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QUALITY AND QUANTITY

A Study of Hegel’s Science of Logic


   Anyone at all acquainted with the Marxist theory of dialectical materialism is familiar with the three laws of motion which express its essence, as set out by Engels on page 62 of The Dialectics of Nature. These laws, we know, provide the scientific theoretical basis for the political leadership of the struggle for the emancipation of the working class through political revolution:


1. All progress takes place through the unity and conflict, interpenetration, and    

    transformation of opposites


2. The law of the transformation of quality into quantity and vice versa


3. The law of the negation of the negation


   To formal, metaphysical thought, the transformation of quality into quantity and quantity back to quality appears impossible; only dialectical logic can reveal this truth, and the best way to train ourselves to think dialectically is by way of a study of Hegel’s Science of Logic.  The great importance of this work is that in it Hegel derives all the concepts proper to scientific thought in strict logical fashion. Starting from the most abstract and universal concept, Being, Hegel derives each concept in turn from the inner content of each preceding concept as a form of motion in accordance with the three laws stated above. However, in studying Hegel’s exposition of dialectical logic, we must take heed of a warning given by Trotsky:


   “Hegel wrote before Darwin and before Marx. Thanks to the powerful impulse given to thought by the French Revolution, Hegel anticipated the general movement of science. But because it was only an anticipation, although by a genius, it received from Hegel an idealistic character. Hegel operated with ideological shadows as the ultimate reality. Marx demonstrated that the movement of these ideological shadows reflected nothing but the movement of material bodies.”  (L. Trotsky, In Defence of Marxism page 66, ISBN 0 902030 18 3)



QUALITY

   

   

  Hegel begins his logic with the consideration of the concept of Being as the universal abstraction, which, he says, is the same as Nothing.  Clearly, since all things and phenomena are forms of being then it must be free of any quality or particularity, since to be like one thing is to be unlike anything else. But, says Hegel, though they are the same, yet they are distinct. Their truth is in their movement, which consists in the vanishing of each into its opposite. This vanishing is a particular kind of motion, and hence when it is completed, when “the vanishing vanishes”, the result is a determinate kind of Being, Determinate Being, which is a definite individual thing with its own individual identity, is a qualitative being. (See page 82, The Science of Logic, Humanity books)


   To consider a thing as a determinate quality is to accept that it has an immanent qualitative limit, a limit which is inherent in its own nature. We must be clear that we are speaking of qualitative, not quantitative limit. As an empirical example, let us take the quality colour, say blue. If the colour alters and becomes progressively paler, a limit will be reached at which it changes to white; if it becomes progressively darker, a limit will be reached at which it becomes black. These are the limits of its quality. Hegel speaks of a thing under such consideration as a something, and another something in which it is in relation as an other.


   “Something and other are, in the first place, both determinate somethings. Secondly, each is equally an other. If of two things we call one A, and the other B, then in the first place B is determined as the other. But A is just as much the other of B. Both are, in the same way, others.” (G.W.F. Hegel, The Science of Logic, page 117, Humanity Books 1999)


  Trotsky once remarked that it is necessary to practice dialectical thought like a pianist does finger exercises. Quotes such as this give us the necessary practice in dialectical thought.  It must be understood that we are here considering a something and an other, without any further determination. Since each is simultaneously both a something and an other, then they are identical, there is no difference between them, they flow into each other and again become a new something. This something is again an other which is identical to a something. This passage of each into the other Hegel calls alteration.


   There is no end to this process and Hegel calls it the qualitative infinite progress, which, for metaphysical thought, which makes something and other totally separate entities existing independently of each other, leads to the spurious infinite. Clearly, if something stands alone unconnected with any other, then it stands outside infinity itself. But if there is a something standing outside infinity, then it is the limitation of infinity, and infinity is therefore another finite something. However many finite things we accumulate, the result is always finite.

   

   The relation between the finite and the infinite can only be understood as the process of alteration of something and other into each other as a continuous process through time. A something, since it is limited, is finite. Hegel explains:


   “In order that the limit which is in something as such should be a limitation, something must at the same time in its own self transcend the limit, it must in its own self be related to the limit and to something which is not. The determinate being of something lies inertly indifferent, as it were, alongside its limit. But something only transcends its limit in so far as it is the accomplished sublation of the limit, is the in-itself as negatively related to it. And since the limit is in the determination itself as a limitation, something transcends its own self.” (Op. Cit. page 132)


   Again, Hegel is speaking of qualitative, not quantitative limit. Clearly, the limit of a something, both qualitatively and quantitatively, is determined by the quality itself. The limit of an island occurs at the point where the land ends and the sea begins. But a limit is a two-sided thing, it is related to what is limited, and to what lies beyond the limit. We notice that the limit is “in the something”, hence in relating to the limit the something must be related to its own self and its beyond, and in this latter sense in “transcends its own self”. The something is related to “what it is and what it is not”. To be related to what it is not is to “transcend the limit”, to be the “accomplished sublation”, to be already partly what it is and partly what it is not. On an earlier page Hegel explains this more generally:


   “Finite things are, but their relation to themselves is that they are negatively self-related and in this very self-relation send themselves away beyond themselves, beyond their being. They are, but the truth of this being is their end … the hour of their birth is the hour of their death.” (Op. Cit., page 129)


  The “negative self-relation” of a finite something is the relation between its identity and its own internal difference, what it is becoming, which negates the identity in the process of transformation. This negative self-relation is the very nature of all material things and must be consciously reflected in thought. All things contradict themselves in this way, so that for a true understanding of the material world external to thought we must consciously reflect this contradiction. Thought must be dialectical. In understanding the “passing beyond” of the finite we come to a true understanding of the infinite. Once again here is Hegel:


   “The falsification of the finite and infinite by the understanding which holds fast to a qualitatively distinct relation between them and asserts that each in its own nature is separate, in fact absolutely separate from each other, comes from forgetting what the Notion of these moments is for the understanding itself. According to this, the unity of the finite and the infinite is not an external bringing together of them, nor an incongruous combination alien to their own nature in which there would be joined together determinations inherently separate and opposed, each having a simple affirmative being independent of the other and incompatible with it; but each is in its own self this unity, and this only as a sublating of its own self in which neither would have the advantage over the other of having an in-itself and an affirmative determinate being. As has already been shown, finitude is only as a transcending of itself; it therefore contains infinity, the other of itself. Similarly, infinity is only as a transcending of the finite; it therefore essentially contains its other and is, consequently, in its own self the other of itself. The finite is not sublated by the infinite as a power existing outside it; on the contrary, its infinity consists in sublating its own self.” (Op. Cit., page145)

   

   Hegel goes on to describe infinity as being-for-self. All finite somethings are determined by some other in cause/effect relation, and this is being-for-other, but there is no being which can be taken as an opposite external to infinity. The opposite which determines it is within itself, the finite, and the finite contains in itself what it is becoming, the infinite. Each is in itself the unity of both. During the course of the motion of matter and thought the finite continually passes beyond to become the infinite. (This is sublation). The infinite, its opposite, only finds being as the finite in every moment.




QUANTITY


   The above study of the dialectical relation between the finite and the infinite according to Hegel provides us with the means to grasp the nature of quality and how it becomes transformed into quanity. However, Hegel was an idealist and he constructed his system of categories and concepts entirely subjectively, as a process of thought. He next proceeds to describe being-for-self as the one, but his derivation of the concept of the one, and the concept of the many from the concept of the one seems unsound; certainly Lenin thought so. In his philosophical notes he refers to this passage in Hegel as follows:


   “In general, all this Being-for-self was, probably, in part required by Hegel to deduce ‘the transition of quality into quantity’ –  quality is determinateness, determinateness for self, the posited, it is the One – this gives the impression of being very farfetched and empty.” (Collected Works, Vol. 38, page 114)


   Whatever we think of Hegel’s subjectively derived conception of the one and the many ones, objective consideration of the world external to thought, empirical knowledge, shows that quantity is exactly that. A heap of stones is one heap and many stones. Quantity is a unity of the one and the many. Hegel goes on to relate quantity to quality:


   “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” (The Science of Logic, page 185)



 Quality is being-for-self, which is infinite. But quantity is the infinite in relation to something. To be a quantity at all it must be a quantity of something, which is finite, so that quantity implies some limit, although no limit is posited. Quantity is a limit which is just as much no limit. Once again we encounter contradiction. We shall encounter it in every moment. “Repulsion” means the division of the one into the many, and “attraction” means the unifying of the many into the one. Hegel describes quantity as the unity of repulsion and attraction:


   “Repulsion is, although negative, still essentially relation; the mutual repulsion and flight is not a liberation from what is repelled and fled from, the one as excluding still remains related to what it excludes. But this moment of relation is attraction and thus is in repulsion itself; it is the negating of that repulsion according to which the ones would be only self-related affirmative beings not excluding one another.” (Op. Cit., page 175)


   The Hegelian system is based on the understanding that each finite something, the self-determined, reflects within itself all that it is in relation to. The whole of which it is a part is reflected within itself. Since it is one of many ones, then these are reflected within itself, so that “the one is many”. But the many which it reflects are united into a one whole through attraction, “the many are one”. In this way the part and the whole are identical.


   “The being-for-self of the one, is, however, essentially the ideality of determinate being and of other. It relates itself not to an other but to itself.  But since being-for-self is fixed as a one, as affirmatively for itself, as immediately present, its negative relation to itself is at the same time a relation to an affirmative being; and since the relation is just as much negative, that to which it relates itself remains determined as a determinate being and an other … is likewise a one. The one is consequently a becoming of many ones.” (Op. Cit., page 167)


   This is simpler than it looks. Hegel is explaining the situation from all possible sides and aspects, leaving nothing out. Each one, in unity and reflecting all the other ones, repels itself from itself, splits and reproduces itself like the amoeba, hence quantity continually reproduces itself.  Although it is quality which it reproduces it is still “indifferent to quality”, because it reproduces all or any quality without reference to it. This is the secret of the transformation of quality into quantity. It is a process which takes place through time, unlike its opposite process, the transformation of quantity into quality, which takes the form of a leap. It is the becoming of the universe itself and everything in it. Next Hagel develops the concepts of repulsion and attraction into continuous and discrete magnitude:


   “Quantity contains the two moments of continuity and discreteness. It is to be posited in both of them as determinations of itself. It is already their immediate unity, that is, quantity is posited at first in only one of its determinations, continuity, and as such is continuous magnitude.” (Op. Cit., page 199)


   If we take an extension 10 inches long, then it is a one and continuous, but it is simultaneously 10 separate inches, and this is a many and discreteness. The 10 inches are repelled from each other, otherwise they could not have separate being, and they are attracted to each other into the one extension, flowing into each other, because they are identical and undifferentiated. It is important to understand that both continuity and discreteness embrace the whole magnitude, although they are distinct moments of the whole, each depending on the other for its being. Clearly, nothing can be continuous unless it is the continuation of a plurality of ones, and there can be no discrete plurality without a continuity as a basis for separation into parts. Hegel next takes a very important step:


   “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 as determinate being, a something, with the one as its determinateness and also as its first negation and limit.” (Op. Cit. page 201)


   Above we discussed the limitation of quality; here we discuss the limitation of quantity. If we take the 10 inch extension as discrete quantity, it is a plurality of one inch extensions. But they are identical, not distinguished from each other, and they flow into each other. Each is sublated into the whole, losing its own individual identity, and passes over into continuous quantity. The whole is then posited as one magnitude, instead of 10 separate magnitudes. But this particular magnitude, this 10 inch extension, is part of infinite, or at least unlimited, extension, quantity. As this part, it excludes all extension external to itself. The crucial thing for dialectical thought is to grasp that it itself excludes all that is beyond it.   It contains within itself its own limitation, its own negation, the point beyond which it is not. The important step we mentioned is the introduction of the concept of limit; here we see, as it were approaching over the horizon, the finite returning, the leap from quantity to quality. We have a little way to go yet.


   The concept of “a” quantity, quantity with a definite limit, is the concept of quantum.


   “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, passes over thirdly, into quantitative infinity. (Op. Cit., page 202)


   Quantum is any quantity with a limit, and its most concrete form is number. Number not only sets the limit, but measures the extent. But quantity as such, like every something or being, contains contradiction, is sundered into two sides or aspects, moments. The moments of quantum are extensive quantum and intensive quantum. Extensive quantum is magnitude, each discrete part lined up externally to the others, 1,2,3 etc. Intensive quantum is degree, such as a reading of a thermometer registering temperature. At 20 degrees, all the lower readings are contained within the final, limiting reading, in this case 20.  What gives the twentieth degree meaning, what determinates it, is its comparison with all those degrees contained within it, and as we have seen, that which has its opposite determinate being absorbed within its own self is being-for-self. But equally intensive quantum, degree, is compared, (is in unity with), to those degrees outside itself, and in this sense is determined by them, is being-for-other. This other has no limit, and is therefore quantitative infinity. This being exists in relation to its opposite, qualitative infinity, which we dealt with above.


   Next Hegel develops the relation between quantum and quantitative infinity:


   “The difference between extensive and intensive quantum is indifferent to the determination of quantum as such. But in general quantum is determinateness posited as sublated, the indifferent limit, the determination of which is equally the negation of itself.” (Op. Cit., page224)


  Hegel is nothing if not thorough. As a cardinal rule, where there is difference there is contradiction, life and movement, cause and effect. Just so that we know that we do not need to take the difference between extensive and intensive quantum into account at this point, he explains that it is “indifferent to the determination of quantum as such”, but taken on their own they are by no means indifferent to each other. Quantum is a determination posited as sublated because it is a limited quantity not now standing alone, but merely a part of something greater, infinite quantity. The limit is indifferent to infinite quantity since no reference was made to it when the limit was set; quantum is limited in-itself.


   The important thing here is that the determination of the limit is at the same time the negation of the limit because to set a limit is to posit what lies beyond, and since there is a beyond it is no limit at all. It is negated. Hegel explains:


   “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 simply affirmative limit, but a limit which becomes.” (Op. Cit., page 225)


   If we take a limited quantity we have by no means wrenched it away from the rest of quantity, clearly it is still a part of quantity and beyond the limit it continues on. Quantum can increase or decrease through change of the limits, it can alter. But, says Hegel, since quantum is in continuity with unlimited quantity it must alter:


   “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 the quantum to impel itself beyond itself and to become other. It consists in undergoing increase or decrease; it is in its own self the externality of the determinateness.” (Ibid.)


   The one, (or something which we discussed in the sphere of quality above), in becoming other, produces its own self, another something. But quantum is different. That which is beyond the limit appears to be infinite, but since it is in unity with quantum which lies outside it, the beyond is also limited and is thus a quantum. The first quantum has reproduced itself, has “repelled itself from itself”. But though it is a quantum, it is not the same quantum, but a greater one, and it is in this sense that quantum must alter. Again, the rest of quantity lies outside it, which again is quantum, and so on to infinity. Quantum is “not simply affirmative limit”, says Hegel, “but a limit which becomes”, and he goes on “… it is now the express character of quantum to impel itself beyond itself and to become other. It consists in undergoing increase and decrease; it is in its own self the externality of the determinateness.” (Ibid.)


  Quantity is not to be thought of as a static amount. It is the infinite unfolding and reproduction of itself from within itself to infinity.


   But for all this quantum is limited, and however many quanta we have, and however it may increase of decrease, the result must be finite. So we are here in the world of the spurious infinite once again, a thought process with which we are all familiar. Hegel demonstrates it with a quote from The Critique of Pure Reason by Emanuel Kant:


   “ … when the subject raises himself in thought above the place he occupies in the world of sense, reaching out to infinity, to stars beyond stars, worlds beyond worlds, systems beyond systems, and then also to the limitless times of their periodic motion, their beginning and duration. Imagination fails before this progress into the infinitely remote, where beyond the most distant world there is still a more distant one, and the past, however remote, has still a remoter past behind it, the future, however distant, a still more distant future beyond it; thought fails in the face of this conception on the immeasurable, just as a dream, in which one goes on and on down a corridor which stretches away endlessly out of site, finishes with falling or fainting.” (Quoted on page 229.)


   A beautiful piece of prose, but while Kant got his art right he got his science wrong. He thought metaphysically, that is, for him all things existed for themselves, in isolation and completely separately. To this way of thinking, the finite exists quite independently of the infinite, and vice versa, and as a result Kant was stuck in the spurious infinite. For dialectical thought any thing, quantum, is self-contradictory. It is finite but only has being in its continual alteration, the transcendence of itself, and is therefore infinite. Again, the infinite only has being in every moment as the finite, quantum. They are united in their opposition, both are finite and infinite. We discovered this relation between the finite and the infinite above when discussing the qualitative finite and infinite.


   When we dealt with the nature of quantity as such we said that it was indifferent to quality; a quantity can be a quantity of any quality. But, as suggested above, Hegel is now bringing the two into a concrete unity:


   “The infinity of quantum has been determined to the stage where it is the negative beyond of quantum, which beyond, however, is contained within the quantum itself. This beyond is the qualitative moment as such. The infinite quantum as the unity of both moments, of the quantitative and qualitative determinateness, is in the first instance a ratio.” (Op.Cit. page 314).


   The infinity of quantum is negative because at first it stands outside it, facing it, something which is “not it.” But this is contradictory, because quantity is continuous; there is no separation between what is inside and what is outside of the quantum. The beyond taken in this way, as including the quantum, and at the same time as related to the quantum as such, gives us two different quantities related to each other as a unity. Such a relation is a ratio.   A ratio is, on the one side, two quanta which are in unity and conflict, and on the other side, a single quantum which is the synthesis of both. For example, consider this ratio;


10:2 = 5


   The quantity 10 is the beyond of quantum, and the quantity 2 is the quantum. The quantum 5 is the synthesis of both, expressing the relation between them, “the unity of the quantitative and qualitative moments”. The crucial question here, is this; why is the beyond the qualitative moment? We must now quote a long paragraph from Hegel, but he is only explaining this one thing as thoroughly as he can:


   “In the ratio, quantum no longer has an indifferent determinateness but is qualitatively determined as simply related to its beyond. It continues itself into its beyond; this, in the first place, is simply another quantum. But they are essentially related to each other not as external quanta, but the determinateness of each consists in this relation to the other. In this their otherness they have thus returned into themselves; what each is, it is in the other; the other constitutes the determinateness of each. The flight of quantum away from and beyond itself has now therefore this meaning, that it changed not merely into an other, or into its abstract other, into its negative beyond, but that in this other it reached its determinateness, finding itself in its beyond, which is another quantum. The quality of quantum, the specific nature of its Notion, is its externality as such, and in ratio the quantum is now posited as having its determinateness in its externality, in another quantum, and as being in its beyond what it is.” (Ibid.)


   The first sentence is a simple statement of the fact which must be explained. We recall that quantity was “indifferent to its determinateness” – a quantity can be a quantity of any quality. But, in ratio, a limited quantity, quantum, is related to its beyond, and this is a relation between something and other, a qualitative relationship. Each is determined by its other as one side of a ratio, and it is this which is their quality. “What each is it is in the other”, says Hagel. Each is a reflection of the other, but is reciprocally reflected in the other in the same way, hence what each “sees” in the other is only itself – it has returned to itself. Such self-determination, being-for-self, is something, a quality. The rest of the paragraph is an explanation of this.


   However, in this moment, ratio, quality remains abstract. It is not any specific quality, just quality itself. Hegel often says of such moments, “it is equal only to itself”. Abstract quality only becomes concretised into determinate quality in the moment of measure.



MEASURE


   “Quantum is now no longer an indifferent or external determination but as such is sublated and is quality, and is that by virtue of which something is what it is; this is the truth of quantum, to be Measure.


   First let us understand how Hegel uses this word. Remember that at the start we said that Hegel derived all the concepts and categories proper to philosophy in strict scientific fashion. The concept measure is directly derived from the concept ratio.


   Consider measurement in practice. We begin with the thing to be measured, then compare it with some standard of measurement which is socially derived. Immediately we have a ratio, a comparison between two quantities, and it is in this sense that Hegel uses the concept measure. It is not one sided, it is always the relation between two quantities, a proportionality. The importance of this can be understood if we reconsider the two sides of quantity, extensive and intensive quantity, as they are contained in the moment of measure. The ratio 10:2 gave the exponent 5. But we get the same exponent with the ratios 20:4, 40:8, or 100:20. Here extensive quantity differs while the exponent, intensive quantity, remains the same.  It is this most decisive moment, when the two sides of the ratio reach the correct ratio, the moment of measure, when the leap from quantity to quality takes place. Let us test Hegel’s logic materialistically by analysis of the world of matter in motion, in this case chemical motion.


  It takes 1 calorie of heat to raise 1 gram of water through 1 degree centigrade. Here the calorie and the gram are the two quanta of the ratio (Extensive) and the degree centigrade is the exponent (Intensive – so much heat per unit weight). We know that if we raise the temperature to 100 degrees there will be a qualitative change, the water will change to steam. In this case the ratio is 100:1 = 100. We could get steam with a ratio of 200:2, 1000:10, or any other extensive quantity provided that the correct intensive quantity is reached. We need not consider ratios greater than 100:1 because such a ratio can only be reached by passing through the moment at which the ratio was 100:1. We call this the limit of measure.


   However, any other changes in extensive quantity, say 50:1 or 1000:25 would not achieve the qualitative leap, so we see that extensive quantity has nothing to do with qualitative change, it is intensive quantity which causes the leap. There is one last consideration to take into account. We know that, when the moment of measure is reached in all such transformations, a further quantitative increase in necessary to bring about the transformation. In the case of heat and water, this is called the latent heat of steam. The water continues to absorb heat without rise in temperature, and this additional heat is transformed into mechanical energy in the form of increased excitation of the water molecules, loosening their bonds and causing an increase in volume, hence the steam engine. This is the moment of invariance, when the two sides of the process are briefly in being simultaneously at the moment of transformation, and it is present as a moment of measure in all transformations, in physical matter, human society, (the class struggle and revolution), and thought.

 
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