SCIENCE OF PHILOSOPHY
Finite, Spurious Infinite, True Infinite
Finite, Spurious Infinite, True InfiniteM. A. Marchetti
A concise exposition of the development of the true infinite is found in Hegel's Encyclopedia Logic (EL92-95). It may be much easier to follow than the one given in the Science of Logic. The following paragraphs are from the Gerates, et al translation of that book, along with some parts of the "Additions" where I felt they were useful. At the end I give my interpretation of the development.
The being that is kept firmly distinct from the determinacy, being-in-itself, would be only the empty abstraction of being. In being-there (Dasein) the determinacy is one with being and is at the same time posited as negation; this determinacy is limit, restriction. Thus, otherness is not something-indifferent outside it, but its own moment. In virtue of its quality, something is first finite and secondly alterable, so that the finitude and alterability belong to its being.
Addition. In being-there negation is still immediately identical with being, and this negation is what we call "limit". Something only is what it is within its limit and by virtue of its limit. We cannot regard limit, therefore, as merely external to being-there; on the contrary, limit totally permeates everything that is there. The interpretation of limit as a merely external determination of being-there is based on a confusion of quantitative with qualitative limit. Here we are dealing first with qualitative limit. When we are considering a piece of land three acres in area, for example, that is its quantitative limit. But, in addition, this piece of land is also a meadow and not a wood or a pond, and this is its qualitative limit.
Something becomes an other, but the other is itself a something, so it likewise becomes an other, and so on ad infinitum.
This infinity is spurious or negative infinity, since it is nothing but the negation of the finite, but the finite arises again in the same way, so that it is no more sublated than not. In other words, this infinity expresses only the requirement that the finite ought to be sublated. This progress ad infinitum does not go beyond the expression of the contradiction, which the finite contains, [i.e.,] that it is just as much something as its other, and [this progress] is the perpetual continuation of the alternation between these determinations, each bringing in the other one.
Addition. If we let something and other, the moments of being-there, fall asunder, the result is that something becomes an other, and this other is itself a something, which, as such, then alters itself in the same way, and so on without end.
What is indeed given is that something becomes another, and the other becomes another quite generally. In its relationship to an other, something is already an other itself vis-a-vis the latter; and therefore, since what it passes into is entirely the same as what passes into it—neither having any further determination than this identical one of being an other—in its passing into another, something only comes together with itself; and this relation to itself in the passing and in the other is genuine Infinity. Or, if we look at it negatively: what is changed is the other, it becomes the other of the other. In this way being is reestablished, but as negation of the negation. It is now being-for-itself.
The following is my interpretation of the development, finalized with some additional comments on the Concept.
We start with something, i.e. a finite thing - a flower, or any finite thing or individual existence, or just the finite in general. The finite has a being, but it also has a determinateness or quality.
The being of something is distinct from its determinateness. In other words, the being of a flower and its scent are two distinct things. The scent is a quality or determinateness of the being of the flower. The flower and its scent are not identical because the flower is much more than just its scent.
If we call the being of the flower its being-in-itself (or being-by-itself), we indicate that being which is distinct and apart from its determinate mode. In other words, being, before any determinations are made of it, is pure being without any determinations, or simple undetermined being - being in general. We will refer to this undetermined being as being-in-itself.
When a determination is made of this being-in-itself, such as the scent of the flower - this quality or scent is something that belongs to the flower - it resides in the being-in-itself of the flower. As being-in-itself, the qualities that it possesses are not yet revealed or manifested - they are implicit. But when being-in-itself is determined then its implicit qualities are made explicit. After a determination of being-in-itself is made it then becomes what is called determinate being.
So there is a difference between being-in-itself and determinate being. Being-in-itself is indeterminate or abstract, while determinate being is being with a qualitative determination. For example, a tract of land is something indeterminate in that we know nothing about its particular qualities. A tract of land may be a pond, a meadow, a forest, etc. The tract of land is thus like being-in-itself until a qualitative determination is made and then we know what type of land it is, which would be its determinate being.
In determinate being, the explicit determinateness is one with the implicit being-in-itself. In other words, the tract of land is implicitly a pond, let us say, before we explicitly determine it is a pond. It is not that the thing-in-itself, may be anything different from what we determine it to be. You might note here that this is Hegel's direct refutation of Kant, who claimed we can't know anything about the thing-in-itself except its appearances. Hegel claims that the appearances are only what the thing-in-itself or being-in-itself is implicitly, which are made explicit when they are determined. Kant artificially separates appearances from the in-itself or implicit, while Hegel comprehends the unity in them.
The determinateness of a thing, however, is also different from the thing-in-itself. The scent of a flower is NOT the flower. A tract of land as a pond is different from tracts of land in general that are not ponds. When the determinateness is considered in its difference from that being-in-itself it is determining, i.e. when it is considered as determinate being as opposed to being-in-itself, then it is considered a negation. Thus determination has two sides: it is identical with what being-in-itself is implicitly, and at the same time it is different from being-in-itself since it is just one determinateness of that being.
Considered as an existent negation, determinateness is called Limit. By calling a tract of land a meadow, it is limited to that qualitative determinateness - it is not a pond or forest, etc. In this case, limit is not a quantitative determination of size, etc., but a qualitative one, since we are using determinateness as the quality of a thing.
In this way, the limit is something that is essential to that which it limits- it is one with what it limits or determines. Therefore, determinateness is both one with and different from its being-in-itself. Likewise limit is one with what it is limiting and as its negation is different from it. (EL92)
In general, something is what it is by its quality, and at the same time something is different from its quality. Something as finite has its limit, and this limit is the negation of the something or something else. So finitude and variability both pertain to its being. The finite is alterable within its own nature as both being and its negation of itself. It is therefore both something and something else.
The negation or other of something is also something. That other something also has a limit, or its negation, i.e. another something. In this way the finite forms a progression of somethings, ad infinitum (EL93). This is the bad or spurious infinite.
The finite is inherently contradictory. It is something and something else. This establishes a progressive alteration showing that the finite as the negation of itself is the infinite, but it is only the spurious infinite - something that ought to be the infinite, but never reaches it because it always retains its contradiction, the finite (EL94).
But the other of something is itself something of which its other is the first something. So both are other to each other. In this otherness they are identical. But that which passes into itself, which has itself outside itself, is just the true infinite (EL95).
The finite has a limit, beyond which it does not find itself. But the infinite has its limit inside itself, i.e. beyond its limit it finds only itself. The infinite is thus the cancellation or negation of its own limit. Since the limit is also a negation, the infinite is in this way the negation of the negation.
We started with something (individuality). It is finite and has some determinate quality. Its determinateness is distinguished from its being-in-itself. This is the distinction between its particularity (determinateness) and universality. This shows itself to be the relation between something and something else, which leads to the spurious infinite. But determinate being is not only something different from being-in-itself since it is implicitly contained within being-in-itself. This identity within difference between the two establishes the infinity of the unity of the universal and particular.
This infinity is a result arrived at by the process of negation of the negation - it does not belong to the immediacy of things. The determinate is the negation of being-in-itself, or the particular is the negation of the universal. The finite is what is negatively related to the infinite. In each case the finite, particular or determinate is IN the infinite, universal or implicit being. At the same time they are Other than the infinite, universal or being-in-itself. This establishes their relationship as identical and different.
The part of their identity is due to the negation of the Otherness (which is itself, negation), i.e. the negation of the negation. This negation of otherness, which results in the identification of the Other with itself, is called Being-for-self. In being-for-self the otherness or determinateness is acknowledged as a negation but it is negated as otherness or as independent of the being that is determined. It is this negation of the independence of the being of the Other that is referred to as being-for-self. This arises directly out of the relation of the infinite as the negation of the negation. Because it involves negation, it does not produce an abstract identity of one thing and another - the resultant unity or truth must be understood as the result plus the process for arriving at it. The original and final stages of the development are different, even though there is a circular return involved in being-for-self. The negation of the negation is a return to the original being, but it includes its absolute negativity, without which it is mere abstract being without negativity.
The concrete universal is that which contains its particularity within itself, where the particulars are also different from it. These particulars, it should be noted, are just universals when they are considered independent of the universal. They are moments of universality when considered in their unity with the universal. When particular determinations are grasped only in their immediacy, they are misconceived as mere singulars or individuals. In fact, they should not be misconstrued as existing independently of the universal of which they are momentary aspects. This is the defect of atomic thinking, or reductionism, which separates the true universality from the particular, and then makes the particular the universal.
It is this unity of the particular and universal that is the true individual or the Concept, which when considered in its immediate being as falling away from the Concept, i.e. as a distinct and externally existing individual, is the existent individual.