Part One of the Encyclopedia of Philosophical Sciences: The Logic
Pure Quantity — Quantum — Degree
Quantity is pure Being, where the mode or character is no longer taken as one with the being itself, but explicitly put as superseded or indifferent.
(1) The expression Magnitude especially marks determinate Quantity, and is for that reason not a suitable name for Quantity in general. (2) Mathematics usually define magnitude as what can be increased or diminished. This definition has the defect of containing the thing to be defined over again: but it may serve to show that the category of magnitude is explicitly understood to be changeable and indifferent, so that, in spite of its being altered by an increased extension or intension, the thing - a house, for example - does not cease to be a house, and red to be red. (3) The Absolute is pure Quantity. This point of view is on the whole the same as when the Absolute is defined to be Matter, in which, though form undoubtedly is present, the form is a characteristic of no importance one way or another. Quantity too constitutes the main characteristic of the Absolute, when the Absolute is regarded as absolute indifference, and only admitting of quantitative distinction. Otherwise pure space, time, etc., may be taken as examples of Quantity, if we allow ourselves to regard the real as whatever fills up space and time, it matters not with what.
The mathematical definition of magnitude as what may be increased or diminished, appears at first sight to be more plausible and perspicuous than the exposition of the notion in the present section. When closely examined, however, it involves, under cover of presuppositions and images, the same elements as appear in the notion of quantity reached by the method of logical development. In other words, when we say that the notion of magnitude lies in the possibility of being increased or diminished, we state that magnitude (or more correctly, quantity), as distinguished from quality, is a characteristic of such kind that the characterised thing is not in the least affected by any change in it.
What then, it may be asked, is the fault which we have to find with this definition? It is that to increase and to diminish is the same thing as to characterise magnitude otherwise. If this aspect then were an adequate account of it, quantity would be described merely as whatever can be altered. But quality is no less than quantity open to alteration; and the distinction here given between quantity and quality is expressed by saying increase or diminution: the meaning being that, towards whatever side the determination of magnitude be altered, the thing still remains what it is.
One remark more. Throughout philosophy we do not seek merely for correct, still less for plausible definitions, whose correctness appeals directly to the popular imagination; we seek approved or verified definitions, the content of which is not assumed merely as given, but is seen and known to warrant itself, because warranted by the free self-evolution of thought. To apply this to the present case: however correct and self-evident the definition of quantity usual in Mathematics may be, it will still fail to satisfy the wish to see how far this particular thought is founded in universal thought, and in that way necessary. This difficulty, however, is not the only one.
If quantity is not reached through the action of thought, but taken uncritically from our generalised image of it, we are liable to exaggerate the range of its validity, or even to raise it to the height of an absolute category. And that such a danger is real, we see when the title of exact science is restricted to those sciences the objects of which can be submitted to mathematical calculation. Here we have another trace of the bad metaphysics (mentioned in § 98n) which replace the concrete idea by partial and inadequate categories of understanding. Our knowledge would be in a very awkward predicament if such objects as freedom, law, morality, or even God himself, because they cannot be measured and calculated, or expressed in a mathematical formula, were to be reckoned beyond the reach of exact knowledge, and we had to put up with a vague generalised image of them, leaving their details or particulars to the pleasure of each individual, to make out of them what he will. The pernicious consequences, to which such a theory gives rise in practice, are at once evident. And this mere mathematical view, which identifies with the Idea one of its special stages, viz., quantity, is no other than the principle of Materialism. Witness the history of the scientific modes of thought, especially in France since the middle of last century. Matter in the abstract is just what, though of course there is form in it, has that form only as an indifferent and external attribute.
The present explanation would be utterly misconceived if it were supposed to disparage mathematics. By calling the quantitative characteristic merely external and indifferent, we provide no excuse for indolence and superficiality, nor do we assert that quantitative characteristics may be left to mind themselves, or at least require no very careful handling. Quantity, of course, is a stage of the Idea: and as such it must have its due, first as a logical category, and then in the world of objects, natural as well as spiritual. Still, even so, there soon emerges the different importance attaching to the category of quantity according as its objects belong to the natural or to the spiritual world. For in Nature, where the form of the Idea is to be other than, and at the same time outside, itself, greater importance is for that very reason attached to quantity than in the spiritual world, the world of free inwardness. No doubt we regard even spiritual facts under a quantitative point of view; but it is at once apparent that in speaking of God as a Trinity, the number three has by no means the same prominence, as when we consider the three dimensions of space or the three sides of a triangle the fundamental feature of which last is just to be a surface bounded by three lines. Even inside the realm of Nature we find the same distinction of greater or less importance of quantitative features. In the inorganic world, Quantity plays, so to say, a more prominent part than in the organic. Even in organic nature, when we distinguish mechanical functions from what are called chemical, and in the narrower sense physical, there is the same difference. Mechanics is of all branches of science, confessedly, that in which the aid of mathematics can be least dispensed with-where indeed we cannot take one step without them. On that account mechanics is regarded, next to mathematics, as the science par excellence; which leads us to repeat the remark about the coincidence of the materialist with the exclusively mathematical point of view. After all that has been said, we cannot but hold it, in the interest of exact and thorough knowledge, one of the most hurtful prejudices, to seek all distinction and determinateness of objects merely in quantitative considerations. Mind to be sure is more than Nature and the animal is more than the plant: but we know very little of these objects and the distinction between them, if a more and less is enough for us, and if we do not proceed to comprehend them in their peculiar, that is, their qualitative character.
Quantity, as we saw, has two sources: the exclusive unit, and the identification or equalisation of these units. When we look therefore at its immediate relation to self, or at the characteristic of self-sameness made explicit by attraction, quantity is Continuous magnitude; but when We look at the other characteristic, the One implied in it, it is Discrete magnitude. Still continuous quantity has also a certain discreteness, being but a continuity of the Many; and discrete quantity is no less continuous, its continuity being the One or Unit, that is, the self-same point of the many Ones.
(1) Continuous and Discrete magnitude, therefore, must not be supposed two species of magnitude, as if the characteristic of the one did not attach to the other. The only distinction between them is that the same whole (of quantity) is at one time explicitly put under the one, at another under the other of its characteristics.
(2)The Antinomy of space, of time, or of matter, which discusses the question of their being divisible for ever, or of consisting of indivisible units, just means that we maintain quantity as at one time Discrete, at another Continuous. If we explicitly invest time, space, or matter with the attribute of Continuous quantity alone, they are divisible ad infinitum. When, on the contrary, they are invested with the attribute of Discrete quantity, they are potentially divided already, and consist of indivisible units. The one view is as inadequate as the other.
Quantity, as the proximate result of Being-for-self, involves the two sides in the process of the latter, attraction and repulsion, as constitutive elements of its own idea. It is consequently Continuous as well as Discrete. Each of these two elements involves the other also, and hence there is no such thing as a merely Continuous or a merely Discrete quantity. We may speak of the two as two particular and opposite species of magnitude; but that is merely the result of our abstracting reflection, which in viewing definite magnitudes waives now the one, now the other, of the elements contained in inseparable unity in the notion of quantity. Thus, it may be said, the space occupied by this room is a continuous magnitude and the hundred men assembled in it form a discrete magnitude.
And yet the space is continuous and discrete at the same time; hence we speak of points of space, or we divide space, a certain length, into so many feet, inches, etc., which can be done only on the hypothesis that space is also potentially discrete. Similarly, on the other hand, the discrete magnitude, made up of a hundred men, is also continuous; and the circumstance on which this continuity depends is the common element, the species man, which pervades all the individuals and unites them with each other.
Quantity, essentially invested with the exclusionist character which it involves, is Quantum (or How Much): i.e. limited quantity.
Quantum is, as it were, the determinate Being of quantity: whereas mere quantity corresponds to abstract Being, and the Degree, which is next to be considered, corresponds to Being-for-self. As for the details of the advance from mere quantity to quantum, it is founded on this: that while in mere quantity the distinction, as a distinction of continuity and discreteness, is at first only implicit, in a quantum the distinction is actually made, so that quantity in general now appears as distinguished or limited. But in this way the quantum breaks up at the same time into an indefinite multitude of quanta or definite magnitudes. Each of these definite magnitudes, as distinguished from the others, forms a unity, while on the other hand, viewed per se, it is a many. And, when that is done, the quantum is described as Number.
In Number the quantum reaches its development and perfect mode. Like the One, the medium in which it exists, Number involves two qualitative/factors or functions; Annumeration or Sum, which depends on the factor discreteness, and Unity, which depends on continuity.
In arithmetic the several kinds of operation are usually presented as accidental modes of dealing with numbers. If necessary and meaning is to be found in these operations, it must be by a principle: and that must come from the characteristic element in the notion of number itself. (This principle must here be briefly exhibited.) These characteristic elements are Annumeration on the one hand, and Unity on the other, of which number is the unity. But this latter Unity, when applied to empirical numbers, is only the equality of these numbers: hence the principle of arithmetical operations must be to put numbers in the ratio of Unity and Sum (or amount), and to elicit the equality of these two modes.
The Ones or the numbers themselves are indifferent towards each other, and hence the unity into which they are translated by the arithmetical operation takes the aspect of an external colligation. All reckoning is therefore making up the tale: and the difference between the species of it lies only in the qualitative constitution of the numbers of which we make up the tale. The principle for this constitution is given by the way we fix Unity and Annumeration.
Numeration comes first: what we may call, making number; a colligation of as many units as we please. But to get a species of calculation, it is necessary that what we count up should be numbers already, and no longer a mere unit.
First, and as they naturally come to hand, Numbers are quite vaguely numbers in general, and so, on the whole, unequal. The colligation, or telling the tale of these, is Addition.
The second point of view under which we regard numbers is as equal, so that they make one unity, and of such there is an annumeration or sum before us. To tell the tale of these is Multiplication. It makes no matter in the process, how the functions of Sum and Unity are distributed between the two numbers, or factors of the product; either may be Sum and either may be Unity.
The third and final point of view is the equality of Sum (amount) and Unity. To number together numbers when so characterised is Involution; and in the first instance raising them to the square power. To raise the number to a higher power means in point of form to go on multiplying a number with itself an indefinite amount of times. Since this third type of calculation exhibits the complete equality of the sole existing distinction in number, viz. the distinction between Sum or amount and Unity, there can be no more than these three modes of calculation. Corresponding to the integration we have the dissolution of numbers according to the same features. Hence besides the three species mentioned, which may to that extent be called positive, there are three negative species of arithmetical operation.
Number, in general, is the quantum in its complete specialisation. Hence we may employ it not only to determine what we call discrete, but what are called continuous magnitudes as well. For that reason even geometry must call in the aid of number, when it is required to specify definite figurations of space and their ratios.
The limit (in a quantum) is identical with the whole of the quantum itself. As in itself multiple, the limit is Extensive magnitude; as in itself simple determinateness (qualitative simplicity), it is Intensive magnitude or Degree.
The distinction between Continuous and Discrete magnitude differs from that between Extensive and Intensive in the circumstance that the former apply !o quantity in general, while the latter apply to the limit or determinateness of it as such. Intensive and Extensive magnitude are not, any more than the other, two species, of which the one involves a character not possessed by the other: what is Extensive magnitude is just as much Intensive, and vice versa.
Intensive magnitude or Degree is in its notion distinct from Extensive magnitude or the Quantum. It is therefore inadmissible to refuse, as many do, to recognise this distinction, and without scruple to identify the two forms of magnitude. They are so identified in physics, when difference of specific gravity is explained by saying that a body with a specific gravity twice that of another contains within the same space twice as many material parts (or atoms) as the other. So with heat and light, if the various degrees of temperature and brilliance were to be explained by the greater or less number of particles (or molecules) of heat and light. No doubt the physicists, who employ such a mode of explanation, usually excuse themselves, when they are remonstrated with on its untenableness, by saying that the expression is without prejudice to the confessedly unknowable essence of such phenomena, and employed merely for greater convenience. This greater convenience is meant to point to the easier application of the calculus: but it is hard to see why Intensive magnitudes, having, as they do, a definite numerical expression of their own, should not be as convenient for calculation as Extensive magnitudes. If convenience be all that is desired, surely it would be more convenient to banish calculation and thought altogether. A further point against the apology offered by the physicists is that to engage in explanations of this kind is to overstep the sphere of perception and experience, and resort to the realm of metaphysics and of what at other times would be called idle or even pernicious speculation. It is certainly a fact of experience that, if one of two purses filled with shillings is twice as heavy as the other, the reason must be, that the one contains, say, two hundred, and the other only one hundred shillings. These pieces of money we can see and feel with our senses: atoms, molecules, and the like, are on the contrary beyond the range of sensuous perception; and thought alone can decide whether they are admissible, and have a meaning. But (as already noticed in § 98, note) it is abstract understanding which stereotypes the factor of multeity (involved in the notion of Being-for-self) in the shape of atoms, and adopts it as an ultimate principle. It is the same abstract understanding which, in the present instance, at equal variance with unprejudiced perception and with real concrete thought, regards Extensive magnitude as the sole form of quantity, and, where Intensive magnitudes occur, does not recognise them in their own character, but makes a violent attempt by a wholly untenable hypothesis to reduce them to Extensive magnitudes.
Among the charges made against modern philosophy, one is heard more than another. Modern philosophy, it is said, reduces everything to identity. Hence its nickname, the Philosophy of Identity. But the present discussion may teach that it is philosophy, and philosophy alone, which insists on distinguishing what is logically as well as in experience different; while the professed devotees of experience are the people who erect abstract identity into the chief principle of knowledge. It is their philosophy which might more appropriately be termed one of identity. Besides it is quite correct that there are no merely Extensive and merely Intensive magnitudes, just as little as there are merely continuous and merely discrete magnitudes. The two characteristics of quantity are not opposed as independent kinds. Every Intensive magnitude is also Extensive, and vice versa. Thus a certain degree of temperature is an Intensive magnitude, which has a perfectly simple sensation corresponding to it as such. If we look at a thermometer, we find this degree of temperature has a certain expansion of the column of mercury corresponding to it; which Extensive magnitude changes simultaneously with the temperature or Intensive magnitude. The case is similar in the world of mind: a more intensive character has a wider range with its effects than a less intensive.
In Degree the notion of quantum is explicitly put. It is magnitude as indifferent on its own account and simple: but in such a way that the character (or modal being) which makes it a quantum lies quite outside it in other magnitudes. In this contradiction, where the independent indifferent limit is absolute externality, the Infinite Quantitative Progression is made explicit-an immediacy which immediately veers round into its counterpart, into mediation (the passing beyond and over the quantum just laid down), and vice versa.
Number is a thought, but thought in its complete self-externalisation. Because it is a thought, it does not belong to perception: but it is a thought which is characterised by the externality of perception. Not only therefore may the quantum be increased or diminished without end: the very notion of quantum is thus to push out and out beyond itself. The infinite quantitative progression is only the meaningless repetition of one and the same contradiction, which attaches to the quantum, both generally and, when explicitly invested with its special character, as degree. Touching the futility of enunciating this contradiction in the form of infinite progression, Zeno, as quoted by Aristotle, rightly says, ‘It is the same to say a thing once, and to say it for ever.’
(1) If we follow the usual definition of the mathematicians, given in § 99, and say that magnitude is what can be increased or diminished, there may be nothing to urge against the correctness of the perception on which it is founded; but the question remains, how we come to assume such a capacity of increase or diminution. If we simply appeal for an answer to experience, we try an unsatisfactory course; because apart from the fact that we should merely have a material image of magnitude, and not the thought of it, magnitude would come out as a bare possibility (of increasing or diminishing) and we should have no key to the necessity for its exhibiting this behaviour. In the way of our logical evolution, on the contrary, quantity is obviously a grade in the process of self-determining thought; and it has been shown that it lies in the very notion of quantity to shoot out beyond itself. In that way, the increase or diminution (of which we have heard) is not merely possible, but necessary.
(2) The quantitative infinite progression is what the reflective understanding usually relies upon when it is engaged with the general question of Infinity. The same thing however holds good of this progression, as was already remarked on the occasion of the qualitatively infinite progression. As was then said, it is not the expression of a true, but of a wrong infinity; it never gets further than a bare ‘ought’, and thus really remains within the limits of finitude, The quantitative form of this infinite progression, which Spinoza rightly calls a mere imaginary infinity (infinitum imaginationis), is an image often employed by poets, such as Haller and Klopstock, to depict the infinity, not of Nature merely, but even of God Himself. Thus we find Haller, in a famous description of God’s infinity, saying:
I heap up monstrous numbers, mountains of millions; I pile time upon time, and world on the top of world; and when from the awful height I cast a dizzy look towards Thee, all the power of number, multiplied a thousand times, is not yet one part of Thee.
Here then we meet, in the first place, that continual extrusion of quantity, and especially of number, beyond itself, which Kant describes as ‘eery’. The only really ‘eery’ thing about it is the wearisomeness of ever fixing, and anon unfixing a limit, without advancing a single step. The same poet however well adds to that description of false infinity the closing line:
These I remove, and Thou liest all before me.
Which means that the true infinite is more than a mere world beyond the finite, and that we, in order to become conscious of it, must renounce that progressus in infinitum.
(3) Pythagoras, as is well known, philosophised in numbers, and conceived number as the fundamental principle of things. To the ordinary mind this view must at first glance seem an utter paradox, perhaps a mere craze. What then, are we to think of it? To answer this question, we must, in the first place, remember that the problem of philosophy consists in tracing back things to thoughts, and, of course, to definite thoughts. Now number is undoubtedly a thought: it is the thought nearest the sensible, or, more precisely expressed, it is the thought of the sensible itself, if we take the sensible to mean what is many, and in reciprocal exclusion. The attempt to apprehend the universe as number is therefore the first step to metaphysics. In the history of philosophy, Pythagoras, as we know, stands between the Ionian philosophers and the Eleatics. While the former, as Aristotle says, never get beyond viewing the essence of things as material (iii), and the latter, especially Parmenides, advanced as far as pure thought, in the shape of Being, the principle of the Pythagorean philosophy forms, as it were, the bridge from the sensible to the supersensible.
We may gather from this, what is to be said of those who suppose that Pythagoras undoubtedly went too far, when he conceived the essence of things as mere number. It is true, they admit, that we can number things; but, they contend, things are far more than mere numbers. But in what respect are they more? The ordinary sensuous consciousness, from its own point of view, would not hesitate to answer the question ‘by handing us over to sensuous perception, and remarking that things are not merely numerable, but also visible, odorous, palpable, etc. In the phrase of modern times, the fault of Pythagoras would be described as an excess of idealism. As may be gathered from what has been said on the historical position of the Pythagorean school, the real state of the case is quite the reverse. Let it be conceded that things are more than numbers; but the meaning of that admission must be that the bare thought of number is still insufficient to enunciate the definite notion or essence of things. Instead, then, of saying that Pythagoras went too far with his philosophy of number, it would be nearer the truth to say that he did not go far enough; and in fact the Eleatics were the first to take the further step to pure thought.
Besides, even if there are not things, there are states of things, and phenomena of nature altogether, the character of which mainly rests on definite numbers and proportions. This is especially the case with the difference of tones and their harmonic concord, which, according to a well-known tradition, first suggested to Pythagoras to conceive the essence of things as number. Though it is unquestionably important to science to trace back these phenomena to the definite numbers on which they are based, it is wholly inadmissible to view the characterisation by thought as a whole as merely numerical. We may certainly feel ourselves prompted to associate the most general characteristics of thought with the first numbers: saying, 1 is the simple and immediate; 2 is difference and mediation; and 3 the unity of both of these. Such associations however are purely external: there is nothing in the mere numbers to make them express these definite thoughts. With every step in this method, the more arbitrary grows the association of definite numbers with definite thoughts. Thus, we may view 4 as the unity of 1 and 3, and of the thoughts associated with them, but 4 is just as much the double of 2; similarly 9 is not merely the square of 3, but also the sum of 8 and 1, of 7 and 2, and so on. To attach, as do some secret societies of modern times, importance to all sorts of numbers and figures, is to some extent an innocent amusement, but it is also a sign of deficiency of intellectual resource. These numbers, it is said, conceal a profound meaning, and suggest a deal to think about. But the point in philosophy is, not what you may think, but what you do think: and the genuine air of thought is to be sought in thought itself, and not in arbitrarily selected symbols.
That the Quantum in its independent character is external to itself, is what constitutes its quality. In that externality it is itself and referred connectively to itself. There is a union in it of externality, i.e. the quantitative, and of independency (Being-for-self)-the qualitative. The Quantum when explicitly put thus in its own self is the Quantitative Ratio, a mode of being which, while, in its Exponent, it is an immediate quantum, is also mediation, viz. the reference of some one quantum to another, forming the two sides of the ratio. But the two quanta are not reckoned at their immediate value: their value is only in this relation.
The quantitative infinite progression appears at first as a continual extrusion of number beyond itself. On looking closer, it is, however, apparent that in this progression quantity returns to itself: for the meaning of this progression, so far as thought goes, is the fact that number is determined by number. And this gives the quantitative ratio. Take, for example, the ratio 2:4. Here we have two magnitudes (not counted in their several immediate values) in which we are only concerned with their mutual relations. This relation of the two terms (the exponent of the ratio) is itself a magnitude, distinguished from the related magnitudes by this, that a change in it is followed by a change of the ratio, whereas the ratio is unaffected by the change of both its sides, and remains the same so long as the exponent is not changed. Consequently, in place Of 2:4, we can Put 3:6 without changing the ratio; as the exponent 2 remains the same in both cases.
The two sides of the ratio are still immediate quanta: and the qualitative and quantitative characteristics still external to one another. But in their truth, seeing that the quantitative itself in its externality is relation to self, or seeing that the independence and the indifference of the character are combined, it is Measure.
Thus quantity by means of the dialectical movement so far studied through its several stages, turns out to be a return to quality. The first notion of quantity presented to us was that of quality abrogated and absorbed. That is to say, quantity seemed an external character not identical with Being, to which it is quite immaterial. This notion, as we have seen, underlies the mathematical definition of magnitude as what can be increased or diminished. At first sight this definition may create the impression that quantity is merely whatever can be altered – increase and diminution alike implying determination of magnitude otherwise - and may tend to confuse it with determinate Being, the second stage of quality, which in its notion is similarly conceived as alterable. We can, however, complete the definition by adding, that in quantity we have an alterable, which in spite of alterations still remains the same. The notion of quantity, it thus turns out, implies an inherent contradiction.
This contradiction is what forms the dialectic of quantity. The result of the dialectic however is not a mere return to quality, as if that were the true and quantity the false notion, but an advance to the unity and truth of both, to qualitative quantity, or Measure. It may be well, therefore, at this point to observe that whenever in our study of the objective world we are engaged in quantitative determinations, it is in all cases Measure which we have in view, as the goal of our operations. This is hinted at even in language, when the ascertainment of quantitative features and relations is called measuring.
We measure, e.g. the length of different chords that have been put into a state of vibration, with an eye to the qualitative difference of the tones caused by their vibration, corresponding to this difference of length. Similarly, in chemistry, we try to ascertain the quantity of the matters brought into combination, in order to find out the measures or proportions conditioning such combinations, that is to say, those quantities which give rise to definite qualities. In statistics, too, the numbers with which the study is engaged are important only from the qualitative results conditioned by them. Mere collection of numerical facts, prosecuted without regard to the ends here noted, is justly called an exercise of idle curiosity, of neither theoretical nor practical interest.
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