On Hegel’s ‘Science of Logic’ : A Realm of Shadows — part fourteen.

David Proud

36 min readJan 21, 2023

‘Are They Shadows’

by Samuel Daniel (1562–1619)

Are they shadows that we see?

And can shadows pleasure give?

Pleasures only shadows be

Cast by bodies we conceive

And are made the things we deem

In those figures which they seem.

But these pleasures vanish fast

Which by shadows are expressed;

Pleasures are not, if they last;

In their passing is their best.

Glory is most bright and gay

In a flash, and so away.

Feed apace then, greedy eyes,

On the wonder you behold;

Take it sudden as it flies,

Though you take it not to hold.

When your eyes have done their part,

Thought must length it in the heart.

In this rapid excursion through Georg Wilhelm Friedrich Hegel’s, (1770–1831), ‘Science of Logic’ we have reached quantitative Infinity. And what is the Notion pf quantitative Infinity? The Understanding sees the Qualitative Something as Quantitative Infinity, an activity wherein Quantum goes beyond itself and remains with itself. The beyond is a Quantum — the Infinitely Great or Infinitely Small Number, depending which way we are counting. This beyond is not only the other of a particular quantum, but of quantum itself.

‘Quantum alters and becomes another quantum; the further determination of this alteration, that it goes on to infinity, lies in that it is positioned as inherently self-contradictory. –Quantum becomes an other; but it continues in its otherness; the other is therefore also a quantum. This latter, however, is the other, not of a quantum, but of the quantum as such, the negative of itself as a limited something, and hence its own unlimitedness, infinity. Quantum is an ought; it implies that it be determined-for-itself, and this being-determined-for-itself is rather the being determined in an other; and, conversely, it is the being-determined in an other as sublated, is indifferent subsisting-for-itself’.

- ‘The Science of Logic’

Quantitative Infinity

This remark, which looks forward to the Infinitely Great/Small Number two steps hence raises a question. Given that Quantum inherently finds its being outside itself and given that this outside (the Qualitative Something) must be expressed as some Number, why does it follow that these numbers must get progressively larger or progressively smaller? Why can’t, for instance, the Qualitative Something alternate from 10 to 11 back to 10 ad infinitum? There are several answers to why an infinite process should be motivated toward a highest or smallest number. First, it is the nature of the Understanding to hazard a complete proposition of the absolute.

In order to complete the infinite set of Quanta, it therefore becomes necessary to head for the Infinitely Great/Small Number that would (if it existed) complete the set. Second, Quantum goes outside itself, but the return trip is not to the original Number but to some other Number ad infinitum. This conclusion is compelled by the lesson learned in the One and the Many. There, the old One from which the new One springs does not go out of existence. Rather, the new One produces yet another One, creating infinitely Many Ones. Through the law of sublation, the same result must occur in Quantitative Infinity, though Hegel nowhere says so explicitly. Finally, there is our old friend the silent fourth, which is entitled to go forward or backward in the logical steps but elects to go forward in order to make the system unfold. For at least these reasons, Quantitative Infinity is purposive activity. It therefore chooses to aim for the goal of completion. Accordingly, Hegel says that Quantitative Infinity is an ought-to-be; it is by implication determined as being for itself, and this being-determinedfor-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 at once acquires within it a double though opposite meaning. Quantum is finite, first, as limited in general; second, as sending itself beyond itself, as being-determined in an other. On the other hand, its infinity, is, first, the unlimitedness of quantum; second, its being-turned-back-into-itself, the indifferent being-for-itself. If we now compare these moments with each other, we find that the determination of quantum’s finitude, its sending itself beyond itself into an other that constitutes its determination, is equally the determination of the infinite; the negation of limit is this same transcendence of determinateness, so that in this negation, in the infinite, quantum has its final determinateness. The other moment of infinity is the for-itself which is indifferent to the limit; but the quantum itself is so limited, as to be indifferent with respect to its limit, and hence with respect to other quanta and its ‘beyond’. In quantum, finitude and infinity (the latter supposedly separate from finitude, as bad infinity) each already possesses within it the moment of the other’.

- ‘The Science of Logic’

Completion, then, is the theme of Quantitative Infinity- one that is doomed to fail. Quantitative Infinity is the impulse to go beyond itself to an other in which its final determination lies. ‘And as the continuity of quantum expresses itself equally in endless extensity and in endless diminution, the progression is interminable either way, though neither the infinitesimal nor the infinite is ever attainable’, explains Andrew Haas.

What is the difference between Qualitative and Quantitative Infinity? In Qualitative Infinity, a and c stood abstractly opposed to each other. Their unity was only in-itself — implicit. Quantitative Infinity, in contrast, continues within itself even as it passes into its beyond. It is a True Infinite. It is possible to quibble with Errol Harris’s, (1908–2009), remark that, to resolve Quantum’s contradiction, ‘the externality of the other must somehow be internalized to produce a true infinity’. At this stage, the extremes each have long since been True Infinites. What Quantum must express is that it is as much its other as it is its own self. Hence, Harris is right that the external must be internalized, but the external must also stay external as it becomes internal. Furthermore, it is already a True Infinite and therefore need not, at this late stage, become one.

The Quantitative Infinite Progress. In Quantitative Infinite Progress, Dialectical Reason denies that Quantitative Infinity is capable of completion. It points out that the complete expression of the infinite set of Quanta requires some Infinitely Great Quantum, but, as it is Quantum, this too must go out of itself. In this dialectical stage, the extremes fall into a Quantitative infinite Spurious Infinity — a senseless modulation Progress toward a new Infinite Great/Small Quantum that is supposed to complete the set of Quanta. This time, the Spurious Infinite takes place within the context of a True Infinite, in the sphere of quantity the limit i.e., Quantum] continues itself into its beyond and hence, conversely, the quantitative infinite too is posited as having quantum within it.

‘The process to infinity is in general the expression of contradiction, here, of the contradiction contained in the quantitative finite or in quantum in general. It is the reciprocal determination of the finite and the infinite that came up for consideration in the sphere of the qualitative, with the difference that, as just indicated, in the sphere of quantity the limit inherently sends itself beyond itself and continues there, and hence, conversely, the quantitative infinite is also posited as having the quantum in it, for in its externality quantum is itself; its externality belongs to its determination’.

- ‘The Science of Logic’

In other words, at the level of Quantity, the Infinite goes beyond itself and stays within itself as it travels into this beyond. The Quantitative Infinite Progress intends but never reaches the Infinitely Greatest/Smallest Number. A qualitative moment prevents Quantitative Infinity from reaching completion. In fact, Hegel emphasizes, there can be no question of getting nearer to the goal of the Infinitely Great/Small, since it is by definition unreachable. This qualitative moment as structurally similar to trauma — a stumbling block, or piece of the Real, which prevents the patient from completing his fantasy as postulated by Jacques Marie Émile Lacan, (1901–1981),

No matter how much the quantum is increased, it shrinks to insignificance.

‘The continuity of quantum with its other brings about the conjunction of the two in the expression of an infinitely great or infinitely small. Since they both still have in them the determination of quantum, they remain alterable and the absolute determinateness which would be a being-for-itself is thus not attained. This being-outside-itself of the determination is posited in the double infinity (posited in the relative opposition of the ‘more’ and the ‘less’) of the infinitely great and the infinitely small. In each, the quantum is maintained in perpetual opposition to its beyond. No matter how much the ‘great’ is enlarged, it shrinks to insignificance; since it refers to the infinite as to its non-being, the opposition is qualitative; the enlarged quantum has gained nothing, therefore, from the infinite; the latter is its nothing now just as before. Or again, the increase in the quantum is not an approximation to the infinite, for the distinction between the quantum and its infinity essentially has also the moment of being non-quantitative. This moment is only the sharpened expression of the contradiction that the quantum ought to be something great, that is, a quantum, and non-finite, that is, not a quantum. — Equally, the infinitely small is, as something small, a quantum and therefore remains absolutely, that is, qualitatively, too great for the infinite and opposed to it. In both, there remains the contradiction of the infinite progress which in them should have reached its goal’.

The Quantitative Infinite Progress is 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.

‘This infinity, which persists in the determination of the beyond of the finite, is to be characterized as the bad quantitative infinity. Like the qualitatively bad infinity, it is the perpetual movement back and forth from one side of the persistent contradiction to the other, from the limit to its non-being, and from the latter back again to the other, the limit. To be sure, the term to which the advance is made in the quantitative progress is not an abstract ‘other’ in general but a quantum which is explicitly posited as different; but this quantum remains opposed to its negation in the same way. Also the progress, therefore, is neither an advance nor a gain but rather a repetition of one and the same move, a positing, a sublating, and then again a positing and a sublating: an impotence of the negative to which what it sublates continuously comes back by its very sublation of it. The two, the positing and the sublation, are so bonded to each other that they absolutely flee from each other and yet, in thus fleeing, they are unable to part but rather become bonded in their very flight from each other’.

- ‘The Science of Logic’

Quantitative Infinite Progress

What is the bond between the two extremes of Quantitative Infinite Progress? Simply that each flees from the other, and in fleeing from each other they cannot become separated but are joined together even in their flight from each other. The alternating Quanta are in a relationship (Verhältnis).

The High Repute of the Progress to Infinity. No admirer of Quantitative Infinity, Hegel complains that in philosophy it has been regarded as ultimate.

‘The bad infinity, especially in the form of the quantitative progress to infinity — this uninterrupted flitting over limits which it is powerless to sublate, and the perpetual falling back into them — is commonly held to be something sublime and a kind of divine service, just as in philosophy it has been regarded as ultimate. This progress has often been exploited in tirades which have been admired as sublime productions. In fact, however, this modern sublimity does not enhance the object, which rather takes flight from it, but bloats the subject who ingests such vast quantities. The poverty of such an irreducibly subjective step by step elevation on the ladder of the quantitative is betrayed by the admission that in that vain labour there is no getting closer to the infinite goal — for the attainment of which, to be sure, quite another line of attack is required’.

With Immanuel Kant, (1724–1804), obviously in mind, Hegel remarks, this modern sublimity does not magnify the object — rather does this take flight — but only the subject which assimilates such vast quantities. In ‘The Critique of Judgment’, Kant defined sublimity as a subjective feeling that one could actually know the thing-in-itself (which is impossible).

‘Therefore, just as the aesthetical Judgement in judging the Beautiful refers the Imagination in its free play to the Understanding, in order to harmonise it with the concepts of the latter in general (without any determination of them); so does the same faculty when judging a thing as Sublime refer itself to the Reason in order that it may subjectively be in accordance with its Ideas (no matter what they are): — i.e. that it may produce a state of mind conformable to them and compatible with that brought about by the influence of definite (practical) Ideas upon feeling. We hence see also that true sublimity must be sought only in the mind of the [subject] judging, not in the natural Object, the judgement upon which occasions this state. Who would call sublime, e.g. shapeless mountain masses piled in wild disorder pon each other with their pyramids of ice, or the gloomy raging sea? But the mind feels itself elevated in its own judgement if, while contemplating them without any reference to their form, and abandoning itself to the Imagination and to the Reason — which although placed in combination with the Imagination without any definite purpose, merely extends it — it yet finds the whole power of the Imagination inadequate to its Ideas’.

- ‘Critique of Judgement’

Hegel also quotes from Kant’s ‘Critique of Practical Reason’ though he often cites from memory soit is more of a paraphrase than a citation:

‘Here are examples of tirades of the kind, which make manifest what this elevation ultimately amounts to. Kant, for example, at the conclusion of the Critique of Practical Reason, deems it as sublime ‘when the subject rises in thought above the place it occupies in the world of the senses, and extends its reach into an unbounded magnitude of worlds beyond worlds and systems of systems and into the limitless times of their periodic motion, their beginning and their continuance. — Imagination fails before this progression into the immeasurably distant, where beyond the most distant world there lies a still more distant one; behind the past, however far back traced, a still more distant past; ahead of the future, however far down projected, yet another future. Thought fails before this representation of the immeasurable, just as in a dream, in which one relentlessly goes on and on down a long corridor without seeing the end of it, and finishes with falling or fainting’.

Hence, the sublime exalts the subject (and not the object). What makes thought succumb to the shock and awe of the Quantitative Infinite Progress is nothing else but the wearisome repetition which makes a limit vanish, reappear, and then vanish again giving only the feeling of the impotence of this infinite or this ought-to-be, which would be master of the finite and cannot.

‘This account, besides capturing all that there is to this quantitative elevation in a wealth of pictorial imagery, deserves praise mainly because of how truthfully it betrays the end result of this elevation: thought succumbs, the upshot is falling and giddiness. What causes thought to succumb, what produces the falling and the giddiness, is nothing else but the boredom of this repetition that makes a limit disappear, come up again, and again disappear, and so lets the rising and the perishing of the one for the other, and of the one into the other, of the here into the there, and the there into the here, perpetuate itself, only conveying the feeling of the impotence of this infinite, this ought, which would want to be master of the finite but cannot’.

- ‘The Science of Logic’

Kant compares the sublime to the withdrawal of the individual into his ego, where absolute freedom opposes the terrors of tyranny and fate. At this moment, Hegel says, the individual knows himself to be equal to himself. This is probably a reference to ‘Critique of Practical Reason’.

‘Now, in order to remove in the supposed case the apparent contradiction between freedom and the mechanism of nature in one and the same action, we must remember what was said in the Critique of Pure Reason, or what follows therefrom; viz., that the necessity of nature, which cannot co-exist with the freedom of the subject, appertains only to the attributes of the thing that is subject to time-conditions, consequently only to those of the acting subject as a phenomenon; that therefore in this respect the determining principles of every action of the same reside in what belongs to past time and is no longer in his power (in which must be included his own past actions and the character that these may determine for him in his own eyes as a phenomenon). But the very same subject, being on the other side conscious of himself as a thing in himself, considers his existence also in so far as it is not subject to time-conditions, and regards himself as only determinable by laws which he gives himself through reason; and in this his existence nothing is antecedent to the determination of his will, but every action, and in general every modification of his existence, varying according to his internal sense, even the whole series of his existence as a sensible being is in the consciousness of his supersensible existence nothing but the result, and never to be regarded as the determining principle, of his causality as a noumenon. In this view now the rational being can justly say of every unlawful action that he performs, that he could very well have left it undone; although as appearance it is sufficiently determined in the past, and in this respect is absolutely necessary; for it, with all the past which determines it, belongs to the one single phenomenon of his character which he makes for himself, in consequence of which he imputes the causality of those appearances to himself as a cause independent on sensibility’.

- ‘Critique of Practical Reason’

_________________________

And furthermore:

‘Also Haller’s description of eternity, which Kant called horrifying, is commonly the object of special admiration, but often for what is precisely not the reason that constitutes its true merit:

I heap up giant numbers,

Pile millions on millions;

Eon upon eon and world upon world,

And when I am on that endless march

And dizzy on that terrifying height

I seek you again.

[The power of numbers, though multiplied a thousandfold,

Is still not even a fraction of you.]

I blot them out and there you are, complete, before me.

‘In stressing the value of this heaping and piling of numbers and worlds as a description of eternity, what is overlooked is that the poet himself declares this so-called terrifying venture into the beyond as futile and hollow, and he concludes that only by giving up this empty infinite progress will the true infinite itself become present to him. There have been astronomers who liked to flatter themselves about the sublimity of their science on the ground that it deals with an immeasurable multitude of stars, with immeasurable spaces and times within which the already vast distances and periods that serve as their units, even when taken many times over, shrink to insignificance. The shallow astonishment to which they surrender themselves, their fatuous hopes of eventually traveling in another life from star to star and in that immensity to make discoveries of always the same kind of things, this they adduce as the main point of excellence of their science — a science which is worthy of admiration rather, not because of such quantitative infinitude but, to the contrary, because of the relations of measure and the laws which reason recognizes in these objects and which, in contrast to that other irrational infinitude, constitute the rational infinite’.

- The Science of Logic’

Notes:

‘Kant calls horrifying’: in the ‘Critique of Pure Reason’

Albrecht, Baron von Haller, (1708–1777), Swiss physiologist and poet, ‘Imperfect Poem on Eternity’, 1736. The stress in the last line is added by Hegel.

-________________________

Of this ego’s self-equality, Hegel agrees that it is the reached beyond; it has come to itself is with itself, here and now. This negative thing — the ego — has determinate reality confronting it as a beyond. In this withdrawal, 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 nonbeing.

How is this so? Recall that Quantitative Infinity stayed within itself, but this “in-itself had no content. All content was in the beyond. Simultaneous with its being-for-self, Quantitative Infinity was pure flight into the beyond and hence a constant modulation between the moment of flight and the moment of return. Ego, it turns out, is the same thing. Here we have the Lacanian view of the subject as suspended between the realm of the Symbolic (i.e., being) and the Real (i.e., nothing).’The subject is nothing but this very split’, explains Eugen Finlk. Kant, in turn, describes the ‘I’ — the universal aspect of personality, which Lacanians insist is not the subject. The subject finds part of its selfhood in its beyond. The subject desires wholeness but cannot achieve it. This is what the Lacanians like Slavoj Žižek, (1949 — ), called symbolic castration. Hegel saw this some 150 years before Lacan. For Hegel, the relation of the subject to its non-being (i.e., the Symbolic realm) “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.

‘The ‘I’, in this solitude, is indeed the attained beyond; it has come to itself, is at home with itself, right here; the absolute negativity which in the progression beyond the quantum of the senses was only in flight, is brought in pure self-consciousness to affirmation and presence. But this pure ‘I’, when held fixed in abstraction and empty of content, has existence in general, the fullness of the natural and the spiritual universe, as a beyond confronting it. The same contradiction reasserts itself that lies at the heart of the infinite progress, namely of a being bent upon itself which is at the same time outside itself, which refers to its other as to its non-being and in this referring remains a longing: for the ‘I’ has fixed itself, on the one side, with its indigent and insufferable emptiness before it, and, on the other side, with a fullness which in being negated is still present as its beyond’.

- ‘The Science of Logic’

Hegel especially complains that Kant equates morality with Quantitative Infinity. The antithesis just described — autonomy v heteronomy — is a qualitative opposition. Kant supposes that the subject, suspended between them, can get closer to moral autonomy but can never reach 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 o£ 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 unattainableness is supposed to be the true sheetanchor and fitting consolation; for morality is supposed to be a struggle, but such it can be only if the will is inadequate to the moral law which thus becomes a sheer beyond for it.

Here is a concise critique of Kant’s doctrine of radical evil. According to Kant, the ego is forever tainted with pathology. It can never finally purge itself of pathology but can only struggle for moral purity. Kant even goes so far as to deduce the immortality of the soul from the very fact that all eternity is required for the soul to reach perfection.Hence, Kant is guilty as charged. He has reduced morality to Quantitative Infinity. With regard to Kant’s opposition of morality and nature, Hegel complains that they are put forth as self-subsistent and mutually indifferent.

‘In this opposition, the ‘I’ and the ‘not-I’, or the will and the moral law and nature and the sensuousness of the will, are presupposed as perfectly self-subsistent and mutually indifferent. The pure will has its own law, which is essentially connected with the senses; nature and the senses, for their part, have their laws which neither stem from nor are conformable to the will, nor are such that, although diverse from the will, would nonetheless be in essential connection with it, but are rather independently determined, finished and complete in themselves. The two are nevertheless both moments of one and the same simple essence, of the ‘I’; the will is determined as the negative with respect to nature so that the will only is to the extent that there is such a thing as a nature which is diverse from it and which it sublates, but by which, in sublating it, it is touched and is itself affected. Nature, also as the sensuous element of the human being, is a self-subsistent system of laws indifferent to limitation through an other; it preserves itself while being limited, comes in connection with the will on its own terms and limits the will of the law just as much as this will limits it. — It is by one single act that the will, in determining itself, sublates the otherness of a nature, and this otherness, in being posited with a determinate existence, resists sublatedness and is not sublated. In the infinite progression the contradiction at work here is not resolved but, to the contrary, it is displayed as unresolved and unresolvable and is declared to be such; the conflict of morality and sense is represented as an absolute relation that exists in and for itself’.

- ‘The Science of Logic’

At the same time, however, both are moments of one and the same being the ego. Hence, the very constitution of the Kantian subject is the Lacanian split subject. Lacanians give Kant the greatest credit for this. This contradiction is never resolved in the infinite progress. On the contrary, it is represented and affirmed as unresolved and unresolvable. This Kantian standpoint is powerless to overcome the qualitative opposition between the finite and infinite.

‘The powerlessness in mastering the qualitative opposition between the finite and infinite and in grasping the idea of the true will which is substantial freedom, this powerlessness takes refuge in magnitude which it used as a middle link, for magnitude is the qualitative as sublated, the distinction that has become indifferent. But since the two members of the opposition still remain in principle qualitatively different, by behaving as quanta in referring to each other they are rather each straight away posited as indifferent to this alteration. Nature is determined through the ‘I’, the senses through the will of the good; the alteration produced in the senses through the will is only a quantitative distinction, one which leaves them be as they are’.

- ‘The Science of Logic’

The subject counts it as nothing that it has supposedly progressed toward the unattainable perfection of pure morality. Kants Antinomy of the Limitation of Time and Space We have seen that Hegel has small regard for Kant’s four antinomies of reason. Now he repeats his conclusion that the antinomies are spurious qualitative infinities, each side of which is a one-sided view of the truth. The antinomy at hand is Kant’s first — the world is (or is not) limited in time and space. Of this antinomy, Henry Allison, (1937 -), remarks, ‘These are the most widely criticized of Kant’s arguments’. This antinomy is the one Kant associates with the category of quantity which is why Hegel discusses it here, (Kant calls time and space ‘the two original quanta of all intuition’).

Hegel’s first proposition about this antinomy is that the world could have been left out of the discussion. Kant could have addressed time as such and space as such. Allison disagrees and thinks that synthesis of a world out of infinite moments is central to Kant’s argument. Hegel also proposes that Kant could have restated his antinomy as follows: (1) there is a limit, and (2) limit must be transcended — two things Hegel says are true of Quantity generally. The Thesis. In terms of time, Kant proves the thesis (the world has a beginning) by showing that the antithesis is impossible. If time has no beginning, then at any given point of time, an ‘eternity’ — an infinite series of temporal measures — has lapsed. The lapse of an infinite series is impossible.

Therefore, time must have a beginning. Hegel proposes that Kant’s proof of the thesis is only the direct assertion of what was to be proved. With regard to time (it has a beginning), the very assertion that time has points presupposes that time is already limited. One point in time is now. It designates the end of the past and the beginning of the future. With regard to the past, now represents a qualitative limit. But why, Hegel implicitly asks, should now be a qualitative limit? Suppose we say that now is a quantitative limit. Time would then continue on from the past, over the now, and into the future, because Quantitative Infinity always partakes as though in media res so to speak. Quantitative Infinity not only must be transcended but is only as the transcending of itself. If time is Quantitative Infinity, then the infinite time series would not have passed away in it, but would continue to flow on.

‘There is only this difference, that the assumed limit is a now which comes at the end of a time just elapsed, whereas the now to be proved is at the beginning of a future time. But this difference is unessential. The now is assumed as the point at which an infinite series of successive states of things is supposed to have flowed away in time, therefore to be an end, a qualitative limit. If this now were considered as only a quantitative limit that flows on and is there, not simply to be surpassed, but as itself self-surpassing, then the infinite time series would not have flowed away in it but would go on flowing, and so the nerve of the proof would fall. On the other hand, if the temporal point is assumed to be the qualitative limit of the past, in which case it is equally the beginning of a future — for each temporal point is in itself the connection of the past and the future — then the point is also the absolute, that is, abstract, beginning of that future, and this is what was to be proved. It is irrelevant that there is already a past preceding its future at this point; for the temporal point is a qualitative limit (and that it is to be taken as qualitative rests on its determination of being completed, elapsed, and therefore of not continuing), and so time is interrupted in it; the past is then without connection to a future that could be called such only in respect to it, and therefore, without this connection to a future, itself only time in general, which has an absolute beginning. But if the future were to stand in connection with the past (as indeed it does) through the now, the given point of time, and in this way were determined as future, then this point of time also, from the other side, would not be a limit; the infinite time series would continue in what was called the future, and would not be, as assumed, completed’.

- ‘The Science of Logic’

A switch from qualitative to quantitative limit destroys Kant’s argument. But, Hegel continues, let us concede that now is a qualitative limit to the past. It would then also be the beginning of the future. But this is precisely the thesis to be proved — that time has a beginning. So what if this beginning was preceded by a deceased past? This does not affect the point. The past is conceived as radically separate from the future. Hence, the very introduction of now — a point in time — presupposes time’s beginning. The antithesis. Kant also proves also proves the antithesis by ruling out the opposite (apagogy). In terms of time, suppose the world has a beginning. Before the beginning, the world does not exist. An existing thing, however, cannot originate from nothing. Nothing comes from nothing. Kant’s argument for the antithesis likewise merely asserts what must be proved, Hegel thinks. Kant’s argument presupposes that, just because the world exists, it must have an antecedent condition which is in time.”This is the very antithesis to be proved. Also, when Kant insists that nothing can come from nothing — when the condition is sought in empty time — this means that the world is taken as temporal and hence limited.

‘Like the others, this apagogical proof contains the direct and unproved assertion of what is supposed to be proved. For it first assumes a beyond of the existent world, an empty time; but it then goes on to continue the existence of this world beyond itself into the empty time, sublates this time thereby, and consequently extends the existence to infinity. The world is an existence; the proof presupposes that this existence arises, and that its arising has an antecedent condition in time. But the antithesis itself is just this, that there is no unconditional existence, no absolute limit, but that the existence of the world always demands an antecedent condition. What was to be proved thus finds itself in the proof as assumption. — Further, the condition is then sought in the empty time, and this is as much as saying that it is assumed as temporal and consequently as existence, and as something restricted. The general assumption, therefore, is that the world presupposes as existence another conditioned existence in time — and so on, therefore, to infinity’.

- ‘The Science of Logic’

Something always precedes the now of the world. There is always a yesterday. All of this, Hegel charges, is presupposed. It is the antithesis itself. Kant’s demonstration of the antithesis in terms of space is likewise rejected. A limit in space implies there must be a void space. We thus have a relation of the finite world to void space. But this is a relation of things to no object. A thing cannot have a relation to nothing — nothing is not a thing. Consequently the world is not limited in space Hegel finds again that Kant has merely restated the proposition — not proved it. Kant assumes that space is not an object, and that, in order to prevent the impossible relation of object to non-object, the object must continue itself as far as space does. This means that Kant thinks space must never be empty — the world must continue into it. Yet this is precisely the antithesis restated.

Hegel concludes this remark by criticizing Kant for subjectivizingcontradiction.For Kant, the four antinomies do not occur in nature. Rather, they occur in consciousness. (Time and space, Kant says, are the very conditions of possibility for subjective intuitions). Of this subjectivization of the first antinomy, Hegel writes that 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.

‘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, by which is meant that the world does not contradict itself within, is not something that sublates itself, but that consciousness alone, in its intuition and in the connection of intuition to understanding and reason, is rather the being which is self-contradictory. It is an excessive tenderness for the world to keep contradiction away from it, to transfer it to spirit instead, to reason, and to leave it there unresolved. In fact, spirit is the one which is strong enough that it can endure contradiction, but it is spirit again which knows how to resolve it. But nowhere does the so-called world — call it the objective, real world, or, in the manner of transcendental idealism, subjective intuition and sense content determined by the category of the understanding — nowhere, however you call it, does it escape contradiction; but it is not capable of enduring it and for that reason it is left to the mercy of the coming and ceasing to be’.

- ‘The Science of Logic’

The so-called world is contradictory, Hegel insists. The world is unable to endure it and is, therefore, subject to coming-to-be and ceasing-to-be.

And so to the Infinity of Quantum. The middle term between Quantitative Infinity and its Infinite Progress is the Infinitely Great and/or Infinitely Small (i.e., the differential δx in the derivative δy/δx) The Infinitely Great/Small is the destination that the Quantitative Infinity implies. It is a Quantum, but 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.

‘The infinite quantum as infinitely great or infinitely small is itself, in itself, the infinite progress; as great or small it is a quantum and at the same time the non-being of quantum. The infinitely great and the infinitely small are, therefore, figurative representations which on closer inspection prove to be but unsubstantial nebulous shadows. In the infinite progress, however, this contradiction is explicitly present and with it that which constitutes the nature of quantum which, as intensive magnitude, has attained its reality and is now posited in its existence as it is in its concept’.

- ‘The Science of Logic’

This should be clear even to non-speculative thinkers. In the Quantitative Infinite Progress, the counting infinitely Great and mathematician aims to reach infinity. That infinitely Small infinity has being is thus presupposed by the counter who aims to reach this end. Yet this end will never be reached.

Infinitely Great and Infinitely Small

It is a non-being. This contradiction — the non-being of infinity — is now explicitly present, and so is the very nature of Quantum. When Quantum reached Intensive Magnitude (Degree), Quantum attained its reality. Degree is unitary, self-related and determinate within itself.

‘Quantum is as degree simple, self-referred, and determined within it. Because the otherness and the determinateness are sublated in it through this simplicity, the determinateness is external to it; it has its determinateness outside it. This, its being-outside-itself, is at first the abstract non-being of quantum in general, the bad infinity. But further, this non-being is also a magnitude; quantum continues in its non-being, for it is precisely in its externality that it has its determinateness, and this, its externality, is itself therefore equally a quantum; the non-being of quantum, the infinity, is thus limited, that is, this beyond is sublated, is itself determined as a quantum which, consequently, in its negation is with itself’.

- ‘The Science of Logic’

As unitary, Degree sublated i.e., negated its otherness and its determinateness. These were external to Degree. This self-externality was the abstract non-being of quantum generally, the spurious infinity. In other words, Degree yielded the Qualitative Something which in turn yielded Quantitative Infinity. If we now examine Quantitative Infinite Progress, we witness each of the extremes erasing itself and establishing its non-being in the other, while expressly continuing itself in the other, so that each was a Quantum as well as not a Quantum. Hence, 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. The in-itself of Quantum is therefore self-externality. Compare the Ought in The Ought, where the in-itself of the Finite was that it must cease to be and become other. Externality determines what Quantum is. The Infinitely Great/Small thus illustrates notion of Quantum. It is not there and yet treated as if it is there: 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.

‘If we now first look at this progress in its abstract determinations as they are displayed before us, what we find in it is the sublating of quantum, but no less also of its beyond; what we find, therefore, is the negation of quantum as well as the negation of this negation. Its truth is the unity of these two negations in which the negations are, but as moments. — This unity is the resolution of the contradiction of which the infinite progress is the expression; its most immediate meaning, therefore, is that of the restoration of the concept of magnitude, of being an indifferent or external limit. On the subject of the infinite progress as such, the only reflection which is usually made is that each quantum, however great or small, can disappear, that it must be possible to transcend it — not, however, that this sublating of the quantum, the beyond, the bad infinite itself, also disappears’.

- ‘The Science of Logic’

How is this claim justified? Why has the Spurious Infinite vanished? Consider what the Infinitely Great/Small is: the end that the Quantitative Infinite Progress could never reach. If we have that end before us, then we do not have the Quantitative Infinite Progress before us. In short, we can take Infinitely Great and Infinitely Small in terms of g — which is isolated from the vanished Quantitative Infinite Progress. This isolation from externality is a sign that Quantity has recaptured its Quality.

We are now in a position to summarize the progress across the realm of Quantum. Quantum (via Quantity) is the negation/sublation of Quality. Considered immediately, as in, say, Continuity or Amount, it is already the first negation — in positivized form. But Quantum is only the first negation in principle. It is posited as a being, and its negation is fixed as the infinite, as the beyond of quantum, which remains on this side as an immediate. But it is the beyond that is overtly the first negation. In the Infinitely Great/Small, quantum is determined in conformity with its Notion, which is different from quantum determined in its immediacy.

‘Even the first sublating, the negation of quality as such whereby the quantum is posited, is in itself the sublation of negation — quantum is sublated qualitative limit, consequently sublated negation — but it is at the same time only in itself; the sublating is posited as an existence, and its negation is then fixed as the infinite, as the side beyond quantum, while the latter remains on its side as an immediate; thus the infinite is determined only as first negation and it is in this way that it appears in the infinite progress. But there is more to it, as has just been shown: there is the negation of negation or what the infinite is in truth. And this we have just seen with the restoration of the concept of quantum. Such a restoration means, in the first place, that to the existence of the quantum there has accrued a more precise determination. What we now have is quantum determined according to its concept, and this quantum is different from the immediate quantum: externality is now the opposite of itself, is posited as a moment of magnitude; quantum, for its part, is posited as having its determinateness in another quantum, through the intermediary of its non-being, of infinity, that is, that it is qualitatively what it is. Yet this comparison of the concept of quantum with its existence belongs more to our reflection, to a relation which is not yet present here. The next determination, rather, which is present here is that quantum has returned to quality, is from now on qualitatively determined. For its defining property, its quality, is externality, the indifference of the determinateness; and quantum is now posited rather as being itself in its externality, of referring to itself therein, of being in simple unity with itself, that is, of being qualitatively determined. — This qualitative being is still more closely determined, namely as being-for-itself; for the very self-reference which quantum has attained has proceeded from mediation, from the negation of the negation. Quantum no longer has infinity, the being-determined-for-itself, outside it, but in it’.

- ‘The Science of Logic’

The Infinitely Great/Small is externality itself, brought inward as a moment of Quantity. For this reason, 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. Because Quantum has brought its externality inward, it is qualitatively what it is.

**‘Ombres portées’ (‘Cast shadows’), 1891, Emile Friant**

Quantum has reverted to quality and is from now on qualitatively determined. Its quality (per Hegel, its peculiarity) is that its determinateness (or content) is external. Quantum is indifferent to outside determination. But the outside is now in: Quantum has infinity, self-determinedness, no longer outside it but within itself. In Infinitely Great and Infinitely Small, Quantum is posited as repelled from itself, with the result that there are two quanta which, however, are sublated, are only as moments of one unity.

‘Quantum is thus posited as repelled from itself, and with that there are two quanta which are however sublated, only moments of one unity, and this unity is the determinateness of quantum. — Quantum, self-referred as indifferent limit and hence qualitatively posited, is the quantitative relation or ratio. — In ratio quantum is external to itself, different from itself; this, its externality, is the reference connecting a quantum to another quantum, each quantum acquiring value only in this connection with its other; and this reference constitutes the determinateness of the quantum which is this unity. In this unity quantum possesses not an indifferent but a qualitative determination; in this its externality has turned back into itself; it is in it what it is’.

- ‘The Science of Logic’

In chapter six Quantum will now appear as double — as Quantitative Ratio. This ratio is the relation between Quality (internality) and Quantity (externality). Quality forms a partnership with Quantity. The middle term between this partnership is Measure — the stuff of chapter seven. Charles Taylor, (1931 — ), expresses his dissatisfaction with Hegel’s entire discussion of Quantum, and we are now in a position to answer his queries. Taylor writes:

‘But one might think that Hegel is a little cavalier in his transitions here. Granted that Quantity is the realm in which things are indifferent to their limit, how does that show that quanta must go beyond themselves, and change? (whatever that mearis). And even if they do so endlessly, even granted Hegel’s dislike for the “bad” infinity of the endless progress, does this show a contradiction requiring resolution by a higher category?’

- Taylor’s ‘Hegel’

The answer to the first question is, since quanta are True Infinites, their very function is to go beyond their limit (while staying what they are). This act is the Quality of the Quantum. But this does not necessarily mean that numbers change and that arithmetic is promiscuous and subjective. Quanta have limits within themselves. Three does not melt into two. If the limit external to a quantum is exceeded, it is exceeded spiritually, not empirically. The answer to the second question is that the bad infinity’s modulation between quanta is itself the higher category. As always, speculative Reason names the autistic modulation of dialectical Reason and underwrites progress to a higher level.

Which brings us to calculus. At the end of chapter five, Hegel inserts three long Remarks, the first two of which are by far the longest Remarks in the Logic. They cover the subject of calculus, which endlessly fascinated Hegel, because the differential δx — the Infinitely Small — embodies his theory that Being is in the process of vanishing. The differential δx stands for the change in the variable x. As such, it is undefinable, because it is supposed to be infinitely close to (but distinguishable from) zero. Yet δx given δy (or δylδx) is fully determinate. This point is important in understanding why the Infinitely Great/Small is a ratio, δx has its being in δy, and vice versa. The two become visible only when brought into conjunction.

In the main, Hegel will criticize nineteenth century calculus for its lingering dependence on geometrical ideas, and for the quantification of δx, which Hegel views as an undefined quality (not a quantity). Future generations of mathematicians would tend to agree. In this regard, Hegel remarks that mathematical necessity is inadequate. Mathematics does nothing but ward off heterogeneous elements — an effort that is itself tainted with heterogeneity.

‘The presentation of no subject matter can be in and for itself as strictly and immanently plastic as is that of thought in its necessary development; nor would any subject matter require such a presentation; in this respect, the science of logic must surpass even mathematics, for no subject matter intrinsically possesses this freedom and independence. The presentation would demand that at no stage of the development should any thought determination or reflection occur that does not directly emerge at that specific stage and does not proceed in it from the preceding determinations — a demand which is also to be found after a fashion in the process of mathematical inference. But I must admit that such an abstract perfection of presentation must generally be renounced; the very fact that the logic must begin with the purely simple, and therefore the most general and empty, restricts it to expressions of this simple that are themselves absolutely simple, without the further addition of a single word; only allowed, as the matter at hand requires, would be negative reflections intended to ward off and keep at bay whatever the imagination or an undisciplined thinking might otherwise adventitiously bring in’.

- ‘The Science of Logic’

The heterogeneous element warded off by mathematicians is the qualitative nature of δx. One commentator views the point of the calculus discussion as follows: Calculus cannot ‘yield the mathematics of nature which Hegel was looking for. [S]uch a mathematics can only take over what is qualitative from experience, it cannot develop it out of itself’. Host-Heino Von Borzeszkowski.The calculus remarks are usually dismissed as ‘digression, at best’, by Geoffrey Reginald Gilchrist Mure, (1893–1979), and a self-indulgence, which I doubt, he is always in control of his material, but suspicions persist: ‘Suspecting Hegel of wishing in part to demonstrate his mastery of mathematics and science to contemporaries and colleagues’, said Clark Butler. Those who prefer scorn to thinking on the matter will find merit in Bertrand Russell’s, (1872–1970), dismissal:

‘Hegel (especially in his Greater Logic) made a quite different use of mathematics. [He] fastened upon the obscurities in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses. It is interesting that some of his worst absurdities in this field were repeated by Engels in the Anti-Dühring, and that, in consequence, if you live in the Soviet Union and take account of what has been done on the principles of mathematics during the last one hundred years, you run a grave risk of being liquidated’.

- ‘Logical Positivism’, in ‘Logic and Knowledge’

Mathematics is not my area of expertise so I will pass by the section on calculus, though I take comfort in the fact that I don’t know how seriously to take Russell’s assessment but other things he that he has said on other topics I know to be absurd so I will assume he is giving voice to absurdities here too. I have found few references to Hegel’s views on the calculus, albeit they seemed prescient for his day, so I cannot do much in the way of cribbing and even if I did I run the risk of exposing myself as a cribber who doesn’t understand what he is cribbing, which as it happens would be the case. Anyway the commentaries digress from the logical progression of the Logic and so analysis of Hegel’s challenging observations on calculus may safely be forborne.