner of speaking'; [2] the word 'cryptic' would seem to be an allusion to the symbolic and profound meaning of geometry, but this is not at all what Leibnitz had in mind, and perhaps as is so often the case with him in so speaking he had only the memory of some esoteric notion, more or less poorly understood. As for the sense in which one should understand the statement that infinitesimal quantities are 'well-founded fictions', Leibnitz declared that 'the infinites and infinitely small are founded in such a way that within the realm of geometry, and even in nature, they may be treated as if they were perfectly real'; [3] indeed, for him, everything that exists in nature in some way implies the consideration of the infinite, or at least of what he believed could be called such. As he said, 'the perfection of the analysis of transcendentals, or of geometry involving the consideration of some infinite would without doubt be all the more important on account of the applications one can make of it to the operations of nature, which introduces the infinite in all that it does'; [4] but perhaps this is only because we cannot have adequate ideas of it, and because it always introduces elements we cannot perceive with complete distinctness. If this is so, then it is necessary not to take too literally such assertions as the following for example: 'Since our method is properly that part of general mathematics that treats of the infinite, one has great need of it in applying mathematics to physics, for as a rule the character of the infinite Author enters into the operations of nature. [5] But if by this even Leibnitz only means that the complexity of natural things goes incomparably beyond the limits of distinct perception, it nonetheless remains that the infinite and infinitely small quantities must have their fundamentum in re; and this foundation is found in the nature of things, at least as conceived by him, and is none other than what he calls the 'law of continuity', which we shall have to examine a little later, and which he regards, rightly or wrongly, as being in short only a particular case of a certain 'law of justice', which is itself ultimately connected to the idea of order and harmony, and which equally finds its application every time a certain symmetry must be observed, as, for example, in the case of combinations and permutations. Now, if the infinite and infinitely small quantities are only fictions, and even admitting that they really are 'well-founded', one might ask oneself this: why use such expressions, which, even if they can be regarded as toleranter verae, are nonetheless incorrect? Here is something which presages, one might say, the 'conventionalism' of modern science, though with the notable difference that the latter is no longer in any way preoccupied with knowing whether the fictions to which it has recourse are 'well-founded' or not, or, according to another expression of Leibnitz, whether they can be interpreted sano sensu [in a reasonable way], or even whether they have any meaning at all. Moreover, since one can do without these fictional quantities and be content with envisaging in their place quantities that can simply be rendered as great or as small as one likes, and which, for that reason, can be said to be indefinitely great or indefinitely small, it would no doubt have been better to do so from the start and thus avoid introducing fictions that, whatever might be their fundamentum in re, are, ultimately, of no practical use, not only with regard to calculation, but even regarding the infinitesimal method itself. The expressions 'indefinitely great' and 'indefinitely small', or what amounts to the same but is perhaps more precise, 'indefinitely increasing' and 'indefinitely decreasing', not only have the advantage of being the only ones that are rigorously exact; they also show clearly that the quantities to which they are applied can only be variable, and not determined, quantities. As a mathematician has rightly said, 'the infinitely small is not a very small quantity, having an actual value capable of being determined; its character is to be eminently variable, and to be able to take on a value less than that of any other one might wish to specify; it would be much better to call them indefinitely small. [6] The use of these terms would have prevented many difficulties and disputes, and there is nothing surprising about this, since it is not a simple question of words, but the replacement of a false idea with a true one, of a fiction with a reality; notably, it would have prevented anyone from taking the infinitesimal quantities to be fixed and determined quantities, for as we said above the word 'indefinite' always carries with it the idea of 'becoming', and consequently of change, or, when it is a question of quantities, of variability; and, had Leibnitz made a habit of using these terms, he would doubtless not have allowed himself to be so easily drawn into the unfortunate comparison concerning the grain of sand. What is more, reducing infinite parva ad indefinite parva [the infinitely small to the indefinitely small] would at any rate have been clearer than reducing them ad incomparabiliter parva [to the incomparably small]; precision would thereby have been gained without any loss of exactitude-quite the contrary. Infinitesimal quantities assuredly are 'not comparable' to ordinary quantities, but this can be understood in more than one way, and indeed it has often enough been taken in other senses than were intended. It is better to say that they are 'unassignable', to use another expression of Leibnitz, for it seems that this term can be rigorously understood only of quantities that are capable of becoming as small as one likes, that is, smaller than any given quantity, and consequently to which one can by no means 'assign' a determined value, however small it might be, and this is indeed the sense of indefinite parva. Unfortunately, it is next to impossible to know whether, in Leibnitz's thought, 'incomparable' and 'unassignable' are truly and completely synonymous; but in any case, it is at the very least certain that a truly 'unassignable' quantity, by reason of the possibility of indefinite decrease that it implies, will thereby be 'incomparable' with respect to any given quantity, and, to extend the idea to different orders of the infinitesimal, even with respect to any quantity in relation to which it can decrease indefinitely, as long as the latter is regarded as possessing at least a relative fixity. If there is one point on which everyone can easily agree, even without going more deeply into questions of principles, it is that the notion of the indefinitely small, at least from the purely mathematical point of view, is perfectly sufficient for infinitesimal analysis, and the 'infinitists' themselves recognize this without great difficulty. [7] In this respect one can thus be content with a definition such as that given by Carnot: 'What is an infinitely small quantity in mathematics? Nothing other than a quantity that can be rendered as small as one likes, without one's being obliged on that account to vary those to which one wants to relate it. [8] But as for the true significance of infinitesimal quantities, the entire matter is not limited to this; for the calculus it matters little that the infinitely small are only fictions, since one can be content with a consideration of the indefinitely small, which raises no logical difficulty; furthermore, since for the metaphysical reasons set out at the beginning we cannot admit a quantitative infinite, whether infinitely great or infinitely small, [9] or indeed any infinite of a determined and relative order, it is quite certain that these can only be fictions and nothing else; but if rightly or wrongly these fictions were introduced into the infinitesimal calculus in the beginning, this is because according to Leibnitz's intention they nevertheless correspond to something, however faulty the manner in which they expressed it. Since we are here concerned with principles and not merely with a method of calculation in and of itself (which would be without interest for us) we should therefore ask what exactly is the value of these fictions, not only from the logical point of view, but also from the ontological point of view, whether they are as 'well-founded' as Leibnitz believed, and whether we can even say with him that they are toleranter verae, and at the very least accept them as such modo sano sensu intelligantur [understood in a reasonable way]. To answer these questions it will be necessary for us to examine more closely his conception of the 'law of continuity', for it was here that he thought to find the fundamentum in re of the infinitely small.