John McTaggart A Commentary on Hegel’s Logic 1910

Quantity |
Pure Quantity

47. Quantity is divided as follows:

I. Quantity. (*Die Quantität*.)

*A*. Pure Quantity. (*Die reine Quantität*.)

*B*. Continuous and Discrete Magnitude. (*Kontinuirliche und diskrete Grosse*.)

*C*. Limitation of Quantity. (*Begrenzunng der Quantität*.)

II. Quantum. (*Quantum*.)

*A*. Number. (*Die Zahl*.)

*B*. Extensive and Intensive Quantum. (*Extensives und intensives Quantum*.)

(a) Their difference. (*Unterschied derselben*.)

(b) Identity of Extensive and Intensive Magnitude. (*Identität der extensiven und intensiven Grösse*.)

(c) The Alteration of Quantum. (*Die Veränderung des Quantums*.)

*C*. The Quantitative Infinity. (*Die Quantitative Unendlichkeit*.)

(a) Its Notion. (*Begriff derselben*.)

(b) The Quantitative Infinite Progress. (*Der Quantitative unendliche Progress*.)

(c) The Infinity of Quantum. (*Die Unendlichkeit des Quantums*.)

III. The Quantitative Ratio. (*Das Quantitative Verhältniss*.)

*A*. The Direct Ratio. (*Das direkte Verhältniss*.)

*B*. The Inverse Ratio. (*Das umgekehrte Verhältniss*.)

*C*. The Ratio of Powers. (*Potenzen Verhältniss*.)

It will be noticed that Quantity is used in an ambiguous manner, since it is the name both of the whole secondary division, and of the first of the tertiary divisions contained in it. The tertiary division might be distinguished if we gave it the name of Undivided Quantity, which, as we shall see, would be appropriate to it.

The treatment of Quantity is not one of the most successful parts of the *Greater Logic*. It occupies a greater space than any of the other eight secondary divisions. Yet the transitions are frequently obscure, and often appear to owe their obscurity to excessive compression. By far the greater part of the 186 pages which are employed on Quantity are occupied with Notes on collateral points. Some of these, indeed, throw additional light on the main argument, but the rest only contain criticisms of Kant’s views on Quantity, and of certain mathematical doctrines. Hegel is never at his best when criticising Kant, and the mathematical discussions are too purely technical to give us much assistance in comprehending the course of the dialectic.

48. Again, were Hegel’s mathematics correct ? Was he right about the mathematics of his own time, and, if so, would he be right about the mathematics of the present day? To answer these questions requires a knowledge of mathematics which I am very far from possessing. Mr Bertrand Russell — one of the few philosophers who are also mathematicians — says: “In Hegel’s day, the procedure of mathematicians was full of errors, which Hegel did not condemn as errors but welcomed as antinomies; the mathematicians, more patient than the philosophers, have removed the errors by careful detailed work on every doubtful point. A criticism of mathematics based on Hegel can, therefore, no longer be regarded as applicable to the existing state of the subject^{[1]}.”

But the value of Hegel’s treatment of Quantity would only be slightly affected by the fact that his criticisms of mathematics were based on ignorance or by the fact that they had been invalidated by the progress of that science. The main object of the dialectic, after all, is to reach the Absolute Idea, and so to demonstrate what is the true nature of reality. Thus the principal function of the lower categories is to lead on to the Absolute Idea. And for this it is only requisite that each of them should logically follow from the one before it, and lead on to the one after it.

Now the question whether Hegel’s various categories of Quantity do perform this function is not affected by any mathematical mistakes which he may have made, nor can it be settled in the negative by any mathematical criticisms. The only question is whether Hegel was justified in starting the dialectic with the category of Pure Being, and whether the validity of the Hegelian categories of Quantity can be shown to be involved in the validity of the category of Pure Being. And this is a question for metaphysics and not for mathematics.

It is true that Hegel’s main aim in the dialectic was not his only aim. He wished, not merely to deduce an absolutely valid conception of reality, but to account for other less valid conceptions, and to range them in the order of their relative validity. He probably believed that the categories with which he deals in the sphere of Quantity were identical with the fundamental notions of mathematics. In so far as they were not so, he must be considered to have failed in his subordinate purpose, and, in so far as he has failed, to have introduced additional obscurity by the fact that he has called his categories by the names of the mathematical notions.

But the purpose in which he may have failed is, as I have said, only of subordinate importance for him. And his failure if there is one^{[2]}, would not be a sign of any metaphysical flaw in his system, but only of mathematical ignorance. If the dialectic process is correct, it will be true of all mathematical conceptions, as of all others, that the way in which we can judge of the degree of their validity will be by means of the dialectic process. If the ideas are themselves stages in that process, the place which they occupy in it will give us their relative validity. If they are not stages in the process, their relative validity can be found by ascertaining the point in the dialectic at which it becomes clear that they are not absolutely valid. For example, if the absolute validity of mathematical ideas implied the absolute validity of the general conception of Quantum, as given in the dialectic, then, as the dialectic transcended Quantum, it would become evident that the mathematical ideas could not be absolutely valid. Thus, even if Hegel’s judgments about mathematics were all wrong, that would not prevent his dialectic from being the foundation of right judgments on the same subject to a person more skilled in mathematics.

49. This stage (*G. L.* i. 212.* Enc*. 99) appears to be identical in content with the last stage of Quality, though expressed with greater immediacy. The two elements, Repulsion and Attraction, which were recognised as inseparable in the final category of Quality, here receive the names of Discreteness and Continuity.

Pure Quantity is a category of the fourth order, while the category immediately preceding it (Relation of Repulsion and Attraction) is of the fifth order. Thus, according to the general method of the dialectic they should not be identical in content. if, however, tile subdivision which produced categories of the fifth order at this point is excessive, as I have maintained above (Section 45), this objection would disappear in an amended dialectic.

But, although Discreteness and Continuity are recognised as inseparable, it is still possible to lay a greater emphasis on one of them than on the other. And we begin, Hegel tells us (*G. L.* i. 213), by laying the greater emphasis on Continuity. The reason appears to be that this element is more characteristic of Quantity, though not more essential to it, than Discreteness. For as long as we had only Repulsion the process remained within Quality, but, as soon as Attraction was added the transition to Quantity to ok place. And there is always a tendency to put most emphasis on the element last reached.

50. (*G. L.* i. 229.) By a somewhat abrupt transition we come to this category, in which Magnitude is to be taken first as Continuous. Here there is as yet no plurality. of Quantities, and the one Quantity is indefinite. A plurality of Quantities would require that they should be Discrete from one another. And, again, no Quantity can be definite unless by its having fixed boundaries — that is to say by being Discrete from the Quantity beyond those boundaries. It is true that, as was said above, all Quantity has an element of Discreteness. But, so far, the only things which are Discrete from one another are the units — the Ones — which are alike Discrete from and Continuous with one another.

Now a One, taken by itself, is not a Quantity at all. For it has no plurality in it. And Ones have no possibility of varying in magnitude. All variations of magnitude are only variations in the number of the Ones. These characteristics are essential to Quantity, and they are not possessed by isolated Ones. And the isolated Ones being, so far, the only Discrete things, we have as yet no definite Quantity, and no plurality of Quantities.

It may appear incorrect to say that a One admits of no plurality. Can we not, it may be asked, conceive an isolated One as consisting of two. halves, four quarters, and so on ? But a One which consists of parts is no longer a *mere *One, which is all that the dialectic has got at present. It is something which, while from one point of view a unit, is, from another point of view, an aggregate of two or four units. And this involves the higher conception of Discrete Magnitude, which has not yet been reached.

In the same way, we may conceive the units of which an aggregate is made up as having magnitude, and as being capable of having different magnitudes, and of varying in magnitude. But we can only do this in so far as we conceive each of them as made up in its turn of parts, and so as not being *mere *Ones.)

The position at present is that we have a plurality of Ones — of the number of which we know nothing — which form a single Quantity. But within this single all-embracing Quantity there are as yet no minor Quantities. Each One is Qualitatively different from each of the others, but all these Qualitative differences are as yet unique. There are no qualities common to more than one One — except, indeed, the quality, if it may be called a quality, of being a One. And this is common to all Ones.

Continuous Magnitude was formed by passing from One to One in virtue of their Continuity. (Continuity, it will be remembered, is what was previously called Attraction. It is the capability, possessed by Ones, of being united in an aggregate.) We now pass to Discrete Magnitude (*G. L.* i. 229). Each One is as really Discrete from all the others as it is continuous with them. Thus a Quantity, less than the whole, can be formed by taking certain Ones together, in virtue of their Continuity, and cutting, them off from all others in virtue of their Discreteness. And this Quantity, being cut off by its Discreteness from the indefinite Quantity beyond it, will be a finite Quantity. In the indefinite Quantity, again, other finite Quantities can be formed, and thus we get a plurality of finite Quantities.

51. In the form of this stage, as presented by Hegel, there appear to be two defects. The first is that no reason is given why we should pass from Pure Quantity to the new stage. The second is that, although Continuous and Discrete Magnitude is not divided into a subordinate triad.. yet there is a distinct dialectic advance within it — namely from Continuous to Discrete Magnitude.

These defects seem to me to be merely a matter of arrangement. Continuous Magnitude is not really a fresh stage, or part of a fresh stage, at all. It is nothing but Pure Quantity, since, as we have seen, it does not permit of definite Quantity, or of a plurality of Quantities.

On the other hand Discrete Magnitude is not merely correlative with Continuous Magnitude. It is distinctly a more advanced conception. It gives us the distinctness and plurality which were lacking before, and it gives them to us by differentiating the relation between Ones — by joining some of them to others, and disjoining them from others again, instead of making the relation uniform.

It is, then, in reality, to Discrete Magnitude that the advance from Pure Quantity is made. This is evident in Hegel’s text, but is misrepresented by his headings. In order that these should correspond with his argument, he should have dealt with Continuous Magnitude under the head of Pure Quantity, and should have made his second stage, s imply Discrete Magnitude, instead of Continuous and Discrete.

It should be remarked that, although the transition to Discrete Magnitude lies in the possibility of breaking off the Quantity at any One, this does not mean that it is merely a possible transition. Continuous Magnitude is that which cannot be broken off at any point. Discrete Magnitude is that which can be broken off at any point. When we are forced to admit the possibility of breaking Magnitude off at any point, this is a necessary transition to the category of Discrete Magnitude.

We can break it off, then, at any point we like. But no reason has been given why we should break it off at one point rather than at another. Nor can any such reason be given until we have passed out of the sphere of Quantity into. Measure. To this point we shall recur later on.

52. (*G. L.* i. 231.) Hegel says that Discrete Magnitude as such is not limited. It is only limited as separated from the Continuous. By this, I conceive, he means that, if the Discrete Magnitude were taken in isolation, its final One would not be a Limit, because it would not divide the Discrete Magnitude from anything else. It is only in so far as it is regarded as in connexion with the indefinite Continuous Magnitude from which it has been carved out, that its final term is to be considered. a Limit. (On Hegel’s use of Limit cp. above, Section 27.)

The Discrete Magnitude, then, shares its Limit with the Continuo us Magnitude outside it. It is thus in a definite relation to that which bounds it, and has itself a definite amount. To definite Quantities Hegel gives the name of Quanta, and so we pass to the second main division of Quantity.

53. (*G. L.* i. 232. *Enc. *101.) In reaching the conception of a limited and definite Quantity we have, according to Hegel reached for the first time the possibility of Number. While Quantity is merely continuous it cannot be numbered. For then there is no intermediate term between the separate Ones and the whole indefinite Quantity. And the separate Ones in their separateness cannot have any Number, since each of them is only One. (But now that we have a definite Quantum, it consists of those Ones which are included between certain Limits, and can therefore be numbered.

54. It may be admitted that, up to this point, there could be no Number of anything less than the whole Quantity. But why could not this have a Number? We do not know how many Ones there are. But this does not prevent them from having a Number, though the dialectic cannot tell us what it is.

Hegel would probably have said that what was infinite could have no Number, and he does not seem to have considered the possibility that there should be a finite number of Ones. But I cannot see that this possibility can be neglected. Each One has — or rather is — a separate Quality. I cannot see anything in the dialectic to exclude the possibility that there should be just twenty such Qualities, and so twenty such Ones, no more and no less.

We must remember that the Ones are not Somethings. The latter had to be infinite in number, since each of them required a fresh Something beyond it. But the Ones have Being for Self, and so avoided, as we saw, this infinite series. Again, if Ones were always divisible into other Ones, their number would necessarily be infinite, but each One is a simple Quality, which is not divisible. Nor does each One involve an endless chain of derivative Ones in the same way, *e.g., *that every relation is related, so that the number of relations is infinite.

It is true that the Number of the whole Quantity of Ones could not have a Limit, in the Hegelian sense, since there would be nothing outside it. But a Limit, in this sense, does not seem necessary, since the Ones which are numbered have Being for Self. They can reciprocally determine each other, and when their natures are given, the number of them is given also.

Thus it seems quite possible that all the Ones, taken together, should have a definite and finite Number. That this possibility should have escaped Hegel may very well, I think, be due to the fact that he did not keep sufficiently in his mind the precise significance of his categories of Quantity.

These categories, like all others in the dialectic, refer only to what is existent. (Cp. above, Section 6.) He is not dealing with the purely abstract conception of quantity, which can be applied to anything which can be thought of at all. His categories of Quantity are attempts to explain the nature of what is existent by the conception of quantities of existent Ones — the nature of each One being, as we saw in the last chapter, a simple and unique Quality.

So far as I can see, he never definitely asserts anything inconsistent with this view of the categories of Quantity — the only view which he is entitled to take — except when he deals with Quantitative Ratio. (Cp. below, Section 66.) But his expressions often suggest that he is thinking rather of abstract quantity than of a Quantity of existent Ones. This may account for his failing to see the possibility of the total number, under the categories of Undivided Quantity, being limited. For of course there is no limit to a purely abstract quantity.

What Hegel says, however, in reaching his category of Number, only requires a verbal correction. For it is true that Hegel’s category of Number is the first point at which any Quantity, less than the whole Quantity of Ones, could have a number.

55. “Quantity is Quantum” says Hegel, “or has a Limit, both as Continuous and as Discrete Magnitude. The difference of these species has here no meaning “ (*G. L.* i. 232). This must not be taken as an assertion that Continuity and Discreteness have no longer meaning as different *moments *in any Quantity. It is only the distinction between Continuous and Discrete *Magnitudes *which has no longer any meaning. And this result was brought about in Limitation of Quantity. For there we saw that a Discrete Magnitude could only be Discrete in so far as it was positively related to that which was outside .it. And this positive relation is what Hegel calls Continuity.

Quantity is now indifferent to its Limit, but not indifferent to having a Limit, for to have a Limit is identical with being a Quantum (*G. L.* i. 2.325 h e distinction seems to be that it is always essential to A Quantum to have a Limit, but never essential to it to have a particular Limit. Of course, if it had a different Limit, it would be a different Quantum. But then there is no reason why it should not be different. This will be .explained when we reach the Quantitative Infinite Progress.

Hegel further says that the Ones which make up any Quantum are indifferent to the Limit, but that the Limit is not indifferent to the Ones (*G. L.* i. 234). As the Limit is that which determines the Quantum to be what it is, it follows that the Ones in a Quantum are indifferent to the Quantum, while the Quantum is not indifferent to them.

This superiority of the units to the aggregate is essential to Quantity, and is implied in all Quantitative statements. When we say, for example, 7 = 5 + 2, we assume that each of the units dealt with will remain unchanged, whether it is combined with more or fewer others. If not, the proposition would not be true. But the aggregates do not remain the same, regardless of the units. If, for example, we take one unit away from 7, what remains is no longer equal to 5 + 2.

56. (*G. L.* i. 252.) Extensive and Intensive Quanta differ from each other in a manner analogous to the difference between Continuous and Discrete Quantity. The distinction between the two pairs of terms is that Extensive and Intensive refer to Quantitative *Limits *only, and, as the Quantum, is identical with Q its Limit, they apply to Quanta, while, since no Quantities except Quanta have Limits, they apply to no Quantities except Quanta. Continuous and Discrete, on the other hand, apply to all Quantities.

We have first Extensive Quantum. This conception is identical with that of Number, except that its determination is now explicitly posited as a plurality (*Vielheit*.) (*G. L.* i. 253).

I do not see why plurality is more explicitly posited in the conception of Extensive Quantum than in that of Number, nor does Hegel give any reason why it should be so. The idea of Extensive Quantum has the same content with the idea of Number. The Extensive Quantum is looked on as primarily a plurality. It is not exclusively a plurality, for, since it is a Quantum, it must be definite, and, being definite, must be Discrete. It is therefore a unity as well as a plurality, but its distinctive mark is plurality. Now this is also the case with Number. A Number is a unity, or it could not be definite. But it is conceived as more essentially a plurality. In Number, as we saw above, the Ones are indifferent to the Quantum, but the Quantum is not indifferent to them. The plurality is thus more essential than the unity.

But since the Quantum is a unity it can also be taken with the greater emphasis on the unity, and when this is done we get the conception of Intensive Quantum (*G. L.* i. 253. *Enc. *103).

The difference between Intensive and Extensive Quantum is thus one of comparative emphasis^{[3]}. Extensive Quantum has a certain unity, but its unity is subordinate to its plurality. It is comparatively Continuous with what is outside it, and comparatively Discrete within itself. Intensive Quantum is more Discrete from the external, more Continuous within, and its unity is therefore greater than that of Extensive Quantum. The Limit of an Intensive Quantum is called its Degree (*G. L*. i. 254.* Enc. *103). The Degree of such a Quantum is rather *Mehrheit* than *Mehreres*, and while it may be spoken of as a Number (*Zahl*), it must not, since it is simple, be regarded as a Sum (*Anzahl*) (*G. L.* i. 254).

57. (*G. L*. i. 255.) The treatment of this point is rather obscure. Hegel says “Extensive and Intensive Magnitudes are thus one and the same determination of Quantum; they are only separated by the fact that one has its Sum inside itself, the other has its Sum outside itself. Extensive Magnitude passes over into Intensive Magnitude, since its plurality falls inherently into a unity, outside which plurality is found. But on the other hand this unity only finds its determination in a Sum, and in a Sum which is regarded as its own; as something which is indifferent to Intensities otherwise determined, it has the externality of the Sum in itself; and thus Intensive Magnitude is as essentially Extensive Magnitude” (*G. L.* i. 256).

Does this mean that !the two terms are strictly correlative — that they stand side, by side in the dialectic process, and that the transition from Intensive to Extensive is of precisely similar nature to the transition from Extensive to Intensive? Or does it mean that Intensive Quantum stands higher on the scale than Extensive, and that the transition from Extensive to Intensive is the transition of the dialectic process, while the transition from Intensive to Extensive only means that what is seen under a higher category can, if we choose, also be regarded under a lower one?

The words quoted above suggest the first of these alternatives. And this is supported by the passage which immediately follows them (*G. L.* i. 257). In this we are told that with this identity we gain a Qualitative Something, since the identity is a unity which is formed by the negation of its differences. This on the whole suggests that the two terms are to be taken as on an absolute equality.

Nevertheless it seems to me that the weight of the evidence is on the whole in favour of the view which finds Intensive Magnitude a more advanced stage of the dialectic process than Extensive Magnitude. To this conclusion I am led by three reasons.

In the first place we cannot safely lay much weight on Hegel’s expressions about the Qualitative Something. For the mention of a Qualitative element here seems very casual. It is dropped as soon as it has been made. We hear nothing more of it while we remain in the division of Quantum. The next mention of a Qualitative element comes in the division which succeeds Quantum — namely Quantitative Relation. And when it comes in there, it is introduced quite independently, with no reference to the passage on p. 257, and in quite a different way. That passage cannot therefore be considered as of much importance.

In the second place, the transition to the next category’ (Alteration of Quantum) does not start from the identity of. Extensive and Intensive Magnitudes, but from the consideration, of Intensive Magnitude taken by itself. This will, I think, be evident when we come to consider the transition, and it would follow that Intensive Magnitude must be above Extensive in the scale of categories, since the possibility of advancing from the intensive a one implies that the Intensive has absorbed the Extensive.

In the third place, this view is supported by several passages. Hegel says (*G. L.* i. 279, 280) that the notion of Quantum reaches its reality as Intensive Magnitude, and is now posited in its Determinate Being m it is in its Notion. This agrees with the *Encyclopaedia, *where he says (*Enc*. 104) that in Degree the notion of Quantum is explicitly posited. Also there is not the slightest doubt that, in the *Encyclopaedia, *Intensive Quantum is higher than Extensive Quantum, since it falls in the third subdivision of Quantity, while Extensive Quantum falls in the second.

58. On the whole, therefore, although the evidence is certainly conflicting, I think that the *Greater Logic *regards Intensive Quantum as higher than Extensive Quantum. We can see why this should. be so. Intensive Quantum emphasises ‘the unity of the Quantum rather than its plurality. In other words, it emphasises the Limit. This carries us further away from the indefinite Quantity with which the treatment of Quantity began. Intensive Quantum is thus the more developed idea of the two.

The necessity of the transition does not lie in any contradiction in Extensive Quantum which forces us to pass to Intensive. The contradiction would lie in denying that a. Quantum which was Extensive was also Intensive. For any Quantum must be Continuous within itself, and Discrete from what is outside it. In virtue of this it is a unity, and so is Intensive. Thus the previous conclusion that the universe is such that the conception of Extensive Quantum is applicable to it, involves that the conception of Intensive Quantum is likewise applicable, and anything else which is involved in the conception of Intensive Quantum.

Hegel’s titles,. then, do injustice to the course of his argument. The real advance is not from the difference between Extensive Quantum and Intensive Quantum to the identity between them. It is rather from Extensive Quantum to intensive Quantum. And thus the two first subdivisions of Extensive and Intensive Quantum should have been (a) Extensive Quantum, (b) Intensive Quantum.

Thus, for the second time in this chapter, we find that Hegel’s titles are misleading. In each case the defect arose from the titles taking as correlative two conceptions, of which his argument shows one to be superior to the other. In the first case it was the Continuous and Discrete; in the second case it was the Extensive and Intensive. It may perhaps be the case that the confusion arose from following in the titles the usage of mathematics, for which each of these pairs is a pair of two correlatives which are strictly on an equality with one another. Should this be the true, explanation, it would add another to the cases in which the consideration of the finite sciences, so far from rendering assistance to the dialectic, has distorted it, and injured its cogency.

59. We now come to the transition to the next category. Of this Hegel says. “The Quantum is the determination posited as transcended, the indifferent limit, the determination which is equally the negation of itself. This discrepancy is developed in, Extensive Magnitude, but it is Intensive Magnitude which is the determinate being of this externality, which constitutes the intrinsic nature of the Quantum. It is posited as its own contradiction, as being the simple determination relating itself to itself, which is the negation of itself, as having its determination, not in itself, but in another Quantum.

“A Quantum is therefore posited as in absolute Continuity, in respect of its Quality, with what is external to it, with its Other. It is therefore not only *possible *that it should go beyond any determination of Magnitude, it is not only *possible *that it should be altered, but it is posited as *necessarily *alterable. The determination of Magnitude continues itself in its Other being in such a way that it has its being only in its Continuity with an Other; it is a limit which is not, but *becomes*” (*G. L*. i. 261. Cp. also *Enc*. 104).

That is to say, there is nothing to decide why, when there is a Quantum, it should be one Quantum, with one Magnitude, rather than another Quantum, with another Magnitude. Magnitudes can only be fixed by non-Quantitative considerations. There is an *a priori *reason why a triangle has three sides, rather than two or four. There is an empirical reason why there are seven apples on this dish, rather than six or eight. But these reasons are not to be found in the nature of three or seven, but in the nature of triangles, or of the distribution of apples.

Now there are no non-Quantitative considerations to determine the Quanta under this category. The only non-Quantitative feature that the Quanta have at all is that each One is a separate and unique Quality, And this obviously can give no reason why some of the Ones should be conjoined in a particular Quantum and others left out. This could only be determined by some general quality, shared by some of the Ones, and not by Others. And this is a conception which the dialectic has not yet reached.

But, it may be objected, why should a reason be wanted at all? Why should it not be an ultimate fact — since some facts must be ultimate — that these seventeen Ones, for example, should be parts of the same Quantum, and that no others should be? This would give a definite Quantum.

I do not think this objection is valid. If this Quantum was an ultimate fact, it would imply that there was some difference between any One inside the Quantum and any One outside it, of a different nature from any difference which could occur between any two Ones inside the Quantum. A, inside the Quantum, cannot differ in the same way from B inside it and from 0 outside it, Now, with the category at present before us, it is impossible that there should be such a difference between differences. Each One differs from every other One precisely in the same way. Each is a separate numerical One, and each is a unique Quality. And there is no other way in w ic any One can differ from another One^{[4]}. Thus, not only can no reason be given for stopping at one point rather than another, but to stop at one point rather than another would introduce a conception (that of different sorts of differences between Ones) positively incompatible with the present category.

60. Hegel expresses this by saying that, while each Quantum has its determination in another Quantum^{[5]}, and stops where the other begins, it is at the same time continuous with this other Quantum — the Ones are just the same on each side of the Limit, and there can be no reason why the Limit should not be put elsewhere, and so add to the Quantum or diminish it. And so we come to

(*G. L.* 261.) Why, it may be asked, did not this conception of the necessary variation of Quantity come before ? Surely it is as true of an Extensive Quantum as of an Intensive Quantum that it is essentially alterable.

I think it is true that, if we had stopped at Extensive Quantum, without going on to Intensive, this conception of Alteration would have necessarily followed from Extensive Quantum. But the more immediately obvious transition — and therefore the one to take first — was the transition to Intensive Quantum. And, if Intensive Quantum was to come in at all, the transition to Alteration of Quantum comes better after it, for the necessity of that transition then becomes far more obvious. As was said in the passage quoted above, it was developed in Extensive Magnitude, but finds its determinate being in Intensive Magnitude.

When we regard a Quantum as, Extensive, we regard the plurality of Ones as the element which is logically prior, and the Quantum as a whole is regarded as dependent on the Ones. Now so long as we refer the Quantum to the Ones, there is a reason for the Quantum being the size it is, and no other, namely that it includes those Ones, and no others. If we go further, and ask why those Ones and no others should be included, no answer could be given, and the conception of Alteration would arise, but so long as we regard the Ones as ultimate in reference to the Quantum, the necessity of Alteration remains in the background.

But with Intensive Quantum it comes at once to the front. For then the unity of the Quantum is the prominent element. And therefore our question — why is it this Quantum, and not a larger or smaller one — cannot be referred back to the Ones which it contains. And therefore the necessity of Alteration, which is due to the impossibility of answering this question follows more obviously and naturally from Intensive Quantum.

This is what Hegel means when he says (*G. L.* i. 253) that a determination of a Quantum through Number (which is a, category previous to Intensive Quantum) does not need another Magnitude, because in Number Quantum has its externality, and its relation to another, inside itself. (If this passage seems to deny all tendency to Alteration in the case of an Extensive Quantum, we must remember the explicit assertion on page 261 that the difference in this respect between Extensive and Intensive is merely a matter of degree.) And again (*G. L.* i. 254) “Degree, therefore, which is simple and in itself, and so has its external Other-being no longer in itself, has that Other-being outside itself, and relates itself to it as to its determination.”

61. We have now come to the end of Extensive and Intensive Quantum, and pass on to the third subdivision of Quantum, which is called

(*G. L.* i. 263.) The first subdivision of Quantitative Infinity, is, as it should be, the restatement of the last subdivision of the preceding triad. The first movement of the Quantum when it passes its Limit is into a Quantity which is simply defined as not being that Quantum. So far, then, it is only Quantity, and no longer Quantum. And as Quantity is only bounded when it is Quantum, this Quantity has no boundaries at all. Thus it is infinite.

Hegel now proceeds to remark on the difference between the Qualitative Infinity, which was one of the triads in Being Determinate, and that Quantitative Infinity with which we are now dealing (*G. L.* i. 264). That which is Qualitatively determined is not *posited *as having, the other in itself. Magnitudes, on the other hand, are posited as being. essentially Alterable — as being, in Hegel’s somewhat peculiar language, “ unequal to, themselves and indifferent to themselves.”

The difference is one which always arises between lower and higher categories in Hegel’s philosophy. The method of the, dialectic changes gradually as the dialectic process advances. (*Cp. Enc. *III, 161, 240.) It becomes more of a. spontaneous advance from category to category, and less of a breaking, down, by negative methods, of the resistance of categories which oppose any movement beyond them. It is thus to be expected, since Quantity comes later than Quality in the process, that the finite in Quantity should lead on to the infinite more expressly and directly than the finite in Quality does.

The transition to the Infinite Quantitative Progress, which now takes place, is analogous to the transition to the Infinite Qualitative Progress. (Cp. above, Section 33.) The Quantum is after all continuous with the indefinite Quantity into which it has passed over. If it were not, it would not have passed over into it. The passage has only taken place because both terms are Quantities, only separated by a Limit to which it is the nature of Quantity to be But the Quantity on the other side of the Limit will also be composed of Ones, and thus the argument is again applicable which originally transformed Quantity into Quantum. The Other Side (*Jenseits*) of the original Quantum is now itself a Quantum. And there fore, like the original Quantum, it is essentially subject to alteration, and will pass the Limit, only thereby to reach a third Quantum, which will be suppressed in its turn, and so on (*G. L.* i. 265)^{[6]}.

62. We now come to

(*G. L.* i. 264. *Enc*. 104.) At this point Hegel inserts an interesting note on the supposed sublimity of the sort of Infinite which is revealed in such a progress as this. Such an Infinite, he says, can produce nothing but weariness (*G. L.* i. 268. *Enc. *104)^{[7]}. This is extremely characteristic of Hegel. When he says that the true Infinite is not the unbounded, but the self-determined, he does not merely change the meaning of a word, but claims for the self-determined all the dignity which is commonly attributed to the unbounded. It is, perhaps, to his deep conviction that true greatness lies in self-limitation, and not in the absence of limitation, that we are to ascribe much of the special reverence which he shows for the ideas of the Greeks, as well as his low opinion of the Romanticism of his own age and country.

We must not forget, however, that Hegel never says that the False Infinite of an Infinite Series is necessarily contradictory, though he does say it is worthless and tedious. But in the present case there is a contradiction, as there was with the Infinite Series in Quality. We had reached the idea of a Quantum, and a Quantum has to be definite. But it can only be definite by having a certain Limit, and by keeping within it. We have seen, however, that any Quantum necessarily passes its Limit, and overflows into a fresh Quantum. It is not, therefore, determined in Magnitude. But it is of the essence of Quantum to be determined, and the dialectic will not permit us to reject the idea of Quantum altogether. In this case, therefore, a contradiction arises.

63. To this argument an objection might be raised. Let us take the Quantum as enlarged till it includes the whole Quantity of Ones. Will it not then be determined, since it is impossible for it to increase beyond this point ? It will not, indeed, have a Limit, in the technical Hegelian sense, but it will have a fixed Magnitude, and this is all that is wanted.

Hegel does not seem to have considered this point. As I said above (Section 54), he would probably have considered that an infinite Quantity of Ones would have no Number, and no definite Magnitude, and he apparently ignored the possibility of the Ones being finite in number. But this possibility, as we saw above, cannot Justifiably be ignored.

It does not, however, remove the contradiction. And this for two reasons. In the first place, the category of Quantum arose from the fact that Quantity, in virtue of its characteristic of Discreteness, could be divided at any point — we could make a Quantum wherever we liked by dividing Quantity. Now if the only way in which we can get a Quantity of a fixed Magnitude is by including all the Ones, then there will only be one such fixed Magnitude, and it will not arise by dividing the total Quantity, but by including it all. This is not a Quantum. For a Quantum is made by dividing the total Quantity, and has always, therefore, other Quanta beyond it. The fixed Magnitude of the whole of Quantity, then, would not be a Quantum, and thus the contradiction would still remain — that it has been proved that there must be determined Quanta, and that no Quantum can be determined.

In the second place, a Quantum would not be determined by the fact that it could increase no further. For its instability works both ways. There is no more reason why it should not be smaller than it is, than why it should not be larger than it is. (Hegel only speaks of the indeterminateness in the one direction, but his arguments apply equally to the other.) Thus, suppose a Quantum could contain all the Ones, the process of Alteration would take place with it as much as with any other, though it could only take place in one direction.

64. How is the contradiction to be avoided? In a very similar way to that in which the same difficulty was met in the case of Qualitative Infinity. That which is outside any Quantum is another Quantum. If we try to find the determination of any Quantum in itself exclusively, then we find that its Limit continually alters, and that the task is endless. But the case is changed if we fully accept the relation of each Quantum to the other which is outside it. No Quantum can determine itself as against another Quantum. But two Quanta can reciprocally determine one another. There is no reason why 7 Ones should not change to 8, or 17 Ones to 16, if we take 7 Ones and 17 Ones as isolated facts, each of which must be determined by itself, or not at all. But if we take these Quanta as related to one another, then there is a reason why 7 Ones should not become 8 — for then the Quantum would bear a different relation to the 17 Ones. And there is a corresponding reason why 17 Ones should not become 16. Thus the Quanta have now some real self-determination, though it is slight; *A* cannot become greater or less, because it would thereby change its relation to *B*. And its relation to B is what it is, not only because B is B, but because *A* is *A*. With this partial self-determination we reach

(*G. L*. i. 279. *Enc. *105) by which is meant the true Infinity of self-determination, as opposed to the False Infinity of an unending progress.

65. It will be noticed that there is a difference between the Quantitative Infinite Progress and the earlier Qualitative Infinite Progress. In. Quality the Something finds its nature only in another Something, which in turn finds its nature in a third, and so on. The Somethings themselves do not change, but fresh Somethings are continually reached in the vain search for a final determination. In Quantity, however, the Infinite Progress is not a Progress of an Infinity of Quanta, but of a single Quantum, which endlessly increases in size as it successively overleaps every Limit.

In Quality no change of anything was possible. The nature of reality was not yet sufficiently complex to allow anything to become different in one respect while remaining the same in others. If a thing is not completely the same it has utterly vanished. It is impossible, therefore, for a Something to change, and the Infinite Progress can only take place by adding fresh Somethings.

In Quantity, however, change is possible. The gradual addition of fresh Ones to a Quantum affords a changing element, while the Ones previously in it afford the permanent element, without which there can be no change.

With this stage of the dialectic the idea of Quality becomes more prominent again. Not only are the Ones each a separate Quality, as they have been all along, but in each Quantum, also, a Qualitative nature begins to develop (*G. L. *i. 281.* Enc. *105). This is most clearly stated in the *Encyclopaedia. “*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 essential character of Quantity was its instability. Now this characteristic begins to disappear. The Quantum can no longer alter without any effect on anything but its own Magnitude. For it is now in relation to some other Quantum, and it cannot alter unless either that other Quantum, or the relation, alters simultaneously. This is the first step (though as yet a very, small one) towards bringing back, on a higher level, the fixity of Quality. With it we pass out of Quantum to the third and last division of Quantity, after some mathematical digressions occupying nearly a hundred pages.

66. (*G. L*. i. 379.* Enc. *105.) The Ratio between two Quanta, says Hegel, is itself a Quantum (*G. L*. i. 380). And it is true that it is a determinate number. But it differs too much from the Quanta, which it relates, to have any claim to the name of Quantum. For they are Quanta of Ones, while the Ratio is not. The Ratio between twelve existent Ones and six existent ,Ones is certainly two, but it is not two existent Ones. Hegel does not seem to see this, and treats all three quantities here as if they were simply terms in abstract arithmetic, in which he is not justified.

67. The first and simplest form of Ratio is called

(*G. L.* i. 381.) The related Quanta are here taken as logically prior, and the Quantum which is their Ratio as logically subsequent. Thus we get, for example, that the Ratio of 7 to 3,5 is 5. The Ratio is called the Exponent.

Now the Quantum which is the Patio is no more determined by the two Quanta of which it is the Ratio than it is by an infinite number of pairs of other Quanta. For example, 5 is equally the Ratio of 6 and 30, of 8 and 40, and so on. It follows that the related Quanta can alter to any extent absolutely, provided that they do not alter relatively. So long as one remains five times the other they may both increase or diminish indefinitely.

Nothing is stable but the Exponent. And therefore Hegel finds it a defect in this category that the Exponent is not sufficiently discriminated from the other Quanta. It cannot be the largest of the three Quanta concerned, but it can be either of the others. We have said that 7 and 3.5 stand to each other in a Ratio, expressed by 5. But we might just as well have said that 5 and 35 stand to each other in a Ratio expressed by 7 (*G. L.* i. 383).

It seems to me that this argument is defective because it ignores the fact, pointed out above, that the Ratio is not a Magnitude of the same sort as the Quanta of which it is a Ratio. They are Quanta made up of existent Ones, using the word One in the special sense in which the dialectic has determined it. But the Ratio is not a Number of Ones, in this sense, at all. Therefore the Ratio and the related Quantum are not interchangeable in the way Hegel asserts.

68. It is on this supposed defect that the transition to the next category is based. Since — this appears to be Hegel’s argument — the Exponent has the stability which the other Quanta do not possess, it must be distinguishable from them. But in Direct Ratio this is not the case, since the Exponent is interchangeable with the related Quanta. We must therefore seek out another sort of Ratio, where the Exponent is marked out by the nature of the relation. Now, if we take three integral numbers, there is a relation between them which has the required definiteness. If one of them is the product of the other two, then it is the largest of the three that will be the product^{[8]}. So we come to

(*G. L.* i. 384) where the Exponent is the product of the two related Quanta. It appears to be called Inverse because the increase of one of the related Quanta involves the diminution of the other.

69. The transition to the next category is extremely obscure. So far as I can understand it, it is as follows (*G. L.* i. 389). Either of the two related Quanta can increase, so long as the other diminishes, the only Limit of this process being that neither of the related Quanta can become larger than the Exponent. Thus either of the related Quanta is implicitly (*an sich*) the Exponent. Hegel calls this “ the negation of the externality of the Exponent.” This means, I believe, that there are no longer necessarily *three *Quanta, but only two, namely the Exponent, connected with one other Quantum, no longer by a third Quantum, but by some non-Quantitative relation. And thus, says Hegel — without giving any further explanation — we reach

(*G. L.* i. 389.) By this he appears to mean only the special relation which exists between two numbers, one of which is the square of the other. It is the square, is the result of the process, which is treated as the Exponent.

70. The transition appears very questionable. It may be admitted that the indefinite approximation of one of the related Quanta to the Exponent brings a Qualitative element into greater prominence, and that the Ratio of Powers has also a relatively prominent Qualitative element. But in other respects they are quite different conceptions. And Hegel gives us no reason for passing at this point from one partially Qualitative relation to another and distinct partially-Qualitative relation. He is satisfied with showing that they are both partially-Qualitative, which is clearly not sufficient.

It is difficult to see, too, why Hegel thought himself justified in considering only those cases where one Quantum was the square of the other, and in excluding cubes and other powers. If, however, he *had *considered those other powers, it would have become evident that the relation between the two Quanta was not yet one which could dispense with a third Quantum. For the question of the power to which one was to be raised to equal the other could only be answered by naming a third Quantum.

Hegel makes the transition to the next category as follows: “Quantity as such appears as opposed to Quality; but Quantity is itself a Quality, a determination in general which relates itself to itself, and which is separated from the other determination, from Quality as such. Yet it is not only a Quality, but the truth of Quality itself is Quantity: Quality has shown itself as going over into Quantity ; Quantity, on the other hand, is in its truth that externality which is turned back on itself, which is not indifferent. So it is Quality itself, in such a way that outside this determination Quality as such is no longer anything .... The Quantum now as indifferent or external determination (so that it is just as much transcended as such, and is the Quality, and is that, through which anything is what it is) is the truth of the Quantum — to be Measure “ (*G. L.* i. 392).

71. We have now reached the end of Hegel’s treatment of Quantitative Ratio. As we have seen, serious objections exist both to the transition from Direct to Inverse Ratio, and to the transition from Inverse Ratio to the Ratio of Powers. But, apart from these, there is a more general objection. The whole triad of Quantitative Ratio is a blind alley. It does not lead, as it professes to lead, to the category of Measure, and the chain of the dialectic cannot be continued through it.

The passage I have quoted above contains the transition from Quantity to Measure. We have therefore before us the way in which the inadequacies of Quantity are, according to Hegel, to be transcended, and in which Quality is to he synthesised with Quantity in Measure. These objects would certainly have been attained if Hegel had succeeded in his attempt to demonstrate that Quantity is Quality. But it seems to me that he has not reached this result by Quantitative Ratio, and that therefore he has neither removed the inadequacies of Quantity, nor synthesised it with Quality.

As to the first. The special characteristic of Quantity was its instability. We saw, to begin with, that it was that which could alter, and yet remain the same. When we reached Alteration of Quantum, we found that it not only could alter, but must alter, and it was to remedy the contradictions thus caused that we were forced to resort to Quantitative Ratio.

Does Quantitative Ratio remove this indifference, even when taken in its highest form, the Ratio of Powers ? Let us pass over the difficulty that the power to which a number is to be raised can only be expressed in another number, which might be any other. Let us confine ourselves, as Hegel does, to squares, arid ignore the Quantitative nature of the index. Has this removed the instability ? If we take 49 Ones as a simple Quantum, it is under the necessity of changing continually. If we take it as the square of 7^{[9]}, has the necessity disappeared ?

Surely it has not. It is true that the Square cannot now change unless the Root changes as well. But the Root is also a Quantum, and so it also will be unstable, and the Square will be unstable with it. The first numbers the Square can change to are no longer 48 and .50 but 36 and 64. But the number of changes of the Square is unbounded except by the total number of Ones, and we have seen that this restriction does not remove the instability of Quantum. And therefore Quantitative Ratio has not removed the contradictions of Quantitative Infinity, nor has it enabled xis to transcend the characteristic nature of Quantity. It is true that the Square and the Root are linked Quanta, but they are still Quanta.

Very closely connected with this is the second defect of the triad. It professes to lead us to Measure, and it must therefore bring back Quality. In the passage quoted above (*G. L.* i. 392) Hegel says that it has done this. Quantity “is Quality itself, in such a way that outside this determination Quality as such is no longer anything.” That is to say, he holds that we have here reached a Synthesis of Quality and Quantity. Now it is true that the introduction of related Quanta has introduced a certain Qualitative element into Quantity. The movements of each separate Quantum are no longer completely arbitrary and unconditioned, and every restriction on the movement means some departure from the typical idea of Quantity. But this does not amount to what Hegel claims to have reached the complete absorption of Quantity. We have got a Quantity, which is more like a Quality than before, but which is still essentially a Quantity and riot a Quality. The test of this is the instability, and the Infinite Progress to which the instability gives rise. Till we have got rid of this, we have not transcended Quantity. For the instability is, as we have seen, the special characteristic of Quantity. Thus we have riot reached a point at which Quantity is transcended, and therefore united with Quality. The category at which we stand is still essentially Quantitative, and does not combine Quantity with Quality. And as Measure certainly has to combine Quantity with Quality, we have not yet got a valid transition to Measure.

72. What then is to be done? We saw reason to think that the transition from Quantum to Quantitative Ratio is valid, and I believe that it is possible to recast the triad of Quantitative Ratio in such a *way *as to make a valid transition to Measure. The Thesis of my proposed triad would be the restatement of the general idea of Quantitative Ratio, as it had been arrived at la the previous category of Infinity of Quantum. It might be called *Quantitative Ratio *as such, or again *Quantitative Ratio in general *(*überhaupt*), either of which would be in accordance with Hegel’s terminology.

The inadequacy of this Thesis would lie in the fact that a Quantum is not fully determined by its Ratio to another unless that other Quantum is determined. Nor can the two Quanta mutually determine one another by their Ratio, for, as we have seen, two Quanta can vary and yet preserve the same Ratio to one another. The second Quantum, then, must be determined by its Ratio to a third, with regard to which the same question will arise, and so on continually. This Infinite Series forms the Antithesis of our triad, and might be called the *Infinite Series of Ratios.*

It will be noticed that this Infinite Series resembles the Infinite Series of Quality rather than the Infinite Series of Quantity. For the Ratios do not continually alter, as the Quanta did in the Series of Quantity. The Infinity comes in through the necessity of going to fresh Ratios to determine those already existing.

Here, as in the two previous cases, the Infinite Series involves a contradiction. The original Quantum is determined. But it can only be determined by a Ratio to a Quantum which is determined otherwise than by a Ratio. But no such Quantum is to be found. Therefore the original Quantum is not determined, and we have a contradiction.

We must pass on, then, to a fresh category, which will remove this contradiction, avid will form the Synthesis of Quantitative Ratio. We have seen that Quantity, however developed, can never, while it remains only Quantity, get rid of the inadequacy which has shown itself once more in the Infinite Series of Ratios. Now the ground of this inadequacy was the necessary instability of *all *Quanta. And this instability, we saw, proceeded from the fact that the differences between all Ones, were so similar that no reason could be assigned why a Quantum should stop at any particular limit rather than another. (Cp. above, Section 59.)

The only way of escaping from our difficulty, therefore, will be to reject this similarity of the differences between the Ones, and to find d a state of’ things in which the natures of the Ones shall link some of them more closely together in a group from which others are excluded. And this can be done only if there are Qualities each of which belongs to several Ones, but not to all, so that each of these forms a bond which binds those Ones which have it into a group from which those which do not have it are excluded.

The instability of Quanta would thus be arrested. For there would be a reason why the Quantum should not increase beyond a certain Limit. Every Quantum is a Quantum of Ones which have a certain common Quality, and beyond a certain Limit there would be no more Ones with that Quality. The Ones outside it would have some other Quality.

We have now reached a category which transcends the inadequacy of Quantitative Ratio, and also of Quantity generally, and so reaches the category of _Measure as defined by Hegel (*G. L.* i. 392) in the passage quoted above. Our argument avoids Hegel’s error of ignoring the difference between a Quantum of Ones and a Quantum (if that name is appropriate to it) which is a Ratio between Quanta.

73. The treatment of Quantity in the *Encyclopaedia is *practically the same as in the *Greater Logic, *except in one point. In the *Greater Logic, *as we have seen, Extensive and Intensive Magnitudes, and the Infinite Progress all fall within the second subdivision, while the third subdivision is completely taken up by Ratio. In the *Encyclopaedia, *the second subdivision (named, as in the *Greater Logic, *Quantum) deals with Extensive Magnitude only. The third subdivision is called Degree, and contains Intensive Magnitude, tile Infinite Progress, and Ratio. This arrangement shows more clearly that an advance is made in passing from Extensive to Intensive Magnitude, but otherwise it seems inferior to the order of the *Greater Logic. *For Intensive Magnitude seems more closely connected with Extensive Magnitude than it is with Ratio. And, again, the Infinite Progress makes manifest the characteristic contradiction inherent in all Quantity. It would seem, therefore, more appropriately placed in the second subdivision, which is the Antithesis of the triad of Quantity, than in the third, which is the Synthesis.

1. *Mind*, 1908, p. 242.

2. Whether there is such a failure or not is left undetermined by Mr Russell’s criticisms, since these do not deal with the main course of the argument but with one of the mathematical Notes.

3. Hegel’s use of the term Intensive Quantum differs considerably from that of most other writers.

4. This argument assumes the principle of the Identity of Indiscernibles, since it would be invalid if Ones could differ in their relations without differing in their nature. But Hegel habitually assumes the truth of this principle. (Cp. Section 6.)

5. The transition from the Quantum is taken by Hegel as being first to an indefinite Quantity. (Cp. below, Section 61.) It would therefore have been better if he had said here that each Quantum was bounded by another *Quantity*.

6. The category of the Notion of Quantitative Infinity, which we have just been considering, corresponds to the category in Quality called Infinity in General, and the Quantitative Infinite Progress corresponds to the Reciprocal Determination of the Finite and Infinite. We saw reason to think (Section 31) that the stage of Infinity in General was a mistake, and that we should have passed, in Quality, direct from Finitude to Reciprocal Determination of the Finite and the Infinite.

I do not, however, think that the Notion of Quantitative Infinity is an invalid category. The argument for the Infinity is quite different in the two cases, and here it seems to be valid. If a Quantum passes its Limit, the first result of that is, as Hegel states it to be, that it becomes an unlimited Quantity. It is a fresh step in the argument to show that this Quantity must still be a Quantum and have a fresh Limit, and so on indefinitely. Thus the passage to the Infinite Progress in Quantity, unlike the passage to the Infinite Progress in Quality, does require a transition through a stage of absence of Limitation.

It was possibly the necessity for such a stage of absence of Limitation in Quantity, which misled Hegel into supposing that it was necessary in Quality as well.

7. His expression in the *Encyclopaedia is “welches Kant als schauderhaft bezeichnet, worin indess das eigentlich Schauderhafte nur die Langweiligkeit sein dürfte*.” in the first edition of his translation Prof. Wallace happily renders this: “which Kant describes as awful. The only really awful thing about it is the awful wearisomeness.” The second edition is, I think, less successful.

8. The related Quanta must be represented by integral numbers, since they consist of indivisible ones. The product therefore must also be integral.

9. I do not recur here to the difficulty that 49 Ones (in the Hegelian sense) are neither the square of 7 Ones, nor of 7, nor of anything else. This is a fresh case of the mistake mentioned above (Section 66).