c. Many Ones. Repulsion

The one and the void make up being-for-itself in its nearest existence. Negation is what each of these moments has for its determination, and each is at the same time posited as an existence. In respect of the former, the one and the void are the relating of negation to negation, as of an other to what is other to it; the one, negation determined as being, the void, negation determined as non-being. But essentially the one is only a relating to itself as relating negation, i.e. it is itself that which the void, lying outside it, is supposed to be. Both are, however, also posited as an affirmative existence, the first as being-for-itself as such, the second as indeterminate existence in general, and as relating themselves to one another as to another existence. The being-for-itself of the one, however, is essentially the ideality both of existence and of the other; its relating is not to an other, but to itself alone. Yet insofar as being-for-itself stands fixed as one, as something that is for itself, as something immediately at hand, its negative relating to itself is at the same time a relating to something that is; and since that relating is quite as much negative, whatever it relates to stays determined as an existence and an other; as essentially a relating to its own self, what is other is not indeterminate negation, is not the void, but is a one just the same. The one is therefore a becoming into many ones.

Properly speaking, however, this is not so much a becoming; for becoming is a passing over from being into nothing, whereas the one becomes only a one. The one, as the term related, contains the negative as relating, and so has that negative upon its own self. In place of becoming there is present, first, the one's own immanent relating; and secondly, insofar as this relating is negative while the one at the same time is something that is, the one repels its own self away from itself. The negative relating of the one to itself is repulsion.

This repulsion, thus as the positing of the many ones yet through the one itself, is the one's own coming-outside-itself, though to such as lie outside it and are themselves nothing but ones. This is repulsion in accordance with the concept, repulsion as it is in itself. Distinct from it is the second repulsion, the one that first hovers before the representation of external reflection, not as a generating of the ones but only as a mutual holding-off of presupposed ones already at hand. It remains to be seen how that repulsion which is in itself determines itself into the second, the external one.

To begin with, it must be settled what determinations the many ones as such possess. The becoming into many, or the being-produced of the many, vanishes immediately as a being-posited; what is produced are ones, not for an other, but relating themselves infinitely to their own selves. The one repels only itself from itself, and hence does not come to be but already is; what is represented as the repelled is likewise a one, something that is; repelling and being-repelled fall to both in like manner, and make no difference.

The ones are thus presupposed over against one another; – posited through the repulsion of the one from its own self; posited beforehand as not posited; their positedness is sublated, they are beings over against one another, as relating themselves only to themselves.

Plurality accordingly appears not as an otherness, but as a determination altogether external to the one. In repelling its own self the one remains a relating to self, just as does that which is at first taken as the repelled. That the ones are others over against one another, gathered up into the determinateness of plurality, is therefore no concern of the ones. Were plurality a relating of the ones themselves to one another, they would limit each other and would have a being-for-other affirmatively upon them. Their relating, – and this they do have through their unity as it is in itself, – as it is here posited, is determined as none at all; it is once more the void posited earlier. It is a limit of theirs, yet external to them, a limit wherein they are supposed not to stand for one another. The limit is that wherein the limited just as much are as are not; the void, however, is determined as pure non-being, and this alone makes up their limit.

The repulsion of the one from its own self is the explication of what the one is in itself; infinity, however, as laid apart, is here infinity that has come outside itself, and outside itself it has come through the immediacy of the infinite, of the one. It is just as much a simple relating of one to one as, rather, the absolute relationlessness of the ones; the former in keeping with the simple affirmative relating of the one to itself, the latter in keeping with that very same relating taken as negative. Or, put otherwise, the plurality of the one is the one's own positing; the one is nothing other than the negative relating of the one to itself, and this relating, hence the one itself, is the many ones. Yet just as much is plurality downright external to the one; for otherness is precisely what the one sublates, and repulsion is the one's relating to itself, its simple equality with its own self. Infinity is what the plurality of the ones amounts to, a contradiction unconstrainedly bringing itself forth.

Remark

Mention was made earlier of Leibniz's idealism. It may be added here that this idealism, setting out from the representing monad which is determined as something that is for itself, got no further than the repulsion we have just examined, and indeed no further than plurality as such, wherein each of the ones is merely for itself, indifferent towards whether others exist or are for themselves, or wherein others are simply not there for the one at all. For itself the monad is the whole self-enclosed world; none of them stands in need of the others; but this inner manifoldness, which the monad has in its representing, alters nothing in its determination of being for itself. Leibniz's idealism takes up plurality immediately as something given and does not comprehend it as a repulsion of the monad; and so it possesses plurality only from the side on which it is abstractly external. The concept of ideality is missing from atomism; atomism does not grasp the one as something which in its own self contains the two moments of being-for-itself and of being-for-it, hence as something ideal, but only as something simply and drily for itself. But it does get beyond the merely indifferent plurality; the atoms come into a further determination over against one another, even if in a way that is strictly inconsistent; whereas in that indifferent independence of the monads, on the contrary, plurality stays put as a rigid fundamental determination, so that their relating falls only within the monad of monads, or within the philosopher who observes them.

C. Repulsion and Attraction

a. Exclusion of the One

The many ones are beings; their existence, or their reference to one another, is non-reference, something external to them; – the abstract void. Yet they themselves are this negative reference to themselves, now as to others that are; – the contradiction already exhibited, infinity, posited in the immediacy of being. What has been repelled by repulsion is now something repulsion finds immediately at hand. Exclusion is what it is in this determination; those many ones which it has neither generated nor posited are the only ones the one repels from itself. Reciprocal, or on every side, such repelling is relative – held within limits by the being which the ones have.

Plurality is, to begin with, otherness not yet posited; as for the limit, it is merely the void, merely that wherein the ones are not. And yet within the limit they too are; the void is where they are, or, what says the same, their repulsion constitutes their common reference.

What stands posited as the existence of the many ones is this reciprocal repulsion; their being-for-itself it is not, for on that they would count as many only when distinguished within a third thing; it is rather their own distinguishing, the distinguishing that keeps them in being. – They negate one another and posit one another as such as are only for-one. Yet at the same time, and equally, they negate this, that they should be only for-one; that ideality of theirs they repel, and they are. – Separated, then, stand the moments which in ideality are joined without qualification. Being for itself, the one is for-one as well, and yet that one for which it is turns out to be the one itself; immediately, therefore, its distinguishing of itself from itself is sublated. In plurality, though, a being belongs to the one that is distinguished; and so being-for-one, determined as it is within exclusion, amounts to a being-for-other. Repelled by an other, each is accordingly sublated and made into something which is not for itself but for-one, and for another one at that.

What the being-for-itself of the many ones now proves to be is their self-preservation, mediated by the repulsion they exercise against one another, wherein each sublates the rest and posits them as a bare being-for-other; and yet that same self-preservation lies just as much in repelling this ideality, so that the ones get posited as not being for-an-other. Such preservation of themselves by way of their negative reference to one another is, however, much rather their dissolution.

The ones not only are, they also preserve themselves through their reciprocal exclusion. First, then, that whereby they were supposed to have the firm hold of their diversity against being negated is their being, and specifically their being-in-itself over against their reference to other; this being-in-itself consists in their being ones. But all are this; in their being-in-itself they are the same, instead of having in it the fixed point of their diversity. Second, their existence and their comportment toward one another, that is, their positing themselves as ones, is the reciprocal negating; but this too is one and the same determination of all of them, through which, accordingly, they rather posit themselves as identical; just as, because they are in themselves the same, the ideality that is supposed to be posited in them by others is their own, and hence something they just as little repel. – By their being and by their positing, therefore, they are only One affirmative unity.

Our comparison is this consideration of the ones: that under both their determinations – as they are, and as they refer to one another – nothing shows itself in them but one and the same, and so their indistinguishability. – Yet what is posited in them within their very reference to one another is also to be looked at. – They are, this is presupposed in that reference, – and they are only insofar as they negate one another and at the same time hold off from themselves this ideality of theirs, their being-negated, that is, negate the reciprocal negating. But since they are only insofar as they negate, their being is negated in the negating of this their negating. Granted, so far as they are, this negating would not negate them, being for them nothing but an externality; the other's negating rebounds off them, brushing their surface and nothing more. Only through the negating of the others, however, do they come back into themselves; as this mediation alone are they, and their return is what constitutes their self-preservation, their being-for-itself. Their negating effects nothing, given the resistance that the beings put up as such, or as negating; hence there is for them no coming back into self, no preserving, and no being.

A moment ago the consideration was made that the ones are the same, each of them being a one just as the other is. Nor is this merely our referring, an external bringing-together; referring is what repulsion itself is; in excluding the ones, the one refers to them, to the ones, which means: to itself. The negative comportment of the ones toward one another is thus nothing but a coming-together-with-itself. Their repelling passes over into this identity, and the identity sublates the diversity and externality which, as excluding, they were rather supposed to maintain over against one another.

This positing of the many ones into One one is attraction.

Remark

Self-subsistence driven to the extreme point of the one that is for itself is self-subsistence abstract and formal, destroying its very self; it is the highest and most stubborn of errors, one which takes itself for the highest truth; – in shapes more concrete it makes its appearance as abstract freedom, as the pure I, and then further as evil. Here is a freedom so far astray as to place its essence in this abstraction, flattering itself that in such being-at-home it gains itself in purity. More determinately, the error of this self-subsistence lies in taking for negative, and behaving negatively toward, that which is its own essence. Comportment negative toward itself is therefore what it is, a comportment which, out to gain its own being, destroys that being, and its whole doing is nothing but the manifestation of how null the doing is. Reconciliation consists in acknowledging that against which the negative comportment is directed to be rather one's own essence, and it comes about only as a letting go of the negativity of one's being-for-itself, instead of holding fast to it.

An old proposition has it that the one is many, and more particularly, that the many are one. On this point the observation must be repeated that the truth of the one and of the many, once put into propositions, takes on a form unsuited to it, and that such truth admits of being grasped and uttered only as a becoming, as a process, repulsion and attraction, not as being, in the manner in which a proposition posits being as a unity at rest. Plato's dialectic in the Parmenides, dealing with the derivation of the many out of the one, namely out of the proposition: one is, was mentioned and recalled above. The inner dialectic of the concept has been set out; easiest to grasp, however, is the dialectic of the proposition that many are one, grasped as external reflection; and here it may well be external, since the object too, the many, is precisely what is external to one another. Compare the many with one another and it emerges at once that each is determined exactly as every other is; each is a one, each is one among the many, each excludes the rest; – so that utterly they are but the same, and utterly there is but One determination at hand. Such is the fact, and the whole business is to take in this simple fact. If the stubbornness of the understanding balks at taking it in, the reason is that difference hovers before it too, and does so with right; but difference is as little dispelled by that fact as, assuredly, that fact holds good regardless of the difference. One might, as it were, console the understanding for its plain taking-in of the fact of unity with the promise that difference will make its entrance again.

b. The One One of Attraction

Repulsion is the self-splintering of the one, at first into many whose negative comportment has no power, since each presupposes the rest as beings; it amounts only to the ought of ideality, whereas in attraction that ideality is realized. Over into attraction repulsion passes, the many ones into One one. At first the two, repulsion and attraction, stand distinguished, repulsion as the reality of the ones, attraction as the ideality posited in them. Attraction refers to repulsion in this manner, that repulsion serves it as presupposition. The matter for attraction is furnished by repulsion. Were there no ones, nothing would be there to attract; if one represents attraction as going on and on, the ones being consumed, an equally unceasing generation of ones is presupposed; the sensuous representation of attraction in space keeps the stream of ones-being-attracted flowing on; where the atoms disappear at the attracting point, another crowd of them steps forth out of the void, and, if one likes, on into infinity. Suppose attraction were represented as accomplished, the many brought to the point of One one: then nothing would be at hand save an inert one, and attracting there would be none. Ideality as existent in attraction has upon itself, further, the determination of its own negation, the many ones to which it is the reference, and attraction cannot be severed from repulsion.

Attracting belongs at the outset to each of the many ones equally, insofar as they are immediately at hand; none enjoys a privilege over another; so there would be an equilibrium in attracting, properly an equilibrium of attraction and repulsion themselves, and an inert repose lacking any existent ideality. Yet a privilege of one such one over another cannot be spoken of here, for it would presuppose a determinate difference between them; attraction is rather the positing of that undifferentiatedness of the ones which is already at hand. Attraction itself is the first positing of a one distinguished from the rest; the rest are merely immediate ones, ones meant to keep themselves in being by repulsion; out of their posited negation, though, the one of attraction emerges, determined accordingly as the mediated, as the one posited as one. Immediate as the first ones are, they do not come back into themselves in their ideality, but possess that ideality in an other.

The One one, by contrast, is ideality realized, ideality posited upon the one; attracting is what it does through the mediation of repulsion, and this mediation it holds within itself as its determination. Hence the attracted ones are not swallowed into it as into a point, which is to say, it does not sublate them abstractly. Since repulsion is contained in its determination, the ones are at the same time preserved within it as many; by its attracting the One one puts, so to speak, something out before itself, and so gains an extent or a filling. Unity of repulsion and attraction in general is thus present in it.

c. The Connection of Repulsion and Attraction

Into the difference of their reference to one another has the difference of one and many determined itself, and that reference falls apart into two references, repulsion and attraction, of which each stands at first self-subsistently outside the other, so that they nevertheless hang together essentially. Their unity, still indeterminate, has yet to emerge more closely.

Repulsion, being the fundamental determination of the one, makes its appearance first, and as immediate – so too its ones, generated by it yet posited at the same time as immediate – and on this account repulsion is indifferent toward attraction, which, repulsion once presupposed in this way, comes to it from without. Attraction, on the other side, is not presupposed by repulsion, so that in repulsion's positing and being it is to have no share, which is to say that repulsion would not already carry upon itself the negation of itself, nor the ones already be negated upon themselves. Taken in this manner, repulsion is before us abstractly for itself, and attraction likewise, over against the ones as beings, has the side of an immediate existence and comes to them, out of its own resources, as an other.

Take mere repulsion, then, thus for itself, and it is the scattering of the many ones into the indeterminate, beyond the sphere of repulsion itself; for repulsion is just this, that the reference of the many to one another be negated; referencelessness, repulsion being taken abstractly, is its determination. But repulsion is not the void merely; the ones, if devoid of reference, are not repelling, not excluding, and this is precisely what makes up their determination. Negative though it be, repulsion is nonetheless essentially reference; the mutual holding-off and fleeing is no liberation from what is held off and fled from, and whatever excludes still stands in connection with that which it excludes. This moment of reference, however, is attraction, which lies accordingly within repulsion itself; attraction negates that abstract repulsion in which the ones would be beings merely referring to themselves, and not excluding ones.

Since, though, the starting point was taken in the repulsion of existent ones, whereby attraction too got posited as stepping up to them from outside, the two, inseparable as they are, are still held apart as distinct determinations; and yet it has turned out that repulsion is not merely presupposed by attraction, but that a back-reference of repulsion to attraction takes place just as much, so that repulsion has in attraction its presupposition no less.

On this determination they are inseparable, and each is at the same time determined as the ought and the restriction of the other. Their ought is their abstract determinateness as something being in itself, a determinateness thereby directed utterly beyond itself, referring to the other, so that each is by means of the other as other; their self-subsistence lies in this, that within such mediation they are posited for one another as an other determining. – Repulsion as positing of the many, attraction as positing of the one, this latter being at once negation of the many, and the former negation of the ideality that the many have in the one, so that attraction likewise is attraction only by means of repulsion, as repulsion is repulsion by means of attraction. That therein, however, mediation with itself through an other is in fact rather negated, and that each of these determinations is mediation of itself with itself, follows from a closer consideration of them and carries them back into the unity of their concept.

First, that each presupposes itself, referring in its presupposition only to itself, is already at hand in the comportment of a repulsion and an attraction that are as yet merely relative.

Relative repulsion is the mutual holding-off of many ones at hand, ones supposed to be found already there as immediate. But that many ones should be is repulsion itself; whatever presupposition it might have is nothing but its own positing. Then too, the determination of being which would accrue to the ones over and above their being posited – that whereby they would be there in advance – belongs equally to repulsion. Repelling is that by which the ones manifest and preserve themselves as ones, by which they are as such. Repulsion itself is their being; repulsion is thus nothing relative over against some other existence, but comports itself throughout to itself alone.

What attraction posits is the one as such, the real one, and against it the many in their existence get determined as merely ideal, as vanishing. Attraction, then, presupposes itself straightaway, namely in the determination whereby the other ones are ideal, ones which otherwise are supposed to be for themselves and to be repelling for others, hence for anything that attracts as well. Over against this determination of repulsion, ideality does not first accrue to them through relation to attraction; ideality is rather presupposed, being the ideality of the ones that is in itself, inasmuch as they, as ones – the one represented as attracting included – are undifferentiated from one another, one and the same.

That each of the two determinations thus presupposes itself, each on its own, further means that each contains the other within itself as a moment. Self-presupposing in general is, in one and the same act, a positing of oneself as the negative of oneself, – repulsion, while what gets presupposed therein is the same as what presupposes, – attraction. Each being in itself a moment only, each thereby passes over out of itself into the other, negates itself upon its own self, and posits itself as its own other. Since the one as such is the coming-out-of-itself, is itself only this, namely to posit itself as its own other, the many, while the many are likewise only this, to collapse together into themselves and to posit themselves as their other, the one, and in precisely this to refer only to themselves, each continuing itself within its other, – there is with this already in itself, undivided, the coming-out-of-itself (repulsion) along with the positing-of-oneself-as-one (attraction). Posited, however, it is in relative repulsion and attraction, that is, in the repulsion and attraction which presuppose immediate, existent ones, that each is this negation of itself upon its own self, and therewith the continuity of itself into its other as well. Repulsion on the part of existent ones is the one's self-preservation by way of the reciprocal holding-off of the others, in such a way that 1) it is upon it that the other ones get negated, which is the side of its existence, or of its being-for-other; that side, though, is thereby attraction, as the ideality of the ones; – and in such a way that 2) the one should be in itself, without reference to the others; yet the in-itself in general has long since passed over into being-for-itself, and beyond that the one is, in itself and by its determination, that very becoming into many. – Attraction on the part of existent ones is the ideality of those ones, and the positing of the one, wherein attraction, as negating and as bringing the one forth, sublates its very self, and, as positing of the one, is upon itself the negative of itself, repulsion.

Herewith the development of being-for-itself is brought to completion and has arrived at its result. The one, as referring to itself infinitely, that is, as posited negation of the negation, is the mediation whereby it thrusts from itself its absolute (that is, abstract) otherness (the many), and, referring to this its non-being negatively, sublating it, is in precisely this nothing but the reference to itself; and the one is nothing but this becoming, wherein the determination that it begins, that is, that it stands posited as an immediate, as something that is, and that as result it should equally have restored itself to the one, to the equally immediate excluding one, has vanished; the process which it is posits and holds it throughout only as something sublated. Sublating, determined at first only as a relative sublating, as the reference to another existent, which for that very reason is itself a differentiated repulsion and attraction, proves equally to pass over into the infinite reference of mediation by way of the negation of the external references of immediates and existents, and to have as its result that very becoming which, its moments having no stability, is the sinking together – or rather the coming-together-with-itself – into simple immediacy. This being, on the determination it has now received, is quantity.

Survey briefly the moments of this transition of quality into quantity, and the qualitative has for its fundamental determination being and immediacy, wherein limit and determinateness stand so identified with the something's being that the something itself vanishes once they are altered; posited thus, it is determined as finite. Owing to the immediacy of this unity, in which difference has vanished, although in itself difference is present in it, in the unity of being and nothing, difference falls as otherness in general outside that unity. This reference to other stands in contradiction with the immediacy wherein qualitative determinateness is reference to itself. Such otherness sublates itself within the infinity of being-for-itself, an infinity that has realized the difference that it has upon and within its own self in the negation of the negation, realized it into the one and the many and their references, and which has raised the qualitative into the genuine unity, a unity, that is, no longer immediate but posited as agreeing with itself.

This unity is accordingly α) being, only as affirmative being, that is, as immediacy mediated with itself through the negation of the negation; being stands posited as the unity that passes through its determinatenesses, limit and so on, which are posited in it as sublated; – β) existence; on such a determination it is negation, or determinateness as a moment of affirmative being, though determinateness that is no longer of the immediate kind but rather reflected into itself, referring not to another but to itself; the utter – the in-itself-being-determined, – the one; otherness as such is itself being-for-itself; – γ) being-for-itself, namely that being which carries itself on through determinateness, in which the one and in-itself-being-determined are themselves posited as sublated. The one is at once determined as having gone out beyond itself and as unity, and so the one, the utterly determinate limit, stands posited as the limit which is none, the limit which, though upon being, is indifferent to it.

Remark

Attraction and repulsion, as is well known, are habitually regarded as forces. This determination of theirs, and the relations bound up with it, are to be compared with the concepts that have emerged for them. – On that representation they are taken to be self-subsistent, so that they do not refer to each other by their nature, i.e. so that neither is 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 being rather something first and in and for itself, while matter, or the determinations of matter, are what these two posit and bring forth. Should it be said that matter has the forces within it, then this unity of theirs is taken to mean a linkage in which the two are at the same time presupposed as free of each other, each of them being in itself.

Kant, as everyone knows, constructed matter out of repulsive and attractive force, or at any rate laid down what he terms the metaphysical elements belonging to such a construction. – Some interest attaches to looking at this construction rather more closely. Remarkable in part is that this metaphysical presentation of an object which appeared to belong, not merely in itself but in its determinations, to experience alone, gave, as an attempt of the concept, at least the initial impetus to the newer philosophy of nature, – to that philosophy which refuses to make nature, as something sensuously given to perception, the ground of science, cognizing nature's determinations instead out of the absolute concept; remarkable in part, too, because people still commonly stop at that Kantian construction and take it for a philosophical beginning and foundation of physics.

Concrete existence of the sort that sensuous matter is remains, to be sure, no object for logic, any more than space and the determinations of space are. Yet 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 called repulsion and attraction because these names lie nearest to hand.

Looked at more closely, the procedure Kant follows in deducing matter from these forces, and which he entitles a construction, hardly merits that title, unless of course every sort of reflection, the analysing sort included, is to go by the name of construction, as later philosophers of nature have indeed called the shallowest ratiocination and the most groundless brew of an arbitrary imagination and a thoughtless reflection, – a brew helping itself especially to the so-called factors of attractive force and repulsive force and parading them everywhere, – a constructing.

Kant's procedure, that is, is at bottom analytic, not constructive. The representation of matter he presupposes, going on to ask what forces are needed if its presupposed determinations are to be preserved. On the one side, accordingly, he calls for attractive force, and does so because through repulsion alone, without attraction, there could properly be no matter. (Foundations of Natural Science, p. 53 f.) Repulsion, on the other side, he likewise derives from matter, and states as its ground that it is because we represent matter to ourselves as impenetrable, since matter presents itself under this determination to the sense of touch, through which it is said to reveal itself to us. Repulsion is on that account thought immediately within the concept of matter, being given along with it; attraction, by contrast, gets attached to that concept only through inferences. But these inferences too rest at bottom on what has just been said, that a matter possessing merely repulsive force would not exhaust what we represent to ourselves under matter. – This, as is evident, is the procedure of a cognition reflecting upon experience, which first perceives determinations in the appearance, then lays these at the base, and for the so-called explaining of them assumes corresponding basic materials or forces that are supposed to bring forth those determinations of the appearance.

As to the difference adduced, namely how repulsive force and how attractive force come to be found in matter by cognition, Kant goes on to remark that attractive force does indeed belong to matter's concept just as much, though it is not contained in it. Kant gives this last expression special emphasis. It is impossible to see, however, what difference is supposed to lie in this; for a determination belonging to the concept of a matter must truly be contained therein.

What makes the difficulty, and produces this empty evasion, is that Kant counts into the concept of matter, one-sidedly and from the outset, nothing but the determination of impenetrability, a determination we are to perceive by way of touch, so that repulsive force, as the warding off of an other from oneself, is said to be given immediately. If, further, matter is not supposed to be able to exist without attractive force, then a representation of matter drawn from perception underlies this assertion; the determination of attraction must accordingly be encounterable in perception as well. It can indeed be well perceived that matter, besides its being-for-itself, which sublates being-for-other (which offers resistance), also has a reference of what is for itself to one another, spatial extension and cohesion, and in rigidity, solidity, a very firm cohesion. For the tearing apart and so forth of a body, explanatory physics requires a force stronger than the attraction of that body's parts towards one another. Out of this perception reflection can derive attractive force just as immediately, or assume it as given, as it did with repulsive force. Indeed, when one examines the Kantian inferences from which attractive force is supposed to follow (the proof of the theorem: that a second fundamental force, a force of attraction, is required for the possibility of matter, loc. cit.), they contain nothing more than this, that by mere repulsion matter would fail to be spatial. Inasmuch as matter is presupposed as space-filling, continuity is ascribed to it, and the force of attraction is assumed as the ground of that continuity.

Granted that such a so-called construction of matter possesses at best an analytic merit, one still curtailed by the impure exposition, the basic thought is always well worth valuing, that of cognizing matter out of these two opposed determinations as its fundamental forces. Kant is chiefly concerned to banish the ordinary mechanical way of representing, which stops short 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, again regarded 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 externally to matter, and never thinks of grasping motion as something inward and comprehending it within matter itself, which is thereby assumed to be for its own part motionless and inert. Before this standpoint there lies nothing but ordinary mechanics, not motion that is immanent and free. – Although Kant does sublate that externality insofar as he makes attraction, the reference of matters to one another insofar as these are assumed to be separate from one another, 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 self-subsistent difference of these two forces which the standpoint of that cognition attributes to them, just as null must every other difference show itself to be that is drawn with regard to their content-determination as something meant to stand fast, since, taken in their truth as they were considered above, the two are moments only, each passing over into the other. – These further determinations of difference I take up as Kant states them.

Attractive force he determines, namely, as a penetrating force, one by which a given matter is able to act immediately on the parts of another even past the surface of contact, while repulsive force counts as a surface force, whereby matters are able to affect one another only within the common surface of contact. The ground adduced for the latter's being only a surface force is the following: »The parts touching each other bound each the other's space of action, and repulsive force cannot move a more distant part except by means of those lying in between; an immediate action of one matter on another crosswise through these, by way of expansive forces (which here means repulsive forces), is impossible.« (see ibid. Erklär. u. Zusätze p. 67.)

It is to be recalled at once that, once nearer or more distant parts of matter are assumed, the difference would likewise arise with regard to attraction, namely that one atom would indeed act upon another, but that a third, more distant one, between which and the first attracting atom the other was situated, would step first of all into the attracting sphere of that intervening atom, which lies nearer to it, so that the first would not exercise an immediate simple action upon the third; from which a mediated acting follows for attractive force just as much as for repulsive force; further, the true penetrating of attractive force would have to consist solely in this, that all parts of matter were attracting in and for themselves, and not that a certain quantity behaved passively and only One atom actively. – Immediately, however, or with regard to repulsive force itself, it is to be noted that in the passage cited there occur parts touching each other, hence a compactness and continuity of a finished matter which does not permit a repelling through itself. But such compactness of matter, where parts touch and are divided by no void any longer, presupposes already the sublatedness of repulsive force; on the sensuous representation of repulsion that governs here, parts touching one another count as parts that do not repel one another. It follows quite tautologically, then, 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. – Reflect, though, that touching parts touch only insofar as they still keep themselves outside one another, and repulsive force turns out by that very fact to lie not on matter's surface merely, but inside the sphere meant to belong to attraction alone.

Kant further adopts the determination that »by the force of attraction matter only occupies a space, without filling it;« (ibid.) »since matter does not fill space by the force of attraction, this force can act through empty space, no matter lying in between setting bounds to it.« – That difference is fashioned about like the earlier one, where a determination was to belong to a matter's concept and yet not be contained in it; matter here is to occupy a space merely, without filling it. In that case repulsion, if we hold to its first determination, is what makes the ones thrust one another away and, in a merely negative fashion, which here means through empty space, refer to one another. Here, however, it is attractive force that keeps space empty; it does not fill space through its reference of the atoms, which is to say, it keeps the atoms in a negative reference to one another. – We see that here, without being conscious of it, Kant runs up against what lies in the nature of the matter, ascribing 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 between 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 left by attractive force disappears. In fact, then, by sublating empty space it sublates the negative reference of the atoms or ones, i.e. the repulsion of them; which is to say, repulsion stands determined as its own opposite.

To this blurring of the differences there is added the further confusion that, as was noted at the outset, the Kantian 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 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, so that the difference of the forces can be attached to this presupposed difference of One central one and of the others as not standing fast over against it. Insofar as they are put to use for explanation, however – for which purpose they get assumed, like repulsive and attractive force elsewhere, in an opposed quantitative ratio, the one waxing as the other wanes – the appearance of motion, for the explanation of which they were assumed, along with its inequality, are supposed to result only out of them. One need only take up the first available exposition of an appearance, say of the unequal velocity a planet has in its orbit around its central body, out of the opposition of those forces, and one soon recognizes the confusion reigning in it, and the impossibility of separating out their magnitudes, so that the one has always just as much to be assumed as increasing which in the explanation is assumed as decreasing, and conversely; to be made intuitable this would require a more extensive exposition than could be given here; but what is needed comes up later, in connection with inverse ratio.

Second Section. Magnitude (Quantity)

The difference between quantity and quality has been indicated already. Whereas quality is determinateness in its first, immediate shape, quantity is determinateness turned indifferent towards being, a limit that just as much is no limit; it is being-for-itself utterly identical with being-for-other, – it is the repulsion of the many ones, a repulsion immediately non-repulsion, their continuity.

Now that what is for itself has been posited so as not to shut out its other but instead to prolong itself affirmatively into it, otherness is present insofar as existence emerges once more along such continuity, while at the same time the determinateness of that existence stands no longer in simple self-reference, belongs no longer as immediate determinateness to the existent something, but has been posited so as to repel itself away from itself and to hold its self-reference qua determinateness rather within another existence (one that is for itself); and since such determinatenesses are at the same time indifferent limits, reflected inwardly, out of all relation, determinateness in general falls outside itself, becomes something utterly external to itself, external no less to the something; a limit of that kind, indifferent on its own account and faced by the something's indifference towards it, makes up the quantitative determinateness of the something.

At the outset a distinction is required between pure quantity and quantity as determinate, that is, the quantum. Taken as the former, quantity is first real being-for-itself gone back into itself, bearing as yet no determinateness upon it; a solid, infinite unity that continues itself within itself.

Second, it moves on to the determinateness that gets posited upon it, one which at once is no determinateness at all, being merely external. It becomes quantum. Quantum is determinateness grown indifferent, i.e. determinateness that reaches out past itself in negating itself; as this otherness of otherness it lapses into the infinite progress. Infinite quantum, however, is that indifferent determinateness sublated, and so the reinstatement of quality.

Third, quantum in qualitative form is the quantitative ratio. Quantum as such merely goes out past itself in general; within the ratio, though, its going out past itself into its otherness takes this form, that the otherness wherein it holds its determination is itself posited as well, is a further quantum; and so its being-returned-into-itself, its self-reference, is present within its otherness.

Underlying such a ratio there remains the externality of quantum; the terms standing in relation are indifferent quanta, having their self-reference within such a being-outside-itself; – with that, the ratio amounts to no more than a formal unity of quality with quantity. The dialectic of the ratio is its passing over into the absolute unity of the two, into measure.

Remark

In the case of something, its limit as quality is essentially its determinateness. If, though, we take limit to mean the quantitative limit, and a field, say, alters this limit of its, it remains a field afterwards as before. Should its qualitative limit be altered instead, this is the 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 still red; were it to change its quality, however, it would cease to be red and would turn blue, and so forth. – That determination of magnitude as quantum which emerged above, namely that a being lies at the base as something enduring, which is indifferent towards the determinateness it has, shows itself in any other example as well.

By the expression magnitude what is understood, as in the examples given, is quantum, not quantity, which is why this name from the foreign tongue has essentially to be used.

The definition given of magnitude in mathematics likewise concerns quantum. Customarily a magnitude is defined as something that admits of being increased or diminished. To increase, though, means making something more great, to diminish, making it less great. There lies in this a difference of magnitude in general from itself, and magnitude would accordingly be that whose magnitude can be altered. To that extent the definition proves clumsy, since the very determination that was to be defined is employed within it. Insofar as the same determination is not to be used in it, the more and the less have to be resolved into an adding-on, as affirmation and indeed, by the nature of quantum, an equally external one, and into a taking-away, as a negation just as external. It is to this external mode both of reality and of negation that the nature of alteration in quantum determines itself generally. Hence in that imperfect expression the chief moment on which everything turns is not to be missed; namely the indifference of the alteration, such that its own more-and-less lies in its very concept, its indifference towards itself.

Chapter 1. Quantity

A. Pure Quantity

Quantity is being-for-itself in sublated shape; the repelling one, whose comportment towards the excluded one had been merely negative, has crossed over into reference with that one, comports itself identically towards the other, and in doing so has forfeited its determination; into attraction, then, being-for-itself has crossed. Into this unity the absolute brittleness of the repelling one has melted away; yet the unity, since it contains that one and is determined at once by the repulsion dwelling within it, is unity with itself as unity of being-outside-itself. Attraction stands in this way as the moment of continuity within quantity.

Continuity is accordingly a simple, self-same reference to self, broken by no limit and no exclusion, yet not an immediate unity but the unity of the ones that are for themselves. Contained in it there is still the outside-one-another of multiplicity, though at the same time as something undifferentiated, unbroken. Multiplicity is posited in continuity as it is in itself; the many are each what the other is, each equal to the other, and multiplicity is therefore simple, differenceless equality. Continuity is this moment of the self-sameness of being-outside-one-another, the self-continuation of the distinguished ones into those distinguished from them. Immediately, therefore, magnitude has in continuity the moment of discreteness, – repulsion as it is merely a moment in quantity. – Continuity is self-sameness, but self-sameness of the many, which nevertheless does not become something excluding; repulsion is what first expands self-sameness into continuity. Discreteness for its part is therefore a discreteness that flows together, whose ones do not have the void, the negative, for their reference, but their own continuity, and do not interrupt this equality with itself in the many.

These two moments, continuity and discreteness, have their unity in quantity, though at first under the form of just one of them, continuity, quantity being the result of that dialectic of being-for-itself which collapsed into the shape of a self-same immediacy. Quantity as such is this simple result insofar as it has not yet developed its moments and posited them upon itself. – At first it contains them, being-for-itself being posited as it is in truth. By its determination, being-for-itself was the self-sublating referring to itself, a perennial coming-outside-itself. But what is repelled is being-for-itself itself; repulsion is accordingly the generative flowing-forth of itself. Since what is repelled is the selfsame, this discerning amounts to unbroken continuity; and since there is a coming-outside-itself, that unbroken continuity is at once a multiplicity, one that just as immediately abides in its equality with itself.

Remark 1

Pure quantity as yet has no limit, or is not yet quantum; even insofar as it becomes quantum, it is not limited by the limit, consisting rather precisely in this, in not being limited by the limit, in having being-for-itself within it as something sublated. One way of putting the presence of discreteness as a moment within it is this: everywhere within itself quantity is utterly the real possibility of the one, while conversely the one, just as utterly, is only as something continuous.

For representation innocent of the concept, continuity readily turns into composition, namely into an external reference of the ones to one another, wherein the one is preserved in its absolute brittleness and exclusion. It has been shown of the one, however, that in and for its own self it passes over into attraction, into its ideality, and that continuity is consequently not external to it but belongs to it itself and is grounded in its essence. It is this externality of continuity to the ones that atomism generally clings to, and the difficulty for representing consists in giving it up. – Mathematics, for its part, throws out any metaphysics wishing to let time consist of temporal points, and space at large, or first of all the line, of spatial points, the surface of lines, space entire of surfaces; ones of that non-continuous kind it refuses to allow. Even where it fixes the magnitude of a surface, say, by having it represented as the sum of lines infinite in number, such discreteness holds good only as a passing representation, and within the infinite multiplicity of those lines, given that the space they are meant to make up is after all a bounded one, the sublatedness of their discreteness is already contained.

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

Quantitas duobus modis a 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 a 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 one wants them, are to be had in space and time, also in matter generally, in light and so forth, even in the I; only, as already noted, quantity is not to be taken to mean quantum. Space, time and so forth are extensions, multiplicities that amount to a going-outside-of-self, a streaming which nonetheless never crosses over into its opposite, into quality or into the one, but which, being a coming-outside-itself, is a perennial self-production of their unity. This absolute being-outside-itself is space, no less utterly unbroken for that, an otherness and once more otherness that stays identical with itself; time is an absolute coming-outside-itself, a generating of the one, the point of time, the now, which straightaway is that point's annihilation and steadily again the annihilation of this passing away; whereby such self-generating of non-being proves no less to be simple equality, identity with self.

As regards 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 thus: Non omnino improbabile est, materiam et quantitatem esse realiter idem. – And in truth these concepts are no further apart than this, that quantity is the pure thought-determination while matter is that same determination in external concrete existence. – To the I likewise the determination of pure quantity belongs, since the I is an absolute othering, a distancing without end or repulsion on every side into the negative freedom of being-for-itself, one that all the same stays utterly simple continuity, – continuity of universality, of being at home with itself, which the infinitely manifold limits, the content of sensations, of intuitions and the like, leave uninterrupted. – Those, on the other hand, who balk at grasping multiplicity as simple unity, and who demand, besides the concept that of the many each is the same as the other, namely one of the many, – for what is under discussion here is not the many as further determined, not green, red and so forth, but the many considered in and for itself, – a representation of this unity too, will find enough of the sort in those continuities, which present the deduced concept of quantity as at hand in simple intuition.

Remark 2

It is into the nature of quantity, namely into its being this simple unity of discreteness and of continuity, that the conflict or the antinomy of the infinite divisibility of space, of time, of matter and the like falls.

The whole of this antinomy lies in this alone, that continuity has to be asserted no less than discreteness. Asserted one-sidedly, discreteness yields infinite or absolute dividedness as its principle, and hence an indivisible; continuity asserted one-sidedly yields, on the contrary, infinite divisibility.

As is well known, Kant's Critique of Pure Reason sets up four (cosmological) antinomies, among which the second concerns the opposition that the moments of quantity constitute.

An important part of the critical philosophy these Kantian antinomies will always be; it was chiefly they that brought down the metaphysics preceding them, and they may be viewed as a principal transition into the philosophy of more recent times, since it was they in particular that helped to establish the conviction that the categories of finitude are null, and this from the side of their content – a road more correct than the formal road of a subjective idealism, for which the one defect of those categories is supposed to lie in their being subjective rather than in what they are in their own selves. Great as its merit is, though, this presentation is highly imperfect: in part obstructed and contorted within itself, in part skewed as regards the result it reaches, a result that takes for granted that cognition possesses no forms of thinking beyond the finite categories. – In both respects a more exact critique is owed to these antinomies, one that will illuminate their standpoint and their method more closely and will also disengage the point on which everything turns from the useless form in which it has been wedged.

Let me note first of all that a shine of completeness was what Kant meant to confer upon his four cosmological antinomies, and he did so by way of the principle of division that he took over from his schema of the categories. Deeper insight into the antinomic nature of reason, or more truly into its dialectical nature, exhibits every concept whatever as a unity of opposed moments, so that the form of antinomic assertions could be conferred on them all. Becoming, existence and the rest, any concept at all, could thus furnish its own peculiar antinomy, and as many antinomies could be set up as there are concepts to be found. – Ancient skepticism spared itself no trouble in pointing out this contradiction, this antinomy, in every concept it met with in the sciences.

Kant, moreover, grasped the antinomy not within the concepts themselves but within cosmological determinations, that is, within a form already concrete. For the antinomy to be had in its purity and treated in its simple concept, the determinations of thought ought not to have been taken in their application to, and their mixture with, the representation of world, space, time, matter and the like, but ought to have been considered purely on their own, apart from that concrete stuff, which contributes neither force nor power here, since it is those determinations alone in which the essence and the ground of the antinomies reside.

Kant gives this concept of the antinomies, that they are »not sophistical artifices but contradictions upon which reason must necessarily strike (to use Kant's expression);« – an important view, this. – »By the natural shine of the antinomies reason, once it sees into the ground of that shine, is indeed no longer taken in, yet is still deceived.« – For the critical resolution, the one proceeding by way of what is called the transcendental ideality of the perceptual world, yields no result but this, that the alleged conflict is made into something subjective, in which of course it goes on being the very same shine, that is, stays as unresolved as it was before. Only in this can its genuine resolution lie: that two determinations which are opposed and yet belong necessarily to one and the same concept have no validity in their one-sidedness, each on its own, but possess their truth solely in their sublatedness, in the unity of their concept.

Looked at more closely, the Kantian antinomies hold nothing beyond the entirely simple categorical assertion of each one of the two opposed moments of a determination, taken on its own in isolation from the other. Yet this simple categorical, or properly assertoric, claim is meanwhile wrapped in a lopsided, twisted scaffolding of ratiocination, by which a shine of proofs is brought forth and the merely assertoric character of the claim is to be concealed and made unrecognizable; as the closer consideration of them will show.

What the antinomy belonging here concerns is the alleged infinite divisibility of matter; it rests upon the opposition of those moments, continuity and discreteness, which the concept of quantity holds within itself.

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

Every composite substance in the world is made up of simple parts, and nothing exists anywhere save the simple or what is put together out of it.

Over against the simple, the atom, there is placed here the composite, a determination that falls far short of the steady or the continuous. – As for the substrate lent to these abstractions, namely substances in the world, it signifies nothing further here than things as they are perceptible by sense, and it bears no influence on what is antinomic in the matter; space or time might have been taken with equal right. – Since the thesis now runs in terms of composition rather than of continuity, it is in truth an analytic or tautological proposition from the start. It belongs to the composite as its immediate determination that it is not one in and for itself, but something linked together externally, something made up of another. And the other of the composite is the simple. Tautological, therefore, is the statement that the composite is made up of the simple. – Once the question is put, of what something is made up, what is being asked for is an other to be specified whose conjunction constitutes that something. Should ink be allowed to be made up of ink again, the sense of the question about being made up of an other has been missed; the question goes unanswered and only repeats itself. Then there is the further question, whether what is being spoken of is to be made up of something at all, or not. The composite, however, is precisely that which is to be a linked thing, made up of an other. – Take the simple, which is to be the other of the composite, merely as something relatively-simple that is once more composite on its own account, and the question stands where it stood. Representation has in view, it may be, only this or that composite, for which likewise this or that something would be specified as its simple, itself a composite on its own account. Here, though, the talk is of the composite as such.

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

»Assume, (thus he begins,) that composite substances were not made up of simple parts; then, were all composition sublated in thought, there would be left no composite part, and, since by the assumption just made there are no simple parts, no simple one either, so that nothing whatever would remain over and accordingly no substance would have been given.« –

Quite correct, this inference: where there is nothing but the composite, and one thinks away everything composite, one has nothing whatever remaining; – granted, but this tautological surplus might have been spared, and the proof might have set out at once from what comes next, namely:

»Either all composition cannot possibly be sublated in thought, or, once it has been sublated, there has to remain something subsisting apart from composition, namely the simple.« »In the first case, though, the composite would again not be made up of substances (since in their case composition is but a contingent relation of substances⁸, without which they, as entities persisting for themselves, must subsist.) – This case, however, runs counter to the presupposition, and so only the second remains: that the substantial composite in the world is made up of simple parts.«

Set down incidentally in a parenthesis is that very ground which makes up the main thing, and as against which everything so far is entirely superfluous. The dilemma is this: either the composite is what remains, or not the composite but the simple. Were the former, the composite, what remains, then what remains would not be the substances, for to these composition is only contingent relation; but substances are what remains, and so what remains is the simple.

Plainly, without the apagogic detour that ground might have been attached straight to the thesis as its proof: 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 the contingency of composition, and the essence is of course the simple. This contingency, though, on which alone everything turns, is not proved but bluntly assumed, assumed in passing and in a parenthesis, as a thing self-evident or a side issue. Self-evident it certainly is that composition is the determination of contingency and externality; but if all that were in question were a contingent being-together in place of continuity, then the trouble of setting up an antinomy over it was not worth taking, or rather none could be set up at all; the assertion that the parts are simple is then, as was recalled, no more than tautological.

Within the apagogic detour, accordingly, we see the very assertion crop up that is supposed to issue from it. More briefly, then, the proof may be put thus:

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

Part of the whole, still, is a look at the concluding proposition; it runs as follows:

»From this it follows immediately that the things of the world are one and all simple entities, that composition is in them merely an outer state, and that the elementary substances have to be thought by reason as simple entities.«

What we see here is the externality, that is, the contingency, of composition adduced as a consequence, though beforehand it had been slipped into the proof in a parenthesis and put to work there.

Kant protests strongly that in the conflicting propositions of the antinomy he is not out for deceptions, so as to conduct (as the saying goes) a lawyer's proof. Less with deception is the proof considered to be charged than with a useless, tortured contortedness, one that serves merely to bring forth the outward shape of a proof and to keep it from standing in full transparency that what was to come forth as a consequence is, in the parenthesis, the very hinge of the proof, and that in general no proof at all is present but only a presupposition.

The antithesis runs:

No composite thing in the world is made up of simple parts, nor does anything simple exist anywhere within it.

The proof is likewise turned apagogic, and in another way is quite as blameworthy as the previous one.

»Posit, so it runs, that a composite thing, as substance, is made up of simple parts. Because every external relation, and with it all composition out of substances, is possible only in space, the space occupied by the composite must be made up of as many parts as the composite is made up of. 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.«

»The simple, accordingly, occupies a space.«

»Now since everything real that occupies a space holds within itself a manifold of things lying outside one another, and is thus 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 use an expression occurring elsewhere in Kant) of faulty procedure.

To begin with, the apagogic turn is a groundless shine. For it is a direct assertion that everything substantial is spatial while space is not made up of simple parts, an assertion turned into the unmediated ground of what is to be proved and with which the whole business of proving is already settled.

This apagogic proof then sets out from the proposition: »that all composition out of substances is an external relation,« only to forget it again straightaway, oddly enough. For the inference runs on: composition is possible only in space, space is not made up of simple parts, and the real occupying a space is consequently composite. Once composition has been assumed to be an external relation, then spatiality too, as the sole medium in which composition is supposed to be possible, is on that very account an external relation for the substances, one that does not concern them and leaves their nature untouched, as little as does anything else that may be inferred from the determination of spatiality. On that very ground the substances ought not to have been placed in space.

It is presupposed, further, that the space into which the substances are here transferred 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 so-called discursive concept. – Out of this Kantian distinction between intuition and concept, as is well known, much mischief with intuiting has grown, and, to be spared the labor of comprehending, people have stretched the worth and the domain of intuiting over all cognition. What belongs here is only this, that space, like intuition itself, must at the same time be comprehended, if, that is, one wants to comprehend at all. Therewith would arise the question 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 in fact concerns, as was recalled, quantity as such, and with it space and time no less.

But once it has been assumed in the proof that space is not made up of simple parts, this should have been ground enough for not transferring the simple into an element unsuited to the determination of the simple. – Here, besides, the continuity of space comes into collision with composition; the one gets confounded with the other, continuity being slipped in where composition belongs, (which in the syllogism yields a quaternio terminorum). Kant's own express determination of space is that space is a single one, its parts resting merely upon limitations, so that they in no way precede the single all-embracing space as its constituents, as it were, out of which its composition might be possible«. (Critique of Pure Reason, 2nd ed., p. 39). Very correctly and determinately is continuity here stated of space, as against composition out of constituents. In the argumentation, by contrast, the transferring of the substances into space is supposed to bring with it a »manifold situated outside one another« and indeed »hence a composite«. Whereas, as was cited, the manner in which a manifold occurs in space is expressly supposed to rule out composition and any constituents preceding the singleness of space.

In the remark appended to the proof of the antithesis, the critical philosophy's otherwise fundamental representation is expressly brought to bear as well: that of bodies we have a concept only as appearances, and that as such they cannot but presuppose space, the condition under which every outer appearance is possible. If, then, by substances nothing is meant but bodies as we see, feel, taste them and so on, then what they are in their concept 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, feeling and so on 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.

Looking with this more exactly at the opposition of thesis and antithesis here, and freeing their proofs of all useless surplus and contortedness, we find that the proof of the antithesis contains – by transferring 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 connected – the assertoric assumption that this connection is contingent, and therewith the assumption of the substances as absolute ones. To the severing of the two moments of quantity, then, and to their direct assertion as simply severed, the entire antinomy reduces itself. Taken by mere discreteness, substance, matter, space, time and the like are divided outright, the one being their principle. Taken by continuity, this one is a sublated one only; the dividing remains divisibility, the possibility of dividing remains as possibility, without any actual arrival at the atom. Even if we halt at the determination given of these oppositions in what has been said, the moment of the atom lies within continuity itself, continuity being outright the possibility of dividing, just as that dividedness, discreteness, for its part sublates every difference among the ones – each simple one being what the other is – and thus contains their likeness and with it their continuity. Because either of the two opposed sides has its other within its own self, and because neither admits of being thought apart from the other, the upshot is that no such determination taken by itself possesses truth; truth belongs only to their unity. Such is the genuine dialectical consideration of these determinations, and such the genuine result.

Far more ingenious and profound than the Kantian antinomy under consideration are those dialectical examples of the old Eleatic school, above all the ones bearing on motion, which likewise have their basis in the concept of quantity and their resolution in it as well. To take them up here too would carry us too far afield; bearing as they do on the concepts of space and of time, they may 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 its own self, and are thus in their own selves the flux of Heraclitus. Worthy they are, on that account, of a more searching consideration than the customary explanation that they are just sophisms; an assertion that holds to empirical perceiving in the way of Diogenes, a way so plain to common human understanding, who, when a dialectician pointed out the contradiction contained in motion, is said to have exerted his reason no further but to have appealed to what the eye sees by silently pacing to and fro – an assertion and a refutation certainly easier to produce than to enter into thoughts, to hold fast the entanglements into which thought leads, thought not fetched from far off but forming itself within ordinary consciousness, and to dissolve them by means of thought itself.

Highly to be praised is the resolution which Aristotle provides for these dialectical formations, a resolution lying in his truly speculative concepts of space, of time, and of motion. To infinite divisibility (which, being represented as though it were carried through, amounts to the same as infinite dividedness, as atoms), on which the most celebrated of those proofs rest, he opposes continuity, applying to time quite as much as to space, so that infinite, that is to say abstract, multiplicity is contained in continuity only in itself, in accordance with possibility. What is actual, as against abstract multiplicity and abstract continuity alike, is the concretion of them, time and space themselves, just as motion and matter are in turn the concrete over against these. Only in itself, or only in accordance with possibility, is the abstract; only as moment of something real does it have its being. Bayle, who in his Dictionaire, art. Zenon, finds Aristotle's resolution of the Zenonian dialectic »pitoyable«, fails to understand what it means for matter to be divisible into infinity only in accordance with possibility; he replies that matter divisible into infinity contains an infinite multitude of parts actually, so that what we have is no infinite en puissance but one existing really and in actuality. – Rather, already the divisibility is itself no more than a possibility, not an existing of the parts, and multiplicity as such is posited within continuity only as a moment, as sublated. – Acuteness of understanding, in which Aristotle too is doubtless unsurpassed, does not reach far enough to grasp and to judge his speculative concepts, any more than the crudity of sensuous representation adduced above reaches far enough to refute Zeno's argumentations; the former labors under the error of taking such thought-things, such abstractions as an infinite multitude of parts, for something, for a truth and an actuality; the latter, sensuous consciousness, cannot be brought past the empirical to thoughts.

Kant's resolution of the antinomy, too, amounts to nothing more than this, that reason ought not to fly beyond sensuous perception but ought to take appearance as it stands. Leaving the content of the antinomy itself to one side, this resolution never reaches the nature of the concept of its determinations, each of which, isolated on its own, is null, being in its own self nothing but the passing over into its other, and which have quantity for their unity and therein their truth.

B. Continuous and Discrete Magnitude

1. The two moments, continuity and discreteness, are what quantity holds within it. In both, as its determinations, quantity is to be posited. – Their immediate unity it already is at once, which is to say that at first it is itself posited in only the one of its determinations, in continuity, and is thus continuous magnitude.

Or, put otherwise: continuity counts, to be sure, as one moment of quantity, and quantity reaches its completion only with the other moment, discreteness. Concrete unity, however, quantity is only to the extent that the moments whose unity it is are differentiated ones. Hence they too must be taken as differentiated, though without being resolved back into attraction and repulsion; each is rather to be kept, in accordance with its truth, within its unity with the other, that is, as the whole. Continuity is nothing but the cohering, solid unity, the unity of the discrete; posited thus, it is no longer moment merely, but quantity entire; continuous magnitude.

2. Continuous magnitude is what immediate quantity is. Yet quantity is nothing immediate at all; immediacy is a determinateness whose sublatedness quantity itself constitutes. In the determinateness immanent to it, then, quantity is to be posited, and this determinateness is the one. Quantity is discrete magnitude.

Discreteness is a moment of quantity, as continuity is, yet it too is quantity entire, precisely because it is moment within quantity, within the whole, and so, being differentiated, does not step out of that whole, out of its unity with the other moment. – Being-outside-one-another in itself 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, by contrast, is that outside-one-another as non-continuous, as broken off. Yet with this multitude of ones there is not present again the multitude of atoms and the void, repulsion as such. Since quantity is what discrete magnitude is, its discreteness is itself continuous. Wherein this continuity in the discrete consists is that the ones are what is equal to one another, or that one and the same unit belongs to them. Discrete magnitude is accordingly the outside-one-another of the many ones as of the equal, not the many ones simply, but posited as the many of a unit.

Remark

Overlooked in the ordinary representations of continuous and of discrete magnitude is the fact that both moments, continuity as well as discreteness, attach to each of these magnitudes, and that what constitutes their difference is nothing but which of the two moments counts as the posited determinateness and which as the one that merely is in itself. Steady magnitudes are space, time, matter and the like, in that they are repulsions of their own selves, a streaming issuing-out-of-self which is at the same time no passing over, no comporting toward something qualitatively other. Absolute possibility belongs to them that the one be posited upon them at any point whatever; not the empty possibility of a bare otherness (one says, for instance, that a tree might just as well be standing where this stone stands), for the principle of the one is contained in their own selves, being the one determination among those out of which they are constituted. Continuity, conversely, is not to be overlooked in [the] discrete magnitude; that moment, as has been shown, is the one taken as unit.

As species of quantity the continuous and the discrete magnitude may indeed be considered, though only insofar as magnitude stands posited under no determinateness foreign to it but under the determinatenesses of its own moments; the customary passage from genus to species brings external determinations to the genus in accordance with some ground of division external to it. Neither the continuous nor the discrete magnitude is as yet a quantum in all this; each is only quantity itself in one of its two forms. Magnitudes they are called, more or less, insofar as they share with quantum this much in general, that of being a determinateness upon quantity.

C. Limitation of Quantity

The one is, in the first place, the principle of discrete magnitude; in the second place, such magnitude amounts to a plurality of ones; in the third place it is essentially continuous, being at once the one as sublated, as unity, the self-continuing as such amid the discreteness of the ones. It stands posited, accordingly, as one magnitude, and what determines it is the one, which upon this positedness and existence counts as an excluding one, a limit upon the unity. Immediately, discrete magnitude as such is not supposed to be limited; but once marked off from the continuous, it is there as an existence and a something, and what determines it is the one, which, standing within an existence, amounts also to first negation and to limit.

Referred to the unity and constituting the negation upon it, this limit is besides that, being a one, referred to itself as well; so it is a limit that encloses and encompasses. What the limit is here does not first mark itself off from the something of its existence; being a one, it is rather this negative point immediately. Yet the being that undergoes limiting here is essentially continuity, and by virtue of continuity it reaches past the limit, past this one, remaining indifferent toward it. So it is that discrete quantity in its reality amounts to one quantity, or quantum – to quantity in the shape of an existence and a something.

Since the one that serves as limit gathers into itself the many ones belonging to discrete quantity, that limit posits them equally as sublated within it; upon continuity in general as such does it stand as limit, and with that the difference between continuous and discrete magnitude counts here for nothing; more precisely, it is limit upon the continuity of the one no less than upon that of the other; into being quanta do both pass over.

Chapter 2. Quantum

Quantum – initially quantity carrying a determinateness or a limit as such – is, in the completeness of its determinateness, number. Quantum becomes distinguished,

secondly, at the outset into extensive quantum, upon which the limit is as limitation of the plurality that exists, and thereafter, this existence passing over into being-for-itself, – into intensive quantum, degree, which, being for itself and therein an indifferent limit, is with equal immediacy outside itself, holding its determinateness upon an other. As this posited contradiction – of being determined thus simply within, while holding its determinateness outside and referring outside itself for it – quantum passes,

thirdly, posited as that which is external to its own self, over into quantitative infinity.

A. Number

Quantity is quantum, that is, it has a limit — and this holds of it as continuous magnitude quite as much as discrete. Between these kinds the difference carries, for the moment, no weight here.

Since quantity is being-for-itself sublated, it stands of itself, in and for its own self, indifferent toward the limit it has. Not indifferent to it, however, is that limit for all that, or the being of a quantum; for within itself, as a moment of its own, quantity holds the one, absolute being-determined, and this one, posited upon quantity's continuity or unity, constitutes its limit — a limit enduring nonetheless as the one which quantity has, generally speaking, become. Principle of the quantum, then, is this one, though the one as the one of quantity. On that account it is, first, continuous, it is unity; secondly it is discrete — a plurality of the ones, in-itself in continuous and posited in discrete magnitude, ones equal among themselves, having that continuity, that same unity. Thirdly, as simple limit this one is negation too of the many ones, a shutting-out of its otherness from itself, a determining of itself over against other quanta. So far, then, the one is a limit α) referring to itself, β) enclosing, and γ) excluding other.

Posited in full in these determinations, quantum is number. Wherein the full positedness lies is the limit's existence as plurality, and therewith its being differentiated from the unity. Hence number shows itself as discrete magnitude, though upon the unity it possesses continuity all the same. That is why it is also quantum in perfect determinateness; for in number the limit is a determinate plurality whose principle is the one, the utterly determinate. Whereas continuity, that wherein the one is merely in itself, as something sublated, – posited as unity, – is the form indeterminateness takes.

Merely as such, quantum is limited in general; abstract, simple determinateness of it is what its limit is. Being number, however, it has this limit posited as manifold within its own self. Within it lie the many ones that constitute its existence, and they lie within it not indeterminately; rather the determinateness of the limit falls into them – other existence, other manys, the limit shuts out, and the ones it holds enclosed are a determinate multitude, the amount, whose other, as against the discreteness that number displays, is the unity, the continuity of that discreteness. The moments of number are made up by amount and unity.

How the many ones out of which the amount consists stand within the limit is what still calls for closer inspection; of the amount the expression holds good that out of the many it consists, since the ones are present in it not as sublated: they are in it, posited merely along with the excluding limit, toward which they behave indifferently. Toward them, though, that limit is not indifferent. In the case of existence, the relation of limit to it had first taken this shape: existence, the affirmative, went on subsisting this side of its limit, the limit — negation — lying outside upon its edge; just so, with the many ones, their breaking off and the shutting out of further ones looks like a determination that falls beyond the ones held enclosed. There, however, the outcome was that limit permeates existence, extends as far as existence extends, and that something is thereby limited in respect of its determination, hence finite. – In the quantitative sphere one pictures the number hundred, for instance, as though the hundredth one alone did the limiting of the many so as to make them a hundred. In one respect this holds; in another, though, no one among the hundred ones enjoys any precedence, all being merely equal; each is quite as much the hundredth; to the limit through which the number is a hundred they therefore all belong; not one of them can the number spare for its determinateness; as against the hundredth one, then, the rest constitute no existence lying beyond the limit, or only within it, no existence at all distinguishable from it. Rather than a plurality over against the enclosing, limiting one, the amount is itself what constitutes this limitation, and that is a determinate quantum; the many constitute a number, one two, one ten, one hundred, and so forth.

Being-determined over against another, the marking off of number from others — that is what the limiting one now is. This marking off does not turn into qualitative determinateness, though, but stays quantitative, falling merely to the comparing external reflection; number stays a one returned into itself, indifferent toward others. An essential determination of number is this indifference of it toward others; it constitutes its being-determined-in-itself and, in one and the same stroke, its own externality. – Thus it is a numerical one, the absolutely determinate, which carries at once the form of simple immediacy, so that reference to another is for it wholly external. Being a one that is number, it further carries the determinateness, insofar as this is reference to another, as moments within its own self, in its difference of unity and amount, and the amount is itself a plurality of ones, which is to say that this absolute externality is within its own self. – The quality of quantum is this contradiction of number, or of quantum generally, within itself, and in the further determinations of that quality the contradiction unfolds.

Remark 1

Spatial magnitude and numerical magnitude tend to be treated as two kinds, on the assumption that spatial magnitude would for itself be determinate magnitude no less than numerical magnitude is; the difference between them would rest solely upon the differing determinations of continuity and discreteness; qua quantum, though, both would occupy one and the same level. What geometry has before it, broadly speaking, is continuous magnitude in the shape of spatial magnitude, and what arithmetic has before it is discrete magnitude in the shape of numerical magnitude. With this unlikeness in their object, however, goes an unlikeness in the manner and completeness of the limitation, or of the being-determined, that each admits. Limitation as such is all that spatial magnitude has; wherever it is to be treated as a quantum utterly determined, number becomes indispensable to it. Spatial figures geometry as such does not measure — it is no art of mensuration; it merely compares them. Even in its definitions the determinations are drawn partly from the equality of sides, of angles, from the equal distance. The circle, for instance, resting as it does purely upon the equality of distance from a mid-point of every point possible within it, calls for no number to determine it. Determinations of this sort, founded upon equality or inequality, are geometrical through and through. Adequate they are not, however, and for others — triangle, quadrangle, e.g. — number becomes requisite, since number carries in its principle, the one, a being-determined-for-itself rather than a being-determined by aid of an other, hence not by comparison. True, spatial magnitude does have in the point the determinateness answering to the one; the point, however, once it comes outside itself, turns into an other, turns into the line; since essentially it is only as a one of space, in the reference it turns into a continuity wherein punctuality — being-determined-for-itself, the one — is sublated. For being-determined-for-itself to be preserved within being-outside-itself, the line has to be represented as a multitude of ones and has to take into itself the limit, the determination of the many ones; which means that the line's magnitude — and likewise that of the remaining spatial determinations — has to be taken as number.

Number and its figures are what arithmetic considers — or rather it does not consider them, it operates upon them. For number is determinateness of the indifferent sort, inert; activating it and bringing it into connection has to come from without. Those modes of connection are the kinds of calculation. Arithmetic lists them one after another, and one plainly hangs upon another. What thread guides their sequence is nevertheless never lifted out in arithmetic. Out of number's own conceptual determination, though, the systematic arrangement falls readily into place — an arrangement to which any account of these elements in the textbooks has a fair claim. Briefly these guiding determinations shall be pointed out here.

Owing to its principle, the one, number is throughout something gathered together from without, an utterly analytic figure harbouring no inner coherence. Being thus a product merely of external making, all calculation comes down to the production of numbers, to a counting, or more determinately: a counting-together. Any diversity in this external producing, which forever does one and the same thing, can rest only upon a difference among the numbers that are to be counted together; and such a difference has itself to be drawn from elsewhere, out of an external determining.

That qualitative difference which constitutes number's determinateness is the one already seen, the difference of unity and amount; to it, therefore, shrinks every conceptual determinateness that the kinds of calculation can exhibit. What difference belongs to numbers qua quanta, on the other hand, is external identity and external difference, equality and inequality — moments of reflection these, to be treated among the determinations of essence, under difference.

One thing more must be sent ahead: numbers admit in general of being produced in two ways, either by gathering together or by parting what has already been gathered; – both taking place within one and the same manner of counting, a gathering-together of numbers answers to what may be styled a positive kind of calculation, a parting to what may be styled a negative one; the determination of the kind of calculation itself owes nothing to this opposition.

These remarks made, the enumeration of the modes of calculation follows herewith. Number's first generation lies in gathering together many as such, each of them posited only as one, – numbering. The ones being external to each other, they present themselves under a sensuous picture, and the operation that generates number consists in a counting off upon fingers, upon dots, and so forth. What four, five, and so forth is admits of nothing but being shown. Where to break off, how much shall be gathered in — that, the limit being external, is contingent, a matter of choice. – Out of the difference of amount and unity, which comes in as the kinds of calculation advance, arises a system of numbers — dyadic, decadic, and so forth; and such a system rests wholly upon the choice as to which amount shall constantly be taken over again as unity.

Numbers that have arisen through numbering are numbered once more; and being posited thus immediately, they are as yet determined without any reference to one another, indifferent toward equality and inequality, of contingent magnitude relative to each other, – hence unequal in general; – adding. – That 7 and 5 amount to twelve is something one finds out by numbering onto the 7 a further 5 ones upon the fingers or otherwise, – whereupon the result is retained in memory, by heart; for nothing inward attaches to it. In the same way one knows that 7 × 5 = 35 by counting off upon the fingers and so forth, one being numbered onto a seven, this done five times over, and the result again retained by heart. All the labour of such numbering, of hitting upon the sums and products, is got rid of by the ready-made one-and-one, or once-one-is-one, which one need only learn by heart.

Kant has (in the Introduction to the Critique of Pure Reason V.) taken the proposition: 7 + 5 = 12, for a synthetic proposition. »One would,« he says, »to begin with certainly think (indeed!) that it is a merely analytic proposition, issuing from the concept of a sum of seven and five in accordance with the principle of contradiction.« Nothing more is meant by the concept of the sum than the abstract determination that these two numbers are to be gathered together, and gathered as numbers, in an external, i.e. conceptless way, – that from seven onward the numbering is to proceed until the ones to be added, five being their fixed amount, are used up; the result goes by the name, familiar enough otherwise, of twelve. »Yet,« Kant goes on, »on closer inspection one finds the concept of the sum of 7 and 5 to contain nothing beyond the union of the two numbers into a single one, whereby nothing whatever is thought as to which single number it is that gathers the two together;« – »let me dissect my concept of such a possible sum as I please, the twelve I shall still not come upon in it.« With the thinking of the sum, with dissection of the concept, the passage from that task to the result has, to be sure, nothing [to] do; »one must step beyond these concepts and take intuition to one's aid, five fingers and so forth, thus adding the units of the five given in intuition to the concept of seven,« he appends. Five, admittedly, is given in intuition, i.e. as a thoroughly external having-been-joined-together of the thought one arbitrarily repeated; but seven is no more a concept than five; concepts that one might step beyond are simply not at hand. By the sum of 5 and 7 is meant the conceptless linking of the two numbers; the numbering carried on thus conceptlessly from seven until the fives run out may be styled a joining-together, a synthesizing, quite as much as the numbering from one onward may – a synthesizing, however, of altogether analytic nature, the coherence being one wholly contrived, with nothing in it, and nothing entering it, that does not lie wholly outwardly before us. As the postulate of prolonging a straight line stands to the postulate of drawing one, so stands the postulate of adding 5 to 7 to the postulate of numbering at all.

Empty as the expression synthesizing is, just as empty is the determination that it takes place à priori. Counting, admittedly, is no determination of sensation — the one thing left over for the à posteriori on Kant's determination of intuition — and counting is doubtless a business carried on upon the ground of abstract intuiting, i.e. an intuiting determined through the category of the one, with every other determination of sensation abstracted from, and concepts abstracted from no less. Vague, on the whole, is what the à priori amounts to; the determination of feeling, as drive, sense, and so forth, harbours the moment of apriority within it quite as much as space and time, as existing — the temporal, the spatial — is determined à posteriori.

Bound up with this, one may add that Kant's claim about the synthetic character of the fundamental propositions of pure geometry holds just as little of substance. Granting as he does that several of them are genuinely analytic, the sole fundamental proposition he brings forward for that representation is the one saying the straight line between two points is the shortest. »My concept of the straight, namely, holds nothing of magnitude but a quality only; wholly added, therefore, is the concept of the shortest, nor can any dissection draw it out of the concept of the straight line; intuition must accordingly be taken to one's aid here, through whose means alone the synthesis is possible.« – Yet here too what is at issue is no concept of the straight in general but the straight line, and that is already something spatial, something intuited. Surely the determination (or the concept, if one likes) of the straight line is none other than this, that it is the utterly simple line, i.e. that in its coming outside itself (the so-called movement of the point) it refers utterly to itself, keeping in its extension no sort of diversity of determination posited, no reference to another point, or to a line beyond it; – the utterly in itself simple direction. Its quality is indeed this simplicity, and should the straight line seem hard to define analytically, that would be owing merely to the determination of simplicity, of self-reference, and merely because reflection, when it determines, has before it first and foremost a manifoldness, a determining through others; taken by itself, though, there is utterly nothing hard in grasping this determination — simplicity of extension within itself, absence of determination through another; – nothing else than this simplicity does Euclid's definition contain. – But the passage now from this quality to the quantitative determination (of the shortest), which was supposed to constitute the synthetic, is altogether analytic and nothing more. Being spatial, the line is quantity as such; the simplest, said of quantum, is the least, and said of a line this comes to the shortest. Geometry may admit such determinations as a corollary to the definition; Archimedes, however, in his books upon sphere and cylinder (see Hauber's transl., p. 4), did the most fitting thing in setting that determination of the straight line up as a fundamental proposition, with the same rightness of sense with which Euclid placed the determination touching parallel lines among the fundamental propositions, since to develop this determination into a definition would equally have called for determinations not immediately native to spatiality but more abstractly qualitative — simplicity, as above, equality of direction, and the like. These ancients also gave their sciences a plastic character, keeping their exposition strictly within the peculiarity of their material, and so shutting out whatever would have been of alien kind for it.

That concept which Kant set up in the synthetic judgments à priori, – the concept of a differentiated that is at the same time inseparable, of an identical that upon its own self is undivided difference, ranks among the great and imperishable things in his philosophy. In intuiting, of course, this concept is likewise present, seeing that it is the concept itself and everything is in itself the concept; but the determinations lifted out in those examples do not exhibit it; number and counting are rather an identity, a producing of an identity, that is utterly external only, a synthesis merely on the surface, a unity of ones, and of ones posited rather as not identical with one another upon themselves, as external instead, sundered for themselves; and in the straight line, the determination of being smallest between two points has for its ground rather the moment of the abstractly identical alone, devoid of difference upon its own self.

From this interruption I turn back to adding itself. Corresponding to it as the negative kind of calculation, subtracting is the equally quite analytic parting into numbers which, as in adding, stand determined relative to one another merely as unequal in general.

2. Next in determination comes the equality of the numbers that are to be numbered. Through such equality they form a unity, and herewith the difference of unity and amount makes its entrance upon number. The task set by multiplication is to count together an amount of units that are themselves an amount. Which of the two numbers gets given as unity and which as amount makes no difference here — whether one says four times three, four being the amount and three the unity, or the other way round, three times four. – Already stated above was that the product is originally found through plain numbering, i.e. counting off upon the fingers and so forth; being able to state the product immediately, later on, rests upon the collection of those products, the multiplication table, and upon knowing that by heart.

By the same determination of difference, division is the negative kind of calculation. Which of the two factors, divisor or quotient, gets determined as unity and which as amount is likewise a matter of indifference. Determined as unity is the divisor, and the quotient as amount, when division's task is pronounced thus, that one would see how often (amount) one number (unity) is contained in a given one; conversely, amount is what the divisor is taken for and unity what the quotient is, when it is said that a number shall be divided into a given amount of equal parts and the magnitude of such a part (the unity) found.

3. The two numbers standing determined relative to one another as unity and amount are, qua number, still immediate toward each other, and therefore unequal in general. Equality of a further sort is that of unity and amount themselves; so completed is the advance toward equality of the determinations that lie within the determination of number. Counting according to this complete equality is raising to a power, (the negative kind of calculation being extraction of roots) – at first, indeed, raising a number to the square, – numbering's perfect being-determined within its own self, where 1) the many numbers being added are the same, and 2) their plurality, their amount, is itself the same as the number posited many times over, the number that is the unity. No other determinations lie in the concept of number that could yield a difference; nor can any further equalizing of the difference which number harbours take place. Raising to powers above the square is a formal continuation, partly – with the even exponents – a mere repetition of squaring, partly – with the odd powers – a place where inequality re-enters; for given the formal equality (with the cube, say, to start with) of the new factor with amount as well as with unity, that factor, qua unity, is something unequal over against the amount (the square, 3 over against 3 · 3); and still more so with the cube of four, where the amount, 3, by which the number that is unity has to be multiplied with itself, differs from that number. – These determinations are in themselves at hand as the concept's essential difference, amount and unity, and they are what has to be equalized if the going-outside-itself is to return completely into itself. In what has just been set out lies further the ground why, on the one hand, solving the higher equations must consist in leading them back to the quadratic, and why, on the other, equations of odd exponents determine themselves only formally, so that precisely where the roots are rational they admit of being found in no other way than through an imaginary expression, i.e. one that is the opposite of what the roots are and express. – Arithmetic's square holds within itself, by what has been stated, the utter-being-determined alone; which is why equations with further formal powers have to be led back to it, just as geometry's right-angled triangle holds the utter-being-determined-within-itself set forth in the Pythagorean theorem, which is why all other geometrical figurations have likewise to be reduced to it for total determination.

Teaching that proceeds by a logically formed judgment deals with the doctrine of powers before the doctrine of proportions; the latter do indeed attach to the difference of unity and amount which constitutes the determination of the second kind of calculation, yet they step outside the one of immediate quantum, wherein unity and amount are moments only; and further determining along these lines stays external to that quantum as well. Number within the ratio is no longer immediate quantum; its determinateness it then has as mediation; the quantitative ratio comes up for consideration in what follows.

Of the further determining of the kinds of calculation here given it may be said that it is no philosophy of them, no laying-out, say, of their inner significance, since in fact it is not an immanent development of the concept. But philosophy must be able to tell apart what by its nature is a material external to itself, and to see that upon such a material the concept's advance can proceed only in external fashion, its moments too being able to exist only in the peculiar form of their externality — here equality and inequality. Telling apart the spheres to which a determinate form of the concept belongs, i.e. wherein it is at hand as concrete existence, is an essential requirement for philosophizing upon real objects, so that the external and contingent may not be disturbed in its peculiarity by ideas, nor these ideas be disfigured and made formal through the material's inadequacy. That externality, however, wherein the concept's moments make their appearance upon that external material, number, is here the adequate form; and inasmuch as they exhibit the object in its understanding, and since besides they carry no speculative demand and therefore look easy, they deserve to find application in the textbooks of the elements.