## 24 Nov 2008

### Hegel, Science of Logic, Vol 1, Bk 1, Sect 2: Magnitude (Quantity), Ch 2 "Quantum," B: Extensive and Intensive Quantum, §§472-478

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[Below is summary. At the end I cite the text in full. My interpretations not informed by a complete read of the text.]

Hegel, Science of Logic

Volume One: The Objective Logic

Book One: The Doctrine of Being

Section 2: Magnitude (Quantity)
B. EXTENSIVE AND INTENSIVE QUANTUM

(a) Their Difference

§ 472

The quantum's determinateness comes from a limit in an amount. However, taken by itself the quantum is a discrete plurality whose limit is a part of it. "Quantum thus then with its limit, which is in its own self a plurality, is extensive magnitude."

§ 473

The opposite of extensive magnitudes are intensive magnitudes. Both extensive and intensive magnitudes are the determinateness of the quantitative limit. The continuity or discreteness of magnitudes, however, determine the magnitude itself without respect to their limits. So we may have a discrete or continuous magnitude without knowing its determinate value, which is determined by its limit. But when we know the value, this is because we know the limit, and in this case we are dealing with extensive and intensive magnitudes.

An extensive magnitude, as a continuous quantity, already contains continuity within itself and its limit. The limit here stands at its entire border, limiting the entire extensive magnitude to a oneness. However, a continuous magnitude that is not taken to be an extensive magnitude also continues without regard for a limit.

Continuity and discreteness are the two sides that together make quantum fully determined and a number. An extensive quantum is different from number because a number's determinateness is expressly posited as a plurality, where an extensive quantum's determinateness is posited as a unity. For example, the number 4 is made up of 4 units. But something that is 4 units long is still a total oneness that takes on a plurality of measure only after number is applied to its extensive magnitude.

§ 474

2. However, a number's determinateness is given on its own, and does not need another number to determine it as determinate. The number 4 is not more or less than 4, but we do not need to be given the numbers outside its limit to know that it is determinate, because it's limit is a part of its value.

§ 475

When the plurality constituting a quantum vanishes so to make the number a unity, it becomes an intensive magnitude. Its limit is now considered unitary, for there are no internal limits of units, and hence this unitary limit is degree.

§ 476

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, because the magnitude is now given as a unitary determination. This determinateness, however, must be expressed by a number. Yet, this number is not an amount but a unitary degree. So when something is at the tenth degree, this is not because it is made up of the sum of ten degrees. It is only one full degree, at the 10th level.

§ 477

3. Because degree does not contain internal units, its external otherness is not within it but rather outside it.

§ 478

The relation between degrees is a "a continuous progress, a flux, which is an uninterrupted, indivisible alteration."

From the text of the original translation:
Quantum
B. EXTENSIVE AND INTENSIVE QUANTUM
(a) Their Difference
§ 472
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.
§ 473
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.
§ 474
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.
§ 475
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.
§ 476
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.
§ 477
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.
§ 478
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.

Hegel. Science of Logic. Transl. A.V. Miller. George Allen & Unwin, 1969.
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