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

'For each ecstatic instant'

by Emily Dickinson (1830 - 1886)

For each ecstatic instant

We must an anguish pay

In keen and quivering ratio

To the ecstasy.

?

For each beloved hour

Sharp pittances of years,

Bitter contested farthings

And coffers heaped with tears.

We have now arrived at Ratio or the Quantitative Relation in Georg Wilhelm Friedrich?Hegel's, (1770 - 1831), 'Science of Logic'. Quantum is an infinite being that changes quantitatively while remaining what it is qualitatively. 'A being which is immediately identical with its respective quality such as to remain the same throughout all its qualitative transformations, is no longer qualitatively but quantitatively determined', explains Herbert Marcuse,?(1898 – 1979). And therefore the infinity of Quantum has been determined to the stage where it is the negative beyond of quantum, and yet which beyond is incorporated within the quantum itself and this beyond is the qualitative moment as such.

'The infinity of quantum has been determined up to the point where it is the negative beyond of quantum, a beyond which quantum, however, has within it. This beyond is the qualitative moment in general. The infinite quantum, as the unity of the two moments, of the quantitative and the qualitative determinateness, is in the first instance ratio'.

- 'The Science of Logic'

At this stage, Quantum is a unity of the qualitative and the quantitative and Quantum in its advanced stage is relation (Verh?ltnis), which is translated as Ratio, although interestingly Beziehung is used for a normal and official?romantic?relationship while?Verh?ltnis?is used for an?affair not that I suppose Hegel was wanting to suggest any such connotation but I like to play with the idea anyway.The relation or Ratio in question is Quality and Quantity and Ratio is the contradiction of externality and self-relation, of the affirmative being of quanta and their negation.

'Ratio in general is:

1. direct ratio. In this, the qualitative moment does not yet emerge explicitly as such; in no other way except still as quantum is quantum posited as having its determinateness in its externality. – In itself, however, the quantitative relation is the contradiction of externality and self-reference, the persistence of quanta and their negation. Such a contradiction next sublates itself:

2. first inasmuch as in indirect or inverse ratio the negation of each of the quanta is as such co-posited in the alteration of the other, and the variability of the direct ratio is itself posited;

3. but in the ratio of powers, the unity, which in its difference refers back to itself, proves to be a simple self-production of the quantum; this qualitative moment itself, finally posited in a simple determination and as identical with the quantum, becomes measure'.

- 'The Science of Logic'

'Its distinct feature is that it is qualitatively determined as simply related to its beyond. In ratio, quantum no longer has a merely indifferent determinateness but is qualitatively determined as simply referring to its beyond. It continues in its beyond, and this beyond is at first just an other quantum. Essentially, however, the two do not refer to each other as external quanta but each rather possesses its determinateness in this reference to the other. In this, in their otherness, they have thus returned into themselves; what each is, that it is in its other; the other constitutes the determinateness of each. – The quantum’s self-transcendence does not now mean, therefore, that quantum has simply changed either into some other or into its abstract other, into its abstract beyond, but that there, in the other, it has attained its determinateness; in its other, which is an other quantum, it finds itself. The quality of quantum, its conceptual determinateness, is its externality as such, and in ratio quantum is now posited as having its determinateness in this externality, in another quantum – as being in its beyond what it is'.

- 'The Science of Logic'

Quantum is continuous with its beyond and the beyond is another Quantum but the relation between Quanta is externally imposed no longer, and these related Quanta have recaptured an integrity that more primitive Quanta were lacking, and as for what this integrity is it is that they have no integrity for in becoming other these Quanta show their true selves, they are as much other as they are themselves. Quantum's flight away from and beyond itself hence has now this meaning, that it transmogrified not merely into an other, or into its abstract other, into its negative beyond, but that in this other it reached its determinateness, discovering itself in its beyond, which is another quantum. Hegel is here implying that Quantum cannot distinguish itself without the aid of the Other hence the Other is as much the stuff of self as it is Other and further hence in distinguishing the other Quantum discovers itself. In this passage Hegel appears to be echoing the Lordship/Bondage, (Herrschaft und Knechtschaft), dialectic in the 'Phenomenology of Spirit' whereby in that dialectic two self-consciousnesses become locked in a life and death struggle. One succeeds and becomes the lord, the other the bondsman (often mistranslated as master and slave. But the master discovers that the other is truly himself. The master is thus reduced to dependency. Likewise, in ratio, Quantum attempts to distinguish itself by expelling the Other, only to find that the Other is as much itself as itself is. Errol?Eustace?Harris,?(1908 – 2009), calls 'The Quantitative Relation or Qualitative Ratio' a chapter that is 'more technical than philosophical' but he thereby downplays its significance, besides which, the Ratio of Powers, with which the chapter ends, is a very clear and cogent demonstration of the qualitative moment in the heart of Quantity.

'Self-consciousness exists in and for itself when, and by the fact that, it so exists for another; that is, it exists only in being acknowledged. The Notion of this its unity in its duplication embraces many and varied meanings. Its moments, then, must on the one hand be held strictly apart, and on the other hand must in this differentiation at the same time also be taken and known as not distinct, or in their opposite significance. The twofold significance of the distinct moments has in the nature of self-consciousness to be infinite, or directly the opposite of the determinateness in which it is posited. The detailed exposition of the Notion of this spiritual unity in its duplication will present us with the process of Recognition'.

.............

'In this experience, self-consciousness learns that life is as essential to it as pure self-consciousness. In immediate self-consciousness the simple 'I' is absolute mediation, and has as its essential moment lasting independence. The dissolution of that simple unity is the result of the first experience; through this there is posited a pure self-consciousness, and a consciousness which is not purely for itself but for another, i.e. is a merely immediate consciousness, or consciousness in the form of thinghood. Both moments are essential. Since to begin with they are unequal and opposed, and their reflection into a unity has not yet been achieved, they exist as two opposed shapes of consciousness; one is the independent consciousness whose essential nature is to be for itself, the other is the dependent consciousness whose essential nature is simply to live or to be for another. The former is lord, the other is bondsman'.

- 'Phenomenology of Spirit'

The quality of Quantum, then, is its externality as such and what is at stake here is not just one Quantum and its beyond, another Quantum, but the relation between these two quanta. Quantum is not only in a Ratio but it is itself posited as a ratio.

'It is quanta that stand to each other in the connection that has now come on the scene. This connection is itself also a magnitude; quantum is not only in relation, but is itself posited as relation; it is a quantum as such that has that qualitative determinateness in itself. So, as relation (as ratio), quantum gives expression to itself as self-enclosed totality and to its indifference to limit by containing the externality of its being-determined in itself: in this externality it is only referred back to itself and is thus infinite within'.

- 'The Science of Logic'

Each extreme therefore has to be taken as a singularity and also as a mediation and the extremes have grown concrete.

Hegel splits into three his chapter on Ratio chapter. First there is Direct Ratio (A/B = C) whereby the qualitative moment is not yet explicit but rather shows the retrogressive mode of having its externality outside itself. Direct Ratio manifests all the defects of the Understanding. Second there is Indirect Ratio, or Inverse Ratio (AB = C), and here, where dialectical Reason holds forth, modulation occurs between the quanta as they negate each other. Third there is the Ratio of Powers (A to the power of 2 = C) and here quantum (A) reproduces itself. When this middle term is posited as a simple determination Measure is arrived at, the unity of Quantity and Quality. At this moment the rightward inclining chapters of Quantity give ground to the centrally inclining chapters of Measure.

The culmination of this chapter, then, is the Ratio of Powers, A to the power of 2 = C and the middle term however is a definition of the Absolute and every proposition of the Understanding and Speculative Reason is a vision of the Absolute. Dialectical Reason in contrast is purely a critique of the Understanding's proposition so we can say that the universe (C) is A to the power of 2, if A that is stands for some thing or Unit. This chapter on Quantitative Relation in effect argues that all things define all other things even while remaining a thing-in-itself and hence a universe of profoundly contextual metonymic things is being delineated.

'Proportions of the Head and a Standing Nude', c. 1488 - 1489, Leonardo da Vinci

Terry Pinkard, (1947 - ), is dismissive with regard to the ratio of powers lamenting that it is 'so idiosyncratic to Hegel's system that it offers little insight to anyone who has not accepted the entire Hegelian outlook - lock, stock and barrel. It is not one of the things that even Hegelians have seen fit to develop, and there is good reason for this lack of interest'. Such criticism disregards the fact that if Hegel is developing a theory of metonymic meaning then this chapter substrate to the concept of Measure should not of necessity be expected to deliver useful benefits in the service of common sense and this chapter of intermediate theorems does indeed further develop what Quality and even freedom are.

Which brings us to the Direct Ratio. In Direct Ratio Direct Ratio is immediate. An infinite being, it has qualitative being-for-self. By way of illustration take 2/7 = C. Hegel calls C the exponent albeit mathematicians would say of C that it is a quotient where with A to the power of 3 = C, 3 is the exponent. What Hegel means by exponent though is simply the relation between two quanta in a ratio and therefore the exponent simply as product is implicitly the unity of Unit and Amount.

'But the determinations that have emerged and which we now have to sum up are not only that this infinite beyond is at the same time some present finite quantum or other, but that its fixity – which makes it with respect to the quantitative moment the infinite beyond that it is, and which is the qualitative moment of being only as abstract self-reference – has developed itself as a mediation of itself with itself in its other, the finite moments of the relation. The general point is that the whole is as such the limit of the reciprocal limiting of the two terms, and that the negation of the negation (and consequently infinity, the affirmative self-relation) is therefore posited. The more particular point is that, as product, the exponent already implicitly is already the unity of unit and amount, whereas each of these two terms is only one of two moments, and for this reason the exponent encloses them in itself and in them it implicitly refers itself to itself'.

- 'The Science of Logic'

In 2/7 = C, C's value survives the increase of 2 and 7. For instance, 4/14 = C. For this reason 2 and 7 have a direct and not an inverse relation. This example demonstrates that C has a being separate and independent from the sides of the ratio, and furthermore,2 and 7 have nothing to do with each other. They were combined by the will of the Direct Ratio mathematician and given C ≠ 0, any pair of numbers could have been selected to represent C. Zero, of course, is not a quantum but is the negation of quantum.

Direct Ratio

Yet Direct Ratio is also a relation of quanta and these quanta are other to the Direct Ratio and so the determinateness of the ratio lies in an other. As a Quantum, C, in C = A/B, is the unity of Unit and Amount, per the law of sublation. Unit stands for Being-for-self and Amount stands for the indifferent fluctuation of the determinateness, the external indifference of quantum.

'The exponent is some quantum or other; however, in referring itself to itself in the otherness which it has within it, it is only a qualitatively determined quantum, for its difference, its beyond and otherness, is in it. This difference in the quantum is the difference of unit and amount – the unit, which is the being-determined-for-itself; the amount, which is the indifferent fluctuation of determinateness, the external indifference of quantum. Unit and amount were at first the moments of quantum; now, in the ratio, in quantum as realized so far, each of its moments appears as a quantum on its own and as determinations of the existence of the quantum, as delimitations against the otherwise external, indifferent determinateness of magnitude'.

- 'The Science of Logic'

Earlier Amount and Unit were moments of Number and hence Hegel refers to the sides of Direct Ratio as less than C, the exponent that determines their being. The integrity of the Direct Ratio thus implicates the a dependence of the sides for Direct Ratio is the Understanding's interpretation of the Infinite Great/Small as portrayed in the Quantitative Infinity diagram. The infinitely Small (δx) was unnameable and when δy is part of a Direct Ratio (C = δy/δx), C is perfectly determinate yet C = δy/δx 'is not a real equation it is rather an incomplete expression in which [G] designates only a ratio of which we do not yet know that other ratio it is equal to', as Alain Lacroix, (1948 - ), explains. The sides of the ratio do not complete the exponent yet the sides of the ratio as in 2/7 = C, can be Numbers on their own. Therefore an infinite regress confronts us and every Number is in turn an Amount and Unit which are in turn Numbers. The sides like δx and δy are never entirely present and Direct Ratio then has its qualitative and quantitative moments. For this reason it is a simple determinateness, a paradox, in virtue of determinatenesses being complex.

'The exponent is this difference as simple determinateness, that is, it has the meaning of both determinations immediately in it. First, it is a quantum and thus an amount. If the one side of the ratio which is taken as unit is expressed in a numerical one, and has only the value of one, then the other, the amount, is the quantum of the exponent itself. Second, it is simple determinateness as the qualitative moment of the sides of the ratio. When the quantum of the one side is determined, the other is also determined by the exponent and it is a matter of total indifference how the first is determined; it no longer has any meaning as a quantum determined for itself but can just as well be any other quantum without thereby altering the determinateness of the ratio, which rests solely on the exponent. The one side which is taken as unit always remains unit however great it becomes, and the other, however great it too thereby becomes, must remain the same amount of that unit'.

- 'The Science of Logic'

Hegel refers to this incompleteness of the sides as their negation and what this means is that the sides of the Ratio are no longer independent, only the exponent of the Ratio lays claim to quality. The sides of the Ratio are the negative to that quality, they embody quantitative difference, but if the sides are incomplete this does not mean that the exponent is complete. Here Hegel calls exponent quotient, C can be reduced to Unit or Amount which means that it too is incomplete. If A/B = C, then A = BC. C can taker the place of A easily enough and upon doing so it becomes one of the sides, incomplete. Hence C is not posited as what it ought to be, the ratio's qualitative unity.

'The exponent ought to be the complete quantum, since the determinations of both sides come together in it; but in fact, even as quotient the exponent only has the value of amount, or of unit. There is nothing available for determining which of the two sides of the relation would have to be taken as the unit or as the amount; if one side, quantum B, is measured against quantum A as unit, then the quotient C is the amount of such units; but if A is itself taken as amount, the quotient C is the unit which is required by the amount A for the quantum B. As exponent, therefore, this quotient is not posited for what it ought to be, namely the determinant of the ratio, or the ratio’s qualitative unity. It is posited as such only to the extent that its value is that of the unity of the two moments, of unit and amount. And since these two sides, as quanta, are indeed present as they should be in the explicated quantum, in the ratio, but at the same time have the value, which is specific to them as the sides of the ratio, of being incomplete quanta and of counting only as one of those qualitative moments, they are to be posited with this negation qualifying them. Thus there arises a more real ratio, one more in accordance with its definition, one in which the exponent has the meaning of the product of the sides. In this determinateness, it is the inverse ratio'.

- 'The Science of Logic'

And so to Inverse Ratio. If Direct Ratio emphasizes Ratio's immediacy, Inverse Ratio emphasizes its incompleteness, which Hegel designates a sign of negativity. Direct Ratio's fault was its failure to be immediate and immune from outside manipulation, an external reflection had to determine whether C was exponent or one of the subordinate sides, therefore in C = A/B, C is exponent. But it is Unit/Amount in A = BC. In the Inverse Ratio, the exponent is some fixed Quantum and the exponent does not shift over to the other side of the equal sign. Apparently we are not to multiply A = BC by 1/C, which would disclose the exponent to be no different from the Unit/Amount, rather we are to consider A as fixed.

Assuredly it is strange that the fault of Direct Ratio, its openness to outside manipulation, becomes the virtue of Inverse Ratio which depends upon outside fixity to differentiate it from Direct Ratio. Yet remember that the Quality of the Quantum is precisely Quantum's openness to outside manipulation which is to say the integrity of Quantum is its lack of integrity. That is what the fixity of the exponent represents, when C = A/B, A and B had a direct relationship, if C remained constant and A is increased, B also increased. As it happens A = BC. If the exponent stays fixed B and C are in an inverse relationship and if B increases, C falls in value. Indeed, B and C can fluctuate wildly and fluctuation is their distinctive character in contrast to the qualitative moment as a fixed limit; they have the character of variable magnitudes, for which the said fixed limit is an infinite beyond.

'With this, however, we have the transition of the inverse ratio into a determination other than the one it had at first. This consisted in the fact that the quantum is immediate but at the same time so connected to an other that the greater it is, the smaller is the other, that it is what it is by virtue of negatively relating to the other; also, a third magnitude is the common restriction on this alteration in magnitude that the two quanta undergo. This reciprocal alteration, as contrasted to the fixed qualitative limit, is here their distinctive property; they have the determination of alterable magnitudes for which that fixed limit is an infinite beyond'.

- 'The Science of Logic'

And yet there is a limit to this inverse relationship and whereas the mathematician can bring it about that B falls and C rises the mathematician cannot compel B to zero. Otherwise the exponent which is supposed to be fixed is annihilated. Geometrically, B and C are in a hyperbolic relation and as one approaches zero, the other approaches an infinitely large number. This resistance of B (or C) to the mathematician's will is an indication that Inverse Ratio has recaptured its Quality. Speculative Reason seizes upon this resistance of B and C and in this refusal of B and C to go to zero, they are equal qualities, immune from outside manipulation at least to this one small extent. Since B = C is a qualitative matter, the in-itself of A = BC is A = BB, or A = B to the power of 2, the Ratio of Powers. Here the first B determines the value of the second B and the second reciprocally determines the value of the first. If A = 16 the mathematician has no choice but to concede that B = {4, -4}.

'Allegory of the liberal arts in the time of the Thirty Years' War', 1636, Bartholomeus Strobel?the Younger?

Each side of the Inverse Ratio, then, limits the other and is simultaneously limited by it. Yet once the side of the ratio achieves its in-itself, its potential, it establishes its independence from the other side. The other magnitude becomes zero. It vanishes.

'This continuity of each in the other constitutes the moment of unity through which the two magnitudes stand in relation – the moment of the one determinateness, of the simple limit which is the exponent. This unity, the whole, constitutes the in-itself of each from which their given magnitude is distinct: this is the magnitude according to which each is only to the extent that it takes from the other a part of their common in-itself which is the whole. But each can take from the other only as much as will make it equal to this in-itself; it has its maximum in the exponent which, in accordance with the stated second determination, is the limit of the reciprocal delimitation. And since each is a moment of the ratio only to the extent that it limits the other and is thereby limited by it, it loses this, its determination, by making itself equal to its in-itself; in this loss, the other magnitude will not only become a zero, but itself vanishes, for what it ought to be is not just a quantum but what it is as such a quantum, namely only the moment of a ratio. Thus each side is the contradiction between the determination as its in-itself, that is, as the unity of the whole which is the exponent, and the determination as the moment of a ratio; this contradiction is infinity again, in a new form peculiar to it'.

- 'The Science of Logic'

Clearly this last point cannot be taken mathematically. If one side (B) of the product (BE) is zero, the other side (also E) must likewise be zero, and B to the power of 2 is no longer equal to A > 0. Rather, the point is that the first B enjoys Being-for-self. It is indifferent to the second B, for whom it is a nothing, a void. But since B = B qualitatively, both B's are zero, they erase themselves and remove their being to a middle term.

Inverse Ratio

Hegel summarizes, the Inverse Ratio as the whole, as exponent, is the limit of the reciprocal limiting of both terms and is therefore posited as negation of the negation, hence as infinity, as an affirmative relation to itself. Therefore the exponent is limit to the sides and the sides are likewise limit to the exponent and the negation of the negation is precisely the refusal of either side to disappear and become zero. A limit is now located in the sides of the ratio, and these sides negate the superiority of the exponent, previously incomplete, the sides now speak for themselves as to what they are within the confines of the externally fixed exponent. Geoffrey Reginald Gilchrist?Mure's,?(1893 – 1979), however offers a dissenting analysis of Direct and Indirect Ratio the sum total of which is as follows: 'In Direct Ratio... the two quanta unified in the constant exponent increase or diminish together. In Indirect Ratio they vary inversely and so in closer relation'. Yet the two sides of the Indirect Ratio are not in closer relation and the function of the Indirect Ratio is to emphasize the qualitative and not the quantitative difference between either side and the exponent.

Of the sides Hegel presents two points. Ratio has an affirmative aspect, immunity or fixity in general, yet, because each side of the ratio cannot be raised to equality with the exponent, each side is, in a sense, fixed. This fixity. the refusal of B or C to equal A means that each side is implicitly the whole of the exponent and Hegel explains that B and C can increase and decrease relatively to each other but they cannot become equal to the exponent. As thus the limit of their reciprocal limiting, the exponent is their beyond to which they infinitely approximate but which they cannot reach.

'The exponent is the limit of the sides of its ratio, within which limit the sides increase and decrease proportionately to each other; but they cannot become equal to this exponent because of the latter’s affirmative determinateness as quantum. Thus, as the limit of their reciprocal limiting, the exponent is (a) their beyond which they infinitely approximate but can never attain. This infinity in which the sides approximate their beyond is the bad infinity of the infinite progression; it is an infinity which is itself finite, that finds its restriction in its opposite, in the finitude of each side and of the exponent itself, and for this reason is only approximation. But (b) the bad infinite is equally posited here as what it is in truth, namely the negative moment in general, in accordance with which the exponent is the simple limit as against the distinct quanta of the ratio: it is the initself to which, as the absolutely alterable, the finitude of the quanta is referred, but which, as the quanta’s negation, remains absolutely different from them. This infinite, which the quanta can only approximate, is then equally found affirmatively present on their side: the simple quantum of the exponent. In it is attained the beyond with which the sides of the ratio are burdened; it is in itself the unity of the two or, consequently, in itself the other side of each side; for each side has only as much value as the other does not have; its whole determinateness thus rests in the other, and this, their being-in-itself, is as affirmative infinity simply the exponent'.

- 'The Science of Logic'

Giacomo Rinaldi asks if A = 16 and B = 1 does not A = C contrary to what Hegel asserts? The replyr is that equality cannot mean numerical equality but rather Hegelis thinking of a qualitative inequality. C cannot repair itself in the way that A can, it can become infinitely large as B becomes infinitely small but it can never be for itself, like the exponent. This fixity, the refusal of B or C to equal A means that each side is implicitly the whole of the exponent and this equality of a given side of the ratio with the exponent supports Mure's contention: 'In Ratio of Powers, where one [i.e., the exponent] is a higher power of the other [i.e., a side of the ratio], they relate, if I follow Hegel, so closely that they are fully equivalent to the exponent, and the total expression is true infinity'. A True Infinite becomes other and remains the same hence the sides become the exponent in the sense that each is fixed. Here fixity stands for quality here yet the same refusal is a negative moment. B and C are constituted by a spurious Infinite that frustrates the mathematician who endeavours to make them equal to the exponent and this resistance to manipulation is the negation of the self-externality of the exponent. This very resistance is the resultant middle term, which is posited as preserving itself and uniting with itself in the negation of the indifferent existence of the quanta, thereby being the determinant of its self-external otherness. Andrew Haas points to Inverse Ratio as evidence that Hegel's Logic does not have a triple structure but contends. He writes, 'in the 'Inverse Ratio ... the second moment has only two sub-moments'. But this overlooks the triadic structure of the chapter as a whole whereby the three moments are Direct, Inverse and Power Ratios.

'Vitruvian Man', 1492, Leonardo da Vinci

And so to the Ratio of Powers. The Ratio of Powers (B2 = 16, for instance) is shown in Ratio of Powers. It is a quantum which, in its otherness, is identical with itself and which determines the beyond of itself.

'Quantum, in positing itself as self-identical in its otherness and in determining its own movement of self-surpassing, has come to be a being-for-itself. As such a qualitative totality, in positing itself as developed, it has for its moments the conceptual determinations of number: the unit and the amount. This last, amount, is in the inverse ratio still an aggregate which is not determined as amount by the unit itself but from elsewhere, by a third determinate aggregate; but now it is posited as determined only by the unit. This is the case in the ratio of powers where the unit, which in it is amount, is at the same time the amount as against itself as unit. The otherness, the amount of units, is the unit itself. The power is an aggregate of units, each of which is this aggregate itself. The quantum, as indifferent determinateness, changes; but inasmuch as the alteration is the raising to a power, the otherness of the quantum is determined purely by itself. – The quantum is thus posited in the power as having returned into itself; it is immediately itself and also its otherness'.

- 'The Science of Logic'

That is to say, given the requirement of B to the power of 2 and the exponent of 16, the one B determines itself and its other and at this point Quantum has reached the stage of being-for-self, Quantum is posited as returned to itself. At an earlier stage we could never tell whether B or C was Unit or Amount whereas now B = B so that Unit is Amount. And for this reason and also because Unit stands for Quality the Ratio of Powers is posited as determined only by the unit. The quantum (B) may undergo alteration, separate and apart from the Ratio of Powers, but in so far as this alteration is a raising to a power this its otherness is limited purely by itself.

Ratio of Powers

Hegel refers to the Ratio of Powers as qualitative yet external which is an apparent contradiction, where the exponent is fixed the variable B is determined by the other B. Hence, its determinateness is external but it is equally internal, this externality is now posited in conformity with the Notion of quantum, as the latter's own self-determining, as its relation to its own self, as its quality and in so far as the externality or indifference of its determining counts the Ratio of Powers is still Quantum.

'The ratio of powers appears at first as an external alteration to which a given quantum is subjected; but it has a closer connection with the concept of quantum, namely, that in the existence into which the quantum has developed in the ratio of powers, quantum has attained that concept, has realized its concept to the fullest. The ratio of powers is the display of what the quantum is implicitly in itself; it expresses its determinateness of quantum or the quality by which it is distinguished from another. Quantum is indifferent determinateness posited as sublated, that is to say, determinateness as limit, one which is just as much no determinateness, which continues in its otherness and in it, therefore, remains identical with itself. Thus is quantum posited in the ratio of powers: its otherness,the surpassing of itself in another quantum, as determined through the quantum itself'.

- 'The Science of Logic'

At this moment it is posited simply or immediately. But also at this moment it has become the other of itself, namely, quality. In going outside itself, Quantum stays within itself, so that in this very externality quantum is self-related and hence is being as quality.

'By being thus posited as it is in conformity to its concept, quantum has passed over into another determination; or, as we can also say, its determination is now also as the determinateness, the in-itself also as existence. It is quantum in so far as the externality or the indifference of its determining (as we say, it is that which can be increased or decreased) is simply accepted and immediately posited; it has become the other of itself, namely quality, in so far as that same externality is now posited as mediated by quantum itself and hence as a moment of quantum, so that in that very externality quantum refers itself to itself – is being as quality'.

- 'The Science of Logic'

B to the power of 2 = 16 has been given as an example of the Ratio of Powers and in it B limits itself and thus has recaptured its Quality but is it not the case that outside forces can erase 16 and choose, say, 25 instead, thereby changing B? Of course Hegel allows for this possibility but nonetheless the Ratio of Powers has a closer connection with the Notion of quantum. In it, Quantum has reached the full extent of its Notion. It expresses the distinctive feature of Quantum. Quantum is the indifferent determinateness, that is, posited as sublated, determinateness as a limit which is equally no limit which continues itself into its otherness and so remains identical with itself therein.

I can hear you asking: why is Quantum a determinateness? Well, recollect that Determinateness was another name for Limit, it stands for a unity of Being and Nothing so Quantum as Number is the unity of Amount (Being) and Unit (Nothing). Number, an early version of Quantum, was indifferent to its Quality, it depended on external reflection to determine which of its parts was Amount and which was Unit but now that indifference is sublated. Number as the Ratio of Powers resists outside manipulation, nonetheless Amount and Unit are indistinguishable precisely because they are equal (5 = 5). Each side of the ratio stays what it is and yet it determines itself in its other, it both remains identical with itself and goes outside itself. Haas contends that the Power Relation is 'a relation of potencies... The mathematician cannot help but speak (always) the words of the philosopher and the sexologist: the decent, respectable power ratio is (also) the indecent language of fornication, the multiple obscenities of the polygamist'. Verh?ltnis does have the alternative meaning of romantic affair (as I noted above), and I am rather in favour of such making of imaginative connections especially sex related and we can at least go along so far as to confirm that the Ratio of Powers is the inseparable unity of two quality-quantities.

Quantum is also said to be the difference of itself from itself.

'If we compare the progressive realization of quantum in the preceding ratios, we find that quantum’s quality of being the difference of itself from itself is simply this: that it is a ratio. As the direct ratio, quantum is this posited difference only in the first instance or immediately, so that the selfreference which it has as exponent, in contrast to its differences, counts only as the fixity of an amount of the unit. In the inverse ratio, as negatively determined, quantum is a relating of itself to itself (to itself as to its negation in which, however, it has its value); as an affirmative self-reference, it is an exponent which, as quantum, is only implicitly in itself the determinant of its moments. But in the ratio of powers quantum is present in the difference as a difference of itself from itself. This externality of determinateness is the quality of quantum and is thus posited, in conformity to the concept of quantum, as quantum’s own determining, as its reference to itself, its quality'.

- 'The Science of Logic'

How so? If we contemplate, BB = 16 evidently the first B is distinguishable from the second B nonetheless, B = B, and so, if the first B is different from the second B, it is different from itself. It is no self-identical entity, of which Hegel is in addition thoroughly critical and to have a selfhood that is different from itself is what it means for Quantum to be a Ratio. At first Direct Ratio showed itself in immediate form, there, its self-relation which it has as exponent, in contrast to its differences, counts only as the fixity of an amount of the unit.

'The exponent of this ratio is no longer an immediate quantum, as in the direct ratio and also in the inverse ratio. In the ratio of powers, the exponent is of an entirely qualitative nature; it is this simple determinateness: that the amount is the unit itself, that the quantum is self-identical in its otherness. And the side of its quantitative nature is to be found in this: that the limit or negation is not an immediate existent, but that existence is posited rather as continuing in its otherness. For the truth of quality is precisely to be quantity, or immediate determinateness as sublated'.

- 'The Science of Logic'

This means that, in Direct Ratio, where Unit is fixed, Amount is fixed and yet the exponent itself was not qualitatively different from Unit or Amount. Direct Ratio was not what it ought to have been and in the Inverse Ratio the exponent is only in principle the determinant of the sides of the ratio, in fact, B and A can fluctuate greatly but they never quite become zero. For this reason the exponent is affirmative in that it has an independence from its sides, that is to say, the Quantum which is exponent relates itself to itself. In the Ratio of Powers, however, self-relation extends to the sides of the Ratio as well as the exponent.

Hegel then summarizes the Quantity's excursions whereby Quantity was at first opposed to Quality, but Quantity was itself a Quality, a purely self-related determinateness distinct from the determinateness of its other, from quality as such. Quantity learned to resist Quality somewhat ironically and in its resistance it showed itself to be a Quality and by repelling its other it became its other. Hence Quantity is in its truth the externality which is no longer indifferent but has returned into itself but Quantity is not just a Quality, it is the truth of quality itself. Mure states: 'In Ratio the endless impotent self-externalizing of Quantity became a self-relation as near to true infinity as Quantity can rise to. It has developed an internal systematicness, which is only thinkable as qualitative'. But this implies that Quantity is not itself always already a True Infinite.

'At first quantity as such thus appears in opposition to quality; but quantity is itself a quality, self-referring determinateness as such, distinct from the determinateness which is its other, from quality as such. Except that quantity is not only a quality, but the truth of quality itself is quantity, and quality has demonstrated itself as passing over into it. Quantity, in its truth, is instead the externality which has returned into itself, which is no longer indifferent. Thus is quantity quality itself, in such a way that outside this determination quality as such would yet not be anything at all. – For the totality to be posited, a double transition is required, not only the transition of one determinateness into the other, but equally the transition of this other into the first, its going back into it. Through the first transition, the identity of the two is present at first only in itself: quality is contained in quantity, but the latter still is only a one-sided determinateness. Conversely, that quantity is equally contained in quality, that it is equally also only as sublated, this results in the second transition, the going back into the first determinateness. This remark regarding the necessity of the double transition is everywhere of great importance for scientific method'.

- 'The Science of Logic'

Without Quantity there could be no Quality.

'Study for the Libyan Sibyl', 1511, Michelangelo

While Measure is in the offing Hegel observes that a double transition was necessary,a chiasmic exchange of properties and not only does one determinateness continue into the other but the other determinateness continues into the original one.That is to say in x to the power of 2 = y, x stays an x and determines the other side of the ratio as x.

Hence Quality is contained in Quantity, but this is still a one-sided determinateness and the converse is also true, Quantity is contained in Quality. This observation on the necessity of the double transition, Hegel contends, is of great importance throughout the whole compass of scientific method, it is evident that within this frame whatever transition may happen no dissolution or vanishing can any longer be expected and the terms remain permanently related. The double transition of Measure, Cinzia Ferrini suggests, was not present in the 1812 version of chapter sixof the Logic, which suspended the reflective trope to the end of Measure,the name of the partnership between Quality and Quantity is Measure.

To conclude: 'A main result of the science of logic is to repudiate quantitative definition of the absolute, and to retrieve qualitative definition', explains Clark Butler. Across the Quantity chapters an exclusive quantitative perspective falls apart. At first Being expelled otherness so that it could be all by itself, independent from the negative, but it discovered that in this mode Being expelled all its content and became Quantity. Quantity stands for the very act of expelling all content, however Quantity discovers that it has an integrity that it cannot expel, a limit that preserves its content within itself. 'This inability to reach its bourne Hegel describes as eine Ohnmacht des Negativen - a weakness of the negative - in that what it abolishes by its own cancelling immediately reasserts itself', contends Harris. Cf 'impotence of the negative'. Theodor W. Adorno, (1903 - 1966), refuses to accept any weakness of the negative and so disagrees with Hegel's entire system but this amounts to a dogmatic insistence upon the self-identity and irreducibility of the negative an irony of which he appears unaware.

This re-appropriation of what is canceled is nevertheless re-appropriation of an other, hence, in Quantitative Infinity, Quantum goes outside itself to a beyond. The infinitely big or small number can never be named yet, in going beyond its limit, Quantum discovers that its own content is beyond the limit. In this sense, Quantum returns to itself when it exports its content to the other and it shows what it is when it shows it is nothing. This return will later be called reflection-into-self which is the mark of Essence but for now it can be noted that the nature of Being has altered, for whereas in the first three chapters Being constituted expelling the negative, now Being constitutes expelling its own self and therefore, in this act of expulsion, accomplishing a return to itself. This return is still implicit and will remain so throughout Measure, the last portion of Being, and the return becomes explicit in Hegel's theory of Reflection.

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I know someone whose ratio certainly has an affirmative aspect, my lady and my muse to whom this article is dedicated, upon whose exquisite proportions I love to dwell. Much of Hegel's Logic may seem rather mysterious, indeed he has sometimes been charged with mysticism, but how are we to distinguish a problem and a mystery? 'When I am dealing with a problem', said Gabriel Marcel, (1889—1973), 'I am trying to discover a solution that can become common property, that consequently can, at least in theory, be rediscovered by anybody at all. But…this idea of a validity for 'anybody at all' or of a thinking in general has less and less application the more deeply one penetrates into the inner courts of philosophy'. A problem can be addressed and solved. But a mystery is 'a problem that encroaches on its own data', as Marcel puts it. The questioner is involved in the question he or she is asking. Hence the mysterious is to be found in questions of freedom, and of love.

Long may my mysterious One my girl of mystery encroach upon my data, a mystery through whom I attain a quantity of an otherwise unattainable understanding of a most profound quality ....

Darkness falls and she will take me by the hand

Take me to some twilight land

Where all but love is grey

Where I can't find my way

Without her as my guide

....

She's a mystery to me

She's a mystery girl

She's a mystery girl

She's a mystery girl

She's a mystery girl

Haunted by her side is the darkness in her eyes

That so enslaves me

But if my love is blind

Then I don't want to see

She's a mystery to me ...

Coming up next:

Measure.

To be continued ...