On Hegel's 'Science of Logic' : A Realm of Shadows - part thirteen.

'The Swimmers (excerpt)

Louis Untermeyer?(1885?– 1977)

Then the swift plunge into the cool, green dark—

The windy waters rushing past me, through me;

Filled with a sense of some heroic lark,

Exulting in a vigor clean and roomy.

Swiftly I rose to meet the feline sea

That sprang upon me with a hundred claws,

And grappled, pulled me down and played with me.

Then, tense and breathless in the tightening pause

When one wave grows into a toppling acre,

I dived headlong into the foremost breaker;

Pitting against a cold and turbulent strife

The feverish intensity of life.

Out of the foam I lurched and rode the wave,

Swimming, hand over hand, against the wind;

I felt the sea's vain pounding, and I grinned

Knowing I was its master, not its slave.

Oh, the proud total of those lusty hours—

The give and take of rough and vigorous tussles

With happy sinews and rejoicing muscles;

The knowledge of my own miraculous powers,

Feeling the force in one small body bent

To curb and tame this towering element.

'Прыжок в воду', 'Jump Into The Water', 1932, Eleazer Langman

In Georg Wilhelm Friedrich Hegel's 'Science of Logic'?we have reached extensive and intensive Quantum. And so what is the difference? In the Number diagram, see previous article, Number's determinateness is isolated in Amount d, e, f. Whereas f is beyond Amount proper yet since Unit is just as much Amount as Amount, f can be included as part of the determinateness which Hegel designates as Amount. g is Number's Unit which can be taken as a plurality, since Amount continues right on through it. Number is nothing but Limit, five is nothing but not six or four. Accordingly, Quantum, with its limit, which Limit is in its own self a plurality, is extensive magnitude.

'We have seen that quantum has its determinateness as limit in amount. In itself quantum is discrete, a plurality which does not have a being different from its limit or its limit outside it. Quantum, thus with its limit which as limit is a plurality, is extensive magnitude'.

- 'The Science of Logic'

If 100 represents a Unit or set of 100 Units extensive Magnitude represents the 100 individuals within the Extensive Magnitude/Extensive Quantum set. Extensive Magnitude is an amount Extensive Magnitude of one and the same unit.

'Extensive magnitude is to be distinguished from continuous magnitude; its direct opposite is not the discrete, but the intensive magnitude. Extensive and intensive magnitudes are determinacies of the quantitative limit itself, whereas quantum is identical with its limit; continuous and discrete magnitudes are, on the contrary, determinations of magnitude in itself, that is, of quantity as such, in so far as in quantum abstraction is made from the limit. – Extensive magnitude has the moment of continuity in it and in its limit, for its many is everywhere continuous; the limit as negation appears, therefore, in this equality of the many as a limiting of the unity. Continuous magnitude is quantity that continues without regard to any limit, and in so far as it is represented with one such limit, the latter is a limitation in general, without discreteness being posited in it. Determined as only continuous magnitude, quantum is not yet determined for itself because the magnitude lacks the one (in which the determinateness-for-itself lies) and number. Similarly, a discrete magnitude is immediately only a differentiated many in general which, if it were to have a limit as a many, would be only an aggregate, that is, would be only indeterminately limited; in order for quantum to be determinate, the many must be concentrated into one and thereby be posited as identical with the limit. Continuous and discrete magnitude, taken as quantum in general, have each posited in it only one of the two sides by virtue of which quantum is fully determined and a number. Taken immediately, this latter is extensive quantum – the simple determinateness which is amount essentially, but the amount of one and the same unit; extensive quantum is distinguished from number only because in the latter the determinateness is explicitly posited as plurality'.

- 'The Science of Logic'

Number, extensive Quantum, in contrast, was plurality - not yet fused by the Understanding into Unit. Dialectical Reason complains that Extensive Magnitude cannot alone account for the determinateness of Number. The many as a plural asunderness or discreteness does not constitute the determinateness as such. There is an externality which constituted the ones as a plurality.

'However, the determinateness through number – how much there is of something – does not require being distinguished from how much there is of something else, as if to the determinateness of one thing belonged how much there is of it and how much there is of an other, for the determinateness of magnitude as such is a limit determined for itself, indifferent and simply self-referring; and in number this limit is posited as enclosed within the one existing for itself: the externality that it has, the reference to other, is inside it. Further, like the many in general, this many of the limit is not internally unequal but continuous; each many is what any other many is; consequently, the many as a many of existents outside one another, or as discrete, does not constitute the determinateness as such. Thus this many collapses for itself into its continuity and becomes simple unity. – Amount is but a moment of number; but, as an aggregate of numerical ones, it does not constitute the determinateness of number; on the contrary, these ones as indifferent and self-external are sublated in number whose being has turned back into itself; the externality that constituted the ones of plurality vanishes in the one as the self-reference of number'.

- 'The Science of Logic'

A counter has determined to stop counting at 100 Units. But there is inherently in Number a being-for-self that resists externality, which now vanishes in the one as a relation of number to its own self. Thus, 100 is many but it is also one. This intensive quality of Number is Intensive Magnitude or Degree. Of Intensive Magnitude (Degree), Hegel writes, the limit of quantum, which as extensive had its real determinateness in the self-external amount, passes over into simple determinateness.

Extensive Magnitude/Extensive Quantum

'The limit of quantum which, as extensive, had its existent determinateness as self-external number, thus passes over into simple determinateness. In this simple determination of limit, quantum is intensive magnitude; and the limit or the determinateness which is identical with quantum is now also posited as simple: it is degree'.

- 'The Science of Logic'

Extensive Magnitude says, I am not a unity. Unity therefore flees the precinct of Extensive Magnitude and takes sanctuary in the temple of Intensive Magnitude. Earlier, Number was Amount - a plurality made into one. Within Amount, each of the Many Ones was equal to any other. None was primus inter pares. If Amount is 100, each One could claim to be the determining 100th. intensive Magnitude Amount did not exhibit determinateness as such (Degree) (Limit), and so Amount collapsed into Unit. Degree, in contrast, is a specific magnitude. It is the 100th One, selected by the counter as the determinant. As such, it is not an aggregate or plural within itself. Rather, it is a plurality only in principle. In Degree, determinate being has returned into being-for-self.

'Degree is thus a determinate magnitude, a quantum, but at the same time it is not an aggregate or several within itself; it is only a plurality; plurality is a severality that has gathered together into simple determination, it is existence that has returned into being-for-itself. It is true that its determinateness must be expressed by a number, which is the being of the quantum as completely determined, but the number is not an amount or a how many times but is rather a onefold, only a degree. When we speak of 10 or 20 degrees, the quantum which has that many degrees – the tenth, the twentieth degree – is not the amount and sum of the degrees; if that were the case, it would be an extensive quantum; it is rather only that one degree, the tenth, the twentieth. It does contain the determinateness found in the number ten or twenty, but not as several ones: the number is there as a sublated amount, as a simple determinateness'.

- 'The Science of Logic'

'Mann auf dem Sprungbrett', 1912, Aksel Waldemar Johannessen, (1880 -1922). 'To disappear into deep water or to disappear toward a far horizon, to become part of depth of infinity, such is the destiny of man that finds its image in the destiny of water'. - 'Water and Dreams', Gaston Bachelard, (1884 - 1962).

The determinateness of Degree must be expressed by a Number. It must be, for example, the 100th One. In this expression, 100 is not Amount. It is only unitary (or a Degree). Now, a single One emerges as primus inter pares over all the other Ones. Upon what meat does this Degree feed that it hath grown so great? Degree has Being-for-self. It resists Continuity in a way that, in the earlier stages of Quantity, the Many Ones could not. But, at the same time, its content is external to itself. If it is the 100th, the 100 is outside of it. Hegel gives the example of a circle with 360 degrees. The determinateness of any one degree derives essentially from the many parts outside it.

'In the circle the unit is called degree because every part of the circle has its determinateness essentially in the many more outside it; it is determined as the unit of only a fixed amount of these units. As a mere spatial magnitude, the degree of the circle is only an ordinary number; considered as degree, it is an intensive magnitude that has meaning only as determined through the amount of degrees into which the circle is divided, just as a number derives its meaning only in the number series'.

- 'The Science of Logic'

One degree of the circle depends on its relation with the others. Justus Hartnack, (1912 - 2005), writes 'If we talk about... a room temperature of 20° C, then the degrees below 20° never formed an extensive magnitude that was absorbed in that degree of temperature'. The opposite is true. The Extensive Magnitude of 20° is precisely all the degrees that 20° excludes. Extensive Magnitude owns plurality - the externality of Degree. Multiplicity 'is interior to extensive quantum... Here number reveals the quality of quantum: determinate indifference... to is own multiplicity (within or without)', explains Andrew?Haas. Although this plurality is likewise One, Extensive Magnitude could not express this truth. It sheds unity to its outside (Degree), just as Degree sheds plurality to its outside. So Degree c and Extensive Magnitude a are doing the same thing - expelling their own content, which we can now interpret as b. Although extensive and intensive quanta are differentiated according to the ways in which they express the multiplicity that forms their other, they are identical insofar as they both are characterized by qualitative indifference.

Intensive Magnitude (Degree)

The expulsion of b yields a middle term. Number as a one excludes from itself the indifference and externality of the amount [ie., the plurality] and is self-relation as relation through itself to an externality.

'In number, quantum is posited in its complete determinateness; but as intensive quantum, in the being-for-itself of number, it is posited as it is according to its concept or implicitly in itself. For the form of self-reference which it has in degree is at the same time the externality of degree to itself. As extensive quantum, number is a numerical plurality and thus has externality inside it. This externality, as plurality in general, collapses into a state of undifferentiatedness and is sublated into the numerical one, the self-reference of number. But quantum has its determinateness as number; it contains this determinateness, as we have said, whether the latter is posited in it or not. Degree, therefore, which as internally simple no longer has this external otherness in it, has it outside it, and refers to it as to its determinateness. A plurality external to it constitutes the determinateness of the simple limit which the degree is for itself. In so far as in the extensive quantum amount was supposed to be found within number, it was sublated there; now, as thus sublated, amount is posited outside number. Number, in being posited as a one, as self-reference reflected into itself, excludes from itself the indifference and the externality of amount and is self-reference as reference to an external through itself'.

- 'The Science of Logic'

The self-erasure of Number is on display. In this middle term, quantum has a reality conformable to its Notion.

'In this, quantum has the reality which is adequate to its concept. The indifference of the determinateness constitutes its quality, that is, a determinateness which is in it as a determinateness external to itself. – Accordingly, degree is a unitary quantitative determinateness among a plurality of such intensities which, though diverse, each being only a simple reference to itself, are at the same time in essential connection with each other, so that each has its determinateness in this continuity with the others. This reference connecting a degree through itself to its other makes ascent and descent on the scale of degrees a continuous progress, a flow, which is an uninterrupted and indivisible alteration; none of the 'more or less' differentiated within it is separate from the others but each has its determinateness only in these others. As a self-referring quantitative determination, each degree is indifferent towards the others; but, in itself, it equally refers to this externality; it is what it is only through the intermediary of this externality; in short, its reference to itself is not an indifferent reference to externality but in this externality it possesses its quality'.

- 'The Science of Logic'

Reality must be taken loosely here, as it gave way to ideality in chapter two. Speculative Reason shows that the soul of Quantum is that Quantum its content is imposed from the outside, even as it has a unitary nature that resists imposition. 'The qualitative aspect of quantity means qualitative opposition, that is, having determinateness in another, by means of its non-being, having its being by virtue of its nothing', explains Haas. And what is the nature of its resistance? Quantum cannot be made to surrender its absolute openness to outside determination. '[E]xternality itself is its specific 'quality,' its 'being-in-itself'. Its quality is thus the very negation of Quality's possibility', explains Giacomo Rinaldi.

Hegel also hints at a shift in the nature of Degree's content that should be made explicit. Extensive Magnitude referred to a set of individuals entirely within the set (but not the set itself). This is what set theory calls a fusion, as opposed to a collection, which adds entity, according to Michael Potter. In our earlier example, it referred to the 100 individuals compounded together. Degree, however, does not refer just to those 100 individuals. It refers to all the other degrees in the universe, degree is a unitary quantitative determinateness among a plurality of such intensities which, though differing from each other, are essentially interrelated so that each has its determinateness in this continuity with the others. A Degree's meaning, then, depends on the infinite set, not on any specific set designated by an outside will. Continuity between Degrees makes possible an ascent and descent in the scale of degrees alteration; none of the various distinct degrees is separate from the others but each is determined only through them'.

The Quality of Quantum

The Quality of the Quantum may in general stand for the dependence of Quantum on otherness for its determinateness, but it also puts on the table the infinite set of Quanta.

Identity of Extensive and Intensive Magnitude.The last section addressed the difference between Extensive and Intensive Magnitude. Ironically, this difference was gathered together in a middle term: the very Quality of Quantum is indifference to Quality, to content. Now we explore this Quality-as-indifference and discover the self-identity lurking within The Quality of Quantum. Our next move is the positive version of Intensive Magnitude, according to which Degree is not external to itself within itself.

'Degree is not inherently external to itself. It is not, however, the indeterminate one, is not the principle of number as such which is not amount except negatively, that is, only in the sense of not being an amount. Intensive magnitude is at first a simple one of many 'more or less'; there are several degrees; but they are not determined either as a unitary one or as a more or less but only as referring to each other as outside each other, or in the identity of the one and the 'more or less than'. Thus, although the several 'more or less' are as such indeed outside the unitary degree, the determinateness of the latter lies nonetheless in its connection with them; the degree thus contains amount. Just as twenty contains as extensive magnitude twenty ones as discrete, the specific degree contains them as continuity, a continuity which simply is this determinate plurality; it is the twentieth degree, and it is this twentieth degree only through the intermediate of this amount which, as such, is outside it'.

- 'The Science of Logic'

If Degree a sheds its content b, c, it equally retains what it sheds. Unlike Number, which has Amount in general, Degree has a particular amount. Degree sheds all the other degrees, with which it is continuous. Thus, 20° sheds all the other degrees, even while it retains for itself the 20 - which uniquely distinguishes 20° from all others. These excluded degrees can be called, collectively, extensive Magnitude (or extensive Quantum) - this time taken negatively. Degree implies 'a reference to a numerical series, to a quantitative manifold; but this latter is posited 'outside' its self-identity ... whereas in the case of Number it was posited 'within' in', explains Haas.

Intensive Magnitude (Degree)

Thanks to the analysis of the Quality of Quantum, we can see that extensive and intensive magnitude are thus one and the same determinateness of quantum; they are only distinguished by the one having amount within itself and the other having amount outside itself.

'Extensive and intensive magnitude are, therefore, one and the same determinateness of quantum; they are distinguished only inasmuch as the one has the amount within and the other has the same without. Extensive magnitude passes over into intensive magnitude because its many collapses in and for itself into oneness and steps outside it. But, conversely, this simple one has its determinateness only in the amount, its amount; indifferent to the otherwise determined intensities, it has the externality of amount in it; thus intensive magnitude is just as essentially extensive magnitude'.

- 'The Science of Logic'

Similarly, we previously saw that Unit and Amount were the same - also Continuous and Discrete Magnitude. Throughout Quantity, the extremes end up being each other - here Intensive and Extensive Magnitude expressly have swapped places. The middle term of the obversely charged extremes is the Qualitative Something. This unity is self-related through the negation of its differences.

'With this identity, the qualitative something comes on the scene; for the identity is the unity that refers back to itself through the negation of its distinct terms; these terms, however, make up the determinateness of the existent magnitude. The something is a quantum, but its qualitative existence is now posited as indifferent to it as it is in itself. One can speak of quantum, number as such, etc., without any mention of a something as their substrate. But the something, self-mediated by virtue of the negation of its determinations, now confronts these as existing for itself, and, since it has a quantum, it confronts them as something which has an extensive and intensive quantum. Its one determinateness which it has as quantum is posited in the distinct moments of unity and amount; it is in itself not only one and the same determinateness, but the positing of it in these differences as extensive and intensive quantum is the return into this unity which, as negative, is the something posited as indifferent to them'.

- 'The Science of Logic'

This is the standard move of Speculative Reason, as developed in and after the True Infinite. It names the very act of the extremes in erasing themselves and stating what they are not. Something appeared in Something - the unity between Quality and Negation. Furthermore, Quality has long been sublated. Why does Hegel use the phrase qualitative something? Why not the quantitative something? Quality was supposed to have stood for independence from otherness. Degree (in both its ummediated and mediated forms) still has its content outside itself. The Qualitative Something is precisely that content - but taken negatively as simply the opposite of Quantum.

Degree depends on that Qualitative Something to define what it is. 'The contradiction of Quantum is that its internal determination rests in a limit which in its very nature posits an external other, on which the precise magnitude of the quantum is as much dependent as it is on what precedes the limit', explains Errol?Eustace?Harris, (1908 – 2009). The Qualitative Something should be taken as the infinite set of Numbers. Degree depends on negating some of these so that it can be some specific Amount. But whatever Amount Degree takes on, the Qualitative Something remains what it is. Such a Something is indifferent to its quantitative limit in Degree. Degree can be increased or decreased without affecting the Something (since the Something is the infinite set of Numbers). That makes the Something qualitative, even though made up of Quanta. The Qualitative Something is therefore substrate to the more primitive quanta.

Extensive Magnitude

Quantum, number as such, and so forth could be spoken of without any mention of its having a something as substrate. But the something now confronts its determinations, through the negation of which it is mediated with itself, as existing for itself and, since it has a quantum, as something which has an extensive and an intensive quantum. As Quantum, the Qualitative Something is subsistent; the original Something of Something was not. Quantum is, after all, a True Infinite. It stays what it is as it becomes something different. In the Qualitative Something, we have a Something that positively resists the transgression of its Limit - which the original Something could not achieve. 'Quantum is infinite self-negation, production, creation, self-expulsion, a coming-out of itself as increase/decrease, that is, the moving, growing, living limit that determines quantity insofar as it infinitely supersedes, determines and takes care of itself)', explains Haas.

Qualitative Something

And so to alteration of Quantum. Middle terms have proved to be names for activities. Accordingly, the Qualitative Something names the inability of Extensive and Intensive Magnitude to stand on their own. These self-erase and transfer their being to other Quanta - to the infinite set of Quanta on which Number's meaning is dependent. The Qualitative Something is contradictory. It is posited as being the simple, self-related determinateness which is the negation of itself, having its determinateness not within itself but in another quantum.

'The difference between extensive and intensive quantum is indifferent to the determinateness of quantum as quantum. But quantum is in general determinateness posited as sublated: the indifferent limit, the determinateness which is just as much the negation of itself. In extensive quantum, this difference is developed; but intensive magnitude is the existence of this externality which quantum is in itself. The difference is posited as the contradiction which it is in itself, of being a simple self-referring determinateness which is the negation of itself, of having its determinateness not in it but in another quantum'.

- 'The Science of Logic'

In other words, if the Qualitative Something is going to express itself, it must express itself in some Quantum. After all, it is the infinite set of Quanta. It has its entire determinateness outside of itself and in these Quanta. If the Qualitative Something has its determinateness outside of itself, then it is in absolute continuity with its externality, with its otherness.

'A quantum, according to its quality, is therefore in absolute continuity with its externality, with its otherness. Consequently, not only can every determinateness of magnitude be transcended, not only can it be altered: that it must alter is now posited. The determination of magnitude continues into its otherness in such a way that it has its being only in this continuity with an other; it is not just a limit that exists but one that becomes.

- 'The Science of Logic'

From this perspective (while admitting that the Qualitative Something is immune from other quanta), the Qualitative Something can both transcend every quantitative determinateness and be altered. In fact, Hegel says its external aspect must alter. More precisely, any Quantum that is proposed to be the final member of the infinite set can be displaced by a larger Quantum. The identity of this final member is therefore bound to alter.

In the Qualitative Something, Quantum reveals its express character of impelling itself beyond itself into its external guise.

'The one is infinite or self-referring negation, and hence the repulsion of itself from itself. Quantum is equally infinite, posited as the self-referring negation; it repels itself from itself. But it is a determinate 'one', the one which has passed over into existence and limit, thus the repulsion of determinateness from itself, not the generation of something that is like itself (as the repulsion of the one) but of its otherness; quantum is now posited in it as sending itself beyond itself. It consists in this, that it increases or decreases; it is within it the externality of determinateness'.

- 'The Science of Logic'

That is, the Qualitative Something is quantitative determinateness. It consists in undergoing increase or decrease: The quantitative determinateness continues itself into its otherness in such a manner that the determination has its being only in this continuity with an other; it is not a simply affirmative limit, but a limit which becomes.When Quantum impels itself beyond itself, it becomes another Quantum. But this new Quantum is a limit which does not stay.

'Thus quantum sends itself beyond itself; this other which it becomes is at first itself a quantum, but a quantum which is not a static limit but one that impels itself beyond itself. The limit which arises in this beyond is therefore only one that again sublates itself and sends itself to a further limit, and so on to infinity'.

- 'The Science of Logic'

The new Quantum becomes yet another Quantum, and so on to infinity.

Alan L. T. Paterson complains that the successor function and induction principle are not accounted for in Hegel's logic. But are not these very ideas the stuff of the Qualitative Something? The successor function is 'the function that sends a natural number to its successor. The induction principle (roughly) says that we can assert a fact about all natural numbers if we know that it is true for the first one (0) and that its truth is preserved when we go from a number to its successor'. Does Hegel have an account of ordinality in his theory of number? Paterson suggests the answer is no, and he may be right. Perhaps the concept of 'more or less' is simply supplied by external reflection of the counter. As Hegel explains elsewhere:

'It has been shown that the quantum as limit of the many is indeterminate in itself and how this external accidental determination becomes a determinacy of the Thing itself through its annihilation as a quantum in differential calculus. Moreover, this is just what will be necessary with respect to things as systems of moments; in other words, the opposition of the moments is not to be considered as this external, quantitative opposition but as opposition as it is in itself-that is, as qualitative opposition, or as determinacy. The quantitative differentiation of the moment of a thing does not affect the concept of the moment or the concept of the thing; but the thing is only the system of its moments, and these only are what they are in relation to each other, and the thing itself is this relation; in that the singular moment changes, it changes its relationship to the others; the whole relation, the thing itself, becomes something else. And it is in truth not a change of the moment that takes place; rather, the life cycle of the Thing itself is expressed since the moment is not on its own account, and its change is wholly determined solely by its relationship, by the being it has in the Thing itself. But this concept of the moment is precisely what the diversity as quantitative does not affect: the determinacy as it is within the Thing itself, or as it is in itself. And the rise and fall on the ladder of degree or of extensive magnitude is only to be regarded as an external indicator. The differentiation of the internal ratio turns the differentiation of the quantum as one of that sort into something quite other than [what] it expresses'.

- 'Jena Logic'

And the rise and fall on the ladder of degree or of extensive magnitude is only to be regarded as an external indicator. This is not to say that externally arranged numbers have no unique qualities. In the Jena System, Hegel gives as an example the fact that 8 = 23 and 9 = 32. By adding one, we travel from 23 to 32. But the addition of one to some number other than 8 reveals no such quality.

Be that as it may the ground is laid to proceed into the river of even murkier waters which is to say Hegel's account of Quantitative Infinity in which one may make it to the other bank refreshed and invigorated or without the required stamina and capacity one may drown.

'Swimmer', 1932, Pablo Picasso. 'I have heard the mermaids singing, each to each./ I do not think that they will sing to me. / I have seen them riding seaward on the waves/ Combing the white hair of the waves blown back/ When the wind blows the water white and black./ We have lingered in the chambers of the sea/ By sea-girls wreathed with seaweed red and brown/ Till human voices wake us, and we drown'. - 'The Love Song of J.?Alfred Prufrock', T. S. Eliot, (1888 – 1965).

_____________________________

'I’m fixed on you, I revel in you'

by Giovanni Francesco Busenello?(1598 – 1659)

I’m fixed on you, I revel in you,

I cling to you, I’m bound to you;

I’m no longer in pain, no longer dying,

oh, my life, oh, my treasure!

I am yours, yours am I,

my hope, admit it, speak:

'You are also my idol'.

Yes, my idol, yes, my heart,

my life, yes!?

Pur ti miro, pur ti godo,

pur ti stringo,?pur t'annodo ;

più non peno, più non moro,

o mia vita, o mio tesoro !

Io son tua, tuo son io,

speme mia, dillo, di' :

'Tu sei pur l'idolo mio'.

Sì, mio ben, sì mio cor,

mia vita, sì !

Dedicated to the One for whom my affections are an intensive magnitude.

'She openeth her mouth with wisdom; and in her tongue?is?the law of kindness'.

- 'Proverbs' 31:26

Coming up next:

Quantitative Infinity.

To be continued ...