PREFACE
Although the present study might, at least at first glance, appear to have only a rather 'specialist' character, the undertaking seemed worthwhile in order to clarify and explain more thoroughly various notions to which we have had recourse on various occasions when we have made use of mathematical symbolism, and this reason alone would suffice to justify it. However, we should add that there are still other, secondary reasons, concerning especially what one could call the 'historical' aspect of the question; the latter, indeed, is not entirely devoid of interest from our point of view inasmuch as all the discussions that have arisen on the subject of the nature and value of the infinitesimal calculus offer a striking example of that absence of principles which characterizes the profane sciences, that is, the only sciences that the moderns know and even consider possible. We have already often noted that most of these sciences, even insofar as they still correspond to some reality, represent no more than simple, debased residues of some of the ancient, traditional sciences: the lowest part of these sciences, having ceased to have contact with the principles, and having thereby lost its true, original significance, eventually underwent an independent development and came to be regarded as knowledge sufficient unto itself, although in truth it so happens that its own value as knowledge is thereby reduced to almost nothing. This is especially apparent with the physical sciences, but as we have explained elsewhere, [1] in this respect modern mathematics itself is no exception if one compares it to what was for the ancients the science of numbers and geometry; and when we speak here of the ancients one must understand by that even those of 'classical' antiquity, as the least study of Pythagorean and Platonic theories suffices to show, or at least should show were it not necessary to take into account the extraordinary
incomprehension of those who claim to interpret them today. Were this incomprehension not so complete, how could one maintain, for example, a belief in the 'empirical' origin of the sciences in question? For in reality they appear on the contrary all the more removed from any 'empiricism' the further back one goes in time, and this is equally the case for all other branches of scientific knowledge. [2]
Mathematicians of modern times, and more particularly still those who are our contemporaries, seem to be ignorant of what number truly is; and by this we do not mean to speak solely of number in the analogical and symbolic sense as understood by the Pythagoreans and Kabbalists, which is all too obvious, but-and this might seem stranger and almost paradoxical-even of number in its simply and strictly quantitative sense. Indeed, their entire science is reduced to calculation in the narrowest sense of the word, [3] that is, to a mere collection of more or less artificial procedures, which are in short only valuable with respect to the practical applications to which they give rise. Basically this amounts to saying that they replace number with the numeral; and furthermore, this confusion of the two is today so widespread that one could easily find it at any moment, even in the expressions of everyday language. [4] Now a numeral is, strictly speaking, no more than the clothing of a number; we do not even say its body, for it is rather the geometric form that can, in certain respects, legitimately be considered to constitute the true body of a number, as the theories of the ancients on polygons and polyhedrons show when seen in the light of the symbolism of numbers; and this, moreover, is in accordance with the fact that all 'embodiment' necessarily implies a 'spatialization'. We do not mean to say, however, that numerals themselves are entirely arbitrary signs, the form of which has been determined only by the fancy of one or more individuals; there must be both numerical
and alphabetical characters-the two of which, moreover, are not distinguished in some languages [5] -and one can apply to the one as well as to the other the notion of a hieroglyphic, that is to say an ideographic or symbolic origin, and this holds for all writing without exception, however obscured this origin might be in some cases due to more or less recent distortions or alterations.
What is certain is that in their notation mathematicians employ symbols the meaning of which they no longer understand, and which are like vestiges of forgotten traditions; and what is more serious, not only do they not ask themselves what this meaning might be, it even seems that they do not want them to have any meaning at all. Indeed, they tend more and more to regard all notation as simple 'convention', by which they mean something set out in an entirely arbitrary manner, but this is a true impossibility, for one never establishes a convention without having some reason for doing so, and for doing precisely that rather than anything else; it is only to those who ignore this reason that the convention can appear as arbitrary, just as it is only to those who ignore the cause of an event that it can appear 'fortuitous'. This is indeed what occurs here, and one can see in it one of the more extreme consequences of the absence of principles, which can even cause the science-or what is so called, for at this point it no longer merits the name in any respect-to lose all plausible significance. Moreover, by the very fact of the current conception of science as exclusively quantitative, this 'conventionalism' has gradually spread from mathematics to the more recent theories of the physical sciences, which thus distance themselves further and further from the reality they intend to
explain; we have emphasized this point sufficiently enough in another work to be able to dispense with further remarks in this regard, and all the more so since we now intend to occupy ourselves more particularly with mathematics alone. From this viewpoint we will only add that when one completely loses sight of the meaning of a notation it becomes all too easy to pass from a legitimate and valid use of it to one that is illegitimate and in fact no longer corresponds to anything, and which can sometimes even be entirely illogical. This may seem rather extraordinary when it is a question of a science like mathematics which should have particularly close ties with logic, yet it is nevertheless all too true that one can find multiple illogicalities in mathematical notions as they are commonly envisaged in our day.
One of the most remarkable examples of these illogical notions, and which we shall consider first and foremost, even though it is certainly not the only one we shall encounter in the course of our exposition, is that of the so-called mathematical or quantitative infinite, which is the source of almost all the difficulties that can be raised against the infinitesimal calculus, or, perhaps more precisely, against the infinitesimal method, for we here have something that, whatever the 'conventionalists' might think, goes beyond the range of a simple 'calculation' in the ordinary sense of the word; and this notion is the source of all difficulties without exception, save those that proceed from an erroneous or insufficient conception of the notion of the 'limit', which is indispensable if the rigor of the infinitesimal method is to be justified and made anything more than a simple method of approximation. As we shall see, moreover, there is a distinction to be made between cases in which the so-called infinite is only an absurdity pure and simple, that is, an idea contradictory in itself, such as that of an 'infinite number', and cases in which it is only employed in an improper way in the sense of indefinite; but it should not be believed because of this that the confusion of the infinite and the indefinite can itself be reduced to a mere question of words, for it rests quite truly with the ideas themselves. What is singular is that this confusion, which had it once been dispelled would have cut short so many discussions, is found in the writings of Leibnitz himself, who is generally regarded as the inventor of the
infinitesimal calculus, although we would rather call him its 'formulator', for his method corresponds to certain realities that, as such, have an existence independent of those who conceive of them and who express them more or less perfectly; realities of the mathematical order, like all other realities, can only be discovered and not invented, while on the contrary it is indeed a question of 'invention' when, as occurs all too often in this field, one allows oneself to be swept away by the 'game' of notation into the realm of pure fantasy. But it would assuredly be quite difficult to make some mathematicians understand this difference, since they willingly imagine that the whole of their science is and must be no more than a fabrication of the human mind', which, if we had to believe them, would certainly reduce their science to a trifling thing indeed. Be that as it may, Leibnitz was never able to explain the principles of his calculus clearly, and this shows that there was something in it that was beyond him, something that was as it were imposed upon him without his being conscious of it; had he taken this into account, he most certainly would not have engaged in any dispute over 'priority' with Newton. Besides, these sorts of disputes are always completely vain, for ideas, insofar as they are true, are not the property of anyone, despite what modern 'individualism' might have to say; it is only error that can properly be attributed to human individuals. We shall not elaborate further on this question, which could take us quite far from the object of our study, although in certain respects it would perhaps not be profitless to make it clear that the role of those who are called 'great men' is to a great extent often a role of 'reception', though they are generally the first to delude themselves as to their own 'originality'.
What concerns us more directly for the moment is this: if we must point out such deficiencies in Leibnitz-deficiencies all the more serious in that they bear above all on questions of principleswhat could be said of those found in other modern philosophers and mathematicians, to whom Leibnitz is certainly superior in spite of everything? This superiority he owes on the one hand to the studies he made of the Scholastic doctrines of the Middle Ages, even though he did not always fully understand them, and on the other hand to certain esoteric data, principally of a Rosicrucian origin or
inspiration, [6] data obviously very incomplete and even fragmentary, which he moreover sometimes applied quite poorly, as we shall presently see in some examples. It is to these two 'sources', to speak as the historians do, that one can definitively relate nearly all that is really valid in his theories, and this also allowed him to react, albeit imperfectly, against the Cartesianism which, in the double domain of philosophy and science, represented the whole ensemble of the tendencies and conceptions that are most specifically modern. This remark suffices, in short, to explain in a few words all that Leibnitz was, and if one seeks to understand him, one must never lose sight of this general information, which we have for this reason deemed worthwhile to set forth at the outset; but it is time to leave these preliminary considerations in order to enter into the examination of the very questions that will allow us to determine the true significance of the infinitesimal calculus.