Hegel’s Science of Logic
The ought as such contains limitation, and limitation contains the ought. Their relation to each other is the finite itself which contains them both in its being-within-self. These moments of its determination are qualitatively opposed; limitation is determined as the negative of the ought and the ought likewise as the negative of limitation. The finite is thus inwardly self-contradictory; it sublates itself, ceases to be. But this its result, the negative as such, is [a] its very determination; for it is the negative of the negative. Thus, in ceasing to be, the finite has not ceased to be; it has become in the first instance only another finite which, however, is equally a ceasing-to-be as transition into another finite, and so on to infinity. But [b] closer consideration of this result shows that the finite in its ceasing-to-be, in this negation of itself has attained its being-in-itself, is united with itself. Each of its moments contains precisely this result; the ought transcends the limitation, that is, transcends itself; but beyond itself or its other, is only the limitation itself. The limitation, however, points directly beyond itself to its other, which is the ought; but this latter is the same duality of being-in-itself and determinate being as the limitation; it is the same thing; in going beyond itself, therefore, it equally only unites with itself. This identity with itself, the negation of negation, is affirmative being and thus the other of the finite, of the finite which is supposed to have the first negation for its determinateness; this other is the infinite.
The infinite in its simple Notion can, in the first place, be regarded as a fresh definition of the absolute; as indeterminate self-relation it is posited as being and becoming. The forms of determinate being find no place in the series of those determinations which can be regarded as definitions of the absolute, for the individual forms of that sphere are immediately posited only as determinatenesses, as finite in general. The infinite, however, is held to be absolute without qualification for it is determined expressly as negation of the finite, and reference is thus expressly made to limitedness in the infinite-limitedness of which being and becoming could perhaps be capable, even if not possessing or showing it-and the presence in the infinite of such limitedness is denied.
But even so, the infinite is not yet really free from limitation and finitude; the main point is to distinguish the genuine Notion of infinity from spurious infinity, the infinite of reason from the infinite of the understanding; yet the latter is the finitised infinite, and it will be found that in the very act of keeping the infinite pure and aloof from the finite, the infinite is only made finite.
The infinite is:
(a) in its simple determination, affirmative as negation of the finite
(b) but thus it is in alternating determination with the finite, and is the abstract, one-sided infinite
(c) the self-sublation of this infinite and of the finite, as a single process — this is the true or genuine infinite.
The infinite is the negation of the negation, affirmation, being which has restored itself out of limitedness. The infinite is, and more intensely so than the first immediate being; it is the true being, the elevation above limitation. At the name of the infinite, the heart and the mind light up, for in the infinite the spirit is not merely abstractly present to itself, but rises to its own self, to the light of its thinking, of its universality, of its freedom.
The Notion of the infinite as it first presents itself is this, that determinate being in its being-in-itself determines itself as finite and transcends the limitation. It is the very nature of the finite to transcend itself, to negate its negation and to become infinite. Thus the infinite does not stand as something finished and complete above or superior to the finite, as if the finite had an enduring being apart from or subordinate to the infinite. Neither do we only, as subjective reason, pass beyond the finite into the infinite; as when we say that the infinite is the Notion of reason and that through reason we rise superior to temporal things, though we let this happen without prejudice to the finite which is in no way affected by this exaltation, an exaltation which remains external to it. But the finite itself in being raised into the infinite is in no sense acted on by an alien force; on the contrary, it is its nature to be related to itself as limitation,— both limitation and as an ought-and to transcend the same, or rather, as self-relation to have negated the limitation and to be beyond it. It is' not in the sublating of finitude in general that infinity in general comes to be; the truth is rather that the finite is only this, through its own nature to become itself the infinite. The infinite is its affirmative determination, that which it truly is in itself. ®
Thus the finite has vanished in the infinite and what is, is only the infinite.
The infinite is; in this immediacy it is at the same time the negation of an other, of the finite. As thus in the form of simple being and at the same time as the non-being of an other, it has fallen back into the category of something as a determinate being in general — more precisely, into the category of something with a limit, because the infinite is determinate being reflected into itself, resulting from the sublating of determinateness in general, and hence is determinate being posited as distinguished from its determinateness. In keeping with this determinateness, the finite stands opposed to the infinite as a real determinate being; they stand thus in a qualitative relation, each remaining external to the other; the immediate being of the infinite resuscitates the being of its negation, of the finite again which at first seemed to have vanished in the infinite.
But the infinite and the finite are not in these categories of relation only; the two sides are determined beyond the stage of being merely others to each other. Finitude, namely, is limitation posited as limitation; determinate being is posited with the determination to pass over into its in itself, to become infinite. Infinity is the nothing of the finite, it is what the latter is in itself, what it ought to be, but this ought-to-be is at the same time reflected into itself, is realised; it is a purely self-related, wholly affirmative being. In infinity we have the satisfaction that all determinateness, alteration, all limitation and with it the ought itself, are posited as vanished, as sublated, that the nothing of the finite is posited. As this negation of the finite the in-itself is determinate and thus, as negation of the negation, is affirmative within itself. But this affirmation as qualitative, is immediate self-relation, is being; and thus the infinite is reduced to the category of a being which has the finite confronting it as an other; its negative nature is posited as the s imply affirmative, hence as the first and immediate negation. The infinite is in this way burdened with the opposition to the finite which, as an other, remains at the same time a determinate reality although in its in-itself, in the infinite, it is at the same time posited as sublated; this infinite is the non-finite-a being in the determinateness of negation. Contrasted with the finite, with the sphere of affirmative determinatenesses, of realities, the infinite is the indeterminate void, the beyond of the finite, whose being-in-itself is not present in its determinate reality.
The infinite as thus posited over against the finite, in a relation wherein they are as qualitatively distinct others, is to be called the spurious infinite, the infinite of the understanding, for which it has the value of the highest, the absolute Truth. The understanding is satisfied that it has truly reconciled these two, but the truth is that it is entangled in unreconciled, unresolved, absolute contradiction; it can only be brought to a consciousness of this fact by the contradictions into which it falls on every side when it ventures to apply and to explicate these its categories.
This contradiction occurs as a direct result of the circumstance that the finite remains as a determinate being opposed to the infinite, so that there are two determinatenesses; there are two worlds, one infinite and one finite, and in their relationship the infinite is only the limit of the finite and is thus only a determinate infinite, an infinite which is itself finite.
This contradiction develops its content into more explicit forms. The finite is real determinate being which persists as such even when transition is made to its non-being, to the infinite; this, as has been shown, has only the first, immediate negation for its determinateness relatively to the finite, just as the finite as opposed to that negation has, as negated, only the significance of an other and is, therefore, still [only] something. When, therefore, the understanding, raising itself above this finite world, ascends to its highest, to the infinite, this finite world remains for it on this side, so that the infinite is only set above or beyond the finite, is separated from it, with the consequence that the finite is separated from the infinite; each is assigned a distinct place — the finite as determinate being here, on this side, and the infinite, although the in-itself of the finite, nevertheless as a beyond in the dim, inaccessible distance, outside of which the finite is and remains.
As thus separated they are just as much essentially connected by the very negation which separates them. This negation which connects them — the somethings reflected into themselves — is the limit of the one relatively to the other, and that, too, in such a manner that each of them does not have the limit in it merely relatively to the other, but the negation is their being-in-itself; the limit is thus present in each on its own account, in separation from the other. But the limit is in the form of the first negation and thus both are limited, finite in themselves. However, each as affirmatively self-related is also the negation of its limit; each thus immediately repels the limit, as its non-being, from itself and, as qualitatively separated from it, posits it as another being outside it, the finite positing its non-being as this infinite and the infinite, similarly, the finite. It is readily conceded that there is a necessary transition from the finite to the infinite — necessary through the determination of the finite — and that the finite is raised to the form of being-in-itself, since the finite, although persisting as a determinate being, is at the same time also determined as in itself nothing and therefore as destined to bring about its own dissolution; whereas the infinite, although determined as burdened with negation and limit, is at the same time also determined as possessing being-in-itself, so that this abstraction of self-related affirmation constitutes its determination, and hence finite determinate being is not present in it. But it has been shown that the infinite itself attains affirmative being only by means of negation, as the negation of negation, and that when this its affirmation is taken as merely simple, qualitative being, the negation contained in it is reduced to a simple immediate negation and thus to a determinateness and limit, which then, as in contradiction with the being-in-itself of the infinite is posited as excluded from it, as not belonging to it, as, on the contrary, opposed to its being-in-itself, as the finite. As therefore each is in its own self and through its own determination the positing of its other, they are inseparable. But this their unity is concealed in their qualitative otherness, it is the inner unity which only lies at their base.
This determines the manner in which this unity is manifested: posited in determinate being, the unity is a changing or transition of the finite into the infinite, and vice versa; so that the infinite only emerges in the finite and the finite in the infinite, the other in the other; that is to say, each arises immediately and independently in the other, their connection being only an external one.
The process of their transition has the following detailed shape. We pass from the finite to the infinite. This transcending of the finite appears as an external act. In this void beyond the finite, what arises? What is the positive element in it? Owing to the inseparability of the infinite and the finite — or because this infinite remaining aloof on its own side is itself limited — there arises a limit; the infinite has vanished and its other, the finite, has entered. But this entrance of the finite appears as a happening external to the infinite, and the new limit as something that does not arise from the infinite itself but is likewise found as given. And so we are faced with a relapse into the previous determination which has been sublated in vain. But this new limit is itself only something which has to be sublated or transcended. And so again there arises the void, the nothing, in which similarly the said determinateness, a new limit, is encountered — and so on to infinity.
We have before us the alternating determination of the finite and the infinite; the finite is finite only in its relation to the ought or to the infinite, and the latter is only infinite in its relation to the finite. They are inseparable and at the same time mutually related as sheer others; each has in its own self the other of itself. Each is thus the unity of itself and its other and is in its determinateness not that which it itself is, and which its other is.
It is this alternating determination negating both its own self and its negation, which appears as the progress to infinity, a progress which in so many forms and applications is accepted as something ultimate beyond which thought does not go but, having got as far as this 'and so on to infinity', has usually reached its goal. This progress makes its appearance wherever relative determinations are pressed to the point of opposition, with the result that although they are in an inseparable unity, each is credited with a self-subsistent determinate being over against the other. The progress is, consequently, a contradiction which is not resolved but is always only enunciated as present.
What we have here is an abstract transcending of a limit, a transcending which remains incomplete because it is not itself transcended. Before us is the infinite; it is of course transcended, for a new limit is posited, but the result is rather only a return to the finite. This spurious infinity is in itself the same thing as the perennial ought; it is the negation of the finite it is true, but it cannot in truth free itself therefrom. The finite reappears in the infinite itself as its other, because it is only in its connection with its other, the finite, that the infinite is. The progress to infinity is, consequently, only the perpetual repetition of one and the same content, one and the same tedious alternation of this finite and infinite.
The infinity of the infinite progress remains burdened with the finite as such, is thereby limited and is itself finite. But this being so, the infinite progress would in fact be posited as the unity of the finite and the infinite; but this unity is not reflected on. Yet it is this unity alone which evokes the infinite in the finite and the finite in the infinite; it is, so to speak, the mainspring of the infinite progress. This progress is the external aspect of this unity at which ordinary thinking halts, at this perpetual repetition of one and the same alternation, of the vain unrest of advancing beyond the limit to infinity, only to find in this infinite a new limit in which, however, it is as little able to rest as in the infinite. This infinite has the fixed determination of a beyond, which cannot be reached, for the very reason that it is not meant to be reached, because the determinateness of the beyond, of the affirmative negation, is not let go. In accordance with this determination the infinite has the finite opposed to it as a being on this side, which is equally unable to raise itself into the infinite just because it has this determination of an other, of a determinate being which perpetually generates itself in its beyond, a beyond from which it is again distinct.
In this alternating determination of the finite and the infinite from one to the other and back again, their truth is already implicitly present, and all that is required is to take up what is before us. This transition from one to the other and back again constitutes the external realisation of the Notion. In this realisation is posited the content of the Notion, but it is posited as external, as falling asunder; all that is required is to compare these different moments which yield the unity which gives the Notion itself; the unity of the infinite and the finite is — as has often been remarked already but here especially is to be borne in mind — the one-sided expression for the unity as it is in truth; but the elimination, too, of this one-sided determination must lie in the externalisation of the Notion now before us.
Taken according to their first, only immediate determination, the infinite is only the beyond of the finite; according to its determination it is the negation of the finite; thus the finite is only that which must be transcended, the negation of itself in its own self, which is infinity. In each, therefore, there lies the determinateness of the other, although according to the standpoint of the infinite progress these two are supposed to be shut out from each other and only to follow each other alternately; neither can be posited and grasped without the other, the infinite not without the finite, nor the latter without the infinite. In saying what the infinite is, namely the negation of the finite, the latter is itself included in what is said; it cannot be dispensed with for the definition or determination of the infinite. One only needs to be aware of what one is saying in order to find the determination of the finite in the infinite. As regards the finite, it is readily conceded that it is the null; but its very nullity is the infinity from which it is thus inseparable. In this way of conceiving them, each may seem to be taken in its connection with its other. But if they are taken as devoid of connection with each other so that they are only joined by 'and', then each confronts the other as self-subsistent, as in its own self only affirmatively present. Let us see how they are constituted when so taken. The infinite, in that case, is one of the two; but as only one of the two it is itself finite, it is not the whole but only one side; it has its limit in what stands over against it; it is thus thefinite infinite. There are present only two finites. It is precisely this holding of the infinite apart from the finite, thus giving it a one-sided character, that constitutes its finitude and, therefore, its unity with the finite. The finite, on the other hand, characterized as independent of and apart from the infinite, is that self-relation in which its relativity, its dependence and transitoriness is removed; it is the same self-subsistence and affirmation which the infinite is supposed to be.
The two modes of consideration at first seem to have a different determinateness for their point of departure, inasmuch as the former is supposed to be only the connection of the infinite and the finite, of each with its other, and the latter is supposed to hold them apart in complete separation from each other; but both modes yield one and the same result: the infinite and the finite viewed as connected with each other — the connection being only external to them but also essential to them, without which neither is what it is — each contains its own other in its own determination, just as much as each, taken on its own account, considered in its own self, has its other present within it as its own moment.
This yields the decried unity of the finite and the infinite — the unity which is itself the infinite which embraces both itself and finitude — and is therefore the infinite in a different sense from that in which the finite is regarded as separated and set apart from the infinite. Since now they must also be distinguished, each is, as has just been shown, in its own self the unity of both; thus we have two such unities. The common element, the unity of the two determinatenesses, as unity, posits them in the first place as negated, since each is supposed to be what it is in its distinction from the other; in their unity, therefore, they lose their qualitative nature-an important reflection for rebutting that idea of the unity which insists on holding fast to the infinite and finite in the quality they are supposed to have when taken in their separation from each other, a view which therefore sees in that unity only contradiction, but not also resolution of the contradiction through the negation of the qualitative determinateness of both; thus the unity of the infinite and finite, simple and general in the first instance, is falsified.
But further, since now they are also to be taken as distinct, the unity of the infinite which each of these moments is, is differently determined in each of them. The infinite determined as such, has present in it the finitude which is distinct from it; the former is the in-itself in this unity, and the latter is only determinateness, limit in it; but it is a limit which is the sheer other of the in-itself, is its opposite; the infinite's determination, which is the in-itself as such, is ruined by the addition of such a quality; it is thus a finitised infinite. Similarly, since the finite as such is only the negation of the in-itself, but by reason of this unity also has its opposite present in it, it is exalted and, so to say, infinitely exalted above its worth; the finite is posited as the infinitised finite.
Just as before, the simple unity of the infinite and finite was falsified by the understanding, so too is the double unity. Here too this results from taking the infinite in one of the two unities not as negated, but rather as the in-itself, in which, therefore, determinateness and limitation are not to be explicitly present, for these would debase and ruin it. Conversely, the finite is likewise held fast as not negated, although in itself it is null; so that in its union with the infinite it is exalted to what it is not and is thereby infinitised in opposition to its determination as finite, which instead of vanishing is perpetuated.
The falsification of the finite and infinite by the understanding which holds fast to a qualitatively distinct relation between them and asserts that each in its own nature is separate, in fact absolutely separate from the other, comes from forgetting what the Notion of these moments is for the understanding itself. According to this, the unity of the finite and infinite is not an external bringing together of them, nor an incongruous combination alien to their own nature in which there would be joined together determinations inherently separate and opposed, each having a simple affirmative being independent of the other and incompatible with it; but each is in its own self this unity, and this only as a sublating of its own self in which neither would have the advantage over the other of having an in-itself and an affirmative determinate being. As has already been shown, finitude is only as a transcending of itself; it therefore contains infinity, the other of itself. ®
Similarly, infinity is only as a transcending of the finite; it therefore essentially contains its other and is, consequently, in its own self the other of itself. The finite is not sublated by the infinite as by a power existing outside it; on the contrary, its infinity consists in sublating its own self.
This sublating is, therefore, not alteration or otherness as such, not the sublating of a something. That in which the finite sublates itself is the infinite as the negating of finitude; but finitude itself has long since been determined as only the non-being of determinate being. It is therefore only negation which sublates itself in the negation. Thus infinity on its side is determined as the negative of finitude, and hence of determinateness in general, as the empty beyond; the sublating of itself in the finite is a return from an empty flight, a negation of the beyond which is in its own self a negative.
What is therefore present is the same negation of negation in each. But this is in itself self-relation, affirmation, but as return to itself, that is through the mediation which the negation of negation is. These are the determinations which it is essential to keep in view; but secondly it is to be noted that they are also posited in the infinite progress, and how they are posited in it, namely, as not yet in their ultimate truth.
In the first place, both the infinite and the finite are negated in the infinite progress; both are transcended in the same manner. Secondly, they are posited one after the other as distinct, each as positive on its own account. We thus compare these two determinations in their separation, just as in our comparison — an external comparing — we have separated the two modes of considering the finite and the infinite: on the one hand in their connection, and on the other hand each on its own account. But the infinite progress expresses more than this; in it there is also posited the connection of terms which are also distinct from each other, although at first the connection is still only a transition and alternation; only a simple reflection on our part is needed to see what is in fact present.
In the first place, the negation of the finite and infinite which is posited in the infinite progress can be taken as simple, hence as separate and merely successive. Starting from the finite, the limit is transcended, the finite negated. We now have its beyond, the infinite, but in this the limit arises again; and so we have the transcending of the infinite. This double sublation, however, is partly only an external affair, an alternation of the moments, and partly it is not yet posited as a single unity; the transcending of each moment starts independently, is a fresh act, so that the two processes fall apart. But in addition there is also present in the infinite progress their connection. First there is the finite, then this is transcended and this negative or beyond of the finite is the infinite, and then this negation is again transcended, so that there arises a new limit, a finite again. This is the complete, self-closing movement which has arrived at that which constituted the beginning; what arises is the same as that from which the movement began that is, the finite is restored; it has therefore united with itself, has in its beyond only found itself again.
The same is the case with the infinite. In the infinite, the beyond of the limit, there arises only another limit which has the same fate, namely, that as finite it must be negated. Thus what is present again is the same infinite which had previously disappeared in the new limit; the infinite, therefore, through its sublating, through its transcending of the new limit, is not removed any further either from the finite-for the finite is only this, to pass over into the infinite-or from itself, for it has arrived at its own self.
Thus, both finite and infinite are this movement in which each returns to itself through its negation; they are only as mediation within themselves, and the affirmative of each contains the negative of each and is the negation of the negation. They are thus a result, and consequently not what they are in the determination of their beginning; the finite is not a determinate being on its side, and the infinite a determinate being or being-in-itself, beyond the determinate being, that is, beyond the being determined as finite. The reason why understanding is so antagonistic to the unity of the finite and infinite is simply that it presupposes the limitation and the finite, as well as the in-itself, as perpetuated; in doing so it overlooks the negation of both which is actually present in the infinite progress, as also the fact that they occur therein only as moments of a whole and that they come on the scene only by means of their opposite, but essentially also by means of the sublation of their opposite.
If, at first, the return into self was considered to be just as much a return of the finite to itself as return of the infinite to itself, this very result reveals an error which is connected with the one-sidedness just criticised: first the finite and then the infinite is taken as the starting point and it is only this that gives rise to two results. It is, however, a matter of complete indifference which is taken as the beginning; and thus the difference which occasioned the double result disappears of itself. This is likewise explicit in the line — unending in both directions — of the infinite progress in which each of the moments presents itself in equal alternation, and it is quite immaterial what point is fixed on or which of the two is taken as the beginning. They are distinguished in it but each is equally only the moment of the other. Since both the finite and the infinite itself are moments of the progress they are jointly or in common the finite, and since they are equally together negated in it and in the result, this result as negation of the finitude of both is called with truth the infinite. Their difference is thus the double meaning which both have. The finite has the double meaning of being first, only the finite over against the infinite which stands opposed to it, and secondly, of being the finite and at the same time the infinite opposed to it. The infinite, too, has the double meaning of being one of these two moments — as such it is the spurious infinite — and also the infinite in which both, the infinite and its other, are only moments. The infinite, therefore, as now before us is, in fact, the process in which it is deposed to being only one of its determinations, the opposite of the finite, and so to being itself only one of the finites, and then raising this its difference from itself into the affirmation of itself and through this mediation becoming the true infinite.
This determination of the true infinite cannot be expressed in the formula, already criticised, of a unity of the finite and infinite; unity is abstract, inert self-sameness, and the moments are similarly only in the form of inert, simply affirmative being. The infinite, however, like its two moments, is essentially only as a becoming, but a becoming now further determined in its moments. Becoming, in the first instance, has abstract being and nothing for its determinations; as alteration, its moments possess determinate being, something and other; now, as the infinite, they are the finite and the infinite, which are themselves in process of becoming.
This infinite, as the consummated return into self, the relation of itself to itself, is being-but not indeterminate, abstract being, for it is posited as negating the negation; it is, therefore, also determinate being for it contains negation in general and hence determinateness. It is and is there, present before us. It is only the spurious infinite which is the beyond, because it is only the negation of the finite posited as real — as such it is the abstract, first negation; determined only as negative, the affirmation of determinate being is lacking in it; the spurious infinite, held fast as only negative, is even supposed to be not there, is supposed to be unattainable. However, to be thus unattainable is not its grandeur but its defect, which is at bottom the result of holding fast to the finite as such as a merely affirmative being. It is what is untrue that is unattainable, and such an infinite must be seen as a falsity. The image of the progress to infinity is the straight line, at the two limits of which alone the infinite is, and always only is where the line — which is determinate being — is not, and which goes out beyond to this negation of its determinate being, that is, to the indeterminate; the image of true infinity, bent back into itself, becomes the circle, the line which has reached itself, which is closed and wholly present, without beginning and end.
True infinity taken thus generally as determinate being which is posited as affirmative in contrast to the abstract negation, is reality in a higher sense than the former reality which was simply determinate; for here it has acquired a concrete content. It is not the finite which is the real, but the infinite. Thus reality is further determined as essence, Notion, Idea, and so on. It is, however, superfluous to repeat an earlier, more abstract category such as reality, in connection with the more concrete categories and to employ it for determinations which are more concrete than it is in its own self. Such repetition as to say that essence, or the Idea, is the real, has its origin in the fact that for untrained thinking, the most abstract categories such as being, determinate being, reality, finitude, are the most familiar.
The more precise reason for recalling the category of reality here is that the negation to which it is opposed as the affirmative is here negation of the negation; as such it is itself opposed to that reality which finite determinate being is. The negation is thus determined as ideality; ideal being [das Ideelle] is the finite as it is in the true infinite — as a determination, a content, which is distinct but is not an independent, self-subsistent being, but only a moment.
['Das Ideale' has a more precise meaning (of the beautiful and its associations) than 'das Ideelle'; the former is not yet appropriate here and for this reason we have used the expression 'ideell'. We do not make this distinction though when speaking of reality; the expressions 'reell' and 'real' are used practically synonymously and no interest is served by giving the words different shades of meaning. - Author's note.]
Ideality has this more concrete signification which is not fully expressed by the negation of finite determinate being. With reference to reality and ideality, however, the opposition of finite and infinite is grasped in such a manner that the finite ranks as the real but the infinite as the 'ideal' [das Ideelle]; in the same way that further on the Notion, too, is regarded as an 'ideal', that is, as a mere 'ideal', in contrast to determinate being as such which is regarded as the real. When they are contrasted in this way, it is pointless to reserve the term 'ideal' for the concrete determination of negation in question; in that opposition we return once more to the one-sidedness of the abstract negative which is characteristic of the spurious infinite, and perpetuate the affirmative determinate being of the finite.
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