The Quantitative Relation or Quantitative Ratio

§ 695

The quantitative infinite and also the difficulties associated 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.

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 quantitative forms intrude 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.

Section Three: Measure

§ 699

Abstractly expressed, in measure quality and quantity are united. Being as such is an immediate identity of the determinateness with itself. This immediacy of the determinateness has sublated itself. Quantity is being which has returned into itself in such a manner that it is a simple self-identity as indifference to the determinateness.

§ 700

But this indifference is only the externality of having the determinateness not in its own self but in an other. Thirdly, we now have self-related externality; as self-related it is also a sublated externality and has within itself the difference from itself-the difference which, as an externality is the quantitative, and as taken back into itself is the qualitative, moment.

§ 701

In transcendental idealism the categories of quantity and quality are followed, after the insertion of relation, by modality, which may therefore be mentioned here.

This category has there the meaning of being the relation of the object to thought. According to that idealism thought generally is essentially external to the thing-in-itself.

In so far as the other categories have only the transcendental character of belonging to consciousness, but to the objective element of it, so modality as the category of relation to the subject, to this extent contains relatively the determination of reflection-into-self; i.e. the objectivity which belongs to the other categories is lacking in the categories of modality; these, according to Kant, do not in the least add to the concept as a determination of the object but only express the relation to the faculty of cognition.

The categories which Kant groups under modality — namely, possibility, actuality and necessity will occur later in their proper place; Kant did not apply the infinitely important form of triplicity — with him it manifested itself at first only as a formal spark of light — to the genera of his categories (quantity, quality, etc.), but only to their species which, too, alone he called categories. Consequently he was unable to hit on the third to quality and quantity.

§ 702

With Spinoza, the mode is likewise the third after substance and attribute; he explains it to be the affections of substance, or that element which is in an other through which it is comprehended. According to this concept, this third is only externality as such; as has already been mentioned, with. Spinoza generally, the rigid nature of substance lacks the return into itself.

§ 703

The observation here made extends generally to those systems of pantheism which have been partially developed by thought.

The first is being, the one, substance, the infinite, essence; in contrast to this abstraction the second, namely, all determinateness in general, what is only finite, accidental, perishable, non-essential, etc. can equally abstractly be grouped together; and this is what usually happens as the next step in quite formal thinking. But the connection of this second with the first is so evident that one cannot avoid grasping it as also in a unity with the latter; thus with Spinoza, the attribute is the whole substance, but is apprehended by the intellect which is itself a limitation or mode; but in this way the mode, the non-substantial generally, which can only be grasped through an other, constitutes the other extreme to substance, the third generally. Indian pantheism, too, in its monstrous fantasies has in an abstract way received this development which runs like a moderating thread through its extravagances; a point of some interest in the development is that Brahma, the one of abstract thought, progresses through the shape of Vishnu, particularly in the form of Krishna, to a third form, that of Siva. The determination of this third is the mode, alteration, coming-to-be and ceasing-to-be-the field of externality in general. This Indian trinity has misled to a comparison with the Christian and it is true that in them a common element of the nature of the Notion can be recognised; but it is essential to gain a more precise consciousness of the difference between them; for not only is this difference infinite, but it is the true, the genuine infinite which constitutes it. This third principle is, according to its determination, the dispersal of the unity of substance into its opposite, not the return of the unity to itself — not spirit but rather the non-spiritual. In the true trinity there is not only unity but union, the conclusion of the syllogism is a unity possessing content and actuality, a unity which in its wholly concrete determination is spirit. This principle of the mode and of alteration does not, it is true, altogether exclude the unity; in Spinozism, for example, it is precisely the mode as such which is untrue; substance alone is true and to it everything must be brought back. But this is only to submerge all content in the void, in a merely formal unity lacking all content. Thus Siva, too, is again the great whole, not distinct from Brahma, but Brahma himself. In other words, the difference and the determinateness only vanish again but are not preserved, are not sublated, and the unity does not become a concrete unity, neither is the disunity reconciled. The supreme goal for man placed in the sphere of coming-to-be and ceasing-to-be, of modality generally, is submergence in unconsciousness, unity with Brahma, annihilation; the Buddhist Nirvana, Nibbana etc., is the same.

§ 704

Now although the mode as such is abstract externality, indifference to qualitative and quantitative determinations, and in essence the external and unessential elements are not supposed to count, it is still, on the other hand, admitted in many cases that everything depends on the kind and manner of the mode; such an admission means that the mode itself is declared to belong essentially to the substantial nature of a thing, a very indefinite connection but one which at least implies that this external element is not so abstractly an externality.

§ 705

Here the mode has the specific meaning of measure. Spinoza’s mode, like the Indian principle of change, is the measureless. The Greek awareness, itself still indeterminate, that everything has a measure — even Parmenides, after abstract being, introduced necessity as the ancient limit by which all things are bounded — is the beginning of a much higher conception than that contained in substance and in the difference of the mode from substance.

§ 706

Measure in its more developed, more reflected form is necessity; fate, Nemesis, was restricted in general to the specific nature of measure, namely, that what is presumptuous, what makes itself too great, too high, is reduced to the other extreme of being brought to nothing, so that the mean of measure, mediocrity is restored. ‘The absolute, God, is the measure of all things’ is not more intensely pantheistic than the definition: ‘The absolute, God, is being,’ but it is infinitely truer. Measure, it is true, is an external kind and manner of determinateness, a more or less, but at the same time it is equally reflected into itself, a determinateness which is not indifferent and external but intrinsic; it is thus the concrete truth of being. That is why mankind has revered measure as something inviolable and sacred.

§ 707

The Idea of essence, namely, to be self-identical in the immediacy of its determined being, is already immanent in measure; so that the immediacy is thus reduced by this self-identity to something mediated, which equally is mediated only through this externality, but is a mediation with itself — that is, reflection, the determinations of which are, but in this being are nothing more than moments of their negative unity. In measure, the qualitative moment is quantitative; the determinateness or difference is indifferent and so is no difference, is sublated. This nature of quantity as a return-into-self in which it is qualitative constitutes that being-in-and-for-itself which is essence. But measure is only in itself or in its Notion essence; this Notion of measure is not yet posited. Measure, still as such, is itself the immediate [seiende] unity of quality and quantity; its moments are determinately present as a quality, and quanta thereof; these moments are at first inseparable only in principle [an sich], but do not yet have the significance of this reflected determination. The development of measure contains the differentiation of these moments, but at the same time their relation, so that the identity which they are in themselves becomes their relation to each other, i.e. is posited. The significance of this development is the realisation of measure in which it posits itself as in relation with itself, and hence as a moment. Through this mediation it is determined as sublated; its immediacy and that of its moments vanishes; they are reflected. Measure, having thus realised its own Notion, has passed into essence.

§ 708

At first, measure is only an immediate unity of quality and quantity, so that: (1), we have a quantum with a qualitative significance, a measure. The progressive determining of this consists in explicating what is only implicit in it, namely, the difference of its moments, of its qualitatively and quantitatively determined being. These moments further develop themselves into wholes of measure which as such are self-subsistent. These are essentially in relationship with each other, and so measure becomes (2), a ratio of specific quanta having the form of self-subsistent measures. But their self-subsistence also rests essentially on quantitative relation and quantitative difference; and so their self-subsistence becomes a transition of each into the other, with the result that measure perishes in the measureless. But this beyond of measure is the negativity of measure only in principle; this results (3), in the positing of the indifference of the determinations of measure, and the positing of real measure — real through the negativity contained in the indifference — as an inverse ratio of measures which, as self-subsistent qualities, are essentially based only on their quantity and on their negative relation to one another, thereby demonstrating themselves to be only moments of their truly self-subsistent unity which is their reflection-into-self and the positing thereof, essence.

§ 709

The development of measure which has been attempted in the following chapters is extremely difficult. Starting from immediate, external measure it should, on the one hand, go on to develop the abstract determination of the quantitative aspects of natural objects (a mathematics of nature), and on the other hand, to indicate the connection between this determination of measure and the qualities of natural objects, at least in general; for the specific proof, derived from the Notion of the concrete object, of the connection between its qualitative and quantitative aspects, belongs to the special science of the concrete. Examples of this kind concerning the law of falling bodies and free, celestial motion will be found in the Encyclopedia. of the Phil. Sciences, 3rd ed., Sections 267 and 270, Remark. In this connection the general observation may be made that the different forms in which measure is realised belong also to different spheres of natural reality. The complete, abstract indifference of developed measure, i.e. the laws of measure, can only be manifested in the sphere of mechanics in which the concrete bodily factor is itself only abstract matter; the qualitative differences of such matter are essentially quantitatively determined; space and time are the purest forms of externality, and the multitude of matters, masses, intensity of weight, are similarly external determinations which have their characteristic determinateness in the quantitative element. On the other hand, such quantitative determinateness of abstract matter is deranged simply by the plurality of conflicting qualities in the inorganic sphere and still more even in the organic world. But here there is involved not merely a conflict of qualities, for measure here is subordinated to higher relationships and the immanent development of measure tends to be reduced to the simple form of immediate measure. The limbs of the animal organism have a measure which, as a simple quantum, stands in a ratio to the other quanta of the other limbs; the proportions of the human body are the fixed ratio of such quanta. Natural science is still far from possessing an insight into the connection between such quantities and the organic functions on which they wholly depend. But the readiest example of the reduction of an immanent measure to a merely externally determined magnitude is motion. In the celestial bodies it is free motion, a motion which is determined solely by the Notion and whose quantitative elements therefore equally depend solely on the Notion (see above); but such free motion is reduced by the living creature to arbitrary or mechanically regular, i.e. a wholly abstract, formal motion.

§ 710

And in the realm of spirit there is still less to be found a characteristic, free development of measure. It is quite evident, for example, that a republican constitution like that of Athens, or an aristocratic constitution tempered by democracy, is suitable only for States of a certain size, and that in a developed civil society the numbers of individuals belonging to different occupations stand in a certain relations to one another; but all this yields neither laws of measure nor characteristic forms of it. In the spiritual sphere as such there occur differences of intensity of character, strength of imagination, sensations, general ideas, and so on; but the determination does not go beyond the indefiniteness of strength or weakness. How insipid and completely empty the so-called laws turn out to be which have been laid down about the relation of strength and weakness of sensations, general ideas, and so on, comes home to one on reading the psychologies which occupy themselves with such laws.

Chapter 1: Specific Quantity A The Specific Quantum B Specifying Measure (a) The Rule (b) Specifying Measure (c) Relation of the Two Sides as Qualities

Remark

§ 736

The exposition here of the connection between the qualitative nature of something and its quantitative determination has its application in the already indicated example of motion. First of all, in velocity as the direct ratio of space traversed and time elapsed, the magnitude of time is taken as denominator while that of space is taken as numerator. If velocity as such is only a ratio of the space and time in a motion, it is immaterial which of the two moments is to be considered as amount or as unit. Space, however, like weight in specific gravity, is an external, real whole as such — hence amount — whereas time, like volume, is the ideal, negative factor, the side of unity. But here there essentially belongs the more important ratio, that which holds between the magnitudes of space and time in free motion; at first, in the still conditioned motion of a falling body where the time factor is determined as a root and the space factor as a square, or in the absolutely free motion of the celestial bodies where the period of revolution is lower by one power than the distance from the sun, the former being a square and the latter a cube. Fundamental relationships of this kind rest on the nature of the interrelated qualities of space and time and on the kind of relation in which they stand, either as a mechanical motion, i.e. as an unfree motion which is not determined by the Notion of the moments of space and time, or as the descent of a falling body, i.e. as a conditionally free motion, or as the absolutely free celestial motion. These kinds of motion, no less than their laws, rest on the development of the Notion of their moments, of space and time, since these qualities as such (space and time) prove to be in themselves, i.e. in their Notion, inseparable and their quantitative relationship is the being-for-self of measure, is only one measure-determination.

§ 737

In regard to the absolute relations of measure, it is well to bear in mind that the mathematics of nature, if it is to be worthy of the name of science, must be essentially the science of measures — a science for which it is true much has been done empirically, but little as yet from a strictly scientific, that is, philosophical point of view. Mathematical principles of natural philosophy-as Newton called his work-if they are to fulfil this description in a profounder sense than that accorded to them by Newton and by the entire Baconian species of philosophy and science, must contain things of quite a different character in order to bring light into these still obscure regions which are, however, worthy in the highest degree of consideration.

It is a great service to ascertain the empirical numbers of nature, e.g., the distances of the planets from one another; but it is an infinitely greater service when the empirical quanta are made to disappear and they are raised into a universal form of determinations of quantity so that they become moments of a law or of measure — immortal services which Galileo for the descent of falling bodies and Kepler for the motion of the celestial bodies, have achieved. The laws they discovered they have proved in this sense, that they have shown the whole compass of the particulars of observation to correspond to them. But yet a still higher proof is required for these laws; nothing else, that is, than that their quantitative relations be known from the qualities or specific Notions of time and space that are correlated.

Of this kind of proof there is still no trace in the said mathematical principles of natural philosophy, neither is there in the subsequent works of this kind. It has already been remarked in connection with the show of mathematical proofs of certain relationships in nature, a show based on the misuse of the infinitely small, that it is absurd to try todemonstrate such proofs on a strictly mathematical basis, i.e. neither empirically nor from the standpoint of the Notion. These proofs presuppose thir theorems, those very laws, from experience; what they succeed in doing is to reduce them to abstract expressions and convenient formulae.

Undoubtedly the time will come when, with a clearer understanding of what mathematics can accomplish and has accomplished, the entire, real merit of Newton as against Kepler — the sham scaffolding of proofs being discarded — will clearly be seen to be restricted to the said transformation of Kepler’s formula and to the elementary analytical treatment accorded to it.

C Being-for-self in Measure Chapter 2 Real Measure A The Relation of Self-Subsistent Measures (a) Combination of Two Measures (b) Measure of a Series of Measure Relations (c) Elective Affinity Remark: Berthollet on Chemical Affinity and Berzelius’s Theory of it

B Nodal Line of Measure Relations Remark: Examples of Such Nodal Lines; the Maxim, ‘Nature Does Not Make Leaps’

§ 774

The system of natural numbers already shows a nodal line of qualitative moments which emerge in a merely external succession. It is on the one hand a merely quantitative progress and regress, a perpetual adding or subtracting, so that each number has the same arithmetical relation to the one before it and after it, as these have to their predecessors and successors, and so on. But the numbers so formed also have a specific relation to other numbers preceding and following them, being either an integral multiple of one of them or else a power or a root. In the musical scale which is built up on quantitative differences, a quantum gives rise to an harmonious relation without its own relation to those on either side of it in the scale differing from the relation between these again and their predecessors and successors. While successive notes seem to be at an ever-increasing distance from the keynote, or numbers in succeeding each other arithmetically seem only to become other numbers, the fact is that there suddenly emerges a return, a surprising accord, of which no hint was given by the quality of what immediately preceded it, but which appears as an actio in distans, as a connection with something far removed. There is a sudden interruption of the succession of merely indifferent relations which do not alter the preceding specific reality or do not even form any such, and although the succession is continued quantitatively in the same manner, a specific relation breaks in per saltum.

§ 775

Such qualitative nodes and leaps occur in chemical combinations when the mixture proportions are progressively altered; at certain points in the scale of mixtures, two substances form products exhibiting particular qualities. These products are distinguished from one another not merely by a more or less, and they are not already present, or only perhaps in a weaker degree, in the proportions close to the nodal proportions, but are bound up with these nodes themselves. For example, different oxides of nitrogen and nitric acids having essentially different qualities are formed only when oxygen and nitrogen are combined in certain specific proportions, and no such specific compounds are formed by the intermediate proportions. Metal oxides, e.g. the lead oxides, are formed at certain quantitative points of oxidation and are distinguished by colours and other qualities. They do not pass gradually into one another; the proportions lying in between these nodes do not produce a neutral or a specific substance. Without having passed through the intervening stages, a specific compound appears which is based on a measure relation and possesses characteristic qualities. Again, water when its temperature is altered does not merely get more or less hot but passes through from the liquid into either the solid or gaseous states; these states do not appear gradually; on the contrary, each new state appears as a leap, suddenly interrupting and checking the gradual succession of temperature changes at these points. Every birth and death, far from being a progressive gradualness, is an interruption of it and is the leap from a quantitative into a qualitative alteration.

§ 776

It is said, natura non facit saltum [there are no leaps in nature]; and ordinary thinking when it has to grasp a coming-to-be or a ceasing-to-be, fancies it has done so by representing it as a gradual emergence or disappearance. But we have seen that the alterations of being in general are not only the transition of one magnitude into another, but a transition from quality into quantity and vice versa, a becoming-other which is an interruption of gradualness and the production of something qualitatively different from the reality which preceded it. Water, in cooling, does not gradually harden as if it thickened like porridge, gradually solidifying until it reached the consistency of ice; it suddenly solidifies, all at once. It can remain quite fluid even at freezing point if it is standing undisturbed, and then a slight shock will bring it into the solid state.

§ 777

In thinking about the gradualness of the coming-to-be of something, it is ordinarily assumed that what comes to be is already sensibly or actually in existence; it is not yet perceptible only because of its smallness. Similarly with the gradual disappearance of something, the non-being or other which takes its place is likewise assumed to be really there, only not observable, and there, too, not in the sense of being implicitly or ideally contained in the first something, but really there, only not observable. In this way, the form of the in-itself, the inner being of something before it actually exists, is transformed into a smallness of an outer existence, and the essential difference, that of the Notion, is converted into an external difference of mere magnitude. The attempt to explain coming-to-be or ceasing-to-be on the basis of gradualness of the alteration is tedious like any tautology; what comes to be or ceases to be is assumed as already complete and in existence beforehand and the alteration is turned into a mere change of an external difference, with the result that the explanation is in fact a mere tautology. The intellectual difficulty attendant on such an attempted explanation comes from the qualitative transition from something into its other in general, and then into its opposite; but the identity and the alteration are misrepresented as the indifferent, external determinations of the quantitative sphere.

§ 778

In the moral sphere, in so far as it is considered under the categories of being, there occurs the same transition from quantity into quality and different qualities appear to be based in a difference of magnitude.

It is through a more or less that the measure of frivolity or thoughtlessness is exceeded and something quite different comes about, namely crime, and thus right becomes wrong and virtue vice. Thus states, too, acquire through their quantitative difference, other things being assumed equal, a distinct qualitative character. With the expansion of the state and an increased number of citizens, the laws and the constitution acquire a different significance. The state has its own measure of magnitude and when this is exceeded this mere change of size renders it liable to instability and disruption under that same constitution which was its good fortune and its strength before its expansion.

C The Measureless

Chapter 3: The Becoming of Essence

A Absolute Indifference B Indifference as an Inverse Ratio of its Factors Remark: Centripetal and Centrifugal Force

C Transition into Essence

§ 803

Absolute indifference is the final determination of being before it becomes essence; but it does not attain to essence. It reveals itself as still belonging to the sphere of being through the fact that, determined as indifferent, it still contains difference as an external, quantitative determination; this is its determinate being, contrasted with which absolute indifference is determined as being only implicitly the absolute, not the absolute grasped as actuality. In other words, it is external reflection which stops short at conceiving the differences in themselves or in the absolute as one and the same, thinking of them as only indifferently distinguished, not as intrinsically distinct from one another. The further step which requires to be made here is to grasp that this reflection of the differences into their unity is not merely the product of the external reflection of the subjective thinker, but that it is the very nature of the differences of this unity to sublate themselves, with the result that their unity proves to be absolute negativity, its indifference to be just as much indifferent to itself, to its own indifference, as it is indifferent to otherness.

§ 804

But we are already familiar with this self-sublating of the determination of indifference; in the development of its positedness, this determination has shown itself to be from every aspect a contradiction. It is in itself the totality in which every determination of being is sublated and contained; it is thus the substrate, but at first only in the one-sided determination of the in-itself, and consequently the differences, namely, the quantitative difference and the inverse ratio of factors, are present in it only in an external manner. As thus the contradiction of itself and its determinedness, of its implicit determination and its posited determinateness, it is the negative totality whose determinatenesses have sublated themselves in themselves and in so doing have sublated this fundamental one-sidedness of theirs, their [merely] implicit being [Ansichsein]. The result is that indifference is now posited as what it in fact is, namely a simple and infinite, negative relation-to-self, its inherent incompatibility with itself, a repelling of itself from itself. The process of determining and being determined is not a transition, nor an external alteration, nor an emergence of determinations in the indifference, but is its own self-relating which is the negativity of itself, of its [merely] implicit being.

§ 805

Now these repelled determinations do not possess themselves, do not emerge as self-subsistent or external determinations, but first, as moments belonging to the implicit unity, they are not expelled from it but are borne by it as the substrate and are filled solely by it; secondly, as determinations which are immanent in the explicated unity, they are only through their repulsion from themselves. The being of the determinations is no longer simply affirmative as in the entire sphere of being, but is now a sheer positedness, the determinations having the fixed character and significance of being related to their unity, each consequently being related to its other and with negation; this is the mark of their relativity.

§ 806

Thus we see that being in general and the being or immediacy of the distinct determinatenesses, no less than the implicit being, has vanished and the unity is being, an immediate presupposed totality such that it is this simple self-relation only as a result of the sublating of this presupposition, and this presupposedness and immediate being is itself only a moment of its repelling, the original self-subsistence and self-Identity is only as the resulting coming together with itself. Being, in its determining, has thus determined itself to essence, a being which, through the sublating of being, is a simple being-with-itself. ®

ESSENCE - Second Part of The Logic