C. QUANTITATIVE INFINITY
(a) Its Notion
Quantum alters and becomes another quantum; the further determination of this alteration, namely, that it goes on to infinity, lies in the circumstance that quantum is established as being immanently self-contradictory. Quantum becomes an other; but it continues itself into its otherness; the other is thus also a quantum. This, however, is not only the other of a particular quantum, but of quantum itself, the negative of quantum as limited; hence it is the unlimitedness of quantum, its infinity. Quantum is an ought-to-be; it is by implication determined as being for itself, and this being-determined-for-itself is rather the being-determined-in-an-other, and, conversely, it is the sublation of being-determined-in-an-other, is an indifferent subsisting for itself.
In this way, finitude and infinity each acquire in themselves a dual, and indeed, an opposite meaning. The quantum is finite, in the first place simply as limited, and secondly, as impelled beyond itself, as being determined in an other. But the infinity of quantum is first, its unlimitedness, and secondly, its returnedness into itself, its indifferent being-for-self. If we now compare these moments with each other, we find that the determination of the finitude of quantum, the impulse to go beyond itself to an other in which its determination lies, is equally the determination of the infinite; the negation of the limit is the same impulsion beyond the determinateness, so that in this negation, in the infinite, quantum possesses its final determinateness. The other moment of infinity is the being-for-self which is indifferent to the limit; but the limiting of quantum itself is such that quantum is explicitly indifferent to its limit, and hence to other quanta and to its beyond. In quantum, finitude and infinity (the spurious infinity supposedly separate from the finite) each already has within it the moment of the other.
The difference between the qualitative and quantitative infinite is that in the former the finite and infinite are qualitatively opposed and the transition of the finite into the infinite, or the relation of each to the other, lies only in the in-itself, in their Notion. Qualitative determinateness, as an immediacy, is related to otherness essentially as to an alien being; it is not posited as having its negation, its other within it. Quantity, on the other hand, is, as such, sublated determinateness; it is posited as being unlike and indifferent to itself, consequently as alterable. Therefore the qualitative finite and infinite stand absolutely, that is abstractly, opposed to each other; their unity is their underlying inner relation; and therefore the finite continues itself into its other only implicitly, not affirmatively. The quantitative finite, on the other hand, is self-related in its infinite, in which it has its absolute determinateness. This their relation is displayed in the first place in the quantitative infinite progress.
(b) The Quantitative Infinite Progress
The progress to infinity is in general the expression of contradiction, here, of that which is implicit in the quantitative finite, or quantum as such. It is the reciprocal determining of the finite and infinite which was considered in the sphere of quality, with the difference that, as just remarked, in the sphere of quantity the limit in its own self dispatches and continues itself into its beyond and hence, conversely, the quantitative infinite too is posited as having quantum within it; for quantum in its self-externality is also its own self, its externality belongs to its determination.
Now the infinite progress is only the expression of this contradiction, not its resolution; but because the one determinateness is continued into its other, the progress gives rise to the show of a solution in a union of both. As at first posed, it is the problem of attaining the infinite, not the actual reaching of it; it is the perpetual generation of the infinite, but it does not get beyond quantum, nor does the infinite become positively present. It belongs to the Notion of quantum to have a beyond of itself. This beyond is first, the abstract moment of the non-being of quantum: the vanishing of quantum is its own act; it is thus related to its beyond as to its infinity, in accordance with the qualitative moment of the opposition. Secondly, however, quantum is continuous with its beyond; quantum consists precisely in being the other of itself, in being external to itself; this externality is, therefore, no more an other than quantum itself; the beyond or the infinite is, therefore, itself a quantum. In this way, the beyond is recalled from its flight and the infinite is attained. But because the infinite now affirmatively present is again a quantum, what has been posited is only a fresh limit; this, too, as a quantum, has again fled from itself, is as such beyond itself and has repelled itself into its non-being, into its own beyond, and as it thus repels itself into the beyond, so equally does the beyond perpetually become a quantum.
The continuity of quantum with its other produces the conjunction of both in the expression of an infinitely great or infinitely small. Since both still bear the character of quantum they remain alterable, and the absolute determinateness which would be a being-for-self is, therefore, not attained. This self-externality of the determination is posited in the dual infinite — which is opposed to itself as a ‘more’ and a ‘less’ — in the infinitely great and infinitely small. In each, the quantum is maintained in perpetual opposition to its beyond. No matter how much the quantum is increased, it shrinks to insignificance; because quantum is related to the infinite as to its non-being, the opposition is qualitative; the increased quantum has therefore gained nothing from the infinite, which is now, as before, the non-being of quantum. In other words, the increase of quantum brings it no nearer to the infinite; for the difference between quantum and its infinity is essentially not a quantitative difference. The expression ‘the infinitely great’ only throws the contradiction into sharper relief; it is supposed to be great, that is, a quantum, and infinite, that is, not a quantum. Similarly, the infinitely small is, as small, a quantum, and therefore remains absolutely, that is, qualitatively, too great for the infinite and is opposed to it. In both, there remains the contradiction of the infinite progress which in them should have reached its goal.
This infinity which is perpetually determined as the beyond of the finite is to be described as the spurious quantitative infinite. Like the qualitative spurious infinite, it is the perpetual movement to and fro from one term of the lasting contradiction to the other, from the limit to its non-being, and from this back again to the limit. It is true that in the quantitative progress the movement is not simply towards an abstract other in general, but towards an explicitly different quantum; but this remains in the same way opposed to its negation. The progress, too, is therefore not a real advance but a repetition of one and the same thing, a positing, a sublating, and then again a positing and again a sublating, an impotence of the negative, for what it sublates is continuous with it, and in the very act of being sublated returns to it. Thus there are two terms, the bond between which is such that they simply flee from each other; and in fleeing from each other they cannot become separated but are joined together even in their flight from each other.
Remark 1: The High Repute of the Progress to Infinity
The spurious infinite, especially in the form of the quantitative progress to infinity which continually surmounts the limit it is powerless to remove, and perpetually falls back into it, is commonly held to be something sublime and a kind of divine worship, while in philosophy it has been regarded as ultimate. This progression has often been the theme of tirades which have been admired as sublime productions. As a matter of fact, however, this modern sublimity does not magnify the object — rather does this take flight — but only the subject which assimilates such vast quantities. The hollowness of this exaltation, which in scaling the ladder of the quantitative still remains subjective, finds expression in its own admission of the futility of its efforts to get nearer to the infinite goal, the attainment of which must, indeed, be achieved by a quite different method.
In the following tirades of this kind it is also stated what becomes of such exaltation and how it finishes. Kant, for example, at the close of the Critique of Practical Reason, represents it as sublime ‘when the subject raises himself in thought above the place he occupies in the world of sense, reaching out to infinity, to stars beyond stars, worlds beyond worlds, systems beyond systems, and then also to the limitless times of their periodic motion, their beginning and duration. Imagination fails before this progress into the infinitely remote, where beyond the most distant world there is a still more distant one, and the past, however remote, has a still remoter past behind it, the future, however distant, a still more distant future beyond it; thought fails in the face of this conception of the immeasurable, just as a dream, in which one goes on and on down a corridor which stretches away endlessly out of sight, finishes with falling or fainting.’
This exposition, besides giving a concise yet rich description of such quantitative exaltation, deserves praise mainly on account of the truthfulness with which it states how it fares finally with this exaltation: thought succumbs, the end is falling and faintness. What makes thought succumb, what causes falling and faintness, is nothing else but the wearisome repetition which makes a limit vanish, reappear, and then vanish again, so that there is a perpetual arising and passing away of the one after the other and of the one in the other, of the beyond in the here and now, and of the here and now in the beyond, giving only the feeling of the impotence of this infinite or this ought-to-be, which would be master of the finite and cannot.
Also Haller’s description of eternity, called by Kant terrifying, is usually specially admired, but often just not for that very reason which constitutes its true merit:
‘I heap up monstrous numbers,
Pile millions upon millions,
I put aeon upon aeon and world upon world,
And when from that awful height
Reeling, again I seek thee,
All the might of number increased a thousandfold
Is still not a fragment of thee.
I remove them and thou nest wholly before me.’
When this heaping and piling up of numbers is regarded as what is valuable in a description of eternity, it is overlooked that the poet himself declares this so-called terrifying journey into the beyond to be futile and empty, and that he closes by saying that only by giving up this empty, infinite progression can the genuine infinite itself become present to him.
There have been astronomers who liked to pride themselves on the sublimity of their science because it had to deal with an innumerable host of stars, with such immeasurable spaces and times in which distances and periods, already vast in themselves, serve as units which, in whatever multiples taken, are again abbreviated to insignificance. The shallow astonishment to which they surrender themselves, the absurd hopes of wandering in another life from one star to another and into immeasurable space to acquire fresh facts of the same kind, this they declare to be a cardinal factor in the excellence of their science — a science which is admirable not on account of such quantitative infinitude but, on the contrary, on account of the relations of measure and the laws which reason recognises in these objects and which are the infinite of reason in contrast to that other, irrational infinite.
To the infinity of outer, sensuous intuition, Kant opposes the other infinite, when ‘the individual withdraws into his invisible ego and opposes the absolute freedom of his will as a pure ego to all the terrors of fate and tyranny, and starting with his immediate surroundings, lets them vanish before him, and even what seems enduring, worlds upon worlds, collapse into ruins, and, alone, knows himself as equal to himself.’
The ego in being thus alone with itself is, it is true, the reached beyond; it has come to itself, is with itself, here and now; the absolute negativity which in the progress beyond the quantum of sense was only a flight, in pure self-consciousness becomes affirmative and present. But this pure ego, because it has fixed itself in its abstraction and emptiness, has determinate reality, the fulness of the universe of nature and mind, confronting it as a beyond. We are faced with that same contradiction which lies at the base of the infinite progress, namely a returnedness-into-self which is at the same time immediately an out-of-selfness, a relation to its other as to its non-being; and this relation remains a longing, because on the one side is the unsubstantial, untenable void of the ego fixed as such by the ego itself, and on the other, the fulness which though negated remains present, but is fixed by the ego as its beyond.
On these two sublimes Kant remarks ‘that admiration (for the first, outer) and reverence (for the second, inner) do indeed stimulate inquiry but cannot be a substitute for their defect’. Thus he declares those exaltations to be unsatisfying for reason, which cannot stop at them and the feelings associated with them, nor can it let the beyond and the void rank as ultimate.
But it is specially in its application to morality that the infinite progress has been taken as ultimate. The just quoted antithesis of finite and infinite in the shape of the manifold world and the ego raised to its freedom, is primarily qualitative. The ego in its self-determining forthwith proceeds to determine nature and to liberate itself therefrom; it thus connects itself through itself with its other which, as an external reality, is manifold and quantitative. The relation to the quantitative becomes itself quantitative; the negative relation of the ego to it, the power of the ego over the non-ego, over sense and outer nature, is consequently so conceived that morality can and ought continually to increase, and the power of sense continually to diminish. But the perfect adequacy of the will to the moral law is placed in the unending progress to infinity, that is, is represented as an absolutely unattainable beyond, and this very unattainableness is supposed to be the true sheet-anchor and fitting consolation; for morality is supposed to be a struggle, but such it can be oily if the will is inadequate to the moral law which thus becomes a sheer beyond for it.
In this opposition, ego and non-ego or the pure will and the moral law, and nature and the sensuousness of the will, are presupposed as completely self-subsistent and mutually indifferent. The pure will has its own appropriate law which stands in an essential relationship to the sphere of sense; and nature and sense on its side has laws which neither stem from nor are conformable to the will nor, although distinct from it, have they even in principle an essential connection with it but are determined independently, are finished and complete in themselves. At the same time, however, both are moments of one and the same simple being, the ego; the will is determined as the negative in relation to nature so that the will only is in so far as there is a sphere distinct from it which it sublates, but with which it thereby comes into contact and by which it is itself affected. Nature itself and nature as the sensuous sphere of man, as an independent system of laws, is indifferent to limitation by an other; it preserves itself in this process of limitation, enters into the relation as an independent factor and limits the will of law just as much as this limits it. The two processes comprise a single act: the self-determining of the will with the sublating of the otherness of nature, and the positing of this otherness as continuing itself as a reality in the process of being sublated, so that the otherness is not sublated. The contradiction involved in this is not resolved in the infinite progress: on the contrary, it is represented and affirmed as unresolved and unresolvable; the conflict of morality and sense is represented as the ultimate, absolute relation.
This standpoint which is powerless to overcome the qualitative opposition between the finite and infinite and to grasp the idea of the true will which is substantial freedom, has recourse to quantity in order to use it as a mediator, because it is sublated quality, the difference which has become indifferent. But since both members of the antithesis remain implied as qualitatively distinct, the fact is rather that each is straightway made indifferent to this alteration because it is as quanta that they are related to each other. Nature is determined by the ego, sense by the will of the good; the alteration produced in sense by the will is only a quantitative difference, one which leaves sense itself unchanged.
In the more abstract exposition of the Kantian philosophy, or at least of its principles, namely in Fichte’s Theory of Science, the infinite progress in the same way constitutes the foundation and the ultimate. In this exposition, the first axiom, ego = ego, is followed by a second, independent of it, the opposition of the non-ego; the relation between the two is also directly assumed as a quantitative difference, that is, the non-ego is partly determined by the ego, and partly not. In this way, the non-ego is continued into its non-being in such wise that in its non-being it remains opposed as something not sublated. Consequently, after the contradictions contained in this have been developed in the system, the final result is that relationship which formed the beginning: the non-ego remains an infinite obstacle, an absolute other; the final relation of the non-ego and the ego to each other is the infinite progress, a longing and aspiration — the same contradiction with which the system began.
Because the quantitative is determinateness posited as sublated it was thought that much, or rather everything, had been gained for the unity of the absolute, for the one substantiality, when opposition generally had been reduced to a merely quantitative difference. That all opposition is only quantitative was for some time a cardinal thesis of recent philosophy; the opposed determinations have the same nature, the same content; they are real sides of the opposition in so far as each of them has within it both determinations, both factors of the opposition, only that on one side one of the factors preponderates, on the other side the other, that is, one of the factors, a material substance or activity, is present in a greater quantity or in an intenser degree in one side than in the other. But in so far as substances or activities are presupposed, the quantitative difference rather confirms and completes their externality and indifference to each other and to their unity. The difference of the absolute unity is supposed to be only quantitative; the quantitative, it is true, is immediate, sublated determinateness, but only the imperfect, as yet only first, negation, not the infinite, not the negation of the negation. When being and thought are represented as quantitative determinations of absolute substance they too, as quanta, become completely external to each other and unrelated as, in a subordinate sphere, do carbon, nitrogen, etc. It is a third, an external reflection, which abstracts from their difference and recognises their unity, but a unity which is inner, implicit only, not for itself. This unity is, therefore, in fact conceived only as a first, immediate unity, or only as being, which in its quantitative difference remains like itself, but does not of itself posit itself as like itself; hence it is not grasped as a negation of the negation, as an infinite unity. Only in the qualitative opposition does the posited infinitude, being-for-self, emerge and the quantitative determination itself pass over into the qualitative, as we shall presently find.
Remark 2: The Kantian Antinomy of the Limitation and Nonlimitation of the World in Time and Space
It was remarked above that the Kantian antinomies are expositions of the opposition of finite and infinite in a more concrete shape, applied to more specific substrata of conception. The antinomy there considered contained the opposition of qualitative finitude and infinitude. In another, the first of the four cosmological antinomics, it is the conflict arising rather from the quantitative limit which is considered. I shall therefore proceed to examine this antinomy here.
It concerns the limitation or non-limitation of the world in time and space. This antithesis could be considered equally well with reference to time and space themselves, for whether time and space are relations of things themselves or are only forms of intuition, the antinomy based on limitation or non-limitation in them is not affected thereby.
The detailed analysis of this antinomy will likewise show that both statements and equally their proofs (which, like those already considered, are conducted apagogically) amount to nothing more than the two simple opposite assertions: (1) there is a limit, and (2) the limit must be transcended.
The thesis is:
‘The world has a beginning in time and is also enclosed within spatial limits.’
That part of the proof which concerns time assumes the opposite:
‘The world has no beginning in time; therefore, up to any given point of time, an eternity has elapsed and consequently an infinite series of successive states of things in the world has passed away. Now the infinity of a series consists precisely in the impossibility of ever completing it by successive synthesis. Therefore an infinite world series which has passed away is impossible and consequently a beginning of the world is a necessary condition of its existence — which was to be proved.’
The other part of the proof which concerns space is based on time. To comprehend a spatially infinite world would require an infinite time and this time must be regarded as having already elapsed in so far as the world in space is to be regarded not as gradually coming to be but as completely given. But it was shown of time in the first part of the proof that it is impossible to assume an infinite time as elapsed.
But it is at once evident that it was unnecessary to make the proof apagogical, or even to carry out a proof at all, since the basis of the proof itself is the direct assertion of what was to be proved. Namely, there is assumed some or any given point of time up to which an eternity has elapsed (eternity here has only the trivial meaning of a simply endless time). Now a given point of time means nothing else than a definite limit in time. In the proof therefore, a limit of time is presupposed as actual; but that is just what was to be proved. For the thesis is, that the world has a beginning in time.
There is only this difference, that the assumed limit of time is a now as end of the time already elapsed, but the limit which is to be proved is a now as beginning of a future. But this difference is immaterial. The now is taken as the point in which an infinite series of successive states of things in the world is supposed to have passed away, therefore as end, a qualitative limit. If this now were considered to be merely a quantitative limit which flows on and which not only must be transcended but is only as the transcending of itself, then the infinite time series would not have passed away in it, but would continue to flow on, and so the argument of the proof would vanish. On the other hand, if the point of time is assumed as a qualitative limit for the past, in which case it is also a beginning for the future (for each point of time is in itself the connection of the past and the future), then it is also an absolute, that is, abstract beginning for the future — and it was this that was to be proved. The fact that its future and this its beginning is already preceded by a past does not affect the argument; because this point of time is a qualitative limit — and that it is to be taken as qualitative is implied in the description of it as completed, elapsed, and therefore as not continuing — therefore in it time is broken off and the past lacks a connection with this time which could only be called future with reference to that past and, consequently, without such connection is only time as such, which has an absolute beginning. But if — as is, then, the case — it were related to the past through the now, the given point of time, and were thus determined as a future, then this point of time, too, regarded from the other side, would not be a limit; the infinite time series would continue itself in what was called future and would not be, as was assumed, completed.
In truth, time is pure quantity; the point of time in which it is supposed to be interrupted, which is employed in the proof, is really only the self-sublating being-for-self of the now. All that the proof does is to represent the absolute limit of time asserted in the thesis as a given point of time, and then straightway to assume it as a completed, that is, abstract point — a popular determination which sensuous conception readily lets pass as a limit, thus allowing as an assumption in the proof what had been put forward as the thing to be proved.
The antithesis runs:
‘The world has no beginning and no limits in space but is infinite with reference both to time and space.’
The proof likewise assumes the opposite:
‘The world has a beginning. Since the beginning is an existence preceded by a time in which the thing is not, there must have been a preceding time in which the world was not, that is, an empty time. Now no originating of anything is possible in an empty time; because no part of such a time possesses in itself and in preference to any other, any distinguishing condition of existence or non-existence. In the world, therefore, many groups of things can indeed begin, but the world itself can have no beginning and with respect to past time is infinite.’
This apagogical proof, like the others, contains the direct and unproved assertion of what it was supposed to prove. That is, it first assumes a beyond of the existing world, an empty time; but it also equally continues the existence of the world beyond itself into this empty time which is thereby sublated, with the result that the existence of the world is continued into infinity. The world is an existence; the proof presupposes that this existence comes into being and that the coming-to-be has an antecedent condition which is in time. But the antithesis itself consists in the very fact that there is no unconditioned existence, no absolute limit, but that the existence of the world always requires an antecedent condition. Thus, what was to be proved is found as an assumption in the proof. Further, the condition is sought in empty time, which means in effect that it is taken as temporal and therefore as an existence and as limited. Altogether, then, the assumption is made that the world as an existence presupposes another conditioned existence in time, and so on, therefore, to infinity.
The proof regarding the infinity of the world in space is the same. Apagogically, the spatial finiteness of the world is assumed; ‘this (the world) would therefore exist in an empty unlimited space and would stand in a relation to it; but such a relation of the world to no object is a nullity’.
Here, too, what was supposed to be proved is directly presupposed in the proof. It is directly assumed that the spatially limited world exists in an empty space and is supposed to stand in a relation to it, that is, there must be a movement out beyond it — on the one hand into the void, into the beyond and non-being of the world, but on the other hand, in order that it be in relation with its beyond, that is, continue itself into it, the beyond must be imagined as filled with the existence of the world. The infinity of the world in space which is asserted in the antithesis is nothing else than, on the one hand, empty space, and on the other the relation of the world to it, that is, the continuity of the world in empty space or the filling of space — which contradiction, namely, space as simultaneously empty and also filled, is the infinite progress of existence in space. This very contradiction, the relation of the world to empty space, is directly made the basis of the proof.
The thesis and antithesis and their proofs therefore represent nothing but the opposite assertions, that a limit is, and that the limit equally is only a sublated one; that the limit has a beyond, with which however it stands in relation, and beyond which it must pass, but that in doing so there arises another such limit, which is no limit.
The solution of these antinomies, as of those previously mentioned, is transcendental, that is, it consists in the assertion of the ideality of space and time as forms of intuition — in the sense that the world is in its own self not self-contradictory, not self-sublating, but that it is only consciousness in its intuition and in the relation of intuition to understanding and reason that is a self-contradictory being. It shows an excessive tenderness for the world to remove contradiction from it and then to transfer the contradiction to spirit, to reason, where it is allowed to remain unresolved. In point of fact it is spirit which is so strong that it can endure contradiction, but it is spirit, too, that knows how to resolve it. But the so-called world (whether it be called an objective, real world or, according to transcendental idealism, a subjective intuition and a sphere of sense determined by the categories of the understanding) is never and nowhere without contradiction, but it is unable to endure it and is, therefore, subject to coming-to-be and ceasing-to-be.
(c) The Infinity of Quantum
1. The infinite quantum as infinitely great or infinitely small is itself implicitly the infinite progress; as great or small it is a quantum and at the same time it is the non-being of quantum. The infinitely great and infinitely small are therefore pictorial conceptions which, when looked at more closely, turn out to be nebulous shadowy nullities. But in the infinite progress, this contradiction is explicitly present and with it that which is the nature of quantum which, as an intensive magnitude, has attained its reality and now in its determinate being is posited as it is in its Notion. It is this identity which we have now to consider.
Quantum as degree is unitary, self-related and determinate within itself. Through this unitary nature, the otherness and determinateness in quantum are sublated, so that the determinatensess is external to it; it has it determinateness outside it. This its self-externality is in the first place the abstract non-being of quantum generally, the spurious infinity. But, further, this non-being is also quantitative and this continues itself into its non-being, for it is in its externality that quantum has its determinateness; this its externality is, therefore, itself equally a quantum., this non-being of quantum, infinity, is thus limited, that is, this beyond is sublated, is itself determined as quantum which, therefore, in its negation is with itself.
But this is what quantum as such is in itself. For it is itself just by being external to itself; externality constitutes that whereby it is quantum and is with itself. In the infinite progress, therefore, the Notion of quantum is posited.
Let us take the progress at first in its abstract determinations as we find them; then in it we have the sublating of quantum, but equally too of its beyond, therefore the negation of quantum as well as the negation of this negation. Its truth is their unity in which they are, but only as moments. It is the resolution of the contradiction of which it is the expression, and its immediate significance is, therefore, the restoration of the Notion of quantity, namely, that quantity is an indifferent or external limit. In the infinite progress as such, the only reflection usually made is that every quantum, however great or small, must be capable of vanishing, of being surpassed; but not that this self-sublating of quantum, the beyond, the spurious infinite itself also vanishes.
Even the first sublation, the negation of quality as such whereby quantum is posited, is in principle [an sich] the sublating of the negation — the quantum is sublated qualitative limit, hence sublated negation — but at the same time it is this only in principle; it is posited as a determinate being, and then its negation is fixed as the infinite, as the beyond of quantum, which remains on this side as an immediate; thus the infinite is determined only as a first negation and it appears as such in the infinite progress. But we have seen that in this something more is present, the negation of the negation, or that which the infinite in truth is. We regarded this previously as the restoration of the Notion of quantity; this restoration means in the first place, that its determinate being has received a more precise determination; we now have quantum determined in conformity with its Notion, which is different from quantum in its immediacy; externality is now the opposite of itself, posited as a moment of quantity itself — quantum is posited as having its determinateness in another quantum by means of its non-being, of infinity; that is, it is qualitatively that which it is. However, this comparison of the Notion of quantum with its determinate being belongs more to our reflection, to a relationship which is not yet present here. The immediately following determination is that the quantum has reverted to quality, is from now on qualitatively determined. For its peculiarity, its quality, is the externality, the indifference of the determinateness; and quantum is now posited as being in fact itself in its externality, as self-related therein, in simple unity with itself, that is, qualitatively determined. This qualitative moment is still more closely determined, namely as being-for-itself; for the self-relation to which it has attained has proceeded from mediation, from the negation of the negation. Quantum has infinity, self-determinedness, no longer outside it but within itself.
The infinite, which in the infinite progress has only the empty meaning of a non-being, of an unattained but sought beyond, is in fact nothing else than quality. Quantum as an indifferent limit goes out beyond itself to infinity; in doing so it seeks nothing else than to be determined for itself, the qualitative moment, which, however, is thus only an ought-to-be. Its indifference to limit, and hence its lack of an explicit determinateness of its own and its passage away from and beyond itself, is that which makes quantum what it is; this its passage into the beyond is to be negated and quantum is to find in the infinite its absolute determinateness.
Quite generally: quantum is sublated quality; but quantum is infinite, goes beyond itself, is the negation of itself. Thus its passage beyond itself is, therefore, in itself the negation of the negated quality, the restoration of it; and thus quantum is explicitly determined as possessing as its own moment, the externality which formerly appeared as a beyond.
Quantum is thus posited as repelled from itself, with the result that there are two quanta which, however, are sublated, are only as moments of one unity, and this unity is the determinateness of quantum. Quantum as thus self-related as an indifferent limit in its externality and therefore posited as qualitative, is quantitative ratio. In the ratio, quantum is external to itself, is distinguished from itself; this its externality is the relation of one quantum to another, each of which has meaning only in this its relation to its other; and this relation constitutes the determinateness of the quantum, which is as such a unity. It has in this unity not an indifferent, but a qualitative, determination; in this its externality it has returned into itself, and in it quantum is that which it is.
Remark 1: The Mathematical Infinite
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