# nLab Science of Logic

under construction

philosophy

### Of physics

• Georg Hegel

Wissenschaft der Logik ( Science of Logic)

Volume 1, The Objective Logic, Part I: The Doctrine of Being; Part II: The Doctrine of Essence.

Volume 2, The Subjective Logic, The Doctrine of the Notion

1812, 1831

on, not quite logic in the usual sense, but more something like logos in the old sense of Heraclitus (e.g. (Heidegger 58)).

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.

# Contents

## Survey

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)

## 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

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

for any type $X$.

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

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

### 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
quality$\sim$ type
determinate being of quality $X$$\vdash X \colon Type$
momentmodality
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 $\emptyset \dashv \ast$
moment of repulsionflat modality $\flat$
moment of attractioncohesion, shape modality $\int$
continuumadjoint modality $\int \dashv \flat$
moment of discretenessflat modality $\flat$
moment of continuitysharp modality $\sharp$
quantityadjoint modality $\flat \dashv \sharp$
vanishing of infinitesimalsreduction modality
being-for-selfreduction modality $Red$
being-for-oneinfinitesimal shape modality $\int_{inf}$
idealityadjoint modality $Red \dashv \int_{inf}$
moment of two negationsdouble negation modality $\not \not$, more generally: bracket type/(-1)-truncation modality
something(-1)-truncation modality, classically double negation modality
measurede Rham coefficients $\flat_{dR} A = fib(\flat A \to A)$ (?)

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

$n$n-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 speach “das Dasein” means “existence” as in “Ach, das Dasein ist doch zwecklos.”

(…)

## Book one Die Lehre vom Sein / The Doctrine of Being

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

The empty type $\emptyset$.

##### C. 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 might think of this as the universal factorization

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

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

Indeed, later it says:

• §174 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

and “determinate being” / Dasein seems to be well interpreted with types expressed as

$\vdash X \colon Type \,.$
###### $\;\;$ 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 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 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 $Red$.

• §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

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

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

###### 2. Momente des Wedrens / Moments of Becoming
• §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 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 : nothing $\dashv$ being

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

• §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 : 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$\;\;\;\dashv$Sein: Vergehen

$\,$

Dasein
becoming :nothing$\;\;\;\dashv$being: 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

#### Second chapter. Deasein / Determinate Being

• §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.
##### A. Dasein as such / Determinate being as such
• §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;
###### a. Dasein überhaupt / Determinant being in general
• §191 From becoming there issues determinate being, which is the simple oneness of being and nothing.

Above we saw that becoming is formalized by the universal unity of opposites of $\emptyset \dashv \ast$, exhibiting any type $X$

$\emptyset \longrightarrow X \longrightarrow \ast \,.$

So determinate being/Dasein is that of types.

§ 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, therefore, as a first, as a starting-point for the ensuing development. It is first of all in the one-sided determination of being; the other determination, nothing, will likewise display itself and in contrast to it.

###### b. Qualität / Quality
• §196 Determinateness thus isolated by itself in the form of being is quality

type

###### 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 $X$ 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.

#### Third chapter. Das Fürsichsein / Being for self

• §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

$\array{ & attraction && repulsion \\ quality : & \int &\dashv& \flat \\ & \bot && \bot \\ quantity : & \flat &\dashv& \sharp \\ & discreteness && continuity }$
##### A. Das Fürsichsein als solches / Being-for-self as such
###### a. Dasein und Fürsichsein / Determinate being and Being-for-self
• §321 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.
###### 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.
###### 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.
###### 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 $(\not X) \coloneqq (X \to \emptyset)$

$\ast \simeq \not \emptyset$.

• §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)

$\int \ast \simeq \underset{\longrightarrow}{\lim}_I \ast$

where the diagram $I$ that the colimit is over is $I = ʃ \ast$ 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

$\flat X$ as well as $\sharp X$ 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

$Ideality \;\colon\; BeingForSelf \dashv BeingForOne$

(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

$Ideality \;\colon\; BeingForSelf \dashv BeingForOne$

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
###### 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

$\array{ \flat X &\longrightarrow& X &\longrightarrow& \sharp X \\ {moment\;of \atop discreteness} && && {moment\;of \atop continuity} }$
###### 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 $X$ has continuity then the shape modality $\int X$ 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 $X$ possesses moment of discreteness= flat modality $\flat X$

• §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

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

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

$\array{ \flat X &\longrightarrow& X &\longrightarrow& \int X \\ repulsion && && { attraction/ \atop cohesion } }$
##### 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

### Third section. The measure.

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

• §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;

## Book two Die Lehre vom Wesen / The doctrine of essence

### Reflection

#### Section 1. Essence as Reflection within Itself

##### Chapter 2 The Essentialities or Determination of Reflection
###### $\;\;$ Remark $A = A$
• §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 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 $A = A$ is, in the first instance, nothing more than the expression of an empty tautology.

The reflector term constructor in an identity type.

###### $\;\;$ 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.

• §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.

## References

Comments on Hegel’s text are for instance in

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

• Inwood, Hegel, 1983

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

• Toposes of laws of motion , transcript of a talk in Montreal, Sept. 1997 (pdf)

Revised on December 4, 2013 03:06:28 by Urs Schreiber (82.113.99.55)