Quantity and Measure in Hegel's 'Science of Logic'

Hegel on Being

Stephen Houlgate

• Publisher: Bloomsbury Academic
• Year: 2021
• ISBN:
  9781350189874
• Edition: 1
• Topic: Filosofia
• Subtopic: Logica in filosofia

Hegel on Being provides an authoritative treatment of Hegel's entire logic of being. Stephen Houlgate presents the Science of Logic as an important and neglected text within Hegel's oeuvre that should hold a more significant place in the history of philosophy. In the Science of Logic, Hegel set forth a distinctive conception of the most fundamental forms of being through ideas on quality, quantity and measure. Exploring the full trajectory of Hegel's logic of being from quality to measure, this two-volume work by a preeminent Hegel scholar situates Hegel's text in relation to the work of Plato, Aristotle, Descartes, Spinoza, Kant, and Frege. Volume II: Quantity and Measure in Hegel's 'Science of Logic' continues the discussion of Hegel's logic of being and considers all aspects of quantity and measure in his logic, including his basic categories of being, writings on calculus, philosophy of mathematics, as well as a comparative study of Hegel and Frege's approach to logic.

PART ONE

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Quantum, Number and the Quantitative Ratio

CHAPTER ONE

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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 “in the form of one of them, of continuity” (SL 154-5 / LS 195).¹ Quantity is initially simple, self-relating being or “self-identical immediacy”; 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 – as the “immediate unity” of continuity and discreteness – continuous being above all (SL 165-6 / LS 209). As continuous, quantity cannot be “composed” of units that are wholly discrete. It is, however, divisible into discrete units, and so, as Hegel puts it, is the “real possibility of the one” (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 “immediate unity”, but what Hegel calls the “concrete unity”, of continuity and discreteness – “the unity of distinct moments” (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 “continuous magnitude” (kontinuierliche Größe). 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 “discrete magnitude” (diskrete Größe). It turns out, therefore, that quantity must generate two different “kinds” or “species” (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 “posited”, and so to the fore, in it.² As we shall see when we reach the idea of the “degree” (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, “accent” (SL 85 / LS 105). Continuous and discrete magnitude may also be said to be distinguished by their different “accents”. 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) – 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 – explicitly at least – immediately itself. This is why each is “concealed” 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 “the whole” (das Ganze) that contains both continuity and discreteness explicitly within itself. As Hegel puts it in the Encyclopaedia Logic, each is “the same whole [ … ] posited first under one of its determinations, and then under the other” (EL 160 / 212-13 [§ 100 R]).

Continuous magnitude, therefore, is the “unity of the discrete”, 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 “heres” and “nows”, since each contains the moment of discreteness or the “principle of the one”; yet, pace Zeno, each is still continuous (see 1: 365). Each is thus an instance of quantity that is “being-outside-one-another [Außereinandersein] continuing itself without negation as an internally self-same connectedness” (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 “without negation”; on the contrary, “discrete magnitude is this outside-one-another [of the ones] as not continuous, as interrupted” (SL 166 / LS 210, emphasis added). Such magnitude is thus discontinuous, broken up into discrete ones, and takes the form of an “aggregate [Menge] of ones”. 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 “quality” 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: “because discrete magnitude is quantity, its discreteness is itself continuous” (SL 166 / LS 210). This is due to the fact that the ones are explicitly “the same as one another”, the same discrete one. This sameness or “homogeneity” 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 “the many of a unity” (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 Hegel’s view, is not simply “composed” 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 “the one for its principle”: 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 “plurality” (Vielheit) of such ones. Third, however, it is “essentially constant [stetig]”, 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 – “self-continuing as such in the discreteness of the ones”. 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 – one continuous being – discrete magnitude must be more than a distinct kind of quantity. It must also take the form of a single, unified quantity: “a magnitude” or “one magnitude” (eine Größe) (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 – it is an aspect of the latter – 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); “it is [thus] posited as one magnitude, and the determinacy of it is the one” (SL 167 / LS 211-12).¹⁰

Hegel adds that this discrete unity must be an “exclusive one” – 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 “edge”. The justification becomes apparent, however, in the remainder of the paragraph: the unity fo...