2 ON MATHEMATICAL NOTATION

WE have often had occasion to remark that in reality most of the profane sciences—the only sciences the moderns know or even consider possible—represent only simple, distorted residues of the ancient, traditional sciences in the sense that the lowest part of these sciences, having ceased to have contact with the principles, and having thereby lost its true, original significance, ended up undergoing an independent development and came to be regarded as a branch of knowledge sufficient unto itself. In this respect, modern mathematics 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, it is necessary to include therein even those of 'classical' antiquity, as the least study of Pythagorean and Platonic theories suffices to show, or at least should 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 for example could one maintain a belief in the 'empirical' origin of the sciences in question? For in reality—and to the contrary—they appear all the more removed from any 'empiricism' the further back one goes in time, and this is moreover equally the case for all other branches of scientific knowledge. Modern mathematicians seem to have become ignorant of what number truly is, for they reduce their entire science to calculation, which for them means a mere collection of more or less artificial processes, and this amounts to saying, in short, that they replace number with the numeral; moreover, this confusion between the two is today so widespread that it can be found everywhere, even in everyday language. 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 in certain respects, can 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. 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 not being distinguished in some languages moreover) and the notion of a hieroglyphic, that is, an ideographic or symbolic origin, can be applied to the one as well as to the other, and this holds for all scripts without exception. What is certain is that mathematicians employ in their notation 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, they even seem not to want them to have any at all. Indeed, they tend more and more to regard all notation as mere 'convention', by which they mean something set out in an entirely arbitrary manner, which in reality is a veritable impossibility, for one never establishes a convention without having some reason for doing so, and for doing precisely that rather than anything else. A convention can appear arbitrary only to those who are ignorant of this reason, and this is exactly what happens in this instance. Likewise, it is all too easy to pass from a legitimate and valid use of a notation to an illegitimate use that no longer corresponds to anything real, and that can even sometimes be completely illogical; this may seem strange when it is a question of a science like mathematics, which should have a particularly close relationship with logic, yet it is nevertheless all too true that one can find many illogicalities in mathematical notions as they are commonly understood. One of the most striking examples of these illogical notions is that of the so-called mathematical infinite, which, as we have amply explained on other occasions, can in reality be no more than the indefinite—and let it not be believed that this confusion of the infinite and the indefinite can be reduced to a mere question of words.

What mathematicians represent by the sign ∞ can in no way be the Infinite understood in its true sense; the sign ∞ is itself a closed figure, therefore visibly finite, just like the circle, which some people have wished to make a symbol of eternity. In fact, the circle can only be a representation of a temporal cycle, indefinite merely in its order, that is to say, of what is properly called perpetuity; and it is easy to see that this confusion of eternity with perpetuity corresponds exactly to that of the infinite with the indefinite. In fact, the indefinite is only a development of the finite; but the Infinite cannot be derived from the finite. Furthermore, the Infinite is no more quantitative than it is determined, for quantity, being only a special mode of reality, is thereby essentially limited. What is more, the idea of an infinite number, that is to say a number greater than all other numbers according to the definition given by mathematicians, is an idea contradictory in itself, for however great a number n might be, the number n + 1 will always be greater in virtue of the law of formation for the indefinite sequence of numbers. This contradiction leads to many others, as various philosophers have noted, although they never saw the full import of this argument, for they believed they could apply to the metaphysical Infinite what applies only to the false mathematical infinite, and thus they fell prey to the same confusion as their adversaries, only in an opposite direction. It is obviously absurd to wish to define the Infinite, for every definition is necessarily a limitation, as the words themselves show clearly enough, and the Infinite is that which has no limits; to seek to place it within a formula, or, in short, to clothe it in a form, is to attempt to place the universal All within one of its minutest parts, which is manifestly impossible. Finally, to conceive of the Infinite as a quantity is not only to limit it, as we have just said, 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 several infinites that coexist without confounding or excluding one another, as well as infinites greater or smaller than other infinites; and, the infinite no longer sufficing, one even invents the 'transfinite', that is, the domain of quantities greater than the infinite: so many words and so many absurdities, even with regard to simple, elementary logic. Here we intentionally speak of 'invention', for if the realities of the mathematical order, like all other realities, can only be discovered and not invented, it is clear that this is no longer the case when, by a 'game' of notation, one allows oneself to be led into the domain of pure fantasy; but how could one hope for mathematicians to understand this difference when they willingly imagine that the whole of their science is and must be no more than a 'construction of the human mind', although if this were true it would of course reduce their science to a mere trifle? What we said concerning the infinitely great, or what is so called, is equally true of what is no less improperly called the infinitely small: however small a number 1/n might be, the number 1/n + 1 will be smaller still; later we shall return to the question of what exactly this notation should be taken to mean. In reality, there is thus neither an infinitely great nor an infinitely small; but one can envisage the sequence of numbers as increasing and decreasing indefinitely in such a way that the so-called mathematical infinite will only be the indefinite, which, let us say again, proceeds from the finite, and is consequently always reducible to it. The indefinite is thus still finite, which is to say limited; even if we do not know its limits, or are incapable of determining them, we do know that they exist, for every indefinitude pertains only to a certain order of things, limited precisely by the existence of other things outside of it. By the same token, one can obviously envisage a multitude of indefinites; one can even add them to each other, or multiply them by each other, which naturally leads to the consideration of indefinites of unequal magnitude, and even different orders of infinitude, in both the increasing direction and the decreasing direction. Once this is understood, we shall be able to see the real significance of the previously mentioned absurdities, which disappear as soon as the so-called mathematical infinite is replaced with the indefinite; but whatever might be obtained thus will of course have no relation to the Infinite, and will always be rigorously null with respect to it; and the same may be said of all ordinary finitude, of which the indefinite is necessarily but an extension. At the same time, these considerations also show in a precise way the impossibility of arriving at synthesis by analysis: however much one adds together an indefinite number of elements successively, one will never obtain the All, because the All is infinite, and not indefinite; it cannot be conceived of as other than infinite, for it could only be limited by something outside of itself, and then it would not be the All. If it can be said that it is the sum of all its elements, this is only on the condition that the word 'sum' be taken in the sense of an integral, which is not calculated by taking its elements one by one; and even were one to suppose that one or more indefinite sequences could be passed through analytically, one would not for that have advanced a single step from the point of view of universality, and one would always be at exactly the same point in relation to the Infinite. Moreover, all of this can be applied analogically to other domains than quantity; and the immediate consequence is that profane science, of which the points of view and methods are exclusively analytical, is by that very fact incapable of transcending certain limitations; here the imperfection is not simply inherent in its present state, as some have wished to believe, but in its very nature, that is, ultimately, in its lack of principles. We have said that the sequence of numbers can be considered indefinite in two directions, the increasing and the decreasing; but this demands some further explanation, for an objection can immediately be raised. True number, what one might call pure number, is essentially whole number; and the sequence of whole numbers, starting from the unit, continues ever to increase, but it progresses entirely in a single direction, and thus the other, opposite direction—that of indefinite decrease—cannot be represented by it. However, one is brought to consider various other kinds of number aside from the whole numbers; these, it is usually said, are extensions of the idea of number, and this is true after a certain fashion; but at the same time these extensions are also distortions, which is what mathematicians seem too easily to forget on account of their 'conventionalism', which causes them to misunderstand the origin and raison d'être of these numbers. In fact, numbers other than whole numbers always appear first and foremost as the representation of the results of operations that would be impossible were one to keep to the point of view of pure arithmetic, which, in all strictness, is the arithmetic of whole numbers alone. Indeed, one does not arbitrarily consider the results of the aforementioned operations thus, instead of regarding them purely and simply as impossible; generally speaking, it is in consequence of the application made of number—discontinuous quantity—to the measurement of magnitudes belonging to the order of continuous quantity. Between these modes of quantity there is a difference of nature such that a correspondence between the two cannot be perfectly established; to remedy this to a certain degree, at least insofar as it is possible, one seeks to reduce, as it were, the intervals of this discontinuity constituted by the sequence of whole numbers, by introducing between its terms other numbers, such as fractional and incommensurable numbers, which would be meaningless apart from this consideration. Moreover, it must be said that in spite of this something of the essentially discontinuous nature of number will inevitably always remain, preventing one from thus obtaining a perfect equivalent to the continuous. The intervals can be reduced as much as one might like—that is, in short, they can be reduced indefinitely—but they cannot be eliminated; thus one is once again brought to consider a certain aspect of the indefinite, and this could find its application in a study of the principles of the infinitesimal calculus, although this is not what we propose to do at present. Under these conditions and with these reservations, one can accept certain of these extensions of the idea of number to which we have just alluded, and give them, or rather restore to them, a legitimate significance; thus, notably, we can consider the inverses of the whole numbers represented by symbols of the form 1/n and forming the indefinitely decreasing sequence, symmetrical to the indefinitely increasing sequence of whole numbers. We must further note that although the symbol 1/n could evoke the idea of fractional numbers, the numbers in question here are not defined as such; it suffices for us to consider the two sequences as constituted by numbers respectively greater and smaller than the unit, that is, by two orders of magnitude having their common limit in the unit, while at the same time both can be regarded as having issued from this unit, which is indeed the primary source of all numbers. Since we have spoken of fractional numbers, we should add in this connection that the definition ordinarily given to them is again absurd: in no way can fractions be 'parts of a unit', as is said, for the true unit is necessarily indivisible and without parts; arithmetically, a fractional number represents no more than the quotient of an impossible division; but this absurdity arises from a confusion of the arithmetical unit with what are called 'units of measurement', which are units only by convention, since in reality they are magnitudes of another sort than number. The unit of length, for example, is only a certain length chosen for reasons foreign to arithmetic, to which one makes the number 1 correspond in order to be able to measure all other lengths by reference to it; but by its very nature as continuous magnitude, all length, even when thus represented numerically by unity, is no less always and indefinitely divisible. Comparing it to other lengths, one might therefore have to consider parts of this unit of measurement, without it in any way being necessary that they be parts of the arithmetical unit; and it is only thus that the consideration of fractional numbers is really introduced, as a representation of the ratios of magnitudes that are not exactly divisible by one another. The measurement of a magnitude is in fact no more than the numerical expression of its ratio to another magnitude of the same species taken as the unit of measurement, or, basically, as the term of comparison; and from this one sees that all measurement is essentially founded on division, something which could give rise to further observations which are important, but beyond our present subject. That said, we can now return to the double numerical indefinitude constituted in the increasing direction by the sequence of whole numbers, and in the decreasing direction by that of their inverses; both sequences start from the unit, which alone is its own inverse, since 1/1 = 1. Moreover, there are as many numbers in one sequence as there are in the other, such that if one considers the two indefinite sets as forming a unique sequence, one could say that the unit occupies the exact mid-point within this sequence of numbers; indeed, for every number n in one sequence, there will correspond another number 1/n in the other, such that n x 1/n = 1, any two inverse numbers multiplied together again producing the unit. To generalize further, if we wished to introduce fractional numbers instead of considering only the sequence of whole numbers and their inverses as we have just done, nothing would be changed in this regard: on one side there would be all the numbers greater than the unit, and on the other all those smaller than the unit; here, again, for any number a/b > 1, there will be a corresponding number b/a < 1 in the other set, and reciprocally, such that a/b × b/a = 1, and there will thus be exactly the same number of terms in each of these two indefinite groups separated by the unit. One can say further that the unit, occupying the mid-point, corresponds to the state of perfect equilibrium, and that it contains in itself all numbers, which proceed from it in pairs of inverse or complementary numbers, each pair, by virtue of its complementarity, constituting a relative unity in its indivisible duality. In what follows we shall further examine the consequences implied by these various considerations.

If one considers the sequence of whole numbers together with that of their inverses, in accordance with what was said above, the first will be indefinitely increasing and the second indefinitely decreasing; one could say that the numbers thus tend on the one side toward the indefinitely great and on the other toward the indefinitely small, understanding by this the very limits of the domain in which one considers these numbers, for a variable quantity cannot but tend toward a limit. The domain in question is, in short, that of numerical quantity taken in every possible extension; this amounts to saying that its limits are not determined by such and such a particular number, however great or small one might suppose it to be, but solely by the nature of number as such. By the same token number, like everything else of a determined nature, excludes all that it is not, and thus there can be no question of any infinite here; moreover, we have just said that the indefinitely great must inevitably be conceived of as a limit, and in this connection one can point out that the expression ‘tend toward infinity’, employed by mathematicians in the sense of ‘increase indefinitely’, is again an absurdity, since the infinite obviously implies the absence of all limits, and since consequently there is nothing toward which it is possible to tend. It goes without saying that the same observations can be applied to modes of quantity other than number, that is, to different kinds of continuous quantity, notably the spatial and the temporal; each of these is likewise capable of indefinite extension within its order, but essentially limited by its very nature, as, moreover, is quantity itself in all its generality; the very fact that there exist things to which quantity is not applicable suffices to demonstrate the contradiction in the idea of the so-called ‘quantitative infinite’.

Furthermore, when a domain is indefinite, we cannot know its limits distinctly, and, consequently, we will not be able to fix them in a precise manner; here, in short, we have the entire difference between indefinitude and ordinary finitude. There thus remains a sort of indeterminacy, but one which is such only from our point of view and not in reality itself, since its limits are no less existent on that account; whether we see them or not in no way changes the nature of things. As far as number is concerned, one could also say that this apparent indeterminacy results from the fact that the sequence of numbers in its entirety is not ‘terminated’ by a certain number, as is always the case with any given portion of the sequence considered in isolation; there is thus no number, however great it might be, that can be identified with the indefinitely great in the sense in which we take it; and parallel considerations naturally apply to the indefinitely small. However, one can at least regard a number as practically indefinite, if one may so express it, when it can no longer be expressed by language or represented in writing, which indeed occurs the moment one considers numbers that go on ever increasing or decreasing; here we have simply a matter of ‘perspective’, if one wishes, but even this is in accordance with the character of the indefinite, which is ultimately nothing other than that of which the limits can be, not done away with—which would be impossible, since the finite can only produce the finite—but simply pushed back to the point of being entirely lost from view.

In this regard certain rather curious questions arise: thus, one could ask why the Chinese language symbolically represents the indefinite by the number ten thousand; the expression ‘the ten thousand beings’, for example, means all beings, which in reality are an indefinite multitude. What is most remarkable is that precisely the same thing occurs in Greek, where a single word likewise serves to express both ideas at once, with a simple difference in accentuation, which is obviously only a quite secondary detail: μύριοι, ‘ten thousand’; μυρίοι, ‘an indefinitude’.^[1] The true reason for this is as follows: the number ten thousand is the fourth power of ten; now according to the formulation of the Tao Te Ching, ‘one produced two, two produced three, three produced all numbers,’^[3] which implies that four, produced immediately after three, is in a way equivalent to the whole set of numbers, and this because, when one has the quaternary, by adding the first four numbers one also has the decad, which represents a complete numerical cycle: 1 + 2 + 3 + 4 = 10; this is the Pythagorean Tetraktys, the significance of which we shall perhaps return to more thoroughly on another occasion. One can further add that this representation of numerical indefinitude has its correspondence in the spatial order: raising a number from one power to the next highest power represents in this order, the addition of a dimension; now, since our space has only three dimensions, its limits are transcended when one goes beyond the third power. In other words, this amounts to saying that elevation to the fourth power marks the very term of its indefinitude, since, as soon as it is effected, one has thereby departed from this extension.

Be that as it may, it is in reality the indefinitely great that mathematicians represent by the sign ∞, as we have said; if the sign did not have this meaning, it would have none at all; and according to the preceding, what is thus represented is not a determined number, but as it were an entire domain, which, moreover, is necessary for it to be possible to envisage inequalities and even different orders of magnitude within the indefinite, as we have already pointed out.

As for the indefinitely small, which can similarly be regarded as embracing everything in the decreasing order that is found to lie outside the limits of our means of evaluation, and which as quantity we are consequently led to consider practically non-existent with respect to us, one can represent it in its own set by the symbol 0—although this is in fact only one of the meanings of zero—without bringing in here the notation of differential or infinitesimal quantity, which essentially finds its justification only in the study of continuous variations; and it must be understood that this symbol no longer represents a determined number for the same reasons as those given for the indefinitely great.

The sequence of numbers such as we have been considering it, extending indefinitely in the two opposite directions of increase and decrease and composed of the whole numbers and their inverses, presents itself in the following form: 0 . . . 1/4, 1/3, 1/2, 1, 2, 3, 4 . . . ∞; two numbers equidistant from the central unit will be inverses or complementaries of one another, thus producing the unit when multiplied together, as we explained earlier: 1/n × n = 1, such that, for the two extremities of the sequence, one would be compelled to write 0 × ∞ = 1 as well. However, since the signs 0 and ∞, the two factors of this product, do not really represent determined numbers, it follows that the expression 0 × ∞ itself constitutes what is called an indeterminate form, and must then be written: 0 × ∞ = n, where n could be any number; but in any case one is thus brought back to ordinary finitude, the two opposed indefinites being so to speak neutralized by one another. Here, once again, one can clearly see that the symbol ∞ most emphatically does not represent the Infinite, for the Infinite can have neither opposite nor complement, and it cannot enter into correlation with anything whatsoever, no more with zero than with the unit or with any number; as the absolute All, it contains Non-Being as well as Being, such that zero itself, whenever it is not regarded as purely nothing, must necessarily be considered to be contained within the Infinite.

In alluding here to Non-Being, we touch on another meaning of zero, quite different from the one we have just been considering, and moreover one that is more important from the point of view of metaphysical symbolism; but in this regard, in order to avoid all confusion between the symbol and that which it represents, it is necessary to make it quite clear that the metaphysical Zero, which is Non-Being, is no more the zero of quantity than metaphysical Unity—which is Being—is the arithmetical unit; what is designated by these terms is so only by analogical transposition, since as soon as one places oneself within the Universal, one is obviously outside of all special domains such as that of quantity. Moreover, it is not insofar as it represents the indefinitely small that zero can be taken as a symbol of Non-Being, but rather insofar as, following another of its mathematical meanings, it represents the absence of quantity, which in its order indeed symbolizes the possibility of non-manifestation, just as the unit symbolizes the possibility of manifestation, since it is the point of departure for the indefinite multiplicity of number, as Being is the principle of all manifestation.

In whatever manner zero is envisaged, it can in no case be taken to be purely nothing, which is all too obvious when it is a question of the indefinitely small; it is true that this is only a derivative sense so to speak, owing to a sort of approximate assimilation of quantities that are negligible for us to the total absence of quantity; but insofar as it is a question of this absence of quantity itself, which is null in this connection, it is quite clear that it cannot be so in all respects, as is apparent in an example like that of the point, which is without extension, that is, spatially null, but which is as we have explained elsewhere nonetheless the very principle of all extension. It is truly strange, moreover, that mathematicians are for the most part inclined to envisage zero as purely nothing, when at the same time it is impossible for them not to regard it as endowed with an indefinite potentiality, since, placed to the right of another, 'significant' digit, it helps to form the representation of a number that, precisely by the repetition of this zero, can increase indefinitely, as for example with the number ten and its successive powers; if zero really were absolutely nothing, this could not be so, and it would even be a useless sign, entirely deprived of real value; here we have yet another inconsistency to add to the list of those we have already pointed out so far.

Returning now to zero considered as a representation of the indefinitely small, what is important is to keep in mind the fact that within the doubly indefinite sequence of numbers, the domain of the latter embraces all that eludes our means of evaluation in a certain direction, just as within the same sequence the domain of the indefinitely great embraces all that eludes our means of evaluation in the other direction. This being said, to speak of numbers less than zero is obviously no more appropriate than to speak of numbers greater than the indefinite; and it is still more unacceptable—if such is even possible—when zero simply represents the absence of quantity, for it is totally inconceivable that a quantity should be less than nothing; this, however, is what is attempted—although in a slightly different sense than the one just discussed—when the consideration of so-called negative numbers is introduced into mathematics, forgetting that these numbers were originally no more than an indication of the result of a subtraction impossible in reality, in which a greater number is taken away from a smaller; but this subject of negative numbers, and the logically contestable consequences it entails, calls for further discussion.

Ultimately, the consideration of negative numbers arises solely from the fact that when a subtraction is arithmetically impossible, its result is nonetheless not devoid of meaning when linked to magnitudes that can be reckoned in two opposite directions, as, for example, distances or times. From this results the geometric representation habitually accorded negative numbers: on a straight line, the distances lying along it are considered to be positive or negative depending on whether they fall in one direction or the other, and a point is chosen to serve as the origin, in relation to which the distances are positive on one side and negative on the other, the origin itself being given a coefficient of zero; the coefficient of each point on the line will thus be the number representing its distance from the origin, and its sign of + or - will simply indicate on which side the point falls on in relation to the origin; with a circle one could likewise designate positive and negative directions of rotation, which would give rise to analogous remarks. Furthermore, as the line is indefinite in both directions, one is lead to consider both a positive and a negative indefinite, represented by the signs +∞, and -∞ respectively, commonly designated by the absurd expressions 'greater infinity' and 'lesser infinity'. One might well ask what a negative infinity would be, or again what could remain were one to take away an infinite amount from something, or even from nothing, since mathematicians regard zero as nothing. In cases such as these one has only to put the matter in clear language in order to immediately see how devoid of meaning they are. We must further add that, particularly when studying the variation of functions, one might next be led to believe that the negative and the positive indefinite merge, such that a moving object, departing from its origin and moving further and further away in the positive direction would return toward the origin from the negative side if the movement were carried on for an indefinite amount of time, or vice versa, whence it would result that the straight line, or what is so considered, would in reality be a closed line, albeit an indefinite one. One could show, moreover, that the properties of the straight line in a plane would be entirely analogous to those of a diameter on the surface of a sphere, and that the plane and the straight line could thus be likened respectively to a sphere and a circle of indefinitely great radius, ordinary circles in the plane then being comparable to the smaller circles on the sphere; without pushing the issue further, we shall only note that here one can grasp the precise limits of spatial indefinitude directly, as it were; if one wishes to maintain some semblance of logic, how then can one still speak of the infinite in all of this? When considering positive and negative numbers as we have just done, the sequence of numbers takes the following form: −∞ ... −4, −3, −2, −1, 0, 1, 2, 3, 4 ... +∞, the order of these numbers being the same as that of the corresponding points on the line, that is, the points having these numbers for their respective coefficients. Although the sequence is just as indefinite in either direction, it is completely different from the one we envisaged earlier: it is symmetric not with respect to 1, but to 0, which corresponds to the origin of the distances; and two numbers equidistant from the central term again reproduce it, but this time by ‘algebraic’ addition—that is, by addition performed while taking account of signs, which in this case would amount, arithmetically speaking, to a subtraction—and not by multiplication. One can immediately see a disadvantage that inevitably results from the artificial—we do not say arbitrary—character of this notation: if one takes the unit as the point of departure, the entire sequence of numbers will immediately follow from it; but, if one takes zero, it is on the contrary impossible to derive any number from it, the reason for this being that in reality the forming of the sequence would then be based on considerations of a geometric rather than an arithmetical order, and also that, in consequence of the difference in nature of the quantities treated in these two branches of mathematics, there can never be a completely rigorous correspondence between arithmetic and geometry, as we have already said. Moreover, the new sequence in no way increases indefinitely in one direction and decreases indefinitely in the other, as was the case with the preceding series; or at least, if one claims to consider it thus, it is only in a most incorrect ‘manner of speaking’. In reality, the sequence increases indefinitely in both directions equally since it is the same sequence of whole numbers that is contained on either side of the central zero; what is called the ‘absolute value’—another rather singular expression, as the quantities in question are always of an essentially relative order—must be taken into consideration only in a purely quantitative respect, the positive or negative signs changing nothing in this regard, since they express no more than differences in ‘situation’, as we have just now explained. The negative indefinite is thus by no means comparable to the indefinitely small; on the contrary, just like the positive indefinite, it belongs with the indefinitely great; the only difference is that it proceeds in another direction, which is perfectly conceivable when it is a question of spatial or temporal magnitudes, but totally devoid of meaning in the case of arithmetical magnitudes, which proceed solely in one direction since they are nothing other than the magnitudes of which the sequence of numbers is composed. Negative numbers are by no means numbers ‘less than zero’, which essentially is but a pure and simple impossibility, and the sign by which they are designated can in no way reverse the order in which they are ranked with respect to their magnitude. Moreover, in order to realize it as clearly as possible, it suffices to note that the point of the coefficient −2, for example, is further from the origin than the point of the coefficient −1, and not less far, as would inevitably be the case were the number −2 in fact less than the number −1; in reality, it is not the distances themselves, insofar as they are capable of being measured, that can be qualified as negative, but only the direction in which they lie; here we have two entirely different things, and it is precisely the confusion of the two that is the source of a large part of the logical difficulties raised by the notation of negative numbers. Among the other bizarre and illogical consequences of this notation, let us draw attention to the question of so-called ‘imaginary’ quantities, introduced in the solving of algebraic equations; these quantities are presented as the roots of negative numbers, which again could answer only to an impossibility; perhaps some other meaning could be assigned to them, whereby they might correspond to something real, but in any case, their theory and application to analytic geometry as presented by contemporary mathematicians hardly appears as anything but a veritable mass of confusions and even absurdities, and as the outcome of a need for excessive and artificial generalizations, a need that does not draw back even at manifestly contradictory propositions; certain theorems concerning the 'asymptotes of a circle', for example, amply suffice to prove that this remark is by no means an exaggeration. One could say, it is true, that this is no longer a question of geometry strictly speaking, but only of algebra translated into geometric terms; but precisely because such translation, as well as its inverse, is possible to a certain degree, it is extended to cases in which it can no longer mean anything, for this is indeed the symptom of an extraordinary confusion of ideas, as well as the extreme result of a 'conventionalism' that goes so far as to cause a loss of the sense of all reality. There is yet more to be said, and before ending we shall now turn to the consequences, also quite contestable, of the use of negative numbers from the point of view of mechanics; indeed, since in virtue of its object the field of mechanics is in reality a physical science, the very fact that it is treated as an integral part of mathematics has not failed to introduce certain distortions. In this regard we shall only say that the so-called 'principles' upon which modern mathematicians build this science such as they conceive of it (and among the various abuses of the word 'principles', this is not the least worthy of remark) are in fact only more or less well-grounded hypotheses, or again, in the most favorable case, only more or less simple, general laws, perhaps more general than others, but still no more than applications of true universal principles in a highly specialized domain. Without entering into excessively long explanations, let us cite, as an example of the first case, the so-called 'principle of inertia', which nothing justifies, neither experience, which on the contrary shows that inertia has no role in nature, nor the understanding, which cannot conceive of this so-called inertia consisting only in a complete absence of properties; rigorously speaking, such a word could only be applied to pure potentiality, but this latter is assuredly something altogether different from the quantified and qualified 'matter' envisaged by physicists. An example of the second instance may be seen in what is called the 'principle of the equality of action and reaction', which is so little a principle as to follow immediately from the general law of the equilibrium of natural forces: whenever this equilibrium is in any way disturbed, it immediately tends to re-establish itself, whence a reaction of which the intensity is equivalent to that of the action that provoked it; it is therefore only a simple, particular case of 'concordant actions and reactions', a principle that does not concern the corporeal world alone, but indeed the totality of manifestation in all its modes and states; and it is precisely on this question of equilibrium that we propose to dwell for a little while. Two forces in equilibrium are usually represented by two opposed 'vectors', that is, by two line segments of equal length, but of opposite directions; if two forces applied to the same point have the same intensity and fall along the same line, but in opposite directions, they are in equilibrium. As they are then without action at their point of application, it is even said that they cancel each other out, although this ignores the fact that if one of the forces is suppressed, the other will immediately act, proving that they were never really annulled in the first place. The forces are characterized by numerical coefficients proportional to their respective intensities, and two forces of opposite direction are given coefficients with different signs, the one positive, the other negative: the one being f, the other -f'. In the case just considered, in which the two forces are of the same intensity, the coefficients characterizing them must be equal with respect to their 'absolute values'; one then has: f = f', from which can be derived the condition for their equilibrium: f - f' = 0, which is to say that the sum of the two forces, or of the two 'vectors' representing them, is null, such that equilibrium is thus defined by zero. Now zero having been incorrectly regarded by mathematicians as a sort of symbol for nothing—as if nothing could be symbolized by something—it seems to follow that equilibrium is the state of non-existence, which is a rather strange conclusion; nonetheless, it is almost certainly for this reason that instead of saying that two forces in equilibrium neutralize one another, which would be more exact, it is said that they annul one another, which is contrary to the reality of things, as we have just made clear by a most simple observation. The true notion of equilibrium is entirely different. In order to understand it, it suffices to point out that all natural forces—and not only mechanical forces, which, let us say again, are no more than a very particular case—are either attractive or repulsive; the first can be considered compressive forces, or forces of contraction, and the second expansive forces, or forces of dilation. Given an initially homogenous medium, it is easy to see that for every point of compression there will necessarily correspond an equivalent expansion at another point, and conversely, such that two centers of force, neither of which could exist without the other, will always have to be considered correlatively. This is what can be called the law of polarity, and it is applicable to all natural phenomena, since it is derived from the duality of the very principles that preside over all of manifestation; in the domain with which physicists occupy themselves, this law is evident above all in electrical and magnetic phenomena. Now if two forces, the one compressive, the other expansive, act upon the same point, then the condition requisite for them to be in equilibrium or to neutralize one another, that is, the condition which, when fulfilled, will produce neither contraction nor dilation, is that the intensities of the two forces be equivalent—we do not say equal, since they are of different species. The forces can be characterized by coefficients proportional to the contraction or dilation they produce, in such a way that if one considers a compressive force and an expansive force together, the first will have a coefficient n > 1, the second a coefficient n' < 1; each of these coefficients will be the ratio of the density of the space surrounding the point in consideration under the action of the corresponding force, to the original density of the same space, which is taken to be homogenous when not subject to any forces, in virtue of a simple application of the principle of sufficient reason. When neither compression nor dilation is produced, the ratio will inevitably equal one, since the density of the space will be unchanged; in order for two forces acting upon a point to be in equilibrium, their resultant must have a coefficient of one. It is easy to see that the coefficient of this resultant is the product—and not the sum, as in the 'classical'

conception—of the coefficients of the two forces under consideration; these two coefficients, n and n', must therefore each be the inverse of the other: n' = 1/n, and we will then have as the condition for equilibrium, (n)(n') = 1; thus equilibrium will no longer be defined by zero, but by the unit. It will be seen that the definition of equilibrium with respect to the unit—which is the only real definition—corresponds to the fact that the unit occupies the mid-point in the doubly indefinite sequence of whole numbers and their inverses, while this central position is as it were usurped by zero in the artificial sequence of positive and negative numbers. Far from being the state of non-existence, equilibrium is on the contrary existence considered in and of itself, independent of its secondary, multiple manifestations; moreover, it is certainly not Non-Being, in the metaphysical sense of the word, for existence, even in this primordial and undifferentiated state, is still the point of departure for all differentiated manifestations, just as the unit is the point of departure for the multiplicity of numbers. As we have just considered it, this unit, in which equilibrium resides, is what the Far-Eastern tradition calls the 'Invariable Middle'; and according to the same tradition, this equilibrium or harmony is the reflection of the 'Activity of Heaven' at the center of each state and of each modality of being. We conclude this study, which makes no claim to be exhaustive, with a 'practical' conclusion; we have shown explicitly why the conceptions of modern mathematicians cannot inspire us with any more respect than do those of the representatives of the other profane sciences; their opinions and views thus have no weight in our eyes, and we need take no account of them in our evaluations of one or another theory, evaluations which, in this domain as well as any other, can be based for us only on the data of traditional knowledge.