Infinity is itself the other of the empty being-other
— Hegel, [[Books/Georg_W_F_Hegel/The_Science_Of_Logic|"Science of Logic"]
The Hegelian ontological impasse ultimately rests upon maintaining that there is a being of the One; or more precisely: that presentation generates structure, that the pure multiple holds in itself the "counting as one". One can also say that Hegel never ceases to write the in-difference of the other and of the Other. In so doing, he renounces that thing for which the ontological can be a situation. This presents itself through two consequences which are tantamount to proof:
– Since it is infinity which articulates the other, the rule, and the Other, it is predictable that the impasse explodes with respect to this concept. The disjunction between the other and the Other – which Hegel seeks to eliminate – reappears in his text in the form of two developments, which are at the same time disjunctive and identical (quality and quantity).
– Since it is mathematics that constitutes the ontological situation, it is necessary for Hegel to debase it. Also, the chapter on quantitative infinity is followed by a gigantic "remark" on mathematical infinity, where Hegel proposes to establish that, in regards to the concept, mathematics represents a state of thought "defective in and of itself" and that its "procedure is unscientific."
THE MATHEME OF INFINITY REVISITED
The Hegelian matrix of the concept of infinity states "concerning qualitative and quantitative infinity, it is essential to note that the finite is not surpassed by a third party, but that it is determinacy, in as much as it is dissolving itself, that surpasses itself."
The notions which architecturally construct the concept are this determinacy [Bestimmtheit"], the starting point of the whole dialectic, and the surpassing [hinausgehen über"]. One easily recognizes here, on the one hand, the initial point of being, and on the other, the itinerary of the dialectical movement which is that which I have also called the "already" [déjà"] and the "yet" [encore"]. It is not an exaggeration to say that all of Hegel rests in this: that the "yet" is immanent to the "already," that everything that is, is already still.
"Something" – a pure term of presentation – is only determinate for Hegel in as much as it can be thought as other than an other. "The extoriority of otherness is in the something's own inwardness." This signifies that the law of the "counting-as-one" is that the counted term possesses in itself the marking-other [la marque-autre"] of its being. Or again: the one can only be said of being insofar as being is its own non-being – is that which is not. For Hegel, there is an identity in becoming of the "there is," [il y a"] (pure presentation) and of "there-is-one" [il y a de l'un"] (structure), of which the mediation is the interiority of the negative. Hegel posits that the "something" must retain the mark of its identity. From this, every point of being is "between" itself and its mark. Determinacy is that which, in order to found the Same, requires that the Other be in the other. There is the origin of infinity.
The analytic is very fine here. If one of the point of being (the counting-as-one of a presented term) – that is to say, its limit, or that which it discerns, results from the fact that it detains the marking-other in interiority (that it is what it is not), the being of that point (inasmuch as one-thing [une-chose"]) is to overstep the limit: "the limit which constitutes the determination of the something but in a way that it is determinate at the same time as its non-being is limitation."
The passage of the pure limit [Grenze"] to the limitation [Schranke"] is the force of an infinity that is directly required by the point of being.
To say that one thing is marked in itself as one has two meanings, because the thing becomes at the same time the gap [l'écart"] between its being and the one-of-its-being. On one of the sides of this gap, it is actually the thing which is one, thus limited by what is not. We have here the stationary result of marking, Grenze, the limit. But on the other side of this gap, the one of the thing is not its being. The thing is, in itself, other than itself. This is Schranke, its limitation. But the limitation is a dynamic result of marking, since the thing, in all necessity, supersedes its limitation. For limitation is the non-being by which the limit occurs. Yet the thing is. Its being is accomplished through the francisement of non-being (that is to say, through the supersession of the limit). The profound root of the movement is that the one – if it marks being in-itself – is superseded by the being which it marks. Hegel has a profound sense that the counting-as-one is a law. But since he wants this law to be, at all costs, a law of being, he transofrms law into an ought. The being of the one consists in the fact that it is imperative to supersede the limitation. The thing is determinate as the "ought-to-be" of that which it is, while not being it. "The being in-itself of determinacy, in its relation to the limit, to itself as limitation, is the ought-to-be."
The one, insofar as it is, is the supersession of its non-being. Thus, being one (determinacy) is accomplished as the francisement of limitation. But at large, it is pure ought-to-be: its being is the imperative of the supersession of its one. The supersession of itself and therefore the dialectic of the finite and the infinite results directly from the fact that the point of being, always discernible, possesses in itself the one. "In the ought-to-be, the transcendence of finitude, that is, infinity begins. The ought-to-be is that which, in the further development, exhibits itself in accordance with the said impossibility as the progress to infinity."
The essence of the Hegelian thesis on infinity, at this stage, is that the point of being, (because always intrinsically discernible) generates from itself the operator of infinity, in other words, the supersession, which combines, as any operator of that type, the step on [le pas-en-plus"] (the yet) – here, the limitation – and the automatism of repetition – (here, the ought-to-be).
In a subtractive ontology one tolerates (even, one demands) that there is the extrinsic, since the counting-as-one cannot be inferred from the inconsistent presentation. In the Hegelian doctrine (which is a generative ontology), all is intrinsic, since the being other is the one-of-being, and all that retains a mark of identity in the form of the interiority of non-being. This results in what, for the subtractive ontology of infinity, is a decision (of the ontology), while for Hegel, it is a law. The fact that the one-essence [essence-une"] of being is to be infinite follows in the Hegelian analysis from the fact that the being of the one is interior to being in general.
With a special genius, Hegel attempts to co-engender the finite and the infinite only from the point of being. The infinite becomes a reason internal to finitude itself, a simple attribute of experience in general, because it is a consequence of the regime of the one, of the gap between [de l'entre-deux"] where the thing lies at the suture of its being-one and its being. Being ought to be infinite: "the finite, is itself, therefore, that sublation [relève"] of itself, the fact of being infinite."