I. From Quality to Quantity

For Hegel there is a clear difference between quality and quantity. A qualitative determination constitutes what something is; when this is altered, therefore, the thing becomes something else. A quantitative determination, by contrast, can be changed without changing the thing itself: ‘a red, whether brighter or paler, is still red’.Footnote ⁶ Yet despite this difference, Hegel thinks that quantity is made necessary by quality. More specifically, it is made necessary by the qualitative category of the one (das Eins).

In the initial stages of his account of quality Hegel distinguishes between ‘determinate being’ (Dasein) and ‘something’ (Etwas). Dasein is rendered determinate by simple negation: it is being that is this, not that. Such being thus never stands alone, but always distinguishes itself from what it is not, from its negation (see SL 111/LS 105-6). The structure of something is subtly different: to be something is not just to be one side of a difference—this, not that—but to be self-relating being. It is to be something of one's own (SL 115/LS 110). Yet something also stands in relation to its negation. The negation of something, however, exhibits the same self-relation as something. Accordingly, it stands apart from something as the latter's other (Anderes) (SL 116/LS 111). There is thus a difference, in Hegel's view, between being the simple, determinate negation of X and being other than X: the latter consists, not just in not being X, in differing from X, but also in being something of its own and relating to itself.Footnote ⁷

In the category of the one (das Eins), being determinate and being self-relating coincide completely, and in this sense quality finds its fullest expression in being one. The one has a determinate identity: it is the one, not anything else. It excludes everything else to such a degree, however, that there is in fact nothing outside it at all. Unlike something, therefore, the one (at least initially) has no other; it hangs alone in the void (das Leere) and in this way is purely self-relating being, or sheer being-for-self (SL 165/LS 168-9).Footnote ⁸

Hegel goes on to show, however, that this void is itself another empty one. The one thus stands in relation to another one after all—and so does that one, and that one, ad infinitum. The one does not just hang alone in the void, therefore, but is one of many ones. Yet in this world of many ones each is the same empty one: there is no qualitative difference between them. This introduces a new logical structure: for in the other ones around it each one encounters and relates to nothing but itself. This in turn means that the one continues beyond itself in the others.

We might think that something (Etwas) continues beyond itself, too, insofar as it relates to another that is itself something. That other, however, is qualitatively different from the first something, precisely because it is other than it. Every one, by contrast, is the very same one: each is the same empty unit. It is only with the one, therefore, that we encounter entities between which there is no qualitative difference at all. Accordingly, only at this point does the idea emerge of being that continues beyond itself; and so only now do we make the transition from quality to a new form of being.

Now in the sphere of quality the limit (Grenze) is where something ceases and another begins (SL 126-7/LS 122-3). With the one, however, we encounter a new form of being that continues beyond its own limit. Where one one ends and another begins, the one itself continues without interruption. In this sense, Hegel argues, the limit that differentiates one one from another is a limit to which the one as such is indifferent. Such a limit to which being has become indifferent is a quantitative limit. The new form of being that arises with the one is thus quantity.

A full account of the transition from quality to quantity would have to examine the relation between the ‘repulsion’ and ‘attraction’ of the many ones (see SL 174-8/LS 179-84).Footnote ⁹ It is clear from Hegel's text, however, that this transition is already made necessary ‘in principle’ (an sich) by the one as such (SL 176/LS 182). For our purposes, therefore, it suffices to say simply that quality gives rise to quantity via the category of the one.

For Hegel, in contrast to Kant, quality and quantity are thus not just two distinct categories; they are related logically because the former makes the latter necessary. Indeed, quantity is itself a distinctive kind of quality: namely, that which continues beyond any limit that is placed on it. This in turn enables a thing with a specific limited magnitude to go beyond that magnitude, and so to get bigger or smaller, without ceasing to be. The familiar fact that magnitudes can be increased or decreased is thus not just an unexplained given, but one that is made intelligible by the derivation of quantity from quality.

To recapitulate: quality consists in being what one is, in being oneself. Yet it is also bound to what is other than it: hence being-in-itself (Ansichsein) is inseparable from being other-related (Sein-für-Anderes) (see SL 119-20/LS 114-16).Footnote ¹⁰ The one (das Eins), however, is purely self-relating: it is just the one it is and so is purely for itself. Indeed, even the others to which it relates are simply further versions of itself: the one is surrounded, not by something other than the one, but simply by many other ones.

The one, therefore, as purely self-relating being, is the most developed form of quality: it is being-oneself taken to its furthest extreme. Its self-relation, however, is profoundly ambiguous: for it is and relates to nothing but itself, yet in so doing it actually continues beyond itself. Insofar as it extends beyond itself, or outside itself, however, it is no longer simply itself, and so is no longer purely qualitative, after all. Rather, it gives rise to a new form of being: quantity. We will now consider how Hegel derives further categories from the simple idea of quantity as such.

II. Quantity as such

As we have seen, quantity is characterized by continuity—by the continuing of the one beyond itself in other ones (SL 187/LS 194-5). As such, quantity necessarily contains many ones or units within itself. These ones are themselves purely self-relating, and so are indifferent and external to one another: they are thus quite discrete units. At the same time, however, they are all identical: each is the same discrete one. In their very discreteness, therefore, they constitute a continuity: the uninterrupted continuing of the discrete one itself. Quantity, for Hegel, is thus ‘the unity of these moments of continuity and discreteness’ (SL 187/LS 195).

Hegel notes, however, that quantity also divides itself into the distinct forms of continuous and discrete magnitude. Where the unity of its two moments is immediate, and so directly to the fore, quantity is itself dominated by unity and continuity. It is thus explicitly continuous, uninterrupted quantity. This is how, for example, the extension of space is to be conceived. Yet where the unity of continuity and discreteness is not immediate, but mediated by their difference, the latter is to the fore and so quantity is itself dominated by difference and discreteness. As such, it is not explicitly continuous, but is broken up into discrete units. Such quantity is thus a discontinuous ‘plurality’ (Menge) (SL 200/LS 210). This is how we conceive of quantity when we ask how many Xs something contains.

Continuous and discrete quantity or magnitude are not, however, absolutely distinct. Continuous magnitude is the continuity of discrete units and so can be interrupted at any point; and the units belonging to discrete magnitude themselves form a continuous unity. Each form of quantity is thus the same unity of continuity and discreteness; but in each just one of the two moments is, in Hegel's word, ‘posited’ as the principal characteristic (SL 200/LS 210).

Here we see an important logical difference between quantity and quality. Qualitative categories, such as something and other, pass over into one another and so prove to be inseparable: something is itself other than its other, and the latter is in turn something. Hegel notes, however, that ‘in the sphere of quality, the differences [ … ] also retain the form of immediate, qualitative being relative to one another’ (SL 123/LS 119). Something may, indeed, be other than its other, but being ‘something’ and being ‘other’ are nonetheless immediately different categories: each, we might say, is immediately itself.

In the sphere of quantity, by contrast, no category is immediately itself and immediately different from its negation. Continuity is already in itself the continuity of the discrete, and discreteness is in itself continuous discreteness. Quantitative determinations do not just pass over into one another, therefore, but each contains its opposite within itself. In this sense, one can say that continuity and discreteness both continue in their opposites. Both are thus infinite, unending, rather than finite, limited determinations.Footnote ¹¹

Insofar as continuity and discreteness are moments of one another, quantity clearly anticipates the logical structure of essence. In the sphere of essence, however, a determination, such as the positive, incorporates its opposite, the negative, as something negated. The positive thus includes the negative as excluded from it (see SL 426-7).Footnote ¹² This is what Hegel means by saying that determinations of essence merely ‘shine’ (scheinen) in their opposites.Footnote ¹³ In quantity, by contrast, continuity is contained affirmatively in discreteness and vice versa. Categories of essence are thus above all the negation of their own negation, and in that sense lack any simple immediacy of their own. Quantitative categories, however, retain a relatively immediate sense, even in their lack of immediacy and independence, since each is affirmatively present as itself in the other. So although quantity does indeed anticipate the sphere of essence, it remains, with quality and measure, a moment of being.

III. Quantum

Now discrete magnitude, as we have seen, comprises a plurality of units: it is the kind of magnitude that prompts us to ask ‘how many?’ This plurality, however, is not just a loose, unconnected multiplicity, but forms a continuous unity. As such a unity, however, it remains characterized by discreteness. In Hegel's view, this means two things. On the one hand, such magnitude encompasses discrete units within itself; on the other hand, however, it is also discrete as a unity. That is to say, the discrete units in discrete magnitude form a unity that is itself discrete—a unity that is, as it were, broken off at either end. Discrete magnitude is not just a general kind of magnitude, therefore, but it takes the form of a single discrete unit, or ‘a magnitude’ (eine Größe) (SL 201/LS 211).

Since this unit is broken off at either end, it is determinately different from other such units. Indeed, it is the limit of such units: where they stop and it begins. The single, discrete unit formed by discrete magnitude must, therefore, be a limited magnitude that excludes other such magnitudes. Accordingly, it is what Hegel calls a quantum. A quantum is thus nothing but ‘quantity with a determinateness or limit’ (SL 202/LS 212). It is quantity, not just as continuous discreteness as such, but as a discrete, limited quantity.

Note that Hegel derives the quantum from discrete magnitude in particular. Discreteness, however, is to be found in both continuous and discrete magnitude. Both kinds of magnitude, therefore, must take, or be capable of taking, the form of quanta. According to Hegel, there is no quantity anywhere that is completely separate from quanta.Footnote ¹⁴

IV. Number

The principal features of the quantum, in Hegel's view, are (a) that it is a discrete unit of quantity, a one, and (b) that it is explicitly determinate and limited. Indeed, it is both of these things together: determinate in being a one. Yet being a one is not enough by itself to render the quantum fully determinate, for in this respect all quanta are the same. Something else is required, therefore, to render a quantum fully determinate and so differentiate it from other quanta. Apart from its being a one, however, there is nothing to a quantum but the ‘plurality’ (Vielheit) of discrete units that it contains. Hegel argues, therefore, that this plurality must be what differentiates two quanta that, as ones, are completely alike.

In order to distinguish a quantum from other quanta, however, this plurality must itself be ‘a determinate plurality’ (eine bestimmte Menge) (SL 203/LS 214).Footnote ¹⁵ It must be this plurality and not that one. Such a determinate plurality Hegel calls an ‘amount’ (Anzahl). When a quantum is understood to contain an amount, it is understood in turn to be a number (Zahl). In contrast to the bare quantum, therefore, the number incorporates not just an undetermined ‘plurality’ of units, but a definite amount of them, and this amount is what differentiates one number from another. The number, however, is not reducible to the amount it contains, but it is also a continuous unity. For Hegel, therefore, ‘amount [Anzahl] and unity [Einheit] constitute the moments of number’ (SL 203/LS 214). Each number is a unity—one number—but it is a unity with its own distinctive amount of units.Footnote ¹⁶

Hegel notes, however, that the two moments of unity and amount, though united in number, are qualitatively distinct from one another (SL 206/LS 217). This is because the amount contains the unique determinacy of a number, whereas its unity does not. This in turn means that the amount, though falling within the unity of the number, is not itself a unity. It is, rather, an aggregate of units that are wholly discrete and external to one another (SL 203/LS 214). This sheer externality without unity is at odds with the fundamental nature of quantity in which discreteness is inseparable from continuity and unity. Nonetheless, it is integral to number. Moreover, it is what determines the specific operations that can be performed with numbers.

Since a number contains units that are wholly external to one another, the latter do not belong together by their nature. It is a matter of indifference to those units, therefore, whether they are collected together as this amount or that, as 3 or as 4: the three units in the number 3 could just as easily be put together with a fourth to make the number 4. This indifference to the way in which they are collected in a number means in turn that units can be taken away from or added to numbers, and that we can thus calculate with numbers. In Hegel's view, therefore, addition and subtraction are not just contingently connected to number, but operations grounded in the logical structure of number itself (see SL 204-12/LS 215-24). Mathematics is thus firmly grounded in logic, just as Frege tried to show (albeit, of course, with a very different conception of logic).Footnote ¹⁷

Two further things should be noted before we continue the story of quantity. First, Hegel's conception of number, as a unity containing an amount of wholly discrete units, closely matches that which was common among the ancient Greeks. Ivor Bulmer-Thomas, for example, maintains that ‘the very first definition of number in the history of Greek mathematics—by Thales according to Iamblichus—was μον $$)(--><$>\ralpha{\vskip-2.3pt\hskip-3pt{\scale75%{\tf="TNR"\char"A9}}\hskip1pt }<$><!--$$ δων σ $$)(--><$>\rupsilon{\vskip-2.3pt\hskip-3.6pt{\scale75%{\tf="TNR"\char"A9}}\hskip1pt <$><!--$$ στημα, “a collection of units” ’, and that ‘Plato's brightest pupil, Eudoxus, defined number as πλ $$)(--><$>\tilde{\reta }<$><!--$$ θος ω $$)(--><$>\rvarrho <$><!--$$ ισμένον, “a determinate multitude” ’.Footnote ¹⁸ It is evident, therefore, that Hegel's presuppositionless logic will not always produce radically new conceptions of categories; in some cases it may simply demonstrate, without taking it for granted from the start, that the ancients were right all along.

Second, the numbers derived at this point in the Logic are the cardinal natural numbers. Fractions are not derived until we reach the idea of a direct ratio between numbers (see SL 245, 315-17/LS 265-6, 351-3); and irrational numbers cannot be conceived until we reach the idea of a quantitative infinite progress.Footnote ¹⁹ Hegel is not simply ignoring these kinds of number, therefore, but in both cases more is required to understand them than the bare idea of number as such.

V. Extensive and Intensive Quantum

Hegel has shown that quantity divides itself into discrete units of itself, or quanta, and that the latter must take the further form of numbers. Numbers, for Hegel, are thus not simply conventional distinctions, or fictions, imposed on the world by human beings, but they are essential features of the world itself. Just as the category of quantity belongs of necessity to thought, so quantity itself belongs to the fabric of being. This means that numbers in turn also belong to being: the world and the things in it are intrinsically susceptible to being counted. And yet it remains true that quantity, and thus number as well, is that to which things are indifferent (at least to a certain degree). Thus, although it belongs to being a thing to have magnitude and to be countable, things do not come with just one number attached to them: I can increase my weight or lose my hair and still remain who I am. As we learn in the section on measure, there is a range of numbers within which a thing has to be confined in order to be what it is; within that range, however, the thing is essentially indifferent to its magnitude and number. Number, for Hegel, is thus a peculiar hybrid that is both intrinsic to being and a matter of indifference to, and so external to, being at one and the same time.

Hegel now shows that quanta and numbers actually come in two different forms: they must be both extensive and intensive. Let us consider first the extensive quantum.

The quantum as such, before it is a number, is simply a discrete, limited unit of quantity. Yet it is not only a unit itself, but also contains many units within it. When that internal manifold is understood to be a specific amount (Anzahl), the quantum is understood to be a number (Zahl). As a number with a specific amount the quantum is clearly distinguished from all other quanta: it is unique. Each number, however, remains a unit—remains one number—and in this respect is the same as all the others. Numbers are thus distinguished by their amounts, but they are all alike in being units that contain an amount at all. In this latter respect, Hegel maintains, all equally are extensive quanta (SL 217/LS 230-1).

Note that, though each number is an extensive quantum, being the latter is, logically, subtly different from being the former. The difference lies in the nature of the amount they contain. A number has a specific and thus quite distinct amount—3 or 4 or 5. To think of a number as an extensive quantum, however, is to disregard the specific amount it contains and to think of it simply as a unit containing some amount, just like all the others.

Hegel maintains that this idea of the extensive quantum now leads directly to that of the intensive quantum or degree. It has to be said that his argument at this point is especially concentrated, even by his standards, but the gist of it, as I understand it, is this.

A number, as we have seen, is a collection, or what we might call a set, of units. These units are all the same—are all one—and so constitute the unity of the number. Yet they also constitute its amount. The unity and amount of a number are thus constituted by the same set of units. As we remarked earlier, however, the two moments of number are also quite distinct from one another. Insofar as the units constitute a unity, therefore, they do not constitute an amount, and vice versa. This in turn means that, although the amount falls within the unity of the number, it is itself a collection of mutually external units without unity—units that can thus be subtracted from or added to numbers.

Recall now that an amount is by definition determinate and specific: it comprises 3, 4 or 5 units and thereby confers a unique identity on to the number concerned. Yet, as we have seen, when the number is conceived as an extensive quantum, like all the others, the amount is no longer specific but just some unspecified amount. An amount that is no longer specific, however, is not actually an amount at all, but merely an indeterminate aggregate (Menge) of units.

This point is crucial and dramatically transforms the extensive quantum: for if the units in that quantum no longer constitute an amount, they must, by virtue of their homogeneity, form a unity, since amount and unity are the only components of the quantum. Furthermore, in ceasing to form an amount, the units also cease being wholly external to one another, as they are in the number: in Hegel's words, ‘the externality, which constituted the ones as a plurality, vanishes [verschwindet]’ (SL 218/LS 232). Yet if the units within the quantum are no longer mutually external, the quantum itself can no longer be extensive. The unity formed by the units, therefore, must be one without inner extension, or a wholly ‘simple unity’ (einfache Einheit).Footnote ²⁰ Such a quantum, without an internal sum of mutually external units, Hegel calls intensive. Every extensive quantum thus mutates, through its inherent logic, into an intensive quantum or degree.

To repeat: the units in the extensive quantum no longer constitute a specific amount, but just form some amount or other. This means, however, that they no longer constitute an amount at all, since the latter is by definition specific. Consequently, the units form a unity as opposed to an amount. Since an amount comprises units that are external to one another, this unity must lack such mutual externality and so be an intensive, rather than extensive, quantum.

As we shall now see, the intensive quantum or degree has a distinctive, and somewhat contradictory, logical structure. Like all quanta, the degree is determinate and limited. We know, however, that a quantum is fully determinate only as a number. Yet, as we have just seen, the degree is a simple unity without an internal amount. The degree cannot, therefore, be a number in the sense in which it has been defined so far, but must be what Hegel calls a ‘simple determinacy’ (einfache Bestimmtheit) (SL 219/LS 233).Footnote ²¹ Such a number, in other words, must have a definite identity but cannot be a collection of mutually external units within itself. It will thus not be a cardinal number like 3, 4 or 5, but a new kind of number altogether: the 3^rd, the 4^th or the 5^th.

Yet there is a further complication to be considered: for numbers, by definition, are rendered determinate by a specific amount of units. Since the degree does not itself contain the amount that gives it its number, Hegel concludes that this amount must fall outside the degree and determine it from the outside (SL 219/LS 233). This does not mean that the tenth degree is surrounded by ten other degrees, but rather that the amount of degrees lying outside it determine it to be the tenth. The number ten contains ten units within itself; the tenth degree, by contrast, has its determining ground in the amount of other degrees there are. This is true of every degree. Together, therefore, they form a ‘scale of degrees’ in which each is what it is only by means of the others (SL 219/LS 234).

The logic of quantity not only makes numbers necessary, therefore, but it also requires that they take two distinct forms. Every number must be extensive and be a cardinal number; but it must also be intensive and so be an ordinal number. The numbers 1, 2, and 3 are thus also to be understood as the first, second, and third in this series. This distinction is not merely a conventional one drawn by us, but belongs, in Hegel's view, to the very nature of numbers and quanta themselves. Quantity, for Hegel, necessarily comes both in definite amounts and in degrees.

VI. Kant and Hegel on Quantity

At this point I interrupt my account of Hegel's conception of quantity in order to compare the latter briefly with the conception of quantity put forward by Kant. The following comments are sketchy, but they will, I hope, highlight some significant similarities and differences between the two philosophers.

1. 1) Categories, for Hegel, are to be considered by themselves: they have their own intrinsic logical structure. For Kant, by contrast, categories should be thought in relation to sensuous intuition. This is because they are nothing but the ‘functions for judging, insofar as the manifold of a given intuition is determined with regard to them’ (CPR B 143; see also CPR B128 and A 245). More precisely, categories are pure concepts through which ‘the synthetic unity of the manifold in intuition’ is thought (CPR A 79/B 105). Yet Kant also accepts that we can consider a category in abstraction from intuition, though ‘one can have no insight at all into the possibility of a thing in accordance with this concept’.Footnote ²² The concept of ‘magnitude’ (Größe), for example, ‘if the restriction to sensible intuition were omitted, would go thus: It is the determination through which many homogeneous elements [vieles Gleichartige] taken together constitute one’.Footnote ²³ Kant thus conceives of the logical structure of magnitude in much the same way as Hegel, namely as a unity of homogeneous units—though he does not derive quantity from the very quality of being one, as Hegel does.

2. 2) Like Hegel, Kant also distinguishes between a quantum and quantity or quantitas. Béatrice Longuenesse explains this difference as follows: ‘“magnitude” (= Größe as quantum) refers to an object immediately given in intuition’; on the other hand, ‘“magnitude” (Größe as quantitas) is a predicate, referring to a quantitative determination of the quantum’. Only the latter, she writes, ‘is strictly speaking an instance of the category of quantity’.Footnote ²⁴ Longuenesse's account of Kant's distinction nicely reflects the fact that Kant understands categories, like all concepts, to be ‘predicates of possible judgments’ through which we understand objects to be determined in a certain way (CPR A 69/B 94). Hegel, however, does not assume from the start that categories are predicates of judgements. In his view, categories are first and foremost concepts that directly disclose ways of being.Footnote ²⁵ Quantity is thus not simply a predicate but a form of being itself. A quantum in turn is not an object to which the predicate ‘quantity’ can be applied, but it is quantity itself in a determinate, limited form. To use Spinozan terminology, a quantum is a mode of quantity itself, rather than an object that falls under the predicate ‘quantity’. This is perhaps the most important difference between the conceptions of quantity entertained by Hegel and Kant.

3. 3) It is not, however, the only difference: for, unlike Hegel, Kant understands number in terms of time. Number, for Kant, is a ‘representation that summarizes the successive addition of one (homogeneous) unit to another’ (CPR A 142/B 182, my emphasis). Yet if we abstract from time and consider the purely logical aspect of number, Kant's conception of the latter turns out to be quite close to Hegel's. Kant claims that the concept of number ‘belongs to the category of allness’, and that ‘allness (totality) is nothing other than a plurality considered as a unity’ (CPR B 111).Footnote ²⁶ This, it seems to me, is not so different from Hegel's idea that ‘amount and unity constitute the moments of number’ (SL 203/LS 214). In this respect, therefore, Hegel and Kant are by no means far apart.

4. 4) Like Hegel, Kant also draws a distinction between extensive and intensive magnitude. This distinction has both a logical and a temporal dimension for Kant. Logically, without regard to time, ‘the magnitude of something, insofar as it is represented as a plurality [Menge] is extensive, [and] as a unity is intensive’.Footnote ²⁷ With regard to time, an extensive magnitude requires the successive synthesis of its manifold parts (CPR A 163/B 204). An intensive magnitude, however, has no parts but is a simple unity; it thus acquires the multiplicity that makes it a specific degree from the fact that ‘its emergence and disappearance can be considered as composite (in time)’.Footnote ²⁸ Whether the degree is the 3^rd, 4^th or 5^th is determined, in other words, not by what lies within it, but by what precedes it (its ‘emergence’) and succeeds it (its ‘disappearance’). This means that, like Hegel, Kant understands a degree to have its distinguishing multiplicity outside itself. Yet a difference between Kant and Hegel remains: for, unlike Hegel, Kant does not derive the degree logically from extensive magnitude.

VII. The Relation of Powers

To conclude this essay, I now return to the logical development of quantity set out by Hegel in the Logic. We have seen quality give rise to quantity, and then seen quantity lead as far as intensive magnitude. Hegel finally goes on to show that quantity itself gives rise to quality. This is because it comes to exhibit the being-for-self or pure self-relation that characterises the highest form of quality, namely the one. This does not mean that quantity simply disappears: Hegel's claim is that quantity as quantity constitutes quality. How does this occur?

Quantity exhibits the form of quality, when one quantum, in relating to another quantum, actually relates to itself and so becomes being-for-self. This logical structure is realized, in Hegel's view, in the ‘ratio’ of powers (Potenzenverhältnis) (SL 321-2/LS 359). A brief comparison of this ratio with two others shows that quantity constitutes quality more specifically by freeing itself—at least in a certain respect—from the idea of an amount (Anzahl).

Each of the three ‘ratios’ Hegel discusses has a so-called ‘exponent’. In mathematics the exponent is the superscript number that indicates how many times a number is to be multiplied by itself: so 2³, 3⁴ and so on.Footnote ²⁹ For Hegel, however, an exponent is something different and is not just found in a ratio of powers. It is found in every kind of ratio and sets a limit to the changes that the quanta in the ratio can undergo. In the direct and inverse ratio the exponent is a fixed amount. So, in a direct ratio of 1:4, the exponent is the simple amount, 4. The number on the left side of the ratio can vary infinitely, but the exponent, 4, limits the changes that can be undergone by the right side. However much the left side increases, therefore, the right side must always increase so that it remains four times as big (see SL 315-17/LS 351-3).

In an inverse ratio, the exponent is once again a fixed amount, but this time it is not just a simple, given amount, but the product of the two quanta in the ratio. So, if one of the quanta has the value of 3 and the other 12, the exponent or product of the two is 36. It is this product, rather than either of its factors, that must remain constant to preserve the ratio. So if 3 increases to 4, 12 does not increase too (as in a direct ratio) but decreases to 9 so that the product remains 36. The two quanta in the ratio are thus not directly but inversely proportional to one another (see SL 317/LS 354).

Note, by the way, that in the first edition of the Logic (whose ‘doctrine of being’ was published in 1812) Hegel understands the exponent of the inverse ratio to be the sum, rather than the product, of the two quanta concerned.Footnote ³⁰ He then corrects his error in the second edition (published 1832). This is thus an interesting example of Hegel's coming to a deeper understanding of a logical structure between the two editions and revising his account accordingly.Footnote ³¹ Hegel takes his logic to be a priori and definitive, but this does not mean that he gets everything right from the start.

Like the direct and inverse ratios, the ratio of powers is also governed by an exponent. This exponent, however, is no longer a fixed amount or quantum, but is ‘of a wholly qualitative nature’ (SL 322/LS 359). More precisely, it consists in the quality of self-relation. In a ratio of powers, a number changes and becomes another number; in so doing, however, it must relate explicitly to itself. This occurs when the number it becomes is in fact its own product, that is, when the new number is just itself raised ‘to a power’. The exponent of the ratio of powers is thus not a specific amount, but requires a number to be multiplied by itself.

Note that in exhibiting the quality of self-relation, the number remains related to another number: 2 relates to 4, 3 to 9, and so on. In the relation of a number to its square, therefore, the quality of self-relation coincides completely with the quantitative relation between two different numbers. Indeed, the identity between quality and quantity is even deeper than this: for precisely by relating to itself in the power to which it has been raised, and so exhibiting quality, the quantum continues beyond itself and so shows itself to be quantitative (see SL 322/LS 359).

VIII. Conclusion

Hegel's account of quantity is complex and detailed, and it is easy to get lost in its complexity. The logical development of quantity, however, has a relatively simple overall structure. Quantity is first made necessary by quality in the form of being-for-self and the one (das Eins): it is the continuity of the one outside itself in other ones. In the ratio of powers, however, quantity returns to being purely for itself, and so to being qualitative, in being quantitative. As John Burbidge points out, therefore, there is a ‘double transition’ from quality to quantity and then from quantity back to quality—a transition that Hegel describes as being ‘of great importance throughout the whole compass of scientific method’ and which we encounter at several points in the Logic.Footnote ³²

Note, however, that the ratio of powers does not just return us to quality by itself (which would lead once again to quantity and so give rise to an endless cycle). It confronts us, rather, with the immediate identity of quantity and quality, of being-outside-oneself and being-for-self: a quantum, in becoming another quantum, thereby constitutes self-relating, qualitative being. This identity of quantity and quality implicitly contains a further category that opens a new sphere of being altogether.

A quantum in the ratio of powers becomes another quantum: 2 becomes 4, 3 becomes 9, and so on. It is also qualitative insofar as it relates to itself and its own product in becoming another. If, however, a quantum is to be fully and explicitly qualitative in being quantitative, it must become wholly self-relating or for itself. This in turn means that it must cease—for the moment, at least—being related to another quantum. It can thus no longer raise itself to a higher power, but must become what Hegel calls a measure (Maß). A measure is still a quantum; but it is one that relates solely to itself—and so stands alone with ‘its own determinacy’—and precisely in so doing constitutes the quality of a thing (SL 333/LS 371).Footnote ³³

A thing, as we know, is indifferent to its quantum or magnitude as such: it can alter its magnitude—and so become bigger or smaller—without ceasing to be what it is.Footnote ³⁴ A thing's measure, by contrast, is the magnitude to which it cannot be indifferent. This is because that measure is the magnitude (or range of magnitudes) that gives the thing its quality and so makes it what it is. The sphere of measure is thus a third sphere of being that is distinct from quality and quantity by being the fusion of both.

Discussion of measure, however, will have to wait for another occasion.Footnote ³⁵ To end, I will simply restate the main thesis of this essay: namely, that Hegel sees an intimate logical connection between quality and quantity that Kant fails to recognize. What grounds this difference between them, in my view, is this: Kant proceeds on the basis of what are for Hegel unwarranted assumptions, including the assumption that quality and quantity are logically (if not epistemically) separate categories. Hegel, by contrast, derives categories without presuppositions from the sheer ‘simplicity of thinking’.Footnote ³⁶ He is thus able to discover a logical connection between quality and quantity that Kant is unable to see. Note, by the way, that Hegel does not discern this connection because he is somehow predisposed to find it. He discerns it—though Kantians find this claim incredible—because he takes less for granted about thought (and being) than Kant does. This, in my view, is the true secret of Hegel.