Hegel’s Logic: An Essay in Interpretation. John Grier Hibben 1902
The idea of quantity, as we have seen, is that aspect of mere being from which the idea of all quality has been eliminated. The category of quantity is described by Hegel from three points of view –
1 Quantity in general (die Quantität).
2. Determinate quantity (das Quantum).
3. Degree (der Grad).
It will be seen in the following exposition that these three aspects of quantity correspond to the three general divisions of quality –
1. Being in general.
2. Determinate being.
3. Self-determined being.
As regards quantity in general, it may be remarked as a matter of terminology that Hegel applies the term magnitude (die Grösse) to determinate quantity rather than to the general notion of quantity. Quantity in general, however, may be considered apart from any reference to definite magnitude, just as quality in general was considered apart from any reference to specific qualities. While quantity in general may be regarded by itself as an essential moment in the evolution of the universal reason, it must not, however, be regarded as an exclusive category.
Hegel has no sympathy with the tendency to reduce all phenomena of the universe to a quantitative basis, including even the phenomena of mind. He insists that a purely mechanical view of the universe, which such a quantitative reduction of all things implies, is by no means a complete or comprehensive view. The mechanical view may seem to suffice in its application to the inorganic world, but it falls short of an adequate explanation when we come to the organic world, and especially when we seek to explain the phenomena of free activity in the sphere of mind.
Inasmuch as the category of quantity is to be regarded as a necessary evolution from the category of being, and also marks a definite characteristic of being, it may be regarded from this point of view, according to the general method of Hegel, as an attribute of the Absolute in one of its manifold phases of manifestation. To define the Absolute merely as quantity would represent, of course, a very one-sided and exceedingly limited conception; but if, on the other hand, it were omitted altogether, the idea of the Absolute would prove wanting so far forth in an essential element of its characterization.
When we come to a more specific inquiry as to the nature of our idea of quantity, we find that it may be conceived from two points of view. Quantity may be either continuous (kontinuirlick) or discrete (diskret). If we regard quantity as an aggregate of many parts, – or, as it may be put, the one which is composed of the many, – and if, moreover, we emphasize the unity into which the many blend, then we have quantity represented as continuous. If, on the other hand, we discount, as it were, in our thought the connecting bond, and emphasize the isolation and reciprocal exclusiveness attaching to the several parts, then quantity will appear as discrete. A line may be taken as an example of continuous quantity. On the other hand, a bushel of apples would be considered as a discrete quantity. The terms, however, “continuous “ and “discrete,” are not mutually exclusive. Quite in keeping with the Hegelian point of view, either one of these terms apart from the other, and excluding the other, represents a relation of unity, and sum or total amount; and of establishing the equality of these two functions.
Thus the simplest arithmetical operation is that of counting. This may be defined as a process which aims to construct an aggregate or sum total by putting together the separate units, one after another. In this operation each unit ranks the same in value as every other. There is no distinction of any kind between them. But it is possible to conceive each unit in question as possessing a value different from every other, – that is, each unit may be conceived as itself an aggregate or sum, possessing varying values, as,,,, etc. When we come to enumerate these sums in order to find the total value in simple units, we are performing the operation of addition.
In multiplication each unit is also an aggregate, but they are all alike and do not vary in value, whereas in addition they are ordinarily unlike. however, multiplication may be represented as a kind of addition.
We may have the following aggregates to count; and we may do this by addition, regarding it merely as a special case in which the aggregates are all alike. Or we can obtain the result directly by taking eight seven times, which is the process of multiplication. In multiplication it is a matter of indifference as to which of the two factors we regard as the aggregate and which the unit.
The process of raising a number to a power is a special case of multiplication. To raise any number to the second power, for instance, the aggregate is taken as many times as it itself contains simple units.
Thus[2]is times, that is, taken times. In such a process there is represented the equality of sum total and unity. To raise a number to a higher power requires only a continued repetition of the process.
Addition, multiplication, and the raising to a power give an exhaustive division of the various modes of arithmetical calculation. The three other processes of subtraction, division, and taking the root of a number do not represent distinct types of arithmetical operations, but are to be regarded merely in the light of inverse operations respectively to addition, multiplication, and the raising to a required power.
As in reference to quantity in general we have found the distinction obtaining between continuous and discrete magnitudes, so in reference to quantum or determinate quantity, there is a similar distinction expressed by the opposite terms of extensive and intensive quantity. Extensive magnitude corresponds to the idea of continuous quantity and intensive magnitude to that of discrete. This correspondence will be seen through the following considerations. Definite magnitude is such only as it possesses a definite quantitative limit. If the magnitude is regarded as a continuous quantity, then the limit is marked simply by the contour of the magnitude itself, – that is, its boundary line of definition. Moreover, from this point of view the separate identity of each part is lost because merged in the whole, which is one and not many, and all included within one and the same limit of circumscription. But if the magnitude is regarded as discrete, then any one of the distinct parts by its position may mark a definite limit. Thus, when we take the temperature of any body, it is the limiting degree which is read off as significant. The quantity of heat which is thus measured is given in terms of intensity or degree (der Grad).
In reference to the intensity of quantitative determination, the various discrete units may be regarded as arranged in order so as to form a series; they therefore do not all count alike. There will always be one which, by its position in the series, will mark the limit, and therefore have a particular significance attaching to it. And as such a series rises or falls, proceeds forwards or backwards, as the case may be, the different units marking the varying limit in every case will indicate corresponding grades of intensity.
As a continuous quantity may be regarded as discrete, so also an extensive magnitude may be conceived as intensive, and an intensive magnitude as extensive. Thus, for instance, the intensity of heat may have an extensive significance as interpreted by the height of the column of mercury. This marks the extent which the mercury, as a whole, has risen in the tube. Hegel illustrates this feature of a change from an intensive to an extensive point of view as seen in the sphere of mind. He draws attention to the fact that a man who has accumulated a certain intensity of mental power is, at the same time, the man who touches life on many sides, so that his capacities have evidently an extensive manifestation as well. This application is somewhat fanciful, it would seem, and should be taken in a figurative rather than in a literal sense, which, however, Hegel himself evidently does not do.
Hegel again enters a protest against those who would subordinate the idea of intensive magnitude to a mere form of extensive magnitude.
He insists that while they are most intimately correlated in thought, nevertheless there is a real distinction between the two that should not be overlooked. The idea of intensity contains an element which is wholly lacking in the bare idea of extension. This, however, must not be interpreted as signifying that the idea of an intensive magnitude is wholly independent of that of extensive magnitude. The one, however, must not be so merged in the other as to lose its individual characteristics completely.
The very concept of quantity itself is such that the limit which is set to it so as to render its quantity a definite amount, or a definite degree of intensity according to the point of view, must be conceived as varying indefinitely without affecting the nature or quality of the magnitude in question. The limit which determines the amount or degree is purely an external determination, and the concept of quantity carries with it the idea of the possibility of pushing out and beyond itself indefinitely. There is no natural or necessary restriction upon a quantitative limit, and therefore the continuous breaking down in our thought of any assigned limit necessitates the conception of an infinite quantitative progression. In this connection Hegel quotes Zeno, who has put this idea in an enigmatical form: “It is the same to say a thing once, and to say it forever.” Such an infinite series gives, however, a false idea of the true significance of infinity. It is false for the same reason that the qualitatively infinite progression is false, as we have already seen. it is what Spinoza calls the imaginary infinity. As an instance of this conception, Hegel quotes the lines of Hailer:[7] –
“Ich bliufe ungeheure Zahien Gebirge Milionen auf,
Ich setze Zeit aid Zeit Und Welt auf Welt za Hauf,
Und wenn ich von der grausen Hoh’
Mit Schwindel wieder nach Dir seh’,
Ist alle Macht der Zahi Vermehrt zu Tausendnial,
Noch nicht em Theil von Dir."[8]
In commenting upon this passage, Hegel remarks: “The same poet, however, well adds to that description of the false infinity the closing line ‘Ich zieb sie ab, und du liegest ganz vor mir.’[9] This means that the true infinite is not to be regarded merely as another world which transcends the finite; and if we are ever to appreciate its significance, we must disabuse our minds of all notions of a progressus in infinitum[10] The doctrine of number, as is well known, was magnified by the ancient Pythagoreans into a complete system of philosophy. While in that school there was an undue exaggeration of the concept of number as expressing the essence of being, it must not be overlooked, however, that Pythagoras touched upon an important truth in his teaching when he insisted that there are certain states of things, certain phenomena of nature, the character of which seem to vary according to a scale of number relations. This may be illustrated in the variations in tone and harmony which, according to common tradition, first suggested to Pythagoras the conception of the essence of all things as number. Hegel, according to his general method, adopts the teachings of this school not in the light of a complete system of philosophy, but merely as one phase among many in the development of the universal reason. The Pythagorean doctrine corresponds roughly, at least, with Hegel’s conception of quantitative relation, which idea marks a natural transition to the third division of quantity, known as measure.
Quantitative relation (das quantitative Verhältniss) may be defined as that relation which obtains between numbers of such a nature that the numbers themselves may vary indefinitely, provided only the relation itself remains constant. Thus the relation of: is the same as that of: 6. In the midst, therefore, of varying quantities, there is a constant which retains its own specific character through a process that may be indefinitely continued without limit. This idea of certain constant features in the midst of quantitative variation would seem to indicate that this constant value has the force of a qualitative character; for, as we have found, it is the quality which remains unchanged in the midst of quantitative alteration. Thus in pushing forward the concept of quantity in the development of all its possible implications, we find between coincident alterations in magnitudes which form a ratio a constant relation obtaining of such a nature that the concept of quantity will not explain it satisfactorily, and we fall back again upon the idea of quality in order to account for it. Thus the idea of quality was found to be partial, and when developed to its utmost limit, carried our thought over into the sphere of quantity. Then the idea of quantity when fully developed brought us back again to that of quality. Is the movement of thought only a circle that merely brings us back to the starting-point? According to Hegel’s method, the incompleteness of thought at this stage is overcome by the dialectic process which combines these two ideas of quality and of quantity into one complete relation representing an advanced and higher point of view. This relation Hegel calls that of qualitative quantity, or of measure (das Maass). This is the third and last stage in the development of the idea of quantity, and represents, as Hegel insists, both the unity and the truth of quality and of quantity combined.