Superphysics
Chapter 2c

# C. QUANTITATIVE INFINITY

by Hegel
##### 17 minutes  • 3433 words

(a) Its Notion

§ 497

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.

§ 498

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.

§ 499

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 § 500

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.

§ 501

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.

§ 502

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.

§ 503

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

§ 504

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.

§ 505

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.’

§ 506

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.

§ 507

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.’

§ 508

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.

§ 509

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.

§ 510

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.’

§ 511

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.

§ 512

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.

§ 513

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.

§ 514

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.

§ 515

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.

§ 516

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.

§ 517

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.