Hegel’s Science of Logic
Qualitative quantity in the first place an immediate, specific quantum. Secondly, this quantum as relating itself to another becomes a quantitative specifying, a sublating of the indifferent quantum. This measure is so far a rule and contains the two moments of measure distinguished; namely, the intrinsic quantitative determinateness and the external quantum. In this distinction, however, these two sides become qualities and the rule becomes a relation between them; consequently measure exhibits itself thirdly, as a relation of qualities. These at first have a single measure, but this is further specified within itself into distinct measures.
1. Measure is the simple relation of the quantum to itself, its own determinateness within itself; the quantum is thus qualitative. At first, as an immediate measure it is an immediate quantum, hence just some specific quantum or other; equally immediate is the quality belonging to it, some specific quality or other. The quantum as this no longer indifferent limit but as a self-related externality, is thus itself quality, and although distinguished from it does not transcend it, neither does the quality transcend the quantum. It is thus the determinateness which has returned into simple identity with itself, one with the specific determinate being, just as this latter is one with its quantum.
If it is desired to make a proposition out of the determination in question, it can be expressed thus: all that exists has a measure. Everything that exists has a magnitude and this magnitude belongs to the nature of the something itself ; it constitutes its specific nature and its being-within-self. Something is not indifferent to this magnitude, so that if this were altered it would continue to be what it is; on the contrary, an alteration of the magnitude would alter the quality of the something. Quantum, as measure, has ceased to be a limit which is no limit; it is now the determination of the thing, which is destroyed if it is increased or diminished beyond this quantum.
A measure taken as a standard in the usual meaning of the word is a quantum which is arbitrarily assumed as the intrinsically determinate unit relatively to an external amount. Such a unit can, it is true, also be in fact an intrinsically determinate unit, like a foot and suchlike original measures; but in so far as it is also used as a standard for other things it is in regard to them only an external measure, not their original measure. Thus the diameter of the earth or the length of a pendulum may be taken, each on its own account, as a specific quantum; but the selection of a particular fraction of the earth's diameter or of the length of the pendulum, as well as the degree of latitude under which the latter is to be taken for use as a standard, is a matter of choice. But for other things such a standard is still more something external. These have further specified the general specific quantum in a particular way and have thereby become particular things. It is therefore foolish to speak of a natural standard of things. Moreover, a universal standard ought only to serve for external comparison; in this most superficial sense in which it is taken as a universal measure it is a matter of complete indifference what is used for this purpose. It ought not to be a fundamental measure in the sense that it forms a scale on which the natural measures of particular things could be represented and from which, by means of a rule, they could be grasped as specifications of a universal measure, i.e. of the measure of their universal body. Without this meaning, however, an absolute measure is interesting and significant only as a common element, and as such is a universal not in itself but only by agreement.
This immediate measure is a simple quantitative determination as, for example, the size of organic beings, of their limbs etc. But everything that exists has a size which makes it what it is, and in general enables it to have an external reality. As a quantum it is an indifferent magnitude open to external determination and capable of increase and decrease. But as a measure it is also distinguished from itself as a quantum, as such an indifferent determination, and is a limitation of that indifferent fluctuation about a limit.
Since the quantitative determinateness of anything is thus twofold — namely, it is that to which the quality is tied and also that which can be varied without affecting the quality — it follows that the destruction of anything which has a measure takes place through the alteration of its quantum. On the one hand this destruction appears as unexpected, in so far as the quantum can be changed without altering the measure and the quality of the thing; but on the other hand, it is made into something quite easy to understand through the idea of gradualness.
The reason why such ready use is made of this category to render conceivable or to explain the disappearance of a duality or of something, is that it seems to make it possible almost to watch the disappearing with one's eyes, because quantum is posited as the external limit which is by its nature alterable, and so alteration (of quantum only) requires no explanation. But in fact nothing is explained thereby; the alteration is at the same time essentially the transition of one quality into another, or the more abstract transition of an existence into a negation of the existence; this implies another determination than that of gradualness which is only a decrease or an increase and is a one-sided holding fast to quantity.
2. The sudden conversion into a change of quality of a change which was apparently merely quantitative had already attracted the attention of the ancients who illustrated in popular examples the contradiction arising from ignorance of this fact; they are familiar under the names of ‘the bald’ and ‘the heap’.
These elenchi are, according to Aristotle's explanation, ways in which one is compelled to say the opposite of what one had previously asserted. The question was asked: does the pulling out of a single hair from the head or from a horse's tail produce baldness, or does a heap cease to be a heap if a grain is removed? An answer in the negative can be given without hesitation since such a removal constitutes only a quantitative difference, a difference moreover which is itself quite insignificant; thus a hair, a grain, is removed and this is repeated, only one of them being removed each time in accordance with the answer given. At last the qualitative change is revealed; the head or the tail is bald, the heap has disappeared. In giving the said answer, what was forgotten was not only the repetition, but the fact that the individually insignificant quantities (like the individually insignificant disbursements from a fortune) add up and the total constitutes the qualitative whole, so that finally this whole has vanished; the head is bald, the purse is empty.
The dilemma, the contradiction which results therefrom, is not a sophism in the usual sense of the word; for such contradiction is not a sham or a deception. The real mistake is committed by the assumed Other (i.e. our ordinary consciousness), the mistake, namely, of assuming a quantity to be only an indifferent limit, i.e. of assuming that it is just a quantity in the specific sense of quantity. This assumption is refuted by the truth to which it is brought — to wit, that quantity is a moment of measure and is connected with quality.
What is refuted is the error of one-sidedly holding fast to the abstract determinateness of quantum. Therefore these examples, too, are not a pointless or pedantic joke but have their own correctness; they are the product of a mentality which is interested in the phenomena which occur in thinking.
Quantum, when it is taken as an indifferent limit, is the aspect of an existence which leaves it open to unsuspected attack and destruction. It is the cunning of the Notion to seize on this aspect of a reality where its quality does not seem to come into play; and such is its cunning that the aggrandisement of a State or of a fortune, etc., which leads finally to disaster for the State to for the owner, even appears at first to be their good fortune.
3. Measure in its immediacy is an ordinary quality with a specific magnitude attaching to it. Now that aspect of the quantum according to which it is an indifferent limit which can be exceeded without altering the quality, is also distinguished from its other aspect according to which it is qualitative and specific. Both are quantitative determinations of one and the same thing; but because of the initial immediacy of measure, this distinction is also to be taken as immediate, and therefore both aspects also have a distinct existence. The existence of measure, then, which is intrinsically determinate magnitude, is in its behaviour towards the existence of the alterable, external aspect, a sublating of its indifference, a specifying of measure.
This is first a rule, a measure which is external with reference to mere quantum. Secondly it is a specific quantity which determines the external quantum, and thirdly both sides, as qualities of a specific quantitative determinateness, are related to one another as one measure.
The rule or standard, which has already been mentioned, is in the first place an intrinsically determinate magnitude which is a unit with reference to a quantum having a particular existence in a something other than the something of the rule; this other something is measured by the rule, i.e. is determined as an amount of the said unit. This comparison is an external act, the unit itself being an arbitrary magnitude which in turn can equally be treated as an amount (the foot as an amount of inches). But measure is not only an external rule; as a specifying measure its nature is to be related in its own self to an other which is a quantum.
Measure is a specific determining of the external, i.e. indifferent magnitude which is now posited by some other existence in general in the measurable something; true, this latter is itself a quantum but, as distinguished from it, it is the qualitative side determining the merely indifferent, external quantum. The measurable something has in it this side of being-for-other to which the indifferent increasing and decreasing is proper. This immanent measuring standard is a quality of the something to which is opposed the same quality in another something; but in the latter something the quality is at first only relative, the quantum having no significance as a measure in relation to the other something which is determined as the standard of measuring.
Something, in so far as it is a measure within itself, has the magnitude of its quality altered from outside itself; it does not accept this externally imposed alteration as an arithmetical amount: its measure reacts against it, behaves towards the amount as an intensive quantum and assimilates it in a characteristic way; it alters the externally imposed alteration, makes this quantum into a different one and through this specifying shows itself to be self-determined in this externality. This specifically assimilated amount is itself a quantum which is also dependent on the other, or is for it only an external amount. Consequently the specified amount is also alterable, but is not therefore a quantum as such but the external quantum specified in a constant manner. The determinate being of measure is thus a ratio, the specific element of which is in general the exponent of this ratio.
When considering intensive, and extensive quantum we found that it is the same quantum which is present, once in the form of intensity and again in the form of extension. In this difference the quantum lying at the base suffers no alteration, the difference being only an outer form. In the specifying measure, on the contrary, the quantum is taken in the first instance in its immediate magnitude, but in the second instance it is taken through the exponent of the ratio in another amount.
The exponent which constitutes the specific element can at first seem to be a fixed quantum, as a quotient of the ratio between the external and the qualitatively determined quantum. But as such, it would be nothing but an external quantum; the exponent here must be understood as nothing else but the qualitative moment itself which specifies the quantum as such. As we have already seen, the strictly immanent qualitative form of the quantum is solely its determination as a power. It must be such a determination which constitutes the ratio and which here, as the intrinsic determination of the quantum, confronts the quantum as externally constituted. The principle of the quantum is the numerical one which constitutes its intrinsic determinedness, and the mode of relation of the numerical one is external; and the alteration which is specified only by the nature of the immediate quantum as such, consists by itself in the addition of such a numerical one and then another and so on. If in this way the alteration of the external quantum is an arithmetical progression, the specifying reaction of the qualitative nature of measure produces another series which is related to the first, increases and decreases with it, but not in a ratio determined by a numerical exponent but in a number of incommensurable ratios, according to a determination of powers.
To cite an example, temperature is a quality in which these two sides of external and specified quantum are distinguished. As a quantum it is an external temperature, and that too, of a body as a general medium, and it is assumed that the alteration of the temperature proceeds on the scale of an arithmetical progression, increasing or decreasing uniformly. On the other hand, the particular bodies in the medium differ in the way they absorb the temperature, for through their immanent measure they determine it as received from outside themselves and the change of temperature in any one of them does not correspond in a direct ratio with that of the medium or of the other bodies among themselves. Different bodies compared at one and the same temperature give the numerical ratios of their specific heats, of their thermal capacities. But the thermal capacities of bodies vary in different temperatures and associated with this is a change in the specific shape. Thus a particular specification is manifested in the increase or decrease of temperature. The ratio of the temperature, taken as external, to the temperature of a specific body which is at the same time dependent on the former temperature, has no fixed exponent; the increase or decrease of this heat does not proceed uniformly with the increase or decrease of the external heat. Here a temperature is assumed which is purely external and whose changes are merely external or purely quantitative. But the temperature is itself the temperature of the air or some other specific temperature. The ratio, therefore, if looked at more closely, would strictly speaking have to be taken not as the ratio of a merely quantitative quantum to a qualitative one, but as the ratio of two specific quanta. In fact, the determining of the specifying ratio has now advanced to the stage where the moments of measure not only consist of a quantitative side and a side qualifying the quantum, both being sides of one and the same quality, but are related to each other as two qualities which are in themselves measures.
1. The qualitative, intrinsically determinate side of the quantum exists only as a relation to the externally quantitative side; as a specifying of the latter it is a sublating of its externality through which quantum as such is. This qualitative side thus has a quantum for its presupposition and its starting point. But this quantum is also qualitatively distinguished from the quality itself; this difference between them is now to be posited in the immediacy of being as such, in which determination measure still is. The two sides are thus qualitatively related and each is on its own account a qualitative determinate being; and the one quantum which at first was only a formal quantum indeterminate in itself, is the quantum of a something and of its quality, and also — now that the connection between them is determined as a measure — the specific magnitude of these qualities. These qualities are related to each other according to their determination as measures which determination is their exponent. But they are already implicitly related to each other in the being-for-self of measure; the quantum in its dual character is both external and specific so that each of the distinct quantities possesses this twofold determination and is at the same time inseparably linked with the other; it is in this way alone that the qualities are determined. They are therefore not only simply determinate beings existing for each other but they are posited as inseparable and the specific magnitude connected with them is a qualitative unity — a single determination of measure in which, in accordance with their Notion, they are implicitly bound up with each other. Measure is thus the immanent quantitative relationship of two qualities to each other.
2. In measure there enters the essential determination of variable magnitude, for measure is quantum as sublated, and therefore no longer what, as quantum, it is supposed to be, but quantum and something else; this something else is the qualitative element and, as we have seen, is nothing else than its relation of powers. In immediate measure this alteration is not yet posited; it is only an arbitrary, single quantum to which a quality is bound. In the specifying of measure (the preceding determination), which is an alteration of the merely external quantum by the qualitative element, there is posited the distinction between the two specific magnitudes and hence generally the plurality of measures in a common, external quantum. It is in this distinguishedness of the quantum from itself that it first shows itself to be a real measure; for it now appears as a determinate being which is both one and the same (e.g. the constant temperature of the medium), and also quantitatively varied (in the different temperatures of the bodies present in the medium). This distinguishedness of the quantum in the different qualities (the different bodies) gives a further form of measure, that in which the two sides are mutually related as qualitatively determinate quanta. This can be called realised measure.
Magnitude, simply as magnitude, is alterable, for its determinateness is a limit which is at the same time no limit, so that the alteration concerns only a particular quantum, the place of which is taken by another. But the genuine alteration is that of the quantum as such; when understood in this way, we have the interesting determination of the variable magnitude in higher mathematics; here we must not stop short at the merely formal determination of variability in general, neither must we introduce any determination other than the simple determination of the Notion according to which the other of quantum is only quality. The genuine determination, therefore, of real variable magnitude is that it is magnitude qualitatively determined, that is, as has been sufficiently demonstrated, determined by a ratio of powers. In this variable magnitude the fact is posited that what counts is not quantum as such but quantum determined in accordance with its other, i.e. qualitatively determined.
The two sides thus related have, in keeping with their abstract aspect as qualities generally, some particular significance — for example, space and time. Taken at first simply as specific magnitudes in their ratio as measures, one of them is an amount which increases and decreases in an external arithmetical progression, the other is an amount which is specifically determined by the first, which for it is unit. Now if each were only just a particular quality, that element of difference would be lacking which, with regard to their character as quantities, would indicate which of them was to be taken as merely externally quantitative and which as varying according to the specifying of its magnitude. If, for example, they are related as root and square, it is immaterial which is regarded as increasing or decreasing merely externally in arithmetical progression, and which, on the other hand, as specifically determining the other quantum.
But the difference between the qualities is not undefined, for as moments of measure the specifying of measure must be present in them. The next determinateness of the qualities themselves is that one is extensive — is in its own self externality — the other is intensive, the being-within-self or negative relatively to the other. Accordingly, the former of the quantitative moments is amount, and the latter is unit; in the simple direct ratio the former is to be taken as the dividend, the latter as divisor, and in the specifying ratio the former as the power or the becoming-other, the latter as the root. In so far as we still count here, i.e. still reflect on the external quantum (which is thus the quite contingent specific magnitude, the empirical amount), the alteration, too, being likewise taken as an external, arithmetical progression, then this falls on the side of the unit, of the intensive quality; the external, extensive side, on the other hand, is to be represented as altering in the specified series. But the direct ratio (like velocity as such, s/t) is here reduced to the merely formal determination which has no existence except as an intellectual abstraction; and even though in the ratio of root and square (as in s = at2) the root is to be taken as an empirical quantum varying in an arithmetical progression, and the other side is to be taken as specified, yet the higher realisation of the qualifying of the quantitative, a realisation more in harmony with the Notion, is that in which both sides are related to each other in higher determinations of powers (as is the case in s3 = at2
1. In the form of specified measure just considered, the quantitative element of both sides is qualitatively determined (both in the ratio of powers); hence they are moments of one measure-determinateness of qualitative nature. At the same time, however, the two sides are so far posited only as immediate, merely different qualities, which do not themselves stand in the same relationship as their quantitative determinatenesses; that is, they cannot be said to have no meaning or existence outside that relationship, as is the case in the determinateness of quantity as a ratio of powers. The qualitative element thus masks itself, specifying not itself but the quantitative determinateness; only in the latter is it posited, remaining on its own account an immediate quality as such which, beside the fact that it explicitly differentiates the magnitude and beside its relation to its other, still has an independent determinate being of its own. Thus space and time, apart from that specification contained in their quantitative determinateness in the descent of a falling body, or in the absolutely free motion, count as space in general and time in general, space having an enduring existence of its own apart from and without time, and time flowing on its own independently of space.
This immediacy of the qualitative element as against its specific measure relation is, however, just as much bound up with a quantitative immediacy and with the indifference of this quantitative aspect in it towards this its relation; the immediate quality has also a merely immediate quantum. Consequently, the specific measure has also a side which is, to begin with, subject to external alteration in the sense of a merely arithmetical progression unaffected by the specific measure and in which falls the external, hence only empirical, determinateness of magnitude. Quality and quantum as thus also appearing outside the specific measure are at the same time correlated with it; immediacy is a moment of those sides which themselves belong to measure. Thus the immediate qualities also belong to measure, are likewise in relation and stand in a quantitative relationship which, as outside the specified determination, the ratio of powers, is itself only a direct ratio and an immediate measure. This conclusion and its import is to be indicated in more detail.
2. Even though the immediately determined quantum as such is, in virtue of its being a moment of measure, established as in itself determinable by the Notion, it is still with reference to the specific measure an externally given quantum. But the immediacy which is thus posited is the negation of the qualitative determination of measure; this has already been demonstrated in respect of the sides of this determination of measure which for that reason appeared as independent qualities. Such negation and the return to the immediate determination of quantity lies in the qualitatively determined relation, for the relation of distinct terms as such implies their correlation as one determinateness which latter, in distinction from the determination of the relation, is here in the sphere of quantity a quantum. As a negation of the distinct qualitatively determined sides, this exponent is a being-for-self or absolute determinedness; but it is such being-for-self only in principle [an sich]; as a determinate being it is a simple immediate quantum which is the quotient or exponent of a direct ratio between the sides of the measure, but in general is the unit appearing as empirical, in the quantitative side of measure. In the motion of failing bodies the spaces traversed are proportional to the squares of the elapsed times, s = at2. This is the specifically determined relation of space and time, a ratio of powers. The other, the direct ratio, would concern space and time as mutually indifferent qualities; it is supposed to be the ratio of the space traversed to the first unit of time. The same coefficient a remains in all the following units of time — the unit being an ordinary quantum, while the amount is determined by the specifying measure. This unit at the same time counts as the exponent of that direct ratio which belongs to the imaginary, spurious velocity i.e. the merely formal velocity which is not specifically determined by the Notion. Such a velocity does not exist here, any more than the one previously mentioned which is supposed to accrue to the falling body at the end of a unit of time. That velocity was ascribed to the first unit of time in the motion of a falling body; but this so-called unit of time is itself only an assumed unit and has as such atomic point no real being.
The beginning of the motion — the alleged smallness of it could make no difference — is straightway a magnitude and one specified by the law of descent of a falling body. The said empirical quantum is attributed to the force of gravity, on the supposition that this force itself has no connection with the actual specification, with the ratio of powers characteristic of a determination of measure. The immediate moment, that in the motion of a falling body the amount of some fifteen spatial units, taken as feet, is traversed in a unit — the so-called first unit — of time, a second, is an immediate measure, like the measurements of the limbs of the human body, the distances and diameters of the planets and so forth. The determination of such a measure falls elsewhere than in the qualitative measure determination itself, here in the law of descent of a falling body; however, the concrete sciences have so far failed to throw any light on the basis of determination of such numbers, which are only immediate and consequently are the empirical embodiment of a measure. Here we are concerned only with the determinateness specified by the Notion, namely, that the said empirical coefficient constitutes the moment of being-for-self in the measure-determination, and that too only in so far as this moment is unexplicated [an sich], and hence an immediacy. The second moment is the developed side of this being-for-self, the specific measure determinateness of the sides. In the motion of falling, a motion which is still half-conditioned and half-free, gravity, according to this second moment, is to be regarded as a force of nature, so that the relationship expressed by the law of descent of a falling body is determined by the nature of space and time, and consequently the said specification, namely, the ratio of powers, falls within gravity; the above-mentioned simple direct ratio expresses only a mechanical relationship between space and time in the [merely] formal velocity which is externally produced and determined.
3. Measure has now acquired the character of a specified quantitative relation which, as qualitative, has in it the ordinary external quantum; but this is not a quantum in general, but essentially a determinant of the relation as such; hence in the sense of an exponent, and by virtue of the immediacy of its determination, of a fixed exponent, namely that of the already mentioned direct ratio between the same qualities whose quantitative relationship is at the same time specifically determined by the ratio. This direct ratio is, so to speak, anticipated and assumed as given in the example used of a measure, namely the law of descent of a falling body; but still, as we remarked, it does not exist in this motion. The fact, however, that the two sides of measure are themselves measures, the one immediate and external, and the other immanently specified, both being contained within the unity of measure itself, means that measure is now further determined, is realised. As this unity, measure contains the relation in which the magnitudes are determined and posited as differently specified by the nature of the qualities; its determinateness is accordingly wholly immanent and self-subsistent, and has at the same time collapsed into the being-for-self of an immediate quantum, the exponent of a direct ratio. The self-determination of the relation is thus negated, for in this its other it has its final, explicit determinateness; and conversely, the immediate measure which is supposed to be in its own self qualitative, possesses in truth such qualitative determinateness only in the other side of the relation. This negative unity is a real being-for-self, the category of a something as a unity of qualities which are related as measures — a completely self-subsistent something. The two sides which have presented themselves as distinct relations also immediately possess a twofold existence; or, to put it more explicitly, a self-subsistent whole of this kind, just because it is a real being-for-self, is at the same time a repulsion into distinct self-subsistent somethings whose qualitative nature and subsistence (materiality) lies in their measure determinateness.
Chapter 2: Real Measure - next section
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