Remark 1

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

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

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

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

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

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

Remark 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The antithesis runs:

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

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

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

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

“Hence the simple occupies a space.”

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

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

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

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

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

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

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

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

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

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

B. Continuous and Discrete Magnitude

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

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

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

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

Remark

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

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

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

C. Limitation of Quantity

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

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

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

Chapter 2. Quantum

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

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

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

A. Number

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

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

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

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

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

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

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

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

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

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

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

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

Remark 1

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

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

Remark 2

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

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

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

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

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

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

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

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

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

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

B. Extensive and Intensive Quantum

1. Their Difference

1. The quantum has its determinateness as limit in the amount. It is something discrete within itself, a many that is bounded; this many, as was shown, has no being for itself that would be distinct from its limit and would keep that limit outside it. For it is precisely within number that plurality makes up the determinateness over against the unit; the one as unit is indeed determined in itself as numerical one, yet as unit it is the indeterminate continuity, undifferentiated within itself; difference, otherness, it holds by way of plurality. Plurality therefore holds the moment of the limit, of negation, within number itself; the difference-in-itself accordingly resides in the amount.

The quantum is thus a manifold, and this plurality is one with its limit; as limit, as determinate quantum, it is a manifold at its own self. So taken it is extensive magnitude.

The extensive magnitude is to be distinguished from the continuous; what stands directly over against it is not the discrete but the intensive magnitude. Extensive magnitude is what lies asunder in its determinateness, or is such insofar as the limit is a manifold; it has the moment of continuity insofar as, upon it and likewise within its limit, this many shows itself as something continuous and the limit shows itself as negation at this equality of the many. Continuous magnitude, however, is quantity carrying itself onward with no regard for a limit, or, insofar as it is represented as having a limit, that limit falls outside that continuity and is a bounding […]ndig[…]in general, without discreteness being posited upon it. — Continuous magnitude is not yet the magnitude truly determined in itself, since it lacks the many ones wherein being-determined-in-itself resides; its limit is therefore outside it, and not yet number. — Just so, discrete magnitude is in its determination immediately no more than a differentiated many in general, which, were it as such supposed to have a limit, would be a mere multitude, that is, something bounded indeterminately and externally. — Insofar as continuous and discrete magnitude are both quantum, however, they are number according to the quantum's true determination, and this is at first as extensive quantum, — the determinateness that is essentially as amount, though as amount of one and the same unit.

2. The extensive quantum is the limit that is manifold within itself. It has the differentiated other at its own self, and for that reason number is what is completely determined at its own self. The determination, by way of number, of how large something is does not require a difference from something else that is large, as though the determinateness of this large thing called for both itself and another large thing; it is a limit determined in itself, and thereby an indifferent limit, related simply to itself. The many of the limit, however, is, like the many in general, nothing unequal within itself but something continuous; each of the many is what the other is; hence its being a many lying asunder, or discrete, does not make up the determinateness as such. This many, then, of its own accord falls back together into its continuity and becomes simple unit. — The many here, however, was not a many for itself in general, but the determination of the many, amount over against the unit. But number is the one of unit and of amount, or the unity that has returned into itself out of the diversity of these determinations. Therein the amount is only a moment, or is sublated; it therefore does not make up the determinateness of number as a multitude of numerical ones; rather these, as indifferent and external to themselves, are sublated in number's having returned into itself; the externality that made up the ones of plurality vanishes in number's reference to itself.

The quantum, which as extensive had its determinateness at the amount external to itself, thus passes over into simple determinateness. Within this simple determination of the limit it is intensive magnitude; and the limit or determinateness as such, which before was as amount, is something simple, the degree.

The degree is thus determinate magnitude, quantum, yet not at the same time multitude, or something plural within its own self; it is only a multiplicity; the multiplicity is the several taken together into the simple determination. Its determinateness is indeed expressed by a number, this being the quantum's being-determined-in-itself, but it is not an amount; it is simple, only One degree. When 10, 20 degrees are spoken of, the quantum that has so many degrees is not their amount and sum; on that reckoning it would be an extensive one; rather it is only one, the tenth, the twentieth degree. It holds the same determinateness that lies in the amount ten, twenty, but it holds them not as several, and is rather number as sublated amount, as simple determinateness.

But this form of reference to itself, which the quantum has attained, is at the same time its becoming-external. As extensive quantum, number has its determinateness at its own self only in numerical plurality; but this plurality, being a many in general, falls together into undifferentiatedness, and as a many external to itself it sublates itself in the one of number, in number's reference to itself. The intensive quantum remains determinate quantum. The determinateness of the quantum, however, is otherness that is external to itself and indifferent. The degree, simple within its own self, which keeps this external otherness no longer in it, keeps it outside it, and refers itself thereto as to its determinateness. There is thus an external plurality; but in such a way that this external element at the same time makes up the simple limit, the determinateness which the degree is for itself. The amount as such thus remains the determinateness of number, but outside the number whose determinateness it is. That the amount, then, insofar as it was supposed to be found within number in the extensive quantum, sublated itself there, determines itself more closely as this: that it has been posited outside number. Since number is one, reference to itself reflected into itself, it thereby shuts the indifference and externality of the amount out of itself, and is reference to itself as reference through itself to something external.

Herein the quantum has the reality conformable to its concept. The quantum is determinate quantity. The determinateness of quantity is indifferent determinateness, which is not as referred to another; it therefore has otherness at its own self, and is in its own self external. So it is amount, the determinate being-differentiated within its own self; the amount makes up a determinate magnitude, and this being-determined, — whether they are three, or four and so forth, falls wholly within number itself; no comparison with others is required for it, nor is it a qualitative difference from something else. Since this externality is an inward externality that refers itself to itself, it is the externality of its own self. It is thus intensive magnitude, simple determinateness, as reference to itself which just as much has its determinateness in something external; the determinateness which at its own self is the determinateness external to itself.

Accordingly, then, the degree is a simple determinateness of magnitude among a multiplicity of intensities which are diverse, yet stand in essential reference to one another, so that each has its determinateness in this continuity with the rest. This reference of the degree through itself to its other makes the ascent and the descent of the scale of degrees a steady advance, a flowing that is an uninterrupted, indivisible alteration. Each of the several that are distinguished therein is thereby not cut off from the others, but has its being-determined only in these others. As a determination of magnitude referring itself to itself, each of the degrees is indifferent toward the rest; yet it is just as much in itself referred to this externality and has its determinateness therein; its reference to itself is therefore just as much the non-indifferent reference to what is external. The external is sublated in the simplicity of the degree; but it is just as much sublated outside the degree as something external too, for it stands in essential reference to the simple determinateness, and so is just as much not external to it.

2. Identity of Extensive and Intensive Magnitude

Intensive magnitude is the amount of extensive magnitude taken together into simplicity; a determinate one that does not have its determinateness at its own self as something plural; the degree is not within itself anything external to itself. Yet it is not merely the indeterminate one, the principle of number in general, which has no amount save the negative one of being no amount at all. — But intensive magnitude has at the same time its determinateness only in an amount. It is a simple one of the several; there are several degrees; determinate, however, they are neither as simple one nor as several, but only within the reference of this being-outside-itself, or within the identity of the one and multiplicity. If, then, the several as such lie outside the simple degree, its determinateness consists in its reference to them; it therefore holds the amount within it. As twenty, taken as extensive magnitude, holds the twenty ones within itself as discrete, so the determinate degree holds them as continuity, being this determinate multiplicity in simple fashion; it is the twentieth degree; and it is the twentieth degree only as this amount. This amount, however, which within the degree is simple, is at the same time externality at its own self; it is amount only as a multitude of numerical ones, which is just as much outside that simplicity of the degree.

The determinateness of intensive magnitude is therefore to be considered from a doubled side. It is first determined by other intensive quanta; it stands in continuity with its otherness, and its determinateness consists in this reference to its otherness. Insofar as it is the simple determinateness, it is thus determined against other degrees; it shuts them out of itself, and has its determinateness in this shutting out.

But secondly it is determined at its own self; to that extent it is so determined in the amount, as in its own amount, not as in the excluded amount, nor in the amount of other degrees. The twentieth degree holds the twenty at its own self; it is not merely determined as differentiated from the nineteenth, the twenty-first and so forth, but its determinateness is its own indifferent amount. Yet since that amount belongs to the degree, and since determinateness has at once essentially the shape of amount, the degree proves to be extensive quantum.

One determinateness of the quantum, then, is all that extensive and intensive magnitude are, and nothing sets them apart save this, that the one has it in simple form, the other in manifold form. Extensive magnitude passes over into intensive magnitude because its many in and for itself falls together into the unit, and, as determinateness of the many, of the numerical ones external to themselves over against the unit, steps outside the unit in number's reference to itself over against this unit. But conversely this simple thing has its determinateness only at the amount, and indeed as its own; for it is at the same time indifferent toward the intensities determined otherwise. Intensive magnitude is thus just as essentially extensive magnitude.

The difference of extensive and intensive magnitude rests on the difference of their moments, of the amount and the unit; magnitude is the one and the other magnitude posited in the determination of the one or the other moment. But because these moments are essential to it, because the determinateness is just as much the determinateness of the many as of something continuous or of a simple reference to itself, as it is of the discrete, of what is external to itself, its being posited in one of them is just as much its being posited in the other; or its existence is this doubled existence, which, however, is a matter of indifference as regards the determinateness of the quantum itself.

Remark

In the ordinary way of representing things, extensive and intensive quantum are wont to be distinguished as kinds of magnitude, as though there were objects that had intensive magnitude only, and others that had extensive magnitude only. There has come in besides the representation belonging to a philosophical natural science, which converted the plural, the extensive, for instance in the fundamental determination of matter, that of filling a space, and likewise in other concepts, into an intensive, in the sense that the intensive, as the dynamic, is the genuine determination, and that density, for instance, or specific space-filling, must essentially be grasped not as a certain multitude and amount of material parts within a quantum of space, but rather as a certain degree of matter's space-filling force.

Determinations of two sorts are to be distinguished here; there occurs the concept of self-subsistent parts subsisting apart from one another, joined into a whole only externally, and, distinct from it, the concept of force. What in the filling of space is on the one side regarded as nothing but a multitude of mutually external atoms is, on the other side, taken for the expression of a simple force lying at the ground. — These relations, of the whole and its parts, and of force and its expression, do not, however, belong here, but will be considered further on. — The other, however, is the quantitative determinateness that occurs in this connection, and in regard to which magnitude is sublated as extensive quantum and converted into degree, as into the determination that alone is supposed truly to be.

As regards this supposed essentiality of the difference it is enough to have shown that the difference is unessential for the determinateness of the quantum itself, that the one form is however essential for the other, and that every existence therefore exhibits its determination of magnitude just as much as extensive as it does as intensive quantum.

Everything serves as an example of this, insofar as it appears in a determination of magnitude. Number itself necessarily has this doubled form immediately at its own self. It is an amount, and to that extent it is extensive magnitude. But it is also a one, a ten, a hundred; to that extent it stands at the transition to intensive magnitude, since within this unit the manifold goes together into something simple. The tenth, the hundredth is this simple thing at its own self, which has its determinateness at the several that fall outside it, and to that extent is properly intensive magnitude. The number is ten, a hundred, and this same number is at once the tenth, the hundredth within the number system; both are the same; either determination can be taken for the other; the tenth number in the number system is ten.

The one in the circle is called degree, because a part of the circle has its determinateness essentially in a several outside it, being determined only as one out of a certain amount of such ones. The degree of the circle is, however, only the principle of the number of a magnitude of the circle, only its one. A quantum of the circle itself is an arc of determinate magnitude, an ordinary number, namely an amount of such ones as are degrees. This number is extensive magnitude, and intensive only insofar as, as was just recalled, number is this in general.

The magnitude of actual objects exhibits its doubled side, that of being extensive and intensive, at the doubled determinations of the object's existence, in one of which the object appears as something external, in the other, however, as something internal. Thus a mass, for instance, is as weight something extensively large insofar as it makes up an amount of pounds, hundredweights and so forth; something intensively large insofar as it exerts a certain pressure; this magnitude of the pressure is something simple, a degree that has its determinateness at a scale of degrees of pressure. In exerting pressure the mass appears as a being-within-itself, as a subject to which the intensive difference of magnitude accrues. — Conversely, whatever exerts this degree of pressure is capable of shifting a certain amount of pounds and so forth from their place, and measures its magnitude thereby.

Or heat has a degree; the degree of warmth, be it the 10th, the 20th and so forth, is a simple sensation, a subjective thing. But this degree is just as much present as extensive magnitude, namely as the expansion of a fluid, of the mercury in the thermometer, of air or of clay and so forth. A taller mercury column, or a slimmer clay cylinder, is how a higher degree of temperature gives itself expression; it warms a larger space just as a lesser degree warms the smaller one.

The higher tone, being the more intensive one, is at the same time a greater multitude of vibrations, or a louder tone, one to which a higher degree is ascribed, makes itself audible within a larger space. — With the more intensive color a larger surface admits of being colored in like manner than with a weaker one; or the brighter, another kind of intensity, is visible farther off than the less bright and so forth.

Just so within the spiritual, the high intensity of character, of talent, of genius is of an equally far-reaching existence, extended effect and many-sided contact. The deepest concept has the most universal significance and application.

3. Alteration of the Quantum

The difference of the extensive and the intensive quantum is a matter of indifference to the determinateness of the quantum at its own self; it is only a difference of the quantum's existence, or the quantum has the determinations that make up the extensive and the intensive as its own moments within itself. But if the quantum is by contrast indifferent toward a difference of its existence, its moments have in exchange entered into an inner opposition. The extensive quantum has, as a one referring itself to itself, passed over into the intensive quantum. The latter, however, which alone is thus to be considered, is the determinateness of magnitude that is simple within itself yet, precisely in this determinateness referring itself to itself, is external to itself, consisting not within itself but in another several.

Intensive magnitude is thus quantum that is for itself, and therein essentially referred to an other. This other is an other of this magnitude; another quantum. Intensive magnitude therefore is only as having its determination in another magnitude. But that it has its determination, its being-in-itself, in another magnitude means that it is not itself but another quantum. Or it passes over essentially into another magnitude.

Intensive magnitude, however, is in general the real quantum. The quantum is the determinateness posited as sublated, the indifferent limit; which is to say that it is that determinateness which no less is the negation of its own self. So the quantum is posited as degree. The degree is the simple determinateness referring itself to itself which is the negation of its own self, since it has its determinateness not at itself but in another quantum; hence, in being this determinate quantum, it is rather essentially not itself but another quantum.

A quantum stands accordingly, in general, in absolute continuity with its externality, with its otherness. Hence not only can every determinateness of magnitude be overstepped, not only can it be altered; it must alter itself. Quanta appeared at first as external to one another, in the determination of numerical ones. But they are not merely external to one another; they are external to themselves. The determination of magnitude therefore carries itself on into its otherness in such wise that its being lies for it solely in that continuity with something else. A quantum is thus itself, and just as essentially not itself but the negation of itself, an other. It is not a limit that is, but one that becomes.

The one is infinite, or the negation referring itself to itself; it is therefore the repulsion of itself away from its own self. The quantum is likewise infinite and repels itself away from its own self. But the quantum is the determinate one, the one that has passed over into existence and into limit. The quantum is thus the repulsion of determinateness away from its own self; that repulsion is therefore not the generating of what is self-same, as the repulsion of the one is, but the generating of its otherness. As the one is not overstepped by any third thing, but repels itself away from itself, so it belongs likewise to the concept of the quantum to send itself out beyond itself and to become an other. It consists in increasing or diminishing itself; it is the externality of determinateness at its own self.

The quantum sends its own self out beyond itself; this other, into which it becomes, is at first itself a quantum; a limit that is not one which merely is, but one which drives itself out past its own self; it continues itself into its otherness; it is external to itself; and this externality of its own self is itself. The limit that has arisen anew in this going-out is thus utterly nothing but one that sublates itself again, and so on into infinity.

C. Quantitative Infinity

1. Its Concept

The quantum alters and becomes another quantum; it is, however, a further determination of this alteration that it goes on into infinity.

The quantum becomes an other; it continues itself into its otherness; the other is therefore also a quantum. But the other is at the same time the other not merely of some quantum, but of the quantum itself. For the quantum is the indifferent determinateness, which is indifferent toward an other, but also toward itself. As its moments have determined themselves in the intensive quantum, it is the determinateness that refers itself not to another but to its own self; yet just as much is this determinateness utterly nothing but the determinateness in an other; the reference to an other is external to it, but it is itself this externality of its own. It is thus the quantum itself which contradicts itself, and so dissolves itself in itself; it is itself accordingly the negation of its own self; the alteration concerns not merely a quantum, but the quantum. The quantum is an ought; what it holds within it is to be determined in itself, and such being-determined-in-itself is in truth the being determined in an other; turned about again, it is that same being-determined-in-an-other in sublated shape; it is indifferent being-determined. It is therefore an other and something external over against its own self; it holds within it this, to be finite, and to go out beyond finitude, beyond the being-determine[…]d in an other, and to be infinite.

With qualitative and quantitative infinity it is essential to note that the finite is not overstepped by any third thing, but that determinateness, as dissolving itself within its own self, goes out beyond itself. But what parts the qualitative from the quantitative infinite is this, that in the first the opposition between what is finite and what is infinite bears a qualitative character, and that the passage out of finitude into infinity, or the way the two refer to one another, resides merely in the in-itself, in their concept. Immediate to begin with, and something that is, is the qualitative determinateness; and it refers itself to otherness essentially as to an other of its own, no positedness attaching to it of holding its negation, its other, at its own self. Magnitude, by contrast, just is determinateness in sublated shape; posited it stands as the negation, as unequal with itself, as the alterable. The qualitative finite and infinite therefore stand absolutely over against each other; their unity is the inward reference lying at the ground; the finite therefore does not continue itself immediately into its other. The quantitative finite, on the contrary, refers itself at its own self into its infinite. Their reference is therefore the infinite progress.

2. The Infinite Progress

The progress into infinity is nothing other than the expression of the contradiction which the quantitatively-finite, or the quantum in general, contains. It is that reciprocal determining of finite and infinite which came under consideration in the qualitative sphere, though with the difference that, as was just recalled, in the quantitative the limit continues itself on its own self into its beyond, and hence conversely the quantitatively-infinite is posited as well, namely the quantum's having its other on its own self. Finite and infinite are, the one the non-being of the other. But because quantitative determinateness is the merely sublated difference, the quantitative is its own self in its being-outside-itself. The quantitatively-infinite is therefore indeed the sublated quantum, not merely as a quantum but as the quantum. But because the quantum continues itself into its sublatedness, the infinite is just as much determined as the opposite of its own self, as quantum.

The quantum, then, is determinateness-in-itself, the determinateness indifferent towards other, which however just as much only is as external to itself. The infinite progress is the expression of this contradiction, not the resolution of it; it stands stock-still within the contradiction, and does not go out beyond it.

Or the progress into infinity is only the task of the infinite, not the attainment of it. It is the perennial producing of the infinite, without ever getting out beyond the quantum itself, and without the infinite's becoming something positive and present. The quantum is such a thing that it lies in its concept to have a beyond of itself. This beyond is first the pure moment of the non-being of the quantum; for the quantum dissolves itself on its own self. Thus it refers itself to its beyond, to its infin ity. This is the qualitative moment of the opposition. But secondly the quantum stands in continuity with this its beyond, which is a not-[suner][…][s] non-being of the quantum; for the quantum […][con]sists precisely in this, in being the other of its own self, in being external to its own self; hence this other, this external, is just as much not an other than the quantum. So the beyond, that is, the infinite, is itself a quantum. The beyond has thereby been summoned back out of its flight, and the infinite reached. But because this thing that has become a this-side is again a quantum, all that has happened is that a new limit has once more been posited. The quantum that has arisen anew has, just because it is quantum, fled from its own self once more, is as such out beyond itself, and has repelled itself away from itself into its non-being; it thus has a perennial beyond. But the quantum at the same time consists precisely in being external to itself. Hence that beyond is itself once again the quantum.

If this, that herein the beyond or the infinite is determined as quantum, and conversely the quantum as infinite, is gathered into one expression, then this joining yields an infinitely great or an infinitely small. But this joining is itself nothing other than merely the false expression of the contradiction, or of the infinite progress. For in it the quantum and its beyond are kept in their absolute determinateness against each other, the one as the non-being of the other. The infinitely great and the infinitely small is represented as a quantum; it is something great or something small; but as quantum it has just as much thrust its beyond away from itself; it is not widened out into the infinite, but kept in perennial opposition to it. The great, widened ever so much, therefore shrivels together into paltriness; for insofar as it refers itself to the infinite as to its non-being, the opposition is, according to this moment, qualitative; the widened quantum has thus wrung nothing from the infinite; the latter is rather, now as before, the non-being of it. Or, the enlarging of the quantum is no drawing nearer to the infinite, for the difference of the quantum and of its infinity has essentially the moment of being a difference that is not quantitative. — In the same way the infinitely small, being something small, is a quantum, and for that reason stays too great for the infinite in an absolute, which is to say a qualitative, manner, and stands over against it.

The infinitely great or small is therefore itself only the infinite progress. This infinity, which is determined as the beyond of the finite, is to be designated the bad quantitative infinity. It is infinity of the progress, and, like the qualitative bad infinity, only the perennial crossing over and back, out of the one member of the abiding contradiction into the other, out of the limit into its non-being, and out of this non-being anew back into that very same thing, the limit. It is not so much a going forward as a repeating of one and the very same, positing, sublating, and positing again and sublating again; an impotence of the negative, to which that which it sublates comes back, through its very own sublating, as something continuous. There are two so knotted together that they simply flee each other; and in fleeing each other they cannot part, but are bound together in their very parting.

Remark 1

The bad infinity is wont, chiefly in the form of the progress of the quantitative into infinity, — this continual overflying of the limit, which is the impotence to sublate it, and the perennial relapse back into it, — to be held for something sublime and for a kind of divine service, just as in philosophy that progress has been looked upon as an ultimate. Tirades of this sort are to be met with everywhere, and they have won admiration as sublime productions. In fact, however, this modern sublimity makes great not the object, which rather escapes it, but only the subject, which devours such great quantities into itself. Yet the poverty of this elevation, which stays subjective and clambers up the ladder of the quantitative, announces itself in this, that amid its futile labor it comes no nearer to the infinite goal, a goal that must be tackled quite otherwise if it is to be reached.

In the following tirades of this sort there is at the same time expressed what such elevation passes over into and where it ceases. Kant for instance adduces as sublime,

“when the subject in thought lifts itself above the place that it occupies in the world of sense, and extends the linkage into the infinitely great, a linkage with stars upon stars, with worlds upon worlds, systems upon systems, and over and above this into the boundless times of their periodic movement, of the beginning and the duration thereof. — Representing succumbs to this going forth into the immeasurably distant, where the most distant world has always yet a more distant one, the past traced back so far has yet a further one behind it, the future carried out ever so far has always yet another before it; thought succumbs to this representation of the immeasurable; as a dream in which someone walks down a long corridor ever further and unforeseeably further, without descrying any end, closes with falling or with dizziness.”

This portrayal, quite apart from crowding the content of the quantitative elevating into a wealth of depiction, deserves praise above all for the truthfulness with which it states how this elevation fares in the end: thought succumbs, the end is falling and dizziness. What makes thought succumb, and brings forth its falling and dizziness, is nothing other than the tedium of that repetition which lets a limit vanish and come up again and vanish again, thus always the one after the other, and the one in the other, in the beyond the this-side, in the this-side the beyond, perennially coming to be and ceasing to be, and which yields nothing but the feeling of the impotence of this infinite or of this ought, which wants to become master over the finite and cannot.

Haller's horrifying description of eternity, so named by Kant, is likewise wont to be particularly admired, but often precisely not on account of that side of it which constitutes its genuine merit:

| “I heap up monstrous numbers, | Pile mountains of millions, | I set time upon time, and world upon world in a heap, | And when from that grisly height | I look with reeling eyes back toward you, | All the might of number, multiplied a thousand times over, | Is not yet one part of you.” | “I take them away, and you lie wholly before | me.”

When value is laid upon that heaping and towering up of numbers and worlds as upon a description of eternity, it is overlooked that the poet himself pronounces this so-called horrifying going-beyond something futile and hollow, and that he closes by saying that only through the giving up of this empty infinite progress does the genuine infinite itself come into presence before him.

As is well known, the astronomers too are fond of priding themselves greatly on the sublimity of their science, and this because it has to do with an immeasurable multitude of stars, with spaces and times so immeasurable that within them distances and periods already so great on their own account serve as units which, taken ever so many times over, shrink back once more to insignificance. The stale astonishment to which they abandon themselves in all this, the tasteless hopes of one day travelling in that other life from one star to another and of gathering the like new knowledge on into the immeasurable, they pass off as a chief moment of the excellence of their science, — a science that is admirable indeed, yet not for the sake of the quantitative infinity that occurs in it, but on the contrary for the sake of the measure-relations and the laws which reason has cognized in these objects, and which are the rational infinite as against that irrational infinity.

To the infinity that refers itself to outer sensuous intuition Kant opposes the other infinity, when

“the individual goes back into its invisible I, and, as a pure I, sets the absolute freedom of its will against all the terrors of fate and of tyranny, beginning with its nearest surroundings, lets these vanish for it, lets likewise what appears as enduring, worlds upon worlds, crash down into rubble, and, alone, cognizes itself as equal to its own self.”

The I in this solitude with itself is indeed the beyond attained; in pure self-consciousness the absolute negativity is brought into presence and is with its own self, that negativity which in the advance out beyond the sensuous quantum does nothing but flee. But in fixing itself in its abstraction and its lack of content, this pure I has existence in general, the fullness of the natural and of the spiritual universe, standing over against it as a beyond. There presents itself the same contradiction that lies at the ground of the infinite progress; namely a having-returned-into-itself which is immediately at the same time a being-outside-itself, a reference to its other as to its non-being. Which reference stays a longing, because the I has fixed for itself its own emptiness on the one side, and the fullness as its beyond.

Kant appends to these two sublimities the remark “that admiration (for the first, the outer) and respect (for the second, the inner) sublimity do indeed spur inquiry on, but cannot make good the want of it.” — He thereby declares those elevations to be unsatisfying for reason, which cannot come to a stop with them and with the sensations bound up with them, and cannot let the beyond and the empty pass for the ultimate.

As an ultimate the infinite progress has been taken above all in its application to morality. The second opposition of the finite and the infinite just adduced, that of the manifold world and of the I raised into its freedom, is at first, in its purity, qualitative. Since the self-determining of the I consists at once in determining nature and in freeing itself from nature, it refers itself through its own self to its other, which as outer existence is something manifold and quantitative. But the determining of something quantitative becomes itself quantitative, and the negative reference of the I to it, the power of the I over the not-I or over sensibility and outer nature, is therefore represented in such a way that morality can and ought to become ever greater, the power of sensibility however ever smaller; the will's complete adequacy to the moral law, however, is displaced into the progress that runs on into infinity, which is to say, represented as an absolute unattainable beyond, and precisely this, that it is an unattainable one, is supposed to be the true anchor and the right consolation.

Within this opposition the I and the not-I, or the pure will and nature and sensibility, are represented as perfectly self-subsistent and indifferent towards each other. The pure will has its own peculiar law, which stands in essential reference to sensibility; likewise nature has laws which are neither drawn from the will nor answerable to it, nor would even, though different from it, have in themselves an essential reference to it, but which are determined altogether on their own account, complete and self-enclosed. Both, however, are simultaneously moments of one and the very same simple essence, of the I; the will is the negative, which consists in sublating nature, and hence only is insofar as there is something distinct from it that is to be sublated by it. It puts itself into a bearing towards sensibility, so as to determine it; it thereby goes out beyond itself, touches sensibility and is thus itself affected by it. Nature and sensibility are presupposed as a self-subsistent system of laws; the being restricted by an other is therefore a matter of indifference to them; nature maintains itself in this being bounded, enters self-subsistently into the reference, and bounds the will just as much as the will bounds nature. — It is one act, that the will determines its own self and sublates the otherness of a nature, and that this otherness is posited, or that it continues itself into its being sublated. The contradiction that lies herein is not resolved in the infinite progress, but on the contrary is set forth and asserted as unresolved and unresolvable; the struggle of morality and of sensibility is represented as the absolute relation that is in and for itself.

The impotence to become master over the opposition of the finite and the infinite takes refuge in magnitude, so as to employ it as the mediatrix, since magnitude is the qualitative sublated, the difference that has grown indifferent. Yet since both members of the opposition lie at the ground as qualitatively diverse, the very fact that in their mutual reference they bear themselves as quanta makes each of them indifferent towards this alteration. Nature is determined by the I; but because this negation contains not the qualitative but only the quantitative difference, it is precisely such a difference as does not touch nature itself, but lets it subsist as what it is.

In the more abstract presentation of the Kantian philosophy, or at any rate of its principles, namely in Fichte's Wissenschaftslehre, the infinite progress makes up in the same way both the foundation and the ultimate. Upon the first principle of that presentation, I = I, there follows a second one independent of it, the opposing of the not-I; the reference of the two is assumed as the quantitative difference, that the not-I is in part determined by the I and in part not. In this manner the not-I continues itself into its non-being as something that stays opposed to its non-being, as something unsublated. After the contradictions lying therein have accordingly been developed, the concluding result is that very relation which was the beginning; the not-I remains an infinite check, an absolutely-other; the last reference of it and of the I to one another is the infinite progress, the same contradiction with which the start was made. The finite, and the finite relation, is supposed to be the absolutely true.

Because the quantitative in general is the negation of determinateness, it was believed that a great deal, or rather everything, stood to be won for the absolute's unity, for the one substantiality, by degrading opposition as such into a difference that is merely quantitative. All opposition is merely quantitative was for a time a chief proposition of the newer philosophy; determinations that stand opposed are of one essence, of one content; — further, each side of the real opposition too has both determinations, both factors, within it; save that on the one side the one factor and on the other side the other is preponderant; and the preponderant was frequently taken also in the sense that in the one side the one factor, a matter or an activity, is at hand in greater amount or in stronger degree than in the other. As for the latter, so far as diverse stuffs or activities are presupposed, what the quantitative difference does is rather to seal and perfect their externality, their indifference towards one another. But as regards the former, that the difference of the absolute unity is supposed to be merely quantitative, the quantitative is indeed immediate determinateness sublated, yet it is the merely imperfect negation; for it is as yet only the first negation, not the infinite one, not the negation of the negation. — Or, in being represented as quantitative determinations of the absolute substance, being and thinking become, as quanta, precisely thereby wholly external and relationless to one another, as carbon, nitrogen and so forth do in a subordinate sphere. It is a third thing, an external reflection, that abstracts from their difference and cognizes their inner unity, a unity that only is in itself. This unity is in this way represented only as a first immediate one, or only as being which in its quantitative difference stays equal to itself, but does not posit itself equal through its own self; or it is not grasped as negation of the negation, as infinite unity. It is only the qualitative opposition that contains the genuine infinity, and the quantitative difference passes over, as will shortly appear more closely, into the qualitative.

Remark 2

It was recalled above that the Kantian antinomies are presentations, in a more concrete shape, of the opposition between finite and infinite, brought to bear upon more special substrates of representation . The antinomy considered above contained rather the opposition of qualitative finitude and infinity. In another, the first among the four cosmological antinomies, what is considered in its conflict is rather the quantitative limit. I shall therefore undertake the examination of this antinomy here.

It concerns, namely, the limitedness or unlimitedness of the world in time and space. — This opposition could with equal right have been considered in respect of time and of space themselves, for nothing in the antinomic character of limitedness or unlimitedness is altered by whether time and space are relations of the things themselves, or are merely forms of intuition.

The closer unfolding of this antinomy will likewise show that the two propositions, and equally their proofs — which, as with the antinomy considered above, are conducted apagogically — run out into nothing but the two simple, mutually opposed assertions: a limit is, and there must be a going out beyond the limit.

The thesis reads:

“The world has a beginning in time, and as regards space it too is enclosed within limits.”

The one part of the proof, the one concerning time, assumes the contrary,

“that as regards time the world has no beginning; then up to every given point in time an eternity has run out, and consequently an infinite series of states of things in the world following upon one another has ela psed. But the infinity of a series consists precisely in this, that through successive synthesis it can never be completed. An infinite elapsed world-series is therefore impossible, and a beginning of the world is accordingly a necessary condition for its existence; which was the thing to be shown.”

The other part of the proof, the one that concerns space, is led back to time. The gathering together of the parts of a world infinite in space would require an infinite time, which would have to count as run out, insofar as the world in space is to be taken not as something becoming but as something completed and given. But of time it was shown in the first part of the proof that to assume an infinite time as run out is impossible.

One sees at once, however, that it was unnecessary to make the proof apagogic, or to conduct a proof at all, since what lies immediately at its own base is the assertion of the very thing that was supposed to be proved. For some one or any given point in time is assumed, up to which an eternity (-- eternity here has only the meagre sense of a badly infinite time) is supposed to have run out. A given point in time signifies nothing other than a determinate limit within time. Within the proof, accordingly, a limit of time is presupposed as something actual; yet this is precisely that which was to be proved. For the thesis consists in this, that the world has a beginning in time.

The only difference that occurs is that the assumed temporal limit is a now as the end of the time previously elapsed, whereas the one to be proved is a now as the beginning of a future. Yet this difference is inessential. Now is assumed as that point wherein an infinite series of states of things in the world, one following upon another, is supposed to have elapsed, hence as an end, as a qualitative limit. Were this now regarded merely as a quantitative limit, one that is to be gone out beyond and that flows, then the infinite series of time would not stand elapsed within it but would go on flowing, and the reasoning of the proof would fall away. But this point in time, assumed as a qualitative limit for the past, is at the same time the beginning for the future, — for in itself every point in time is the reference of past and future, — and indeed it is an absolute beginning for that future. For it makes no difference to the matter that before this future of its and before the beginning of that future a past already is; since this point in time is a qualitative limit, — and to assume it as qualitative lies in the determination of the completed, of what has run out, hence of what does not continue itself, — time is thereby broken off in it, and the past that is being spoken of stands without reference to the time which could be called future only in regard to this past, and which is therefore only time as such, time having an absolute beginning. But did it stand, — (as it indeed does —) through the now, the given point in time, in a reference to the past, were it in fact future, then from the other side this point in time too would be no limit, the infinite series of time would carry itself on into that which was called future, and would not, as was assumed, be completed.

In truth time is pure quantity; the point in time employed in the proof, at which time was supposed to be interrupted, is rather only the self-sub lating being-for-itself of the now. The proof achieves nothing beyond setting before representation, as a given point in time, that absolute limit of time which the thesis asserts, and flatly assuming it, a popular determination which sensuous representing readily lets pass as a limit, thereby letting hold good within the proof, as an assumption, that which had earlier been put forward as what was to be proved.

The antithesis states:

“The world has neither a beginning nor limits in space, but is infinite in view of time as well as of space.”

The proof posits the contrary:

“Let the world have a beginning. Since the beginning is an existence before which there goes a time wherein the thing is not, then a time wherein the world was not must have gone before, that is, an empty time. But in an empty time no coming-to-be of any thing whatever is possible; because no part of such a time has in itself, ahead of another, any distinguishing condition of existence over that of non-existence. Hence many a series of things can indeed begin within the world, but the world itself can take no beginning, and in respect of elapsed time it is infinite.”

This apagogic proof contains, like the others, nothing beyond the direct and unproven assertion of the very thing it was meant to prove. For it first assumes a beyond of worldly existence, an empty time; but then it just as much continues worldly existence out beyond itself into this empty time, thereby sublates that time, and thus carries existence on into infinity. The world is an existence; and the proof presupposes of this existence that it comes to be, and that coming-to-be has a preceding condition in time. But in this precisely the antithesis itself consists, that no existence is unconditioned and no limit absolute, and that essential existence always demands a preceding condition. This condition is at the same time itself conditioned; it is sought in empty time, which amounts to saying that it is itself assumed as temporal and hence as existence, and as something limited. Quite generally, then, the assumption has been made that the world as existence presupposes another existence, and so on into infinity.

The proof with regard to the world's infinity in space is just the same. In apagogic fashion the spatial finitude of the world is assumed; “it would thus find itself in an empty unlimited space and would have a relation to it; yet a relation of this sort, of the world to no object, is nothing.”

What was supposed to be proved is here likewise directly presupposed in the proof. For it is directly assumed that the limited spatial world should find itself in an empty space and have a relation to it, which is to say that it must be gone out beyond, on the one side into the void, into the world's beyond and non-being, while on the other side it thereby stands in relation to that void, hence continues itself into it, and that the beyond is to be represented as filled with worldly existence. What the antithesis asserts, the infinity of the world in space, is nothing other than, on the one part, empty space and, on the other part, the relation of the world to it, which is to say the continuity of the world within it, or the filling of it; and this contradiction, space at once as empty and at once as filled, is the infinite progress of existence within space. But this contradiction itself, the world's relation to empty space, is what is directly assumed in the proof.

Thesis and antithesis and their proofs therefore present nothing but the opposed assertions that a limit is, and that this same limit is no less merely a sublated one; namely, that the limit has a beyond with which it stands in reference, towards which one is to go out beyond it, but wherein there arises once more such a limit as is none.

The resolution of these antinomies is, like that of the one above, transcendental, which is to say that it lies in asserting the ideality of space and of time, as forms of intuition, in this sense: that the world in its own self is not in contradiction with itself, is not something self-sublating, but that consciousness, in its intuiting and in the reference of intuition to understanding and reason, is an essence contradicting itself.

3. Infinity of the Quantum

1. The infinite quantum, as infinitely great or infinitely small, is itself the infinite progress; it is quantum as a great or a small, and is non-being of quantum as infinite. The infinitely great and the infinitely small are therefore images of representation which on closer consideration show themselves to be null mist and shadow. But the infinite progress expresses nothing other than the nature of quantum, which as intensive magnitude has attained its reality.

Quantum, returned into itself, is simple, referred to itself and as determined in itself. But since through this simplicity otherness and determinateness are sublated at its own self, determinateness is external to it; it has its absolute determinateness rather outside it. This being-outside-itself of quantum is at first the abstract non-being of quantum as such, bad infinity. But further it is also a great, quantum continues itself into its non-being, for it has its determinateness precisely in its externality; this externality of quantum is therefore just as much itself quantum, only another quantum, which however sublates itself once more as the first did.

Quantum is therefore something determined in itself; but this determinateness of its it has outside itself, so that it sublates itself; conversely, in its being-outside-itself it has returned into itself, its being-outside-itself is just as much sublated.

This circle is the genuine element that is posited in the infinite progress. There is present quantum and its beyond. First, quantum sublates itself, it is at its own self the going out beyond its limit; the beyond is infinity, but it is bad infinity, for second, quantum continues itself into it. This beyond, the non-being of quantum, infinity itself is limited, and a quantum is posited anew, which is to say, this beyond is itself sub lated. Quantum is precisely itself through its being-external; this is exactly what makes up the determinateness of quantum, or what quantum is. In the infinite progress, therefore, there is the concept of quantum as it is in itself; and there is present in the progress the sublating of quantum, but just as much of its beyond; or the negation of quantum as well as the negation of this negation.

The going out beyond quantum is the negation of quantum, the infinite; but a new quantum is posited, and this is the negation of the infinite, of this bad infinite which counts for representation as an absolute, as a last term that does not sublate itself again and out beyond which nothing further could be gone. The truth of the infinite progress is therefore that quantum and its beyond are posited, but that they are posited as sublated. Its truth is accordingly their unity, wherein they are, but as moments.

This is thus the true resolution of the contradiction whose expression the infinite progress is. It consists in nothing other than the restoration of the concept of magnitude, namely that magnitude is an indifferent or external limit. In the infinite progress as such, reflection is usually directed only upon this, that every quantum, be it ever so great or small, vanishes, that it must be possible to go out beyond it; but not upon this, that this sublating of quantum, the beyond, the bad infinite, itself vanishes too. This happens, however, in that quantum continues itself into its negation, in that out beyond every quantum, into its sublating, a new quantum is posited. The first sublating is indeed in itself the sublating of negation, — for quantum is sublated limit, — but it is at the same time only in itself this; for this infinite is fixed as the beyond of quantum, which still remains subsisting as a this-side; or quantum is taken only as something immediate, and the infinite only as the first negation. But in the infinite progress there is more present, — than merely the sublating of immediate quantum, or than merely a first sublating; therein this bad infinite too is, — through the new limiting, sublated; there is therefore present in it the negation of negation, or that which the infinite in truth is. — But the concept of quantum is not merely restored, it has also received its closer determination; there has arisen the quantum determined by its concept, which is distinct from immediate quantum.

2. The beyond of quantum has, namely, a more determinate, positive significance than merely that of the non-being of quantum; and so likewise does the sublating of this beyond, and the uniting of it with quantum itself.

Quantum as indifferent limit is determined at its own self; this self-referring being-determined is the having-vanished of its externality, which it has at its own self; this externality thereby steps outside quantum; its going out beyond itself is its essential moment, it refers itself through its own self to its externality; but this externality makes up its being determined in itself, and the nature of its being-determined-in-itself consists in this externality. The beyond of quantum is therefore not the mere non-being, the empty, indeterminate negation of quantum. Rather quantum goes out beyond itself just insofar as it is indifferent limit; it sublates this indifference and posits the being-in-itself of that indifference as an infinite beyond, as that wherein it is negated, wherein it is not its own self but the externality of its own self. But this externality is rather the opposite of itself; it is an absolute moment of magnitude itself; for quantum is not itself in its immediacy, but is essentially a going out beyond itself; this going out beyond itself, this externality of its, therefore belongs to quantum itself.

But its going out beyond itself is the sublating of its indifference towards the external, which is its determinateness, and it thereby posits this determinateness as its own self. It sublates its beyond, its negation, which is to say, it sublates the externality of its being-determined; unites it with itself and thereby makes itself determined in itself.

Each of the moments of the movement of the infinite progress is the opposite of its own self; for the infinite progress is the posited contradiction. First, quantum goes out beyond itself; this therefore means 1) it sublates itself, posits its negation, its beyond, and 2) it thereby posits rather its absolute being-determined, that which it is in itself. Second, this infinite is determined again, a new limit is posited; this accordingly means 1) the being-in-itself of quantum is sublated, it only withdraws an indifferent quantum once more, 2) the negation of quantum, the beyond of quantum, is sublated, its going out beyond itself is thus taken back into quantum itself. Both sides express this, that quantum is negated and that the negation of quantum is negated; there is therefore posited its infinite reference to its own self, or its being-determined-in-itself. Infinity, which was only the bad one and a beyond of quantum, belongs to quantum, quantum is itself infinite.

In this restoration of quantum, quantum is sublated as indifferent limit, as this perennial going out beyond itself. The indifference and externality of quantum therefore vanishes only insofar as the beyond of quantum is sublated. Quantum no longer has infinity, being-determined-in-itself, outside itself. The limit is therefore sublated as indifferent or as sublated. It has thus become qualitative once more.

The infinite, then, which in the infinite progress has only the empty significance of a non-being, of a beyond, is in fact nothing other than quality. Quantum is indifferent limit; it goes out beyond itself into infinity; in doing so it seeks nothing other than being-determined-in-itself, the qualitative moment. But this qualitative moment is not a beyond of quantum, it lies within quantum itself. For precisely this going out beyond itself, or the beyond, the negation of quantum, is what makes quantum into quantum; this is its determinateness in itself; its very indifference is its determination itself.

Or, quantum is sublated quality; but quantum is infinite, goes out beyond itself, it is the negation of itself. It is therefore the negation of negated quality, or it is the restoration of that quality.

But the quantum that is sublated as indifferent limit and qualitatively determined is the quantitative ratio. In the ratio quantum is external to itself, distinct from its own self; but this externality of its, the reference to the other quantum, at the same time makes up its determinateness; therein it has not an indifferent but a qualitative determination; in its externality it has returned into itself.