THE NOTATION OF NEGATIVE NUMBERS
If we now return to the second and more important of the two mathematical senses of zero, namely that of zero considered as a representation of the **indefinitely small**, this is because 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 these means of evaluation in the other direction. This being so, to speak of numbers 'less than zero' is obviously no more appropriate than to speak of numbers 'greater than the indefinite', and it is all the more unacceptable-if such is pos-sible-when zero is taken in its other sense as purely and simply representing the **absence of quantity**, for it is totally inconceivable that a quantity should be less than nothing. In a certain sense, however, this is precisely what is done when one introduces the consideration of so-called **negative numbers** to mathematics, forgetting as a result of modern 'conventionalism' that these numbers were originally no more than an indication of the result of a subtraction that is in fact impossible, in which a greater number is taken away from a smaller; besides, we have already pointed out that all generalizations or extensions of the idea of number arise only from the consideration of operations that are impossible from the point of view of pure arithmetic; but this conception of negative numbers, and the consequences it entails, demand some further explanation.
We said earlier that the sequence of whole numbers is formed starting from the **unit**, and not from zero; indeed, the unit being fixed, the entire sequence of numbers is inferred from it in such a
way that one could say that it is already implied and contained in principle within the initial unit [1] whereas it is obvious that no number can be derived from zero. Passage from zero to the unit cannot be made in the same way as passage from the unit to other numbers, or from any given number to the next, and to suppose the passage from zero to the unit possible is to have already implicitly posited the unit. [2] Finally, to place zero at the beginning of the sequence of numbers as if it were the first in the sequence, can mean only one of two things: either one admits, contrary to what has already been established, that zero really is a number, and consequently that its ratios with respect to other numbers are of the same order as the ratios of these numbers are to each other-which is not the case, since zero multiplied or divided by a given number is always zeroor this is a simple device of notation, which can only lead to more or less inextricable confusions. In fact, the use of this device is never justified except to permit the introduction of the notation of negative numbers, and if such notation doubtless offers certain advantages for the convenience of calculation-an entirely 'pragmatic' consideration, which is not in question here and which is even without any real importance from our point of view-it is easy to see that it is not without grave logical difficulties. The first of these is precisely the conception of negative quantities as 'less than zero', an affirmation which Leibnitz ranked among the affirmations that are only *toleranter verae*, but which in reality is, as we were just saying, entirely devoid of meaning. 'To affirm an isolated negative quantity as less than zero,' says Carnot, 'is to veil the science of mathematics, which should be a science of the obvious, in an impenetrable cloud, and to thrust oneself into a labyrinth of paradoxes, each more bizarre than the last. [3] On this point we may follow his judgment,
which is above suspicion and is certainly not exaggerated; moreover, one should never forget in using this notation of negative numbers that it is a matter of nothing more than a simple **convention**.
The reason for this convention is as follows: when a given subtraction is arithmetically impossible, its result is nonetheless not devoid of meaning when this subtraction is linked to magnitudes that can be reckoned in **two opposite directions**, as, for example, with distances measured on a line, or angles of rotation around a fixed point, or again the time elapsed in moving from a certain instant toward either the past or the future. From this results the geometric representation habitually accorded negative numbers: taking an entire straight line, indefinite in both directions, and not in one only, as was the case earlier, the distances along the line are considered positive or negative depending on whether they fall one way 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. For each point on the line there is a number corresponding to the measurement of its distance from the origin, which, in order to simplify our language, we can call its coefficient; once again, the origin itself will naturally have zero for its coefficient, and the coefficients of all the other points on the line will be numbers modified by the signs $+$ and $-$, which in reality simply indicate on which side the point falls in relation to the origin. On a circumference one could likewise designate positive and negative directions of rotation, and starting from an initial position of the radius, one would take each angle to be positive or negative according to the direction in which it lies, and so on analogously. But to keep to the example of the straight line, two points equidistant from the origin, one on either side, will have the same number for their coefficients, but with contrary signs, and in all cases, a point that is further than another from the origin will naturally have a greater coefficient; thus it is clear that if a number $n$ is greater than another number $m$, it would be absurd to say, as is ordinarily done, that $-n$ is smaller than $-m$, since on the contrary it represents a greater distance. Moreover the sign thus placed in front of a number cannot really modify it in any way with regard to quantity, since it represents nothing with respect to the measurements of distances themselves,
but only the **direction** in which these distances are traversed, which, properly speaking, is an element of a qualitative, and not a quantitative, order. [4]
Moreover, as the line is indefinite in both directions, one is led to envisage both a positive and a negative indefinite, represented by the signs $+\infty$ and $-\infty$ 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; one has only to put these matters in clear language in order to see immediately how devoid of meaning they are. We must further add that particularly when studying the variation of functions, one is then led to believe that the negative and the positive indefinite merge in such a way that a moving object departing from the origin and moving further and further away in the positive direction would return to the origin from the negative side, or inversely, if the movement were followed for an indefinite amount of time, whence it would result that the straight line, or what would then be considered as such, would in reality be a **closed line**, albeit an indefinite one. Furthermore, one could show that the properties of a straight line in a plane would be entirely analogous to those of a great circle, or diametrical circle 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, and consequently of indefinitely small curvature, ordinary circles in the plane then being comparable to the smaller circles on the sphere; for this analogy to be rigorous, one would further have to suppose a 'passage to the limit', for it is obvious that however great a radius might become through indefinite increase, it always describes a sphere and not a plane, and that the sphere only tends to be merged with the plane, and its great
circle [or diameter] with lines, such that plane and line are limits, in the same way that a circle is the limit of a regular polygon with an indefinitely increasing number of sides. Without pushing the issue further, we shall only remark that through considerations of this sort, one can as it were directly grasp the precise limits of spatial indefinitude; how then, if one wishes to maintain some appearance of logic, can one still speak of the infinite in all this?
When considering positive and negative numbers as we have just done, the sequence of numbers takes the following form: $-\infty \ldots-4,-3,-2,-1,0,1,2,3,4 \ldots+\infty$, 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, which, moreover, is the mark of the real origin of the sequence thus formed. Although the sequence is equally indefinite in both directions, it is completely different from the one we envisaged earlier, which contained the whole numbers and their inverses: this one is symmetric not with respect to the unit, but with respect to **zero**, which corresponds to the origin of the distances; and if two numbers equidistant from this central term are to return to it, it will not be by multiplication, as in the case of inverse numbers, but by **'algebraic' addition**, that is, effected while taking account of signs, which in this case would amount to a subtraction, arithmetically speaking. Moreover, we can by no means say of the new sequence that it is indefinitely increasing in one direction and indefinitely decreasing in the other, as we could of the preceding, or at least, if one claims to consider it thus, this is only a most incorrect 'manner of speaking', as is the case when one envisages numbers 'less than zero'. 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-must only be taken into consideration in a purely quantitative respect, the positive or negative signs changing nothing in this regard, since, in reality, they express no more than differences in **'situation'**, as we have just explained. The negative indefinite is therefore by no means comparable to the indefinitely small; on the contrary, it belongs with the indefinitely great as does the positive indefinite; the only difference, which is not one of
a quantitative order, 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 for arithmetical magnitudes, for which such a progression is necessarily unique since it cannot be anything other than that of the very sequence of whole numbers.
Among the bizarre or illogical consequences of the notation of negative numbers, we shall further draw attention to the consideration of so-called **'imaginary' quantities** which were introduced in the solving of algebraic equations and which, as we have seen, Leibnitz ranked at the same level as infinitesimal quantities, namely as what he called 'well-founded fictions'. These quantities, or what are so called, are presented as the roots of negative numbers, although in reality this again only corresponds to a pure and simple **impossibility**, since, whether a number is positive or negative, its square is necessarily always positive by virtue of the rules of algebraic multiplication. Even if one could manage to give these 'imaginary' quantities some other meaning, thereby making them correspond to something real-a possibility we shall not examine here-it is nonetheless quite certain that their theory and application to analytic geometry as it is presented by contemporary mathematicians never appears as anything but a veritable web of confusions and even absurdities, and as the product of a need for excessive and entirely artificial generalizations, which need does not retreat even before manifestly contradictory propositions; certain theorems concerning the 'asymptotes of a circle', for example, amply suffice to prove that this remark is by no means exaggerated. It is true that one could say that this is no longer a question of geometry properly speaking, but, like the consideration of a **'fourth dimension'** of space, [5] only of algebra translated into geometric language; but precisely because such a translation, as well as its inverse, is possible and legitimate to a certain degree, some people would also like to extend it to cases where it can no longer mean anything, and this is indeed quite serious, for it is the symptom of an extraordinary confusion of ideas, as well as the extreme result of a 'conventionalism' taken so far as to cause some people to lose all sense of reality.
[1] *This point was established in the preceding chapter concerning the origin of number.*
[2] *In the metaphysical sense, the unit (the principle of number) must be posited prior to the mathematical concept of 'nothing' (zero).*
[3] *Lazare Carnot, 'Réflexions sur la Métaphysique du Calcul Infinitésimal' (1797). The author refers to the French mathematician and philosopher of the Revolution.*
[4] *The direction (sign) is a qualitative property, not a quantitative one, of the spatial magnitude.*
[5] *A further example of artificial generalization from algebraic formalism into inappropriate geometric language.*