Magnitude (Quantity)
§ 387
The difference between quantity and quality has been stated. Quality is the first, immediate determinateness, quantity is the determinateness which has become indifferent to being, a limit which is just as much no limit, being-for-self which is absolutely identical with being-for-other — a repulsion of the many ones which is directly the non-repulsion, the continuity of them.
§ 388
Because that which is for itself is now posited as not excluding its other, but rather as affirmatively continuing itself into it, it is otherness in so far as determinate being again appears in this continuity and its determinateness is at the same time no longer in a simple self-relation, no longer an immediate determinateness of the determinately existent something, but is posited as self-repelling, as in fact having the relation-to-self as a determinateness in another something (which is for itself; and since they are at the same time indifferent, relationless limits reflected into themselves, the determinateness in general is outside itself, an absolutely self-external determinateness and an equally external something; such a limit, the indifference of the limit within itself and of the something to the limit, constitutes the quantitative determinateness of the something.
§ 389
In the first place, pure quantity is to be distinguished from itself as a determinate quantity, from quantum. As the former, it is in the first place real being-for-self which has returned into itself and which as yet contains no determinateness: a compact, infinite unity which continues itself into itself.
§ 390
Secondly, this develops a determinateness which is posited in it as one which is at the same time no determinateness, as only an external one. It becomes quantum. Quantum is indifferent determinateness, that is, a self-transcending, self-negating determinateness; as this otherness of otherness it relapses into the infinite progress. But the infinite quantum is the indifferent determinateness sublated, it is the restoration of quality.
§ 391
Thirdly, quantum in a qualitative form is quantitative ratio. Quantum transcends itself only generally: in ratio, however, its transition into its otherness is such that this otherness in which it has its determination is at the same time posited, is another quantum. Thus quantum has returned into itself and in its otherness is related to itself.
§ 392
At the base of this ratio there is still the externality of quantum; the quanta which are related to each other are indifferent, that is, they have their self-relation in such self-externality. The ratio is thus only a formal unity of quality and quantity. Its dialectic is its transition into their absolute unity, into Measure.
Remark: Something's Limit as Quality
Quantity
A. PURE QUANTITY
§ 395
Quantity is sublated being-for-self; the repelling one which related itself only negatively to the excluded one, having pa**ed over into relation to it, treats the other as identical with itself, and in doing so has lost its determination: being-for-self has pa**ed over into attraction. The absolute brittleness of the repelling one has melted away into this unity which, however, as containing this one, is at the same time determined by the immanent repulsion, and as unity of the self-externality is unity with itself. Attraction is in this way the moment of continuity in quantity.
§ 396
Continuity is, therefore, simple, self-same self-relation, which is not interrupted by any limit or exclusion; it is not, however, an immediate unity, but a unity of ones which possess being-for-self. The asunderness of the plurality is still contained in this unity, but at the same time as not differentiating or interrupting it. In continuity, the plurality is posited as it is in itself; the many are all alike, each is the same as the other and the plurality is, consequently, a simple, undifferentiated sameness. Continuity is this moment of self-sameness of the asunderness, the self-continuation of the different ones into those from which they are distinguished.
§ 397
In continuity, therefore, magnitude immediately possesses the moment of discreteness — repulsion, as now a moment in quantity. Continuity is self-sameness, but of the Many which, however, do not become exclusive; it is repulsion which expands the selfsameness to continuity. Hence discreteness, on its side, is a coalescent discreteness, where the ones are not connected by the void, by the negative, but by their own continuity and do not interrupt this self-sameness in the many.
§ 398
Quantity is the unity of these moments of continuity and discreteness, but at first it is so in the form of one of them, continuity, as a result of the dialectic of being-for-self, which has collapsed into the form of self-identical immediacy. Quantity is, as such, this simple result in so far as being-for-self has not yet developed its moments and posited them within itself. It contains them to begin with as being-for-self posited as it is in truth. The determination of being-for-self was to be a self-sublating relation-to-self, a perpetual coming-out-of-itself. But what is repelled is itself; repulsion is, therefore, the creative flowing away of itself. On account of the self-sameness of what is repelled, this distinguishing or differentiation is an uninterrupted continuity; and because of the coming-out-of-itself this continuity, without being interrupted, is at the same time a plurality, which no less immediately remains in its self-identicalness.
Remark 1: The Conception of Pure Quantity
Remark 2: The Kantian Antinomy of the Indivisibility and the Infinite Divisibility
B Continuous and Discrete Magnitude
Remark: The Usual Separation of These Magnitudes
§ 432
In the usual ideas of continuous and discrete magnitude, it is overlooked that each of these magnitudes contains both moments, continuity and discreteness, and that the distinction between them consists only in this, that in one of the moments the determinateness is posited and in the other it is only implicit. Space, time, matter, and so forth are continuous magnitudes in that they are repulsions from themselves, a streaming forth out of themselves which at the same time is not their transition or relating of themselves to a qualitative other. They possess the absolute possibility that the one may be posited in them at any point — not the empty possibility of a mere otherness (as when it is said, it is possible that a tree might stand in the place of this stone), but they contain the principle of the one within themselves; it is one of the determinations which constitute them.
§ 433
Conversely, in discrete magnitude continuity is not to be overlooked; this moment is, as has been shown, the one as unity.
Continuous and discrete magnitude can be regarded as species of quantity, provided that magnitude is posited, not under any external determinateness, but under the determinatenesses of its own moments; the ordinary transition from genus to species allows external characteristics to be attributed to the former according to some external basis of cla**ification. And besides, continuous and discrete magnitude are not yet quanta; they are only quantity itself in each of its two forms. They are perhaps, called magnitudes in so far as they have in common with quantum simply this-to be a determinateness in quantity.
C. LIMITATION OF QUANTITY
§ 434
Discrete magnitude has first the one for its principle; secondly, it is a plurality of ones; and thirdly, it is essentially continuous; it is the one as at the same time sublated, as unity, the continuation of itself as such in the discreteness of the ones. Consequently, it is posited as one magnitude, the determinateness of which is the one which, in this posited and determinate being is the excluding one, a limit in the unity. Discrete magnitude as such is immediately not limited; but as distinguished from continuous magnitude it is a determinate being, a something, with the one as its determinateness and also as its first negation and limit.
§ 435
This limit, which is related to the unity and is the negation in it, is also, as the one, self-related; it is thus the enclosing, encompa**ing limit. Limit here is not at first distinguished from its determinate being as something, but, as the one, is immediately this negative point itself. But the being which here is limited is essentially a continuity, by virtue of which it pa**es beyond the limit, beyond this one, to which it is indifferent. Real discrete quantity is thus a quantity, or quantum — quantity as a determinate being and a something.
§ 436
Since the one which is a limit includes within itself the many ones of discrete quantity, it equally posits them as sublated within it; and because it is a limit of continuity simply as such, the distinction between continuous and discrete magnitude is here of no significance; or, more correctly, it is a limit to the continuity of the one as much as of the other; both undergo transition into quanta.
Quantum
§ 437
Quantum, which to begin with is quantity with a determinateness or limit in general is, in its complete determinateness, number. Quantum differentiates itself secondly, into (a) extensive quantum, in which the limit is a limitation of the determinately existent plurality; and (b) intensive quantum or degree, the determinate being having made the transition into being-for-self. Intensive quantum as both for itself and at the same time immediately outside itself — since it is an indifferent limit — has its determinateness in an other. As this manifest contradiction of being determined simply within itself yet having its determinateness outside it, pointing outside itself for it, quantum posited as being in its own self external to itself, pa**es over thirdly, into quantitative infinity.
A. NUMBER
§ 438
Quantity is quantum, or has a limit, both as continuous and as discrete magnitude. The difference between these two kinds has here, in the first instance, no immediate significance.
§ 439
The very nature of quantity as sublated being-for-self is ipso facto to be indifferent to its limit. But equally, too, quantity is not unaffected by the limit or by being, a quantum; for it contains within itself as its own moment the one, which is absolutely determined and which, therefore, as posited in the continuity or unity of quantity, is its limit, but a limit which remains what it has become, simply a one.
§ 440
This one is thus the principle of quantum, but as the one of quantity. Hence, first, it is continuous, it is a unity; secondly, it is discrete, a plurality of ones, which is implicit in continuous, or explicit in discrete magnitude, the ones having equality with one another, possessing the said continuity, the same unity. Thirdly, this one is also a negation of the many ones as a simple limit, an excluding of its otherness from itself, a determination of itself in opposition to other quanta. Thus the one is [a] self-relating, [b] enclosing and [c] other-excluding limit.
§ 441
Quantum completely posited in these determinations is number. The complete positedness lies in the existence of the limit as a plurality and so in its distinction from the unity. Consequently, number appears as a discrete magnitude, but in the unity it equally possesses continuity. It is, therefore, also quantum in its complete determinateness, for its principle the one, the absolutely determinate. Continuity, in which the one is present only in principle, as a sublated moment — posited as a unity — is the form of indeterminateness.
§ 442
Quantum, merely as such, is limited generally; its limit is an abstract simple determinateness of it. But in quantum as number, this limit is posited as manifold within itself. It contains the many ones which constitute its determinate being, but does not contain them in an indeterminate manner, for the determinateness of the limit falls in them; the limit excludes other determinate being, that is, other pluralities and the ones it encloses are a specific aggregate, the amount — which is the form taken by discreteness in number — the other to which is the unit, the continuity of the amount. Amount and unit constitute the moments of number.
§ 443
As regards amount, we must see more closely how the many ones of which it consists are present in the limit; it is correct to say of amount that it consists of the many, for the ones are in it not as sublated but as affirmatively present, only posited with the excluding limit to which they are indifferent. This, however, is not indifferent to them. In the sphere of determinate being, the relation of the limit to it was primarily such that the determinate being persisted as the affirmative on this side of its limit, while the limit, the negation, was found outside on the border of the determinate being; similarly, the breaking-off [in the counting] of the many ones and the exclusion of other ones appears as a determination falling outside the enclosed ones. But in the qualitative sphere it was found that the limit pervades the determinate being, is coextensive with it, and consequently that it lies in the nature of something to be limited, that is, finite. In the quantitative sphere a number, say a hundred, is conceived in such a manner that the hundredth one alone limits the many to make them a hundred. In one sense this is correct; but on the other hand none of the hundred ones has precedence over any other for they are only equal — each is equally the hundredth; thus they all belong to the limit which makes the number a hundred and the number cannot dispense with any of them for its determinateness. Hence, relatively to the hundredth one, the others do not constitute a determinate being that is in any way different from the limit, whether they are outside or inside it. Consequently, the number is not a plurality over against the enclosing, limiting one, but itself constitutes this limitation which is a specific quantum; the many constitute a number, a two, a ten, a hundred, and so on.
§ 444
Now the limiting one is the number as determined relatively to other numbers, as distinguished from them. But this distinguishing does not become a qualitative determinateness but remains quantitative, falling only within the comparing external reflection; the number, as a one, remains returned into itself and indifferent to others. This indifference of a number to others is an essential determination of it and constitutes the implicit determinedness of the number, but also the number's own externality. Number is thus a numerical one as the absolutely determinate one, which at the same time has the form of simple immediacy and for which, therefore, the relation to other is completely external. Further, one as a number possesses determinateness (in so far as this is a relation to other) as the moments of itself contained within it, in its difference of unit and amount; and amount is itself a plurality of ones, that is, this absolute externality is in the one itself. This contradiction of number or of quantum as such within itself is the quality of quantum, in the further determinations of which this contradiction is developed.
Remark 1: The Species of Calculation in Arithmetic; Kant's Synthetic Propositions a priori
Remark 2: The Employment of Numerical Distinctions for Expressing Philosophical Notions
B Extensive and Intensive Quantum
(a) Their Difference
(b) Identity of Extensive and Intensive Magnitude
Remark 1: Examples of This Identity
Remark 2: The determination of degree as applied by Kant to the soul
(c) Alteration of Quantum
C Quantitative Infinity
(a) Its Notion
(b) The Quantitative Infinite Progress
Remark 1: The High Repute of the Progress to Infinity
Remark 2: The Kantian Antinomy of the Limitation and Nonlimitation of the World
(c) The Infinity of Quantum
Remark 1: The Specific Nature of the Notion of the Mathematical Infinite
Remark 2: The Purpose of the Differential Calculus Deduced from its Application
Remark 3: Further Forms Connected With the Qualitative Determinateness of Magnitude
Chapter 3 The Quantitative Relation or Quantitative Ratio
A The Direct Ratio
B Inverse Ratio
C The Ratio of Powers
Remark
§ 695
In the Remarks above on the quantitative infinite, it was shown that this infinite and also the difficulties a**ociated with it have their origin in the qualitative moment which makes its appearance in the sphere of quantity, and also how the qualitative moment of the ratio of powers in particular is the source of various developments and complexities. It was shown that the chief obstacle to a grasp of the Notion of this infinite is the stopping short at its merely negative determination as the negation of quantum, instead of advancing to the simple affirmative determination which is the qualitative moment. The only further remark to be made here concerns the intrusion of quantitative forms into the pure qualitative forms of thought in philosophy. It is the relationship of powers in particular which has been applied recently to the determinations of the Notion. The Notion in its immediacy was called the first power or potence; in its otherness or difference, in the determinate being of its moments, the second power; and in its return into itself or as a totality, the third power. It is at once evident that power as used thus is a category which essentially belongs to quantum — these powers do not bear the meaning of the potentia, the dynamis of Aristotle. Thus, the relationship of powers expresses determinateness in the form or difference which has reached its truth, but difference as it is in the particular Notion of quantum, not as it is in the Notion as such. In quantum, the negativity which belongs to the nature of the Notion is still far from being posited in the determination proper to the Notion; differences which are proper to quantum are superficial determinations for the Notion itself and are still far from being determined as they are in the Notion. It was in the infancy of philosophic thinking that numbers were used, as by Pythagoras, to designate universal, essential distinctions - and first and second power, and so on are in this respect not a whit better than numbers. This was a preliminary stage to comprehension in the element of pure thought; it was not until after Pythagoras that thought determinations themselves were discovered, i.e., became on their own account objects for consciousness. But to retrogress from such determinations to those of number is the action of a thinking which feels its own incapacity, a thinking which, in Opposition to current philosophical culture which is accustomed to thought determinations, now also makes itself ridiculous by pretending that this impotence is something new, superior, and an advance.
§ 696
There is as little to be said against the expression power when it is used only as a symbol, as there is against the use of numbers or any other kind of symbols for Notions - but also there is just as much to be said against them as against all symbolism whatever in which pure determinations of the Notion or of philosophy are supposed to be represented.
§ 697
Philosophy needs no such help either from the world of sense or from the products of the imagination, or from subordinate spheres in its own peculiar province, for the determinations of such spheres are unfitted for higher spheres and for the whole. This unfitness is manifest whenever categories of the finite are applied to the infinite; the current determinations of force, or substantiality, cause and effect, and so on, are likewise only symbols for expressing, for example, vital or spiritual relationships, i.e. they are untrue determinations for such relationships; and still more so are the powers of quantum and degrees of powers, both for such and for speculative relationships generally.
§ 698
If numbers, powers, the mathematical infinite, and suchlike are to be used not as symbols but as forms for philosophical determinations and hence themselves as philosophical forms, then it would be necessary first of all to demonstrate their philosophical meaning, i.e. the specific nature of their Notion. If this is done, then they themselves are superfluous designations; the determinateness of the Notion specifies its own self and its specification alone is the correct and fitting designation. The use of those forms is, therefore, nothing more than a convenient means of evading the task of grasping the determinations of the Notion, of specifying and of justifying them.