Hegel on being. Volune 2, Quantity and measure in Hegel's 'Science of Logic'

449 Pages·2021·10.58 MB·english

by  Stephen Houlgate

HEGEL ON BEING
VOLUME 2
i
ii
HEGEL ON BEING

QUANTITY AND
MEASURE IN
HEGEL’S SCIENCE
OF L OGIC
VOLUME 2
Stephen Houlgate
iii
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iv
CONTENTS
P REFACE TO V OLUME 2 ix
NO TE ON R EFERENCES AND A BBREVIATIONS xi
LI ST OF AB BREVIATIONS xiii
Part One: Quantum, Number and the Quantitative Ratio
1 Quantum and Number 3
Continuous and discrete magnitude 3
The limiting of quantity: the quantum 5
Number 10
Amount and unity in number 14
The indifference of numbers 17
Hegel and the Greeks on number 20
2 Excursus: Hegel on the Operations of Arithmetic 25
Numbering and addition 25
Kant and Hegel on “7 + 5 = 12” 27
Multiplication and “raising to a power” 30
3 Excursus: Hegel and Frege – Similarities and Differences 33
Frege’s project 36
Similarities and differences between Frege and Hegel 42
Logicism 42
The objectivity of logic 43
v
vi CONTENTS
Logic and psychology 45
Proof 49
Logic and truth 53
Logic as “analytic” 54
4 Excursus: Frege’s Assumptions 59
Formal and speculative logic 59
Logic and arithmetic 63
“Function” and “argument” in Frege’s thought 68
Concept and object 72
The universal quantifi er 74
The concept . . . 80
The extension of a concept 84
Frege’s context principle 87
5 Excursus: Hegel and Frege on Number 91
Frege’s critique of the traditional (Greek) conception of number 91
Hegel on “something” and the “one” 97
Frege on numbers, objects and concepts 102
Frege’s defi nition of number 112
The criterion of identity for numbers 112
Frege’s direct defi nition of number 120
Frege’s defi nitions of 0 and 1 132
6 Extensive and Intensive Magnitude 139
Extensive magnitude 139
From extensive to intensive magnitude 140
Intensive magnitude 145
The identity of extensive and intensive magnitude 148
The changing quantum 152
Conclusion 157
7 Quantitative Infi nity 159
The quantitative infi nite progress 160
The true infi nity of the quantum 163
The truly infi nite quantum and the direct ratio 166
8 Excursus: Hegel and Kant’s First Antinomy 171
Kant’s fi rst antinomy: the thesis, its proof and Hegel’s critique 171
Kant’s fi rst antinomy: the antithesis, its proof and Hegel’s critique 175
The resolution of the fi rst antinomy: quantity, space and time 176
CONTENTS vii
9 The Quantitative Relation or Ratio (V erh ä ltnis ) 181
The direct ratio 181
The transition to the inverse ratio 184
The inverse ratio 187
Further complexity in the inverse ratio 190
The transition to the ratio of powers 197
The ratio of powers 202
The transition to measure 206
10 Excursus: Hegel on Differential Calculus 209
The philosophical justifi cation of calculus 209
Problems with the common conception of calculus 211
The true quantitative infi nite and the mathematical infi nite 214
, fractions and series 219
as the relation between the “elements” of quantities 222
Newton and Euler 225
as a derived function or “derivative” (of a power-function) 228
Finding derivatives by expanding (or “developing”) binomials 231
Derived functions as “coeffi cients” 234
as a relation with no “real meaning” 236
Part Two: Measure
11 Specifi c Quantity 245
Measure as a specifi c quantum 246
The specifying measure and specifi c heat 250
The realized measure: Galileo and Kepler 257
Quality and quantity in the realized measure 263
Being-for-self in measure: the “empirical coeffi cient” 267
The transition to real measure 272
12 Real Measure 277
Real measure as the relation between two ratios: specifi c gravity 278
Real measure as the combination of two measures 283
Real measure as a series of measures 287
Measures as exclusive unities 297
Elective affi nities 300
The “more or less” 302
Elective affi nity and “intensity” 307
Complications: Hegel and Berthollet 310
viii CONTENTS
13 The Nodal Line and the Measureless 315
The nodal line of measure-relations 317
Jumps in nature (and examples) 323
The measureless 328
14 Indifference 337
Absolute indifference 337
Indifference as the inverse ratio of its factors 340
Indifference and its Dasein 346
The contradiction in indifference 349
Centripetal and centrifugal force 353
Hegel and Spinoza 356
15 From Measure to Essence 359
Indifference and essence 359
Essence as “non-immediacy” 362
Essence and being 364
16 Conclusion 369
NO TES 373
B IBLIOGRAPHY 415
I NDEX 421
PREFACE TO VOLUME 2
In volume 1 of H egel on Being I fi rst explained the purpose and method of
Hegel’s logic. Its distinctive feature, in my view, is that it is (or is intended to
be) the free, presuppositionless derivation of categories that belong to both
thought and being. It is thus both a logic and an ontology at the same time.
Since this logic is to have no systematic presuppositions, it must begin with a
category that is utterly indeterminate, namely pure being. It then proceeds by
considering what further categories, if any, are to be derived from this
indeterminate starting point. In Part Two of volume 1 we saw that a whole
series of categories are derived from being, including s omething , fi nitude ,
infi nity and the o ne (E ins ). These categories of “quality” constitute, for Hegel,
the most basic categories of thought, but also the most basic ways of being. We
usually just take it for granted that there are “things” in the world and that they
will at some point cease to be. Hegel, however, shows why there must be
“things”, and why they must cease to be, by showing that being itself makes it
necessary logically that there be fi nite things.
At the end of his account of quality, Hegel argues that being also makes
quantity necessary. It does so via the quality of being o ne . The one, Hegel
claims, proves to be one among many but thereby to continue beyond itself,
and in so doing it gives rise to being that consists specifi cally in the continuity
of discrete units. This being is quantity. For Hegel, therefore, quantity is not
just a contingent feature of things, but they must have quantity because – besides
being fi nite – they are ones , and being one logically entails being many-in-being-
one and thus being quantitative.
Note that with the logical transition from quality to quantity, what it is to be
qualitative does not itself change. It still comprises all the categories considered
in volume 1. Quality gives rise to quantity, however, by proving no longer just
ix