INFINITE AND INDEFINITE

Proceeding in a manner inverse to that of profane science, and in accordance with the unchanging perspective of all traditional science, we must before all else set forth the principle that will allow us almost immediately to resolve the difficulties to which the infinitesimal method has given rise, without letting ourselves be led astray by potentially interminable discussions, as indeed happens in the case of those modern philosophers and mathematicians who, by the very fact that they lack this principle, have never provided a satisfactory and definitive solution to these difficulties. This principle is the very idea of the Infinite, understood in its only true sense, which is the purely metaphysical sense, and on this subject, moreover, we have only summarily to recall what we have already expressed more completely elsewhere: [1] the Infinite is properly that which has no limits, for 'finite' is obviously synonymous with 'limited'; one cannot then correctly apply this term to anything other than that which has absolutely no limits, that is to say the universal All, which includes in itself all possibilities and consequently cannot be limited by anything in any way; the Infinite, thus understood, is metaphysically and logically necessary, for not only does it not imply any contradiction, not enclosing within itself anything negative, but it is on the contrary its negation that would be contradictory. Furthermore, there can obviously be only one Infinite, for two supposedly distinct infinites would limit and therefore inevitably exclude one another; consequently, every time the term 'infinite' is used in any sense other than that which we have just mentioned, we can be assured a priori that this use is necessarily improper, for it amounts in short either to ignoring the metaphysical Infinite altogether, or to supposing another Infinite alongside it. It is true that the Scholastics admitted what they called the infinitum secundum quid [the infinite in a certain respect], and that they carefully distinguished it from the infinitum absolutum [the absolute infinite], which alone is the metaphysical Infinite; but we can see here only an imperfection in their terminology, for although this distinction allowed them to escape the contradiction of a plurality of infinites understood in the proper sense, the double use of the word infinitum nonetheless certainly risked causing multiple confusions, and besides, one of the two meanings was then altogether improper, for to say that something is infinite only in a certain respect-and this is the exact significance of the expression infinitum secundum quid-is to say that in reality it is not infinite at all. [2] Indeed, it is not because a thing is not limited in a certain sense or in a certain respect that one can legitimately conclude that it is limited in no way at all, the latter being necessary for it to be truly infinite; not only can it be limited in other respects at the same time, but we can even say that it is of necessity so, inasmuch as it is a determined thing, which, by its very determination, does not include every possibility, and this amounts to saying that it is limited by that which lies outside of it; if, on the contrary, the universal All is infinite, this is precisely because there is nothing that lies outside of it. [3] Therefore every determination, however general one supposes it to be and however far one extends the term, necessarily excludes the true notion of the infinite; [4] a determination, whatever it might be, is always a limitation, since its essential character is to define a certain domain of possibilities in relation to all the rest, and by that very fact to exclude all the rest. Thus it is truly 'nonsense' to apply the idea of the infinite to any given determination, as for example, in the instance we are considering more particularly here, to quantity or to one or another of its modes. The idea of a 'determined infinite' is too manifestly contradictory for us to dwell upon any longer, although this contradiction has most often escaped the profane thought of the moderns; and even those whom one might call 'semi-profane, [5] like Leibnitz, were unable to perceive it clearly. In order to bring out the contradiction still further we could say in other fundamentally equivalent terms that it is obviously absurd to wish to define the Infinite, since a definition is in fact nothing other than the expression of a determination, and the words themselves show clearly enough that what is subject to definition can only be finite or limited. To seek to place the Infinite within a formula, or, if one prefer, to clothe it in any form whatsoever is, consciously or unconsciously, to attempt to fit the universal All into one of its minutest parts, and this is assuredly the most manifest of impossibilities. What we have just said suffices to establish, without leaving room for the slightest doubt and without necessitating any other considerations that there cannot be a mathematical or quantitative infinite, and that this expression does not even have any meaning, because quantity is itself a determination. Number, space, and time, to which some people wish to apply the notion of this so-called infinite, are determined conditions, and as such can only be finite; they are but certain possibilities, or certain sets of possibilities, beside and outside of which there exist others, and this obviously implies their limitation. In this instance still more can be said: to conceive of the Infinite quantitatively is not only to limit it, but in addition it is to conceive of it as subject to increase and decrease, which is no less absurd; with similar considerations one quickly finds oneself envisaging not only several infinites that coexist without confounding or excluding one another, but also infinites that are larger or smaller than others; and finally, the infinite having become so relative under these conditions that it no longer suffices, the 'transfinite' is invented, that is, the domain of quantities greater than the infinite. Here, indeed, it is properly a matter of 'invention', for such conceptions correspond to no reality. So many words, so many absurdities, even regarding simple, elementary logic, yet this does not prevent one from finding among those responsible some who even claim to be 'specialists' in logic, so great is the intellectual confusion of our times! We should point out that just now we did not merely say 'to conceive of a quantitative infinite', but 'to conceive of the Infinite quantitatively', and this calls for a few words of explanation. By this expression we wanted to allude more particularly to those who are called 'infinitists' in contemporary philosophical jargon; indeed, all the discussions between 'finitists' and 'infinitists' clearly show that at least both have in common this completely false idea that the metaphysical Infinite is akin to the mathematical infinite, if they do not purely and simply identify the two. [6] Thus they all equally ignore the most elementary principles of metaphysics, since it is on the contrary precisely the conception of the true, metaphysical Infinite that alone allows us to reject absolutely every 'particular infinite', if one may so express it, such as the so-called quantitative infinite, and to be assured in advance that, wherever it is encountered, it can only be an illusion; we shall then only need to ask what could have brought about this illusion in order to be able to replace it with a notion closer to the truth. In short, every time it is a question of a particular thing, of a determined possibility, we can be certain a priori that it is limited by that very fact, and, we can say, limited by its very nature, and this holds equally true in the case where, for whatever reason, we cannot actually reach its limits; but it is precisely this impossibility of reaching the limits of certain things, and sometimes even of conceiving of them clearly, that causes the illusion that these things have no limits, at least among those for whom the metaphysical principle is lacking; and, let us say it again, it is this illusion and nothing more that is expressed in the contradictory assertion of a 'determined infinite'. In order to rectify this false notion, or rather to replace it with a true conception of things, [7] we must here introduce the idea of the indefinite, which is precisely the idea of a development of possibilities the limits of which we cannot actually reach; and this is why we regard the distinction between the Infinite and the indefinite as fundamental to all questions in which the so-called mathematical infinite appears. Without doubt this is what corresponds in the intention of its authors to the Scholastic distinction between the infinitum absolutum and the infinitum secundum quid. It is certainly unfortunate that Leibnitz, who had borrowed so much from Scholasticism, had neglected or not been aware of this, for however imperfect the form in which it was expressed, it would have allowed him to respond quite easily to certain objections raised against his method. In contrast to this, it seems that Descartes had indeed tried to establish the distinction in question, but he was very far from having expressed or even conceived of it with sufficient precision, since according to him the indefinite is that of which we do not perceive the limits, and which in reality could be infinite, although we could not affirm it to be so, whereas the truth is that we can on the contrary affirm that it is not so and that it is by no means necessary to perceive its limits in order to be certain that they exist. One can thus see how vague and confused are all such explanations, and always as a result of the same lack of principle. Descartes indeed said: 'And for us, seeing things in which, in a certain sense, [8] we note no limits, we cannot ascertain thereby that they are infinite, but we shall only consider them to be indefinite. [9] And he gives as examples the extension and divisibility of bodies; he does not contend that these things are infinite, but he does not seem to want to deny it formally either, and all the more so since he had just declared that he did not wish to 'entangle himself in disputes over the infinite,' which is rather too easy a way to brush aside the difficulties, even if he does say a little later that 'although we shall observe properties that seem to us not to have any limits, we do not fail to recognize that this proceeds from our lack of understanding and not from their nature. [10] In short, he wishes with good reason to reserve the name infinite for what has no limits; but on the one hand he appears not to know with the absolute certitude that is implied in all metaphysical knowledge, that what has no limits cannot be anything but the universal All, and on the other hand the very notion of the indefinite needs to be much more precise; had it been so, a great number of subsequent confusions would doubtless not have been as readily produced. [11] We say that the indefinite cannot be infinite because it always implies a certain determination, whether it is a question of extension, duration, divisibility, or some other possibility; in a word, whatever the indefinite may be, and according to whatever aspect it is considered, it is still of the finite and can only be of the finite. No doubt, its limits may be extended until they are found to be out of our reach, at least insofar as we seek to reach them in a certain manner that we can call 'analytical', as we shall explain more thoroughly in what follows; but they are by no means abolished thereby, and in any case, if limitations of a certain order can be abolished, others possessing the same nature as the first will still remain, for it is by virtue of its nature, and not simply by some more or less exterior or accidental circumstances, that every particular thing is finite, whatever the degree to which certain limits can be extended. In this regard one might point out that the sign ∞, by which mathematicians represent their so-called infinite, is itself a closed figure, therefore visibly finite, just like the circle, which some people have wished to make a symbol of eternity, while it can in fact only be a figure of a temporal cycle, indefinite merely in its order, that is to say, of what is properly called perpetuity; [12] and it is easy to see that this confusion of eternity with perpetuity, so common among modern Westerners, is closely related to that of the Infinite and the indefinite. In order to better understand the idea of the indefinite and the manner in which it is formed from the finite taken in its ordinary sense, one can consider an example such as that of the sequence of numbers: here, it is obviously never possible to stop at a determined point, since after every number there is always another that can be obtained by adding a unit; consequently, the limitation of this indefinite sequence must be of an order other than that which applies to a definite set of numbers taken between any two determined numbers; it must derive not from particular properties of certain numbers, but rather from the very nature of number in all its generality, that is to say from the determination that, essentially constituting this nature, makes number at once what it is and not anything else. One could make exactly the same observation if it were no longer a question of number but of space or time likewise considered in every possible extension to which they are subject. [13] Any such extension, as indefinite as one conceives it to be and as it in fact is, will never in any way take us out of the finite. Indeed, whereas the finite necessarily presupposes the Infinite-since the latter is that which comprehends and envelops all possibilities-the indefinite on the contrary proceeds from the finite, of which it is in reality only a development and to which it is consequently always reducible, for it is obvious that whatever process one might apply, one cannot derive from the finite either anything more or anything other than that which was already potentially contained therein. To take again the example of the sequence of numbers, we can say that this sequence, with all the indefinitude it implies, is given to us by its law of formation, since it is from this very law that its indefinitude immediately results; now this law consists in the following, that given any number, one can form the next by adding a unit. The sequence of numbers is therefore formed by successive additions of the unit to itself, indefinitely repeated, which is basically only the indefinite extension of the process of formation for any arithmetical sum; and here one can see quite clearly how the indefinite is formed starting from the finite. This example, moreover, owes its particular clarity to the discontinuous character of numerical quantity; but, to take things in a more general fashion applicable to all cases, it would suffice to insist on the idea of 'becoming' that is implied by the term 'indefinite', and this we expressed above in speaking of the development of possibilities, a development that in itself and in its whole course always consists of something unfinished; [14] the importance of the consideration of 'variables' as they concern the infinitesimal calculus will give to this last point its full significance.