A. Pure Quantity

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

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

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

Remark 1

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

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

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

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

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

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

Remark 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The antithesis runs:

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

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

»Posit, so it runs, that a composite thing, as substance, is made up of simple parts. Because every external relation, and with it all composition out of substances, is possible only in space, the space occupied by the composite must be made up of as many parts as the composite is made up of. Now space is made up not of simple parts but of spaces. Each part of the composite must therefore occupy a space.« »The very first parts of everything composite, however, are simple.«

»The simple, accordingly, occupies a space.«

»Now since everything real that occupies a space holds within itself a manifold of things lying outside one another, and is thus composite, composite indeed out of substances, the simple would be a substantial composite. Which is self-contradictory.«

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

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

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

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

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

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

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

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

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

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

B. Continuous and Discrete Magnitude

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

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

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

Discreteness is a moment of quantity, as continuity is, yet it too is quantity entire, precisely because it is moment within quantity, within the whole, and so, being differentiated, does not step out of that whole, out of its unity with the other moment. – Being-outside-one-another in itself is what quantity is, and continuous magnitude is that being-outside-one-another as prolonging itself without negation, as a cohesion self-same within itself. Discrete magnitude, by contrast, is that outside-one-another as non-continuous, as broken off. Yet with this multitude of ones there is not present again the multitude of atoms and the void, repulsion as such. Since quantity is what discrete magnitude is, its discreteness is itself continuous. Wherein this continuity in the discrete consists is that the ones are what is equal to one another, or that one and the same unit belongs to them. Discrete magnitude is accordingly the outside-one-another of the many ones as of the equal, not the many ones simply, but posited as the many of a unit.

Remark

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

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

C. Limitation of Quantity

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

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

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

Chapter 2. Quantum

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

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

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

A. Number

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

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

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

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

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

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

Remark 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Remark 2

Pythagoras, as is well known, set forth relations of reason or philosophemes in numbers, and more recent times too have seen philosophy make use of numbers and of the forms of their connection, powers and so on, whether to regulate thoughts by them or to express thoughts through them. – Pedagogically considered, number has been held for the object best suited to inner intuition, and the calculative occupation with number's relations for that activity of spirit in which spirit brings before intuition its ownmost relations and, generally, essence in its fundamental relations. – How far so high a worth can accrue to number issues from number's concept, in the shape in which that concept has come out.

Number, we saw, is the absolute determinateness of quantity, its element being difference become indifferent; – determinateness in itself, posited at the same time wholly and only externally. An analytic science is what arithmetic is, since every linking and every difference occurring at its object lies not within the object itself but is inflicted upon it from wholly without. No concrete object belongs to arithmetic, none possessing inner relations in itself, relations at first hidden from knowledge, not given along with the immediate representation of the object, but such as have first to be brought out by the exertion of cognition. Far from containing the concept, and with it the task set for conceptualizing thinking, arithmetic is that thinking's opposite. Owing to the indifference of what gets linked toward the linking, a linking that wants necessity, thinking finds itself here caught up in an activity that is at once the extreme divestment of its own self, in the violent activity of moving within thoughtlessness and of linking what is capable of no necessity. The abstract thought of externality itself: that is the object.

Being this thought of externality, number is likewise the abstraction from sensuous manifoldness; out of the sensuous it has kept nothing but the abstract determination of externality as such; the sensuous is thereby brought, in number, nearest of all to thought; the pure thought of thought's own divestment is what number is.

Hence spirit, in rising above the sensuous world and knowing its own essence, may light upon number, this inward, abstract externality, as its choice, while it casts about for an element in which to have its pure representation, the expression of its essence, and before it seizes thought itself as that element and wins the purely spiritual expression for thought's presentation. That is why, early in the history of science, we find number put to use for the expression of philosophemes. The final stage of imperfection in grasping the universal, grasping it still encumbered with the sensuous, is what number makes up. Definite consciousness the ancients had of number's standing midway between the sensuous and thought. Aristotle reports of Plato (Metaphys. I.5.) his saying that, alongside the sensuous and the ideas, the mathematical determinations of things stand between the two, marked off from the sensuous in being invisible (eternal) and unmoved, and from the ideas in being a many and an alike, whereas the idea is simply and solely self-identical and within itself one. – Cited in Malchi Vita Pythagorae ed. Rittershus. p. 30 f. is a fuller and thoroughly thought-out reflection on this by Moderatus of Cadiz; that the Pythagoreans hit upon numbers he puts down to their inability as yet to seize the fundamental ideas and first principles clearly in reason, those principles being hard to think and hard to utter; for purposes of designation numbers do good service in instruction; therein they imitated, among others, the geometers, who, unable to render the corporeal in thoughts, avail themselves of figures and say, this is a triangle, intending nonetheless that the drawing which meets the eye not be taken for the triangle, but that only the thought of the triangle be represented thereby. It was in this way that the Pythagoreans pronounced the thought of unity, of selfsameness and likeness, together with the ground of concord, of coherence and of the conservation of all things, of what is self-identical, to be one, and so on. – Needless to remark, the Pythagoreans went over from the numerical expression to the expression in thought as well, to the explicit categories of like and unlike, of limit and of infinity; already regarding those numerical expressions it is reported (ibid. in the notes to p. 31. 1. 5. out of a Vita of Pythagoras in Photius p. 722) that a distinction was drawn by the Pythagoreans between the monas and the one; for the thought they took the monas, but for the number the one; likewise the two for the arithmetical item, the dyas (for so it must presumably be read there) for the thought of the indeterminate. – In the first place these ancients discerned quite rightly how inadequate the numerical forms are for determinations of thought, and quite as rightly they went on to require, in place of that first stopgap, the expression proper to thoughts; how much further along in their reflecting they were than those who nowadays reckon it something praiseworthy, thorough even and profound, to set in the place of determinations of thought numbers themselves and numerical determinations, such as powers, then the infinitely large, the infinitely small, one divided by the infinite and further determinations of the sort, themselves often no better than a perverse mathematical formalism, and so to revert to that impotent childhood.

Since the expression was adduced above that number stands between the sensuous and thought, in that number has this at once from the sensuous, namely to bear the many, the mutual outsideness, upon it, one must remark that this many itself, the sensuous as taken up into thought, is that category belonging to the sensuous of being external in its own self. Transposed into this very element of being-outside-itself, the further, concrete, true thoughts, what is most alive and most mobile, comprehended only in relating, turn into determinations dead and immobile. The richer thoughts grow in determinateness and therewith in relation, the more confused on the one side, and on the other the more arbitrary and drained of sense, does their presentation in forms of the sort numbers are become. The one, the two, the three and the four, henas or monas, dyas, trias, tetraktys, still lie near the wholly simple abstract concepts; but where numbers are to pass over into concrete relations, wanting to keep them still near the concept is futile.

Should the determinations of thought now be designated by one, two, three, four for the movement of the concept, through which alone the concept is concept, then here is the hardest thing exacted of thinking. In the element of its opposite, of relationlessness, thinking moves; the labor of derangement is its business. Comprehending that one is three, say, and three one, is a demand so hard because the one is what is relationless and so does not display upon its own self that determination whereby it passes over into what is opposed to it, being rather just this, to shut out and refuse such relation outright. Conversely the understanding exploits this against speculative truth (against the truth, for instance, deposited in the doctrine that goes by the name of the Trinity) and counts that truth's determinations, which make up one unity, so as to parade it as sheer nonsense, – which is to say, the understanding itself perpetrates the nonsense of turning what is relation outright into something relationless. In the name Trinity it was of course not reckoned upon that the one and number would be looked on by the understanding as the content's essential determinateness. Contempt toward the understanding is what that name gives voice to, though the understanding has firmly established its vanity of holding to the one and to number as such and has pitted them against reason.

To take numbers, geometrical figures, as has often been done with the circle, the triangle etc., for mere symbols (the circle standing, say, for eternity, the triangle for the Trinity) – is on the one side innocuous enough; foolish, on the other side, is supposing that more gets expressed thereby than thought is able to grasp and to express. If in such symbols, as in others engendered by fantasy in the mythologies of peoples and in poetry at large, beside which the fantasy-less geometrical figures cut a meagre figure anyway, and if in these too a profound wisdom, a profound meaning is supposed to reside, then thinking's one and only concern is to bring out into the open the wisdom that lies therein and lies not in symbols alone but in nature and in spirit as well; within symbols the sensuous element still leaves truth clouded and shrouded; only in the form of thought does truth grow wholly manifest to consciousness; the meaning is nothing other than the thought itself.

But fetching in mathematical categories with a view to determining from them something for the method or the content of philosophical science shows itself essentially perverse for this reason: that insofar as mathematical formulas signify thoughts and differences of the concept, it is rather philosophy that has first to state, determine and justify this their signification. Philosophy, in its concrete sciences, has to take what is logical out of logic, not out of mathematics; taking refuge, for philosophy's logical dimension, in the shapes which the logical assumes in other sciences, many of them mere intimations of it and others stuntings of it as well, can be nothing but a stopgap of philosophical incapacity. Merely applying such borrowed formulas is in any case an external comportment; the application would itself have to be preceded by a consciousness of their worth as much as of their meaning; but such consciousness is yielded by thinking consideration alone, not by the authority these formulas carry out of mathematics. Consciousness of that kind about them is logic itself, and this consciousness strips their particular form away, makes that form superfluous and useless, sets it right, and alone procures for them their warrant, their sense and their worth.

How matters stand with the use of number and of calculation, insofar as it is supposed to make up a pedagogical mainstay, emerges of itself from what has gone before. A non-sensuous object is number, and a non-sensuous business the occupation with number and its combinations; spirit is thereby held to reflection into itself and to an inward abstract labor, a matter of great, yet one-sided, importance. On the other side, after all, since nothing underlies number but external, thoughtless difference, that business turns thoughtless and mechanical. The straining of effort consists chiefly in holding fast to what is void of concept and in combining it without concept. The content is the empty one; the solid substance of ethical and spiritual life and of its individual shapings, the noblest nourishment with which education is to rear the youthful spirit, is to be crowded out by the contentless one; where such exercises are made the main thing and the main occupation, no other effect can follow than that spirit gets hollowed out and blunted in form as in content. So very external, and hence mechanical, a business is calculating that machines have been built which carry out arithmetical operations to perfection. Knew one nothing of the nature of calculating beyond this single circumstance, there would lie in it the verdict upon what that bright idea came to, of making calculation the chief means of cultivating spirit and of racking spirit into perfecting itself as a machine.

B. Extensive and Intensive Quantum

a. Their Difference

1. Its determinateness as limit the quantum has, as was seen just now, in the amount. Discrete within itself it is, a many possessing no being that would differ from its limit and hold that limit outside it. Thus taken along with its limit, a limit manifold in its own self, the quantum is extensive magnitude.

Extensive magnitude must be kept apart from the continuous; over against the extensive there stands directly not the discrete but the intensive. Determinatenesses of the quantitative limit itself are what extensive and intensive magnitude are, the quantum being identical with its limit; whereas continuous and discrete magnitude are determinations of magnitude in itself, i.e. of quantity as such, once abstraction is made in the quantum from the limit. – Upon its own self and within its limit extensive magnitude bears the moment of continuity, seeing that its many is throughout a continuous many; to that extent the limit, qua negation, shows itself at this equality of the many, as a delimiting of the unit. Quantity carrying itself onward with no regard for a limit is continuous magnitude, and where it is represented as having one, that limit is a delimiting in general, without discreteness being posited in it. Merely as continuous magnitude, the quantum has yet to be truly determined for itself, lacking as it does the one, wherein being-determined-for-itself resides, and lacking number too. Discrete magnitude, likewise, is immediately no more than a differentiated many in general, one which, were it as such to have a limit, would amount to a mere multitude, i.e. to something bounded indeterminately; for it to be as determinate quantum there is required that gathering of the many into one whereby the many get posited as identical with the limit. Of the two sides, each magnitude, the continuous and the discrete alike, has as quantum in general only one posited upon it, the side through which it is completely determined and is as number. Immediately, number is extensive quantum, – the simple determinateness which essentially is as amount, though as amount of one and the selfsame unit; from number it differs in this alone, that in number the determinateness stands expressly posited as plurality.

2. Yet no difference from some other large thing is needed for determining, by way of number, how large something is, as though this large thing required for its determinateness both itself and another large thing, since determinateness of magnitude generally is a limit determined for itself, indifferent, related simply to itself; and within number such a limit stands posited as shut up inside the one that is for itself, having externality, the reference-to-other, within its own self. Moreover, this many of the limit itself, like the many in general, is nothing unequal within itself but rather something continuous; whatever one of the many is, so is the other; hence its being a many that lies apart or discrete does not make up the determinateness as such. Of its own accord, therefore, this many falls back together into its continuity and turns into simple unit. – Only a moment of number is the amount; but it does not, as a multitude of numerical ones, constitute number's determinateness; on the contrary, those ones, indifferent and external to themselves, stand sublated once number has gone back into itself; whatever externality made up the ones of plurality disappears within the one, taken as the reference of number to itself.

So the limit of the quantum, which qua extensive had its existent determinateness as amount external to itself, passes over into simple determinateness. Within this simple determination of the limit, intensive magnitude is what the quantum is; and the limit or determinateness identical with the quantum now stands posited likewise as something simple, – the degree.

Determinate magnitude, quantum, is accordingly what degree is, yet not at once a multitude, or something plural within its own self; a multiplicity is all it is; and the multiplicity is the several gathered up into the simple determination, existence gone back into being-for-itself. By a number its determinateness must indeed find expression, number being the complete being-determined of the quantum, yet not as amount but simply, as one single degree. Where 10, 20 degrees get spoken of, the quantum having so many degrees is the tenth, the twentieth degree, not their amount and sum; on that reckoning it would be an extensive one; rather it is only a single one, the tenth, the twentieth degree. Whatever determinateness lies in the amount ten, twenty it does hold, yet holds it not as several ones but is number qua sublated amount, qua simple determinateness.

3. Within number the quantum stands posited in its full determinateness; qua intensive quantum, though, as within its being-for-itself, what it is posited as is what it is according to its concept, or in itself. For the form of reference-to-self that belongs to it in degree is at once its own being-external-to-itself. Qua extensive quantum, number is numerical plurality, and thus carries externality inside it. Such externality, being many in general, falls together into undifferentiatedness and sublates itself within the one of number, number's reference to itself. Amount, though, is how the quantum has its determinateness; amount it contains, as was shown just now, notwithstanding that amount is no longer posited upon it. Degree, therefore, simple within its own self and no longer holding this external otherness in it, holds that otherness outside it, referring itself thereto as to its determinateness. What makes up the determinateness of the simple limit that degree is for itself is a plurality external to degree. That the amount, insofar as it was supposed to be found in number within the extensive quantum, sublated itself there, thus works out to this: outside number is where it stands posited. Posited as one, as reference-to-self reflected into itself, number shuts out of itself the indifference and externality of amount, and is reference to itself as reference through itself to something external.

Herein the quantum possesses the reality conformable to its concept. Its quality is constituted by the indifference of the determinateness; that is, by the determinateness which upon its own self is a determinateness external to itself. – Degree, accordingly, is a simple determinateness of magnitude among a multiplicity of such intensities, diverse from one another, each a merely simple reference to itself, yet standing at once in essential reference to one another, such that within this continuity with the rest each one gets its determinateness. This reference of degree through itself to its other turns ascent and descent along the scale of degrees into a steady advance, a flowing that is alteration uninterrupted and indivisible; of the several ones distinguished therein not one is cut off from the rest, each having its being-determined only in them. Qua determination of magnitude referring itself to itself, every degree is indifferent toward the others; but no less is it in itself referred to this externality, being only by means of that externality what it is, its reference to itself being in one stroke the non-indifferent reference to what is external, and having therein its quality.

b. Identity of Extensive and Intensive Magnitude

Nothing external to itself lies within degree. Still, degree is not on that account the indeterminate one, the principle of number generally, whose sole amount is the negative one of being no amount whatever. To begin with, what intensive magnitude comes to is a simple one of the several; degrees there are, several of them; determinate, however, they are neither qua simple one nor qua several, but solely within the reference of this being-outside-itself, that is, within the identity of one and multiplicity. Granted that the several as such fall outside the simple degree, still it is in the degree's reference to them that its determinateness consists; the amount is accordingly harbored within it. Just as twenty, taken extensively, keeps the twenty ones within itself in discrete fashion, so the determinate degree keeps them as continuity, being in simple fashion this determinate multiplicity; such a degree is the twentieth; and the twentieth degree it is only by mediation of that amount, which qua amount falls outside it.

Two sides, then, call for consideration in the determinateness of intensive magnitude. Other intensive quanta determine it, and with its otherness it stands in continuity, its determinateness consisting precisely in that reference to the otherness. Now insofar as, first, it is the simple determinateness, its determination is one against other degrees; those it shuts out of itself, and in this shutting out its determinateness lies. Secondly, however, determinate it is in its own self; such it is in the amount, taken as its own amount, not in an amount excluded from it, nor in the amount belonging to other degrees. Within its own self the twentieth degree keeps the twenty; being marked off from the nineteenth, the twenty-first etc. is not all that determines it, for its amount is what its determinateness consists in. Once the amount counts as belonging to the degree, though, and the determinateness has essentially the shape of amount, extensive quantum is what the degree turns out to be.

One and the same determinateness of the quantum, then, is what extensive and intensive magnitude both are; nothing sets them apart save this, that in the former the amount lies within, in the latter that very amount lies without. Because its many, in and for itself, falls together into the unit, and the many then steps outside that unit, extensive magnitude passes over into the intensive. Conversely, though, what this simple thing owes its determinateness to is nothing but the amount, and indeed the amount as its own; indifferent as it is toward intensities determined otherwise, it bears the externality of amount upon its own self; and so intensive magnitude proves no less essentially to be extensive.

Along with this identity the qualitative something makes its entrance; for the identity is a unity referring itself to itself by way of the negation of its differences, while those differences are what the existent determinateness of magnitude consists in; hence this negative identity is something, and indeed a something to which its quantitative determinateness is a matter of indifference. A quantum is what something is, yet by now qualitative existence, such as it is in itself, stands posited as indifferent toward that quantum. Talk of quantum, of number as such etc. had been possible with no something to serve them as substrate. By now, however, something steps over against these determinations of its own, mediated with itself through their negation, as existing for itself, and, in possessing a quantum, as one and the same possessor of an extensive and of an intensive one. That one determinateness which something has qua quantum stands posited in the differentiated moments of unit and of amount; nor is that determinateness merely in itself one and the same, for its being posited in these differences, as extensive and as intensive quantum, amounts to the return into this unity, a unity which, being negative, is the something posited as indifferent toward it.

Remark 1

In the ordinary way of representing things, extensive and intensive quantum are apt to get distinguished as kinds of magnitude, as though certain objects possessed intensive magnitude only, others extensive only. To this there has been added the representation belonging to a philosophical natural science that converted the plural, the extensive, in the fundamental determination of matter, say, of filling a space, and likewise in other concepts, into an intensive, in the sense that the intensive, qua dynamic, is the genuine determination, and that density or specific filling of space, for example, must essentially be grasped not as a certain multitude and amount of material parts occupying a quantum of space, but rather as a certain degree of matter's space-filling force.

Determinations of two sorts require distinguishing here. Where one speaks of the conversion of the mechanical manner of consideration into the dynamic, there occur the concept of self-subsistent parts subsisting outside one another, joined into a whole only externally, and, distinct from it, the concept of force. What in the filling of space counts on the one side 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. – Now the relation of whole to parts, and that of force to its expression, which here come to confront each other, belong not yet in this place but will be considered further on. This much may be recalled at once: that the relation of force to its expression, corresponding as it does to the intensive, is at first indeed the truer of the two as against the relation of whole and parts; but that force is on that account no less one-sided than the intensive, and the expression, the externality of the extensive, is from force just as inseparable, so that in both forms alike, the intensive and the extensive, one and the same content is at hand.

The other determinateness cropping up here is the quantitative as such, sublated as extensive quantum and converted into degree, this being taken for the determination that alone should truly be; though it has already been shown that this degree contains the former quite as much, so that each form is essential to the other, and hence that every existence exhibits its determination of magnitude no less as extensive than as intensive quantum. Anything whatever therefore serves as an example of this, insofar as it makes its appearance in a determination of magnitude. Number itself necessarily bears this doubled form immediately upon it. An amount it is, and to that extent extensive magnitude; but it is also one, a ten, a hundred; to that extent it stands upon the transition to intensive magnitude, since within this unit the manifold goes together into what is simple. One is extensive magnitude in itself, being representable as any amount of parts one pleases. Thus the tenth, the hundredth is this simple, intensive thing, whose determinateness lies at the several that fall outside it, i.e. at the extensive. Ten, a hundred is what number is, and simultaneously the tenth, the hundredth within the number system; the determinateness is in both cases the same.

Within the circle the one goes by the name of degree, because what the part of the circle owes its determinateness to is essentially the several lying outside it, the part being determined as one out of a closed amount of such ones. Regarded as mere magnitude of space, the circle's degree is nothing but an ordinary number; regarded as degree, it is intensive magnitude, having sense only insofar as it is determined by the amount of degrees the circle is divided into, much as number in general has its sense only within the number series.

A more concrete object's magnitude exhibits its doubled side, that of being extensive and intensive, at the doubled determinations of the object's existence, in the one of which the object appears as something external, in the other as something internal. A mass, for instance, is as weight something extensively large insofar as it makes up an amount of pounds, hundredweights etc.; something intensively large insofar as it exerts a certain pressure; and the magnitude of that pressure is a simple thing, a degree whose determinateness lies at a scale of degrees of pressure. In exerting pressure the mass shows itself 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 etc. from their place, and measures its own magnitude thereby.

Or heat has a degree; the degree of warmth, whether the 10th, the 20th etc. is a simple sensation, subjective in kind. Yet just as much is this degree at hand as extensive magnitude, namely as the expansion of a fluid, of the mercury in the thermometer, of air or of clay etc. As a taller column of mercury, or as a narrower cylinder of clay, a higher degree of temperature expresses itself; a larger space it warms in the same way in which a lesser degree warms only the smaller space.

Being the more intensive one, the higher tone is at the same time a greater multitude of vibrations, or a louder tone, one to which a higher degree gets ascribed, makes itself audible across a larger space. – A larger surface can be coloured in like manner with the more intensive color than with a weaker one; or what is brighter, another sort of intensity, is visible from farther off than what is less bright etc.

Just so within the realm of spirit: the high intensity of character, of talent, of genius goes together with an existence equally far-reaching, an effect equally extended, a contact equally many-sided. The deepest concept has the most universal significance and application.

Remark 2

A peculiar use has Kant made of the application of the determinateness of intensive quantum to a metaphysical determination of the soul. Where he criticizes the metaphysical propositions concerning the soul – paralogisms of pure reason, as he names them – he comes upon the consideration of the inference drawn from the soul's simplicity to its permanence. Against that inference he sets (Critique of Pure Reason, p. 414) »that, even granting the soul this simple nature, seeing that it contains no manifold outside one another and therefore no extensive magnitude, one could nonetheless as little deny it, as one could any existing thing, intensive magnitude, i.e. a degree of reality as regards the whole of its faculties, indeed the whole of what constitutes existence, a degree that may decrease through all the infinitely many smaller degrees, whereby the supposed substance might be turned into nothing, though not through partition, yet through a gradual slackening (remissio) of its powers; for even consciousness possesses at all times a degree open to still further lessening, and consequently so does the faculty of being conscious of oneself, and so do all the remaining faculties.« – Within rational psychology, such as that abstract metaphysics was, the soul counts not as spirit but as something that merely immediately is, as a soul-thing. Kant is thus within his right in applying to it the category of quantum, »as to anything existing whatever«, and, insofar as this immediate being stands determined as simple, that of intensive quantum. To spirit, admittedly, being does belong, though of a wholly other intensity than that of intensive quantum – rather of an intensity wherein the form of the merely immediate being, and every category thereof, count as sublated. It was not only the removal of the category of extensive quantum that had to be conceded; that of quantum in general had to be removed. Yet something further still awaits recognition: how, in the eternal nature of spirit, existence, consciousness, finitude are, and how they proceed from it without spirit thereby turning into a thing.

c. The Alteration of the Quantum

Towards the determinateness of quantum as such, the difference between extensive and intensive quantum is a matter of indifference. Quantum in general, however, just is determinateness in the shape of being posited as sublated – limit grown indifferent, determinateness that is no less its own negation. Within extensive magnitude this difference stands developed, while intensive magnitude constitutes the existence of that externality which quantum inwardly is. Posited within itself as quantum's contradiction, the difference amounts to this: to be the simple determinateness relating itself to itself which negates its own self, and to hold its determinateness not at it but in a further quantum.

A quantum accordingly stands posited, by its quality, in absolute continuity with what is external to it, with its otherness. Every determinateness of magnitude, therefore, not merely can be overstepped, not merely can be altered; it stands posited that alter itself it must. The determination of magnitude continues itself into its otherness in such fashion that only within this continuity with an other does it possess its being; a limit that is it therefore is not, but one that becomes.

Infinite is the one, or the negation that relates to itself, and hence repulsion of it away from its own self. Quantum too is infinite, posited as negativity relating to itself; away from itself it repels itself. Yet a determinate one it is, the one gone over into existence and into limit, and so repulsion of determinateness away from its own self – no generating of what is self-same, such as the repulsion of the one gives, but generating of its otherness; at it itself there now stands posited the task to send [itself] out beyond itself and to turn into an other. Increasing or diminishing itself is what it consists in; it is the externality of determinateness at it itself.

Beyond its own self, then, quantum sends itself; this other, into which it becomes, is at first a quantum likewise, yet no less a limit that, rather than merely being, drives itself out past itself. Hence the limit arisen anew in this going-out is utterly nothing else than one which sublates itself once more and passes itself along to a further one, and so on into infinity.

C. Quantitative Infinity

a. Its Concept

Quantum alters and turns into another quantum; that this alteration presses on into infinity – its further determination – has its ground in quantum's being set down as contradicting itself at it itself. – An other is what quantum becomes; but into its otherness it continues itself; a quantum, then, is what the other is too. This, however, is the other not of some quantum merely, but of the quantum itself, the negative of quantum as something limited, and thus quantum's unlimitedness, its infinity. Quantum is an ought; what it holds within it is to be determined for itself, and such being-determined-for-itself is rather a being determined in an other; conversely again it is that being-determined-in-an-other sublated, an indifferent subsisting-for-itself.

Thereby finitude and infinity each acquire at once, at it itself, a doubled and indeed opposed signification. Finite the quantum is, first as something limited in general, second as the sending of itself out past itself, as the being determined in an other. Its infinity, on the other hand, is first its not-being-limited, second its having-returned-into-itself, that indifferent being-for-itself. Set these moments side by side at once, and it emerges that quantum's determination of finitude – the sending of itself past itself to an other wherein its determination is supposed to lie – counts equally as determination of the infinite; negating the limit is that very same passing out beyond determinateness, so that in this negation, in the infinite, quantum would hold its last determinateness. Infinity's other moment is the being-for-itself indifferent towards the limit; quantum itself, however, is limited in just this way, that towards its limit, and thereby towards other quanta and towards its own going-beyond, it is what is for itself indifferent. In the case of quantum, finitude and the (bad) infinity supposed to stand apart from it each already carry the other's moment at it.

Qualitative and quantitative infinite part company in this, that with the former the opposition of finite and infinite is qualitative, and that the passing of the finite into the infinite, or the relating of the two to one another, lies only in the in-itself, in their concept. As immediate is qualitative determinateness, and it relates to otherness essentially as to a being other than itself; that it has its negation, its other, at it itself is not posited of it. Magnitude, by contrast, is as such determinateness sublated; posited it is as unequal with itself, indifferent towards its own self, and therefore as the alterable. Absolutely, i.e. abstractly, do the qualitative finite and infinite therefore stand over against each other; their unity is the inward relation lying at the ground; only in itself, then, and not at it, does the finite continue itself into its other. The quantitative finite, conversely, relates itself at it itself into its infinite, wherein its absolute determinateness is supposed to reside. What first exhibits this relation of theirs is the quantitatively infinite process.

b. The Quantitative Infinite Progress

Contradiction generally is what the progress into infinity expresses – here the contradiction contained in the quantitatively finite, in quantum as such. It is that reciprocal determining of finite and infinite already considered within the qualitative sphere, save for this difference, just recalled, that in the quantitative the limit sends and continues itself at it itself into its beyond, so that conversely the quantitatively infinite too stands posited as having quantum at it itself; for quantum in its being-outside-itself is at once its own self, its externality belonging to its determination.

Merely the expression of this contradiction is the infinite progress, not its resolution; yet because the one determinateness is continuous into its other, it brings about an apparent resolution in a unification of the two. In its first positing it is the task of the infinite, not the reaching of it – the perennial generating of the infinite, without ever getting past quantum itself, and without the infinite becoming anything positive and present. It lies in quantum's concept to have a beyond of itself. Such a beyond is, first, the abstract moment of quantum's non-being; quantum dissolves in its own self; so does it relate to its beyond as to its infinity, following the qualitative moment of the opposition. Second, however, quantum stands in continuity with that beyond; for quantum consists precisely in being the other of its own self, external to itself; this external, accordingly, is just as much not an other than quantum; hence the beyond, the infinite, is itself a quantum. Called back thereby out of its flight is the beyond, and the infinite is reached. But since what has come over to this side is a quantum again, nothing has been posited except a new limit once more; and this limit, being quantum, has fled from its own self again, is as such out past itself, and has repelled itself away into its non-being, into that beyond of its own self, which becomes quantum just as perennially as quantum thrusts itself away from itself towards the beyond.

Quantum's continuity into its other yields the linking of the two in the expression of an infinitely great or an infinitely small. Because both still bear the determination of quantum at them, alterable they remain, and the absolute determinateness that would amount to a being-for-itself is thus not reached. In the doubled infinite, opposed to itself according to more and less, in the infinitely great and the infinitely small, this being-outside-itself of the determination stands posited. At each of them quantum stays preserved in perennial opposition against its beyond. Widen the great as far as you will, it collapses into insignificance; in relating to the infinite as to its own non-being it renders the opposition qualitative; nothing, then, has the widened quantum wrested from the infinite, which stays, now as before, its non-being. Or again, magnifying the quantum brings no approach to the infinite, since the difference between quantum and its infinity carries essentially also the moment of not being a quantitative difference. Only the contradiction driven into narrower compass is thereby expressed: a great it is supposed to be, i.e. a quantum, and infinite, i.e. no quantum. – Likewise the infinitely small, being small, is a quantum, and stays therefore absolutely, i.e. qualitatively, too great for the infinite, standing opposed to it. Preserved in both is the contradiction of the infinite progress that was supposed to have found its goal in them.

Such infinity, persistently determined as the finite's beyond, deserves the name of the bad quantitative infinity. Like the qualitative bad infinity it is the perennial passing across and back from one member of the persisting contradiction over to the other, from limit to limit's non-being, and from that back once more to the very same, to the limit. In the quantitative progress, what is advanced to is indeed no abstract other in general, but a quantum posited as diverse; still, it stands in opposition to its negation just the same. Not an advancing and getting further on, therefore, is this progress, but a repeating of one and the very same – positing, sublating, positing again and sublating again; an impotence of the negative, which through its own sublating sees what it sublates return as something continuous. Two are so knotted together that they flee one another utterly; and in fleeing one another they cannot part, but are tied together in their mutual flight.

Remark 1

Chiefly in the form of the progress of the quantitative into infinity – this ongoing overflying of the limit, which is the impotence to sublate it, and the perennial relapse into it – is the bad infinity apt 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. Many a tirade has this progress served, tirades that have won admiration as sublime productions. What this modern sublimity in fact magnifies, however, is not the object, which rather flees away, but only the subject, which gulps down such great quantities into itself. The paltriness of this elevation, remaining as it does subjective and climbing the ladder of the quantitative, announces itself in the very fact that amid futile labor it confesses not to come nearer the infinite goal, a goal that, if it is to be reached, must of course be tackled quite otherwise.

In the following tirades of this kind there is expressed at the same time what such elevation passes over into and where it ceases. Kant for instance presents as sublime (Kr. d. pract. V. Schl.) »the case where the subject in thought lifts itself above the place it occupies in the world of sense and widens the linkage into the infinitely great, a linkage with stars above stars, with worlds above worlds, systems above systems, and beyond that into boundless times of their periodic motion, of its beginning and its endurance. – Representing succumbs to this advance into the immeasurably remote, where the remotest world still has one remoter yet, the past traced back so far has yet a further one behind it, the future carried out ever so far has still always another before it; thought succumbs to this representation of the immeasurable; as a dream in which one walks a long passage ever further and unforeseeably further, without descrying an end, closes with a fall or with dizziness.«

Besides compressing the content of quantitative elevation into a richly furnished picture, this presentation deserves praise above all for the truthfulness with which it states how this elevation fares at the end: thought succumbs, the end is falling and dizziness. What makes thought succumb, and brings forth its falling and the dizziness, is nothing other than the tedium of the repetition, which lets a limit vanish and appear again and vanish again – so always the one for the other, and each 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 no more than the feeling of impotence in this infinite or this ought, which wants to become master over the finite and cannot.

Haller's horrifying description of eternity, as Kant called it, is likewise apt to be particularly admired, though often precisely not on account of that side which constitutes its genuine merit:

»Monstrous numbers I heap up, mountains of millions I raise, Time upon time I set, and world on world I lay, And when from that grim height With reeling eyes I look back to you, All the power of number, multiplied a thousandfold, Is not yet one part of you.«

»I take them away, and you lie wholly before me.« When value is set upon that heaping and towering of numbers and worlds as upon a description of eternity, then it goes unnoticed that the poet himself pronounces this so-called horrifying going-beyond futile and hollow, and that he closes by saying that only through the abandoning of this empty infinite progress does the genuine infinite itself come into presence before him.

There have been astronomers who liked to pride 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 in themselves serve as units which, taken ever so many times over, shrink back once more into insignificance. The stale astonishment to which they abandon themselves in this, the insipid hopes of travelling one day in that other life from one star to another and of acquiring the like new knowledge onward into the immeasurable – these they passed off as a chief moment in their science's excellence, a science admirable not on account of any such quantitative infinity, but on the contrary on account of the ratios of measure and the laws that reason cognizes in these objects, laws which are the rational infinite as against that irrational infinity.

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

»the individual goes back into its invisible I, and sets the absolute freedom of its will, as a pure I, 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, collapse into rubble, and, alone, cognizes itself as equal to itself.«

The I in this solitude with itself is indeed the beyond attained; it has come to its own self, is with itself, on this side; in pure self-consciousness the absolute negativity is brought to affirmation and presence, the very negativity which in that advance beyond the sensuous quantum does nothing but flee. But in fixing itself in its abstraction and lack of content, this pure I finds existence in general, the whole 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 relating to its other as to its own non-being; and such relating stays a longing, because the I has fixed for itself its own contentless, untenable emptiness on the one hand, and on the other, as its beyond, that fullness which in the negation nonetheless stays present.

To these two sublimities Kant appends the remark »that admiration (for the first, the outer) and respect (for the second, the inner) sublimity do indeed stimulate inquiry, but cannot make good the want of it«. – He thereby declares those elevations unsatisfying for reason, which cannot come to rest with them and with the sentiments bound up with them, nor let the beyond and the empty pass for the ultimate.

As an ultimate, however, the infinite progress has been taken above all in its application to morality. The second opposition of finite and infinite just adduced, that of the manifold world and of the I raised into its freedom, is at first qualitative. The self-determining of the I aims at once at determining nature and at freeing itself from nature; thus through its own self it relates to its other, which as outer existence is something manifold and also something quantitative. The relation to something quantitative itself becomes quantitative; the negative relation of the I to it, the power of the I over the not-I, 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 ever smaller. As for the will's complete adequacy to the moral law, that gets displaced into the progress running on into infinity, i.e. represented as an absolute unattainable beyond, and just this unattainability is supposed to yield the true anchor and the right consolation; for morality is supposed to be as struggle, and struggle there is only where the will fails to match the law, the law thereby being utterly a beyond for it.

Within this opposition, I and not-I – or the pure will and the moral law, and again the will's nature and sensibility – count as presupposed, each perfectly self-subsistent and indifferent towards the other. The pure will has its own peculiar law standing in essential relation to sensibility; and nature and sensibility have for their part laws that are neither drawn from the will nor answerable to it, nor would even, though different from it, have in themselves an essential relation to it, but are determined altogether for themselves, complete and closed within themselves. Both, however, are at once moments of one and the same simple essence, namely of the I; over against nature the will stands determined as the negative, so that it only is insofar as something distinct from it is there for sublating by it, something by which it is in this very act touched and itself affected. To nature, and to nature as the sensibility of the human being, taken as a self-subsistent system of laws, the being limited by an other is a matter of indifference; it maintains itself in this being limited, enters self-subsistently into the relation, and sets a limit to the will of the law no less than that will sets one to it. – It is one act, that the will determines itself and sublates the otherness of a nature, and that this otherness is posited as existent, continues itself into its being sublated, and is not sublated. The contradiction lying in this 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 sensibility is represented as the absolute relation that is in and for itself.

The impotence to master the qualitative opposition of the finite and the infinite, and to grasp the idea of the genuine will, of substantial freedom, takes refuge in magnitude, so as to employ it as the mediatrix, magnitude being the qualitative sublated, difference grown indifferent. Yet since both members of the opposition remain lying at the ground as qualitatively diverse, the very fact that in their mutual relation they behave as quanta rather posits each of them at once as indifferent towards this alteration. Nature is determined by the I, sensibility by the will of the good; the alteration brought about at nature by that will is only a quantitative difference, one that lets nature subsist as what it is.

Within the more abstract presentation of the Kantian philosophy, or at any rate of its principles – Fichte's Wissenschaftslehre – the infinite progress makes up, in the same way, both foundation and ultimate. Upon the first principle of that presentation, I=I, there follows a second independent of it, the opposing of the not-I; the relation of the two is at once likewise assumed as a quantitative difference, in 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 so as to stay opposed to that non-being, as something unsublated. After the contradictions lying therein have accordingly been developed within the system, the final result is that very relation which was the beginning; the not-I remains an infinite check, an absolute other; the last relation of it and the I to one another is the infinite progress, longing and striving – the very contradiction with which the start was made.

Since the quantitative is determinateness in the shape of being posited as sublated, people believed that much – rather, that everything – had been gained for the absolute's unity, for the One substantiality, once opposition as such was demoted to a difference merely quantitative. All opposition is merely quantitative was for a time a chief proposition of the more recent philosophy; one essence, one content belongs to the opposed determinations, which are real sides of the opposition insofar as each of them carries both its determinations, both factors, in it, save that on the one side the one factor and on the other the other is preponderant, in the one side the one factor, a matter or an activity, being at hand in greater amount or in stronger degree than in the other. Wherever different materials or activities are presupposed, what the quantitative difference does is rather to confirm and complete their externality, their indifference towards each other and towards the unity they have. Merely quantitative, so it is said, is the difference of the absolute unity; the quantitative is indeed immediate determinateness sublated, yet only imperfectly so, being as yet the first negation only, not the infinite one, not negation of the negation. – In being represented as quantitative determinations of absolute substance, being and thinking too become, as quanta, perfectly external and relationless to each other, 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 merely in-itself and not equally for-itself. Thus in fact this unity gets represented merely as a first immediate one, or merely as being that in its quantitative difference stays equal to itself without positing itself equal through its own self; grasped as negation of the negation, as infinite unity, it accordingly is not. Only in qualitative opposition does the posited infinity, being-for-itself, come forth, and the quantitative determination itself passes over, as will shortly appear more closely, into the qualitative.