PART ONE
Quantum, Number and the Quantitative Ratio
CHAPTER ONE
Quantum and Number
CONTINUOUS AND DISCRETE MAGNITUDE
According to speculative logic, quantity is the unity of continuity and discreteness, but, as we saw in volume 1, it is first of all âin the form of one of them, of continuityâ (SL 154-5 / LS 195).1 Quantity is initially simple, self-relating being or âself-identical immediacyâ; yet it also contains a multiplicity of discrete ones; it is thus self-relating being that continues through the many ones within it. Accordingly, quantity is initially â as the âimmediate unityâ of continuity and discreteness â continuous being above all (SL 165-6 / LS 209). As continuous, quantity cannot be âcomposedâ of units that are wholly discrete. It is, however, divisible into discrete units, and so, as Hegel puts it, is the âreal possibility of the oneâ (SL 155 / LS 196).
Yet quantity is not in fact just the unity of continuity and discreteness, but it also contains their difference. Indeed, the difference between them is built into their unity: for quantity unites continuity and discreteness. In truth, therefore, quantity is not merely the âimmediate unityâ, but what Hegel calls the âconcrete unityâ, of continuity and discreteness â âthe unity of distinct momentsâ (SL 165-6 / LS 209-10).
If, however, continuity and discreteness are to be explicitly distinct moments of quantity, the former cannot simply contain the latter (as is initially the case), but must also stand in contrast to it. That in turn means that quantity cannot just be continuous-in-being-discrete, but must also be continuous-rather-than-discrete. Such overtly one-sided quantity Hegel calls âcontinuous magnitudeâ (kontinuierliche GröĂe). Equally, discreteness cannot simply be a moment absorbed in the continuity of quantity. It, too, must stand in contrast to continuity and so constitute quantity governed by discreteness rather than continuity. Such quantity Hegel names âdiscrete magnitudeâ (diskrete GröĂe). It turns out, therefore, that quantity must generate two different âkindsâ or âspeciesâ (Arten) of itself (SL 167 / LS 211). Since each is a form of quantity, it unites both continuity and discreteness within itself; yet each differs from the other because a different moment of quantity is âpositedâ, and so to the fore, in it.2 As we shall see when we reach the idea of the âdegreeâ (Grad), not all quantity has to belong to one or the other of these two kinds.3 The difference between the two is nonetheless inherent in quantity itself.
In his account of determinate being (Dasein), Hegel argues that reality and negation are both forms of quality. Reality, however, is quality with an affirmative, rather than a negative, âaccentâ (SL 85 / LS 105). Continuous and discrete magnitude may also be said to be distinguished by their different âaccentsâ. This similarity between the two forms of quantity and of quality is not accidental but reflects a deeper parallel between them: for continuous and discrete magnitude are forms of quantity that is no longer simple immediacy or being (Sein), but explicitly determinate being (Dasein) â being that contains a determinate difference explicitly within itself.4
The difference between the two forms of quantity and those of quality should not, however, be overlooked. Reality and negation are qualitatively different from one another, so each is â explicitly at least â immediately itself. This is why each is âconcealedâ in the other: each is hidden in and by the immediacy of the other. Continuous and discrete magnitude, by contrast, are forms of quantity and so are not qualitatively different. There is, indeed, a difference between them, but each is âthe wholeâ (das Ganze) that contains both continuity and discreteness explicitly within itself. As Hegel puts it in the Encyclopaedia Logic, each is âthe same whole [ ⊠] posited first under one of its determinations, and then under the otherâ (EL 160 / 212-13 [§ 100 R]).
Continuous magnitude, therefore, is the âunity of the discreteâ, but it is nonetheless marked by continuity above all (SL 166 / LS 210). It is the kind of quantity that is exhibited by the continuous extension of space and continuous succession of time. Space and time can be divided into âheresâ and ânowsâ, since each contains the moment of discreteness or the âprinciple of the oneâ; yet, pace Zeno, each is still continuous (see 1: 365). Each is thus an instance of quantity that is âbeing-outside-one-another [AuĂereinandersein] continuing itself without negation as an internally self-same connectednessâ (SL 166-7 / LS 210-11).
Discrete magnitude, by contrast, is marked by the separateness of the unit or one. It is explicitly distinct from continuous magnitude and so lacks the overt continuity that characterizes the latter. It does not, therefore, extend smoothly and âwithout negationâ; on the contrary, âdiscrete magnitude is this outside-one-another [of the ones] as not continuous, as interruptedâ (SL 166 / LS 210, emphasis added). Such magnitude is thus discontinuous, broken up into discrete ones, and takes the form of an âaggregate [Menge] of onesâ. We encounter this kind of quantity, not in the continuous extension of space, but when we are prompted explicitly to ask how many units there are.5 Discrete magnitude is thus a different kind of quantity from continuous magnitude; one might even say that it is a different âqualityâ of quantity (provided one bears in mind that it does not differ qualitatively from its counterpart in the way reality differs from negation). Continuous magnitude is precisely continuous and uninterrupted and so forms an explicit unity of ones. Discrete magnitude, by contrast, is discontinuous and takes the form of an aggregate of separate ones.
Yet this aggregate is not just a plurality of atoms in a void, atoms that repel one another and are purely for themselves; that plurality belongs to the sphere of quality, not quantity. Discrete magnitude, as a kind of quantity, is an aggregate of ones that form a continuity in their discreteness: âbecause discrete magnitude is quantity, its discreteness is itself continuousâ (SL 166 / LS 210). This is due to the fact that the ones are explicitly âthe same as one anotherâ, the same discrete one. This sameness or âhomogeneityâ significantly qualifies their discreteness, for it means that they are not merely separate and discrete, but constitute continuous self-same being: the discrete ones together form a unity. Such unity is not projected by us onto the ones, but belongs to the ones themselves: it is the unity they form through the continuing of their discreteness. The unity still differs from that found in continuous magnitude, because it is constituted by explicitly discontinuous, discrete ones. Yet it means that the latter are not just isolated ones, but âthe many of a unityâ (das Viele einer Einheit): discrete magnitude is a continuous, unified aggregate, rather than a mere aggregate, of discrete ones.6 In that respect, even discrete magnitude, in Hegelâs view, is not simply âcomposedâ of wholly discrete units; the ones in such magnitude are discrete moments, rather than quite separate parts, of it.7
THE LIMITING OF QUANTITY: THE QUANTUM
At the start of 1.2.1.C Hegel again outlines the main features of discrete magnitude. First, he notes, such magnitude has âthe one for its principleâ: what distinguishes it from continuous magnitude is the explicit presence of the discrete, discontinuous one (SL 167 / LS 211). Second, it is, or contains, a âpluralityâ (Vielheit) of such ones. Third, however, it is âessentially constant [stetig]â, since each one is the same discrete one.8 In discrete magnitude, the ones are thus not just discontinuous, but also form a continuity and unity â âself-continuing as such in the discreteness of the onesâ. Discrete magnitude is continuous after all, thanks to the constancy of the discreteness within it.9
Hegel goes on to argue, however, that since the many ones it contains constitute a unity â one continuous being â discrete magnitude must be more than a distinct kind of quantity. It must also take the form of a single, unified quantity: âa magnitudeâ or âone magnitudeâ (eine GröĂe) (SL 167 / LS 211). The reason why this should be is as follows. The unity constituted by discrete magnitude is, as a unity, continuous; yet it is what discrete magnitude proves to be â it is an aspect of the latter â and so must differ from the unity found in continuous magnitude. It differs from the latter by being a unity that is itself discrete. Discrete magnitude does, indeed, form a unity; but, in contrast to continuous magnitude (which extends without interruption), it forms an explicitly discrete unity. As discrete, this unity must be something (Etwas) with an identity of its own, indeed something that is a discrete one (Eins); âit is [thus] posited as one magnitude, and the determinacy of it is the oneâ (SL 167 / LS 211-12).10
Hegel adds that this discrete unity must be an âexclusive oneâ â a one that shuts out what it is not. It is not immediately clear what justifies this additional claim, since the one in the sphere of quantity is supposed to have lost its negative, exclusive âedgeâ. The justification becomes apparent, however, in the remainder of the paragraph: the unity fo...