Hegel’s Science of Logic
1. We have seen that quantum has its determinateness as limit in amount. Within itself quantum is discrete, a plurality which has no being distinct from its limit, nor is the limit external to it. Quantum thus then with its limit, which is in its own self a plurality, is extensive magnitude.
Extensive and continuous magnitude are to be distinguished from each other; the direct opposite of the former is not discrete but intensive magnitude. Extensive and intensive magnitudes are determinatenesses of the quantitative limit itself, whereas quantum is identical with its limit; continuous and discrete magnitudes, on the other hand, are 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 present within itself and in its limit, for its many is altogether continuous; the limit as negation appears, therefore, in this equality of the many as a limiting of the oneness. Continuous magnitude is quantity as continuing itself without regard to any limit and in so far as it is conceived as having a limit, this is simply a limitation free from any posited discreteness. Quantum as only continuous magnitude is not yet truly determined as being for itself because it lacks the one (in which being-for-selfness is implied) and number. Similarly, a discrete magnitude is immediately only a differentiated many in general; were this as such supposed to have a limit, it would be only an aggregate, that is, would be only indefinitely limited; before it can be a specific quantum, the many must be compressed into a one and thereby posited as identical with the limit. Continuous and discrete magnitude, taken simply as quanta have each posited in it only one of the two sides which together make quantum fully determined and a number. This latter is immediately an extensive quantum — the simple determinateness which is essentially an amount, but an amount of one and the same unit; extensive quantum is distinguished from number only by this, that in number the determinateness is expressly posited as a plurality.
2. However, the determinateness of something in terms of number does not require it to be distinguished from another numerically determined something, as if both were necessary to the determinateness of the first; and this is because the determinateness of magnitude as such is a limit determinate by itself, indifferent and related simply to itself; and in number the limit is posited as included in the one, which is a being-for-self, and it has within itself the externality, the relation to other. Further, this many of the limit itself is, like the many as such, not unequal within itself but continuous; each of the many is the same as the others; consequently, the many as a plural asunderness or discreteness does not constitute the determinateness as such. This many, therefore, spontaneously collapses into its continuity and becomes a simple oneness. Amount is only 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 which has returned into itself; the externality which constituted the ones as a plurality vanishes in the one as a relation of number to its own self.
Consequently the limit of quantum, which as extensive had its real determinateness in the self-external amount, passes over into simple determinateness. In this simple determination of the limit, quantum is intensive magnitude; and the limit or determinateness which is identical with the quantum is now also thus posited as unitary — degree.
The degree is thus a specific magnitude, a quantum; but at the same time it is not an aggregate or plural within itself, it is a plurality only in principle (eine Mehrheit), for plurality has been brought together into a simple, unitary determination, determinate being has returned into being-for-self. The determinateness of degree must, it is true, be expressed by a number, the completely determined form of quantum, but the number is not an amount but unitary, only a degree. When we speak of ten or twenty degrees, the quantum that has that number of degrees is the tenth or twentieth degree, not the amount and sum of them — as such, it would be an extensive quantum — but it is only one degree, the tenth or twentieth. It contains the determinateness implied in the amount ten or twenty, but does not contain it as a plurality but is number as a sublated amount, as a unitary determinateness.
3. In number, quantum is posited in its complete determinateness; but as intensive quantum, as in number's being-for-self, it is posited as it is in its Notion or in itself. That is to say, the form of self-relation which it has in degree is at the same time the externality of the degree to its own self. Number, as an extensive quantum, is a numerical plurality and so has the externality within itself. This externality, as simply a plurality, collapses into undifferentiatedness and sublates itself in the numerical one, in its self-relation. Quantum, however, has its determinateness as an amount; it contains this, as we have already seen, even though the amount is no longer posited in it. Degree, therefore, which, as in its own self unitary, no longer has within itself this external otherness, has it outside itself and relates itself to it as to its determinateness. A plurality external to the degree constitutes the determinateness of the simple limit which the degree is for itself. In extensive quantum amount, in so far as it was supposed to be present in the number, was so only as sublated; now it is determined as placed outside the number. Number as a one, being posited as self-relation reflected into itself, excludes from itself the indifference and externality of the amount and is self-relation as relation through itself to an externality.
In this, quantum has a reality conformable to its Notion. The indifference of the determinateness constitutes its quality, that is, the determinateness which is in its own self a self-external determinateness. Accordingly, degree is a unitary quantitative determinateness among a plurality of such intensifies which, though differing from each other, each being only a simple self-relation, are at the same time essentially interrelated so that each has its determinateness in this continuity with the others. This relation of degree through itself to its other makes ascent and descent in the scale of degrees a continuous progress, a flux, which is an uninterrupted, indivisible alteration; none of the various distinct degrees is separate from the others but each is determined only through them. As a self-related determination of quantity, each degree is indifferent to the others; but it is just as much implicitly related to this externality, it is only through this externality that it is what it is; its relation to itself is, in short, the non-indifferent relation to externality, and in this it has its quality.
Degree is not external to itself within itself. Nevertheless, it is not the indeterminate one, the principle of number as such, which is not amount save in the negative sense of not being any particular amount. Intensive magnitude is primarily a unitary one of a plurality; there are many degrees, but they are determined neither as a simple one nor as a plurality, but only in the relation of this self-externality, or in the identity of the one and the plurality. if, therefore, the many as such are indeed outside the simple, unitary degree, nevertheless the determinateness of the degree consists in its relation to them; it thus contains amount. just as twenty, as an extensive magnitude, contains the twenty ones as discrete within it, so does the specific degree contain them as the continuity which this determinate plurality simply is; it is the twentieth degree, and is the twentieth degree only by virtue of this amount, which as such is outside it.
The determinateness of intensive magnitude is, therefore, to be considered from two sides. Intensive magnitude is determined by other intensive quanta and is continuous with its otherness, so that its determinateness consists in this relation to its otherness. Now in the first place, in so far as it is a simple determinateness it is determinate relatively to other degrees; it excludes them from itself and has its determinateness in this exclusion. But, secondly, it is determinate in its own self; this it is in the amount as its own amount, not in the amount as excluded, nor in the amount of other degrees. The twentieth degree contains the twenty within itself; it is not only determined as distinguished from the nineteenth, twenty-first, and so on, but its determinateness is its own amount. But in so far as the amount is its own — and the determinateness is at the same time essentially an amount — the degree is an extensive quantum.
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 magnitude passes over into intensive magnitude because its many spontaneously collapse into oneness, outside which the many stand. But conversely, this unitary degree has its determinateness only in the amount, and that too in its own amount; as indifferent to the differently determined intensifies it has within itself the externality of the amount; and so intensive magnitude is equally essentially an extensive magnitude.
With this identity, the qualitative something makes its appearance, for the identity is the unity which is self-related through the negation of its differences; these differences, however, constitute the determinate being of the quantitative determinateness; this negative identity is therefore a something, and a something which is indifferent to its quantitative determinateness. Something is a quantum; but now the qualitative determinate being, as it is in itself, is posited as indifferent to quantum. 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 these 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. Its one determinateness which it has as quantum is posited in the differentiated moments of unit and amount; this determinateness is not only in itself one and the same, but its positing in these differences as extensive and intensive quantum is the return into this unity which, as negative, is the something posited as indifferent to it.
Remark 1: Examples of This Identity
Extensive and intensive quantum are usually distinguished in the ordinary conception of them as kinds of magnitude, as if some objects had only intensive, others only extensive magnitude. In addition, we have the conception of a philosophical science of Nature in which what is a plurality or extensive — for example, in the fundamental property of matter to occupy space, and in other concepts too — is converted into something intensive, meaning thereby that the intensive aspect as dynamic is the true determination; density, or the specific filling of space, for example, must essentially be understood not as a certain aggregate and amount of material parts in a quantum of space, but as a certain degree of the space-filling force of matter.
There are two kinds of determinations to be distinguished here. In what has been called the conversion of the mechanical into the dynamic point of view, there occurs the concept of separately existing, independent parts, which are only externally combined into a whole, and the concept of force which is distinct from this. In the occupation of space, what is regarded on the one hand as only an aggregate of atoms external to one another, is on the other hand regarded as the expression of an underlying simple force. But these relations of whole and parts, of force and its expression, which here stand opposed to each other, do not belong in this section; they will be considered in their proper place later on. But this much may be remarked here, that though the relation of force and its expression which corresponds to intensive magnitude is in the first instance truer than that of whole and parts, yet this does not make force, as intensive, any less one-sided; also expression, the externality of extensive magnitude, is equally inseparable from force; so that one and the same content is equally present in the two forms, both in intensive and in extensive magnitude.
The other determinateness which occurs here is the quantitative as such, which, as extensive quantum, is sublated and transformed into degree, the supposedly true determination; but it has been shown that degree equally contains the former determinateness, so that the two forms are essential to each other; consequently, every existence exhibits its quantitative character just as much as an extensive as an intensive quantum.
Consequently everything, in so far as it manifests a quantitative character, serves as an example of this. Number itself necessarily has this double form immediately within it. It is an amount in so far as it is an extensive magnitude; but number is also one, a ten, a hundred, and as such it is on the threshold of transition into an intensive magnitude, seeing that in this unity the plurality has become simple. One is in itself an extensive magnitude, it can be represented as an arbitrary amount of parts. Thus the tenth, the hundredth, is this simple, intensive magnitude which has its determinateness in the plurality lying outside it, that is, in extensive magnitude. Number is a ten, a hundred and at the same time the tenth, hundredth, in the system of numbers; both are the same determinateness.
In the circle the one is called degree because the determinateness of any part of the circle derives essentially from the many parts outside it; that is, it is determined as only one of a fixed amount of such ones. As a mere spatial magnitude, the degree of the circle is only an ordinary number; taken as degree, it is intensive magnitude which has a meaning only as determined by the amount of degrees into which the circle is divided, just as number generally has meaning only in the number series.
The magnitude of a more concrete object exhibits its dual aspects of being extensive and intensive, in the dual determinations of its real being, in one of which it appears as an outer being but in the other as an inwardness. Thus, for example, a mass as weight is an extensive magnitude, in so far as it constitutes an amount of pounds, hundredweights, etc., and an intensive magnitude in so far as it exerts a certain pressure; the magnitude of the pressure is a simple number, a degree, which is specified by its place in a scale of degrees. As exerting pressure, mass is manifested as a being-within-self, as a subject to which belongs a difference of intensive magnitude. Conversely, that which exerts this degree of pressure is capable of displacing a certain amount of pounds, etc., and its magnitude is measured by this.
Again, heat has a degree; this degree, whether it be the tenth, twentieth and so on, is a simple sensation, something subjective. But this degree is equally present as an extensive magnitude, as the expansion of a fluid, of mercury in a thermometer, of air, or sound, and so on. A higher degree of temperature expresses itself as a longer column of mercury, or as a narrower sound cylinder; it heats a larger space in the same way as a lower degree heats only a smaller space.
The higher note is, as more intensive, at the same time a greater number of vibrations, and a louder note, to which we ascribe a higher degree, is audible in a larger space. A larger surface can be coloured with a more intensive colour than with a weaker colour used in the same way; or again a brighter object (another kind of intensity) is visible at a greater distance than one less bright, and so forth.
Similarly in the spiritual sphere, high intensity of character, of talent or genius, is bound up with a correspondingly far-reaching reality in the outer world, is of widespread influence, touching the real world at many points. The profoundest Notion also has the most universal significance and application.
Remark 2: The determination of degree as applied by Kant to the soul
The determinateness of intensive quantum has been applied by Kant in a peculiar way to a metaphysical determination of the soul. In his criticism of metaphysical propositions about the soul, which he calls paralogisms of pure reason, he comes to consider the inference from the simplicity of the soul to its permanence' He counters this argument by saying 'that even if we admit the simple nature of the soul since, namely, it does not contain a plurality of separate parts and therefore no extensive magnitude, yet we cannot deny to it, any more than to any other existent thing, an intensive magnitude, that is, a degree of reality in respect of all its faculties, indeed, in respect of all that constitutes its existence; this degree can diminish through all the infinitely many smaller degrees so that although the postulated substance cannot be reduced to nothing by division (into parts), it can be so reduced by a gradual lessening (remissio) of its powers. For consciousness itself has, at any moment, a degree which can always be diminished, and the same must therefore also be true of its faculty of being aware of itself and thus of all the other faculties. In rational psychology, which is an abstract metaphysics, the soul is considered not as spirit but as a merely immediate being, as a soul thing. Kant thus has the right to apply the category of quantum to it 'as to any other existent thing', and in so far as this immediate being is determined as simple, to apply to it the category of intensive quantum. Being does, of course, belong to spirit, but its intensity is wholly different from that of intensive quantum; indeed, its intensity is such that in it the form of merely immediate being and all its categories are sublated. What should have been admitted was the elimination not only of the category of extensive quantum but that of quantum altogether. But a further advance has still to be made, namely, to understand how existence, consciousness, finitude, is in the eternal nature of spirit and proceeds from it without spirit thereby becoming a thing.
The difference between extensive and intensive quantum is indifferent to the determinateness of quantum as such. But in general quantum is determinateness posited as sublated, the indifferent limit, the determinateness which is equally the negation of itself. In extensive magnitude this difference is developed; but intensive magnitude is the existence of this externality which quantum is within itself. This difference, as internally self-contradictory, 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.
A quantum, therefore, in accordance with its quality, is posited in absolute continuity with its externality, with its otherness. Therefore, not only can it transcend every quantitative determinateness, not only can it be altered, but it is posited that it must alter. 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.
The one is infinite or self-related negation, hence the repulsion of itself from itself. The quantum, too, is infinite and is posited as self-related negativity; it repels itself from itself. But the quantum is a determinate one, the one which has passed over into determinate being and limit; it is, therefore, the repulsion of the determinateness from itself, not the producing of that which is the same as itself as in the repulsion of the one, but the producing of its otherness; it is now the express character of quantum to impel itself beyond itself and to become an other. In consists in undergoing increase or decrease; it is in its own self the externality of the determinateness.
Thus quantum impels itself beyond itself; this other which it becomes is in the first place itself a quantum; but it is quantum as a limit which does not stay, but which impels itself beyond itself. The limit which again arises in this beyond is, therefore, one which simply sublates itself again and impels itself beyond to a further limit, and so on to infinity.
C. Quantitative Infinity - next section
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