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Science of Logic

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The English translation is by A. V. Miller in 1969. More recently George di Giovanni has published a new translation, Cambridge University Press, 2010.

Here the objective logic is not quite logic in the usual sense, but rather something like logos in the old sense of Heraclitus (e.g. (Heidegger 58, Lambek 82)) or the Nous in the sense of Anaxagoras (§54, PoS pref. §55) or metaphysics (§85). Something closer to the usual notion of logic is what Hegel calls the “subjective logic”, though of course he also talks about both being the same.

Note that Hegel included an abbreviated version of The Science of Logic as the first part of The Encyclopedia of the Philosophical Sciences, followed there by The Philosophy of Nature and The Philosophy of Mind. This first part is often referred to as the Shorter Logic.

With Hegel you’re getting what seems to us today a curious package. The whole dynamic (or ‘dialectic’) of the unfolding of the Logik is prior to any actual thinking, realised in concrete humans. In fact, the world, and that part of it which is human thought, is the Idea (or Spirit) realising itself. I say ‘curious’, but in a way I’m hearing echoes of this in things Urs is suggesting, as though the universe worked itself out according to type theory. For Hegel one isn’t to be a Kantian, where what one theorises about the universe is just how the universe is taken up by the human understanding, with the further idea that this will be limited in certain ways, e.g., no access to the thing-in-itself. For Hegel, the human mind itself is part of the universe and as such part of the unfolding of the Idea/Spirit. [][\cdots] It’s a dizzying picture which tends either to delight or revolt people. [[ See at absolute idealism ]] (Corfield)

Contents

Topics of Volume One: The objective logic

The following extracts some paragraphs from the text, with comments on how to possibly think about this in terms of homotopy type theory, along the lines that William Lawvere has been suggesting over the decades.

The paragraph numbers refer to the numbers as given in the English translation at Hegel-by-hypertext Hegel’s science of logic.

Before we get to the content, here some remarks.

Heraclitus and the Logos of Becoming

Hegel wrote (according to this):

Heraclitus is the one who first declared the nature of the infinite and first grasped nature as in itself infinite, that is, its essence as process. The origin of philosophy is to be dated from Heraclitus. His is the persistent Idea that is the same in all philosophers up to the present day, as it was the Idea of Plato and Aristotle.”

“… there is no proposition of Heraclitus which I have not adopted in my logic.”

Triads of Opposites and their Unity

Hegel famously invokes opposite to the extent of paradoxes, as sources for new phenomena via their synthesis. The whole text is structured by triads of chapters each with triads of sections, etc. see Inwood 83, p. 263 for a diagram.

Formalization of Unity of Opposites

On p. 11 of Cohesive toposes and Cantor’s Lauter Einsen William Lawvere proposes that these triads of unity of opposites are captured by adjoint pairs of idempotent monads/comonads (“adjoint cylinders”), such as

\flat \dashv \sharp

flat modality \dashv sharp modality

This gives unifying triples of the form

X X X opposite1 unity opposite2. \array{ \flat X &\longrightarrow& X &\longrightarrow& \sharp X \\ \\ opposite\;1 && unity && opposite\;2 } \,.

for any type XX.

Several examples of this appear below.

Notice that indeed a fair bit of structure follows from maps of this form.

For instance for the points-to-pieces transform induced by the shape modality \dashv flat modality dichotomy \int \dashv \flat, we have, as discussed at tangent cohesion – Cohesive and differential refinement

dRΩA dRΣA θ A dRΩA A dRΣA θ A A A, \array{ && \int_{dR} \Omega A && \longrightarrow && \flat_{dR}\Sigma A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \flat \int_{dR} \Omega A && && A && && \int \flat_{dR}\Sigma A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{\int \theta_A}} \\ && \flat A && \longrightarrow && \int A } \,,

Formalization dictionary

Hegel’s “Science of Logic” may seem rather mysterious. Over the decades, William Lawvere had suggested, more or less explicitly, that parts of it are usefully understood as being about – or conversely as being formalized and hence interpreted by – aspects of categorical logic. For instance Lawvere suggested that the recurring notion of “unity and identity of opposites” is usefully thought of in terms of certain adjunctions, as discussed in Formalization of Unity of Opposites.

In view of this one may notice that modern foundations of constructive mathematics via type theory and in particular via homotopy type theory may offer more opportunities like this to give Hegel’s intuitions a formalized home or incarnation in a useful way.

The following table lists proposals for possible such identifications. The content below means to provide for each keyword commented passages in Science of Logic to support this identification and illuminate it. But of course this remains just a proposal and subject at least to debate.

Hegel’s logicmodal homotopy type theory
momentmodality
unity of oppositesadjoint modality
spherelevel
AufhebungAufhebung
groundantecedent
entering into existenceterm introduction
immediacy of reflectionreflector term in identity type
all things are differentintensional identity
being, One(context of) unit type
nothingempty type
becomingadjoint modality *\emptyset \dashv \ast
DaseinAufhebung of becoming by sharp modality \sharp
moment of repulsionflat modality \flat
moment of attractioncohesion, shape modality \int
qualityadjoint modality attraction \dashv repulsion = \int \dashv \flat
moment of discretenessflat modality \flat
moment of continuitysharp modality \sharp
quantityadjoint modality \flat \dashv \sharp
vanishing of infinitesimalsreduction modality
being-for-selfreduction modality \Re
being-for-oneinfinitesimal shape modality inf\int_{inf}
idealityadjoint modality inf\Re \dashv \int_{inf}
realityadjoint modality inf inf\int_{inf} \dashv \flat_{inf}
moment of two negationsdouble negation modality ¬¬\not \not, more generally: bracket type/(-1)-truncation modality
something(-1)-truncation modality, classically double negation modality
measure (= gauge)quality \dashv quantity

Notice that the above involves the first two stages in the tower of n-truncation modalities:

nnn-truncation modality
-2unit type modality
-1(-1)-truncation modality, classically double negation modality

On the translation of the terms

The orginal German is at times maybe more evocative than the established English translations

  • “für sich sein”, which standard sources translate as “being-for-self”, really means to be alone and undisturbed. One says: “Ich gehe jetzt in mein Büro, ich muss mal für mich sein um mich zu konzentrieren.” (I’ll retreat to my office to be alone and undisturbed.)

  • “für eins sein” which Hegel uses, is not proper German and probably wasn’t even at his time, but it is clearly meant to rhyme with “für sich sein”, and the similar phrase that does exist is “für einander sein”, which means: to be available for others.

  • “Dasein”, which in the standard translations appears as “determinate being” is really much more immediate “being there”, “existing”. “Wann sollen wir da sein?” means “When are we supposed to be there?” In more formal speech “das Dasein” means “existence” as in “Ach, das Dasein ist doch zwecklos.” (Note that di Giovanni has ‘existence’ here.)

(…)

Introduction

Allgemeiner Begriff der Logik

  • §53 Die Logik ist sonach als das System der reinen Vernunft, als das Reich des reinen Gedankens zu fassen. Dieses Reich ist die Wahrheit, wie sie ohne Hülle an und für sich selbst ist. Man kann sich deswegen ausdrücken, daß dieser Inhalt die Darstellung Gottes ist, wie er in seinem ewigen Wesen vor der Erschaffung der Natur und des endlichen Geistes ist.

  • §53 Accordingly, logic is to be understood as the system of pure reason, as the realm of pure thought. This realm is truth as it is without veil and in its own absolute nature. It can therefore be said that this content is the exposition of God as he is in his eternal essence before the creation of nature and a finite mind.

  • §54 Anaxagoras wird als derjenige gepriesen, der zuerst den Gedanken ausgesprochen habe, daß der Nus, der Gedanke, das Princip der Welt, daß das Wesen der Welt als der Gedanke bestimmt ist. Er hat damit den Grund zu einer Intellektualansicht des Universums gelegt, deren reine Gestalt die Logik seyn muß. Es ist in ihr nicht um ein Denken über etwas, das für sich außer dem Denken zu Grunde läge, zu thun, um Formen, welche bloße Merkmale der Wahrheit abgeben sollten; sondern die nothwendigen Formen und eigenen Bestimmungen des Denkens sind der Inhalt und die höchste Wahrheit selbst.

  • §54 Anaxagoras is praised as the man who first declared that Nous, thought, is the principle of the world, that the essence of the world is to be defined as thought. In so doing he laid the foundation for an intellectual view of the universe, the pure form of which must be logic. What we are dealing with in logic is not a thinking about something which exists independently as a base for our thinking and apart from it, nor forms which are supposed to provide mere signs or distinguishing marks of truth; on the contrary, the necessary forms and self-consciousness of thought are the content and the ultimate truth itself.

Allgemeine Einteilung der Logik

  • §85 Die objektive Logik tritt damit vielmehr an die Stelle der vormaligen Metaphysik, als welche das wissenschaftliche Gebäude über die Welt war, das nur durch Gedanken aufgeführt seyn sollte.—Wenn wir auf die letzte Gestalt der Ausbildung dieser Wissenschaft Rücksicht nehmen, so ist erstens unmittelbar die Ontologie, an deren Stelle die objektive Logik tritt,—der Theil jener Metaphysik, der die Natur des Ens überhaupt erforschen sollte;—das Ens begreift sowohl Seyn als Wesen in sich, für welchen Unterschied unsere Sprache glücklicherweise den verschiedenen Ausdruck gerettet hat.—Alsdann aber begreift die objektive Logik auch die übrige Metaphysik insofern in sich, als diese mit den reinen Denkformen die besondern, zunächst aus der Vorstellung genommenen Substrate, die Seele, die Welt, Gott, zu fassen suchte, und die Bestimmungen des Denkens das Wesentliche der Betrachtungsweise ausmachten.

  • §85 The objective logic, then, takes the place rather of the former metaphysics which was intended to be the scientific construction of the world in terms of thoughts alone. If we have regard to the final shape of this science, then it is first and immediately ontology whose place is taken by objective logic — that part of this metaphysics which was supposed to investigate the nature of ens in general; ens comprises both being and essence, a distinction for which the German language has fortunately preserved different terms. But further, objective logic also comprises the rest of metaphysics in so far as this attempted to comprehend with the forms of pure thought particular substrata taken primarily from figurate conception, namely the soul, the world and God; and the determinations of thought constituted what was essential in the mode of consideration.

Book one Die Lehre vom Sein / The Doctrine of Being

\;\; Womit muss der Anfang der Wissenschaft gemacht werden?

  • §121 Was somit über das Seyn ausgesprochen oder enthalten seyn soll, in den reicheren Formen des Vorstellens von Absolutem oder Gott, dieß ist im Anfange nur leeres Wort, und nur Seyn; dieß Einfache, das sonst keine weitere Bedeutung hat, dieß Leere ist also schlechthin der Anfang der Philosophie.

  • §121 Consequently, whatever is intended to be expressed or implied beyond being, in the richer forms of representing the absolute or God, this is in the beginning only an empty word and only being; this simple determination which has no other meaning of any kind, this emptiness, is therefore simply as such the beginning of philosophy.

  • §122 Diese Einsicht ist selbst so einfach, daß dieser Anfang als solcher, keiner Vorbereitung noch weiteren Einleitung bedarf; und diese Vorläufigkeit von Raisonnement über ihn konnte nicht die Absicht haben, ihn herbeizuführen, als vielmehr alle Vorläufigkeit zu entfernen.

  • §122 This insight is itself so simple that this beginning as such requires no preparation or further introduction; and, indeed, these preliminary, external reflections about it were not so much intended to lead up to it as rather to eliminate all preliminaries.

\;\; Allgemeine Einteilung des Seins

First section: Determinateness (Quality)

First chapter

From The Shorter Logic: * §86 Pure being constitutes the beginning, because it is pure thought as well as the undetermined, simple immediate, and the first beginning cannot be anything mediated and further determined.

  • §87 Now this pure being is a pure abstraction and thus the absolutely negative which, when likewise taken immediately, is nothing.

  • §88 Conversely, nothing, as this immediate, self-same category, is likewise the same as being. The truth of being as well as of nothing is therefore the unity of both; this unity is becoming.

A. Sein / Being
  • §132 Being, pure being, [] it has no diversity within itself nor any with a reference outwards.

This is the unit type *\ast.

Indeed, later this is called “Das Eins” which is maybe indeed better translated as “The Unit” instead of as “The One” as commonly done.

B. Nichts / Nothing
  • §133 Nothing, pure nothing: it is simply equality with itself, complete emptiness,

The empty type \emptyset.

C. Werden Becoming
  • §134 Pure Being and pure nothing are, therefore, the same. What is the truth is neither being nor nothing, but that being — does not pass over but has passed over — into nothing, and nothing into being. But it is equally true that they are not undistinguished from each other, that, on the contrary, they are not the same, that they are absolutely distinct, and yet that they are unseparated and inseparable and that each immediately vanishes in its opposite. Their truth is therefore, this movement of the immediate vanishing of the one into the other: becoming, a movement in which both are distinguished, but by a difference which has equally immediately resolved itself.

According to the formalization of such unity of opposites as above we identify this becoming (following Lawvere 91) as the universal factorization

X * nothing becoming being \array{ \emptyset &\longrightarrow& X &\longrightarrow& \ast \\ \\ nothing && becoming && being }

of the factorization of the unique function from the empty type to the unit type through any other type XX.

Indeed, later in §174 it says:

  • there is nothing which is not an intermediate state between being and nothing.

Also, below it says

  • §222 Being and nothing in their unity, which is determinate being
1. Unity of Being and Nothing
\;\; Remark 1 The opposition of being and nothing in ordinary thinking
\;\; Remark 2: Defectiveness of the Expression “Unity, Identity of Being and Nothing”
  • §152 But the third in which being and nothing subsist must also present itself here, and it has done so; it is becoming. In this being and nothing are distinct moments; becoming only is, in so far as they are distinguished.

In view of the above it seems that “moment” is well translated with modality.

\;\; Remark 3 The isolating of these abstractions
\;\; Remark 4 Incomprehensibility of the beginning
  • §171 It is impossible for anything to begin, either in so far as it is, or in so far as it is not; for in so far as it is, it is not just beginning, and in so far as it is not, then also it does not begin. If the world, or anything, is supposed to have begun, then it must have begun in nothing, but in nothing — or nothing — is no beginning; for a beginning includes within itself a being, but nothing does not contain any being. Nothing is only nothing. In a ground, a cause, and so on, if nothing is so determined, there is contained an affirmation, a being. For the same reason, too, something cannot cease to be; for then being would have to contain nothing, but being is only being, not the contrary of itself.

  • §174 Das Angeführte ist auch dieselbe Dialektik, die der Verstand gegen den Begriff braucht, den die höhere Analysis von den unendlich-kleinen Größen giebt. Von diesem Begriffe wird weiter unten ausführlicher gehandelt.—Diese Größen sind als solche, bestimmt worden, die in ihrem Verschwinden sind, nicht vor ihrem Verschwinden, denn als dann sind sie endliche Größen;—nicht nach ihrem Verschwinden, denn alsdann sind sie nichts. Gegen diesen reinen Begriff ist eingewendet und immer wiederholt worden, daß solche Größen entweder Etwas seyen, oder Nichts; daß es keinen Mittelzustand (Zustand ist hier ein unpassender, barbarischer Ausdruck) zwischen Seyn und Nichtseyn gebe.—Es ist hierbei gleichfalls die absolute Trennung des Seyns und Nichts angenommen. Dagegen ist aber gezeigt worden, daß Seyn und Nichts in der That dasselbe sind, oder um in jener Sprache zu sprechen, daß es gar nichts giebt, das nicht ein Mittelzustand zwischen Seyn und Nichts ist. Die Mathematik hat ihre glänzendsten Erfolge der Annahme jener Bestimmung, welcher der Verstand widerspricht, zu danken.

  • §174 The foregoing dialectic is the same, too, as that which understanding employs the notion of infinitesimal magnitudes, given by higher analysis. A more detailed treatment of this notion will be given later. These magnitudes have been defined as such that they are in their vanishing, not before their vanishing, for then they are finite magnitudes, or after their vanishing, for then they are nothing.

Vanishing of infinitesimal objects is expressed by the reduction modality \Re.

  • §174 there is nothing which is not an intermediate state between being and nothing.

The universal factorization for unity of opposites of the empty type \dashv unit type adjoint modality

X * nothing becoming being \array{ \emptyset &\longrightarrow& X &\longrightarrow& \ast \\ \\ nothing && becoming && being }

of the factorization of the unique function from the empty type to the unit type through any other type XX.

2. Momente des Werdens / Moments of Becoming
  • §176 Das Werden, Entstehen und Vergehen, ist die Ungetrenntheit des Seyns und Nichts; nicht die Einheit, welche vom Seyn und Nichts abstrahirt; sondern als Einheit des Seyns und Nichts ist es diese bestimmte Einheit, oder in welcher sowohl Seyn als Nichts ist. Aber indem Seyn und Nichts, jedes ungetrennt von seinem Anderen ist, ist es nicht. Sie sind also in dieser Einheit, aber als verschwindende, nur als Aufgehobene. Sie sinken von ihrer zunächst vorgestellten Selbstständigkeit zu Momenten herab, noch unterschiedenen, aber zugleich aufgehobenen.

  • §176 Becoming is the unseparatedness of being and nothing, not the unity which abstracts from being and nothing; but as the unity of being and nothing it is this determinate unity in which there is both being and nothing. But in so far as being and nothing, each unseparated from its other, is, each is not. They are therefore in this unity but only as vanishing, sublated moments. They sink from their initially imagined self-subsistence to the status of moments, which are still distinct but at the same time are sublated.

  • §177 Nach dieser ihrer Unterschiedenheit sie aufgefaßt, ist jedes in derselben als Einheit mit dem Anderen. Das Werden enthält also Seyn und Nichts als zwei solche Einheiten, deren jede selbst Einheit des Seyns und Nichts ist; die eine das Seyn als unmittelbar und als Beziehung auf das Nichts; die andere das Nichts als unmittelbar und als Beziehung auf das Seyn; die Bestimmungen sind in ungleichem Werthe in diesen Einheiten.

  • §177 Grasped as thus distinguished, each moment is in this distinguishedness as a unity with the other. Becoming therefore contains being and nothing as two such unities, each of which is itself a unity of being and nothing; the one is being as immediate and as relation to nothing, and the other is nothing as immediate and as relation to being; the determinations are of unequal values in these unities.

An archetypical description of the unity of opposites. Here:

becoming/Werden : nothing \dashv being

\;\;\; empty type \dashv unit type

\;\;\; *\emptyset \dashv \ast

This is also the interpretation in (LawvereComo, p. 11).

X* \emptyset \longrightarrow X \longrightarrow \ast
  • §178 Das Werden ist auf diese Weise in gedoppelter Bestimmung; in der einen ist das Nichts als unmittelbar, d. i. sie ist anfangend vom Nichts, das sich auf das Seyn bezieht, das heißt, in dasselbe übergeht, in der anderen ist das Seyn als unmittelbar d. i. sie ist anfangend vom Seyn, das in das Nichts übergeht,—Entstehen und Vergehen.

  • §178 Becoming is in this way in a double determination. In one of them, nothing is immediate, that is, the determination starts from nothing which relates itself to being, or in other words changes into it; in the other, being is immediate, that is, the determination starts from being which changes into nothing: the former is coming-to-be and the latter is ceasing-to-be.

\;\; nothing \dashv being :\;\colon\; ceasing

3. Sublating of Becoming
  • §180 The resultant equilibrium of coming-to-be and ceasing-to-be is in the first place becoming itself. But this equally settles into a stable unity. Being and nothing are in this unity only as vanishing moments; yet becoming as such is only through their distinguishedness. Their vanishing, therefore, is the vanishing of becoming or the vanishing of the vanishing itself. Becoming is an unstable unrest which settles into a stable result.

  • §181 This could also be expressed thus: becoming is the vanishing of being in nothing and of nothing in being and the vanishing of being and nothing generally; but at the same time it rests on the distinction between them. It is therefore inherently self-contradictory, because the determinations it unites within itself are opposed to each other; but such a union destroys itself.

  • §182 This result is the vanishedness of becoming, but it is not nothing; as such it would only be a relapse into one of the already sublated determinations, not the resultant of nothing and being. It is the unity of being and nothing which has settled into a stable oneness. But this stable oneness is being, yet no longer as a determination on its own but as a determination of the whole.

  • §183 Becoming, as this transition into the unity of being and nothing, a unity which is in the form of being or has the form of the onesided immediate unity of these moments, is determinate being.

Dasein
Werden :Nichts\;\;\;\dashvSein: Vergehen

\,

Dasein
becoming :nothing\;\;\;\dashvbeing: ceasing
  • §187 The more precise meaning and expression which being and nothing receive, now that they are moments, is to be ascertained from the consideration of determinate being as the unity in which they are preserved. Being is being, and nothing is nothing, only in their contradistinction from each other; but in their truth, in their unity, they have vanished as these determinations and are now something else. Being and nothing are the same; but just because they are the same they are no longer being and nothing, but now have a different significance. In becoming they were coming-to-be and ceasing-to-be; in determinate being, a differently determined unity, they are again differently determined moments. This unity now remains their base from which they do not again emerge in the abstract significance of being and nothing.

moment \leftrightarrow modality

Notice that all this has a striking resemblance to the following lines from the Tao Te Ching (English translation following Xiao-Gang Wen here):

The nameless nonbeing is the origin of universe;

The named being is the mother of all observed things.

Within nonbeing, we enjoy the mystery of the universe.

Among being, we observe the richness of the world.

Nonbeing and being are two aspects of the same mystery.

From nonbeing to being and from being to nonbeing is the gateway to all understanding.

Second chapter. Dasein / Determinate Being

A. Dasein as such / Determinate being as such
  • §188 Daseyn ist bestimmtes Seyn; seine Bestimmtheit ist seyende Bestimmtheit, Qualität.

  • §188 In considering determinate being the emphasis falls on its determinate character; the determinateness is in the form of being, and as such it is quality. Through its quality, something is determined as opposed to an other, as alterable and finite; and as negatively determined not only against an other but also in its own self. This its negation as at first opposed to the finite something is the infinite; the abstract opposition in which these determinations appear resolves itself into the infinity which is free from the opposition, into being-for-self.

The first sentence here is made up by the translator, in the original it says:

  • Daseyn ist bestimmtes Seyn;

Di Giovanni has

  • Existence is determinate being;

In any case, by the discussion at Becoming we have that “being” is a moment of the adjunction (*)(\emptyset \dashv \ast) and the discussion at Relation between repulsion and attraction we have that “quality” is the adjunction ()(\int \dashv \flat). Therefore it seems that

  • types have “being” in the presence of (*)(\emptyset \dashv \ast)

    • types moreover have “existence”/Dasein in the further presence of ()(\int \dashv \flat).

For more on this see at Remark on reality as opposite to ideality.

a. Dasein überhaupt / Determinant being in general
  • § 191 From becoming there issues determinate being, which is the simple oneness of being and nothing. Because of this oneness it has the form of immediacy. Its mediation, becoming, lies behind it; it has sublated itself and determinate being appears

Above we saw that becoming is formalized by the universal unity of opposites of nothing \dashv being, i.e. *\emptyset \dashv \ast, exhibiting any type XX as intermediate (via \emptyset-unit and *\ast-counit of a comonad)

X*. \emptyset \longrightarrow X \longrightarrow \ast \,.

Now by § 191 determinate being is the sublation of this unity of opposites. By the discussion at Aufhebung – Examples – Aufhebung of Becoming this is given by the level of the flat modality \dashv sharp modality-opposition ()(\flat \dashv \sharp), Dasein:

Dasein *. \array{ \flat \; &\dashv& \;\;\sharp \\ \vee \; &\nearrow_{\mathrlap{Dasein}}& \;\;\vee \\ \emptyset \; &\dashv& \;\;\ast } \,.
  • § 194 Determinate being corresponds to being in the previous sphere

Here “sphere” is level.

So \sharp is the version of *\ast (being) in the next level, which indeed it is by the above.

b. Qualität / Quality
  • §196 Determinateness thus isolated by itself in the form of being is quality
c. Etwas / Something
  • §208 In determinate being its determinateness has been distinguished as quality; in quality as determinately present, there is distinction — of reality and negation. Now although these distinctions are present in determinate being, they are no less equally void and sublated. Reality itself contains negation, is determinate being, not indeterminate, abstract being. Similarly, negation is determinate being, not the supposedly abstract nothing but posited here as it is in itself, as affirmatively present [als seiend], belonging to the sphere of determinate being.

    Thus quality is completely unseparated from determinate being, which is simply determinate, qualitative being.

Dasein, quality, type, something

  • §209 This sublating of the distinction is more than a mere taking back and external omission of it again, or than a simple return to the simple beginning, to determinate being as such. The distinction cannot be omitted, for it is. What is, therefore, in fact present is determinate being in general, distinction in it, and sublation of this distinction; determinate being, not as devoid of distinction as at first, but as again equal to itself through sublation of the distinction, the simple oneness of determinate being resulting from this sublation. This sublatedness of the distinction is determinate being’s own determinateness; it is thus being-within-self: determinate being is a determinate being, a something.

  • §209 Dieß Aufgehobenseyn des Unterschieds ist die eigne Bestimmtheit des Daseyns; so ist es Insichseyn; das Daseyn ist Daseyendes, Etwas.

  • §210 Something is the first negation of negation, as simple self-relation in the form of being.

  • §211 Something is the negation of the negation in the form of being;

  • §212 This mediation with itself which something is in itself, taken only as negation of the negation, has no concrete determinations for its sides; it thus collapses into the simple oneness which is being.

Here “double negation” is plausibly matched with the double negation modality.

Concerning “something”: if XX is a type, then by propositions-as-types there is something of this type if the type is inhabited. But classically this is expressed by by its double negation modality. Hence: there is something of some quality/type if that is a double-negation modal type.

B. Die Endlichkeit / Finitude.
a. Etwas und ein Anderes. / Something and an Other
  • §221 Being-for-other and being-in-itself constitute the two moments of the something.

something : Being-for-other \dashv being-in-itself

  • §222 Being and nothing in their unity, which is determinate being

Notice that above this unity is called becoming.

  • Dieß führt zu einer weitern Bestimmung. Ansichseyn und Seyn-für-Anderes sind zunächst verschieden; aber daß Etwas dasselbe, was es an sich ist, auch an ihm hat, und umgekehrt, was es als Seyn-für-Anderes ist, auch an sich ist,—dieß ist die Identität des Ansichseyns und Seyns-für-Anderes, nach der Bestimmung, daß das Etwas selbst ein und dasselbe beider Momente ist, sie also ungetrennt in ihm sind.—Es ergiebt sich formell diese Identität schon in der Sphäre des Daseyns, aber ausdrücklicher in der Betrachtung des Wesens und dann des Verhältnisses der Innerlichkeit und Äußerlichkeit, und am bestimmtesten in der Betrachtung der Idee, als der Einheit des Begriffs und der Wirklichkeit.
C. Die Unendlichkeit
a. Das Unendliche Überhaupt
b. Wechselwirkung des Endlichen und Unendlichen
c. Die affirmative Unendlichkeit
Der Übergang
  • §305 Die Idealität kann die Qualität der Unendlichkeit genannt werden; aber sie ist wesentlich der Proceß des Werdens und damit ein Übergang, wie des Werdens in Daseyn, der nun anzugeben ist. Als Aufheben der Endlichkeit, d. i. der Endlichkeit als solcher und ebenso sehr der ihr nur gegenüberstehenden, nur negativen Unendlichkeit ist diese Rückkehr in sich, Beziehung auf sich selbst, Seyn. Da in diesem Seyn Negation ist, ist es Daseyn, aber da sie ferner wesentlich Negation der Negation, die sich auf sich beziehende Negation ist, ist sie das Daseyn, welches Fürsichseyn genannt wird.

  • §305 Ideality can be called the quality of infinity; but it is essentially the process of becoming, and hence a transition — like that of becoming in determinate being — which is now to be indicated. As a sublating of finitude, that is, of finitude as such, and equally of the infinity which is merely its opposite, merely negative, this return into self is self-relation, being. As this being contains negation it is determinate, but as this negation further is essentially negation of the negation, the self-related negation, it is that determinate being which is called being-for-self.

Compare “ideality” as “quality of infinity” with

quality: ideality: inf \array{ quality \colon \int \dashv \flat \\ ideality \colon \Re \dashv \int_{inf} }

Third chapter. Das Fürsichsein / Being for self

  • §318 Im Fürsichseyn ist das qualitative Seyn vollendet;

  • §318 In being-for-self, qualitative being finds its consummation;

  • §319 Being-for-self is first, immediately a being-for-self — the One.

    Secondly, the One passes into a plurality of ones — repulsion — and this otherness of the ones is sublated in their ideality — attraction.

    Thirdly, we have the alternating determination of repulsion and attraction in which they collapse into equilibrium, and quality, which in being-for-self reached its climax, passes over into quantity.

Here we have a second-order unity of opposites: quantity itself is

quantity : discreteness \dashv continuity

and by the above we take the

continuum : repulsion \dashv attraction

to be quality, then we get from the adjoint triple

shape modality \dashv flat modality \dashv sharp modality

the duality of dualities

attraction repulsion quality: quantity: discreteness continuity \array{ & attraction && repulsion \\ quality : & \int &\dashv& \flat \\ & \bot && \bot \\ quantity : & \flat &\dashv& \sharp \\ & discreteness && continuity }
A. Das Fürsichsein als solches / Being-for-self as such
a. Dasein und Fürsichsein / Determinate being and Being-for-self
  • §321 Das Fürsichseyn ist, wie schon erinnert ist, die in das einfache Seyn zusammengesunkene Unendlichkeit; es ist Daseyn, insofern die negative Natur der Unendlichkeit, welche Negation der Negation ist, in der nunmehr gesetzten Form der Unmittelbarkeit des Seyns, nur als Negation überhaupt, als einfache qualitative Bestimmtheit ist.

  • §321 But being, which in such determinateness is determinate being, is also at once distinct from being-for-self, which is only being-for-self in so far as its determinateness is the infinite one above-mentioned; nevertheless, determinate being is at the same time also a moment of being-for-self; for this latter, of course, also contains being charged with negation. Thus the determinateness which in determinate being as such is an other, and a being-for-other, is bent back into the infinite unity of being-for-self, and the moment of determinate being is present in being-for-self as a being-for-one.

b. Sein-für-Eines / Being-for-one
  • §322 To be ‘for self’ and to be ‘for one’ are therefore not different meanings of ideality, but are essential, inseparable moments of it.
Anmerkung
  • §324 Die Idealität kommt zunächst den aufgehobenen Bestimmungen zu, als unterschieden von dem, worin sie aufgehoben sind, das dagegen als das Reelle genommen werden kann. So aber ist das Ideelle wieder eins der Momente und das Reale das andere;

  • §324 But thus the ideal is again one of the moments, and the real the other;

hence another unity of opposites is idealityrealityideality \dashv reality.

für sich sein für eins sein ideality inf reality inf inf \array{ & \text{für sich sein} && \text{für eins sein} \\ ideality & \Re &\dashv& \int_{inf} \\ & \bot && \bot \\ reality & \int_{inf} &\dashv& \flat_{inf} }

See also at The One and the Many.

We might interpret this as follows: the ( inf inf)(\int_{inf} \dashv \flat_{inf})-adjunction is that which, by the discussion at differential cohesion, makes all types XX have an associated structured étale topos (Sh(X),𝒪 X)(Sh(X), \mathcal{O}_X). In a sense this gives the type XX a “reality” as a topos.

Hence in view of the previous disucssion at Existence As Such it seems we have the following

  • types have “being” in the presence of (*)(\emptyset \dashv \ast)

    • types moreover have “existence”/Dasein in the further presence of ()(\int \dashv \flat).

      • types moreover have “reality” in the further presence of ( inf inf)(\int_{inf} \dashv \flat_{inf})

in other words we have the following situation, in view of p. 7 of Some Thoughts on the Future of Category Theory:

reinesSein Dasein Realitaet being existence reality purebeing determinatebeing id inf id inf * \array{ reines\;Sein && Dasein && Realitaet \\ being && existence && reality \\ pure\,being && determinate\,being \\ \\ && && \Re & \subset & id \\ && && \bot && \bot \\ && \int & \subset & \int_{inf} & \subset & id \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & \flat_{inf} \\ \bot & & \bot && \\ \ast & \subset& \sharp }

where “\subset” denotes inclusion of modal types

c. Eins
  • §328 Being-for-self is the simple unity of itself and its moment, being-for-one.

  • §329 The moments which constitute the Notion of the one as a being-for-self fall asunder in the development. They are: (1) negation in general, (2) two negations, (3) two that are therefore the same, (4) sheer opposites, (5) self-relation, identity as such, (6) relation which is negative and yet to its own self.

If we translate “moment” as modality then here the double negation modality comes to mind.

Notice that the empty type and the unit type are the modal types for the double negation modality.

B. Eins und Vieles. / The One and the Many
  • Die Idealität des Fürsichseyns als Totalität schlägt so fürs erste in die Realität um, und zwar in die festeste, abstrakteste, als Eins.
a. Das Eins an ihm selbst
b. Das Eins und das Leere / The One and the Void
  • §335 The one is the void as the abstract relation of the negation to itself.
\;\; Remark: Atomism
  • §337 The one in this form of determinate being is the stage of the category which made its appearance with the ancients as the atomistic principle, according to which the essence of things is the atom and the void.

Eins: atom, infinitesimally thickened point

c. Viele Eins. Repulsion. / Many ones. Repulsion.
  • §340 The one and the void constitute the first stage of the determinate being of being-for-self. Each of these moments has negation for its determination and is at the same time posited as a determinate being; according to the former determination the one and the void are the relation of negation to negation as of an other to its other: the one is negation in the determination of being, and the void is negation in the determination of non-being.

Das Eins (the One): *\ast unit type

Das Leere (the void): \emptyset empty type ( leere Menge !)

Negation (¬X)(X)(\not X) \coloneqq (X \to \emptyset)

*¬\ast \simeq \not \emptyset.

  • §342 the one repels itself from itself. The negative relation of the one to itself is repulsion.

  • §343 This repulsion as thus the positing of many ones but through the one itself, is the one’s own coming-forth-from-itself but to such outside it as are themselves only ones. This is repulsion according to its Notion, repulsion in itself. The second repulsion is different from it, it is what is immediately suggested to external reflection: repulsion not as the generation of ones, but only as the mutual repelling of ones presupposed as already present.

To see a formalization of “the one repels itself from itself”, suppose we have a shape modality \int but without the assumption that it preserves finite product types. (This is what the term “shape” really refers to).

Then given just the empty type \emptyset and the unit type *\ast, there is one new type to be formed (since necessarily \int \emptyset \simeq \emptyset) and this is

* \int \ast

Below we see that this, being a discrete type, is what Hegel describes with “repulsion”: The points in *\int \ast do not attract/cohese, they are different and repel.

At the same time, being a discrete type it is necessarily a homotopy colimit of copies of the unit type (see here)

*lim I* \int \ast \simeq \underset{\longrightarrow}{\lim}_I \ast

where the diagram II that the colimit is over is I=ʃ*I = ʃ \ast itself.

For a similar argument see Lawvere’s Cohesive toposes and Cantor’s Lauter Einsen). On p. 6 there is suggested that the unity of opposites “all elements of a set are indistinguishable and yet distinct” is captured by the fact that both

X\flat X as well as X\sharp X have the same image under \flat.

\;\; Remark: The Monad of Leibniz
  • §348 We have previously referred to the Leibnizian idealism. We may add here that this idealism which started from the ideating monad, which is determined as being for itself, advanced only as far as the repulsion just considered, and indeed only to plurality as such, in which each of the ones is only for its own self and is indifferent to the determinate being and being-for-self of the others; or, in general, for the one, there are no others at all. The monad is, by itself, the entire closed universe; it requires none of the others. But this inner manifoldness which it possesses in its ideational activity in no way affects its character as a being-for-self. The Leibnizian idealism takes up the plurality immediately as something given and does not grasp it as a repulsion of the monads. Consequently, it possesses plurality only on the side of its abstract externality.

    The atomistic philosophy does not possess the Notion of ideality; it does not grasp the one as an ideal being, that is, as containing within itself the two moments of being-forself and being-for-it, but only as a simple, dry, real being-for-self.

    It does, however, go beyond mere indifferent plurality; the atoms become further determined in regard to one another even though, strictly speaking, this involves an inconsistency; whereas, on the contrary, in that indifferent independence of the monads, plurality remains as a fixed fundamental determination, so that the connection between them falls only in the monad of monads, or in the philosopher who contemplates them.

To summarize, in §322 we get a clear prescription:

To be ‘for self’ and to be ‘for one’ are therefore not different meanings of ideality, but are essential, inseparable moments of it.

So we are to find an adjoint modality that expresses

Ideality:BeingForSelfBeingForOne Ideality \;\colon\; BeingForSelf \dashv BeingForOne

(or possibly the other way around).

The complaint about Leibniz in §348, makes pretty clear what this is about:

The atomistic philosophy does not possess the Notion of ideality; it does not grasp the one as an ideal being, that is, as containing within itself the two moments of being-forself and being-for-it, but only as a simple, dry, real being-for-self.

Here “atoms” really refers to the decomposition of the continuum into points (atoms of space, as in monad in nonstandard analysis) because in §337 it says:

The one in this form of determinate being is the stage of the category which made its appearance with the ancients as the atomistic principle, according to which the essence of things is the atom and the void.

But “The one” (The unit) with its repulsion of many we claimed before is well modeled by what \flat produces, the underlying points, the atoms of space.

So in conclusion the statement here is that it is a defect of both the ancients as well as of Leibniz to consider atoms/monads/points which have no way to look outside of themselves into interaction with others, that instead one needs to characterized atoms/monads/points by the above adjoint modality which expresses Ideality.

In conclusion, Eins (“The One”/“The Unit”) is a notion of atom which is similar to what the ancients and Leibniz called atom/monad, only that it improves on that by keeping an additional “moment” which the ancients and Leibniz forgot to retain.

Now in William Lawvere’s Toposes of Laws of Motion “atom” is proposed to refer to, essentially, infinitesimally thickened points. Indeed, the “infinitesimal thickening” of the point has something to do with the point “coming out of itself”and interacting with other points.

So possibly the adjoint modality given by reduction modality \dashv infinitesimal shape modality captures some of this well.

Here is a cartoon of an infinitesimally thickened point with its infinitesimal antennas reaching out to test what’s going on around

\array{ -- \bullet -- }

and here is the reduced point, all by itself/for itself

. \array{ \bullet } \,.

Notice that in superalgebra one says “soul” for these “antennas” and “body” for what remains. Therefore it seems plausible to conclude that the formalization of the unity of opposites

Ideality:BeingForSelfBeingForOne Ideality \;\colon\; BeingForSelf \dashv BeingForOne

is the adjoint modality given by reduction modality \dashv infinitesimal shape modality. The “Ideality” of infinitesimal extension gives the Eins, the atom-of-space, its dual character of containing a reduced point for-itself and at the same time an infinitesimal thickening that extends beyond that.

C. Repulsion und Attraktion
a. Ausschlißen des Eins.
b. Das Eine Eins der Attraktion
c. Die Beziehung der Repulsion und der Attraktion
  • §369 Die Repulsion daseyender Eins ist die Selbsterhaltung des Eins durch die gegenseitige Abhaltung der andern, so daß 1) die anderen Eins an ihm negirt werden, dieß ist die Seite seines Daseyns oder seines Seyns-für-Anderes; diese ist aber somit Attraktion, als die Idealität der Eins;—und daß 2) das Eins an sich sey, ohne die Beziehung auf die andere; aber nicht nur ist das Ansich überhaupt längst in das Fürsichseyn übergegangen, sondern an sich, seiner Bestimmung nach, ist das Eins jenes Werden zu Vielen.—Die Attraktion daseyender Eins ist die Idealität derselben, und das Setzen des Eins, worin sie somit als Negiren und Hervorbringen des Eins sich selbst aufhebt, als Setzen des Eins das Negative ihrer selbst an ihr, Repulsion ist.

  • §369 The repulsion of the determinately existent ones is the self-preservation of the one through the mutual repulsion of the others, so that (1) the other ones are negated in it-this is the side of its determinate being or of its being-for-other; but this is thus attraction as the ideality of the ones; and (2) the one is in itself, without relation to the others; but not only has being-in-itself as such long since passed over into being-for-self, but the one in itself, by its determination, is the aforesaid becoming of many ones. The attraction of the determinately existent ones is their ideality and the positing of the one, in which, accordingly, attraction as a negating and a generating of the one sublates itself, and as a positing of the one is in its own self the negative of itself, repulsion.

  • §370 Damit ist die Entwickelung des Fürsichseyns vollendet und zu ihrem Resultate gekommenen.

  • §370 With this, the development of being-for-self is completed and has reached its conclusion.

Since Dasein and Fürsichsein both are qualitative being/are quality (§188, §318, §321) the above gives us the unity of opposites

quality:attractionrepulsion quality \;\colon\; attraction \dashv repulsion
Remark: The Kantian Construction of Matter from the Forces of Attraction and Repulsion
  • §374 Kant, as we know, constructed matter from the forces of attraction and repulsion, or at least he has, to use his own words, set up the metaphysical elements of this construction.

Not about actual forces in matter so much as about what makes the points in the continuum both stay apart (repulsion) and at the same time hang together (attraction/cohesion).

Second section. The magnitude

First chapter. Die Quantität / The quantity

A. Die reine Quantität / Pure quantity
  • §398 Quantity is the unity of these moments of continuity and discreteness

By unity of opposites and since the flat modality matches the “moment of discreteness” this is the duality with the sharp modality

X X X momentofdiscreteness momentofcontinuity \array{ \flat X &\longrightarrow& X &\longrightarrow& \sharp X \\ {moment\;of \atop discreteness} && && {moment\;of \atop continuity} }
On attraction / cohesion
  • §395 Attraction is in this way the moment of continuity in quantity.

attraction is what holds stuff together, hence this is the idea of cohesion

if XX has continuity then the shape modality X\int X is the result of letting things collaps under their cohesion/attraction

On discreteness and repulsion
  • §397 In continuity, therefore, magnitude immediately possesses the moment of discreteness — repulsion, as now a moment in quantity.

continuous object XX possesses moment of discreteness= flat modality X\flat X

  • §398 Quantity is the unity of these moments of continuity and discreteness,

By the formalization of unity of opposites this must mean that “moment of continuity” is the right adjoint modality to the flat modality. This is the sharp modality \sharp. Therefore their unity of opposites is

quantity:X X X momentofdiscreteness momentofcontinuity quantity \;\colon\; \array{ \flat X &\longrightarrow& X &\longrightarrow& \sharp X \\ \\ {moment\;of \atop discreteness} && && {moment\;of \atop continuity} }

Notice that byLawvere’s Cohesive Toposes and Cantor’s “lauter Einsen” precisely this unity of opposites is that characteristic of cardinality (Mengen/Kardinalen).

we also have

X X X repulsion attraction/cohesion \array{ \flat X &\longrightarrow& X &\longrightarrow& \int X \\ repulsion && && { attraction/ \atop cohesion } }
B. Kontinuirliche und diskrete Größe.
On the continuum
  • §400 Mathematics, on the other hand, rejects a metaphysics which would make time consist of points of time; space in general — or in the first place the line — consist of points of space; the plane, of lines; and total space of planes. It allows no validity to such discontinuous ones. Even though, for instance, in determining the magnitude of a plane, it represents it as the sum of infinitely many lines, this discreteness counts only as a momentary representation, and the sublation of the discreteness is already implied in the infinite plurality of the lines, since the space which they are supposed to constitute is after all bounded.

The continuum.

Diese Antinomie besteht allein, darin daß die Diskretion eben so sehr als die Kontinuität behauptet werden muß. Die einseitige Behauptung der Diskretion giebt das unendliche oder absolute Getheiltseyn, somit ein Untheilbares zum Princip; die einseitige Behauptung der Kontinuität dagegen die unendliche Theilbarkeit.

On space, time, matter
  • §432 Space, time, matter, and so forth are continuous magnitudes
C. Begrenzung der Quantität

Second chapter. Quantum

Third section. The measure.

  • §699 Im Maaße sind, abstrakt ausgedrückt, Qualität und Quantität vereinigt.

  • §699 Abstractly expressed, in measure quality and quantity are united

unity of opposites

measure:quantityquality measure \colon quantity \dashv quality
  • §703 The observation here made extends generally to those systems of pantheism which have been partially developed by thought. The first is being, the one, substance, the infinite, essence; in contrast to this abstraction the second, namely, all determinateness in general, what is only finite, accidental, perishable, non-essential, etc. can equally abstractly be grouped together; and this is what usually happens as the next step in quite formal thinking. But the connection of this second with the first is so evident that one cannot avoid grasping it as also in a unity with the latter;

A. The Specific Quantum

  • §714 Ein Maaß, als Maaßstab im gewöhnlichen Sinne, ist ein Quantum, das als die an sich bestimmte Einheit gegen äußerliche Anzahl willkürlich angenommen wird. Eine solche Einheit kann zwar auch in der That an sich bestimmte Einheit seyn, wie Fuß und dergleichen ursprüngliche Maaße; insofern sie aber als Maaßstab zugleich für andere Dinge gebraucht wird, ist sie für diese nur äußerliches, nicht ihr ursprüngliches Maaß.—So mag der Erddurchmesser, oder die Pendellänge, als specifisches Quantum für sich genommen werden. Aber es ist willkürlich, den wievielsten Theil des Erddurchmessers oder der Pendellänge und unter welchem Breitengrade man diese nehmen wolle, um sie als Maaßstab zu gebrauchen. Noch mehr aber ist für andere Dinge ein solcher Maaßstab etwas Äußerliches. Diese haben das allgemeine specifische Quantum wieder auf besondere Art specificirt, und sind dadurch zu besondern Dingen gemacht. Es ist daher thöricht, von einem natürlichen Maaßstab der Dinge zu sprechen. Ohnehin soll ein allgemeiner Maaßstab nur für die äußerliche Vergleichung dienen; in diesem oberflächlichsten Sinne, in welchem er als allgemeines Maaß genommen wird, ist es völlig gleichgültig, was dafür gebraucht wird. Es soll nicht ein Grundmaaß in dem Sinne seyn, daß die Naturmaaße der besondern Dinge daran dargestellt und daraus nach einer Regel, als Specifikationen Eines allgemeinen Maaßes, des Maaßes ihres allgemeinen Körpers, erkannt würden. Ohne diesen Sinn aber hat ein absoluter Maaßstab nur das Interesse und die Bedeutung eines Gemeinschaftlichen, und ein solches ist nicht an sich, sondern durch Übereinkommen ein Allgemeines.

  • §714 A measure taken as a standard in the usual meaning of the word is a quantum which is arbitrarily assumed as the intrinsically determinate unit relatively to an external amount. Such a unit can, it is true, also be in fact an intrinsically determinate unit, like a foot and suchlike original measures; but in so far as it is also used as a standard for other things it is in regard to them only an external measure, not their original measure. Thus the diameter of the earth or the length of a pendulum may be taken, each on its own account, as a specific quantum; but the selection of a particular fraction of the earth’s diameter or of the length of the pendulum, as well as the degree of latitude under which the latter is to be taken for use as a standard, is a matter of choice. But for other things such a standard is still more something external. These have further specified the general specific quantum in a particular way and have thereby become particular things. It is therefore foolish to speak of a natural standard of things. Moreover, a universal standard ought only to serve for external comparison; in this most superficial sense in which it is taken as a universal measure it is a matter of complete indifference what is used for this purpose. It ought not to be a fundamental measure in the sense that it forms a scale on which the natural measures of particular things could be represented and from which, by means of a rule, they could be grasped as specifications of a universal measure, i.e. of the measure of their universal body. Without this meaning, however, an absolute measure is interesting and significant only as a common element, and as such is a universal not in itself but only by agreement.

Here Maßstab is translated as “standard”. It can also, maybe better, be translated as “gauge”. Therefore by (699) we have

attraction repulsion quality: gauge quantity: discreteness continuity \array{ & & attraction && repulsion \\ & quality : & \int &\dashv& \flat \\ gauge & \bot & \bot && \bot \\ & quantity : & \flat &\dashv& \sharp \\ & & discreteness && continuity }

B. Specifying measure

(a) The Rule
  • {725} §725 Die Regel oder der Maaßstab, von dem schon gesprochen worden, ist zunächst als eine an sich bestimmte Größe, welche Einheit gegen ein Quantum ist, das eine besondere Existenz ist, an einem andern Etwas, als das Etwas der Regel ist, existirt,—an ihr gemessen, d. i. als Anzahl jener Einheit bestimmt wird. Diese Vergleichung ist ein äußerliches Thun, jene Einheit selbst eine willkürliche Größe, die ebenso wieder als Anzahl (der Fuß als eine Anzahl von Zollen) gesetzt werden kann. Aber das Maaß ist nicht nur äußerliche Regel, sondern als specifisches ist es dieß, sich an sich selbst zu seinem Andern zu verhalten, das ein Quantum ist.

  • §725 The rule or standard [[ gauge ]], which has already been mentioned, is in the first place an intrinsically determinate magnitude which is a unit with reference to a quantum having a particular existence in a something other than the something of the rule; this other something is measured by the rule, i.e. is determined as an amount of the said unit. This comparison is an external act, the unit itself being an arbitrary magnitude which in turn can equally be treated as an amount (the foot as an amount of inches). But measure is not only an external rule; as a specifying measure its nature is to be related in its own self to an other which is a quantum.

(b) Specifying Measure
(c) Relation of the two Sides as Qualities

Book two Die Lehre vom Wesen / The doctrine of essence

Reflection

Section 1. Essence as Reflection within Itself

Chapter 1 Illusory Being
A The essential and the unessential
B Illusory being
C Reflection
Chapter 2 The Essentialities or Determination of Reflection
\;\; Remark A=AA = A
  • §863 Thus the essential category of identity is enunciated in the proposition: everything is identical with itself, A = A.

The reflector(!) term constructor in an identity type. This is more explicit below at Identity.

A Identity
  • §869 Essence is therefore simple identity with self.

  • §869 This identity-with-self is the immediacy of reflection.

The reflector(!) term constructor in an identity type. Below this is called te First original law of thought.

\;\; Remark 1: Abstract identity
\;\; Remark 2: First original law of thought
  • §875 In this remark, I will consider in more detail identity as the law of identity which is usually adduced as the first law of thought.

    This proposition in its positive expression A=AA = A is, in the first instance, nothing more than the expression of an empty tautology.

The reflector term constructor in an identity type.

B Difference
\;\; (a) Absolute difference
\;\; (b) Diversity
\;\; Remark: The Law of Diversity
  • §903 All things are different; or: there are no two things like each other.

Reminiscent of identity types in intensional type theory.

C Contradiction
  • §903 When all the conditions of a fact are present, it enters into Existence.

  • §1035 The fact emerges from the ground. It is not grounded or posited by it in such a manner that ground remains as a substrate; on the contrary, the positing is the movement of the ground outwards to itself and its simple vanishing.

    [..][..]

    This immediacy that is mediated by ground and condition and is self-identical through the sublating of mediation, is Existence.

term introduction in natural deduction

(…)

  • §1190 Die Wirklichkeit als selbst unmittelbare Formeinheit des Innern und Äußern ist damit in der Bestimmung der Unmittelbarkeit gegen die Bestimmung der Reflexion in sich; oder sie ist eine Wirklichkeit gegen eine Möglichkeit. Die Beziehung beider auf einander ist das Dritte, das Wirkliche bestimmt ebenso sehr als in sich reflektirtes Seyn, und dieses zugleich als unmittelbar existirendes. Dieses Dritte ist die Nothwendigkeit.

(..)

  • §1190 Actuality as itself the immediate form — unity of inner and outer is thus in the determination of immediacy over against the determination of reflection-into-self; or it is an actuality as against a possibility. Their relation to each other is the third term, the actual determined equally as a being reflected into itself, and this at the same time as a being existing immediately. This third term is necessity.

  • Die Wirklichikeit ist formell, insofern sie als erste Wirklichkeit nur unmittelbare, unreflektirte Wirklichkeit, somit nur in dieser Formbestimmung, aber nicht als Totalität der Form ist. Sie ist so weiter nichts als ein Seyn oder Existenz überhaupt. Aber weil sie wesentlich nicht bloße unmittelbare Existenz, sondern, als Formeinheit des Ansichseyns oder der Innerlichkeit, und der Äußerlichkeit ist, so enthält sie unmittelbar das Ansichseyn oder die Möglichkeit. Was wirklich ist, ist möglich.

  • § 1192 Actuality is formal in so far as, being primary actuality, it is only immediate, unreflected actuality, and hence is only in this form-determination but not as the totality of form. As such it is nothing more than a being or Existence in general. But because it is essentially not a mere immediate Existence but exists as form-unity of being-within-self or inwardness and outwardness, it immediately contains the in-itself or possibility. What is actual is possible.

so

Wirklichkeit:Ansichsein/innerouter Wirklichkeit \colon Ansichsein/inner \dashv outer

Book three Die Lehre vom Begriff / The doctrine of the notion

References

A good survey is in

  • Paul Redding, section 3.2 of Georg Wilhelm Friedrich Hegel, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.) (web)

and a comprehensive set of lectures is at

Hegel himself expand on the relation of the Science of Logic to the Tao Te Ching in

  • Georg Hegel, Lectures on the Philosophy of Religion. Volume II: Determinate Religion. Edited by Peter C. Hodgson; translated by R.F. Brown, P.C. Hodgson, and J.M. Stewart, with the assistance of J.P. Fitzer and H.S. Harris. Berkeley: University of California Press, 1995 (orig. 1987). (Translation of: Vorlesungen über die Philosophie der Religion.) This extract (pp. 556-561) is from the Lectures of 1827; A. Immediate Religion, or Nature Religion; 1. The Religion of Magic; c. The State Religion of the Chinese Empire and the Dao. (web)

Further comments on Hegel’s text include

  • Martin Heidegger, Hegel and the Greeks, Conference of the Academy of Sciences at Heidelberg, July 26, 1958 (web)

  • Inwood, Hegel, 1983

  • Joachim Lambek, The Influence of Heraclitus on Modern Mathematics, In Scientific Philosophy Today: Essays in Honor of Mario Bunge, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co.

Proposals for formalizing some of Hegel’s thoughts in categorical logic have been put forward by William Lawvere in several places, for instance in

Related commentary is in

Revised on December 2, 2014 22:09:35 by Urs Schreiber (82.113.106.174)