Remark

Leibniz's idealism was spoken of a short while ago. It may be added here that this idealism, starting from the representing monad, from being-for-itself, got no further in the further determination of this being-for-itself than the repulsion just now considered, and indeed no further than plurality, in which each of the ones is only for itself, indifferent towards the existence and the being-for-itself of others, or in which others are not there for the one at all. For itself the monad is the whole self-enclosed world; not one of them stands in need of the rest. The inner manifoldness that it has in its representing is of no concern to us here; for it changes nothing in its determination of being for itself; the monad, since the manifoldness is an ideal one, remains referred to itself alone, the alterations unfold within it and are no references of the monads to one another; what, according to the real determination, is taken as a reference of the monads to one another is an independent and merely simultaneous becoming. Leibniz's idealism, moreover, picks up plurality immediately as something giv en, and does not comprehend it as a repulsion of the monad. Hence it has plurality only from the side of its absolute externality, not from the side on which the monad's reference to itself, being negative, is just as much itself plurality; — the two moments that repulsion holds together within itself. Atomism, for its part, is without the concept of ideality; the one it fails to grasp as a something having within its own self both moments, that of being-for-itself and that of being-for-it; hence not as something ideal, but only as something simply and immediately for itself. Against that, it does pass beyond the merely indifferent plurality; the atoms do after all enter into a further determination over against one another, even if not through repulsion itself; whereas in that indifferent independence of the monads plurality, which is their fundamental determination, as was already recalled above, falls, if anywhere, only within the monad of monads or within the philosopher who observes them, and is not a determination of the monads in themselves. Or precisely insofar as plurality is not a determination of the monads in themselves, insofar as they are not others for one another, this determination belongs to appearance alone, is external to their essence, and their truth is only the substance which is One.

3. Mutual Repulsion

1. Repulsion makes up the reference of the one to itself, but is just as much its coming-outside-itself. This coming-outside-itself, the plurality of the ones, is the repulsion of the one from itself; hence not a determination external to the one, not distinct from repulsion as simple reference to itself. Considered more closely, the one refers to itself as to something immediate; but immediacy is being; repulsion, however, as the negation that refers itself to itself, is not immediacy or being. The one therefore refers to itself at the same time as to its absolute non-being; it is a repelling of itself away from itself; what has been repelled is on the one side indeed the one itself, but just as much its non-being. This repelled thing, itself as one, is something immediate, and at the same time is determined as the non-being of what refers itself to itself; or as an absolutely other. Plurality at first contained no otherness; the limit was only the void, or only that wherein the ones are not. But they are in the limit as well; they are within the void, or their repulsion is their common reference.

The repulsion of the one, then, in being a repelling of itself away from itself, is at the same time a repelling of the one as of an other away from itself, and with that a mutual repelling of the many ones.

The many in this way stand, as repelling one another, in reference to one another; they preserve themselves in repulsion as beings for themselves; their reference consists in negating their reference.

This mutual repulsion is what first makes up the existence of the many ones; for it is not their being-for-itself, which would be differentiated only within a third, but their own self-preserving differentiating. More closely determined, insofar as therein each preserves itself against the others, it is a mutual excluding. Or this reference is a merely relative repulsion. They negate one another, that is, they posit one another as such as are only for-one. But they negate at the same time, and just as much, this, to be only for-one; they repel this ideality of theirs.

2. In this existence of the many ones there thus come apart the moments that in ideality are united without qualification. In its being-for-itself the one is indeed also for-one, so much so that this sublatedness of otherness is its reference to itself. But at the same time being-for-one, as it stands determined in relative repulsion, in excluding, is a being-for-other. Each is repelled by the other, sublated, and made into something that is not for itself but for-one. Its being-for-one accordingly falls not only within the one as such itself, but within another one as well, and is being-for-other.

The being-for-itself of the many ones is hereby their repulsion against one another, whereby they so preserve themselves that they mutually sublate one another and posit the others as a mere being-for-other. But at the same time repulsion consists in repelling this ideality, and in positing itself as not being for-an-other. Yet the two are once more one and the same reference; mutual repulsion is mutual sublating, each preserves itself only in positing the others as a being-for-other, as a non-existence, and just as much only in sublating this, being for an other.

3. Being-for-other is to that extent as much sublated as it is present. But it is posited and sublated in different respects. The ones are immediate; they refer themselves to one another repellingly, sublatingly; they thus mutually bring the being-for-itself of the others down to being-for-other; this moment therefore has its place in reference to others. But the one sublates this its being-for-other; this moment is its reference to itself. The one is being-for-other only within others; but this sublatedness of the one is no concern of the one; within it the others are not as existent, immediate others, but only as sublated, and through this it refers itself to itself.

The one was repulsion in that it repels itself away from itself, and since what is repelled is accordingly only the one itself, it is thereby immediate return into itself. But this repelling has passed over into the repulsion of others and of being-for-other away from itself. The one preserves itself for itself only by referring itself negatively to others, and, this negation being mutual, by sublating the being-for-one that it therein receives. Repulsion, the repelling of the one away from itself, has thus passed over into a repelling of the others, into the positing of the others as being only for-one, and with that into the sublating of its being-for-other, into attraction.

C. Attraction

Repulsion is the self-splintering of the one, at first into many, and then, for the sake of their immediacy, into others. But since the ones are simply many and equally others, no difference among them is thereby present, and the absolute being-determined of the one on its own self is not yet realized. For the one as the ideal, which is just as much for itself as it is for one, both in a single identity, collapses, for the sake of this lack of difference, into the immediacy of being. Because in this ideality no genuine other is present, no genuine sublating of otherness takes place either, and hence no real ideality. The latter now comes to be in attraction. Repulsion does indeed contain others; but since the many ones are one and all others to themselves, their repulsion holds itself in equilibrium; they themselves sublate their mutual being-for-one. They repel repulsion, or otherness.

Now in that the one ceases to be the merely simple reference of negation to itself, and arrives at a determinate difference within itself, it becomes totality, or the identity of ideality and reality. The absolute being-determined has then reached its summit, it has gone back into itself; and quality, the immediate being-determined through an other, or otherness as such, becomes something indifferent; at this unity compact within itself, quality becomes quantity.

1. One One

Repulsion makes the many ones into beings-for-other. But it is the many to which this repelling accrues, and it accrues to them as ones. Yet as ones they are infinite reference to themselves, and as such they just as much repel this being-for-other, or that repelling. This repulsion of repulsion is thus, as self-sublating, attraction.

Here, however, the difference mentioned enters in; the one, namely, posits the other ones as being-for-other, and — insofar as this repelling were mutual — sublates its being-for-other, which it would therein receive; the being-for-other of the others, however, it preserves.

Attraction, namely, is repulsion of repulsion. The one posits the other ones ideally, as being-for-other, but just as much sublates this being-for-other again. Thereby the return of the one into itself is posited, or that same infinite reference to itself which the one in itself is. But with this there are two sorts of one present; namely the immediate one, or the one as it is in itself, and then the one that out of its dispersion, out of plurality, returns into itself.

This one may be called the real one insofar as it returns into itself out of plurality and out of being-for-other, and has this moment upon it, though as a sublated one; or insofar as the moment of being-for-one which it contains in its ideality is no longer merely this abstract moment, but the immediate ones make it up. The other one, by contrast, is this immediate one that does not return into itself, that essentially is as something sublated, and remains in being-for-other.

That first one is the attracting one; the one that gives itself its moment of being-for-one at the immediate ones. These become attracted. They are immediate; but the one is essentially this, not to be an immediate being; for it is rather the negation that refers itself to itself. Since, then, they are immediate, they are only unequal to themselves, others on their own selves.

With this there is also present the otherness that is in itself, and the earlier, merely external otherness has vanished. The immediate one is only as something sublated, which is only for-other. But the being-for-itself which is only for-other is precisely otherness on its own self.

Further, the attracting one, which sublates being-for-other within itself and returns out of it into itself, is precisely thereby no longer the simple being-for-itself, but that which has otherness too as a moment within its own self.

The attracting one, then, as returning into itself out of plurality, determines itself as one, it is one as not being many, one One.

2. Equilibrium of Attraction and Repulsion

Being-for-itself, which has determined itself as one, first loses itself as plurality in absolute externality, and preserves itself therein not so much according to its immediacy, — insofar as the many are also ones, — as it restores itself out of it into one One.

This one that has returned into itself is not merely the simple reference to itself, but the reference to itself as to sublated otherness. — Further, otherness as it occurs here is not the immediate otherness of existence as such, but the one's own otherness, plurality. Being-for-itself, according to its having become out of existence, is indeed already in itself sublated otherness; but here it had once more to posit its other upon its own self, in order to be also in being-for-itself as such what it in itself is. Otherness, however, has within it a form other than in existence. Because being-for-itself is infinite reference to itself, otherness upon it is only plurality, itself as other.

But in that being-for-itself has thus sublated its immediacy and is being-for-itself that is for itself, being-determined has indeed within it made itself into absolute being-determined upon its own self, into the absolutely qualitative; but in this reality it has already gone out beyond quality. The one is only one One insofar as within it plurality, that is, the one itself, is sublated. — Or the one has, as one One, gone together with itself; instead of being exclusive, therefore, it has posited itself in continuity.

Attraction, namely, or the one One considered more closely: it is determined on its own self, for it is not one among the many, it has sublated plurality with in itself; it is therefore not something determinate over against an other, but has the other, and the reference to it, upon its own self. As one One, however, its absolute determinateness has likewise gone back into immediacy, and refers itself as exclusive to the many, as over against others, as over against its non-being, which would itself be immediate. But it is only one One; the many are not at all, they have sublated themselves; thus they are posited into one with the one, and the latter is no longer one as such.

The one One is in itself attraction, sublated repulsion; but this one itself begins by being something immediate; it is a one, and its reflection into itself consists precisely in sublating immediacy. The repulsion of repulsion sublates only its own being-for-other, but preserves the being-for-other of the others; yet an own being of that sort, one that would distinguish itself from others, presupposes an original, an immediate differentiatedness of the ones, which is not present. Repulsion is therefore a being-for-one of the many as such, and insofar as it is repulsion of repulsion, it is just as much a preserving of the many ones whose being-for-one is repelled by themselves. All are therefore equally attracting, all posit one another in the same manner as being-for-other, and repel the same, sublate it in their infinite reference within themselves. The many ones are thus preserved.

— Already in the sensuous representation of spatial attraction the stream of points coming to be attracted continues on; in place of the atoms that vanish in the one attracting point, another multitude steps forth out of the nothing. This becoming does not go back into the result of the one One in such a way that only the one One and nothing else would be; in that manner only the initial determination, the one and the void, would be posited, and the reality of the one, the returning into itself out of the many, would have vanished. Rather, in that it becomes itself as one One through this return, it is exclusive, one One over against many, and it thereby preserves them just as much. But the preservation of the many means nothing else than that they are attracting, that they sublate their being-for-other.

Attraction and repulsion are in this way not merely in equilibrium; they are in fact identical and indistinguishably the same. Repulsion appears at first as the excluding of the others; but this excluding is a positing of them as beings-for-other. Attraction, however, is the same, for it consists precisely in the self-preservation of the one against the others, in the sublating of them, in the positing of them as beings-for-other. Repulsion, conversely, is further the sublating of this being-for-other, through it the one preserves itself in sublating its being-negated; but attraction is precisely this sublating of its being-for-other, which sublating preserved it. Sensuous representation alone keeps up the difference between attraction and repulsion, in that it holds fast to one immediate point and lets the immediacy of the others vanish, though in fact just as much come to be again.

As repulsion repels itself, so attraction attracts itself, or is attraction of attraction. For according to its determination it is the positing of the many ones as ideal, and thereby the becoming of one One, which is to remain for itself and to sublate its being-for-other . But among the many ones that are to be sublated, all ones are comprised; attraction sublates just as much the one one whose becoming it is supposed to be. Or conversely, in that as the becoming of the one One it sublates the being-for-other of the one, it just as much sublates the positing whereby the ones come to be being-for-other, which is to say, once again, that it sublates itself.

This identity of repulsion and attraction accordingly has the result that the infinite reference of the one to itself is its being-for-other; its being-for-itself is infinite negation of itself, infinite being-outside-itself, and this being-outside-itself is conversely infinite having-returned into itself.

In itself the one is only this infinite reference to itself, whose result is the identity of attraction and repulsion; — the one is nothing apart from repulsion and attraction. But insofar as the one has received the shape of immediacy, these appear as references of the one, so that outside them it would preserve itself for itself; as if its being-for-an-other were distinguished from its being-for-itself, or rather from its being-in-itself, its infinite reference to itself. Taken, however, as in itself, distinguished from its negative reference, the one is the immediate one, the many. But just as immediately the many collapses into one, or the many is the negation of itself. For of the many each is one, or each is a many, or each distinguishes itself without qualification from the others and shuts them out from itself. Yet precisely in this they are equal to one another; each has entirely and utterly the same determinations that the other has; in this, that the one of the many is not what the other is, they are the same.

The formerly relative repulsion and attraction, which was only a reference of the ones, from which their immediacy, as reference to itself, was distinguished, is therefore in fact absolute repulsion and attraction; repulsion and attraction which are identical. What is present is that the one, as referring itself infinitely to itself, refers itself to its absolute otherness, and in referring itself to this its non-being, refers itself precisely therein to itself, and that the one is itself only this referring. Its immediacy, its being, is rather its otherness, and this its being-outside-itself is its being.

Remark

Attraction and repulsion are, as is well known, customarily viewed as forces. On this representation they are taken to be self-subsistent, so that they do not refer to one another by their nature, that is, so that neither is meant to be merely a moment passing over into its opposite, but is instead to persist firmly over against the other. They are further represented as coming together in a third, in matter; yet in such a way that this becoming-one does not count as their truth, each of them being rather something first and in and for itself, while matter is what they posit and bring forth. If it is said that matter has these forces within itself, then by this unity of theirs a linkage is understood in which the two are at the same time presupposed as free of one another, each being in itself.

Kant, as is well known, has constructed matter out of the repulsive and attractive force, or at least has laid down, as he expresses it, the metaphysical elements belonging to this construction. — To throw a closer light on this construction will not be without interest. This metaphysical presentation of an object which seemed to belong, not only itself but in its determinations, to experience alone, is on the one part remarkable because it has at least given the impetus to the newer philosophy of nature, — the philosophy which does not make nature, as something sensuously given to perception, the ground of science, but cognizes nature's determinations out of the absolute concept; on the other part, too, because people still frequently stop at that Kantian construction and take it for a philosophical beginning and foundation of physics.

A concrete existence such as sensuous matter does not, to be sure, yet belong here, any more than space and the determinations of space do. But at the base of attractive and repulsive force too, insofar as they are regarded as forces of sensuous matter, there lie the pure determinations of the one and the many considered here, together with their references to one another, which I have likewise called repulsion and attraction.

The way in which Kant deduces matter out of these forces, and to which he gives the name of a construction, does not, looked at more closely, deserve that name, unless indeed every kind of reflection, even the analysing kind, is called a construction, as later philosophers of nature have of course called the shallowest ratiocination and the most groundless brew of an arbitrary imagination and a thoughtless reflection, — a brew which helped itself especially to the so-called factors of attractive force and repulsive force and paraded them everywhere, — a constructing.

Kant's procedure is at bottom analytic, not constructive. He presupposes the representation of matter, and then asks what forces belong to it if its presupposed determinations are to be preserved. On the one part, accordingly, he calls for attractive force on this ground, because through repulsion alone, without attraction, there could properly be no matter. (Foundations of Natural Science, p. 53 f.) Repulsion, on the other part, he likewise derives from matter, and states as its ground that we represent matter to ourselves as impenetrable, since matter presents itself under this determination to the sense of feeling, through which it is said to reveal itself to us. Repulsion is thus thought immediately within the concept of matter, because it is thereby immediately given; attraction, by contrast, is attached to that concept through inferences. But at the base of these inferences lies what has just been said, namely that a matter which had merely repulsive force would not exhaust what we represent to ourselves under matter.

This, as is evident, is the procedure of the ordinary cognition that reflects upon experience, which first perceives determinations in the appearance, then lays these at the base, and for the so-called explaining of them assumes basic materials and also forces which are supposed to bring forth those determinations of the appearance.

With regard to the difference adduced, as to how repulsive force and how attractive force come to be found in matter by cognition, Kant remarks still further that attractive force, with respect to the concept of matter, does indeed belong to it just as much, although it is not contained therein. Kant singles out this last expression. It is not to be seen, however, what difference is supposed to lie in this; for a determination which belongs to the concept of a matter must truly be contained therein. —

What creates the difficulty, and brings on this empty evasion, consists in this, that Kant reckons into the concept of matter merely the determination of impenetrability, one that we are said to perceive through feeling, on account of which repulsive force, as the warding off of an other from oneself, is said to be immediately given. But if matter is not supposed to be able to exist without attractive force, then a representation of matter drawn from perception lies at the base of this; the determination of attraction must accordingly in fact likewise be drawn from perception and therefore be encounterable in it. It can indeed be well perceived that matter, besides its being-for-itself, which sublates being-for-other, also has a reference of what is for itself to one another, spatial extension and cohesion. Out of this perception reflection can derive attractive force just as immediately, or assume it as given, as it did with repulsive force. In fact, if the inferences from which attractive force is supposed to be derived are examined, (see the proof of the proposition: that a force of attraction as second fundamental force is demanded by the possibility of matter, loc. cit.) they contain nothing but this, that through mere repulsion matter would not be spatial. Inasmuch as matter is presupposed as space-filling, continuity is thereby ascribed to it, and the force of attraction is assumed as the ground of that continuity.

Even if this so-called construction of matter had at most an analytic merit, one still curtailed by the impure exposition, the basic thought remains always well worth valuing, that of cognizing matter out of these two opposed determinations as its fundamental forces. Kant is concerned above all to banish the ordinary mechanical way of representing, which stops at the one determination, impenetrability, at punctiformity that is for itself, and turns the opposed determination, the reference of matter within itself or of several matters, regarded again as particular ones, to one another, into something external; — the way of representing which, as Kant says, will otherwise allow no moving forces except through pressure and impact, hence only through action from outside. This externality of cognition always presupposes motion as already at hand, and never thinks of grasping it as something inward and comprehending it itself and within matter, but assumes matter for its part as motionless and as inert. Now although Kant does sublate this externality insofar as he makes attraction, the reference of matters to one another insofar as these are assumed to be different, or of matter in general in its being-outside-itself, into a force of matter itself, on the other side his two fundamental forces nevertheless remain, within matter, external and self-subsistent over against one another.

As null as was the difference of these two forces which was attributed to them with regard to cognition, just as null must every other difference show itself to be that is drawn in respect of their content-determination, since they, as they were considered above in their truth, are only moments which vanish into one another. I take up these further determinations of difference as Kant states them.

He determines attractive force as a penetrating force, repulsive force as a surface force. The ground adduced for the latter's being only a surface force is the following: „The parts touching one another bound each the space of action of the other, and repulsive force cannot move a more distant part except by means of those lying in between; an immediate action of one matter upon another passing crosswise through these, by way of expansive forces (which here means repulsive forces), is impossible.“

I will not dwell on the point that, once nearer or more distant parts of matter are assumed, with regard to attraction there would likewise arise the difference that one atom would indeed act upon another, but that a third, more distant one, such that the other lay between it and the first attracting atom, would step first of all into the attracting sphere of the intervening atom that is nearer to it, so that the first would not exercise an immediate simple action; from which a mediated action could be developed for attractive force just as much as for repulsive force; — further, that the true penetrating of attractive force would in general have to consist solely in this, that all parts of matter were attracting in and for themselves, and not that a certain multitude behaved passively and only One atom actively. But I note immediately, with regard to repulsive force, that in the passage cited there occur parts touching one another, hence a compactness and continuity of a finished matter which does not permit a repelling through itself; this compactness of matter, however, in which parts touch one another and are no longer separated by the void, already presupposes the sublatedness of repulsive force. Parts touching one another are, according to the sensuous representation of repulsion that prevails here, to be taken as parts which do not repel one another. It follows, then, quite tautologically, that where the non-being of repulsion is assumed, no repulsion can take place. From this, however, nothing further follows for a determination of repulsive force.

In the same way it is natural to sensuous representation, presupposing as it does one attracting point and others which do not attract but are only attracted, to assume that the former brings something about with its attracting, and lays a thickness about itself as a sphere, so that within this sphere, because it stands under the dominion of that point's attraction, repulsion is sublated and can therefore take place only outside, against the surface of this sphere. — On the one part the surface appears as that which still stands in relation to an other not referred to it. On the other part, however, repulsion is itself within that sphere of attraction. Those atoms, that is, or material parts which are there for representation as attracted, are for it in fact just as much repelled ones as well (-- since we let repulsion count as removal, attraction as approach, to a determinate point —). For the attracted ones, if they were only this, would have vanished into the point of attraction, and there would be only this atom, not an attracted one distinguished from it, and hence not parts touching one another, that is, parts at the same time held apart. But insofar as such parts are assumed, repulsion is in fact not excluded from that sphere of attraction, but is present within it.

Kant adopts, further, the additional determination that by the force of attraction matter only occupies a space, without filling it. „Because matter does not fill space by the force of attraction, this force can act through empty space, since no matter lying in between sets bounds to it.“ — This difference is constituted about like the earlier one, where a determination was to pertain to the concept of a matter and yet not be contained in it. By the force of attraction matter is not to fill space, but is, in respect of this force, to relate itself to itself through empty space; — it is thus not to be seen how it is supposed to occupy space if space is empty. But further it is repulsion, if we hold to its first determination, through which the ones thrust one another away and only negatively, which here means, through empty space, refer to one another. Here, however, attractive force keeps space empty for itself, it fills space through its reference of the atoms in no way, which is to say, it keeps the atoms in a negative reference to one another. — We see accordingly that here Kant unconsciously runs up against what lies in the nature of the matter, in the nullity of the difference between repulsion and attraction, in that he ascribes to attractive force precisely what, according to the first determination, he ascribed to the opposed force. In the course of the business of fixing the difference of the two forces, it had come about that the one had passed over into the other. — Through repulsion, by contrast, matter is supposed to fill a space, so that through repulsion the empty space which attractive force leaves disappears. In fact, therefore, in sublating empty space it sublates the negative reference of the atoms or ones, that is, their repulsion; or repulsion is posited as the opposite of itself.

To this blurring of the differences just exhibited there is added the further confusion that, as was noted at the very outset, this exposition of the opposed forces is analytic, and that throughout the whole presentation matter, which is first supposed to be derived from its elements, already turns up as finished and constituted. In the definition of surface force and penetrating force both are assumed as moving forces by which matters are to be able to act in the one way or the other. — They are therefore exhibited here as forces not by which matter first comes about, but by which it, already finished, is merely set in motion. But insofar as the talk is of forces by which different matters act upon one another and set each other in motion, this is something quite other than the determination and reference which they were supposed to have as the moments of matter.

The same opposition, as attractive and repulsive force, is constituted in a further determination by centripetal and centrifugal force. These seem to afford an essential difference, since within their sphere One one, a centre, stands fast, over against which the other ones comport themselves as not being for themselves. Insofar as they are put to use for explanation, however — for which purpose they are assumed, as repulsive and attractive force are elsewhere, in an opposed quantitative ratio, so that the one waxes as the other wanes — the appearance and its inequality are supposed to result only out of them. One need only take up the first available exposition of an appearance, for instance of the unequal velocity which a planet has in its orbit around its central body, out of the opposition of those forces, and one soon recognizes the confusion reigning within it, and the impossibility of separating out their magnitudes, so that the one has always just as good a claim to be assumed as increasing which in the explanation is assumed as decreasing, and conversely.

3. Transition to Quantity

The qualitative has for its fundamental determination being and immediacy, in which being and nothing are one; limit and determinateness are so identical with the being of the something that with their alteration the something itself vanishes. Because this unity is an immediate one, wherein the difference has disappeared, a difference which is nonetheless in itself at hand within it in the unity of being and nothing, that difference falls, as otherness, outside the unity. This reference to an other, however, contradicts the immediacy in which qualitative determinateness stands. It sublates this otherness, sublates itself, in the infinity of being-for-itself, which is the reference of being-determined to itself, being-determined in itself.

In this equality with itself the qualitative, which at first had the other as something external, has raised itself to its genuine unity. But its determinateness, immediacy, has at the same time vanished.

Being-for-itself is at first only the concept of the infinite reference of the negative to itself, without at the same time containing the negative as a real difference within this unity, so that through this simple unity it itself collapses again into immediacy and has the other as a many outside it. But this many is itself a one, or the one is plurality within itself. The movement of being-for-itself has consisted in realizing itself, or in positing within itself the otherness sublated in it, and thereby in exhibiting itself as identity with itself in otherness.

What is now present, then, is the one which is in unity with itself, yet not immediately, but rather in this, that it refers to its non-being, and thereby to itself; its infinite reference to itself through its non-being. The one is accordingly expanded into unity; otherness has become a limit which, in its negation returned into itself, is no longer determinateness as reference to an other, and hence is an indifferent limit. The immediate unity of the qualitative with itself has thus passed over into unity with itself through its otherness. This unity, in which otherness is taken back into itself and determinateness is thereby indifferent, the sublated quality, is quantity.

Second Section. Magnitude

(Quantity.)

The difference of quantity from quality has just been stated. Quality is the first and immediate determinateness; quantity is determinateness that has grown indifferent towards being, a limit which just as much is no limit.

Being has received the determination of having its simple equality with itself in its otherness and only through the sublating of its otherness.

Otherness and determinateness, insofar as determinateness comes forth again in this sphere, is therefore no longer an immediate, abiding determinateness, but a sublated one, something that does not stand in simple reference to itself but is rather something utterly external to itself. Quantity is determinateness returned infinitely into itself; it is no longer being as reference to an other and as the non-being of an other; determinateness has sublated itself in its otherness, with which it is in unity; and quantity is the indifference of determinateness. — But insofar as determinateness comes on the scene again as distinct from this unity of its own, it comes on as what it is in truth, namely utterly only as in unity with its otherness. As quality it was supposed to be a determinateness that is, standing in simple reference with itself; but as quantity it is determinateness merely sublated, external, not being within itself but in the other.

But at the outset pure quantity must be marked off from itself as determinate quantity, from quantum.

Quantity is first real being-for-itself returned into itself, which as yet has no determinateness upon it; the solid infinite unity.

This passes over, second, into determinateness, but into one which is at the same time none, being merely external. It becomes quantum. Quantum is the determinateness grown indifferent, in other words, determinateness that goes out beyond itself, negating itself; as this otherness of otherness it becomes infinite. Infinite quantum, however, is indifferent determinateness sublated, or it is the reinstatement of quality.

Third, quantum in qualitative form is the quantitative ratio. Quantum merely goes out beyond itself in general; in the ratio, however, it goes out beyond itself into its otherness in such a way that it has its determination in this otherness, and is hence at the same time returned into itself, and the reference to itself is present in its otherness. In the ratio, therefore, quantum has returned into quantity, which is thereby at the same time determined as quality.

At the base of this ratio there still lies the indifference of quantum, or it is a merely formal unity of quality and quantity. The movement of the ratio is its passing over into their absolute unity, into measure.

Remark

In qualitative being the limit appeared at first as something which, distinguished from the being-within-itself of the something, is an external limit, towards which the something itself is indifferent. But this externality of the limit sublated itself at once, and the limit showed itself to be one with the being-within-itself of the something, and to be determinateness. That limit, however, was not yet the quantitative limit; for the being-within-itself of the something is as yet only immediate, and over against it the other maintains itself; it is not yet the infinite being-returned-into-itself of quantity, in which otherness has sublated itself in and for itself. In the case of something, therefore, its limit is essentially its determinateness.

If accordingly we understand by limit the quantitative limit, and a field, for instance, alters its limit, namely the quantitative one, it remains a field afterwards as before. But if its qualitative limit is altered, then this is its determinateness by virtue of which it is a field, and it becomes meadow, forest, and so forth. — A red that is more intense or fainter is always red; but if it changes its quality, it ceases to be red; it would become blue, and so forth. — The true and determinate concept of magnitude, as it has here emerged, that an abiding being lies at the base which is indifferent towards the determinateness that it has, results in every other example as well.

A magnitude is customarily defined as something that admits of being increased or diminished. But to increase means to make something more great, to diminish, to make it less great, and the more in more great, and the less in less great — resolves itself again in this way. There lies in this a difference of magnitude in general from itself, and magnitude would accordingly be that whose magnitude admits of being altered. The definition shows itself to be clumsy for this reason, that within it the very determination is employed which was to be defined. Yet in this imperfect expression the chief moment on which everything turns is not to be mistaken; namely the indifference of the alteration, the fact that its own more and less lies in its very concept; its indifference towards itself.

Chapter 1. Quantity

A. Pure Quantity

1. Magnitude is being-for-itself in sublated form; the repelling one, which conducted itself merely negatively towards another, has passed over into reference with it, conducts itself identically towards the other, and has thereby lost its determination. Being-for-itself has become attraction; but attraction has not itself remained the becoming of the many into one, for the difference of one one from others has likewise vanished and this becoming has passed into rest. Attraction and repulsion are sublated in a unity, or have sunk down to moments. The one is in reference to itself, through attraction, and at the same time to itself as to an other, through repulsion. The one, as this one which has just as much coalesced with the ones that repel one another, has thus acquired, so to speak, a breadth, and has expanded itself into unity. The absolute brittleness of the repelling one has melted into this unity, which, however, since it contains that one, is at the same time determined through the repulsion dwelling within it, and thus as unity of being-outside-itself is unity with itself. In this way attraction has become the moment of continuity in magnitude.

Continuity is therefore simple self-same reference to self, interrupted by no limit and no exclusion, yet not an immediate unity but the unity of the ones that are for themselves. Contained in it, therefore, is the outside-one-another of plurality, but at the same time as one that is undifferentiated, uninterrupted. Plurality is posited in continuity as it is in itself; namely, the many are each of them what the others are, each equal to the other, and plurality is therefore simple, differenceless equality. Continuity is this moment of the self-sameness of being-outside-one-another.

2. Immediately, then, magnitude has in continuity the moment of discreteness. Continuity is self-sameness, but self-sameness of the many, which nevertheless does not become something excluding; and it is repulsion that first stretches this self-sameness out into continuity. Discreteness is therefore for its part a discreteness that flows together, whose ones do not have the void, the negative, for their reference, and do not interrupt continuity, the equality with itself in the many. The difference of repelling is therefore present only as distinguishability.

3. Magnitude, as the unity of these moments, of continuity and discreteness, may be called quantity; for with the expression magnitude what lies nearer to representation is the immediate side of it, and bounded magnitude, the quantum, whereas quantity recalls rather what is reflected and the concept of it.

Quantity is thus being-for-itself as it is in truth. It was the self-sublating referring to itself, a perennial coming-outside-itself. But what is repelled is it itself; repulsion is therefore the generative flowing-forth of its own self. For the sake of the selfsameness of what is repelled, this discerning is unbroken continuity; and for the sake of the coming-outside-itself, this continuity, without being interrupted, is at the same time plurality, which just as immediately remains in its equality with itself.

Remark 1

Pure quantity has as yet no limit, or is not yet quantum; — even insofar as it becomes quantum, it is not limited by the limit, for it consists precisely in this, in not being limited by the limit, in having being-for-itself within it as something sublated. That it is discreteness sublated can also be expressed thus: that quantity is utterly, everywhere within it, the real possibility of the one, but conversely that the one just as utterly is only as something continuous.

For representation devoid of the concept, continuity readily becomes composition, namely an external reference of the ones to one another, in which the one is preserved in its absolute brittleness and exclusion. It has shown itself in the case of the one, however, that of its own self, in and for itself, it goes over into attraction, into its ideality, and that continuity is accordingly nothing external to it but belongs to the one itself and has its ground in its essence. It is this externality of continuity for the ones that atomism in general clings to, and to abandon it and to go into the concept, into the inner, is what constitutes the difficulty for representing.

The concept of pure quantity as against mere representation is what Spinoza, to whom it mattered above all, has in mind when he (Eth. P. I. Prop. XV. Schol.) speaks of quantity in the following way:

Quantitas duobus modis à nobis concipitur, abstrakte scilicet sive superficialiter, prout nempe ipsam imaginamur; vel ut substantia, quod a solo intellectu fit. Si itaque ad quantitatem attendimus, prout in imaginatione est, quod saepe et facilius à nobis fit, reperietur finita, divisibilis et ex partibus conflata, si autem ad ipsam, prout in intellectu est, attendimus, et eam, quatenus substantia est, concipimus, quod difficillime fit, — infinita, unica et indivisibilis reperietur. Quod omnibus, qui inter imaginationem et intellectum distinguere sciverint, satis manifestum erit.

More determinate examples of pure quantity, if such are wanted, are to be had in space and time, also in matter generally, in light and so forth, even in the I; only, as already remarked, what is to be understood by them is not the quantum, or magnitude in general insofar as this points first of all to the quantum. Space, time and so forth are extensions, pluralities, which are a going-outside-of-self, a streaming that nevertheless does not pass over into its opposite, into quality or into the one, but which, as coming-outside-itself, are a perennial self-producing. Space is this absolute being-outside-itself, which is just as much utterly uninterrupted, an otherness-and-again-otherness that is identical with itself; time is an absolute coming-outside-itself, a becoming-nothing that continually is again the becoming-nothing of this passing away; so that this self-generating of non-being amounts just as much to simple equality, to identity with itself.

As concerns matter as quantity, among the seven propositions preserved from Leibniz's first dissertation (l. page of vol. I of his works) there is one bearing on this, the second, which runs as follows: Non omnino improbabile est, materiam et quantitatem esse realiter idem. — In fact these concepts differ no further than in this, that quantity is the pure thought-determination, whereas matter is the same determination in external concrete existence. — To the I as well the determination of pure quantity belongs, since it is an absolute becoming-other, an infinite distancing or repulsion on all sides into the negative freedom of being-for-itself, which yet remains utterly simple continuity. — Those, by contrast, who balk at grasping plurality as simple unity, and who, besides the concept that of the many each is the same as the other, namely one of the many, — for what is spoken of here is not the many as further determined, not green, red and so forth, but the many considered in and for itself, — demand a representation of this unity as well, will find enough of the kind in those continuities, whose simple intuition gives immediately the deduced concept of quantity.

Remark 2

Into the nature of quantity, namely its being this simple unity of discreteness and of continuity, there falls the conflict or the antinomy of the infinite divisibility of space, time, matter and so forth.

This antinomy rests on nothing else than that discreteness has to be maintained no less than continuity. Maintained one-sidedly, discreteness yields as principle the infinite or absolute dividedness, and thus something indivisible; continuity maintained one-sidedly yields, by contrast, infinite divisibility.

Kant's Critique of Pure Reason, as everyone knows, sets out four (cosmological) antinomies, of which the second bears on the opposition constituting the moments of quantity.

An important part of the critical philosophy these Kantian antinomies will always remain; they above all brought down the metaphysics that went before, and they may count as a chief transition into the more recent philosophy. Great as their merit is, however, their presentation is highly imperfect: in part hampered and contorted within itself, in part skewed with respect to its result. Their remarkableness earns them a closer critique, one that will both throw more light on their standpoint and method and release the cardinal point, the one on which everything turns, from the useless form into which it has been squeezed.

Let me note first that Kant meant to lend his four cosmological antinomies a shine of completeness by way of the principle of division which he borrowed from his schema of the categories. Yet the deeper insight into the antinomic, or more truly the dialectical, nature of reason takes every concept whatever as a unity of opposed moments, on which the form of antinomic claims could be conferred. Becoming, existence and so forth, and any other concept, could accordingly yield its own peculiar antinomy, so that as many antinomies might be set out as there are concepts set out.

Kant, moreover, took up the antinomy not within the concepts themselves, but within cosmological determinations, which are an already concrete form. To have the antinomy pure and handle it in its simple concept, the thought-determinations ought to have been taken not in their application to, and their mixing with, the representation of world, space, time, matter and so forth, but ought to have been considered purely by themselves, apart from that concrete material, which contributes neither force nor power here, seeing that they by themselves make up the essence and the ground of the antinomies.

Kant gives this concept of the antinomy, that they are “not sophistical artifices, but contradictions on which reason must of necessity strike (in Kant's own wording);” — a view of importance. — “Reason, once it has seen into the ground of the antinomies' natural shine, is indeed taken in by it no longer, yet it is deceived still.” — For the critical resolution, namely that through what is called the transcendental ideality of the perceived world, arrives at no result beyond making the alleged conflict into something subjective, where it of course persists as the very same shine, that is, is left as unresolved as it was. Their genuine resolution can lie in this alone, that two determinations, being opposed and yet both necessary to one and the same concept, cannot hold good in their one-sidedness, each by itself, but possess their truth only in their sublatedness.

Looked at more closely, the Kantian antinomies hold nothing beyond the quite simple categorical claim put forward for each of the antinomy's two opposed moments. Wrapped about this simple categorical, or properly assertoric, claim, however, is a skewed and twisted scaffolding of ratiocination, by which a shine of proofs is produced and the merely assertoric character of the claim is to be concealed and rendered unrecognizable; as will become plain when they are looked at more closely.

The antinomy pertaining here bears on the alleged infinite divisibility of matter; it is founded on the opposition between the moments continuity and discreteness that the concept of quantity encloses within it.

Its thesis, in Kant's presentation, runs thus:

Each composite substance in the world is composed of simple parts, and there exists nothing anywhere save the simple, or what is put together out of it.

Placed here over against the simple, the atom, is the composite, a determination that lags far behind the steady or continuous. — As for the substrate lent to these abstractions, namely empirical substances in the world, which here signifies nothing more than things as they are perceptible to sense, it has no bearing on what is antinomic itself; space and time might have been taken with equal right. — And since the thesis speaks of composition rather than of continuity, it is properly an analytic or tautological proposition. Its immediate determination is just this, that the composite is not in and for itself but merely something linked externally, and is made up of an other. — But the other of the composite is the simple. Tautological, therefore, is the proposition that the composite is made up of the simple. — Once one asks of what something is made up, what is being called for is an other whose conjunction would constitute that something. Let ink be made up of ink once more, and the sense of the question about being made up has been missed; it goes unanswered. The one question still left is whether that of which we are speaking is to be made up of something or not. The composite, however, is precisely that which is not immediate, not in and for itself, but a mediated, a linked thing, made up of an other. Should it therefore be made up of composites again, the question stands: of what is the composite made up? just as it stood; for it lies in the composite itself. — Take the simple, which is the other of the composite and the very thing being asked after, merely for a relatively-simple that is on its own account composite again, and the answer turns once more into the answer that ink is made up of ink, so that the question is only repeated. Before representation there hovers, as a rule, only this or that composite, for which likewise this or that something would be given as its simple, itself perhaps a composite again on its own account. Here, though, the talk is of the composite as such. Neither, then, can it be asked afresh of what the simple is made up, on the ground that it is itself a composite; for the simple is no composite, but rather the other of the composite.

Kant's proof of the thesis takes, as do all his proofs of the remaining antinomic propositions, the roundabout way, one that will show itself to be quite superfluous, of being apagogic.

“Suppose, he begins, that composite substances were not composed of simple parts; then, were all composition sublated in thought, no composite part would remain, and, there being (by the assumption just laid down) no simple parts either, no simple one, hence nothing whatever would be left over, and accordingly no substance would have been given.” —

The inference is quite correct: where there is nothing but the composite, and one thinks all the composite away, nothing whatever is left over; — that will be granted, but this tautological surplus might have been dispensed with, and the proof might have started straight off with what comes next:

“Either it is impossible for all composition to be sublated in thought, or, once it has been sublated, there must remain something subsisting free of all composition, that is, the simple.”

“In the first case, however, the composite would again not be made up of substances (because in their case composition is but a contingent relation ³ of the substances, with out which these, as beings persisting for themselves, must subsist.) — Since this case contradicts the presupposition, only the second is left: namely that the substantial composite in the world is made up of simple parts.”

That ground, set down by the way in a parenthesis, is in fact the main thing, against which all that has gone before is wholly superfluous. The dilemma runs: either the composite is what abides, or not, but rather the simpl[e]. Were it the former, the composite, that abides, then what abides would not be the substances, since for these composition is only a contingent relation; but substances are what abides, and hence they are simple.

Plainly, without the apagogic detour that ground might have been appended straight away, as its proof, to the thesis: composite substance is made up of simple parts, because composition is a merely contingent relation of the substances, external to them therefore and of no concern to the substances themselves. — Grant that the contingency of composition holds good, and the essence is of course the simple. This contingency, though, on which alone everything turns, is not proved but flatly assumed, assumed in passing and in a parenthesis, as a thing that goes without saying or a side issue. That composition is the determination of contingency and externality does indeed go without saying; but by composition continuity was to be understood, and continuity is surely not to be disposed of in a parenthesis.

Within the apagogic detour, then, we see the very claim turn up that is supposed to issue from it. More briefly the proof may be put like this:

Assume that composite substances were not made up of simple parts. Now all composition can be sublated in thought, (being only a contingent relation;) hence once it is sublated no substances would be left over, unless they were made up of simple parts. Substances, however, we must have, since we assumed them; not everything is to vanish on us, something is to be left over, for we presupposed a persisting thing of that sort and called it substance; this something must therefore be simple.

One more thing belongs to the whole, a look at the concluding proposition; it runs as follows:

“From this it follows at once that the things of the world are without exception simple beings, that composition is in them merely an outer state, and that reason is bound to think the elementary substances as beings that are simple.”

Here the contingency of composition is adduced as a consequence, after having been earlier slipped into the proof parenthetically and put to work there.

Kant protests vigorously that in the conflicting propositions of the antinomy he is not after deceptions, so as to conduct (as the saying goes) a lawyer's proof. What the proof considered is to be charged with is not so much deception as a useless, tortured contortedness, needed only in order to produce the outward shape of a proof and to prevent its standing in full transparency that what was to come forth as a consequence was, in the parenthesis, the hinge of the proof.

The antithesis runs:

No composite thing in the world is composed of simple parts, and there exists nothing simple anywhere within it.

The proof is likewise given an apagogic turn, and is quite as blameworthy as the previous one, though in another way.

“Posit, so it runs, that a composite thing (as substance) is made up of simple parts. Since every external relation, and hence all composition out of substances, is possible only in space, the space it occupies must be made up of as many parts as the composite itself. Now space is made up not of simple parts but of spaces. Each part of the composite must therefore occupy a space.”

“The very first parts of everything composite, however, are simple.”

“Hence the simple occupies a space.”

“Now since everything real that occupies a space encloses within itself a manifold lying outside one another, and is hence composite, composite indeed out of substances, the simple would be a substantial composite. Which is self-contradictory.”

This proof may be called a whole nest (to borrow an expression occurring elsewhere in Kant) of faulty procedure.

To begin with, the apagogic turn is a shine wholly without ground. For the assumption that everything substantial is spatial, whereas space does not consist of simple parts, is a direct claim which makes up the immediate ground of what is to be proved and with which the whole business of proving is already done.

This apagogic proof then starts from the proposition: “that all composition out of substances is an external relation,” only, oddly enough, to forget it again at once. The inference runs on, namely, that composition is possible only in space, that space is not made up of simple parts, and that the real occupying a space is consequently composite. Composition having once been assumed as an external relation, spatiality, as the only element in which composition is held to be possible, is on that very account itself an external relation, one that does not concern the substances and leaves their nature untouched, as little as does everything else that may still be inferred from the determination of spatiality.

It is presupposed, further, that the space into which the substances are here transposed is not made up of simple parts; the ground being that space is an intuition, that is, on Kant's determination, a representation such as can be given only through a single object, and not a discursive concept. — From this Kantian distinction of intuition and concept, as everyone knows, much mischief with intuiting has arisen, and to be spared the labor of comprehending, people have stretched its worth and its domain into infinity. What belongs here is only this, that space, like intuition itself, must at the same time be comprehended; if, that is, one means to comprehend at all. Therewith the question would arise whether space, even were it as intuition simple continuity, would not have to be grasped according to its concept as made up of simple parts, or else space would enter the very antinomy into which substance alone was transposed. Grasped abstractly, the antinomy does in fact concern, as was recalled, quantity in general, and with it space and time no less.

But once the proof has assumed that space is not composed of simple parts, that ought to have been reason enough not to transpose the simple into an element unsuited to the determination of the simple.

In the remark to the proof of the antithesis the critical philosophy's otherwise fundamental representation is expressly brought to bear as well: that only as appearances do we have a concept of bodies, and that as such they cannot but presuppose space, the condition of the possibility of every outer appearance. If by the substances nothing is meant but bodies as we see, feel and taste them and so forth, then what they are in thinking is not really in question; what is at issue is only what sense perceives. Briefly put, then, the proof of the antithesis came to this: the whole experience of our seeing, our feeling and so forth shows us nothing but the composite; not even the best microscopes and the finest blades have yet let us strike upon anything simple. So reason too is not to want to strike upon anything simple.

If, then, we look more exactly at the opposition of this thesis and antithesis and free their proofs of all useless surplus and contortedness, the proof of the antithesis contains, — by transposing the substances into space, — the assertoric assumption of continuity, just as the proof of the thesis contains, — by assuming composition to be the mode in which the substantial is referred, — the assertoric assumption of the contingency of this reference, and with it of the absolute ones. To the severing of quantity's two moments, then, and to their direct assertion insofar as they are severed, the whole antinomy reduces itself. Taken by mere discreteness, substance, matter, space, time and so forth are divided outright, the one being their principle. Taken by continuity, this one is only a sublated one; the dividing remains divisibility, the possibility of dividing remains as possibility, without any actual arrival at the atom. — But so taken, continuity itself contains the moment of the atom; just as that dividedness has sublated every difference among the ones, — for each simple one is that which the other is, — and so likewise contains their absolute likeness and with it their continuity. Because each of the two opposed sides has its other on its own self, and neither admits of being thought apart from the other, it follows that no one of these determinations, taken by itself, possesses truth, but only their unity. Such is the genuine dialectical consideration of them, and such the genuine result.

Far more ingenious and profound than the Kantian antinomy just considered are those dialectical instances of the old Eleatic school, above all the ones bearing on motion, which are likewise founded on the concept of quantity and find their resolution in it. To take them up here as well would carry us too far afield; belonging as they do more closely to the concepts of space and time, they are to be handled where those are treated and in the history of philosophy. The highest honor do they do to the reason of those who devised them; the pure being of Parmenides is their result, for they exhibit the dissolution of all determinate being within itself, and are thus on their own selves the flux of Heraclitus. For that reason they merit a more searching treatment than the usual account of them as being just sophisms; an assertion that clings to perception after the fashion of Diogenes, so plain to ordinary human understanding, who, when a dialectician exhibited the contradiction contained in motion, is said to have strained his reason no further but to have pointed to what the eye sees by silently pacing back and forth, — an assertion and refutation certainly easier to produce than their genuine cognition and resolution, which presupposes an insight into the dialectical nature of the concepts.

Kant's resolution of the antinomy lies in this alone, that reason is not to fly beyond sensuous perception but is to take appearance as it stands. This resolution leaves the content of the antinomy itself lying to one side; it fails to reach the nature of the concept, which is essentially the unity of opposites, each of them, isolated on its own, null and in its own self nothing but the passing over into its other, as here quantity is this unity and therein the truth of the two determinations that make up the antinomy.

B. Continuous and Discrete Magnitude

1. The two moments, continuity and discreteness, are contained in quantity. At first it is their immediate unity. Thus it stands itself under the determination of continuity, and is continuous magnitude.

Or: continuity is at first, to be sure, only one of quantity's moments, and quantity reaches completion only with the other, discreteness. But continuity is just as essentially the whole as well; for only as unity of the discrete is it the cohering, solid unity. Continuity is therefore not moment alone, but no less quantity entire; and this, in that immediate and itself continuous unity, is not so much quantity as magnitude; — hence continuous magnitude.

2. Quantity, taken as immediate, is continuous magnitude. Quantity, however, is in no way something immediate; or, immediacy is a determinateness, a quality belonging to it, whose sublatedness quantity itself is. It therefore passes out of immediacy or indeterminateness into determinateness; but the determinateness immanent to it is the one. — Or: quantity taken immediately, continuous magnitude, is not quantity as such, but quantity as determinate; the genuine determinateness of quantity, however, is the one, and quantity is as discrete magnitude.

Discreteness is in general a moment of quantity, yet is itself the whole of quantity too, since quantity is essentially mediated, negative within itself, standing in the determinateness of the one, an at first indeterminate plurality of ones. Being-outside-one-another is what quantity is, and continuous magnitude is that being-outside-one-another as prolonging itself without negation, as a cohesion self-same within itself. Discrete magnitude is that outside-one-another as non-continuous, as broken off. With this multitude of ones, however, there is not present once more the multitude of atoms and the void. Rather, since discrete magnitude is quantity, the continuity sublated in it is itself continuous. Wherein this continuity in the discrete lies is that the ones are equal to one another, or that they have one and the same unit. Discrete magnitude is accordingly the outside-one-another of the many ones as of the equal, not the many ones simply, but as the many of a unit.

Remark

In the ordinary representation of continuous and of discrete magnitude it is overlooked that each of these magnitudes has both moments on it, continuity as well as discreteness, and that their difference is constituted solely by which of the two moments counts as the determination lying at the ground, a determination that is not, however, the only one present in such a magnitude. Continuous magnitude, though, does not have discreteness on it in such a way that it would be made up of ones, for the ones are sublated within it, but has it as being-outside-one-another; it is not mere likeness with itself, but is that which has essentially sublated and preserved the one within it, the likeness of the being-outside-itself of repulsion. Space, time, matter and so forth are quantities which have a steady magnitude, in that they are repulsions of their own selves, a streaming issuing-out-of-self that is not a passing over into an other. They have the absolute possibility that the one be posited upon them at any point whatever; they have this possibility not as the empty possibility of a bare otherness (as one says, it would be possible that a tree stood in this stone's place) but rather they contain the principle of the one on their own selves.

Conversely, continuity is not to be overlooked on discrete magnitude; this moment, as has been shown, is the one as unit.

Continuous and discrete magnitude may be considered species of magnitude, but only insofar as magnitude is posited under no determinateness external to it, but under the determinateness of its own moments. In the ordinary passage from genus to species one lets external determinations come to the genus according to some ground of division external to it. — Further, however, continuous magnitude passes over into discrete magnitude, because the former is indeed magnitude in one determination, but immediacy or continuity is not the peculiar, immanent determinateness of quantity; rather this is the one. Or magnitude has a real determination only as discrete, for with that difference or otherness enters upon it on its own self. Continuous magnitude is only steady, undifferentiated on its own self, differentiated only as against the discrete magnitude standing over against it. — Yet the real determination is not yet complete in discrete magnitude as such, in the ones, which are steady through their unit; there belongs to it in addition the determination of this their continuity by the one.

C. Limitation of Quantity

Discrete magnitude has, in the first place, the one for its principle; in the second place it is essentially continuous, being the one at once as sublated, as unity, the one made broad, so to speak, the one carried on. Insofar as the one, or the many ones, are just as essentially and immediately unity, however, nothing is thereby posited but quantity at large, or, insofar as the one stands sublated in the unity and as many ones sinks together into the unity, continuous quantity. But this latter has conversely gone over into discrete magnitude, and continuity is the moment sublated within the one. On the one hand, then, the one is indeed widened out into unity, and this unity has not vanished but is on the contrary essentially at hand, yet it stands posited along with a negation; upon the unity the one becomes limit. Continuity is an essential moment and carries the negation upon it, yet is at the same time marked off from this its negation, which in that determination is limit. This limit, besides being referred to the unity and being the negation upon it, is also referred to itself; being what it is in itself, namely one, it is a limit that encloses and takes in. What the limit is here does not first mark itself off from the being-within-itself, from the something of its existence; being one, it is immediately this negative point itself. On the other side, the being that undergoes limiting is here essentially continuity, which reaches out past the limit and past this one. Discrete quantity in its truth is therefore one quantity, or quantum.

Or magnitude is at first the immediate unity of continuity and discreteness. As quantity it is the unity of these moments returned into itself; as this their negative unity it carries upon it the difference which, in immediate or continuous magnitude, had merely vanished or was merely possible.

First, this negative unity is not only the unity of continuity and discreteness as abstract moments, but the unity of these same taken as continuous and discrete magnitude. Between continuous and discrete magnitude there is at large no genuine difference. — Secondly, however, this negative unity is not a determinateness into which magnitude goes over, but one which magnitude has upon its own self; it is the one, wherein quantity posits itself as in its own determinateness. Since quantity is at large sublated quality, since upon its own self it is infinite, there is present in its movement no going over into absolute otherness; rather its determining consists just as much only in the stepping forth of the moments already at hand within it.

Chapter 2. Quantum

Quantum is real quantity, as existence is real being. It is at first quantity with a determinateness or limit at large, but in its complete determinateness it is number. Quantum marks itself off

secondly into extensive and intensive quantum, whose difference is on the one hand a matter of indifference, so that one and the same numerical determinateness is at hand in the one manner just as much as in the other. On the other hand, however, therein lies the difference of quantum upon its own self, which

thirdly, as being in itself external to its own self, goes over into quantitative infinity.

A. Number

Quantity is quantum, or has a limit. Insofar as continuous and discrete magnitude are looked upon as kinds of magnitude, quantum is the one as well as the other taken as limited; or each of them has a limit; upon the continuous the limit stands as limit of continuity; upon the discrete as negation upon the plurality, which for itself is at large an undifferentiated multitude. But the difference of these kinds carries no significance here any longer.

At first, as negative unity of the difference, of continuity and of discreteness, quantity is a being-within-itself in which the difference is sublated, or which marks itself off from the difference. Quantity is in itself being-for-itself sublated; it is therefore already in and for its own self indifferent toward its limit.

But as little as something has a limit marked off from its being-within-itself, so little is that the case here. The limit is that whereby something severs itself from an other and refers itself to its own self; through its limit, then, something is within itself and not within others; its limit is therefore its being-within-itself. To quantity at large it is not immediately a matter of indifference to have the limit, or to be a quantum; for it holds the one, absolute being-determined, within its own self, as a moment of its own.

This one is the principle of quantum; it is not, however, the abstract one, but the one as the one of quantity. On that account it is first continuous; it is unity; secondly it is discrete, and thereby is within itself a plurality of ones, which however have likeness with one another, that continuity, the same unity. Thirdly, this one is negation of continuity and of discreteness; and since these make up its moments, it is thus the negation of its own self; but since it just as immediately is, this negation of itself is at the same time a shutting out of its non-being from itself, a determining of itself over against other quanta. So far, the one is a limit that refers itself to itself, that encloses, and that shuts an other out.

It has been said that the moments of continuity and of discreteness are held within the limiting one. Insofar as in this limiting the one is what determines, or is the whole at large in the form of discreteness, continuity is at hand as the unity of the many ones; it is the one, insofar as the one is the principle, or the many are all ones. This unity thereby marks itself off at the same time from the many as such. But continuity is also the indeterminate of plurality at large, and so far the one stands upon it as limit. The many as discrete many or as ones admit of no limiting, for as beings-for-themselves they hold the limit within them as a sublated moment, and are absolute negativity over against it. A multitude as such is no limit upon the many themselves, it is a determination wholly external to them. The limit is upon them only insofar as they are the many that are alike in this, that they are many; this their continuity is the indeterminate being upon which the negation stands as limit. At the same time, however, it is not limit upon continuity insofar as continuity is as the unity, for the unity makes up precisely the moment marked off from the many, from the discrete and therewith from the negative at large.

The quantum appears therefore in its being-determined-in-itself not as continuous but as discrete magnitude, as showed itself in the transition to it as well. Quantum as limited continuous magnitude is an indeterminate limit; for such a limit does not hold the continuous as many ones, and hence not in the form of being-determined-in-its-own-self either. — The moments of continuity and discreteness, however, since they are within quantum as their unity, are themselves the being-determined-in-itself that makes up their unity. Continuity is as unity, and equally as many ones. Discreteness, or the difference, is furthermore therein not merely the indeterminate difference of plurality at large, but the determinate difference of the unity over against plurality. This, however, is at the same time not a merely qualitative difference, for the many are ones, they have the same unity. — Furthermore, the many is not marked off from the limit or from the limiting one; it makes up the continuity as well as the discreteness of the enclosing one itself, for it is itself continuous and discrete; quantum, or the limit of quantity as such, is itself quantity.

Quantum thus determined upon its own self is number. It is quantum in its determinateness, because it is only a comportment of the one, of what is absolutely determined-in-itself, toward its own self, a comportment which in its difference from itself, that is, in being-determined as determined through an other, stays like itself, or wherein this difference is just as immediately a sublated one.

Number has first the one as its principle, and so far the one is the continuous one, or the unity. Furthermore, this unity is repell[e]d from itself; it is as many ones; but these many themselves make up only the one, insofar as the one is what limits. The many of number make up quantum; plurality is a moment of the limiting one; the many that are set apart and enclosed by the limit are not outside their limit; this limit is the one itself, and this one is quantity and the discrete or the continuous itself, which the many are. These many make up the amount of number. In one respect the amount marks itself off from the one taken as the unity, yet at the same time it is only an amount of such unities. In another respect it is not a plurality over against the enclosing, limiting one; rather the amount itself makes up this limitation, which is a determinate quantum; the many make up one number, one two, one ten, one hundred and so forth.

Number has therefore for its moments the unity and the amount, and is itself the unity of the two. The former makes up the moment of continuity, the latter that of discreteness, as they are within quantum in the shape of number. The unity marks itself off from the amount, and at the same time the two are joined in number itself as the negative one, in the ten, in the hundred, which is quite as much unity itself as it is this amount.

The limiting one is being-determined over against an other, the marking off of number from others. But this marking off does not turn into qualitative determinateness; it stays quantitative, falling only to the comparing external reflection; number itself stays returned into itself and indifferent toward the other, or is not referred to it.

This indifference of number toward an other is number's essential determination; it makes up its being-determined-in-itself, but at the same time its own externality. — As regards the first, quantity itself is not indifferent toward the limit; it has the limit upon its own self in its moment of discreteness. But this limit is not the reference to an other as other; it is indifferent toward that. This indifference consists in the negation of quantity, the one, being infinitely referred to itself and having otherness upon its own self as sublated; furthermore, the peculiar repulsion of the one that is for itself has sublated itself as well. The one of number is so far a numerical one; an absolutely in-and-for-itself determined one, which at the same time has the form of immediacy, and to which the reference to an other is therefore wholly external. As a one that is number, it has furthermore the determinateness, insofar as this is reference to an other, within its own self, in its difference of unity and of amount. This difference, however, is at the same time quantitative, since the amount is plurality of unities, and plurality is the discrete moment of number itself, or its one.

But quite as much is quantity itself the sublated determinateness, the difference grown external. The one is the principle of number as numerical one, that is, as an indifferent one to which the reference to an other is wholly external. Number, however, is the reference of this one; it is the unity that as many ones returns into itself. But because they are numerical ones, this reference and return into self is to them quite as much something indifferent. The limit of quantum consists in the amount, in the plurality external to itself, which has for its principle or unity the indifferent one. Number in this way is being-determined-in-itself, yet the being-determined-in-itself of externality, or a being-determined-in-itself that is just as immediately the complete externality of being-determined. Quantity is infinity within itself. Number, more precisely, is this infinity as determined in itself within its own self, and as an equally absolute sublatedness or externality of being-determined.

Remark 1

Spatial magnitude and numerical magnitude are commonly regarded as two kinds, as though spatial magnitude were for itself just as much determinate magnitude as numerical magnitude is; their difference would rest solely upon the differing determinations of continuity and discreteness; as quantum, however, they would stand upon the same level. Geometry does indeed have, broadly speaking, continuous magnitude in spatial magnitude for its object, and arithmetic discrete magnitude in numerical magnitude. But with this unlikeness of object there goes no likeness in the manner and completeness of the limiting, or of the being-determined. Science essentially considers the determinatenesses of these objects insofar as they are quanta and comport themselves on this side. The manner of limiting, however, is in the two objects likewise different. Spatial magnitude has only a limiting at large; insofar as it is to be considered as a quantum determined in itself, it stands in need of number. Geometry, too, does not consider spatial figures according to a magnitude determined in and for itself; it does not measure them; is no art of mensuration; but only compares them, that is, it considers them only as relative quanta, according to a determination of magnitude that they have toward others. In its definitions as well the determinations are drawn in part from the equality of sides, of angles, from the equal distance. Thus the circle, resting as it does solely upon the equality of the distance of all points possible within it from a midpoint, calls for no number to determine it. These determinations, founded upon equality or inequality, are geometrical through and through. But they do not suffice, and for others, the triangle, the quadrangle for instance, number becomes requisite, number holding within it the being-determined-in-itself, not the being-determined by the aid of an other, hence not by comparison.

Number, however, holds this determinateness in itself, because the one is its principle. Spatial magnitude does indeed have in the point the determinateness answering to the one; but the point, insofar as it comes outside itself and becomes an other, turns into the line; because it essentially is only as a one of space, in the reference it turns into a continuity wherein punctuality, the being-determined-in-itself, the one, is sublated. Insofar as the being-determined-in-itself is to preserve itself within being-outside-itself, the line must be represented as a multitude of ones, and the limit must take up into itself the determination of the many ones, that is, the line's magnitude — and likewise that of the other determinations of space — must be taken as number.

Remark 2

As is well known, Pythagoras set forth relations of reason or philosophemes in numbers, and in more recent times calculating has been taken as equivalent in meaning to thinking, or, as it has been put more precisely, to pure real thinking. — In pedagogical respect, too, number has been held for the object best suited to inner intuiting, and the calculating occupation with number's relations for that activity of spirit wherein spirit brings its ownmost relations, and the fundamental relations of essence at large, before intuition. — How far number may be entitled to this high worth follows from its concept, as that concept has emerged.

Number is the absolute determinateness of quantity; its element is the difference grown indifferent. It is therefore determinateness in itself, determinateness at the same time posited wholly and only externally. Arithmetic is on that account an analytic science, because all the linkings and differences occurring upon its object do not already lie within the object itself but are inflicted upon it wholly from without. It has no concrete object that would possess inner relations in itself, relations at first hidden from knowledge, not given in the immediate representation of the object, but such as would first have to be brought out by the labor of cognition. On the contrary, the object's relations are put into it purely by reflection itself; in its calculating business reflection therefore has to do only with such determinations as it has put in. Because these references accordingly hold no genuine otherness within them, arithmetic has nothing to do with what is opposed; it does not have the task of the concept at all; it moves only along the thread of its own identity, and comports itself in its activity purely analytically.

On account of the indifference of what gets linked toward the linking, a linking that wants necessity, thinking finds itself here in an activity that is at the same time the utmost divestment of its own self, in the violent activity of moving within thoughtlessness and of linking what is capable of no necessity. For the object, number, is only the thought, and the abstract thought, of externality itself. In every other concrete object thinking is likewise external to itself, yet such an object is at the same time upon its own self something inwardly linked and necessary; thinking accordingly finds essential references within it; number, by contrast, has for its principle what is essentially without reference.

On account of this pure externality and this want of determination of its own, thinking has in number an infinitely determinable matter that offers no resistance through references peculiar to it. Number is at the same time the abstraction from all sensuous manifoldness, and has kept nothing of the sensuous but the abstract determination of externality itself. Through this abstraction it lies, so to speak, nearest to thought; it is only the pure thought of thought's own divestment.

Spirit, which raises itself above the sensuous world and knows its essence, may therefore, while it searches for an element to serve its pure representation, the expression of its essence, hit upon choosing number, this inward, abstract externality, before it grasps thinking itself as this element and wins for its presentation the purely spiritual expression. That is why in the history of science we see number put to use for the expression of philosophemes, before thinking had found the expression that holds only the abstract thought itself. Number makes up the last stage of this expression's imperfection; with number, thinking, which has already left sensuous representation behind for its presentation, leaves behind altogether even the pure thought of externality.

Now since thinking lays its determinations down in this element, they sink therein, on account of the element's nature just considered, immediately into want of concept; or the thoughts become, within it as the thoughtless, something thoughtless. The thoughts, what is most alive, most mobile, grasped only in relating, become in this element of being-outside-itself dead, motionless determinations. The richer in determinateness and in reference the thoughts grow, the more confused on the one side and the more arbitrary and drained of sense on the other does their presentation in numbers become. The one, the two, the three, the four, as henas or monas, dyas, trias, tetraktys, still lie very near to simple concepts; but where numbers are to go over into further relations of the concept, wanting to keep them still near the concept is in vain.

But even if the concept is held fast only in the one, two, three, four, if these are to be thought and set in movement, this is the hardest movement of thinking; for instead of having to do purely with itself and being at home with itself, thinking has at the same time immediately to struggle with its own divestment. It moves in the element of its opposite, of referencelessness; its business is the labor of derangement. To grasp, for instance, that one is three and three one, is so hard a demand because the one, which is dominant in number, is what is without reference, and hence does not display upon its own self the determination whereby it goes over into what is opposed to it, but is rather just this, to shut out and refuse such a reference outright.

Since, then, thought purifies itself of sensuous stuff, the last stage is that the sensuous, the external, becomes for it the pure thought of this externality, becomes number, and that it takes number for the element and matter of its own self. But it has still to overcome this abstract thoughtlessness too, and to grasp its determinations in its own immediate form, namely as being, becoming and so forth, as essence, identity and so forth.

As for the view of common calculating itself, that it is thinking, because it „is a determination of the relative plurality, or of the determinable „repeatability of one and the same in an other, through the absolute unity of the identical,“ then to that extent calculating is of course thinking. But reading, writing and so forth is quite as much thinking; for in these too there is a determination of a relative many through an identity. Calculating has, as has come out, on the one side the advantage over other functions of thinking or of consciousness in the abstractness of its matter or element; but on the other side it stands behind them through the conceptlessness of the one, which is indeed something purely identical with itself and repeating itself in an other, namely in the many, yet is therein to hold itself essentially as without reference and to remain external to its own other, so that the genuine, namely the comprehending, unity of thinking is to be absent within it.

What is to be made of the use of number and of calculating, insofar as it is supposed to make up a chief pedagogical foundation, follows of itself from what has gone before. Number is a non-sensuous object, and the occupation with it and its combinations a non-sensuous business; spirit is thereby held to reflection into itself and to an inward abstract labor. On the other side, however, since external, thoughtless difference lies at the ground of number, that business turns at the same time into a thoughtless, mechanical business, and the straining of powers consists chiefly in killing the liveliness of spirit, in suppressing the concept, in holding fast what is void of concept and in combining it without concept. Because calculating is so very external, and hence mechanical, a business, machines have, as is well known, been made that carry out the arithmetical operations most perfectly. Were one to know of the nature of calculating this single circumstance alone, there would lie in it the verdict upon what it comes to when calculating is made spirit's chief business, and spirit is laid on the rack of perfecting itself into a machine.