nLab
Science of Logic

under construction

This page is about

and about the further body of text that this is a part of, which consists of the later

which itself subsumes, in outline, the earlier

that, in turn, is claimed (§50, §51) to be the derivation of the concept of Science that the Science of Logic develops.

Together these texts lay out what has come to be called Hegel’s system (§1805) (following Spinoza's system (§1287); Hegel himself speaks of the system of pure reason §53b). Notice that these texts overlap. The first part of the Encyclopedia of the Philosophical Sciences , widely known as the “Shorter Logic”, covers the material of the Science of Logic in more condensed but also slightly re-worked form. Similarly the topics of the Phenomenology of the Spirit re-appear in condensed and slightly re-worked form inside the third piece of the Encyclopedia.

Here the objective logic is not logic in the usual sense – that usual sense of logic (in fact Aristotle's logic) Hegel calls instead the subjective logic. The objective logic is rather like the logos in the old sense of Heraclitus (HistPhil, Heidegger 58, Lambek 82)) or the Nous in the sense of Anaxagoras (§54, PoS pref. §55) or just metaphysics (§85), which Hegel views as the “substantial” aspect of logic §48.

Hegel’s system is meant to be the revelation of a dynamical ontological process (a diagram is below) of intrinsic oppositions and unifications (“Bewegung des Wissens”, “Werden des Seins”) by which the logos develops from nothingness over various stages of determinations of being (Seinsbestimmungen) to, roughly, the Idea, a kind of absolute of Plato’s doctrine of ideas §55. The idea then appears in the Essence (erscheint im Wesen), next externalizes itself in the form of Nature, where, eventually, embodied now as the soul PN§308 of nature (nous) and of its life forms, it grows into the concious and then self-concious Spirit. Eventually by introspection the Spirit recovers this dynamical process of being inside itself (and so the system reiterates §1812, §1814).

Other philosophical systems, Hegel claims (see also (Lect Hist Phil, Result)), are subsumed by and find their place in his more comprehensive and “scientific” system, notably

Also there is some serious structuralism (e.g. §1065a) in the Science of Logic.

Where Hegel speaks of his system as being science “Wissenschaft” (e.g. PdGVorrede§5), and phenomenology and where he refers to its dialectic method §62, §63 this is to claim that the process is indeed being systematically derived and proven (§50, EncPreface1stEd) in fact being observed PdGPreface15 as the Arbeit des Begriffs (PhenVorr) – but by “supersensuous inner intuition” §1786a. This observation = speculari is what the term speculative philosophy for this school of thought refers to. Therefore Hegel is speaking to some extent in a mystic or gnostic tone, revealing truths by the way of a seer, following the similarly mysterious second part of Plato’s Parmenides dialogue, §357. One may hence argue that this is not so much philosophy as mysticism (§3, Enc§82d, Russell 45, Copleston 71, Stanfield 14), gnosticism (Bauer 1835) or hermeticism (Magee 01). Indeed, in PhenGeistVorrede and in §8, §9, §10, §11, §50, §51, §52, §53a, §53b Hegel explains that the Phenomenolgy of the Spirit is to be regarded as the “first part of the system” in that it establishes the necessary spiritual background to embark on the intellectual project of the Science of Logic, that a spirit which is to see and follow the speculative dialectics and the “work on the notion” that is the content of the Science of Logic needs to previously have worked itself through the stages all the way starting from sensual perception, conciousness, reason, via moral and intellectual education, through various stages of religion, to finally achieve the stage of “absolute knowledge”. Apparently not actually expecting that all his readers prepare themselves accordingly, Hegel at times pauses to comment on the incomprehensibility of his very text (e.g. the Incomprehensibility of the beginning, the life of the Notion in the Idea and the nature of the Soul §1648, or matter being the unity of space and time PN§261b), much like the mystic Meister Eckhart did in his texts on the union of the soul with god. Hegel explicitly acknowledges accomplishments of gnosticism and mysticism for philosophy (EncPreface2ndEd), but indicates there that he regards his system as more refined. In the preliminaries to his Lectures on the Philosophy of Religion Hegel states that philosophy and religion have the same subject, fall together, both are “worship” of the “eternal truth in its very objectivity”, just by different means.

In any case, this means that, in an ironic meta-contradiction, the harder Hegel tries, by his own account, to make philosophy a science, to make it logical and to root it in observation, the further he may seem to depart from what is “commonly” understood by these very words and to indulge in the opposite activity.

Accordingly, after a phase of great popularity in the philosophical community of the 19th century (“German idealism”, §316), Hegel’s system has variously (see here for examples) been rejected and outright ridiculed, famously so by Bertrand Russell in Logic as the Essence of Philosophy and in A History of Western Philosophy, as being obfuscating and in fact nonsensical (e.g. already Grassmann 1844, p. xv whose introduction however speaks entirely in a Hegel-like tone, then later for instance Carnap 32). Following the lead of Russell, the whole field of analytic philosophy defined itself – in what is known as the “revolt against idealism” – to a large extent in opposition to Hegels absolute idealism (and more generally to “continental philosophy”), and aimed, in contrast, for undisputable clarity of argument, optimally by use of formalized predicate logic. Indeed Hegel’s system clearly defies any attempt to formalize it in predicate logic. Hegel was aware of this, but insisted:

§1798 formal thinking lays down for its principle that contradiction is unthinkable; but as a matter of fact the thinking of contradiction is the essential moment of the Notion.

However, there is more to formal logic than plain predicate logic. Foundational systems of categorical logic and of type theory (which happens to have its roots in (Russell 08)) subsume first-order logic but also allow for richer category-theoretic universal constructions such as notably adjunctions and modal operators (see at modal type theory). That adjunctions stand a good chance of usefully formalizing recurring themes of duality (of opposites) in philosophy was observed in the 1980s (Lambek 82) notably by William Lawvere. Since then, Lawvere has been proposing (review includes Rodin 14), more or less explicitly and apparently (Lawvere 95) inspired by (Grassmann 1844), that at least some key parts of Hegel’s Logic, notably his concepts of unity of opposites, of Aufhebung (sublation) and of abstract general, concrete general and concrete particular as well as the concepts of objective logic and subjective logic as such (Law94b) have an accurate, useful and interesting formalization in categorical logic. Not the least, the concept and terminology of category, modality, theory and doctrine matches well under this translation from philosophy to categorical logic.

Lawvere also proposed formalizations in category theory and topos theory of various terms appearing prominently in Hegel’s Philosophy of Nature, such as the concept of intensive or extensive quantity and of cohesion. While, when taken at face value, these are hardly deep concepts in physics, and were not at Hegel’s time, in Lawvere’s formalization and then transported to homotopy type theory (as cohesive homotopy type theory), they do impact on open problems in fundamental physics and even in pure mathematics (see also at Have professional philosophers contributed to other fields in the last 20 years?), a feat that the comparatively simplistic mathematics that is considered in analytic philosophy seems to have little chance of achieving.

Lawvere 92: It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.

nPOV

Therefore, while going through Hegel’s text, this page here attempts to spell out as much as seems possible the translation of the system to a category-theoretic or modal type-theoretic formalization (an “nPOV”). The way this formalization works in general is surveyed below in Formalization in categorical logic / in Modal type theory; a dictionary version of the formalization that we arrive at is in The formalization dictionary; and diagram showing the resulting process is in Survey diagram.

Contents

Formalization in Categorical logic / in Modal homotopy type theory

We introduce and survey here the formalization of Hegel’s Logic in categorical logic/type theory, in the vein of the above remark, which is discussed incrementally in the main text below.

We survey the matching of the formalization to Hegel’s text in.

  1. Formalization dictionary

  2. Survey diagram

All this appears below in three stages,

  1. in the Seinslogik at around §714, where the modalities are implemented in a bare homotopy type theory;

  2. then “reflected” in the Wesenslogik around §989, where the type theory is now equipped with a type universe and where every modality ModMod now has a “reflection” Mod'Mod' in the type universe;

  3. then “externalized” in Nature, given by a model ρ\rho of this modal type theory on a particular (infinity,1)-topos H\mathbf{H}, where now every abstract modality ModMod has a representation ρ(Mod)\rho(Mod) as an actual (infinity,1)-comonad on H\mathbf{H}.

Subjectivity

Concept

Homotopy type theory as such (UFP 13) is a logic of types, of (mathematical) concepts (Martin-Löf 73, 1.1, Ladyman&Presnel 14). (References which recall that the modern “type” is a contraction of “type of mathematical concepts” include for instance also (Sale 77, p. 6).

With the univalence axiom for weakly Tarskian type universes included – which says that this essence appears properly reflected within itself – then its interpretation via categorical semantics is in elementary homotopy toposes (Shulman 12a, Shulman 12b, Shulman 14). These are the models of homotopy type theory. Conversely, homotopy type theory is the internal language of homotopy toposes, hence the latter are its “externalization”. This way homotopy type theory overlaps much with (higher) categorical logic. See at relation between type theory and category theory for more background on this.

Accordingly, since it is more immediately readable, we display mostly categorical expressions in the following, instead of the pure type theoretic syntax.

Judgement

The earliest formulation of a logic of concepts is arguably Aristotle's logic, which famously meant to reason about the relation of concepts such as “human” and “mortal”. We consider now a natural formalization of at least the core intent of Aristotle’s logic in dependent homotopy type theory.

Formalizations of Aristotle’s logic in categorical logic or type theory has previously been proposed in (LaPalmeReyes-Macnamara-Reyes 94, 2.3) and in (Pagnan 10, def. 3.1). The formalization below agrees with these proposal in the identification of the Aristotlean judgement “All BB are AA” with the type-theoretic judgement “f:BA\vdash f \colon B\to A”, and with the identification of syllogisms with composition of fuch function terms.

All BB are AA.

If CC is a concept, a type, then a judgement

c:C c \colon C

says that cc is an instance of the concept CC, or that cc is a term of type CC.

For instance if \mathbb{N} is the concept of natural numbers, then the judgement n:n \colon \mathbb{N} says that nn is a natural number. Clearly here the “concept” \mathbb{N} may just as well be thought of as the set of all its instances.

Given concepts/types AA and BB, there is the concept of maps between them, the function type BAB\to A. In the categorical semantics this is the internal hom.

The judgement that there is a function, hence an instance ff of the concept of functions

f:BA f \colon B \longrightarrow A

says that ff is a rule that takes instances/terms of BB to instances/terms of AA. At least if this is a monomorphism f:BAf \colon B \hookrightarrow A (so that the corresponding a:Af 1(a)a\colon A \vdash f^{-1}(a) is a proposition) then this says in words that ff wittnesses the fact that

All instances of BB are instances of AA.

or for short just

All BB are AA

hence that if AA is das Allgemeine (general, universal) concept then BB is das Besondere (special, particular) concept.

This formalization of Aristotle’s “All BB are AA” in categiorical logic/type has been proposed in (LaPalmeReyes-Macnamara-Reyes 94, 2.3), where it is attributed to William Lawvere, and in (Pagnan 10, def. 3.1).

Notice that the choice of ff here is an important part of the formalization which is missing in Aristotle’s informal logic and causing ambiguity there.

For instance all natural numbers are real numbers, but there are many inequivalent subgroup inclusions \mathbb{Z}\hookrightarrow \mathbb{R} realizing this. For the purposes of prequantum field theory these choices correspond to the choice of Planck's constant (see the discussion there).

Similarly, once we have that the informal “All BB are AA.” is formalized by a map of types, we see further refinement of the ancient logical notion.

First, the meaning of BB,AA may dependend on some context CC. Leaving that implicit is arguably the greatest source of ambiguity in Aristotle’s logic. But it is easily fixed while staying true to the original intention: in general BB and AA are to be taken as CC-dependent types. Then the intended meaning of All BB are AA. is expressed by the dependent product over the function type formed in context CC

f:C(BA). f\colon \underset{C}{\prod} (B \longrightarrow A) \,.

Second, if ff is not a monomorphism it still expresses the fact that for every instance of BB there is a corresponding instance of AA. Hence in general, we should further specify if ff is an n-truncated morphisms. This is a general phenomenon in passing to higher homotopy types: the (epi, mono) factorization system on homotopy 0-types refines to a tower of (n-epi, n-mono) factorization systems for all natural numbers nn.

Individual EE is BB.

There is the unit type

E=* E = \ast

of which there is a unique instance, das Einzelne (individual). As a concept, this may be regarded as the concept of pure being: since any two instances of the concept EE just purely are, there is no distinction and hence there is a unique instance.

Hence a function of the form EBE \longrightarrow B is equivalently an instance/term bb of BB. In words this says that

The individual bb is an instance of the general concept AA.

of for short just

Individual bb is BB.

There is the identity type b=bb = b, which expresses the concept that bb is equivalent to itself.

The single introduction rule for identity types gives for all XX the statement that there is indeed an instance of this concept

r b:b=b. r_b \colon b = b \,.

The categorical semantics of (b=b)(b = b) is the loop space object Ω bB\Omega_b B, which is canonically a pointed object via the constant loop id n:*Ω bBid_n \colon \ast \to \Omega_b B.

Under composition of loops, this object canonically carries the structure of an infinity-group.

Proposition

In any homotopy topos H\mathbf{H}, the operation of forming loop space objects constitutes an equivalence of (infinity,1)-categories

Grp(H)BΩH 1 */ Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}} \mathbf{H}^{\ast/}_{\geq 1}

between infinity-groups in H\mathbf{H} and pointed connected objects in H\mathbf{H}.

The inverse equivalence B\mathbf{B} is called delooping. See at looping and delooping for more.

Now in homotopy type theory and in elementary (infinity,1)-toposes, the definition of infinity-groups as grouplike A-infinity algebras is not available, since the latter is not a finitary concept. But by prop. 1 the concept also has a simple finitary equivalent incarnation, which is available in homotopy type theory: we may identify an infinity-group GG with its pointed connected delooping type BG\mathbf{B}G.

Indeed this is most useful: homotopy type theory in the context of BG\mathbf{B}G is the infinity-representation theory of GG:

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}H
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

(NSS, dcct)

Some B 1B_1 is B 2B_2.

In order to formalize judgements of intersection of concepts of the form

Some B 1B_1 is B 2B_2.

it is necessry to specify a context. Regard both B 1B_1 and B 2B_2 as dependent types witnessed by display maps

f i:B iCf_i \colon B_i \longrightarrow C

to a common context CC. Then the product type in context, hence, in the categorical semantics, the homotopy fiber product

B 1×CB 2 B_1 \underset{C}{\times} B_2

is the type whose terms are the “some” instances of B 1B_1 which are also instances of B 2B_2, and vice versa. Indeed, the fiber product canonically sits in the homotopy pullback diagram

B 1×CB 2 B 2 B 1 C \array{ B_1 \underset{C}{\times} B_2 &\longrightarrow& B_2 \\ \downarrow && \downarrow \\ B_1 &\longrightarrow& C }

and if we read B 1×CB 2B_1 \underset{C}{\times} B_2 as “some B 1B_1” then according to the previous paragraph the top morphism expresses that “all of these particular B 1B_1 (but not necessarily all of B 1B_1 itself) are B 2B_2”.

Example

(principal infinity-bundles as judgements)

Specifically if C=BGC = \mathbf{B}G here is pointed via a map from B 2=E=*B_2 = E = \ast, and connected, hence equivalently the delooping of its infinity-group GG of loops, then (writing now BB for B 1B_1) a map of types

c:BBG c \colon B \longrightarrow \mathbf{B}G

may be thought of as a cocycle on BB with coefficients in GG, representing a class in the nonabelian cohomology of BB. See at cohomology for more on this general concept of cohomology.

In this case the homotopy fiber of ff is the GG-principal infinity-bundle PBP \to B classified by ff, fitting into the homotopy pullback square.

P E B c BG \array{ P &\longrightarrow& E \\ \downarrow && \downarrow \\ B &\stackrel{c}{\longrightarrow} & \mathbf{B}G }

Via the above translation this is an Aristotlean judgement of the form “Some BB are EE” in the context of BG\mathbf{B}G.

Summary

In summary we have that basic judgements in Aristotle’s logic, when some implicit assumptions are made explicit and the broad intention is retained, are naturally taken to be formalized in type theory as combinations of a type former and a judgement asserting a term of that type, as follows.

Aristotle's logicformal syntaxtype theory
conceptCCtype
judgementc:Cc \colon Ctyping judgement
All BB are AA.f:BAf \colon B \longrightarrow A function type
Some B 1B_1 is B 2B_2.s:B 1×AB 2s \colon B_1 \underset{A}{\times} B_2product type
Individual EE is BB.e:EBe \colon E \to B.unit type/global element
Deduction

The figure EBAE-B-A

Functions may be composed. Given b:EBb \colon E \to B and f:BAf \colon B \to A, then their composite is a function fe:EAf e \colon E \to A. In type theory this is an example of natural deduction (cut elimination), in words this is a syllogism

All BB are AA.

Individual EE is BB.

Hence

Individual EE is AA.

f :BA b :EB fb :EA \begin{aligned} f & \colon B \longrightarrow A \\ b & \colon E \longrightarrow B \\ f b & \colon E \longrightarrow A \end{aligned}

The figure BBAB-B-A

Analogously, the categorical semantics for

Some B 1B_1 is B 2B_2.

All B 2B_2 is AA.

Hence

Some B 1B_1 is AA.

(all in some context CC) is given by the horizontal composite in diagrams of the form

B 1×CB 2 B 2 A B 1 C. \array{ B_1 \underset{C}{\times} B_2 &\longrightarrow& B_2 &\longrightarrow& A \\ \downarrow && \downarrow & \swarrow \\ B_1 &\longrightarrow& C } \,.

The method (absolute Idea)

What is not present in such bare homotopy type theory is determination of further qualities of types. For instance for synthetically formalizing physics one needs that types have topological and moreover differential geometric qualities to them. Some externalizations of homotopy type theory exhibit these, others do not. We now consider adding axioms to homotopy type theory that narrow it in on those models that do exhibit further quality in addition to the pure being of types.

Moments

A general abstract way to express a kind of quality carried by types is to posit a projection operation \bigcirc that projects out the moment of pure such quality.

For instance for formalizing realistic physics one needs to determine bosonic and fermionic moments (we come to this below), and one way of doing so is by considering a projection operation that projects every space of fields to its purely bosonic body (lemma 1 below).

Generally, for XX a type, then X\bigcirc X is to be the result of projecting out some pure quality of XX. This being a projection means that XX\bigcirc X \simeq \bigcirc \bigcirc X. For this to be constructive, we need to specify a specific comparison map that gives this equivalence. Hence we say a moment projection is an operation \bigcirc on the type system together with natural functions XXX \to \bigcirc X such that (XX)\bigcirc(X \to \bigcirc X) is an equivalence X\bigcirc \stackrel{\simeq}{\longrightarrow} \bigcirc \bigcirc X.

In categorical semantics this means essentially that \bigcirc is an idempotent monad on the type system H\mathbf{H}.

Alternatively we may ask for a comparison map the other way around, XX\Box X \longrightarrow X, such that (XX)\Box(\Box X \longrightarrow X) is an equivalence. In categorical semantics this means essentially that \Box is an idempotent comonad.

Definition

A moment on (or in) a type system H\mathbf{H} is

or

Given a moment, we write

H ,H \mathbf{H}_{\bigcirc, \Box} \hookrightarrow \mathbf{H}

for the inclusion of its image, which we think of as the collection of those types that exhibit the moment purely (conversely these define the kind of moment as whatever quality it is that they all exhibit purely).

Remark

This is a language construct natural and familiar also from the point of view of computational trinitarianism, see at monad (in computer science).

Further, it makes sense to refer to moments ,\Box,\;\bigcirc also as modal operators or just modalities for short, and speak of type theory equipped with such operators as modal type theory, a type-theoretic refinement of modal logic. In this language the types in H \mathbf{H}_{\Box} are the \Box-_modal types_.

Remark

A moment \bigcirc or \Box may be thought of as encoding a concept of similarity: as the operator projects out some details of the quality of a type and only retains some pure moment, it coarse-grains the nature of a type to some extent. Hence two types XX and YY may be different but “similar with respect to \bigcirc-quality” if their images under \bigcirc are equivalent:

(Xsimilar Y)((X)=(Y)) (X \,similar_{\bigcirc}\, Y) \coloneqq (\bigcirc(X) \,= \, \bigcirc(Y))

(where on the right we have an identity type of the type universe).

Example of this are made explicit below as example 4, example 5.

Proposition

The category H \mathbf{H}_{\bigcirc} is equivalently the Eilenberg-Moore category of \bigcirc.

This is a standard fact in category theory, see at idempotent monad – Algebras.

Remark

Prop. 2 means that we may naturally make sense of “pure qualtity” also for (co-)monads that are not idempotent, the pure types should be taken to be the “algebras” over the monad.

A single moment \Box or \bigcirc may be interpreted as most anything, since it is not further determined yet. One now observes that there is an intrinsic, self-propelling way to further determine such abstract moments, by asking for their opposite and for their negative moments.

Opposites

Definition

(unity of opposites)

The opposite of a moment \bigcirc, def. 1 is, if it exists, another moment \Box in adjunction with it,

  1. either \bigcirc left adjoint to \Box and such that there is an adjoint triple

    H H H \mathbf{H}_{\Box}\simeq H_{\bigcirc} \stackrel{\hookrightarrow}{\stackrel{\longleftarrow}{\hookrightarrow}} \mathbf{H}

    which we denote by

    \Box \dashv \bigcirc
  2. or right adjoint to it with

    H H H \mathbf{H}_{\Box}\simeq \mathbf{H}_{\bigcirc} \stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}} \mathbf{H}

    which we denote by

    . \bigcirc \dashv \Box \,.

(We always display adjoint functors with the left adjoint on top and its right adjoint beneath.)

We say that the adjunction itself is the unity of opposites, and we indicate this by labels as in

momentunityofoppositesoppositemoment. \stackrel{moment}{} \Box \stackrel{unity \atop {of\,opposites}}{\dashv} \bigcirc \stackrel{opposite\;moment}{} \,.
Remark

In categorical semantics an opposition of moments, def. 2,

It is fairly familiar from the practice of category theory that adjunctions express oppositions. The following example is drawn from arithmetic and is meant to illustrate this in a familar context, but the actual examples that we will be concerned with are more fundamental.

Example

Consider the two inclusions even,odd:(,<)(,<)even, odd \colon (\mathbb{Z},\lt ) \hookrightarrow (\mathbb{Z},\lt) of the even and the odd integers, i.e. the maps n2nn \mapsto 2 n and n(2n+1)n \mapsto (2n+1), respectively.

oddeven \mathbb{Z} \stackrel{\overset{even}{\hookrightarrow}}{\stackrel{\longleftarrow}{\underset{odd}{\hookrightarrow}}} \mathbb{Z}

Both are adjoint to the operation of forming the floorfloor of the result of dividing by two, this is right adjoint to the inclusion of even numbers, and left adjoint to the inclusion of odd numbers.

even odd \array{ \stackrel{even}{} & \Box &\dashv& \bigcirc & \stackrel{odd}{} }

This has been considered in (Lawvere 00)

Example

Consider the inclusion ι:(,<)(,<)\iota \colon (\mathbb{Z}, \lt) \hookrightarrow (\mathbb{R}, \lt) of the integers into the real numbers, both regarded linear orders. This inclusion has a left adjoint given by ceilingceiling and a right adjoint civen by floorfloor. The composite CeilingιceilingCeiling \coloneqq \iota ceiling is an idempotent monad and the composite FloorιfloorFloor \coloneqq \iota floor is an idempotent comonad on \mathbb{R}. Both express a moment of integrality in an real number, but in opposite ways, each real number xx\in \mathbb{R} sits in between its floor and celling

Floor(x)<x<Ceiling(x). Floor(x) \lt x \lt Ceiling(x) \,.
ceiling floor. \array{ \stackrel{ceiling}{} &\bigcirc &\dashv& \Box & \stackrel{floor}{} } \,.

This example highlights that:

Remark

There is an opposition between the two kinds of opposition here:

  1. ()(\Box \dashv \bigcirc) – Here are two different opposite “pure moments” .

  2. ()(\bigcirc \dashv \Box) – Here is only one pure moment, but two opposite ways of projecting onto it.

Determinate negation

If X\Box X is a pure moment found inside XX, then it makes to ask for its complement moment or its negative

Definition

The negative of a comonadic moment \Box is what remains after taking away the piece of pure \Box-quality, hence is the cofiber of the counit map:

¯(X)cofib(XX). \overline{\Box}(X) \coloneqq cofib(\Box X \to X) \,.

The intuitive meaning suggests to ask whether this kind of negation of determinations is faithful in that there is no \Box-moment left in the negative ¯\overline{\Box}, hence whether

¯*. \Box \overline{\Box} \simeq \ast \,.

In general there is no reason for this to be the case. But if \Box also has an opposite in the sense of def. 2, then one of the two opposite moments is left adjoint, hence preserves cofibers, and then a little more may be said.

Consider the case of an opposition of the form \bigcirc \dashv \Box. In view of remark 5 then both \bigcirc and \Box express the same pure moment, just opposite ways of projecting onto it. Therefore in this situation it makes sense to alternatively ask that there is no \bigcirc-moment left in the ¯\overline{\Box}.

Definition

Given a unity of opposite moments \bigcirc \dashv \Box, def. 2, we say this has determinate negation if \Box and \bigcirc both restrict to 0-type and such that there

  1. **\bigcirc \ast \simeq \ast;

  2. \Box \longrightarrow \bigcirc is epi.

Proposition

For an opposition with determinate negation, def. 4, then on 0-types there is no \bigcirc-moment left in the negative of \Box-moment:

*. \bigcirc \Box \simeq \ast \,.
Proof

Given that \bigcirc, being a left adjoint, preserves colimits, hence cofibers, the first condition in def. 4 gives that

¯X=cofib(XX)cofib(XX). \bigcirc \overline{\Box} X = \bigcirc cofib(\Box X \to X) \simeq cofib(\Box X \to \bigcirc X) \,.

Now the second condition and the fact that epiness is preserved by pushout say that this result receives an epimorphism from the terminal object. But this forces it to be the terminal obect itself.

The proof of prop. 3 depends crucially on the restriction to 0-types. At the other extreme, on stable types the intuition that \bigcirc-moment is complementary to ¯\overline{\Box}-moment is verified in the following sense:

Proposition

For opposite moments of the form \bigcirc \dashv \Box, def. 2, then for stable types XX the hexagons

¯X ¯X ¯X X ¯X X X \array{ && \overline{\bigcirc} X && \longrightarrow && \overline{\Box} X \\ &\nearrow& &\searrow& & \nearrow && \searrow \\ \overline{\bigcirc} \Box X && && X && && \bigcirc \overline{\Box} X \\ &\searrow& &\nearrow & & \searrow && \nearrow \\ && \Box X && \longrightarrow && \bigcirc X }

are homotopy exact in that

  1. both squares are homotopy Cartesian, hence are fracture squares;

  2. the boundary sequences are long homotopy fiber sequences.

In particular every stable type is the fibered direct sum of its pure \bigcirc-moment and its pure ¯\overline{\Box}-moment:

X(X)¯X(¯X). X \simeq (\bigcirc X) \underset{\bigcirc \overline{\Box} X}{\oplus} (\overline{\Box} X) \,.

In this form this has been highlighted in (Bunke-Nikolaus-Völkl 13) in the context of our prop. 7 below. See at differential hexagon for the proof.

Accidence

Definition

For moments of the form ()(\bigcirc \dashv \Box), say that a type JJ is accidence for \bigcirc if \bigcirc is equivalently the localization at the terminal morphism J*J \to \ast.

This implies in particular that J*\bigcirc J \simeq \ast.

Progression

We have seen how to formalize determination of qualities of types together with their opposite and their negative determination. But so far these determinations are abstract in that when interpreting them in models they could come out as all kinds of very different-natured (co-)monads. What is missing is something that bases these determinations on a concrete ground with respect to which they would gain actual meaning.

Indeed, there are natural ways in which determinations of qualities may progress from given ones to further ones: on the one hand a given unity of oppositions may itself have an opposite and hence exhibit a higher order “opposition of oppositions”, on the other hand a given unity of oppositions may be “resolved” inside one new quality, which then in turn may have its own opposite and negative in turn, and so on.

Progression I: Higher order opposition

Given a concrete particular moment (i.e. an interpretation of the moment in categorical semantics), then adjoints to it are a property of the moment, not a choice to made. Abstractly we may specify that moments proceed to further moments this way by positing further adjoints.

Definition

(opposition of unities of oppositions)

Given one opposition \Box \dashv \bigcirc, we say that on opposition of oppositions is a further left adjoint \lozenge \dashv \Box, which we may think of as constituting a system of adjoint of this form:

unity2 oppositionofunities unity1 . \array{ \lozenge &\overset{unity\;2}{\dashv}& \Box \\ \bot &\stackrel{opposition \atop {of\;unities}}{}& \bot \\ \Box &\underset{unity\;1}{\dashv}& \bigcirc } \,.

In principle this may go on, but in models one finds that there are essentially no examples with a fourth adjoint that do not degenerate to the ambidextrous situation where \lozenge \simeq \bigcirc.

This shows further how oppositions serve to further determine moments: while a bare \Box-operator has all kinds of unrelated interpretations in models, asking it to be in opposition with a \bigcirc-moment considerably constrains its possible interpretations, further asking it to participate in an opposition-of-oppositions constraints it much more still, and asking for yet one more opposition tends to overconstrain it such as to degenerate.

Progression II: Resolution of oppositions

There is another way for a system of moments to proceed, not by adding further oppositions, but by resolving them.

Definition

(resolution of unity of opposites)

Given an essential subtopos \Box \dashv \bigcirc then one may ask if it sits inside a bigger essential subtopos, we write

2 2 1 1 \array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee && \vee \\ \Box_1 &\dashv& \bigcirc_1 }

to indicate that the image of 1\bigcirc_1 is contained in the image of 2\bigcirc_2, and we say that 2\bigcirc_2 is at a higher level or in a higher sphere than 1\bigcirc_1.

If in addition 1< 2\Box_1 \lt \bigcirc_2 then this means that the opposing moments of 1 1\Box_1 \dashv \bigcirc_1 both are of purely 2\bigcirc_2 nature, and hence we say that 2\bigcirc_2 resolves or lifts or sublates or is Aufhebung of this (unity of) oppostions. We might indicate this by:

2 2 1 1 \array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee && \vee \\ \Box_1 & &\dashv& \bigcirc_1 }

Dually there may be Aufhebung of the form

2 2 1 1 \array{ & \Box_2 &\dashv& \bigcirc_2 \\ & \vee && \vee \\ \Box_1 &\dashv& & \bigcirc_1 }

Notice that for oppositions of the other form, given

2 2 1 1 \array{ \bigcirc_2 &\dashv& \Box_2 \\ \vee && \vee \\ \bigcirc_1 &\dashv& \Box_1 }

then resolution in the sense that 2 1 1\bigcirc_2 \Box _1 \simeq \Box_1 and 2 1 1\Box_2 \bigcirc_1 \simeq \bigcirc_1 is automatic.

These two kinds of progression, higher order opposition, def. 6, and resolution of oppositions, def. 7, may alternate to produce processes of oppositions of moments and of their resolutions of the form

b b a a a a \array{ \vdots &\dashv& \vdots \\ \bot && \bot \\ \lozenge_b &\dashv& \Box_b \\ \vee && \vee \\ \lozenge_a &\dashv& \Box_a \\ \bot && \bot \\ \Box_a &\dashv& \bigcirc_a }

Objective Logic

By the above discussion, we are led to add to homotopy type theory the axiom that there are various moments \bigcirc or \Box. But which?

In the existing literature on modal logic it is tradition to consider unspecified idempotent (co-)monads to the formal system and have them acquire the intended meaning only via a specific choice of interpretation in a model. But here we are after developing genuine theory that works across all its possible interpretations, and hence we want an axiomatic determination of moments.

We observe now that there is a canonical starting point of two opposing moments that are secretly present in plain homotopy type theory. This hence constitutes a ground from which naturally a progression, as above, of determinations of further moments emanates.

The ground

The beginning

We observe now that in plain type theory already by itself carries one non-trivial unity of opposites.

There is the unit type *\ast. As a concept, this is the concept with a unique instance (up to equivalence). As such this may be thought of as the concept of “pure being”: an instance of this concept just purely is, without having any further qualities, and hence two instances have no distinctions between them, both just purely are, nothing else, and so they are indistinguishlably the same.

Every type has a unique map X*X\to \ast to that. Hence there is the monad which sends every type to *\ast and this is a moment

=* \bigcirc = \ast

according to def. 1.

Example

Every type is similar to every other, in the formal sense of def. 2, with respect to the *\ast-moment. This says that all things are similar at least in that they are at all (§906).

Dually there is the empty type \emptyset. As a concept, this is the concept with no instance. As such this may be thouhgt of as the concept of “not being”: since any instance of that concept would at least be (namely be an instance of the concept), but there is no such instance.

The empty type is such that it has a unique map X\emptyset \to X to any other type XX, hence the comonad which sends every type to the empty type, and this is a moment

=. \Box = \emptyset \,.

It is immediate that:

Proposition

In plain homotopy type theory there is a unity of opposites, def. 2,

*. \emptyset \dashv \ast \,.

We also call this the initial opposition.

Remark

It may be suggestive to think of this initial opposition in one of the following ways.

  1. The initial opposition of prop. 9 is (leaving context extension notationally implicit) the adjunction between dependent sum and dependent product over the context given by the empty type

    ()(). \underset{\emptyset}{\sum}(-) \;\vdash\; \underset{\emptyset}{\prod}(-) \,.
  2. The initial opposition of prop. 9 is the Cartesian productdahsv\dahsvinternal hom-adjunction of the empty type

    (()×)(()). \left( (-) \times \emptyset \right) \;\dashv\; \left( \emptyset \to (-) \right) \,.

On the other hand, the Cartesian productdahsv\dahsvinternal hom-adjunction of the unit type

(()×*)(*()) \left( (-) \times \ast \right) \;\dashv\; \left( \ast \to (-) \right)

is the identity moment, in opposition with itself:

idid. id \dashv \id \,.

This trivially resolves the initial opposition. Moreover, the negative, def. \ref, of idid is *\ast:

id¯=* \overline{id} = \ast

Also on pointed objects, where the negative of *\ast is defined, it is idid

So that we find

id id =*¯ * =id¯. \array{ id &\dashv& id & = \overline{\ast} \\ \vee && \vee \\ \emptyset &\dashv& \ast & = \overline{id} } \,.

See also at Setzende Reflexion

Notice for completeness that the negative \ref of \emptyset is the maybe monad. (This is however not a moment in the sense of def. 1 since it is not idempotent.)

Now we may find a progression of further moments by considering the resolution of this unity and then opposites to this resolution, and so forth.

Remark

Every essential subtopos level \Box \dashv \bigcirc contains the initial opposition of prop. 9 as the minimal one:

*. \array{ \Box &\dashv& \bigcirc \\ \vee && \vee \\ \emptyset &\dashv& \ast } \,.

We are to demand that this provides a resolution, def. 7 of the initial opposition *\empty \dashv \ast, prop. 9, in that

. \bigcirc \emptyset \simeq \emptyset \,.

In the categorical semantics this says equivalently that ()(\Box \dashv \bigcirc) is a dense subtopos.

Double negation

Proposition

The smallest dense subtopos of a topos is that of local types with respect to double negation loc ¬¬\sharp \coloneqq loc_{\neg \neg}.

(Johnstone 02, corollary A4.5.20).

Therefore we may add the demand that the resolution of (*)(\emptyset \dashv \ast) be by loc ¬loc_{\neq \neg} (Lawvere 91, p. 8, Shulman 15). This equivalently means to demand that the double negation subtopos is essential .

Thus we have found the first step in the process by demanding resolution of the initial opposition. We will denote this by

=loc ¬¬ * \array{ \flat &\dashv& \sharp & = loc_{\neg \neg} \\ \vee && \vee \\ \emptyset &&\dashv& \ast }
Proposition

The double negation subtopos is Boolean topos.

(Johnstone 02, lemma A4.5.21)

This means that ()(\flat \dashv \sharp) is naturally regarded as being the ground topos of the topos formed by the ambient type system, with the corresponding adjoint triple

H ΓH \mathbf{H}_{\sharp} \stackrel{\hookrightarrow}{\stackrel{\stackrel{\Gamma}{\longleftarrow}}{\hookrightarrow}} \mathbf{H}

regarded as the termimal geometric morphism whose direct image Γ\Gamma forms global points (aka global sections).

Therefore we label the resolution of the initial opposition as “ground” for “ground topos”(base topos).

=loc ¬¬ ground *. \array{ \flat &\dashv& \sharp & = loc_{\not\not} \\ \vee &\stackrel{ground}{}& \vee \\ \emptyset &\dashv& \ast } \,.
Cohesive substance
Quantity

This means then that \flat is the operation of taking global points and regarding the collection of them as equipped with discrete structure. Hence \flat is the moment of pure discreteness.

This in turn means that \sharp is the moment of pure continuity (co-discreteness).

discreteness =loc ¬¬ continuity ground * \array{ \stackrel{discreteness}{} & \flat &\dashv& \sharp &= loc_{\not\not}& \stackrel{continuity}{} \\ & \vee &\stackrel{ground}{}& \vee \\ & \emptyset &\dashv& \ast & }

We may hence also say that X\flat X is the “point content” of XX. If we regard the equivalence class of X\flat X then this is the cardinality of the point content of XX, the Größe of the point content, the discrete quantity of XX.

content discreteness =loc ¬¬ continuity ground * \array{ & && \flat && \stackrel{content}{} \\ & && \bot & \\ \stackrel{discreteness}{} & \flat &\dashv& \sharp & = loc_{\not \not}& \stackrel{continuity}{} \\ & \vee &\stackrel{ground}{}& \vee \\ & \emptyset &\dashv& \ast & }

The types XX that are fully determined by their moment of continuity are those for which XXX \to \sharp X is a monomorphism. In categorical semantics these are the concrete objects or equivalently the separated presheaves for \sharp: they are determined by their global points. These are the codomains of those functions which in thermodynamics one calls intensive quantities, functions whose value is genuinely given by their restriction to all possible points.

Contrary to that, objects which have purely the negative moment of continuity ¯\overline{\sharp} are codomains for “fuctions” which vanish on points and receive their contribution only from regions that extend beyond a single point. In thermodynamics these are called extensive quantities, (e.g. differential forms in positive degree). This concept of extension is precisely that which gave the name to Hermann Grassmann’s Ausdehnungslehre that introduced the concept of exterior differential form.

In summary, we have found that ()(\flat \dashv \sharp) expresses quantity, discrete quantity and continuous quantity, and that the latter is further subdivided into intensive and extensive quantity.

content discrete ¯/ quantity continuous(intensive/extensive) ground * \array{ & && \flat & && \stackrel{content}{} \\ & && \bot & && \\ \stackrel{discrete}{} & \overline{\flat}/\flat &\stackrel{quantity}{\dashv}& \sharp & & & \stackrel{continuous\; (intensive/extensive)}{} \\ & \vee &\stackrel{ground}{}& \vee \\ & \emptyset &\dashv& \ast & }
Quality

Proceeding, we next demand a second order opposition, def. 6, of the above opposition ()(\flat \dashv \sharp), hence we posit a moment ʃ&#643; such that

ʃ . \array{ &#643; &\dashv& \flat \\ \bot && \bot \\ \flat &\dashv& \sharp } \,.

We ask this to have definite negation, def. 4. This means that

  1. ʃ**&#643; \ast \simeq \ast — the shape of the point is trivial;

  2. ʃ\flat \to &#643; is epi on 0-types — the points-to-pieces transform is onto.

Together this are the axioms of cohesion as considered in (Lawvere 07). (There it is additionally asked that ʃ&#643; preserves binary Cartesian products.)

The intuition is that positing these qualites on a type system makes it, or rather its types XX, behave like a cohesive substance where points X\flat X are separate but held together by a cohesive attraction which, when the opposing repulsion is removed and only pure ʃ&#643;-moment is retained, makes them collapse to the components ʃX&#643; X.

In the more refined categorical semantics of homotopy toposes \flat modulates locally constant infinity-stacks. The above adjunction then expresses the central statement of higher Galois theory (dcct):

XGrpd ʃXGrpd \frac{X \to \flat Grpd_\infty}{&#643; X\to Grpd_\infty}

saying that locally constant \infty-stacks on XX are equivalent to infinity-permutation representations of ʃX&#643; X, and that ʃX&#643; X therefore is the fundamental infinity-groupoid of XX, the shape of XX, both in the intuitive as well as in the technical sense of algebraic topology.

Therefore we further add labels as follows.

shape ʃ content discrete ¯ / quantity =loc ¬¬ continuous(intensive/extensive) ground * \array{ \stackrel{shape}{} && & &#643; &\stackrel{}{\dashv}& \flat & & & \stackrel{content}{} \\ && & \bot &\stackrel{}{}& \bot \\ \stackrel{discrete}{} &\overline{\flat}& / & \flat &\stackrel{quantity}{\dashv}& \sharp & = loc_{\not\not} & & \stackrel{continuous\; (intensive/extensive)}{} \\ && & \vee &\stackrel{ground}{}& \vee \\ && & \emptyset &\dashv& \ast & }

This means that in the presence of the further opposition ʃ&#643; \dashv \flat the types XX which already had an underlying point content X\flat X now also have a shape determined by these points sticking together via a cohesive attraction. This is a qualitative aspect of the types in addition to their quantitative moments X\flat X and X\sharp X.

Example

By remark 2 the shape modality ʃ&#643; determines a concept of similarity of types. This is a well known one: ʃ encodes that two types have the same shape. XX and YY may look like different differential geometric spaces, but (Xsimilar ʃY)(X \, similar_{&#643;} Y) holds if they have the same shape.

In the standard model given by smooth infinity-groupoids, discussed in some detail around theorem 1 below there is for instance the circle S 1S^1 and the cylinder S 1×(0,1)S^1 \times (0,1) over it, both regarded as smooth manifolds in the standard way. As such they are not equal (not diffeomorphic), but clearly they are similar in some sense. The shape modality makes one such sense precise: ʃ(S 1)ʃ(S 1×(0,1)B&#643; (S^1) \simeq &#643; (S^1 \times (0,1) \simeq B \mathbb{Z} and hence

S 1similar ʃ(S 1×(0,1)). S^1 \, similar_{&#643;} \, (S^1 \times (0,1)) \,.

For instance there are now types for which X=*\flat X = \ast and yet they may be very different from the point *\ast themselves, hence while quanitatively these do not differ from the point, they must have some quality that distinguish them from the point. Hence this unity of opposites is geometric quality. In standard models this geometric quality is for instance topology or smooth structure or formal smooth structure or supergeometric structure.

Therefore we write:

shape ʃ quality content discrete ¯ / quantity =loc ¬¬ continuous(intensive/extensive) ground * \array{ \stackrel{shape}{} && & &#643; &\stackrel{quality}{\dashv}& \flat & & & \stackrel{content}{} \\ && & \bot &\stackrel{}{}& \bot \\ \stackrel{discrete}{} &\overline{\flat}& / & \flat &\stackrel{quantity}{\dashv}& \sharp & = loc_{\not\not} & & \stackrel{continuous\; (intensive/extensive)}{} \\ && & \vee &\stackrel{ground}{}& \vee \\ && & \emptyset &\dashv& \ast & }

Since with (ʃ)(&#643; \dashv \sharp) we have arrived at an opposition of the form (dash)(\bigcirc \dash \Box), we should ask for further determination of these qualities by demanding via def. 5 that ʃ&#643; has an accidence \mathbb{R}:

ʃ=loc . &#643; = loc_{\mathbb{R}} \,.

In view of the above interpretation of (ʃ)(&#643;\dashv \flat) via higher Galois theory, this comes with a clear meaning: this produces the A1-homotopy theory for 𝔸 1=\mathbb{A}^1 = \mathbb{R}. We may think of \mathbb{R} as being the continuum, i.e. the real line which is the model for the geometric paths that make ʃX&#643;X be the fundamental infinity-groupoid of XX.

Gauge (Measure)

With the concepts given by ()(\flat \dashv) and by (ʃ)(&#643; \dashv \flat) thus understood, it remains to find which concept the full unity of unities of opposites

ʃ \array{ &#643; &\dashv& \flat \\ \bot && \bot \\ \flat &\dashv& \sharp }

expresses.

Recall that the Brown representability theorem from stable homotopy theory:

Proposition

stable homotopy types EE are equivalently generalized cohomology theories E E^\bullet via

E (X)=[X,S]. E^\bullet(X) = [X,S] \,.
Proposition

For the moments (ʃ)(&#643; \dashv \flat) the exact hexagon of prop. 4

ʃ¯X ¯X ʃ¯X X ʃ¯X X ʃX \array{ && \overline{&#643;} X && \stackrel{}{\longrightarrow} && \overline{\flat} X \\ & \nearrow & & \searrow & & \nearrow_{} && \searrow \\ \overline{&#643;} \flat X && && X && && &#643; \overline{\flat} X \\ & \searrow & & \nearrow & & \searrow && \nearrow_{} \\ && \flat X && \longrightarrow && &#643; X }

exhibits cohesive stable homotopy types XX as differential generalized cohomology theories.

Moreover, the existence of \sharp means that the mapping stacks into these coefficients have differential concretification to moduli stacks of differential cocycles.

The first statement is the key insight in (Bunke-Nikolaus-Völkl 13). For more amplification of this point see at Differential cohomology is Cohesive homotopy theory.

Here the moments appearing in the hexagon have the following interpretation.

connectionformsontrivialbundles deRhamdifferential curvatureforms curvature deRhamtheorem flatdifferentialforms geometricbundleswithconnection rationalizedbundle topol.class Cherncharacter geometricbundleswithflatconnection comparison/regulatormap shapeofbundle \array{ && {{connection\;forms}\atop{on\;trivial\;bundles}} && \stackrel{de\;Rham\;differential}{\longrightarrow} && {{curvature}\atop{forms}} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}} \\ {{flat}\atop{differential\;forms}} && && {{geometric\;bundles}\atop{with\;connection}} && && {{rationalized}\atop{bundle}} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{geometric\;bundles}\atop{with\;flat\;connection}} && \underset{comparison/regulator\;map}{\longrightarrow} && {{shape}\atop{of\;bundle}} }

Now, cocycles in differential cohomology are the mathematical incarnation of physical fields in (stable) higher gauge theory (e.g. Freed 00). Hence the existence of the opposing moments ʃ&#643; \dashv \flat \dashv \sharp means that types carry gauge measure.

From the gauge theoretic perpective the \flat-moment is that exhibited by flat infinity-connections, its negative ¯\overline{\flat} moment is that exhibited by infinity-connections given by just differential form data. For ordinary differential cohomology, differential K-theory etc. this is the “rational” aspect.

Hence in summary we have found determinations as follows.

shape loc = ʃ quality / ¯ content(flat/rational) gaugemeasure discrete ¯ / quantity =loc ¬¬ continuous(intensive/extensive) ground * \array{ \stackrel{shape}{} && loc_{\mathbb{R}} = & &#643; &\stackrel{quality}{\dashv}& \flat & / & \overline{\flat}& \stackrel{content\;(flat/rational)}{} \\ && & \bot &\stackrel{gauge\;measure}{}& \bot \\ \stackrel{discrete}{} &\overline{\flat}& / & \flat &\stackrel{quantity}{\dashv}& \sharp & = loc_{\not\not} & & \stackrel{continuous\; (intensive/extensive)}{} \\ && & \vee &\stackrel{ground}{}& \vee \\ && & \emptyset &\dashv& \ast & }
Elastic substance
Étalé

Continuing the process, we posit a furrther opposition of moments lifting the previous ones.

& ʃ \array{ \Im &\dashv& \& \\ \vee && \vee \\ &#643; &\dashv& \flat }

Since this are oppositions of the form \bigcirc \dashv \Box, Aufhebung is automatic here and not a further axiom.

To see what these new moments mean, observe that now

XXʃX X \to \Im X \to &#643; X

is an factorization of the full shape projection through a finer approximation. Hence in addition to an intrinsic concept of path (a 1-morphism in the fundamental infinity-groupoid ʃX&#643; X) there is now an intrinsic concep of small path.

Accordingly, what were locally constant infinity-stacks in the higher Galois theory encoded by \flat now become coverings that are constant on small scales. This is the concept of étale morphism as being a formally étale morphism with a condition of smallness on its fibers.

Hence we find that this further determination is that of the moment of being étalé.

infinitsimalshape infinitesimalquality & étalé shape loc = ʃ quality / ¯ content(flat/rational) gaugemeasure discrete ¯ / quantity =loc ¬¬ continuous(intensive/extensive) ground * \array{ \stackrel{infinitsimal \atop shape}{} && & \Im &\stackrel{infinitesimal \atop quality}{\dashv}& \& & && \text{étalé} \\ && & \vee && \vee \\ \stackrel{shape}{} && loc_{\mathbb{R}} = & &#643; &\stackrel{quality}{\dashv}& \flat & / & \overline{\flat}& \stackrel{content\;(flat/rational)}{} \\ && & \bot &\stackrel{gauge\;measure}{}& \bot \\ \stackrel{discrete}{} &\overline{\flat}& / & \flat &\stackrel{quantity}{\dashv}& \sharp & = loc_{\not\not} & & \stackrel{continuous\; (intensive/extensive)}{} \\ && & \vee &\stackrel{ground}{}& \vee \\ && & \emptyset &\dashv& \ast & }
Infinitesimal

Proceed by positing a further opposition dashb\Re \dashb \Im.

To see what this moment means, observe that the “small shape” obtained above is representable by passing to pure \Re-moments

UXUX \frac{U \to \Im X}{\Re U \to X}

This has been understood in the 60s, in the context of crystalline cohomology, to be the characterization of paths that are so small that they are infinitesimal. The negative Re¯\overline{Re}-moment is that of infinitesimal objects, the pure \Re-moment is that of “reduced objects” (“real” objects), those without infinitesimal extension.

In summary this gives:

infinitesimal/reduced ¯ / infinitesimalshape infinitesimalquality & étalé shape loc = ʃ quality / ¯ content(flat/rational) gaugemeasure discrete ¯ / quantity =loc ¬¬ continuous(intensive/extensive) ground * \array{ \stackrel{infinitesimal/reduced}{} &\overline{\Re} & / & \Re &\dashv& \Im & & & \\ && & \bot && \bot \\ \stackrel{infinitesimal \atop shape}{} && & \Im &\stackrel{infinitesimal \atop quality}{\dashv}& \& & && \text{étalé} \\ && & \vee && \vee \\ \stackrel{shape}{} && loc_{\mathbb{R}} =& &#643; &\stackrel{quality}{\dashv}& \flat & / & \overline{\flat}& \stackrel{content\;(flat/rational)}{} \\ && & \bot &\stackrel{gauge\;measure}{}& \bot \\ \stackrel{discrete}{} &\overline{\flat}& / & \flat &\stackrel{quantity}{\dashv}& \sharp & = loc_{\not\not} & & \stackrel{continuous\; (intensive/extensive)}{} \\ && & \vee &\stackrel{ground}{}& \vee \\ && & \emptyset &\dashv& \ast & }

More in detail, we may ask just how small these small paths are. Hence we demand more generally an infinite tower

ʃ<= (0)< (1)< (2)< (3)<<id &#643; \lt \Im = \Im_{(0)} \lt \Im_{(1)} \lt \Im_{(2)} \lt \Im_{(3)} \lt \cdots \lt id

of infinitesimal shape modalities, yielding a further factorization of the shape unit as

X (3)X (2)X (1)XXʃX. X \to \cdots \to \Im_{(3)}X \to \Im_{(2)}X \to \Im_{(1)}X \to \Im X \to &#643; X \,.
Differential

In total, so far these are the axioms of differential cohesion (dcct). Using these one may naturally axiomatize local diffeomorphism (def. 13 below), jet bundles and related concepts.

Hence with these moments posited, types now now qualities of synthetic differential geometry. On top of just cohesively sticking to each other, the terms in the types now may feel a tighter differential connectedness, we have now a rigidly elastic substance.

Solid substance
Grading

Proceed to a new level of oppositions

R . \array{ \rightsquigarrow &\dashv& \R \\ \vee && \vee \\ \Re &\dashv& \Im } \,.

This gives for each type XX a factorization

XXX \Re X \longrightarrow \stackrel{\rightsquigarrow}{X} \longrightarrow X

of the comparison map of the reduction modality \Re.

This hence means now that the purely \rightsquigarrow-types are in between reduced type and unreduced types, hence they are reduced in some sense, but possibly not properly. Hence there are now two kinds of infinitesimals, and the \rightsquigarrow-types have no extension by infinitesimals of one kind, but possibly infinitesimal extension of the other kind is left.

Hence there is now a kind of grading on the infinitesimals and righsquigarrow\righsquigarrow quotients out everything not in degree 0.

The geometric quality of our formal substance that this encodes so far may hence be thought of as akin to Kapranov's noncommutative geometry, which is about ordinary spaces which however may have exotic noncommutative infinitesimal thickenings. We will find that the next two determinations in the progression of the moments refines this further to something of the quality of supergeometry, where the infinitesimal thickening satisfies some strong constraints.

Cyclic grading

The moments proceed by a further higher-oder opposition

R \array{ \rightrightarrows &\dashv& \rightsquigarrow \\ \bot && \bot \\ \rightsquigarrow &\dashv& \R }

For this to have non-degenerate models one finds that infinitesimals in degree 0 must be allowed to map to products of infinitesimals in non-vanishing degree. This means that the grading is not by a free group, but for instance by a finite cyclic group /n\mathbb{Z}/n\mathbb{Z}-grading. The minimal choice is /2\mathbb{Z}/2\mathbb{Z}-grading.

Super

We are to require that this level provides Aufhebung of the previous oppositions, def. 7, in that

. \rightsquigarrow \Im \simeq \Im \,.

By adjunction this implies that

. \Re \rightrightarrows \simeq \Re \,.

This says that the reduced part of the even-graded part is the same as the reduced part of the original, hence that odd-grade is removed by reduction, hence that odd-graded moment is nilpotent. This makes the graded-algebras be actually superalgebras.

By the Pauli exclusion principle/spin-statistics theorem, this is what characterizes fermions: the purely fermionic part is the negative moment ¯\overline{\rightsquigarrow}.

We indicate this notationally by

e¯ \e \coloneqq \overline{\rightsquigarrow}

We may determine this further. Since ()(\rightrightarrows \dashv \rightsquigarrow) is again an opposition of the form ()(\bigcirc \dashv \Box), we may ask for a type that is co-accidence for \rightsquigarrow according to def. 5. We write 0|1\mathbb{R}^{0|1} for this type

coloc 0|1. \rightsquigarrow \;\simeq\; coloc_{\mathbb{R}^{0|1}} \,.

In summary we now have arrived at the following process of determinations.

/ e bosonic/fermionic coloc 0|1= R super infinitesimal/reduced ¯ / infinitesimalshape infinitesimalquality & étalé shape loc = ʃ quality / ¯ content(flat/rational) gaugemeasure discrete ¯ / quantity =loc ¬¬ continuous(intensive/extensive) ground * \array{ && & \rightrightarrows &\dashv& \rightsquigarrow & / & \e & \stackrel{bosonic/fermionic}{} \\ && & \bot && \bot \\ && coloc_{\mathbb{R}^{0|1}}=& \rightsquigarrow &\dashv& \R \\ && & \vee &\stackrel{super}{}& \vee \\ \stackrel{infinitesimal/reduced}{} &\overline{\Re} & / & \Re &\stackrel{}{\dashv}& \Im & & & \\ && & \bot && \bot \\ \stackrel{infinitesimal \atop shape}{} && & \Im &\stackrel{infinitesimal \atop quality}{\dashv}& \& & && \text{étalé} \\ && & \vee && \vee \\ \stackrel{shape}{} && loc_{\mathbb{R}} =& &#643; &\stackrel{quality}{\dashv}& \flat & / & \overline{\flat}& \stackrel{content\;(flat/rational)}{} \\ && & \bot &\stackrel{gauge\;measure}{}& \bot \\ \stackrel{discrete}{} & \overline{\flat} & / & \flat &\stackrel{quantity}{\dashv}& \sharp & = loc_{\not\not} & & \stackrel{continuous\; (intensive/extensive)}{} \\ && & \vee &\stackrel{ground}{}& \vee \\ && & \emptyset &\dashv& \ast & }

Prop. 4 here gives a decomposition of types into their purely even-graded part and their purely fermionic part

e(X) X e(X) X. \array{ && && && \stackrel{}{\e(X)} \\ & & & & & \nearrow_{} && \searrow \\ && && X && && \stackrel{\rightrightarrows}{\e(X)} \\ & & & & & \searrow && \nearrow_{} \\ && && && \stackrel{\rightrightarrows}{X} } \,.

A substance subject to the Pauli exclusion principle given by the above super-grading is yet more rigid than just by the elasticity we had before: it exhibits solidity.

\,

We conclude the process at this point. One may explore it further by continuing it with further resolutions and further oppositions, but for the applications to physics that we consider below the three stages beyond the ground that we have so far turn out to be sufficient.

Objectivity

It is a familiar tought in our age, in view of the intimate relation between physics and mathematics, that theories of physics have a natural mathematical formulation, that it is compelling to consider them also just from within mathematics itself. Famous examples include the formalization of classical mechanics by symplectic geometry, the formalization of Einstein gravity by pseudo-Riemannian geometry/Cartan geometry, the close relation of quantum mechanics and quantum field theory to representation theory (Wigner classification) and more recently the identification of local topological field theory with the theory of symmetric monoidal (infinity,n)-categories.

Hence while mathematics is part of the subjective logic in that it admits the freedom to consider any mathematical structure whatsoever, this suggests to identify among these the “objective” mathematical structures which are theories of physics and as such express a more objective reality than random mathematical structures do.

Classical mechanics (Mechanism)

Process

Given a logic of concepts as above, with its basic constructs of judgements of the form f:XYf \colon X \to Y, among the most natural structures to consider are correspondences, which go from a type X 1X_1 to a type X 2X_2 via an intermediate type YY by maps

Y i o X 1 X 2. \array{ && Y \\ & {}^{\mathllap{i}}\swarrow && \searrow^{\mathrlap{o}} \\ X_1 && && X_2 } \,.

This is the immediate generalization of a relation as we pass from homotopy 0-types to general homotopy types and thereby allow monomorphisms to be replaced by general maps.

Now one observe that a correspondence is naturally interpreted as a process :

every instance/term ycoloYy\colo Y may be thought of as a process under which i(y):X 1i(y) \colon X_1 turns into o(y):X 2o(y) \colon X_2.

In traditional mathematical physics this is familiar from the concept of Lagrangian correspondences which serves to encoce much of classical mechanics.

Physical law

Or rather, classical mechanics is encoded by prequantized Lagrangian correspondences, the prequantization expressing the prequantum bundle, an action functional and hence the laws of motion.

By the discussion there, a prequantized Lagrangian correspondence is itself again just a correspondence, but now in context, hence between dependent types, namely depending on a type of phases.

A detailed discussion of how classical field theory is formalized via correspondences in cohesive homotopy type theory in the context of a type of phases is ar

From the dicussion there one finds a picture of sliced correspondences interpreted as classical mechanics as follows.

spaceoftrajectories initialvalues Hamiltonianevolution phasespaceincoming actionfunctional phasespaceoutgoing prequantumbundle in prequantumbundle out 2groupofphases. \array{ && {{space\,of} \atop {trajectories}} \\ & {}^{\mathllap{{initial}\atop {values}}}\swarrow && \searrow^{\mathrlap{{Hamiltonian} \atop {evolution}}} \\ {{phase\,space} \atop {incoming}} && \swArrow_{{action} \atop {functional}} && {{phase\,space} \atop {outgoing}} \\ & {}_{\mathllap{{prequantum}\atop {bundle}_{in}}}\searrow && \swarrow_{\mathrlap{{prequantum} \atop {bundle}_{out}}} \\ && {{2-group} \atop {of\,phases}} } \,.
Quantum mechanics ((quantum-)Chemism)

Recall from remark 6 that the initial opposition gave rise also to the maybe monad, as the negative of the empty moment: ¯=maybe\overline{\empty} = maybe.

The negative of id\id is *\ast.

The opposite of *\ast is \empty.

The negative of \empty is maybemaybe.

While maybemaybe is not idempotent, by remark 3 we may still ask for the types which are pure with respect to it in that they they are objects in its Eilenberg-Moore category. These are precisely the pointed types.

On pointed types the smash product yields a symmetric monoidal structure which is not Cartesian, and we enter the realm of linear type theory in the generality of dependent linear type theory. As discussed there, dependent sum and dependent product here now naturally yield the concept of secondary integral transforms, across correspondences, which in view of the above interpretation of correspondences as spaces of trajectories are really path integrals. Developing this one finds that correspondences in linear homotopy type theory give rise to formalization of quantization and quantum mechanics.

For details see at Quantization via Linear homotopy types.

Boundary conditions (Teleology)

In this context a boundary condition is given by a (prequantized) correspondence which on one end is just the unit type

Y b o * X. \array{ && Y \\ & {}^{\mathllap{b}}\swarrow && \searrow^{\mathrlap{o}} \\ \ast && && X } \,.

For more on this see at

The idea

Including in homotopy type theory the progression of modal operators that we have found above

id id / e bosonic/fermionic solidity coloc 0|1= R rheonomic super infinitesimal/reduced ¯ / elasticity infinitesimalshape infinitesimalquality & étalé shape loc = ʃ quality / ¯ content(flat/rational) cohesion gaugemeasure discrete ¯ / quantity =loc ¬¬ continuous(intensive/extensive) ground * \array{ &&& && & id &\dashv& id \\ &&& && & \vee && \vee \\ &&& && & \rightrightarrows &\dashv& \rightsquigarrow & / & \e & \stackrel{bosonic/fermionic}{} \\ \stackrel{solidity}{} &&& && & \bot && \bot \\ &&& && coloc_{\mathbb{R}^{0|1}} = & \rightsquigarrow &\dashv& \R && & \stackrel{rheonomic}{} \\ &&& && & \vee &\stackrel{super}{}& \vee \\ &&& \stackrel{infinitesimal/reduced}{} &\overline{\Re} & / & \Re &\dashv& \Im & & & \\ \stackrel{elasticity}{} &&& && & \bot && \bot \\ &&& \stackrel{infinitesimal \atop shape}{} && & \Im &\stackrel{infinitesimal \atop quality}{\dashv}& \& & && \text{étalé} \\ &&& && & \vee && \vee \\ &&& \stackrel{shape}{} && loc_{\mathbb{R}} = &&#643; &\stackrel{quality}{\dashv}& \flat & / & \overline{\flat}& \stackrel{content\;(flat/rational)}{} \\ \stackrel{cohesion}{} &&& && & \bot &\stackrel{gauge\;measure}{}& \bot \\ &&& \stackrel{discrete}{} &\overline{\flat} & / & \flat &\stackrel{quantity}{\dashv}& \sharp & = loc_{\not\not} & & \stackrel{continuous\; (intensive/extensive)}{} \\ &&& && & \vee &\stackrel{ground}{}& \vee \\ &&& && & \emptyset &\dashv& \ast & }

makes its term model richer: there are now true propositions and generally terms that may be constructed which are not constructible in plain homotopy type theory. These terms reflect the idea that is induced by these determinations, in that every interpretation of this modal type theory has to realize (externalize) these terms and make these propositions true.

We now indicate some of these new constructions.

Maurer-Cartan forms

Let GG be a an ∞-group type. This means that there is specified a pointed connected type BG\mathbf{B}G and an equivalence GΩBGG\simeq \Omega \mathbf{B}G with its loop space object. We say that BG\mathbf{B}G is the delooping of GG. Notice that all this happens internal to the ambient cohesive homotopy type theory, which makes BG\mathbf{B}G have the interpretation of the moduli ∞-stack of cohesive GG-principal ∞-bundles, instead of just the bare homotopy type of the classifying space

BGʃBG. B G \simeq &#643; \mathbf{B}G \,.

This richer geometric structure is what the boldface in BG\mathbf{B}G is meant to remind us of.

Definition

Denote the first and second homotopy fiber of the comparison map B𝔾B𝔾\flat \mathbf{B}\mathbb{G} \to \mathbf{B}\mathbb{G} of the flat moment of this as follows.

Gθ G dRBGBGBG. G \stackrel{\theta_G}{\longrightarrow} \flat_{dR}\mathbf{B}G \longrightarrow \flat \mathbf{B}G \longrightarrow \mathbf{B}G \,.

This double homotopy fiber θ G\theta_G has the interpretation of being the Maurer-Cartan form on GG.

Differential cohomology

Let 𝔾\mathbb{G} be an abelian ∞-group type. The group of phases.

This being abelian just means that there is specified a delooping type B𝔾\mathbf{B} \mathbb{G} and an equivalence 𝔾ΩB𝔾\mathbb{G}\simeq \Omega \mathbf{B} \mathbb{G} with its loop space object, and that with B 0𝔾𝔾\mathbf{B}^0 \mathbb{G} \coloneqq \mathbb{G} we have inductively that B n𝔾\mathbf{B}^n \mathbb{G} is itself equipped with the structure of an abelian ∞-group.

For the present purpose we will assume in addition that 𝔾\mathbb{G} is 0-truncated, which makes it simply an abelian group.

Definition

A Hodge filtration is a compatible system of filtrations of B 2𝔾\flat \mathbf{B}^2\mathbb{G} of the form

Ω cl 2 dRB 2𝔾B 2𝔾. \mathbf{\Omega}^{2}_{cl} \to \cdots \to \flat_{dR} \mathbf{B}^2 \mathbb{G} \to \flat \mathbf{B}^2 \mathbb{G} \,.

with 0-truncated extensive Ω cl 2\mathbf{\Omega}^{2}_{cl}.

Definition

Given a Hodge filtration, write B𝔾 conn\mathbf{B}\mathbb{G}_{conn} for the homotopy fiber product

B𝔾 connB𝔾× dRB 2𝔾Ω cl 2 \mathbf{B}\mathbb{G}_{conn} \coloneqq \mathbf{B}\mathbb{G}\underset{\flat_{dR}\mathbf{B}^2\mathbb{G}}{\times} \mathbf{\Omega}^2_{cl}

of the Maurer-Cartan form θ B𝔾\theta_{\mathbf{B}\mathbb{G}} with the last Holdge filtration stage.

Proposition

The decomposition of B𝔾 conn\mathbf{B}\mathbb{G}_{conn} into its (ʃ,¯)(&#643;, \overline{\flat})-moments according to prop 4 reproduces the defining Cartesian sqare of def. 11:

Ω cl 2 B𝔾 conn dRB 2𝔾 B𝔾 dRB𝔾 B𝔾 \array{ && \mathbf{\Omega}^2_{cl} \\ & \nearrow && \searrow \\ \mathbf{B}\mathbb{G}_{conn} && && \flat_{dR}\mathbf{B}^2\mathbb{G} \\ & \searrow && \nearrow && \searrow \\ && \mathbf{B}\mathbb{G} && && \flat_{dR}B \mathbb{G} \\ && & \searrow && \nearrow \\ && && B \mathbb{G} }
WZW terms

A map

c:BGB p+2𝔾 \mathbf{c} \colon \mathbf{B}G \longrightarrow \mathbf{B}^{p+2} \mathbb{G}

is equivalently a cocycle of degree p+2p+2 in the group cohomology of GG.

Definition

Given a group cocycle c\mathbf{c} and a Hodge filtration, then a refinement of the Hodge filtration along the group cocycle is a chose of 0-trucated extensive Ω flat 1(,𝔤)\mathbf{\Omega}^1_{flat}(-,\mathfrak{g}) fitting into a square

Ω flat 1(,𝔤) μ Ω cl 2 dRBG dRc dRB 2𝔾. \array{ \mathbf{\Omega}^1_{flat}(-,\mathfrak{g}) &\stackrel{\mu}{\longrightarrow}& \mathbf{\Omega}^2_{cl} \\ \downarrow && \downarrow \\ \flat_{dR} \mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}& \flat_{dR} \mathbf{B}^2 \mathbb{G} } \,.

Given this, write

G˜G× dRBGΩ flat 1(,𝔤) \tilde G \coloneqq G \underset{\flat_{dR}\mathbf{B}G}{\times} \mathbf{\Omega}^1_{flat}(-,\mathfrak{g})
Example

For GG 0-truncated, then the canonical choice is Ω flat 1(,𝔤)= dRBG\mathbf{\Omega}^1_{flat}(-,\mathfrak{g}) = \flat_{dR}\mathbf{B}G. With this one has G˜G\tilde G \simeq G.

On the other extreme, for G=B p+1𝔾G = \mathbf{B}^{p+1}\mathbb{G} then the canonical choice is Ω flat 1(,𝔤)=Ω cl p+2\mathbf{\Omega}^1_{flat}(-,\mathfrak{g}) = \mathbf{\Omega}^{p+2}_{cl}. With this one has G˜B p+1𝔾 conn\tilde G \simeq \mathbf{B}^{p+1}\mathbb{G}_{conn}.

This means that in general G˜\tilde G is a homotopy fiber product of GG with B p+1𝔾 conn\mathbf{B}^{p+1}\mathbb{G}_{conn}, hence that a map to out of some Σ\Sigma is a pair of a map ΣG\Sigma \to G and of (p+1)(p+1)-form data on Σ\Sigma. This is the kind of field content of higher gauged WZW models.

Proposition

Given a group cocycle c:BGB p+2\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+2} and a form refinement μ\mu as in def. 12, then there exists an essentially unique prequantization

L WZW:G˜B conn p+1 \mathbf{L}_{WZW} \;\colon\; \tilde G \longrightarrow \mathbf{B}^{p+1}_{conn}

of μ(θ G)\mu(\theta_G) whose underlying B p𝔾\mathbf{B}^p\mathbb{G}-principal ∞-bundle is \Omega \\mathbf{c}.

We call this the WZW term whose curvature is μ(θ G)\Omega \\\mu(\theta_G).

VV-Manifolds

See also at geometry of physics -- manifolds and orbifolds.

Definition

Given X,YHX,Y\in \mathbf{H} then a morphism f:XYf \;\colon\; X\longrightarrow Y is a local diffeomorphism if its naturality square of the infinitesimal shape modality

X X f f Y Y \array{ X &\longrightarrow& \Im X \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}} \\ Y &\longrightarrow& \Im Y }

is a pullback square.

Remark

The abstract definition 13 comes down to being the appropriate synthetic differential supergeometry-version of the traditional statement that ff is a local diffeomorphism if the diagram of tangent bundles

TX X Tf f TY Y \array{ T X &\longrightarrow& X \\ \downarrow^{\mathrlap{T f}} && \downarrow^{\mathrlap{f}} \\ T Y &\longrightarrow& Y }

To see this, notice by the discussion at synthetic differential geometry that for DD an infinitesimally thickened point, then for any XHX \in \mathbf{H} the mapping space [D,X][D,X] is the jet bundle of XX with jets of order as encoded by the infinitesimal order of DD. In particular if 𝔻 1(1)\mathbb{D}^1(1) is the first order infinitesimal interval defined by the fact that its algebra of functions is the algebra of dual numbers C (𝔻 1(1))=(ϵ)/(ϵ 2)C^\infty(\mathbb{D}^1(1)) = (\mathbb{R} \oplus \epsilon \mathbb{R})/(\epsilon^2), and XX is a smooth manifold, then

[𝔻 1(1),X]TX [\mathbb{D}^1(1), X]\simeq T X

is the ordinary tangent bundle of XX. Now use that the internal hom [D,][D,-] preserves limits in its second argument, and that, by the hom-adjunction, H(U,[D,X])H(U×D,X)\mathbf{H}(U, [D,X]) \simeq \mathbf{H}(U \times D, X) and finally use that H(U×D,X)H((U×D),X)H(U,X)\mathbf{H}(U \times D, \Im X)\simeq \mathbf{H}(\Re(U \times D), X)\simeq \mathbf{H}(U,X).

Let now VHV \in \mathbf{H} be given, equipped with the structure of a group (infinity-group).

Definition

A VV-manifold is an XHX \in \mathbf{H} such that there exists a VV-atlas, namely a correspondence of the form

U V X \array{ && U \\ & \swarrow && \searrow \\ V && && X }

with both morphisms being local diffeomorphisms, def. 13, and the right one in addition being an epimorphism, hence an atlas.

Proposition

If f:XYf \;\colon\; X \longrightarrow Y is a local diffeomorphism, def. 13, then so is image f:XY\stackrel{\rightsquigarrow}{f}\colon \stackrel{\rightsquigarrow}{X} \longrightarrow \stackrel{\rightsquigarrow}{Y} under the bosonic modality.

Proof

Since the bosonic modality provides Aufhebung for \Re\dashv \Im we have \rightsquigarrow \Im \simeq \Im. Moreover \Im \rightsquigarrow \simeq \Im anyway. Finally \rightsquigarrow preserves pullbacks (being in particular a right adjoint). Hence hitting a pullback diagram

X X f f Y Y \array{ X &\longrightarrow& \Im X \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}} \\ Y &\longrightarrow& \Im Y }

with \rightsquigarrow\;\; yields a pullback diagram

X X f f Y Y \array{ \stackrel{\rightsquigarrow}{X} &\longrightarrow& \Im \stackrel{\rightsquigarrow}{X} \\ \downarrow^{\mathrlap{\stackrel{\rightsquigarrow}{f}}} && \downarrow^{\mathrlap{\Im \stackrel{\rightsquigarrow}{f}}} \\ \stackrel{\rightsquigarrow}{Y} &\longrightarrow& \Im \stackrel{\rightsquigarrow}{Y} }
Corollary

The bosonic space X\stackrel{\rightsquigarrow}{X} underlying a VV-manifold XX, def. 14, is a V\stackrel{\rightsquigarrow}{V}-manifold

Frame bundles
Definition

Given XHX \in \mathbf{H}, its infinitesimal disk bundle T infXXT_{inf} X\to X is the pullback of the unit of the infinitesimal shape modality along itself

T infX X X X. \array{ T_{inf} X &\stackrel{}{\longrightarrow}& X \\ \downarrow && \downarrow \\ X &\longrightarrow& \Im X } \,.

Given a point x:*Xx \;\colon\; \ast \to X, then the infinitesimal neighbourhood *𝔻 xX\ast \to \mathbb{D}_x \to X of that point is the further pullback of the infinitesimal disk bundle to this point:

𝔻 x T infX X * x X X. \array{ \mathbb{D}_x &\longrightarrow & T_{inf} X &\stackrel{}{\longrightarrow}& X \\ \downarrow && \downarrow && \downarrow \\ \ast &\stackrel{x}{\longrightarrow} & X &\longrightarrow& \Im X } \,.

More generally, for kk \in \mathbb{N} then the kkth order infinitesimal disk bundle is

T (k)X (k)X X X \array{ T_{(k)} X &\stackrel{}{\longrightarrow}& \Im_{(k)} X \\ \downarrow && \downarrow \\ X &\longrightarrow& \Im X }

and accordigly the kkth order infinitsimal neighbourhood is

𝔻(k) x T (k)X (k)X * x X X. \array{ \mathbb{D}(k)_x &\longrightarrow & T_{(k)} X &\stackrel{}{\longrightarrow}& \Im_{(k)}X \\ \downarrow && \downarrow && \downarrow \\ \ast &\stackrel{x}{\longrightarrow} & X &\longrightarrow& \Im X } \,.

It is natural not to pick any point, but to collect all infinitesimal disks around all the points of a space:

Definition

The relative flat modality is the operation rel\flat^{rel} that sends XHX \in \mathbf{H} to the homotopy pullback

rel X X X. \array{ \flat^{rel} &\longrightarrow& X \\ \downarrow && \downarrow \\ \flat X &\longrightarrow& \Im X } \,.

More generally, for any kk \in \mathbb{N} then the order kk relative flat modality is the pullback in

(k) rel (k)X X X. \array{ \flat^{rel}_{(k)} &\longrightarrow& \Im_{(k)} X \\ \downarrow && \downarrow \\ \flat X &\longrightarrow& \Im X } \,.
Definition

The general linear group GL(V)GL(V) is the automorphism infinity-group of the infinitesimal neighbourhood 𝔻 e V\mathbb{D}^V_e, def. 15, of the neutral element e:*𝔻 e VVe \colon \ast \to \mathbb{D}^V_e \to V:

GL(V)Aut(𝔻 e V). GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V_e) \,.
Proposition

For XX a VV-manifold, def. 14, then its infinitesimal disk bundle T infXXT_{inf} X \to X, def. 15, is associated to a GL(V)GL(V)-principal Fr(X)XFr(X) \to X – to be called the frame bundle, modulated by a map to be called τ X\tau_X, producing homotopy pullbacks of the form

T infX V/GL(V) X τ X BGL(V)Fr(X) * X τ X BGL(V). \array{ T_{inf} X &\longrightarrow& V/GL(V) \\ \downarrow && \downarrow \\ X &\stackrel{\tau_X}{\longrightarrow}& \mathbf{B} GL(V) } \;\;\; \array{ Fr(X) &\longrightarrow& \ast \\ \downarrow && \downarrow \\ X &\stackrel{\tau_X}{\longrightarrow}& \mathbf{B} GL(V) } \,.
Definition

A framing of a VV-manifold is a trivialization of its frame bundle, prop. 11, hence a diagram in H\mathbf{H} of the form

X * BGL(V) \array{ X && \longrightarrow && \ast \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}GL(V) }
Remark

It is useful to express def. 18 in terms of the slice topos H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)}. Write VFrameH /BGL(V)V\mathbf{Frame}\in \mathbf{H}_{/\mathbf{B}GL(V)} for the canonical morphism *BGL(V)\ast \to \mathbf{B}GL(V) regarded as an object in the slice. Then a framing as in def. 18 is equivalently a morphism

ϕ:τ XVFrame \phi \colon \tau_X \longrightarrow V\mathbf{Frame}

in H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)}.

Proposition

The group object VV, canonically regarded as a VV-manifold, carries a canonical framing, def. 18, ϕ li\phi_{li}, induced by left translation.

GG-Structure

See also at geometry of physics -- G-structure and Cartan geometry.

Definition

Given a homomorphism of groups GGL(V)G \longrightarrow GL(V), a G-structure on a VV-manifold XX is a lift c\mathbf{c} of the frame bundle τ X\tau_X of prop. 11 through this map

X G τ X BGL(V). \array{ X && \stackrel{}{\longrightarrow} && G \\ & {}_{\mathllap{\tau_X}}\searrow &\swArrow& \swarrow \\ && \mathbf{B}GL(V) } \,.
Remark

As in remark 9, it is useful to express def. 19 in terms of the slice topos H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)}. Write GStrucH /BGL(V)G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(V)} for the given map BGBGL(V)\mathbf{B}G\to \mathbf{B}GL(V) regarded as an object in the slice. Then a GG-structure according to def. 19 is equivalently a choice of morphism in H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)} of the form

c:τ XGStruc. \mathbf{c} \;\colon\; \tau_X \longrightarrow G\mathbf{Struc} \,.

In other words, GStrucH /BGL(v)G\mathbf{Struc} \in \mathbf{H}_{/\mathbf{B}GL(v)} is the moduli stack for GG-structures.

Example

A choice of framing ϕ\phi, def. 18, on a VV-manifold XX induces a G-structure for any GG, given by the pasting diagram in H\mathbf{H}

X * BGL(V) \array{ X &\longrightarrow& \ast &\longrightarrow& \\ & \searrow & \downarrow & \swarrow \\ && \mathbf{B}GL(V) }

or equivalently, via remark 9 and remark 10, given as the composition

c li:τ XϕVFrameGStruc. \mathbf{c}_{li} \;\colon\; \tau_X \stackrel{\phi}{\longrightarrow} V\mathbf{Frame} \longrightarrow G\mathbf{Struc}\,.

We call this the left invariant GG-structure.

Definition

For XX a VV-manifold, then a G-structure on XX, def. 19, is integrable if for any VV-atlas VUXV \leftarrow U \rightarrow X the pullback of the GG-structure on XX to VV is equivalent there to the left-inavariant GG-structure on VV of example 7, i.e. if we have an correspondence in the double slice topos (H /BGL(V)) /GStruc(\mathbf{H}_{/\mathbf{B}GL(V)})_{/G\mathbf{Struc}} of the form

τ U τ V τ X c li c GStruc. \array{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && \swArrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} } \,.

The GG-structure is infintesimally integrable if this holds true at at after restriction along the relative shape modality relUU\flat^{rel} U \to U, def. 16, to all the infinitesimal disks in UU:

τ relU τ V τ X c li c GStruc. \array{ && \tau_{\flat^{rel}U} \\ & \swarrow && \searrow \\ \tau_V && \swArrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} } \,.

Finally, the GG-structure is order kk infinitesimally integrable if this holds for the order-kk relative shape modality (k) rel\flat^{rel}_{(k)}.

Definition

Consider an infinity-action of GL(V)GL(V) on VV which linearizes to the canonical GL(V)GL(V)-action on 𝔻 e V\mathbb{D}^V_e by def. 17. Form the semidirect product GL(V)VGL(V) \rtimes V. Consider any group homomorphism GGL(V)G\to GL(V).

A (GGV)(G\to G\rtimes V)-Cartan geometry is a VV-manifold XX equipped with a GG-structure, def. 19. The Cartan geometry is called (infinitesimally) integrable if the GG-structure is so, according to def. 20.

Remark

For VV an abelian group, then in traditional contexts the infinitesimal integrability of def. 20 comes down to the torsion of a G-structure vanishing. But for VV a nonabelian group, this definition instead enforces that the torsion is on each infinitesimal disk the intrinsic left-invariant torsion of VV itself.

Traditionally this is rarely considered, matching the fact that ordinary vector spaces, regarded as translation groups VV, are abelian groups. But super vector spaces regarded (in suitable dimension) as super translation groups are nonabelian groups. Therefore super-vector spaces VV may carry intrinsic torsion, and therefore first-order integrable GG-structures on VV-manifolds are torsion-ful.

Indeed, this is a phenomenon known as the torsion constraints in supergravity. Curiously, as discussed there, for the case of 11-dimensional supergravity the equations of motion of the gravity theory are equivalent to the super-Cartan geometry satisfying this torsion constraint. This way super-Cartan geometry gives a direct general abstract route right into the heart of M-theory.

Definite forms
Definition

Given a group cocycle c:BGB p+2𝔾\mathbf{c} \colon \mathbf{B}G\to\mathbf{B}^{p+2}\mathbb{G} with WZW term, prop. 9, of the form

L WZW V:VB p+1𝔾 \mathbf{L}_{WZW}^V \colon V \longrightarrow \mathbf{B}^{p+1}\mathbb{G}

and given a VV-manifold XX we say that an integrable globalization of L WZW V\mathbf{L}_{WZW}^V over XX is a WZW on on XX

L WZW X:XB p+1𝔾 conn \mathbf{L}_{WZW}^X \;\colon\;X \longrightarrow \mathbf{B}^{p+1}\mathbb{G}_{conn}

such that there is a VV-atlas for XX

U V X \array{ && U \\ & \swarrow && \searrow \\ V && && X }

which extends to a correspondence between L WZW\mathbf{L}_{WZW} and L WZW X\mathbf{L}_{WZW}^X

U V X L WZW V L WZW X B p+1𝔾 conn. \array{ && U \\ & \swarrow && \searrow \\ V && \swArrow && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}^V}}\searrow && \swarrow_{\mathrlap{\mathbf{L}_{WZW}^X}} \\ && \mathbf{B}^{p+1}\mathbb{G}_{conn} } \,.

Accordingly, as in def. 20 we say that L WZW X\mathbf{L}_{WZW}^X is an infinitesimally integrable globalization if this correspondence exists after restriction along the inclusion relUU\flat^{rel} U \to U of the infinitesimal disks in XX and such that

  1. the induced section of the associated [𝔻(1) V,B p+1𝔾 conn][\mathbb{D}(1)^V,\mathbf{B}^{p+1}\mathbb{G}_{conn}]-fiber infinity-bundle is definite on the restriction L WZW 𝔻\mathbf{L}_{WZW}^{\mathbb{D}} of L WZW V\mathbf{L}_{WZW}^V to the infinitesimal disk;

  2. also the underlying cocycle is definite, in that the infinitesimal disk bundle lifts to an 𝔻 V(1)\mathbb{D}^V(1)-gerbe (for the induced group structure on 𝔻 V(1)\mathbb{D}^V(1)).

If B p+1𝔾 conn\mathbf{B}^{p+1}\mathbb{G}_{conn} had no higher gauge transformations, then this would already ensure that such a globalization globalizes L WZW V\mathbf{L}_{WZW}^V locally cohesively, but here in higher differential geometry this property becomes genuine structure and hence we need to demand it. There is an axiomatic way to say this (see dcct for details) and if this is imposed then we say that L WZW X\mathbf{L}_{WZW}^X is a definite globalization of L WZW V\mathbf{L}_{WZW}^V.

Proposition

There is a canonical (∞,1)-functor from (infinitesimally integrable) definite globalizations of L WZW X\mathbf{L}_{WZW}^X over a VV-manifold XX to (infinitesimally integrable) GG-structures on XX, def. 19, for

G=intens(Stab GL(V)(L WZW 𝔻 V)) G = intens(\mathbf{Stab}_{GL(V)}(\mathbf{L}_{WZW}^{\mathbb{D}^V}))

the intensification (in the sense above) of the stabilizer ∞-group of the restriction of L WZW V\mathbf{L}_{WZW}^V along the inclusion of the typical infinitesimal disk 𝔻 VV\mathbb{D}^V \to V.

Nature

Given a theory of physics, made sufficiently precise in formal logic, then an interpretation of the theory by a model “is” nature as predicted by this theory.

For instance if we considered Einstein gravity to be the theory of pseudo-Riemannian manifolds subject to some energy condition, then a model for this theory is one concrete particular spacetime.

Above we saw that cohesive (elastic) homotopy type theory contains Cartan geometry, hence in particular pseudo-Riemannian geometry in its idea, as well as gauge theory and hence we accordingly find models of nature here.

Recall specifically that

  1. From prop. 9 we have that group cocycles c:BVB p+2𝔾\mathbf{c}\colon \mathbf{B}V \longrightarrow \mathbf{B}^{p+2}\mathbb{G} of degree p+2p+2 induce WZW terms in that degree and hence the WZW sigma model prequantum field theory on the worldvolume of a p-brane propagating on the “model spacetimeVV.

  2. A second cocycle on the infinity-group extension classified by c\mathbf{c} yields a type of p˜\tilde p-brane on which these pp-branes may end;

  3. This structure is naturally generalized to VV-manifolds XX equipped with definite globalizations of these WZW terms, defining pp-branes propagating on XX.

  4. The definite globalization of the WZW term L WZW\mathbf{L}_{\mathrm{WZW}} induces a Stab(L WZW)Stab(\mathbf{L}_{WZW}) structure on XX and the requirement that this be infinitesimally integrable is a torsion constraint on XX.

We now find an externalization of the idea such that

  1. There is a canonical bouquet of higher group cocycles and their ∞-group extensions emanating from the unique 0-truncated purely fermionic type – the superpoint.

  2. The resulting branes and their intersection laws are those seen in string theory;

  3. The resulting spacetimes are superspacetimes as in the relevant supergravity theories;

  4. The resulting torsion constraints, namely the supergravity torsion constraints, imply, in the maximally extended situation, the Einstein equations of motion of 11-dimensional supergravity, specifically of d=4 N=1 supergravity arising in the guise of M-theory on G2-manifolds.

This is a “theory of everything” in the sense of modern fundamental physics, which is beeing argued to have viable phenomenology, see at G2-MSSM for more on this. Even if it turns out that there are no models in this theory which match quantitative measurements in experiment, it is noteworthy that the qualitative structure of this theory is that of Einstein-Yang-Mills-Dirac-Higgs theory and hence matches faithfully the qualitative features of nature that is in experiment. Given our starting point above this is maybe not to be lightly dismissed.

Externalization
Theorem

The cohesive+elastic+solid homotopy type theory above has a faithful (i.e. non-degenerate) categorical semantics in the homotopy topos SuperFormalSmoothGrpdSuperFormalSmooth\infty Grpd of super formal smooth infinity-groupoids.

We now spell out the construction of this model and indicate the proof of this statement.

Definition

Write

There are then “semidirect product” sites CartSpInfinPointCartSp \rtimes InfinPoint and CartSpSuperPointCartSp \rtimes SuperPoint (whose objects are Cartesian products of the given form inside synthetic differential supergeometry and whose morphisms are all morphisms in that context (not just the product morphisms)).

Set then

SmoothGrpdSh (CartSp) Smooth \infty Grpd \coloneqq Sh_\infty(CartSp)

for the collection of smooth ∞-groupoids;

FormalSmoothGrpdSh (CartSpInfPoint) FormalSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes InfPoint)

for the collection of formal smooth ∞-groupoids (see there) and finally

SuperFormalSmoothGrpdSh (CartSpSuperPoint) SuperFormalSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint)

for that of super formal smooth ∞-groupoids.

Proposition

The sites in question are alternatingly (co-)reflective subcategories of each other (we always display left adjoints above their right adjoints)

*CartSpCartSpInfPointCartSpSuperPoint. \ast \stackrel{\longleftarrow}{\hookrightarrow} CartSp \stackrel{\hookrightarrow}{\longleftarrow} CartSp\rtimes InfPoint \stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}} CartSp \rtimes SuperPoint \,.

Here

Passing to (∞,1)-categories of (∞,1)-sheaves, this yields, via (∞,1)-Kan extension, a sequence of adjoint quadruples as follows:

Δ: Grpd SmoothGrpd FormalSmoothGrpd SuperFormalSmoothGrpd \array{ & && && &\longleftarrow& \\ & && &\hookrightarrow& &\hookrightarrow& \\ & &\longleftarrow& &\longleftarrow& &\longleftarrow& \\ \Delta \colon & \infty Grpd &\hookrightarrow& Smooth \infty Grpd &\hookrightarrow& FormalSmooth \infty Grpd &\hookrightarrow& SuperFormalSmooth \infty Grpd \\ & &\longleftarrow& &\longleftarrow& \\ & &\hookrightarrow& }

the total composite labeled Δ\Delta is indeed the locally constant infinity-stack-functor.

Forming adjoint triples from these adjoint quadruples gives idempotent (co-)monads

ʃ* &#643; \dashv \flat \dashv \ast
& \Re \dashv \Im \dashv \&
\rightrightarrows \dashv \rightsquigarrow \dashv \Re

satisfying the required inclusions of their images.

Proof

All the sites are ∞-cohesive sites, which gives that we have an cohesive (infinity,1)-topos. The composite inclusion on the right is an ∞-cohesive neighbourhood site, whence the inclusion SmoothGpdSuperFormalSmoothGrpdSmooth\infty Gpd\hookrightarrow SuperFormalSmooth\infty Grpd exhibits differential cohesion.

With this the rightmost adjoint quadruple gives the Aufhebung of \Re \dashv \Im by R\rightsquigarrow \dashv \R and the further opposition \rightrightarrows \dashv \rightsquigarrow.

Remark

The model in def. 23 admits also the refinement of the infinitesimal shape modality to an infinite tower

ʃ<= (0)< (1)< (2)< (3)< &#643; \lt \Im = \Im_{(0)} \lt \Im_{(1)} \lt \Im_{(2)} \lt \Im_{(3)} \lt \cdots

characterizing kkth order infinitesimals. Let

*=InfPoint (0)InfPoint (1)InfPoint (2)InfPoint (3)InfPoint \ast = InfPoint_{(0)} \hookrightarrow InfPoint_{(1)} \hookrightarrow InfPoint_{(2)} \hookrightarrow InfPoint_{(3)} \hookrightarrow \cdots InfPoint

be the stratification of InfPointInfPoint by its full subcategories on those objects whose coresponding Weil algebras/local Artin algebras are of the form V\mathbb{R} \oplus V with V k=0V^k = 0. Each of these inclusions has coreflection, given by projection onto the quotient by the ideal V kV^k, as kk ranges

Proposition

The model in def. 23 verifies the required determinate negations

  1. determinate negations I:

    • ʃ**&#643; \ast \simeq \ast;

    • ʃ\flat \to &#643; is epi restricted to 0-types;

  2. determinate negations II:

    • ʃloc &#643; \simeq loc_{\mathbb{R}} for SmoothMfsSuperFormalSmoothGrpd\mathbb{R} \in SmoothMfs \hookrightarrow SuperFormalSmooth\infty Grpd the ordinary real line;

    • coloc 0|1\rightsquigarrow \simeq coloc_{\mathbb{R}^{0|1}} for 0|1SuperMfdsSuperFormalSmoothGrpd\mathbb{R}^{0|1} \in SuperMfds \hookrightarrow SuperFormalSmooth\infty Grpd the ordinary odd line.

Proof

The first three items follow with the discussion at ∞-cohesive site.

Proposition

The model in def. 23 verfies the required Aufhebungen

  1. \sharp \emptyset \simeq \emptyset;

  2. \rightsquigarrow \Im \simeq \Im.

Proof

For the statement \sharp \emptyset \simeq \emptyset consider the following:

Since the site SS of HSuperFormalSmoothGrpds\mathbf{H} \coloneqq SuperFormalSmooth\infty Grpds has a terminal object *\ast, it follows that for XHX\in \mathbf{H} any sheaf X:𝒮 opSetX \colon \mathcal{S}^{op}\to Set then

XX(*) \flat X \simeq X(\ast)

(where we may leave the constant re-embedding implicit, due to it being fully faithful).

Moreover, for every object U𝒮U\in \mathcal{S} there exists a morphism i:*Ui \colon \ast \to U hence for every XHX\in \mathbf{H} and every UU there exists a morphism i *:X(U)Xi^\ast \colon X(U)\to \flat X. This means that if X\flat X \simeq \emptyset then X(U)X(U) \simeq \emptyset for all U𝒮U \in \mathcal{S} and hence XX\simeq \emptyset.

We now show that this condition is equivalent to the required Aufhebung:

Generally, given a topos equipped with a level of a topos given by an adjoint modality ()()(\Box\dashv \bigcirc) \coloneqq (\flat \dashv \sharp), then the condition \sharp \emptyset \simeq \emptyset is equivalent to (X)(X)(\flat X \simeq \emptyset) \Leftrightarrow (X \simeq \emptyset).

Because: in a topos the initial object \emptyset is a strict initial object, and hence (X)(X)(X \simeq \emptyset) \simeq (X \to \emptyset). Therefore in one direction, assuming \sharp \emptyset \simeq \emptyset then

(X) (X) (X) (X) (X). \begin{aligned} (X \simeq \emptyset) & \simeq (X \to \emptyset) \\ & \simeq (X \to \sharp \emptyset) \\ & \simeq (\flat X \to \emptyset) \\ & \simeq (\flat X \simeq \emptyset) \end{aligned} \,.

Conversely, assuming that (X)(X)(\flat X \simeq \emptyset) \Leftrightarrow (X \simeq \emptyset), then for all XX

(X) (X) (X) (X) (X) \begin{aligned} (X\to \emptyset) & \simeq (X\simeq \emptyset) \\ & \simeq (\flat X \simeq \emptyset) \\ & \simeq (\flat X \to \emptyset) \\ & \simeq (X\to \sharp \emptyset) \end{aligned}

and hence by the Yoneda lemma \emptyset \simeq \sharp \emptyset.

\,

Second, for the statement \rightsquigarrow \Im \simeq \Im consider the following:

For any XHX \in \mathbf{H} and any U×D sCartSpSuperInfPointHU \times D_s\in CartSp \rtimes SuperInfPoint \hookrightarrow \mathbf{H} we have by adjunction natural equivalences

H(U×D s,X) H(U×D s,X) H((U×D s),X) H(U,X) H((U×D s),X) H(U×D s,X). \begin{aligned} \mathbf{H}(U \times D_s , \stackrel{\rightsquigarrow}{\Im X}) & \simeq \mathbf{H}(\stackrel{\rightrightarrows}{U \times D_s} , \Im X) \\ &\simeq \mathbf{H}(\Re(\stackrel{\rightrightarrows}{U \times D_s}) , X) \\ & \simeq \mathbf{H}(U, X) \\ & \simeq \mathbf{H}(\Re(U \times D_s), X) \\ & \simeq \mathbf{H}(U \times D_s, \Im X) \end{aligned} \,.

Here the crucial step is the observation that on representables, by construction, the reduced part of the even part is the reduced part of the original object.

But observe that

Proposition

While, due to prop. 14, in the model of def. 23

  • the opposition ʃ&#643; \dashv \sharp has determinate negation in the sense of def. 4;

on the other hand

  • the opposition \rightrightarrows \dashv \rightsquigarrow does not have definite negation in the sense of def. 4.
Proof

The definition would require that

0|2 0|2 \stackrel{\rightsquigarrow}{\mathbb{R}^{0|2}} \longrightarrow \stackrel{\rightrightarrows}{\mathbb{R}^{0|2}}

is an epimorphism. But this is equivalent to the point inclusion

*𝔻(1) \ast \longrightarrow \mathbb{D}(1)

into the formal dual of the algebra of dual numbers.

Space-Time-Matter

We discuss now how in the externalization of the theory given by theorem 1 there naturally appears spacetime from the idea.

The progressive system of moments above, yields, by prop. 14, two god-given objects:

real linesuperpoint
= 1|0\mathbb{R} = \mathbb{R}^{1\vert 0} 0|1\mathbb{R}^{0\vert 1}
ʃloc &#643; \simeq loc_{\mathbb{R}}coloc 0|1\rightsquigarrow \simeq coloc_{\mathbb{R}^{0\vert 1}}

Both have familiar structure of an abelian group object, \mathbb{R} being the additive group, hence there are arbitrary deloopings B n 0|1\mathbf{B}^n \mathbb{R}^{0|1} and B n\mathbf{B}^{n}\mathbb{R}.

Given two types, there are the judgements in which these appear as subject and as predicate, in the sense discussed above.

There are no non-trivial judgements with (a delooping of) \mathbb{R} as the subject and (a delooping of) 0|1\mathbb{R}^{0|1} as the predicate. But there turn out to be some exceptional judgements with subject 0|q\mathbb{R}^{0|q} and predicate B d\mathbf{B}^d \mathbb{R}.

By example 1 this leads to the deduction of the object which is the homotopy fibers of the corresponding maps. From these one obtains further judgements, then further objects, and so forth. This way a “bouquet” of objects is induced from the initial ones.

We now discuss how this bouquet first of all yields super Minkowski spacetime as an extension of the superpoint PN§256b and then further the extended super Minkowski spacetimes arising from super p-brane condensates (FSS).

Minkowski spacetime

Consider first the superpoint 0|1\mathbb{R}^{0|1}.

Remark

This is the unique 0-truncated object which is

  1. purely negative to bosonic moment;

  2. purely opposite to bosonic moment;

in that

e( 0|1) 0|1 \e(\mathbb{R}^{0|1})\simeq \mathbb{R}^{0|1}
0|1*. \stackrel{\rightrightarrows}{\mathbb{R}^{0|1}} \simeq \ast \,.

Since 0|1\mathbb{R}^{0|1} (and the other objects obtained in a moment) are contractible as super Lie groups, we may conveniently discuss them as super Lie algebras. Regarding 0|1\mathbb{R}^{0\vert 1} as a super Lie algebra, then its Chevalley-Eilenberg algebra is freely generated from a (1,odd)(1,odd)-bigraded element dθd\theta

CE( 0|1)=( dθ,d CE=0). CE(\mathbb{R}^{0\vert 1}) = \left( \wedge^\bullet \langle d\theta \rangle, d_{CE} = 0 \right) \,.

It is evident that

Proposition

The second super Lie algebra cohomology of 0|1\mathbb{R}^{0\vert 1} is

H 2( 0|1,)= H^{2}(\mathbb{R}^{0\vert 1}, \mathbb{R}) = \mathbb{R}

represented by the 2-cocycles of the form

dθdθCE( 0|1). d\theta\wedge d\theta \in CE(\mathbb{R}^{0\vert 1}) \,.

The Lie algebra extension

1|1 0|1 B 1 \array{ \mathbb{R}^{1|\mathbf{1}} \\ \downarrow \\ \mathbb{R}^{0|1} &\stackrel{}{\longrightarrow}& \mathbf{B}^1 \mathbb{R} }

classified by this is the super translation group in 1-dimension.

This is the worldline of the superparticle.

There are no further non-trivial cocycles here giving further extensions.

Hence next consider the Cartesian product of the initial superpoint with itself.

0|2= 0|1× 0|1. \mathbb{R}^{0|2} = \mathbb{R}^{0|1}\times \mathbb{R}^{0|1} \,.
Remark

This is still purely of negative bosonic moment in that e( 0|2) 0|2e(\mathbb{R}^{0|2}) \simeq \mathbb{R}^{0|2}, but it is no longer has purely no moment opposed to bosonic moment (witnessing that the fermionic opposition is not complete, lemma 16), instead

0|2𝔻(1) \stackrel{\rightrightarrows}{\mathbb{R}^{0|2}} \simeq \mathbb{D}(1)

is the first-order infinitesimal interval (the formal dual of the “algebra of dual numbers”).

Proposition

The second super Lie algebra cohomology of 0|2\mathbb{R}^{0\vert 2} is

H 2( 0|2,) 3 H^2(\mathbb{R}^{0\vert 2}, \mathbb{R}) \simeq \mathbb{R}^3

represented by the cocycles of the form

a 11dθ 1dθ 1+a 22dθ 2dθ 2+a 12dθ 1dθ 2. a_{11} \, d\theta_1 \wedge d\theta_1 + a_{22} \, d\theta_2 \wedge d\theta_2 + a_{12} \, d\theta_1 \wedge d\theta_2 \,.

The extension classified by this

2,1|2 0|2 B 3 \array{ \mathbb{R}^{2,1|\mathbf{2}} \\ \downarrow \\ \mathbb{R}^{0|2} &\stackrel{}{\longrightarrow}& \mathbf{B} \mathbb{R}^3 }

is 3-dimensional super Minkowski spacetime.

Proof

This follows by inspection of the real spin representations in dimension 3, see the details spelled out at spin representation – via division algebras – Example d=3).

Now the old brane scan gives:

Proposition
H 3( 2,1|2)= H^3(\mathbb{R}^{2,1\vert \mathbf{2}}) = \mathbb{R}

represented by the 3-cocycle which, as a left invariant super differential form on 2,1|2\mathbb{R}^{2,1\vert \mathbf{2}} is the WZW term in the Green-Schwarz action functional for the super 1-brane in 3d.

𝔰𝔱𝔯𝔦𝔫𝔤 hetonG 2 2,1|2 B 2 \array{ \mathfrak{string}_{het \, on \, G_2} \\ \downarrow \\ \mathbb{R}^{2,1\vert \mathbf{2}} &\stackrel{}{\longrightarrow}& \mathbf{B}^2 \mathbb{R} }

A definite globalization, of this 3-cocycle over a 3|2\mathbb{R}^{3\vert \mathbf{2}}-manifold requires, by def. 22, that the tangent bundle is a bundle of super Lie algebras and that the cocycle extends to a definite form. This imposes G-structure for GG the Lorentz group (or rather its spin group double cover).

Proposition

The joint stabilizer of GL( 2,1|2)GL(\mathbb{R}^{2,1\vert 2}) of the Lie bracket and the 3-cocycle is the spin group Spin(2,1)Spin(2,1), the double cover of the Lorentz group SO(2,1)SO(2,1).

There are two inequivalent ways to embed Spin(2,1)GL( 2,1|2)Spin(2,1) \to GL(\mathbb{R}^{2,1\vert 2}). We write 2,1|2\mathbb{R}^{2,1\vert \mathbf{2}} and 2,1|2¯\mathbb{R}^{2,1\vert \overline{\mathbf{2}}} to distinuish 3d super Minkowski spacetime equipped with these two actions.

This is one special case of a more general statement which we come to as prop. 22 below.

Consider then 2,1|4\mathbb{R}^{2,1\vert 4}

2,1|4 0|2 0|2 B 3. \array{ && \mathbb{R}^{2,1\vert 4} \\ & \swarrow && \searrow \\ \mathbb{R}^{0 \vert 2} && && \mathbb{R}^{0\vert 2} \\ & \searrow && \swarrow \\ && \mathbf{B} \mathbb{R}^3 } \,.
Proposition

There is a 1-dimensional space of Spin(2,1)Spin(2,1)-invariant 2-cocycles on 2,1|2+2¯\mathbb{R}^{2,1\vert \mathbf{2} + \overline{\mathbf{2}}}. The Lie algebra extension classified by that is 4d super Minkowski spacetime

3,1|4 2,1|2+2¯ B \array{ \mathbb{R}^{3,1\vert 4} \\ \downarrow \\ \mathbb{R}^{2,1\vert \mathbf{2}+\overline{\mathbf{2}}} &\longrightarrow& \mathbf{B}\mathbb{R} }
Proof

By inspection of the real spin representations in dimension 4.

Now the old brane scan gives:

Proposition
H 4( 3,1|4)= H^4(\mathbb{R}^{3,1\vert \mathbf{4}}) = \mathbb{R}

represented by the 4-cocycle which, as a left invariant super differential form on 3,1|2\mathbb{R}^{3,1\vert \mathbf{2}} is the WZW term in the Green-Schwarz action functional for the super 2-brane in 4d.

𝔪2𝔟𝔯𝔞𝔫𝔢 onG 2 3,1|4 B 3. \array{ \mathfrak{m}2\mathfrak{brane}_{on\,G_2} \\ \downarrow \\ \mathbb{R}^{3,1\vert \mathbf{4} } &\stackrel{}{\longrightarrow}& \mathbf{B}^3 \mathbb{R} } \,.
Lorentz symmetry

Notice that so far we have obtained 3-dimensional and 4-dimensional Minkowski spacetime and the WZW-term for the superstring and the membrane propagating on it without assuming knowledge of the Lorentz group. In fact we assumed nothing but the presence of the real line \mathbb{R} and the odd line 0|1\mathbb{R}^{0|1} and we have simply investigated their cohomology.

The following proposition shows that the Lorentz group, in fact its universal cover by the pseudo-Riemannian spin group is deduced from this.

Proposition

Let d1,1,N\mathbb{R}^{d-1,1,N} be super Minkowski spacetime in dimension d{3,4,6,10}d \in \{3,4,6,10\} and let ϕΩ 3( d1,1|N)\phi \in \Omega^{3}(\mathbb{R}^{d-1,1|N}) the corresponding 3-form characterizing the super-1-brane (superstring) in this dimension, according to the brane scan . Then the stabilizer subgroup of both the super Lie bracket and the cocycle is the Spin group Spin(d1,1)Spin(d-1,1):

Stab GL( d1,1|N)([,],ϕ)Spin(d1,1)GL( d1,1|N). Stab_{GL(\mathbb{R}^{d-1,1|N})}([-,-], \phi) \simeq Spin(d-1,1) \hookrightarrow GL(\mathbb{R}^{d-1,1|N}) \,.
Proof

It is clear that the spin group fixes the cocycle, and by the discussion at spin representation it preserves the bracket. Therefore it remains to be seen that the Spin group already exhausts the stabiizer group of bracket and cocycle. For that observe that the 3-cocycle is

(ψ,ϕ,v)η([ψ,ϕ],v), (\psi,\phi, v) \mapsto \eta( [\psi,\phi], v ) \,,

where η(,)\eta(-,-) is the given Minkowski metric, and that the bilinear map

[,]:SSV [-,-]\colon S \otimes S\to V

is surjective. This imples that if gGL( d1,1|N)g \in GL(\mathbb{R}^{d-1,1|N}) preserves both the bracket and the cocycle for all ψ,ϕS\psi, \phi \in S and vVv \in V to

η([g(ψ),g(ϕ)],g(v))=η(g([ψ,ϕ]),g(v))=η([ψ,ϕ],v) \eta( [g(\psi),g(\phi)], g(v) ) = \eta( g([\psi,\phi]), g(v) ) = \eta( [\psi,\phi], v )

then it preserves the Minkowski metric for all w,vw,v

η(g(w),g(v))=η(w,v). \eta(g(w), g(v)) = \eta(w,v) \,.

This means that 2,1|2\mathbb{R}^{2,1|2}-manifolds XX equipped with the 3-cocycle as a definite form such that the resulting G-structure according to prop. 13 also preserves the the group structure on 2,1|2\mathbb{R}^{2,1|2}, then this is equivalent to equipping XX with Lorentzian orthogonal structure, hence with super-pseudo-Riemannian metric, hence with a field-configuration for 3d supergravity.

Fundamental branes

The brane bouquet that we find…

𝔪2𝔟𝔯𝔞𝔫𝔢 onG 2 3,1|4 2,1|2¯ 𝔰𝔱𝔯𝔦𝔫𝔤 IIAonG 2 2,1|2+2¯ 3 0|2+2¯ 2,1|2 0|2 0|1 0|1 \array{ && \mathfrak{m}2\mathfrak{brane}_{on\,G_2} \\ && \downarrow \\ && \mathbb{R}^{3,1|\mathbf{4}} && \mathbb{R}^{2,1|\overline{\mathbf{2}}} \\ && \downarrow & \nearrow& & \searrow \\ \mathfrak{string}_{IIA\,on\,G_2} &\longrightarrow& \mathbb{R}^{2,1|\mathbf{2}+\overline{\mathbf{2}}} && && \mathbb{R}^{3} \\ && \downarrow &\searrow& & \nearrow \\ && \mathbb{R}^{0|\mathbf{2}+\overline{\mathbf{2}}} && \mathbb{R}^{2,1|\mathbf{2}} \\ && && \downarrow \\ && && \mathbb{R}^{0|\mathbf{2}} \\ && & \swarrow && \searrow \\ && \mathbb{R}^{0|1} && && \mathbb{R}^{0|1} }

this is equivalently the physics coming from M-theory on G2-manifolds, given by the extensions that emanate from 32 copies of the smallest superpoint:

𝔪5𝔟𝔯𝔞𝔫𝔢 𝔪2𝔟𝔯𝔞𝔫𝔢 p𝔡2p𝔟𝔯𝔞𝔫𝔢 10,1|32 𝔰𝔱𝔯𝔦𝔫𝔤 IIA 9,1|16+16¯ 0|16+16¯ 0|1 0|1 \array{ && && \mathfrak{m}5\mathfrak{brane} \\ && && \downarrow \\ && && \mathfrak{m}2\mathfrak{brane} \\ && && \downarrow \\ && \underset{p}{\coprod} \mathfrak{d}{2p}\mathfrak{brane} && \mathbb{R}^{10,1|\mathbf{32}} \\ && \downarrow && \downarrow \\ && \mathfrak{string}_{IIA} &\longrightarrow & \mathbb{R}^{9,1|\mathbf{16}+\overline{\mathbf{16}}} \\ && && \downarrow \\ && && \mathbb{R}^{0|\mathbf{16}+\overline{\mathbf{16}}} \\ && & \swarrow && \searrow \\ && \mathbb{R}^{0|1} && \cdots && \mathbb{R}^{0|1} }

These are branches of The brane bouquet of string theory, see there for more. By prop. 9 each branch here gives the WZW form for the corresponing Green-Schwarz super p-brane sigma model.

Gravity

Above we have found two interlocking ingredients arising from the axiomatics:

  1. abstract generals – Given any group object VV, then there is an abstract general concept of VV-manifolds XX, def. 14. Given furthermore a WZW term L WZW V\mathbf{L}_{WZW}^V on VV, then there is an abstract general concept of definite globalizations of this term over these manifolds XX, 22 inducing G-structures on XX, prop. 13.

  2. concrete individuals – We have found concrete individual VVs: extended super Minkowski spacetimes, prop. 18, prop. 20 emanating from the objects which represent the moments ʃ&#643; and \rightrightarrows, and we have further found individual L WZW V\mathbf{L}_{WZW}^V: the super p-brane WZW terms, prop. 21 etc., forming The brane bouquet.

Plugging the concrete individuals into the general abstract theory, we hence obtain particular phenomena.

(1) relU d1,1|N X L WZW brane L WZW X B p+2𝔾 conn \array{ && \flat^{rel}_{(1)} U \\ & \swarrow && \searrow \\ \mathbb{R}^{d-1,1|\mathbf{N}} && \swArrow && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}^{brane}}}\searrow && \swarrow_{\mathrlap{\mathbf{L}_{WZW}^X}} \\ && \mathbf{B}^{p+2}\mathbb{G}_{conn} }

Specifically there is 11-dimensional super Minkowski spacetime V= 10,1|32V = \mathbb{R}^{10,1\vert \mathbf{32}} carrying the WZW term L WZW M2\mathbf{L}_{WZW}^{M2} for the M2-brane, in some sense the endpoint of the bouquet of super-spacetimes. The KK-compactification of this on a 7-dimensional G2-manifold yields the 4-dimensional super-Minkowski spacetime discussed above, with the WZW term for the super 2-brane in 4d.

The 11-dimensional super-Minkowski spacetime is special in many ways, one of which is that in this dimension the equations of motion of 11-dimensional supergravity on a (Spin(10,1)Iso( 10,1|32))(Spin(10,1)\hookrightarrow Iso(\mathbb{R}^{10,1\vert \mathbf{32}}))-super Cartan geometry XX modeled on 10,1|32\mathbb{R}^{10,1\vert \mathbf{32}} are already captured by just a constraint on the torsion tensor. But by remark 11 this means that in dimension 11 the equations of motion of supergravity have an immediate axiomatization in our objective logic.

equivalent to just the condition that the of XX is at each point and to first infinitesimal order the intrinsic torsion of 10,1|32\mathbb{R}^{10,1\vert \mathbf{32}}

Proposition

First-order integrable (Spin(10,1)Iso( 10,1|32))(Spin(10,1)\hookrightarrow Iso(\mathbb{R}^{10,1\vert \mathbf{32}}))-super-Cartan geometries, def. 21, on 10,1|32\mathbb{R}^{10,1\vert\mathbf{32}}-manifolds XX, def. 14, which are first-order integrable with respect to the intrinsic left-invariant torsion of 10,1|32\mathbb{R}^{10,1\vert \mathbf{32}}, remark 11, are equivalent to vacuum solutions to the equations of motion of 11-dimensional supergravity, i.e. to solutions for which the field strength of the gravitino and of the supergravity C-field vanishes identically, hence to solutions to the ordinary vacuum Einstein equations in 11d.

Proof

(Howe 97) shows that imposing (on some chart) dE a+ω a bE bψ¯Γ aψ=0\mathbf{d} E^a + \omega^{a}{}_b \wedge E^b - \bar \psi \Gamma^a \psi = 0 implies (and hence is equivalent to) the equations of motion of 11d supergravity. These equations (see e.g. D’Auria-Fré 82, p. 31) then show that furthermore requiring dψ+12ω abΓ abψ=0\mathbf{d} \psi + \tfrac{1}{2}\omega_{a b} \Gamma^{a b}\psi = 0 (and hence requiring the full supertorsion tensor to be that of super-Minkowski spacetime) puts the field strength of the gravitino and of the supergravity C-field to 0.

Remark

Vacuum Einstein solutions as in prop. 23, are considered notably in the context of M-theory on G2-manifolds (e.g. Acharya 02, p. 9). See also at M-theory on G2-manifolds – Details – Vacuum solution and torsion constraints.

Proposition

Given a definite globalization L WZW X\mathbf{L}_{WZW}^X of a super pp-brane WZW term L WZW V\mathbf{L}_{WZW}^V, then the stabilizer infinity-group of L WZW\mathbf{L}_{WZW} is the integrated BPS charge algebra of this solution of supergravity.

See at BPS charge – Formalization in higher differential geometry.

Formalization dictionary

We here survey in the form of a dictionary which concepts in Hegel’s Logic we are formalizing by which structure in modal type theory according to the general procedure laid out above. Each entry of the dictionary is equipped with citations pointing to those paragraphs in Hegel’s texts, as well as to parts of the secondary literature, which we argue, below, as suggesting and supporting this identification.

Of course, as with the formalization of anything that is vague and informal, this identification involves choices that are subjective, based on one’s sense of the mathematics, the philosophy and notably the mystic speculative poetry that is the very nature of Hegel’s work. Hence this table should be regarded as a proposal whose purpose is not to claim a formalization set in stone, but to provide a convenient survey and summary of what is happening in the bulk text below.

Our paragraph numbering is as follows:

  • for the Science of Logic itself (which has no paragraph numbering in the original), we follow the English translation by Miller as found online at Hegel-by-Hyper-Text;

    §xyz for the Science of Logic

  • for the Encyclopedia of the Philosophical Sciences where the original does have paragraph numbering in the main text, but not in the preface and introduction, we follow the original numbering in the main text (and still need to figure out what to do about the rest);

    EL§xyz for the Shorter Logic;

    PN§xyz for the Philosophy of Nature;

    PG§xyz for the Philosophy of the Spirit

(Since we are considering formalization as opposed to speculation, we are necessarily inside the doctrine of the Notion, and hence the following table does not show that as an item, but shows the subsections of the subjective logic as items in boldface. From this perspective the objective logic is one of these subsections and displayed accordingly.)

The formalization dictionary
Hegel’s logicmodal homotopy type theory
subjektive Begriffslogiktype theory/natural deduction§1280, Law94b
Begriff, concept, notiontype§1280, MaLö73, Sale 77, Se94 LaPr14
UrteiljudgementMaLö96
Schluss (syllogism)cut elimination, natural deductionGe35, RMR 94, 2.3
Grund (unvermittelt)antecedent§1021
das Begründete (vermittelt)succedent/consequent§1021, §1035
entering into existenceterm introduction§1033, §1035
momentmodality, modal operator (idempotent monad (co-)monad)
unity of oppositesadjoint modality (moment \dashv co-moment)§908, Law91, Law94, Law96
negativecofiber of counit of a comonad§911, §938
unity of opposite unities of oppositesadjoint triple adjoint modality (moment comoment comoment cocomoment)\left(\array{moment &\dashv& comoment \\ \bot && \bot \\ comoment &\dashv& cocomoment }\right)
spherelevel§194
AufhebungAufhebung, inclusion of adjoint modality in higher levelLaw89
Objektive Logik
Seinslogikmodal type theoryLaw94b
being, Oneunit type *\ast§86, §132, §1663
nothingempty type \emptyset§133
becomingadjoint modality (*)(\emptyset \dashv \ast)§134, §152, §176, §177, §180, Law91
everything is an intermediate stage between nothing and beingcomparison map via counit/uni-factorization (X*)(\emptyset \to X \to \ast),§174
DaseinAufhebung of becoming via sharp modality ( *)\left(\array{\flat &\dashv&\sharp \\ \vee && \vee \\ \emptyset &\dashv& \ast}\right)§182, §183, §187, §191, §194
moment of repulsionflat modality \flat§342, Law94
moment of attractioncohesion, shape modality ʃ&#643;§395
qualityadjoint modality (attraction \dashv repulsion) = (ʃ&#643; \dashv \flat)§369, §370, §342
\to Etwas§1056
moment of discretenessflat modality \flat
moment of continuitysharp modality \sharp
quantityadjoint modality (discreteness \dashv continuity) = (\flat \dashv \sharp)§398, Law94
extensive quantity\sharp-anti-modal types
intensive quantity\sharp-separated objects
measure (= gauge), unity of quantity and qualitycohesion adjoint triple adjoint modality (attraction quality repulsion discreteness quantity continuity)=(ʃ )\left(\array{ attraction &\stackrel{quality}{\dashv}& repulsion \\ \bot && \bot \\ discreteness &\stackrel{quantity}{\dashv}& continuity } \right) = \left(\array{ &#643; &\dashv& \flat \\ \bot && \bot \\ \flat &\dashv& \sharp }\right) §699, §708, §714, §725
non-being of infinitesimalsreduction modality \Re§174, §404
being-for-oneinfinitesimal flat modality &\& §322
being-for-selfinfinitesimal shape modality &\& §305
ideality (inf. quality)unity of opposites (& \Im \dashv \& )§305, §322
Aufhebung of finiteness( & ʃ )\left(\array{\Im &\dashv& \& \\ \vee && \vee \\ &#643; &\dashv& \flat }\right)§304 , §305
realityadjoint modality (\Re \dashv \Im)§304 , §305
unity of ideality and realitydifferential cohesion adjoint triple adjoint modality ( &)\left( \array{ \Re &\dashv& \Im \\ \bot&\stackrel{}{}& \bot && \\ \Im &\dashv& \& } \right)§304, §324, EL§214, §1636
absolute indifferenceadjoint modality (ididid \dashv id)§803, §808, §812
Wesenslogikhomotopy type theory
Wesen, essencethe ambient (∞,1)-category, ambient (∞,1)-topos§803, §812, §828
essence opposing itself, Bewegung von Nichts zu Nichtsididid \dashv id§813, §823, §835, §839
id id =*¯ * =id¯ \array{ id &\dashv& id & = \overline{\ast} \\ \vee && \vee \\ \vdots && \vdots \\ \vee && \vee \\ \emptyset &\dashv& \ast & = \overline{id} } Setzende Reflexion, §853b
essence appears as reflected in itselfobject classifier = type of types = universe TypeType§816, §834, §850, §1037
essence as infinite return-into-selfcumulative hierarchy of universe levels Type 1<Type 2<Type 3<Type_1 \lt Type_2 \lt Type_3 \lt \cdots §860b
Scheintype universe/object classifier§833, §818
das Aeusserethe ambient category of being H\mathbf{H}§1149, §1163b
das Innerethe internal type universe TypeHType \in \mathbf{H}§1149, §1163b
das Absolute, absolute Wirklichkeitunivalence, unity of inner and outer type universe (AB)(A=B)(A \stackrel{\simeq}{\to} B)\simeq (A = B)§1149, §1159, §1163c, §1187
Dingobject of the category§1065a
Möglichkeit, possibilitypossibility monad W=W * W\lozenge_W = W^\ast \sum_W = dependent sum followed by context extension§1191 etc.
Notwendigkeit, necessitynecessity comonad =W * W\Box = W^\ast \prod_W = dependent product followed by context extension§1191 etc.
Zufälligkeit, randomness, contingencyfunction monad WW *\prod_W W^\ast = context extension followed by dependent product§1191 etc., TorMcCar10a, TorMcCar10b, Ver14
eigentliche Wirklichkeit, unity of possibility and necessityadjoint pair of (co-)monads ()(\lozenge \dashv \Box), hence local cartesian closure§1160 (with §1159), §1190, §1192
everything is identical with itselfterm introduction for identity types§863, §875
all things are differentintensional identity§903
absoluter Widerspruch / absolute contradictionadjoint modality (false \dashv true) = (*\emptyset \dashv \ast)§931 §934
Aufhebung des WiderspruchsAufhebung of absolute contradiction via sharp modality ( *)\left(\array{\flat &\dashv& \sharp \\ \vee && \vee \\ \emptyset &\dashv& \ast}\right)§943, §944, §945
abs. Grundbase topos of sharp-modal types§945
Formshape modality ʃ&#643;§973a
Inhaltflat modality \flat§989
Matter/(gauge-)Fields(ʃ)(&#643; \dashv \flat)§989, §1068
\to Ding§1048
Substanzthe whole differential cohesive (∞,1)-topos/cohesive homotopy type theory§1235, §1238, §1281
Accidenztype 𝔸\mathbb{A} exhibiting a moment, as in ʃsomeqloc 𝔸&#643; \someq loc_{\mathbb{A}}§1236, §1237, §1238
Objektivität des Begriffs
der mechanische Proceßcode execution§1529b, §1552b, §1572
der absolute Mechanismusclassical mechanics
das Gesetzlaws of nature, equations of motion(§1572, §1575)
der Chemismusin guise of (quantum-)chemistry: quantum mechanics§1579a
die Teleologieboundary conditions for the laws of nature§1597e
Ideethe term model of the above modal type theory, in particular the true propositions§1630b, §1631, §1633, §1634
Naturmodel (representation, categorical semantics) of the above modal type theoryPN§192, PN§193b, EL§244, §1782, §1817
Raumétale stacks, being the models of infinitesimal shape \Im-modal typesPN§254a
Raum-Zeitbrane bouqtePN§256b
Lichtbosonic modality \rightsquigarrowPN§276
Körper der Starrheitfermionic modality e\ePN§279
Außereinander der Materieunity of opposites (e\e \dashv \rightsquigarrow)PN§290
opposition of Außereinander and Schwere(ρ(e) ρ() ρ() ρ())\left(\array{ \rho(\e) &\dashv& \rho(\rightsquigarrow) \\ \bot && \bot \\ \rho(\rightsquigarrow) &\dashv& \rho(\,) }\right)PN§262c, PN§290

Survey diagram

The following diagram means to show the development of the system as formalized above. Unities of opposites at a given stage are shown as adjoint modalities organized horizontally. As discussed above, there are different ways in which passage happens vertically:

  1. further determinations of being (these are the transitions within each book of the system)

    1. second order unity of opposites, coming from an adjoint triple, arranges to a square of adjoints, with adjunctions going vertically. This is progression by opposition. The name of the given second order unity is indicated in the middle of these squares.

    2. Aufhebung, by which a new adjoint modality appears whose modal types include the modal types of the previous stage, indicated by a vertical inclusion sign \vee.

  2. reiteration of the system of adjoint modalities (these are the transitions between the books of the system)

    1. First there is the plain system of modalities, starting from the smallest subcategories *\emptyset \dashv \ast and ending at the maximal subcategories, the whole category itself ididid \dashv id.

    2. Then, passing to the Essence, this system appears reflected, namely as type names in the type universe. Now all the previous modalities repeat in quotation marks (following standard notation as discussed for instance at propositional extensionality).

    3. Next, passing to Nature, the system externalizes or represents itself via a model ρ\rho (as discussed at relation between type theory and category theory). Now each modality ()(-) re-appears as its representation ρ()\rho(-) in that model.

Weg des Wissens, Bewegung des Seyns

(§808, §809)

Geist ρ(id) ρ(id) starreMaterie ρ(e) Aussereinandersein ρ() Licht Festigkeit ρ() Schwere ρ() ρ() Weichheit ρ() Elastizitaet ρ() Haerte ρ(&) Natur ρ(ʃ) ρ() Kohaesion Dichtigkeit ρ() ρ() ρ() ρ(*) Entaeusserung Idee Substanz id id Form ʃ Inhalt Existenz Ding abs.Grund AufhebungdesWiderspruchs falsch abs.Widerspruch * wahr (AB) abs.Wirklichkeit (A=B) dasAeussere H Type dasInnere Reflexionsbestimmungen Reflexion Schein Type 0Type 1Type 2 Erscheinung Wesen WW * WW * Zufaelligkeit Moeglichkeit W * W W * W Notwendigkeit An&Fuersichsein id id ObjektiveLogik AufhebungderDifferenzen NichtseinderInfinitesimalen Realitaet SeinderInfinitesimalen Fuersichsein inf.QualitaetIdealitaet/ & Fuereinessein DieKategorienSeinsbestimmungen Sein AufhebungderEndlichkeit Etwas Ansichsein Attraktion ʃ Qualitaet Repulsion SeinfuerAnderes Eichmass Dasein Diskretion Quantitaet Kontinuitaet AufhebungdesWerdens reinesSein Nichts Werden * Sein SubjektiveLogik Begriff \array{ Geist && && && \\ \\ && && && && \rho(id) & \dashv & \rho(id) \\ && && && && \vee & & \vee \\ && && && &\stackrel{starre \atop Materie}{}& \rho(\e) & \stackrel{Aussereinandersein}{\dashv} & \rho(\rightsquigarrow) & \stackrel{Licht}{} \\ && && && && \bot & \stackrel{{Festigkeit} }{} & \bot \\ && && && && \rho(\rightsquigarrow) & \stackrel{Schwere}{\dashv} & \rho(\,\,) \\ && && && && \vee & & \vee \\ && && && && \rho(\Re) & \stackrel{Weichheit}{\dashv} & \rho(\Im) \\ && && && && \bot & \stackrel{Elastizitaet}{} & \bot \\ && && && &\stackrel{}{}& \rho(\Im) &\stackrel{Haerte}{\dashv} & \rho(\& ) & \stackrel{}{} \\ && && && && \vee && \vee \\ Natur && && && & & \rho(&#643;) &\stackrel{}{\dashv}& \rho(\flat) & \\ && && && && \bot &\stackrel{Kohaesion}{}& \bot \\ && && && \stackrel{Dichtigkeit}{} && \rho(\flat) &\dashv& \rho(\sharp) & \\ && && &&&& \vee && \vee \\ && &&&&\stackrel{}{}&& \rho(\emptyset) &\dashv& \rho(\ast) & \\ \\ && && && && & Entaeusserung \\ Idee \\ && && && \stackrel{Substanz}{} && 'id' &\dashv& 'id' \\ && && && && \vdots && \vdots \\ && && && &\stackrel{Form}{} & '&#643;' & \stackrel{}{\dashv}& '\flat' & \stackrel{Inhalt}{} \\ && && Existenz && && \bot && \bot \\ && && \stackrel{Ding}{} && && '\flat' &\dashv& '\sharp' & \stackrel{abs.\,Grund}{} \\ && && &&&& \vee &\stackrel{Aufhebung \atop {des\;Widerspruchs}}{}& \vee \\ && &&&&\stackrel{}{}&\stackrel{falsch}{}& '\emptyset' &\stackrel{abs.\,Widerspruch}{\dashv}& '\ast' & \stackrel{wahr}{} \\ && \\ && \\ && && && && (A \stackrel{\simeq}{\to} B) & \stackrel{abs.\,Wirklichkeit}{\simeq} & ('A' = 'B') & \\ && && && &\stackrel{das\;Aeussere}{}& \mathbf{H} & & Type & \stackrel{das\;Innere}{} \\ && \stackrel{Reflexionsbestimmungen}{} \\ && && && && & Reflexion \\ \\ &&&& Schein && &\stackrel{}{} & &\stackrel{\stackrel{\stackrel{\vdots}{Type_2}}{Type_1}}{Type_0}& & \stackrel{}{} \\ \\ && && && && & Erscheinung \\ && \\ && && Wesen && & \stackrel{}{} & \sum_W W^\ast & \stackrel{}{\dashv} & \prod_W W^\ast & \stackrel{Zufaelligkeit}{} \\ && && && &\stackrel{Moeglichkeit}{}& W^\ast \sum_W &\stackrel{}{\dashv}& W^\ast \prod_W & \stackrel{Notwendigkeit}{} \\ \\ && && && \\ \\ && && &&\stackrel{An\& Fuersichsein}{}&& id &\stackrel{}{\dashv}& id \\ {Objektive \atop Logik}{} && && &&&& \vee &\stackrel{{Aufhebung \atop {der\;Differenzen}}}{}& \vee \\ && && &&& \stackrel{{Nichtsein\;der}\atop Infinitesimalen}{}& \Re &\stackrel{Realitaet}{\dashv}& \Im & \stackrel{{Sein\;der}\atop{Infinitesimalen}}{} \\ && && \stackrel{}{} &&&& \bot &\stackrel{}{}& \bot \\ && && &&\stackrel{Fuersichsein}{} && \Im &\stackrel{Idealitaet /}{\stackrel{inf.\,Qualitaet}{\dashv}}& \& &&\stackrel{Fuer-eines-sein}{} \\ && \stackrel{Seinsbestimmungen}{\stackrel{Die\;Kategorien}{}}&& Sein &&&& \vee &\stackrel{{Aufhebung \atop {der\;Endlichkeit}}}{}& \vee \\ && && \stackrel{Etwas}{} &&\stackrel{Ansichsein}{}&\stackrel{Attraktion}{}& &#643; &\stackrel{Qualitaet}{\dashv}& \flat & \stackrel{Repulsion}{} & \stackrel{Sein-fuer-Anderes}{} \\ && && &&&& \bot &\stackrel{Eichmass}{}& \bot \\ && && &&\stackrel{Dasein}{}&\stackrel{Diskretion}{}& \flat &\stackrel{Quantitaet}{\dashv}& \sharp & \stackrel{Kontinuitaet}{} \\ && && &&&& \vee &\stackrel{Aufhebung \atop {des\;Werdens}}{}& \vee \\ && &&&&\stackrel{reines\;Sein}{}&\stackrel{Nichts}{}& \emptyset &\stackrel{Werden}{\dashv}& \ast & \stackrel{Sein}{} \\ && && && \\ \\ {Subjektive \atop Logik}{} && && Begriff && }

Introduction and Prefaces

Vorrede zur ersten Ausgabe / Preface of first edition

§3 Indem so die Wissenschaft und der gemeine Menschenverstand sich in die Hände arbeiteten, den Untergang der Metaphysik zu bewirken, so schien das sonderbare Schauspiel herbeigeführt zu werden, ein gebildetes Volk ohne Metaphysik zu sehen, – wie einen sonst mannigfaltig ausgeschmückten Tempel ohne Allerheiligstes. – Die Theologie, welche in früheren Zeiten die Bewahrerin der spekulativen Mysterien und der obzwar abhängigen Metaphysik war, hatte diese Wissenschaft gegen Gefühle, gegen das Praktisch-Populäre und gelehrte Historische aufgegeben.

§3 Philosophy [Wissenschaft] and ordinary common sense thus co-operating to bring about the downfall of metaphysics, there was seen the strange spectacle of a cultured nation without metaphysics – like a temple richly ornamented in other respects but without a holy of holies. Theology, which in former times was the guardian of the speculative mysteries and of metaphysics (although this was subordinate to it) had given up this science in exchange for feelings, for what was popularly matter-of-fact, and for historical erudition.

§8 Was nun auch für die Sache und für die Form der Wissenschaft bereits in sonstiger Rücksicht geschehen seyn mag; die logische Wissenschaft, welche die eigentliche Metaphysik oder reine spekulative Philosophie ausmacht, hat sich bisher noch sehr vernachlässigt gesehen. Was ich unter dieser Wissenschaft und ihrer Standpunkte näher verstehe, habe ich in der Einleitung vorläufig angegeben. Die Nothwendigkeit, mit dieser Wissenschaft wieder einmal von vorne anzufangen, die Natur des Gegenstandes selbst, und der Mangel an Vorarbeiten, welche für die vorgenommen Umbildung hätten benutzt werden können, mögen bei billigen Beurtheilern in Rücksicht kommen, wenn auch eine vieljährige Arbeit diesem Versuche nicht eine größere Vollkommenheit geben konnte. —Der wesentliche Gesichtspunkt ist, daß es überhaupt um einen neuen Begriff wissenschaftlicher Behandlung zu thun ist. Die Philosophie, indem sie Wissenschaft seyn soll, kann, wie ich anderwärts erinnert Phänomenologie des Geistes, Vorr. zur ersten Ausg.—Die eigentliche Ausführung ist die Erkenntniß der Methode, und hat ihre Stelle in der Logik selbst, habe, hierzu ihre Methode nicht von einer untergeordneten Wissenschaft, wie die Mathematik ist, borgen, so wenig als es bei kategorischen Versicherungen innerer Anschauung bewenden lassen, oder sich des Raisonnements aus Gründen der äußern Reflexion bedienen. Sondern es kann nur die Natur des Inhalts seyn, welche sich im wissenschaftlichen Erkennen bewegt, indem zugleich diese eigne Reflexion des Inhalts es ist, welche seine Bestimmung selbst erst setzt und erzeugt.

§8 Now whatever may have been accomplished for the form and content of philosophy in other directions, the science of logic which constitutes metaphysics proper or purely speculative philosophy, has hitherto still been much neglected. What it is exactly that I understand by this science and its standpoint, I have stated provisionally in the Introduction. The fact that it has been necessary to make a completely fresh start with this science, the very nature of the subject matter and the absence of any previous works which might have been utilised for the projected reconstruction of logic, may be taken into account by fair-minded critics, even though a labour covering many years has been unable to give this effort a greater perfection. The essential point of view is that what is involved is an altogether new concept of scientific procedure. Philosophy, if it would be a science, cannot, as I have remarked elsewhere, borrow its method from a subordinate science like mathematics, any more than it can remain satisfied with categorical assurances of inner intuition, or employ arguments based on grounds adduced by external reflection. On the contrary, it can be only the nature of the content itself which spontaneously develops itself in a scientific method of knowing, since it is at the same time the reflection of the content itself which first posits and generates its determinate character.

§9 Der Verstand bestimmt und hält die Bestimmungen fest; die Vernunft ist negativ und dialektisch, weil sie die Bestimmungen des Verstands in Nichts auflöst; sie ist positiv, weil sie das Allgemeine erzeugt, und das Besondere darin begreift. Wie der Verstand als etwas Getrenntes von der Vernunft überhaupt, so pflegt auch die dialektische Vernunft als etwas Getrenntes von der positiven Vernunft genommen zu werden. Aber in ihrer Wahrheit ist die Vernunft Geist, der höher als Beides, verständige Vernunft, oder vernünftiger Verstand ist. Er ist das Negative, dasjenige, welches die Qualität sowohl, der dialektischen Vernunft, als des Verstandes ausmacht;—er negirt das Einfache, so setzt er den bestimmten Unterschied des Verstandes, er löst ihn eben so sehr auf, so ist er dialektisch. Er hält sich aber nicht im Nichts dieses Resultates, sondern ist darin ebenso positiv, und hat so das erste Einfache damit hergestellt, aber als Allgemeines, das in sich konkret ist; unter dieses wird nicht ein gegebenes Besonderes subsumirt, sondern in jenem Bestimmen und in der Auflösung desselben hat sich das Besondere schon mit bestimmt. Diese geistige Bewegung, die sich in ihrer Einfachheit ihre Bestimmtheit, und in dieser ihre Gleichheit mit sich selbst giebt, die somit die immanente Entwickelung des Begriffes ist, ist die absolute Methode des Erkennens, und zugleich die immanente Seele des Inhalts selbst. —Auf diesem sich selbst konstruirenden Wege allein, behaupte ich, ist die Philosophie fähig, objektive, demonstrirte Wissenschaft zu seyn.

§9 The understanding determines, and holds the determinations fixed; reason is negative and dialectical, because it resolves the determinations of the understanding into nothing; it is positive because it generates the universal and comprehends the particular therein. Just as the understanding is usually taken to be something separate from reason as such, so too dialectical reason is usually taken to be something distinct from positive reason. But reason in its truth is spirit which is higher than either merely positive reason, or merely intuitive understanding. It is the negative, that which constitutes the quality alike of dialectical reason and of understanding; it negates what is simple, thus positing the specific difference of the understanding; it equally resolves it and is thus dialectical. But it does not stay in the nothing of this result but in the result is no less positive, and in this way it has restored what was at first simple, but as a universal which is within itself concrete; a given particular is not subsumed under this universal but in this determining, this positing of a difference, and the resolving of it, the particular has at the same time already determined itself. This spiritual movement which, in its simple undifferentiatedness, gives itself its own determinateness and in its determinateness its equality with itself, which therefore is the immanent development of the Notion, this movement is the absolute method of knowing and at the same time is the immanent soul of the content itself. I maintain that it is this self-construing method alone which enables philosophy to be an objective, demonstrated science

§10 In dieser Weise habe ich das Bewußtseyn in der Phänomenologie des Geistes darzustellen versucht. Das Bewußtseyn ist der Geist als konkretes und zwar in der Äußerlichkeit befangenes Wissen; aber die Formbewegung dieses Gegenstandes beruht allein, wie die Entwickelung alles natürlichen und geistigen Lebens, auf der Natur der reinen Wesenheiten, die den Inhalt der Logik ausmachen. Das Bewußtseyn, als der erscheinende Geist, welcher sich auf seinem Wege von seiner Unmittelbarkeit und äußerlichen Konkretion befreit, wird zum reinen Wissen, das sich jene reinen Wesenheiten selbst, wie sie an und für sich sind, zum Gegenstand giebt. Sie sind die reinen Gedanken, der sein Wesen denkende Geist. Ihre Selbstbewegung ist ihr geistiges Leben, und ist das, wodurch sich die Wissenschaft konstituirt, und dessen Darstellung sie ist.

§10 It is in this way that I have tried to expound consciousness in the Phenomenology of Spirit. Consciousness is spirit as a concrete knowing, a knowing too, in which externality is involved; but the development of this object, ®like the development of all natural and spiritual life, rests solely on the nature of the pure essentialities which constitute the content of logic.

Consciousness, as spirit in its manifestation which in its progress frees itself from its immediacy and external concretion, attains to the pure knowing which takes as its object those same pure essentialities as they are in and for themselves. They are pure thoughts, spirit thinking its own essential nature. Their self-movement is their spiritual life and is that through which philosophy constitutes itself and of which it is the exposition.

§11 Es ist hiermit die Beziehung der Wissenschaft, die ich Phänomenologie des Geistes nenne, zur Logik angegeben.—Was das äußerliche Verhältniß betrifft, so war dem ersten Theil des Systems der Wissenschaft, (Bamberg und Würzburg bei Göbhard 1807). Dieser Titel wird der zweiten Ausgabe, die auf nächsten Ostern erscheinen wird, nicht mehr beigegeben werden.—An die Stelle des im Folgenden erwähnten Vorhabens eines zweiten Theils, der die sämmtlichen andern philosophischen Wissenschaften enthalten sollte, habe ich seitdem die Encyklopädie der philosophischen Wissenschaften, voriges Jahr in der dritten Ausgabe, ans Licht treten lassen (Anmerkung zur zweiten Ausgabe), der die Phänomenologie enthält, ein zweiter Theil zu folgen bestimmt, welcher die Logik und die beiden realen Wissenschaften der Philosophie, die Philosophie der Natur und die Philosophie des Geistes, enthalten sollte, und das System der Wissenschaft beschlossen haben würde. Aber die nothwendige Ausdehnung, welche die Logik für sich erhalten mußte, hat mich veranlaßt, diese besonders ans Licht treten zu lassen; sie macht also in einem erweiterten Plane die erste Folge zur Phänomenologie des Geistes aus. Späterhin werde ich die Verarbeitung der beiden genannten realen Wissenschaften der Philosophie folgen lassen.—Dieser erste Band der Logik aber enthält als erstes Buch die Lehre vom Seyn; das zweite Buch, die Lehre vom Wesen, als zweite Abtheilung des ersten Bandes; der zweite Band aber wird die subjektive Logik, oder die Lehre vom Begriff enthalten.

§11 In the foregoing there is indicated the relation of the science which I call the Phenomenology of Spirit, to logic. As regards the external relation, it was intended that the first part of the System of Science which contains the Phenomenology should be followed by a second part containing logic and the two concrete [realen] sciences, the Philosophy of Nature and the Philosophy of Spirit, which would complete the System of Philosophy. But the necessary expansion which logic itself has demanded has induced me to have this part published separately; it thus forms the first sequel to the Phenomenology of Spirit in an expanded arrangement of the system. It will later be followed by an exposition of the two concrete philosophical sciences mentioned. This first volume of the Logic contains as Book One the Doctrine of Being; Book Two, the Doctrine of Essence, which forms the second part of the first volume, is already in the press; the second volume will contain Subjective Logic or the Doctrine of the Notion.

Allgemeiner Begriff der Logik

§33 Es fühlt sich bei keiner Wissenschaft stärker das Bedürfniß, ohne vorangehende Reflexionen, von der Sache selbst anzufangen, als bei der logischen Wissenschaft. In jeder andern ist der Gegenstand, den sie behandelt, und die wissenschaftliche Methode von einander unterschieden; so wie auch der Inhalt nicht einen absoluten Anfang macht, sondern von andern Begriffen abhängt, und um sich herum mit anderem Stoffe zusammenhängt. Diesen Wissenschaften wird es daher zugegeben, von ihrem Boden und dessen Zusammenhang, so wie von der Methode nur lemmatischer Weise zu sprechen, die als bekannt und angenommen vorausgesetzten Formen von Definitionen und dergleichen ohne weiteres anzuwenden, und sich der gewöhnlichen Art des Raisonnements zur Festsetzung ihrer allgemeinen Begriffe und Grundbestimmungen zu bedienen.

§33 In no science is the need to begin with the subject matter itself, without preliminary reflections, felt more strongly than in the science of logic. In every other science the subject matter and the scientific method are distinguished from each other; also the content does not make an absolute beginning but is dependent on other concepts and is connected on all sides with other material. These other sciences are, therefore, permitted to speak of their ground and its context and also of their method, only as premises taken for granted which, as forms of definitions and such-like presupposed as familiar and accepted, are to be applied straight-way, and also to employ the usual kind of reasoning for the establishment of their general concepts and fundamental determinations.

§34 Die Logik dagegen kann keine dieser Formen der Reflexion oder Regeln und Gesetze des Denkens voraussetzen, denn sie machen einen Theil ihres Inhalts selbst aus und haben erst innerhalb ihrer begründet zu werden. Nicht nur aber die Angabe der wissenschaftlichen Methode, sondern auch der Begriff selbst der Wissenschaft überhaupt gehört zu ihrem Inhalte, und zwar macht er ihr letztes Resultat aus; was sie ist, kann sie daher nicht voraussagen, sondern ihre ganze Abhandlung bringt dieß Wissen von ihr selbst erst als ihr Letztes und als ihre Vollendung hervor. Gleichfalls ihr Gegenstand, das Denken oder bestimmter das begreifende Denken, wird wesentlich innerhalb ihrer abgehandelt; der Begriff desselben erzeugt sich in ihrem Verlaufe, und kann somit nicht vorausgeschickt werden. Was daher in dieser Einleitung vorausgeschickt wird, hat nicht den Zweck, den Begriff der Logik etwa zu begründen, oder den Inhalt und die Methode derselben zum voraus wissenschaftlich zu rechtfertigen, sondern, durch einige Erläuterungen und Reflexionen, in raisonnirendem und historischem Sinne, den Gesichtspunkt, aus welchem diese Wissenschaft zu betrachten ist, der Vorstellung näher zu bringen.

§34 Logic on the contrary, cannot presuppose any of these forms of reflection and laws of thinking, for these constitute part of its own content and have first to be established within the science. But not only the account of scientific method, but even the Notion itself of the science as such belongs to its content, and in fact constitutes its final result; what logic is cannot be stated beforehand, rather does this knowledge of what it is first emerge as the final outcome and consummation of the whole exposition. Similarly, it is essentially within the science that the subject matter of logic, namely, thinking or more specifically comprehensive thinking is considered; the Notion of logic has its genesis in the course of exposition and cannot therefore be premised. Consequently, what is premised in this Introduction is not intended, as it were, to establish the Notion of Logic or to justify its method scientifically in advance, but rather by the aid of some reasoned and historical explanations and reflections to make more accessible to ordinary thinking the point of view from which this science is to be considered.

The problem of circularity of foundations. The traditional approach in mathematics is (see e.g. Trimble 13 Are logical foundaitons circular?) to presuppose first order logic as accepted a priori (i.e. not itself formalized in another system, for one needs to start somewhere), and then formulate all theory, notably set theory, in terms of first order logic.

But other such starting points are possible. For instance taking instead type theory or homotopy type theory as the foundation, then it first of all contains first-order logic (in its restriction to types which are mere propositions) but second it already contains a kind of constructive set theory “natively”. See at homotopy type theory FAQ.

Here §34 is asking for something like an intrinsic justification of such an otherwise a priori foundation “out of itself”. Common attitude is that this is circular, hence nonsensical, in that one needs to start somewhere, at least one needs to assume that there is general agreement on first-order logic (see e.g. Trimble 13). Hegel’s attitude is that, yes, it is circular, but, no, this does not mean that it is non-sensical.

§48 Ganz ohne Rücksicht auf metaphysische Bedeutung aber wird dasjenige betrachtet, was gemeinhin unter Logik verstanden wird. Diese Wissenschaft, in dem Zustande, worin sie sich noch befindet, hat freilich keinen Inhalt der Art, wie er als Realität und als eine wahrhafte Sache in dem gewöhnlichen Bewußtsein gilt. Aber sie ist nicht aus diesem Grunde eine formelle, inhaltsvoller Wahrheit entbehrende Wissenschaft. In jenem Stoffe, der in ihr vermißt [wird], welchem Mangel das Unbefriedigende derselben zugeschrieben zu werden pflegt, ist ohnehin das Gebiet der Wahrheit nicht zu suchen. Sondern das Gehaltlose der logischen Formen liegt vielmehr allein in der Art, sie zu betrachten und zu behandeln. Indem sie als feste Bestimmungen auseinanderfallen und nicht in organischer Einheit zusammengehalten werden, sind sie tote Formen und haben den Geist in ihnen nicht wohnen, der ihre lebendige konkrete Einheit ist. Damit aber entbehren sie des gediegenen Inhalts, – einer Materie, welche Gehalt an sich selbst wäre. Der Inhalt, der an den logischen Formen vermißt wird, ist nichts anderes als eine feste Grundlage und Konkretion dieser abstrakten Bestimmungen; und ein solches substantielles Wesen pflegt für sie außen gesucht zu werden. Aber die logische Vernunft selbst ist das Substantielle oder Reelle, das alle abstrakten Bestimmungen in sich zusammenhält und ihre gediegene, absolut-konkrete Einheit ist. Nach dem also, was eine Materie genannt zu werden pflegt, brauchte nicht weit gesucht zu werden; es ist nicht Schuld des Gegenstandes der Logik, wenn sie gehaltlos sein soll, sondern allein der Art, wie derselbe gefaßt wird.

§48 But what is commonly understood by logic is considered without any reference whatever to metaphysical significance. This science in its present state has, it must be admitted, no content of a kind which the ordinary consciousness would regard as a reality and as a genuine subject matter. But it is not for this reason a formal science lacking significant truth. Moreover, the region of truth is not to be sought in that matter which is missing in logic, a deficiency to which the unsatisfactoriness of the science is usually attributed. The truth is rather that the insubstantial nature of logical forms originates solely in the way in which they are considered and dealt with. When they are taken as fixed determinations and consequently in their separation from each other and not as held together in an organic unity, then they are dead forms and the spirit which is their living, concrete unity does not dwell in them. As thus taken, they lack a substantial content — a matter which would be substantial in itself. The content which is missing in the logical forms is nothing else than a solid foundation and a concretion of these abstract determinations; and such a substantial being for them is usually sought outside them. But logical reason itself is the substantial or real being which holds together within itself every abstract determination and is their substantial, absolutely concrete unity. One need not therefore look far for what is commonly called a matter; if logic is supposed to lack a substantial content, then the fault does not lie with its subject matter but solely with the way in which this subject matter is grasped.

The substantial aspect of logic.

§50 In der Phänomenologie des Geistes habe ich das Bewußtsein in seiner Fortbewegung von dem ersten unmittelbaren Gegensatz seiner und des Gegenstandes bis zum absoluten Wissen dargestellt. Dieser Weg geht durch alle Formen des Verhältnisses des Bewußtseins zum Objekte durch und hat den Begriff der Wissenschaft zu seinem Resultate. Dieser Begriff bedarf also (abgesehen davon, daß er innerhalb der Logik selbst hervorgeht) hier keiner Rechtfertigung, weil er sie daselbst erhalten hat; und er ist keiner anderen Rechtfertigung fähig als nur dieser Hervorbringung desselben durch das Bewußtsein, dem sich seine eigenen Gestalten alle in denselben als in die Wahrheit auflösen.

§50 In the Phenomenology of Mind, I have exhibited consciousness in its movement onwards from the first immediate opposition of itself and the object to absolute knowing. The path of this movement goes through every form of the relation of consciousness to the object and has the Notion of science of its result. This Notion therefore (apart from the fact that it emerges within logic itself) needs no justification here because it has received it in that work; and it cannot be justified in any other way than by this emergence in consciousness, all the forms of which are resolved into this Notion as into their truth.

§51 Der Begriff der reinen Wissenschaft und seine Deduktion wird in gegenwärtiger Abhandlung also insofern vorausgesetzt, als die Phänomenologie des Geistes nichts anderes als die Deduktion desselben ist. Das absolute Wissen ist die Wahrheit aller Weisen des Bewußtseins, weil, wie jener Gang desselben es hervorbrachte, nur in dem absoluten Wissen die Trennung des Gegenstandes von der Gewißheit seiner selbst vollkommen sich aufgelöst hat und die Wahrheit dieser Gewißheit sowie diese Gewißheit der Wahrheit gleich geworden ist.

§51 The Notion of pure science and its deduction is therefore presupposed in the present work in so far as the Phenomenology of Spirit is nothing other than the deduction of it. Absolute knowing is the truth of every mode of consciousness because, as the course of the Phenomenology showed, it is only in absolute knowing that separation of the object from the certainty of itself is completely eliminated: truth is now equated with certainty and this certainty with truth.

§52 Die reine Wissenschaft setzt somit die Befreiung von dem Gegensatze des Bewußtseyns voraus. Sie enthält den Gedanken, insofern er eben so sehr die Sache an sich selbst ist, oder die Sache an sich selbst, insofern sie ebenso sehr der reine Gedanke ist. Als Wissenschaft ist die Wahrheit das reine sich entwicklende Selbstbewußtseyn, und hat die Gestalt des Selbst, daß das an und für sich seyende gewußter Begriff, der Begriff als solcher aber das an und für sich seyende ist.

§52 Thus pure science presupposes liberation from the opposition of consciousness. It contains thought in so far as this is just as much the object in its own self, or the object in its own self in so far as it is equally pure thought. As science, truth is pure self-consciousness in its self-development and has the shape of the self, so that the absolute truth of being is the known Notion and the Notion as such is the absolute truth of being.

§53a Dieses objektive Denken ist denn der Inhalt der reinen Wissenschaft. Sie ist daher so wenig formell, sie entbehrt so wenig der Materie zu einer wirklichen und wahren Erkenntniß, daß ihr Inhalt vielmehr allein das absolute Wahre, oder wenn man sich noch des Worts Materie bedienen wollte, die wahrhafte Materie ist,—eine Materie aber, der die Form nicht ein Äußerliches ist, da diese Materie vielmehr der reine Gedanke, somit die absolute Form selbst ist.

§53a This objective thinking then, is the content of pure science. Consequently, far from it being formal, far from it standing in need of a matter to constitute an actual and true cognition, it is its content alone which has absolute truth, or, if one still wanted to employ the word matter, it is the veritable matter — but a matter which is not external to the form, since this matter is rather pure thought and hence the absolute form itself.

§53b 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.

§53b 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.

To wit, the Logic ends with the appearance of the idea in §1636 and then nature appears from that in PN§192 “as the idea in the form of otherness”.

§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.

§55 Um dies in die Vorstellung wenigstens aufzunehmen, ist die Meinung auf die Seite zu legen, als ob die Wahrheit etwas Handgreifliches sein müsse. Solche Handgreiflichkeit wird zum Beispiel selbst noch in die Platonischen Ideen, die in dem Denken Gottes sind, hineingetragen, als ob sie gleichsam existierende Dinge, aber in einer anderen Welt oder Region seien, außerhalb welcher die Welt der Wirklichkeit sich befinde und eine von jenen Ideen verschiedene, erst durch diese Verschiedenheit reale Substantialität habe. Die Platonische Idee ist nichts anderes als das Allgemeine oder bestimmter der Begriff des Gegenstandes; nur in seinem Begriffe hat etwas Wirklichkeit; insofern es von seinem Begriffe verschieden ist, hört es auf, wirklich zu sein, und ist ein Nichtiges; die Seite der Handgreiflichkeit und des sinnlichen[44] Außersichseins gehört dieser nichtigen Seite an.

§55 To get some idea of this one must discard the prejudice that truth must be something tangible. Such tangibility is, for example, imported even into the Platonic Ideas which are in God’s thinking, as if they are, as it were, existing things but in another world or region; while the world of actuality exists outside that region and has a substantial existence distinct from those Ideas and only through this distinction is a substantial reality. The Platonic Idea is the universal, or more definitely the Notion of an object; only in its Notion does something possess actuality and to the extent that it is distinct from its Notion it ceases to be actual and is a non-entity; the side of tangibility and sensuous self-externality belongs to this null aspect. But on the other side, one can appeal to the conceptions of ordinary logic itself; for it is assumed, for example, that the determinations contained in definitions do not belong only to the knower, but are determinations of the object, constituting its innermost essence and its very own nature. Or, if from given determinations others are inferred, it is assumed that what is inferred is not something external and alien to the object, but rather that it belongs to the object itself, that to the thought there is a correspondent being.

Plato’s doctrine of ideas

§61 Was solchen Inhalt betrifft, so ist schon oben der Grund angegeben worden, warum er so geistlos ist. Die Bestimmungen desselben gelten in ihrer Festigkeit unverrückt und werden nur in äußerliche Beziehung miteinander gebracht. Dadurch, daß bei den Urteilen und Schlüssen die Operationen vornehmlich auf das Quantitative der Bestimmungen zurückgeführt und gegründet werden, beruht alles auf einem äußerlichen Unterschiede, auf bloßer Vergleichung, wird ein völlig analytisches Verfahren und begriffloses Kalkulieren. Das Ableiten der sogenannten Regeln und Gesetze, des Schließens vornehmlich, ist nicht viel besser als ein Befingern von Stäbchen von ungleicher Länge, um sie nach ihrer Größe zu sortieren und zu verbinden, – als die spielende Beschäftigung der Kinder, von mannigfaltig zerschnittenen Gemälden[47] die passenden Stücke zusammenzusuchen. – Man hat daher nicht mit Unrecht dieses Denken dem Rechnen und das Rechnen wieder diesem Denken gleichgesetzt. In der Arithmetik werden die Zahlen als das Begrifflose genommen, das außer seiner Gleichheit oder Ungleichheit, d.h. außer seinem ganz äußerlichen Verhältnisse keine Bedeutung hat, das weder an ihm selbst noch dessen Beziehung ein Gedanke ist. Wenn auf mechanische Weise ausgerechnet wird, daß drei Viertel mit zwei Dritteln multipliziert ein Halbes ausmacht, so enthält diese Operation ungefähr soviel und sowenig Gedanken als die Berechnung, ob in einer Figur diese oder jene Art des Schlusses statthaben könne.

§61 Regarding this content, the reason why logic is so dull and spiritless has already been given above. Its determinations are accepted in their unmoved fixity and are brought only into external relation with each other. In judgments and syllogisms the operations are in the main reduced to and founded on the quantitative aspect of the determinations; consequently everything rests on an external difference, on mere comparison and becomes a completely analytical procedure and mechanical calculation. The deduction of the so-called rules and laws, chiefly of inference, is not much better than a manipulation of rods of unequal length in order to sort and group them according to size — than a childish game of fitting together the pieces of a coloured picture puzzle. Consequently, this thinking has been equated, not incorrectly, with reckoning, and reckoning again with this thinking. In arithmetic, numbers are regarded as devoid of any concrete conceptual content, so apart from their wholly external relationship they have no meaning, and neither in themselves nor in their interrelationships are thoughts. When it is calculated in mechanical fashion that three-fourths multiplied by two-thirds makes one-half, this operation contains about as much and as little thought as calculating whether in a logical figure this or that kind of syllogism is valid.

mechanical nature of formal logic

§62 Das Einzige, um den wissenschaftlichen Fortgang zu gewinnen – und um dessen ganz einfache Einsicht sich wesentlich zu bemühen ist –, ist die Erkenntnis des logischen Satzes, daß das Negative ebensosehr positiv ist oder daß das sich Widersprechende sich nicht in Null, in das abstrakte Nichts auflöst, sondern wesentlich nur in die Negation seines besonderen Inhalts, oder daß eine solche Negation nicht alle Negation, sondern die Negation der bestimmten Sache, die sich auflöst, somit bestimmte Negation ist; daß also im Resultate wesentlich das enthalten ist, woraus es resultiert, – was eigentlich eine Tautologie ist, denn sonst wäre es ein Unmittelbares, nicht ein Resultat. Indem das Resultierende, die Negation, bestimmte Negation ist, hat sie einen Inhalt. Sie ist ein neuer Begriff, aber der höhere, reichere Begriff als der vorhergehende; denn sie ist um dessen Negation oder Entgegengesetztes reicher geworden, enthält ihn also, aber auch mehr als ihn, und ist die Einheit seiner und seines Entgegengesetzten. – In diesem Wege hat sich das System der Begriffe überhaupt zu bilden – und in unaufhaltsamem, reinem, von außen nichts hereinnehmendem Gange sich zu vollenden.

§62 All that is necessary to achieve scientific progress — and it is essential to strive to gain this quite simple insight — is the recognition of the logical principle that the negative is just as much positive, or that what is self-contradictory does not resolve itself into a nullity, into abstract nothingness, but essentially only into the negation of its particular content, in other words, that such a negation is not all and every negation but the negation of a specific subject matter which resolves itself, and consequently is a specific negation, and therefore the result essentially contains that from which it results; which strictly speaking is a tautology, for otherwise it would be an immediacy, not a result. Because the result, the negation, is a specific negation, it has content. It is a fresh Notion but higher and richer than its predecessor; for it is richer by the negation or opposite of the latter, therefore contains it, but also something more, and is the unity of itself and its opposite. It is in this way that the system of Notions as such has to be formed — and has to complete itself in a purely continuous course in which nothing extraneous is introduced.

dialectic

§63 Wie würde ich meinen können, daß nicht die Methode, die ich in diesem Systeme der Logik befolge – oder vielmehr die dies System an ihm selbst befolgt –, noch vieler Vervollkommnung, vieler Durchbildung im einzelnen fähig sei; aber ich weiß zugleich, daß sie die einzige wahrhafte ist. Dies erhellt für sich schon daraus, daß sie von ihrem Gegenstande und Inhalte nichts Unterschiedenes ist; – denn es ist der Inhalt in sich, die Dialektik, die er an ihm selbst hat, welche ihn fortbewegt. Es ist klar, daß keine Darstellungen für wissenschaftlich gelten können, welche nicht den Gang dieser Methode gehen und ihrem einfachen Rhythmus gemäß sind, denn es ist der Gang der Sache selbst.

§63 I could not pretend that the method which I follow in this system of logic — or rather which this system in its own self follows — is not capable of greater completeness, of much elaboration in detail; but at the same time I know that it is the only true method. This is self-evident simply from the fact that it is not something distinct from its object and content; for it is the inwardness of the content, the dialectic which it possesses within itself, which is the mainspring of its advance. It is clear that no expositions can be accepted as scientifically valid which do not pursue the course of this method and do not conform to its simple rhythm, for this is the course of the subject matter itself.

Das, wodurch sich der Begriff selbst weiterleitet, ist das vorhin angegebene Negative, das er in sich selbst hat; dies macht das wahrhaft Dialektische aus. Die Dialektik, die als ein abgesonderter Teil der Logik betrachtet und in Ansehung ihres Zwecks und Standpunkts, man kann sagen, gänzlich verkannt worden, erhält dadurch eine ganz andere Stellung. – Auch die Platonische Dialektik hat selbst im Parmenides, und anderswo ohnehin noch direkter, teils nur die Absicht, beschränkte Behauptungen durch sich selbst aufzulösen und zu widerlegen, teils aber überhaupt das Nichts zum Resultate. Gewöhnlich sieht man die Dialektik für ein äußerliches und negatives Tun an, das nicht der Sache selbst angehöre, in bloßer Eitelkeit als einer subjektiven Sucht, sich das Feste und Wahre in Schwanken zu setzen und aufzulösen, seinen Grund habe oder wenigstens zu nichts führe als zur Eitelkeit des dialektisch behandelten Gegenstandes.

Kant hat die Dialektik höher gestellt – und diese Seite gehört unter die größten seiner Verdienste –, indem er ihr den Schein von Willkür nahm, den sie nach der gewöhnlichen Vorstellung hat, und sie als ein notwendiges Tun der Vernunft darstellte. Indem sie nur für die Kunst, Blendwerke vorzumachen und Illusionen hervorzubringen, galt, wurde schlechthin vorausgesetzt, daß sie ein falsches Spiel spiele und ihre ganze Kraft allein darauf beruhe, daß sie den Betrug verstecke; daß ihre Resultate nur erschlichen und ein subjektiver Schein seien. Kants dialektische Darstellungen in den Antinomien der reinen Vernunft verdienen zwar, wenn sie näher betrachtet werden, wie dies im Verfolge dieses Werkes weitläufiger geschehen wird, freilich kein großes Lob; aber die allgemeine Idee, die er zugrunde gelegt und geltend gemacht hat, ist die Objektivität des Scheins und Notwendigkeit des Widerspruchs, der zur Natur der Denkbestimmungen gehört: zunächst zwar in der Art, insofern diese Bestimmungen von der Vernunft auf die Dinge an sich angewendet werden; aber eben, was sie in der Vernunft und in Rücksicht auf das sind, was an sich ist, ist ihre Natur. Es ist dies Resultat, in seiner positiven Seite aufgefaßt, nichts anderes als die innere Negativität derselben, als ihre sich selbst bewegende Seele, das Prinzip aller natürlichen und geistigen Lebendigkeit überhaupt. Aber sowie nur bei der abstrakt-negativen Seite des Dialektischen stehengeblieben wird, so ist das Resultat nur das Bekannte, daß die Vernunft unfähig sei, das Unendliche zu erkennen; – ein sonderbares Resultat, indem das Unendliche das Vernünftige ist, zu sagen, die Vernunft sei nicht fähig, das Vernünftige zu erkennen. In diesem Dialektischen, wie es hier genommen wird, und damit in dem Fassen des Entgegengesetzten in seiner Einheit oder des Positiven im Negativen besteht das Spekulative. Es ist die wichtigste, aber für die noch ungeübte, unfreie Denkkraft schwerste Seite. Ist solche noch darin begriffen, sich vom sinnlich-konkreten Vorstellen und vom Räsonieren[52] loszureißen, so hat sie sich zuerst im abstrakten Denken zu üben, Begriffe in ihrer Bestimmtheit festzuhalten und aus ihnen erkennen zu lernen. Eine Darstellung der Logik zu diesem Behuf hätte sich in ihrer Methode an das obenbesagte Einteilen und in Ansehung des näheren Inhalts an die Bestimmungen, die sich für die einzelnen Begriffe ergeben, zu halten, ohne sich auf das Dialektische einzulassen. Sie würde der äußeren Gestalt nach dem gewöhnlichen Vortrag dieser Wissenschaft ähnlich werden, sich übrigens dem Inhalte nach auch davon unterscheiden und immer noch dazu dienen, das abstrakte, obzwar nicht das spekulative Denken zu üben, welchen Zweck die durch psychologische und anthropologische Zutaten populär gewordene Logik nicht einmal erfüllen kann. Sie würde dem Geiste das Bild eines methodisch geordneten Ganzen geben, obgleich die Seele des Gebäudes, die Methode, die im Dialektischen lebt, nicht selbst darin erschiene.

§71 At first, therefore, logic must indeed be learnt as something which one understands and sees into quite well but in which, at the beginning, one feels the lack of scope and depth and a wider significance. It is only after profounder acquaintance with the other sciences that logic ceases to be for subjective spirit a merely abstract universal and reveals itself as the universal which embraces within itself the wealth of the particular — just as the same proverb, in the mouth of a youth who understands it quite well, does not possess the wide range of meaning which it has in the mind of a man with the experience of a lifetime behind him, for whom the meaning is expressed in all its power. Thus the value of logic is only apprehended when it is preceded by experience of the sciences; it then displays itself to mind as the universal truth, not as a particular knowledge alongside other matters and realities, but as the essential being of all these latter.

general abstract, concrete general

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.

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.

Vorbegriff (Enzyklopädie)

Erste Stellung des Gedankens zur Objektivität

Zweite Stellung des Gedankens zur Objektivität

Dritte Stellung des Gedankens zur Objektivität

EL§61 If we are to believe the Critical philosophy, thought is subjective, and its ultimate and invincible mode is abstract universality or formal identity. Thought is thus set in opposition to Truth, which is no abstraction, but concrete universality. In this highest mode of thought, which is entitled Reason, the Categories are left out of account. The extreme theory on the opposite side holds thought to be an act of the particular only, and on that ground declares it incapable of apprehending the Truth. This is the Intuitional theory.

abstract general, concrete general

In Hibben-Luft, p. 143 is says about the Shorter Logic:

Particularity and individuality are related as “abstract” and “concrete”, respectively. The particular is the “abstract individual”. The individual is the “concrete particular”. The universal is their union, and may be either “abstract” or “concrete”. The so-called “concrete universal” is Hegel’s gold standard for conceptual thought [[]].

Näherer Begriff und Einteilung der Logik

Das Logische hat der Form nach drei Seiten:

α) die abstrakte oder verständige,

β) die dialektische oder negativ-vernünftige,

γ) die spekulative oder positiv-vernünftige.

Abstrakte oder Verständige Logik
Dialektische oder negativ-vernünftige Logic
Dialektische oder positive-vernünfitge Logik

Enc§82a Im gemeinen Leben pflegt der Ausdruck Spekulation in einem sehr vagen und zugleich untergeordneten Sinn gebraucht zu werden, so z. B., wenn von Heirats- oder Handelsspekulationen die Rede ist, worunter dann nur so viel verstanden wird, einerseits daß über das unmittelbar Vorhandene hinausgegangen werden soll und andererseits daß dasjenige, was den Inhalt solch Spekulationen bildet, zunächst nur ein Subjektives ist, jedoch nicht ein solches bleiben, sondern realisiert oder in Objektivität übersetzt werden soll.

Enc§82b Es gilt von diesem gemeinen Sprachgebrauch hinsichtlich der Spekulationen dasselbe, was früher von der Idee bemerkt wurde, woran sich dann noch die weitere Bemerkung schließt, daß vielfältig von solchen, die sich schon zu den Gebildeteren rechnen, von der Spekulation auch ausdrücklich in der Bedeutung eines bloß Subjektiven gesprochen wird, in der Art nämlich, daß es heißt, eine gewisse Auffassung natürlicher oder geistiger Zustände und Verhältnisse möge zwar, bloß spekulativ genommen, sehr schön und richtig sein, allein die Erfahrung stimme damit nicht überein, und in der Wirklichkeit könne dergleichen nicht zugelassen werden. Dagegen ist dann zu sagen, daß das Spekulative seiner wahren Bedeutung nach weder vorläufig noch auch definitiv ein bloß Subjektives ist, sondern vielmehr ausdrücklich dasjenige, welches jene Gegensätze, bei denen der Verstand stehenbleibt (somit auch den des Subjektiven und Objektiven), als aufgehoben in sich enthält und eben damit sich als konkret und als Totalität erweist.

Enc§82c Ein spekulativer Inhalt kann deshalb auch nicht in einem einseitigen Satz ausgesprochen werden. Sagen wir z. B., das Absolute sei die Einheit des Subjektiven und des Objektiven, so ist dies zwar richtig, jedoch insofern einseitig, als hier nur die Einheit ausgesprochen und auf diese der Akzent gelegt wird, während doch in der Tat das Subjektive und das Objektive nicht nur identisch, sondern auch unterschieden sind.

Enc§82d Hinsichtlich der Bedeutung des Spekulativen ist hier noch zu erwähnen, daß man darunter dasselbe zu verstehen hat, was früher, zumal in Beziehung auf das religiöse Bewußtsein und dessen Inhalt, als das Mystische bezeichnet zu werden pflegte. Wenn heutzutage vom Mystischen die Rede ist, so gilt dies in der Regel als gleichbedeutend mit dem Geheimnisvollen und Unbegreiflichen, und dies Geheimnisvolle und Unbegreifliche wird dann, je nach Verschiedenheit der sonstigen Bildung und Sinnesweise, von den einen als das Eigentliche und Wahrhafte, von den anderen aber als das dem Aberglauben und der Täuschung Angehörige betrachtet. Hierüber ist zunächst zu bemerken, daß das Mystische allerdings ein Geheimnisvolles ist, jedoch nur für den Verstand, und zwar einfach um deswillen, weil die abstrakte Identität das Prinzip des Verstandes, das Mystische aber (als gleichbedeutend mit dem Spekulativen) die konkrete Einheit derjenigen Bestimmungen ist, welche dem Verstand nur in ihrer Trennung und Entgegensetzung für wahr gelten. Wenn dann diejenigen, welche das Mystische als das Wahrhafte anerkennen, es gleichfalls dabei bewenden lassen, daß dasselbe ein schlechthin Geheimnisvolles sei, so wird damit ihrerseits nur ausgesprochen, daß das Denken für sie gleichfalls nur die Bedeutung des abstrakten Identischsetzens hat und daß man um deswillen, um zur Wahrheit zu gelangen, auf das Denken verzichten oder, wie auch gesagt zu werden pflegt, daß man die Vernunft gefangennehmen müsse.

speculation is mysticism

Enc§82e Nun aber ist, wie wir gesehen haben, das abstrakt verständige Denken so wenig ein Festes und Letztes, daß dasselbe sich vielmehr als das beständige Aufheben seiner selbst und als das Umschlagen in sein Entgegengesetztes erweist, wohingegen das Vernünftige als solches gerade darin besteht, die Entgegengesetzten als ideelle Momente in sich zu enthalten. Alles Vernünftige ist somit zugleich als mystisch zu bezeichnen, womit jedoch nur so viel gesagt ist, daß dasselbe über den Verstand hinausgeht, und keineswegs, daß dasselbe überhaupt als dem Denken unzugänglich und unbegreiflich zu betrachten sei.”

Allgemeine Einteilung des Seins

§126 Diese Eintheilung ist hier, wie in der Einleitung von diesen Eintheilungen überhaupt erinnert worden, eine vorläufige Anführung; ihre Bestimmungen haben erst aus der Bewegung des Seyns selbst zu entstehen, sich dadurch zu definiren und zu rechtfertigen. Über die Abweichung dieser Eintheilung von der gewöhnlichen Aufführung der Kategorien, nämlich als Quantität, Qualität, Relation und Modalität, was übrigens bei Kant nur die Titel für seine Kategorien seyn sollen, in der That aber selbst, nur allgemeinere, Kategorien sind,—ist hier nichts zu erinnern, da die ganze Ausführung das überhaupt von der gewöhnlichen Ordnung und Bedeutung der Kategorien Abweichende zeigen wird.

§126 At this stage, this division is, as was remarked of these divisions generally in the Introduction, a preliminary statement; its determinations have first to arise from the movement of being itself and in so doing define and justify themselves. As regards the divergence of this classification from the usual presentation of the categories, namely, as quantity, quality, relation and modality — these moreover with Kant are supposed to be only titles for his categories though they are, in fact, themselves categories, only more general ones — this calls for no special comment here, as the entire exposition will show a complete divergence from the usual arrangement and significance of the categories.

category (philosophy)

§127 Nur dieß kann etwa bemerkt werden, daß sonst die Bestimmung der Quantität von der Qualität aufgeführt wird,—und dieß—wie das Meiste—ohne weiteren Grund. Es ist bereits gezeigt worden, daß der Anfang sich mit dem Seyn als solchem macht, daher mit dem qualitativen Seyn. Aus der Vergleichung der Qualität mit der Quantität erhellt leicht, daß jene die der Natur nach erste ist. Denn die Quantität ist die schon negativ gewordenen Qualität; die Größe ist die Bestimmtheit, die nicht mehr mit dem Seyn Eins, sondern schon von ihm unterschieden, die aufgehobene, gleichgültig gewordenen Qualität ist. Sie schließt die Veränderlichkeit des Seyns ein, ohne daß die Sache selbst, das Seyn, dessen Bestimmung sie ist, durch sie verändert werde; da hingegen die qualitative Bestimmtheit mit ihrem Seyn Eins ist, nicht darüber hinausgeht, noch innerhalb desselben steht, sondern dessen unmittelbare Beschränktheit ist. Die Qualität ist daher, als die unmittelbare Bestimmtheit die erste und mit ihr der Anfang zu machen.

§127 This only perhaps can be remarked, that hitherto the determination of quantity has been made to precede quality and this as is mostly the case — for no given reason. It has already been shown that the beginning is made with being as such, therefore, with qualitative being. It is easily seen from a comparison of quality with quantity that the former by its nature is first. For quantity is quality which has already become negative; magnitude is the determinateness which is no longer one with being but is already differentiated from it, sublated quality which has become indifferent. It includes the alterableness of being, although the category itself, namely Being, of which it is the determination, is not altered by it. The qualitative determinateness, on the other hand, is one with its being: it neither goes beyond it nor is internal to it, but is its immediate limitedness. Quality therefore, as the immediate determinateness, is primary and it is with it that the beginning must be made.

§128 Das Maaß ist eine Relation, aber nicht die Relation überhaupt, sondern bestimmt der Qualität und Quantität zu einander; die Kategorien, die Kant unter der Relation befaßt, werden ganz anderwärts ihre Stelle nehmen. Das Maaß kann auch für eine Modalität, wenn man will, angesehen werden; aber indem bei Kant diese nicht mehr eine Bestimmung des Inhalts ausmachen, sondern nur die Beziehung desselben auf das Denken, auf das Subjektive, angehen soll, so ist dieß eine ganz heterogene, hierher nicht gehörige Beziehung.

§128 Measure is a relation, but not relation in general, for it is the specific relation between quality and quantity; the categories which Kant includes under relation will come up for consideration in quite another place. Measure can also, if one wishes, be regarded as a modality; but since with Kant modality is supposed no longer to constitute a determination of the content, but to concern only the relation of the content to thought, to the subjective element, it is a quite heterogeneous relation and is not pertinent here.

§129 Die dritte Bestimmung des Seyns fällt innerhalb des Abschnittes, der Qualität, indem es sich als abstrakte Unmittelbarkeit zu einer einzelnen Bestimmtheit gegen seine anderen innerhalb seiner Sphäre herabsetzt.

§129 The third determination of being falls within the section Quality, for as abstract immediacy it reduces itself to a single determinateness in relation to its other determinatenesses within its sphere.

First section. Bestimmtheit (Qualität) / Determinateness (Quality)

First chapter

From the shorter Logic:

EL§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.

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

{EL#88} EL§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. See also §1663.

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 in def. 2 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

which points to the Aufhebung of this duality via the sharp modality.

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.

Mathematically, the vanishing of infinitesimal objects is exactly what 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. Aufheben des Werdens / Sublating of Becoming

§180 Das Gleichgewicht, worein sich Entstehen und Vergehen setzen, ist zunächst das Werden selbst. Aber dieses geht eben so in ruhige Einheit zusammen. Seyn und Nichts sind in ihm nur als verschwindende; aber das Werden als solches ist nur durch die Unterschiedenheit derselben. Ihr Verschwinden ist daher das Verschwinden des Werdens, oder Verschwinden des Verschwindens selbst. Das Werden ist eine haltungslose Unruhe, die in ein ruhiges Resultat zusammensinkt.

§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 Dieß könnte auch so ausgedrückt werden: Das Werden ist das Verschwinden von Seyn in Nichts, und von Nichts in Seyn, und das Verschwinden von Seyn und Nichts überhaupt; aber es beruht zugleich auf dem Unterschiede derselben. Es widerspricht sich also in sich selbst, weil es solches in sich vereint, das sich entgegengesetzt ist; eine solche Vereinigung aber zerstört sich.

§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 Dieß Resultat ist das Verschwundenseyn, aber nicht als Nichts; so wäre es nur ein Rückfall in die eine der schon aufgehobenen Bestimmungen, nicht Resultat des Nichts und des Seyns. Es ist die zur ruhigen Einfachheit gewordene Einheit des Seyns und Nichts. Die ruhige Einfachheit aber ist Seyn, jedoch ebenso, nicht mehr für sich, sondern als Bestimmung des Ganzen.

§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 Das Werden so Übergehen in die Einheit des Seyns und Nichts, welche als seyend ist, oder die Gestalt der einseitigen unmittelbaren Einheit dieser Momente hat, ist das Daseyn.

§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.

By the discussion around §177 the unity of opposites of nothing and being is to be expressed by the adjunction

* \emptyset \dashv \ast

between the modalities which are constant on the empty type/initial object and on the unit type/terminal object, respectively.

To exhibit Aufhebung of this duality we are now to produce another such adjunction of the form ()(\flat \dashv \sharp) which characterizes a higher level of a topos, and such that both \emptyset and *\ast are \sharp-modal types.

This is the case for \flat the flat modality and \sharp the sharp modality over a cohesive site, this is discussed at Aufhebung – over cohesive sites.

So the first step to further determination in the Proceß is this:

Dasein AufhebungdesWerdens reinesSein Nichts Werden * Sein \array{ && && &&\stackrel{Dasein}{}&& \flat &\stackrel{}{\dashv}& \sharp & \\ && && &&&& \vee &\stackrel{Aufhebung \atop {des\;Werdens}}{}& \vee \\ && &&&&\stackrel{reines\;Sein}{}&\stackrel{Nichts}{}& \emptyset &\stackrel{Werden}{\dashv}& \ast & \stackrel{Sein}{} }
Dasein
Werden :Nichts\;\;\;\dashvSein: Vergehen

\,

Dasein
becoming :nothing\;\;\;\dashvbeing: ceasing

§187 Der nähere Sinn und Ausdruck, den Seyn und Nichts, indem sie nunmehr Momente sind, erhalten, hat sich bei der Betrachtung des Daseyns, als der Einheit, in der sie aufbewahrt sind, zu ergeben. Seyn ist Seyn, und Nichts ist Nichts nur in ihrer Unterschiedenheit von einander; in ihrer Wahrheit aber, in ihrer Einheit, sind sie als diese Bestimmungen verschwunden, und sind nun etwas anderes. Seyn und Nichts sind dasselbe; darum weil sie dasselbe sind, sind sie nicht mehr Seyn und Nichts, und haben eine verschiedene Bestimmung; im Werden waren sie Entstehen und Vergehen; im Daseyn als einer anders bestimmten Einheit sind sie wieder anders bestimmte Momente. Diese Einheit bleibt nun ihre Grundlage, aus der sie nicht mehr zur abstrakten Bedeutung von Seyn und Nichts heraustreten.

§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 als solches / 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 (ʃ)(&#643; \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 (ʃ)(&#643; \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,

something: type, object

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

§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.

§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.

B. Die Endlichkeit / Finitude.
a. Etwas und ein Anderes. / Something and an Other

§221 Seyn-für-Anderes und Ansichseyn machen die zwei Momente des Etwas aus.

§221 Being-for-other and being-in-itself constitute the two moments of the something.

Hence a unity of opposites:

Ansichsein Etwas SeinfuerAnderes. \array{ Ansichsein &\stackrel{Etwas}{\dashv}& Sein-fuer-Anderes } \,.

This will repeat in the Wesenslogik with Etwas replaced by Ding, see §1048.

momentunitycomoment
SeinslogikAnsichseynEtwasSein-fuer-Anderes
WesenslogikExistenzDing

Notice that there in the discussion of Ding is a comment hidden that concerns the Etwas here:

§1056 Die Qualität ist die unmittelbare Bestimmtheit des Etwas; das Negative selbst, wodurch das Seyn Etwas ist.

§1056 Quality is the immediate determinateness of something, the negative itself through which being is something.

This says that the the “immediate determinateness” of the adjunction in §221 is the adjunction shape modality\dashvflat modality that we identify with quality. We display this in the Process.

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

This is the Aufhebung discussed around §182, §183

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.

For the last one, see below §1636.

C. Die Unendlichkeit
a. Das Unendliche Überhaupt
b. Wechselwirkung des Endlichen und Unendlichen
c. Die affirmative Unendlichkeit

§304 In Beziehung auf Realität und Idealität wird aber der Gegensatz des Endlichen und Unendlichen so gefaßt, daß das Endliche für das Reale gilt, das Unendliche aber für das Ideelle gilt; wie auch weiterhin der Begriff als ein Ideelles und zwar als ein nur Ideelles, das Daseyn überhaupt dagegen als das Reale betrachtet wird.

With reference to reality and ideality, however, the opposition of finite and infinite is grasped in such a manner that the finite ranks as the real but the infinite as the ‘ideal’ [das Ideelle]; in the same way that further on the Notion, too, is regarded as an ‘ideal’, that is, as a mere ‘ideal’, in contrast to determinate being as such which is regarded as the real.

Notice here from Der quantitative unendliche Progress and §530 that “infinite” refers much to the infinite progression that in mathematics is referred to as sequences and series, and that what “ideal” ( ideell ) about them is that they need not converge to any finite value, but be regarded as sequences — such as formal power series. Hence if we take the “infinite” and the “infinitesimal” to go together – as also in §502 – then §304 gives that “the infinitesimal ranks as the ideal”, whereas “the reduced ranks as the real”. See also the discussion below §305.

This way we may think of the “ideal” here as related to the idealization involved in the concept of infinitesimals, which are “not real” in an evident sense and of the “real” hence of the finite, the non-infinitesimal.

Now the reduction modality \Re is the operation that makes all infinitesimal vanish, i.e. it expresses precisely the “vanishing of infinitesimals” in §174. Hence its modal types are precisely those without infinitesimal extension. It is perfectly plausible to think of these as the real types.

Moreover, the \Re-anti-modal types are indeed precisely those that consist entirely only of infinitesimals (the infinitesimally thickened points), hence those which are “ideel” but not “real” in the sense of §304.

We discuss below §305 the adjunction that \Re-participates in and the unity of opposites that this may be thought to express.

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.

If here we take the “infinite” and the “infinitesimal” to go together – as in §502 – then the above would give also that ideality is the quality of the infinitesimal.

Now we have from §699 that quality, being the oppisite moment of quantity, is the duality of opposites formalized by shape modality and flat modality

quality:ʃ \array{ quality \colon &#643; \dashv \flat }

This indeed has a direct infinitesimal analog, namely the adjunction between the infinitesimal shape modality \Im and infinitesimal flat modality &\& .

This clearly suggests to translate “ideality is the quality of the infinite” in §305 as the unity of opposites which is expressed by the adjunction & \Im \dashv \&

infqualityideality & ʃ quality \array{ & & \Im &\stackrel{ideality}{\stackrel{inf\, quality}{\dashv}}& \& \\ && \vee && \vee \\ && &#643; &\stackrel{quality}{\dashv}& \flat }

as part of the Proceß. From §322 we see what the moments here are to be called:

being-for-self inf.qualityideality/ & being-for-one attraction ʃ quality repulsion \array{ & \text{being-for-self} & \Im &\stackrel{ideality /}{\stackrel{inf.\, quality}{\dashv}}& \& & \text{being-for-one} \\ && \vee && \vee \\ & attraction & &#643; &\stackrel{quality}{\dashv}& \flat & repulsion }

In fact, this provides Aufhebung for quality, as indicated, which we may read as the Aufhebung of the finite as it passes into the infinite (which we read as expressed via the infinitesimal).

Moreover, this extends to an adjoint triple &\Re \dashv \Im \dashv \& , with the reduction modality \Re (from the discussion below §304) on the left

This hence gives a second order unity of opposites

ideality/inf.quality & \array{ && \Re &\dashv& \Im \\ && \bot && \bot \\ && \Im &\stackrel{{ideality/ \atop {inf.\,quality}}}{\dashv}& \& }

and hence exibits a dual moment of ideality, which, by the discussion below §304, is related to reality.

Now by §324 the opposite moment of ideality is indeed supposed to be reality. So we should write

reality ideality/inf.quality & \array{ && \Re &\stackrel{reality}{\dashv}& \Im \\ && \bot && \bot \\ && \Im &\stackrel{{ideality/ \atop {inf.\,quality}}}{\dashv}& \& }

and interpret not just the reduction modality alone as being about reality, but as being just one moment of it, the other moment being expresed by the infinitesimal shape modality \Im .

This happens to make good sense: the modal types of \Im in context XX are the étale spaces over XX, exhibiting étale groupoids (see the discussion at differential cohesion for details). In terms of geometry this is what characterizes among all generalized geometric objects those that are manifolds, orbifolds and generally, geometric stacks. These are indeed “the real spaces” as opposed to non-étale spaces such as generic moduli stacks, in that a “real space” such as a spacetime is an geometric stack, while some “abstract”, hence maybe “ideal” space such as that of “all electromagnetic field configurations” (a moduli stack) of not an étale groupoid. (See also the discussion of this point at higher geometry).

Therefore the infinitesimal shape-modal types certainly qualify as one “aspect of reality” in any mathematical description of physics (see also at geometry of physics), and so we conclude that reading the adjunction as

reality: reality \colon \Re \dashv \Im

makes good sense.

Notice that – and this is of course precisely what the second-order duality with ideality:&ideality \colon \Im \dashv \& expresses – while this is all about reality, in the above sense, it is so only via ideal (ideelle) infinitesimals. This is of course in a way just the big insight of Leibniz when formulating differential calculus in terms of infinitesimals (today: synthetic differential geometry): in order to express the physical reality that is described, notably, by differential equations, it is most useful to consider the idealized concept of infinitesimals, itself without reality, but nevertheless serving to characterize reality.

The view that the concept of infinitesimals are closely related to reality is also expressed in Cohen83, secion 19:

Für diesen Höhepunkt kritischer Naturerkenntnis bildet die Charakteristik der infinitesimalen Größe als intensiver die notwendige Vermittlung; denn die kritische Bedeutung der Realität wird vorzugsweise an der infinitesimalen Intensität durchgeführt.

In conclusion, we add to the Proceß the following piece

vanishingofinfinitesimals reality ideality/inf.quality & ʃ quality \array{ & \stackrel{vanishing \atop {of\, infinitesimals}}{} & \Re &\stackrel{reality}{\dashv}& \Im \\ && \bot && \bot \\ & & \Im &\stackrel{ideality/ \atop {inf.\, quality} }{\dashv}& \& \\ && \vee && \vee \\ && &#643; &\stackrel{quality}{\dashv}& \flat }

Moreover, here the bottom step upward is an Aufhebung in the mathematical sense that &ʃʃ\& \circ &#643; \simeq &#643;. In view of this, we learn from §305 “Die Idealität … Als Aufheben der Endlichkeit” that we might label this as

vanishingofinfinitesimals reality Idee ideality/inf.quality & Aufhebungder.Endlichkeit ʃ quality \array{ & \stackrel{vanishing \atop {of\, infinitesimals}}{} & \Re &\stackrel{reality}{\dashv}& \Im \\ && \bot &\stackrel{Idee}{}& \bot \\ & & \Im &\stackrel{ideality/ \atop {inf.\, quality} }{\dashv}& \& \\ && \vee &\stackrel{Aufhebung \atop {der.\, Endlichkeit}}{}& \vee \\ && &#643; &\stackrel{quality}{\dashv}& \flat }

§316 Anmerkung 2. Der Satz, daß das Endliche ideell ist, macht den Idealismus aus. Der Idealismus der Philosophie besteht in nichts anderem, als darin, das Endliche nicht als ein wahrhaft Seyendes anzuerkennen. Jede Philosophie ist wesentlich Idealismus, oder hat denselben wenigstens zu ihrem Princip, und die Frage ist dann nur, inwiefern dasselbe wirklich durchgeführt ist. Die Philosophie ist es so sehr als die Religion; denn die Religion anerkennt die Endlichkeit ebenso wenig als ein wahrhaftes Seyn, als ein Letztes, Absolutes, oder als ein Nicht-Gesetztes, Unerschaffenes, Ewiges. Der Gegensatz von idealistischer und realistischer Philosophie ist daher ohne Bedeutung. Eine Philosophie, welche dem endlichen Daseyn als solchem wahrhaftes, letztes, absolutes Seyn zuschriebe, verdiente den Namen Philosophie nicht; Principien älterer oder neuerer Philosophien, das Wasser, oder die Materie oder die Atome sind Gedanken, Allgemeine, Ideelle, nicht Dinge, wie sie sich unmittelbar vorfinden, d. h. in sinnlicher Einzelnheit, selbst jenes thaletische Wasser nicht; denn, obgleich auch das empirische Wasser, ist es außerdem zugleich das Ansich oder Wesen aller anderen Dinge; und diese sind nicht selbstständige, in sich gegründete, sondern aus einem Anderen, dem Wasser, gesetzte, d. i. ideelle. Indem vorhin das Princip, das Allgemeine, das Ideelle genannt worden, wie noch mehr der Begriff, die Idee, der Geist, Ideelles zu nennen ist, und dann wiederum die einzelnen sinnlichen Dinge als ideell im Princip, im Begriffe, noch mehr im Geiste, als aufgehoben sind, so ist dabei auf dieselbe Doppelseite vorläufig aufmerksam zu machen, die bei dem Unendlichen sich gezeigt hat, nämlich daß das eine Mal das Ideelle das Konkrete, Wahrhaftseyende ist, das andere Mal aber ebenso sehr seine Momente das Ideelle, in ihm Aufgehobene sind, in der That aber nur das Eine konkrete Ganze ist, von dem die Momente untrennbar sind.

idealism

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 : attraction \dashv repulsion

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: ʃ quantity: discreteness continuity \array{ & attraction && repulsion \\ quality : & &#643; &\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. Seyn-für-eines / Being-for-one

§322 Seyn-für-eines – Dieß Moment drückt aus, wie das Endliche in seiner Einheit mit dem Unendlichen oder als Ideelles ist.

Für-sich-seyn und Für-Eines-seyn sind also nicht verschiedene Bedeutungen der Idealität, sondern sind wesentliche, untrennbare Momente derselben.

§322 Being-for-one – This moment expresses the manner in which the finite is present in its unity with the infinite, or is an ideal being [Ideelles].

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

By the discussion below §305 the inclusion of flat modal types into infinitesimal flat modality modal types

& \array{ \& \\ \vee \\ \flat }

reflects the Aufhebung which is the passage from the finite to the infinitesimal.

So we may pronounce, in the Proceß the infinitesimal flat modality as Seyn-fuer-eines, being-for-one.

Below §305 we find the adjunction which plausibly captures the unity of opposites called ideality, and hence by §322 what its moments are to be called.

Fuersichsein Idealitaet & Fuereinssein \array{ Fuersichsein & \Im &\stackrel{Idealitaet}{\dashv}& \& & Fuereinssein }

See also the discussion below §348

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 we have another unity of opposites is idealityrealityideality \dashv reality. See also at The One and the Many.

It seems that in Science of Logic there is no name given to this unity. But in the shorter logic there is:

EL§214 Die Idee kann als die Vernunft, (dies ist die eigentliche philosophische Bedeutung der Vernunft), ferner als das Subjekt-Objekt, als die Einheit des Ideellen und Reellen, des Endlichen und Unendlichen, der Seele und des Leibs, als die Möglichkeit, die ihre Wirklichkeit an ihr selbst hat, als das, dessen Natur nur als existierend begriffen werden kann, u.s.f gefaßt werden; weil in ihr alle Verhältnisse des Verstandes, aber in ihrer unendlichen Rückkher, und Identität in sich enthalten sind.

EL§214 The Idea may be described in many ways. It may be called reason; (and this is the proper philosophical signification of reason); subject-object; the unity of the ideal and the real, of the finite and the infinite, of soul and body; the possibility which has its actuality in its own self; that of which the nature can be thought only as existent, etc.

realideal
finiteinfinitesimal
bodysoul

(Regarding “soul and body” see also the comments at The monad of Leibniz, where the similarity to the standard terminology “soul” and “body” for (super-)infinitesimals is pointed out.)

But see below The idea, where the idea is given as the unity of the notion and the real. See the discussion below §1683.

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 ʃ&#643; 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 ʃ&#643; \emptyset \simeq \emptyset) and this is

ʃ * &#643; {}_{\ast}

Below we see that this, being a discrete type, is what Hegel describes with “repulsion”: The points in ʃ*&#643; \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)

ʃ *lim I* &#643; {}_{\ast} \simeq \underset{\longrightarrow}{\lim}_I \ast

where the diagram II that the colimit is over is I=ʃ *I = &#643; {}_{\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. (This happens to fit well with EL§214) 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.

Remark: The unity of the One and the Many

§357 It is an ancient proposition that the one is many and especially that the many are one. We may repeat here the observation that the truth of the one and the many expressed in propositions appears in an inappropriate form, that this truth is to be grasped and expressed only as a becoming, as a process, a repulsion and attraction-not as being, which in a proposition has the character of a stable unity. We have already mentioned and recalled the dialectic of Plato in the Parmenides concerning the derivation of the many from the one, namely, from the proposition: the one is. The inner dialectic of the Notion has been stated; it is easiest to grasp the dialectic of the proposition, that the many are one, as an external reflection; and it may properly be grasped externally here inasmuch as the object too, the many, are mutually external. It directly follows from this comparison of the many with one another that any one is determined simply like any other one; each is a one, each is one of the many, is by excluding the others — so that they are absolutely the same, there is present one and only one determination. This is the fact, and all that has to be done is to grasp this simple fact. The only reason why the understanding stubbornly refuses to do so is that it has also in mind, and indeed rightly so, the difference; but the existence of this difference is just as little excluded because of the said fact, as is the certain existence of the said fact in spite of the difference. One could, as it were, comfort understanding for the naive manner in which it grasps the fact of the difference, by assuring it that the difference will

b. Das eine Eins der Attraktion
c. Die Beziehung der Repulsion und der Attraktion

§361 The difference of the one and the many is now determined as the difference of their relation to one another, with each other, a relation which splits into two, repulsion and attraction, each of which is at first independent of the other and stands apart from it, the two nevertheless being essentially connected with each other. Their as yet indeterminate unity is to be more precisely ascertained

So we are looking now for a unity of opposites of the form

attractionrepulsion. attraction \dashv repulsion \,.

The natural choice is shape modality \dashv flat modality ʃ&#643; \dashv \flat. For instance for n\mathbb{R}^n a Cartesian space then ʃ n*&#643; \mathbb{R}^n \simeq \ast is a One into which all the points of the space have collapsed, whereas n\flat \mathbb{R}^n is the many Ones out of which this space consist.

§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.

§372 This unity is, therefore, [[]] determinate being

Since determinate being, Dasein and Fürsichsein both are qualitative being/are quality (§188, §318, §321) the unity of opposites from §361 should be quality

quality:attractionrepulsion quality \;\colon\; attraction \dashv repulsion

By §361 we put

attraction ʃ quality repulsion. \array{ attraction & &#643; & \stackrel{quality}{\dashv} & \flat & repulsion } \,.

Continuing the example from §361 this makes perfect sense: given a Cartesian space n\mathbb{R}^n then the opposition between

  1. the single One ʃ n&#643; \mathbb{R}^n obtained by having all its points collapse under the attraction of its cohesion;

  2. the many Ones n\flat \mathbb{R}^n obtained by having the points of n\mathbb{R}^n repel each other against their cohesive attraction

exhibits exactly the cohesive (continuous, smooth) quality of n\mathbb{R}^n, the quality that distinguishes it from the bare set n\flat \mathbb{R}^n of its underlying points, as well as from the bare contractible homotopy type X\int X obtained from it.

Notice that later when Nature has appeared, the unity of attraction and repulsion becomes gravity PN§204.

§372 This unity is, therefore, [a] being, only as affirmative, that is immediacy, which is self-mediated through negation of the negation; being is posited as the unity which pervades its determinatenesses, limit, etc., which are posited in it as sublated; [b] determinate being: in such determination it is the negation or determinateness as a moment of affirmative being, yet determinateness no longer as immediate, but as reflected into itself, as related not to an other but to itself; a being determined simply in itself-the one; the otherness as such is itself a being-for-self; [c] being-for-self, as that being which continues itself right through the determinateness and in which the one and the intrinsic determinedness is itself posited as sublated. The one is determined simultaneously as having gone beyond itself, and as unity; hence the one, the absolutely determined limit, is posited as the limit which is no limit, which is present in being but is indifferent to it.

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 (yet) 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).

But later when Nature has appeared, the unity of attraction and repulsion indeed becomes gravity PN§204.

Second section. The magnitude

First chapter. Die Quantität / The quantity

A. Die reine Quantität / Pure quantity

§396 Continuity is, therefore, simple, self-same self-relation, which is not interrupted by any limit or exclusion; it is not, however, an immediate unity, but a unity of ones which possess being-for-self. The asunderness of the plurality is still contained in this unity, but at the same time as not differentiating or interrupting it. In continuity, the plurality is posited as it is in itself; the many are all alike, each is the same as the other and the plurality is, consequently, a simple, undifferentiated sameness. Continuity is this moment of self-sameness of the asunderness, the self-continuation of the different ones into those from which they are distinguished.

This kind of continuity is expressed by the sharp modality \sharp.

§397 In continuity, therefore, magnitude immediately possesses the moment of discreteness — repulsion, as now a moment in quantity. Continuity is self-sameness, but of the Many which, however, do not become exclusive; it is repulsion which expands the selfsameness to continuity. Hence discreteness, on its side, is a coalescent discreteness, where the ones are not connected by the void, by the negative, but by their own continuity and do not interrupt this self-sameness in the many.

An opposite, and so this is the flat modality in which all points lose their cohesive attraction and repel each other to isolated point.s

§398 Die Quantität ist die Einheit dieser Momente, der Kontinuität und Diskretion

§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} }

Hence we add to the Proceß the unity of opposites

discreteness quantity continuity \array{ discreteness & \flat &\stackrel{quantity}{\dashv}& \sharp & continuity }

EL§99 Die Quantität ist das reine Sein, an dem die Bestimmtheit nicht mehr als eins mit dem Sein selbst, sondern als aufgehoben oder gleichgültig gesetzt ist.

EL§99 Quantity is pure Being, where the mode or character is no longer taken as one with the being itself, but explicitly put as superseded or indifferent.

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 ʃX&#643; X is the result of letting things collapse 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 ʃX repulsion attraction/cohesion \array{ \flat X &\longrightarrow& X &\longrightarrow& &#643; 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

§437 Quantum, which to begin with is quantity with a determinateness or limit in general is, in its complete determinateness, number. Quantum differentiates itself secondly, into (a) extensive quantum, in which the limit is a limitation of the determinately existent plurality; and (b) intensive quantum or degree, the determinate being having made the transition into being-for-self. Intensive quantum as both for itself and at the same time immediately outside itself — since it is an indifferent limit — has its determinateness in an other. As this manifest contradiction of being determined simply within itself yet having its determinateness outside it, pointing outside itself for it, quantum posited as being in its own self external to itself, passes over thirdly, into quantitative infinity.

extensive and intensive quantity

A. Die Zahl

natural number object (a discrete object)

§441 Quantum completely posited in these determinations is number. The complete positedness lies in the existence of the limit as a plurality and so in its distinction from the unity. Consequently, number appears as a discrete magnitude, but in the unity it equally possesses continuity.

real number object

B. Extensives und Intensives Quantum

extensive and intensive quantity

In modern thermodynamics

  • an extensive quantity is one expressed by a differential form in positve degree (for instance a mass density 3-form on Euclidean 3-space) – this is close to the “extension” of differential forms in Grassmann?’s Ausdehnungslehre;

  • an intensive quantity is one expressed by a differential form of degree 0, namely by a function (for instance a temperator function on that Euclidean 3-space).

The existence and difference between these two concepts is neatly encoded by the sharp modality \sharp:

an intensive quantity object such as the smooth sheaf \mathbb{R} of real numbers is characterized by being a concrete object, witnessed by the fact that the unit of the sharp modality is a monomorphism

\mathbb{R} \hookrightarrow \sharp \mathbb{R}.

On the other hand extensive quantities such as given by the sheaves Ω p\mathbf{\Omega}^p of differential forms in positive degree p1p \geq 1 are \sharp-anti-modal objects? Ω p*\sharp \mathbf{\Omega}^p \simeq \ast.

C. Die quantitative Unendlichkeit
a. Begriff derselben
b. Der quantitative unendliche Progreß

§500 The progress to infinity is in general the expression of contradiction, here, of that which is implicit in the quantitative finite, or quantum as such. It is the reciprocal determining of the finite and infinite which was considered in the sphere of quality, with the difference that, as just remarked, in the sphere of quantity the limit in its own self dispatches and continues itself into its beyond and hence, conversely, the quantitative infinite too is posited as having quantum within it; for quantum in its self-externality is also its own self, its externality belongs to its determination.

§501 Now the infinite progress is only the expression of this contradiction, not its resolution; but because the one determinateness is continued into its other, the progress gives rise to the show of a solution in a union of both. As at first posed, it is the problem of attaining the infinite, not the actual reaching of it; it is the perpetual generation of the infinite, but it does not get beyond quantum, nor does the infinite become positively present. It belongs to the Notion of quantum to have a beyond of itself. This beyond is first, the abstract moment of the non-being of quantum: the vanishing of quantum is its own act; it is thus related to its beyond as to its infinity, in accordance with the qualitative moment of the opposition. Secondly, however, quantum is continuous with its beyond; quantum consists precisely in being the other of itself, in being external to itself; this externality is, therefore, no more an other than quantum itself; the beyond or the infinite is, therefore, itself a quantum. In this way, the beyond is recalled from its flight and the infinite is attained. But because the infinite now affirmatively present is again a quantum, what has been posited is only a fresh limit; this, too, as a quantum, has again fled from itself, is as such beyond itself and has repelled itself into its non-being, into its own beyond, and as it thus repels itself into the beyond, so equally does the beyond perpetually become a quantum.

§502 Die Kontinuität des Quantums in sein Anderes bringt die Verbindung beider in dem Ausdruck eines Unendlich-Großen oder Unendlich-Kleinen hervor. Da beide die Bestimmung des Quantums noch an ihnen haben, bleiben sie veränderliche und die absolute Bestimmtheit, die ein Für-sichseyn wäre, ist also nicht erreicht. Dieß Außersichseyn der Bestimmung ist in dem gedoppelten Unendlichen, das sich nach dem Mehr und Weniger entgegengesetzt ist, dem Unendlich-großen und Kleinen, gesetzt. An jedem selbst ist das Quantum im perennirenden Gegensatze gegen sein Jenseits erhalten. Das Große noch so sehr erweitert, schwindet zur Unbeträchtlichkeit zusammen; indem es sich auf das Unendliche als auf sein Nichtseyn bezieht, ist der Gegensatz qualitativ; das erweiterte Quantum hat daher dem Unendlichen nichts abgewonnen; dieses ist vor wie nach das Nichtseyn desselben. Oder, die Vergrößerung des Quantums ist keine Näherung zum Unendlichen, denn der Unterschied des Quantums und seiner Unendlichkeit hat wesentlich auch das Moment ein nicht quantitativer Unterschied zu seyn. Es ist nur der ins Engere gebrachte Ausdruck des Widerspruchs; es soll ein Großes d. i. ein Quantum, und unendlich, d. i. kein Quantum seyn.—Eben so das Unendlichkleine ist als Kleines ein Quantum und bleibt daher absolut d. h. qualitativ zu groß für das Unendliche, und ist diesem entgegengesetzt. Es bleibt in beiden der Widerspruch des unendlichen Progresses erhalten der in ihnen sein Ziel gefunden haben sollte.

§502 The continuity of quantum with its other produces the conjunction of both in the expression of an infinitely great or infinitely small. Since both still bear the character of quantum they remain alterable, and the absolute determinateness which would be a being-for-self is, therefore, not attained. This self-externality of the determination is posited in the dual infinite — which is opposed to itself as a ‘more’ and a ‘less’ — in the infinitely great and infinitely small. In each, the quantum is maintained in perpetual opposition to its beyond. No matter how much the quantum is increased, it shrinks to insignificance; because quantum is related to the infinite as to its non-being, the opposition is qualitative; the increased quantum has therefore gained nothing from the infinite, which is now, as before, the non-being of quantum. In other words, the increase of quantum brings it no nearer to the infinite; for the difference between quantum and its infinity is essentially not a quantitative difference. The expression ‘the infinitely great’ only throws the contradiction into sharper relief; it is supposed to be great, that is, a quantum, and infinite, that is, not a quantum. Similarly, the infinitely small is, as small, a quantum, and therefore remains absolutely, that is, qualitatively, too great for the infinite and is opposed to it. In both, there remains the contradiction of the infinite progress which in them should have reached its goal.

infinitesimal

####### Anmerkung 1

####### Anmerkung 2

c. Die Unendlichkeit des Quantums

§530 1. The infinite quantum as infinitely great or infinitely small is itself implicitly the infinite progress;

infinitesimal and limit of a sequence

####### Anmerkung 1

In Rücksicht der Erhaltung des Verhältnisses im Verschwinden der Quantorum findet sich (anderwärts, wie bei Carnot, Réflexions sur la Métaphysique du Calcul Infinitésimal.) der Ausdruck, daß vermöge des Gesetzes der Stätigkeit die verschwindenden Größen noch das Verhältniß, aus dem sie herkommen, ehe sie verschwinden, behalten. —Diese Vorstellung drückt die wahre Natur der Sache aus, insofern nicht die Stätigkeit des Quantums verstanden wird, die es im unendlichen Progreß hat, sich in sein Verschwinden so zu kontinuiren, daß im Jenseits seiner wieder nur ein endliches Quantum, ein neues Glied der Reihe entsteht; ein stätiger Fortgang wird aber immer so vorgestellt, daß die Werthe durchloffen werden, welche noch endliche Quanta sind.

In demjenigen Übergange dagegen, welcher in das wahrhafte Unendliche gemacht wird, ist das Verhältniß das stätige; es ist so sehr stätig und sich erhaltend, daß er vielmehr allein darin besteht, das Verhältniß rein herauszuheben, und die verhältnißlose Bestimmung, d. i. daß ein Quantum, welches Seite des Verhältnisses ist, auch außer dieser Beziehung gesetzt, noch Quantum ist, verschwinden zu machen. —Diese Reinigung des quantitativen Verhältnisses ist insofern nichts anders, als wenn ein empirisches Daseyn begriffen wird. Dieß wird hierdurch so über sich selbst erhoben, daß sein Begriff dieselben Bestimmungen enthält, als es selbst, aber in ihrer Wesentlichkeit und in die Einheit des Begriffes gefaßt, worin sie ihr gleichgültiges, begriffloses Bestehen verloren haben.

Third chapter Das quantitative Verhältnis (quantitative ratio)

Third section. The measure.

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

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

(Repeated in §708, below.)

So by the formalization of unity of opposites we have

measure:quantityquality measure \colon quantity \dashv quality

and since quantity and quality are already themselves unities of opposites, we find that Maß (Eichmaß) is the second-order adjunction

Qualitaet Eichmass Quantitaet \array{ Qualitaet && \int &\dashv& \flat \\ Eichmass && \bot && \bot \\ Quantitaet && \flat &\dashv& \sharp }

See the Proceß diagram.

Notice that in PN§202b it says

The truly philosophical science of mathematics as theory of magnitude would be the science of measures, but this already presupposes the real particularity of things, which is only at hand in concrete nature.

§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;

§708 Das Maaß ist zunächst unmittelbare Einheit des Qualitativen und Quantitativen, so daß (1) erstens ein Quantum ist, das qualitative Bedeutung hat, und als Maaß ist. Dessen Fortbestimmung ist, daß an ihm, dem an sich bestimmten, —der Unterschied seiner Momente, des qualitativen und quantitativen Bestimmtseyns, hervortritt. Diese Momente bestimmen sich weiter selbst zu Ganzen des Maaßes, welche insofern als Selbstständige sind; indem sie sich wesentlich aufeinander beziehen, wird das Maaß (2) zweitens Verhältniß von specifischen Quantis, als selbstständigen Maaßen. Ihre Selbstständigkeit beruht aber wesentlich zugleich auf dem quantitativen Verhältnisse und dem Größenunterschiede; so wird ihre Selbstständigkeit ein Übergehen in einander. Das Maaß geht damit im Maaßlosen zu Grunde.—Dieß Jenseits des Maaßes ist aber die Negativität desselben nur an sich selbst; es ist dadurch (3) drittens die Indifferenz der Maaßbestimmungen, und als reell mit der in ihr enthaltenen Negativität das Maaß gesetzt, als umgekehrtes Verhältniß von Maaßen, welche als selbstständige Qualitäten wesentlich nur auf ihrer Quantität und auf ihrer negativen Beziehung aufeinander beruhen, und damit sich erweisen, nur Momente ihrer wahrhaft selbstständigen Einheit zu seyn, welche ihre Reflexion-in-sich und das Setzen derselben, das Wesen, ist.

§708 At first, measure is only an immediate unity of quality and quantity, so that: (1), we have a quantum with a qualitative significance, a measure. The progressive determining of this consists in explicating what is only implicit in it, namely, the difference of its moments, of its qualitatively and quantitatively determined being. These moments further develop themselves into wholes of measure which as such are self-subsistent. These are essentially in relationship with each other, and so measure becomes (2), a ratio of specific quanta having the form of self-subsistent measures. But their self-subsistence also rests essentially on quantitative relation and quantitative difference; and so their self-subsistence becomes a transition of each into the other, with the result that measure perishes in the measureless. But this beyond of measure is the negativity of measure only in principle; this results (3), in the positing of the indifference of the determinations of measure, and the positing of real measure — real through the negativity contained in the indifference — as an inverse ratio of measures which, as self-subsistent qualities, are essentially based only on their quantity and on their negative relation to one another, thereby demonstrating themselves to be only moments of their truly self-subsistent unity which is their reflection-into-self and the positing thereof, essence.

§709 Die Entwickelung des Maaßes, die im Folgenden versucht worden, ist eine der schwierigsten Materien; indem sie von dem unmittelbaren, äußerlichen Maaße anfängt, hätte sie einer Seits zu der abstrakten Fortbestimmung des Quantitativen (einer Mathematik der Natur) fortzugehen, anderer Seits den Zusammenhang dieser Maaßbestimmung mit den Qualitäten der natürlichen Dinge anzuzeigen, wenigstens im Allgemeinen; denn die bestimmte Nachweisung des aus dem Begriffe des konkreten Gegenstandes hervorgehenden Zusammenhangs des Qualitativen und Quantitativen gehört in die besondere Wissenschaft des Konkreten; wovon Beispiele in der Encykl. der philos. Wissensch. 3te Aufl. . 267 u. 270 Anm. das Gesetz des Falles und das der freien himmlischen Bewegung betreffend, nachzusehen sind. Es mag hierbei dieß überhaupt bemerkt werden, daß die verschiedenen Formen, in welchen sich das Maaß realisirt, auch verschiedenen Sphären der natürlichen Realität angehören. Die vollständige, abstrakte Gleichgültigkeit des entwickelten Maaßes d. i. der Gesetze desselben kann nur in der Sphäre des Mechanismus Statt haben, als in welchem das konkrete Körperliche nur die selbst abstrakte Materie ist; die qualitativen Unterschiede derselben haben wesentlich das Quantitative zu ihrer Bestimmtheit; Raum und Zeit sind die reinen Äußerlichkeiten selbst, und die Menge der Materien, Massen, Intensität des Gewichts, sind ebenso äußerliche Bestimmungen, die an dem Quantitativen ihre eigenthümliche Bestimmtheit haben.

§709 The development of measure which has been attempted in the following chapters is extremely difficult. Starting from immediate, external measure it should, on the one hand, go on to develop the abstract determination of the quantitative aspects of natural objects (a mathematics of nature), and on the other hand, to indicate the connection between this determination of measure and the qualities of natural objects, at least in general; for the specific proof, derived from the Notion of the concrete object, of the connection between its qualitative and quantitative aspects, belongs to the special science of the concrete. Examples of this kind concerning the law of falling bodies and free, celestial motion will be found in the Encyclopedia. of the Phil. Sciences, 3rd ed., Sections 267 and 270, Remark. In this connection the general observation may be made that the different forms in which measure is realised belong also to different spheres of natural reality. The complete, abstract indifference of developed measure, i.e. the laws of measure, can only be manifested in the sphere of mechanics in which the concrete bodily factor is itself only abstract matter; the qualitative differences of such matter are essentially quantitatively determined; space and time are the purest forms of externality, and the multitude of matters, masses, intensity of weight, are similarly external determinations which have their characteristic determinateness in the quantitative element.

Regarding this, see also below §714.

First chapter. Die specifische Quantität.

A. Das specifische Quantum / 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.

This concept of Maßstab in §714 is very explicitly that of Eichmaß, a choice that is made (durch Übereinkommen). The English translation that captures this maybe more properly than “standard” is “gauge”.

This aspect is further amplified below in §725, which states that this choice is the choice of Einheit (unit), i.e. Maßeinheit. Mathematically, indeed, the choice of units is precisely a choice of gauge as in gauge theory. See at physical unit for more on this. By the discussion there (see also at torsor), this is indeed all about ratios, just as stated in §708 above.

(Observe also that §709 above said that to develop a theory of measure, hence a theory of gauge, is to develop a “mathematics of nature”. Moreover, by Philosophy of Nature §202 “The truly philosophical science of mathematics as theory of magnitude would be the science of measures”.)

Therefore by (§699) we may label this part of the Proceß as follows:

.
attraction repulsion quality: ʃ gauge quantity: discreteness continuity \array{ & & attraction && repulsion \\ & quality : & &#643; &\dashv& \flat \\ gauge & \bot & \bot && \bot \\ & quantity : & \flat &\dashv& \sharp \\ & & discreteness && continuity }

This is striking, as at the same time precisely this adjoint triple is also an abstract axiomatization of (higher) gauge theory in physics via cohesion. This is discussed further at differential cohomology hexagon.

Moreover, below after transition from the Seinslogik to the Wesenslogik, find gauge fields again, with the above structure repeated in the Wesenslogik, see the discussion around §714.

B. Specificirendes Maß / Specifying measure
(a) Die Regel / The Rule

§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.

Choice of Einheit unit is choice of gauge, as in gauge theory. See at physical unit for more on this.

(b) Das specificirende Maß / Specifying Measure
(c) Verhältnis beider Seiten als Qualitäten / Relation of the two Sides as Qualities
C. Das Fürsichseyn im Maaße

Chapter two. Das reale Maaß

Chapter three. Das Werden des Wesens.

A. Die absolute Indifferenz
B. Die Indifferenz als umgekehrtes Verhältniß ihrer Faktoren.
C. Übergang in das Wesen.

§803 Die absolute Indifferenz ist die letzte Bestimmung des Seyns, ehe dieses zum Wesen wird;

§803 Absolute indifference is the final determination of being before it becomes essence; but it does not attain to essence.

Under the formalization of unity of opposites, “absolute indifference” is plausibly the name of the trivial adjunction ididid\dashv id which describes a difference that is none, hence an indifference.

Wesen:idid. Wesen \colon id \dashv id \,.

This fits well with the idea in §803 that the process of determination comes to an end in the Wesen, because indeed (idid)(id \dashv id) is the topmost level in the mathematical sense.

Wesen : id id \array{ Wesen &\colon& id &\dashv& id \\ && \vee && \vee \\ && \vdots && \vdots }

See also §812

Die Lehre vom Wesen / The doctrine of essence

§807 Die Wahrheit des Seins is das Wesen.

§807 The truth of being is essence.

§808 Diese Bewegung, als Weg des Wissens vorgestellt, so erscheint dieser Anfang vom Seyn und der Fortgang, der es aufhebt und beim Wesen als einem Vermittelten anlangt, eine Thätigkeit des Erkennens zu seyn, die dem Seyn äußerlich sey und dessen eigene Natur nichts angehe.

§809 Aber dieser Gang ist die Bewegung des Seyns selbst. Es zeigte sich an diesem, daß es durch seine Natur sich erinnert, und durch dieß Insichgehen zum Wesen wird.

Bewegung des Seins from pure being to essence. The Proceß.

§812 Das Wesen aber, wie es hier geworden ist, ist das, was es ist, nicht durch eine ihm fremde Negativität, sondern durch seine eigne, die unendliche Bewegung des Seyns. Es ist An-und-Fürsichseyn; absolutes Ansichseyn, indem es gleichgültig gegen alle Bestimmtheit des Seyns ist, das Andersseyn und die Beziehung auf anderes schlechthin aufgehoben worden ist.

§812 But essence as it has here come to be, is what it is, through a negativity which is not alien to it but is its very own, the infinite movement of being. It is being that is in itself and for itself; it is absolute being-in-itself in that it is indifferent to every determinateness of being, and otherness and relation-to-other have been completely sublated.

So Essence is the ultimate Aufhebung, hence the topmost level

Wesen : id id \array{ Wesen &\colon& id &\dashv& id \\ && \vee && \vee \\ && \vdots && \vdots }

See the Proceß.

§813 weil es Abstoßen seiner von sich oder Gleichgültigkeit gegen sich, negative Beziehung auf sich ist, setzt es sich somit sich selbst gegenüber und ist nur insofern unendliches Fürsichsein, als es die Einheit mit sich in diesem seinem Unterschiede von sich ist.

§813 Because it is self-repelling or indifferent to itself, negative self-relation, it sets itself over against itself and is infinite being-for-self only in so far is as it is at one with itself in this its own difference from itself.

“die Einheit mit sich in diesem seinem Unterschiede von sich”

So indeed the Essence is in opposition to itself, which is what the trivial unity of opposites ididid \dashv id expresses See also §839

§815 Das Wesen steht zwischen Sein und Begriff und macht die Mitte derselben und seine Bewegung den Übergang vom Sein in den Begriff aus. Das Wesen ist das Anundfürsichsein, aber dasselbe in der Bestimmung des Ansichseins; denn seine allgemeine Bestimmung ist, aus dem Sein herzukommen oder die erste Negation des Seins zu sein. Seine Bewegung besteht darin, die Negation oder Bestimmung an ihm zu setzen, dadurch sich Dasein zu geben und das als unendliches Fürsichsein zu werden, was es an sich ist. So gibt es sich sein Dasein, das seinem Ansichsein gleich ist, und wird der Begriff.

§815 Essence stands between being and Notion; it constitutes their mean, and its movement is the transition from being into the Notion. Essence is being-in-and-for-itself, but in the determination of being-in-itself; for the general determination of essence is to have proceeded from being, or to be the first negation of being. Its movement consists in positing within itself the negation or determination, thereby giving itself determinate being and becoming as infinite being-for-self what it is in itself. It thus gives itself its determinate being that is equal to its being-in-itself and becomes Notion.

Der Begriff ist das daseiende Wesen.

§816 Das Wesen scheint zuerst in sich selbst, oder ist Reflexion

§816 At first, essence shines or shows within itself, or is reflection

With the translation from §803,§812 of “Das Wesen” = terminal Aufhebung = “the ambient category”, the original German here translates really to:

The ambient category appears within itself

See also §834, §1036, §1037 below.

The Rezk-Lurie theorem states that the very characterization of an infinity-topos is that it (is presentable with universal colimits and) has a type of types = object classifier = universe (in the mathematical sense!). This type of types is indeed nothing but the small reflection of the full \infty-topos inside it self.

(Indeed, §850 below highlights that “Reflexion” here does not mean the usual “to reflect on a topic” (but means “Reflexion überhaupt”, for what it’s worth).)

There is of course not just one object classifier/type of types but a cumulative hierarchy of them (see also at universe polymorphism)

Type 1Type 2Type 3Type 4 Type_1 \subset Type_2 \subset Type_3 \subset Type_4 \subset \cdots

By the above this should correspond to an infinite Reflexion of the Wesen inside itself, and sure enough, this is what §860b below says.

Notice the following characteristic properties of infinity-topos in relation to WdL:

  1. identity types, characterized by their term introduction rule via the reflector refl:(A=A)refl: (A = A), w

    This matches neatly with section 1, chapter 2 §863, §903.

  2. object classifier=type of types=universe=self-relfecton

    This matches neatly with section 2 §1037, also §834 etc.

  3. local cartesian closure, equivalently incarnated in the base change adjoint triples whose associated monad/comonad adjoint pairs are nothing but the possibility\dashvnecessity-adjunction as discussed at Possible worlds via homotopy type theory

    This matches neatly with section 3 §1160.

In summary the formalization dictionary gives fairly accurately that in Das Wegen Hegel speak about locally Cartesian closed (∞,1)-categories with object classifier.

This is pretty close to being the proposed definition of elementary (∞,1)-topos, as in (Shulman 12).

Hence we formalize Wesen by \infty-topos.

This fits well also with Phen§760: “der Begriff des Wesens ist die absolute Abstraktion, welche reines Denken ist”, “the notion of being is the absolute abstract, which is pure thought”

Phen§760 Denn der Begriff des Wesens, erst indem er seine einfache Reinheit erlangt hat, ist er die absolute Abstraktion, welche reines Denken und damit die reine Einzelnheit des Selbsts, so wie um seiner Einfachheit willen das Unmittelbare oder Sein ist. Was das sinnliche Bewußtsein genannt wird, ist eben diese reine Abstraktion, es ist dies Denken, für welches das Sein das Unmittelbare ist.

Phen§760 For only when the notion of Being has reached its simple purity of nature, is it both the absolute abstraction, which is pure thought and hence the pure singleness of self, and immediacy or objective being, on account of its simplicity. What is called sense-consciousness is just this pure abstraction; it is this kind of thought for which being is the immediate.

Section 1.Das Wesen als Reflexion in ihm selbst. / Essence as Reflection within Itself

§817 Das Wesen ist erstens Reflexion. Die Reflexion bestimmt sich; ihre Bestimmungen sind ein Gesetztseyn, das zugleich Reflexion in sich ist; es sind

zweitens diese Reflexions-Bestimmungen oder die Wesenheiten zu betrachten.

Drittens macht sich das Wesen als die Reflexion des Bestimmens in sich selbst, zum Grunde, und geht in die Existenz und Erscheinung über.

Der Schein / Illusory Being

§818 Das Wesen aus dem Sein herkommend scheint demselben gegenueber zu stehen; dies unmittelbare Sein ist zunaechst das Unwesentliche. Allein es ist zweitens mehr als nur unwesentliches, es ist wesenloses Sein, es ist Schein. Drittens, dieser Schein ist nicht ein aeusserliches, dem Wesen anderes, sondern er ist sein eigner Schein. Das Erscheinen des Wesens in ihm selbst ist die Reflexion.

§818 Essence that issues from being seems to confront it as an opposite; this immediate being is, in the first instance, the unessential. But secondly, it is more than merely unessential being, it is essenceless being, it is illusory being. Thirdly, this illusory being is not something external to or other than essence; on the contrary, it is essence’s own illusory being. The showing of this illusory being within essence itself is reflection.

type universe TypeHType \in \mathbf{H}., see also §833

§827 Es ist die Unmittelbarkeit des Nichtseins, welche den Schein ausmacht; dies Nichtsein aber ist nichts anderes als die Negativität des Wesens an ihm selbst. Das Sein ist Nichtsein in dem Wesen. Seine Nichtigkeit an sich ist die negative Natur des Wesens selbst. Die Unmittelbarkeit oder Gleichgültigkeit aber, welche dies Nichtsein enthält, ist das eigene absolute Ansichsein des Wesens. Die Negativität des Wesens ist seine Gleichheit mit sich selbst oder seine einfache Unmittelbarkeit und Gleichgültigkeit. Das Sein hat sich im Wesen erhalten, insofern dieses an seiner unendlichen Negativität diese Gleichheit mit sich selbst hat; hierdurch ist das Wesen selbst das Sein. Die Unmittelbarkeit, welche die Bestimmtheit am Scheine gegen das Wesen hat, ist daher nichts anderes als die eigene Unmittelbarkeit des Wesens; aber nicht die seiende Unmittelbarkeit, sondern die schlechthin vermittelte oder reflektierte Unmittelbarkeit, welche der Schein ist, - das Sein nicht als Sein, sondern nur als die Bestimmtheit des Seins, gegen die Vermittlung; das Sein als Moment.

§827 It is the immediacy of non-being that constitutes illusory being; but this non-being is nothing else but the negativity of essence present within it. In essence, being is non-being. Its intrinsic nothingness is the negative nature of essence itself. But the immediacy or indifference which this non-being contains is essence’s own absolute being-in-itself. The negativity of essence is its equality with itself or its simple immediacy and indifference. Being has preserved itself in essence in so far as the latter in its infinite negativity has this equality with itself; it is through this that essence itself is being. The immediacy of the determinateness in illusory being over against essence is consequently nothing other than essence’s own immediacy; but the immediacy is not simply affirmative [seiend], but is the purely mediated or reflected immediacy that is illusory being-being, not as being, but only as the determinateness of being as opposed to mediation; being as a moment.

A Das Wesentliche und das Unwesentliche / The essential and the unessential
B Der Schein / Illusory being

§823 1. Das Sein ist Schein. Das Sein des Scheins besteht allein in dem Aufgehobensein des Seins, in seiner Nichtigkeit; diese Nichtigkeit hat es im Wesen, und außer seiner Nichtigkeit, außer dem Wesen ist er nicht. Er ist das Negative gesetzt als Negatives.

§823 1. Being is illusory being. The being of illusory being consists solely in the sublatedness of being, in its nothingness; this nothingness it has in essence and apart from its nothingness, apart from essence, illusory being is not. It is the negative posited as negative.

Notice that here the text begins to say Nichtigkeit (\sim“ nothingness” for the result of Aufhebung of all determinations. According to §803 this ultimate Aufhebung is the trivial unity of opposites iddashbidid \dashb id. The idid-modal operator encodes no determination and hence encodes “nothing”, but this is rather different from the sense of Nichts, Nichtsein (again \sim nothing) that is at the beginning of the Seinslogik.

We will read Nichtigkeit (\sim “nothing”) here in the Wesenslogik in this sense of §823. This seems to match well with its use here, in particular with the repeated statemens §835 that movement of the essence is not like that of being from something (some determination, really) to something else (some other determination, oppose of sublated), but moves from nothing to nothing. This is hence speaking of ididid \dashv id. See also §827

§827 Es ist die Unmittelbarkeit des Nichtseins, welche den Schein ausmacht; dies Nichtsein aber ist nichts anderes als die Negativität des Wesens an ihm selbst. Das Sein ist Nichtsein in dem Wesen. Seine Nichtigkeit an sich ist die negative Natur des Wesens selbst. Die Unmittelbarkeit oder Gleichgültigkeit aber, welche dies Nichtsein enthält, ist das eigene absolute Ansichsein des Wesens. Die Negativität des Wesens ist seine Gleichheit mit sich selbst oder seine einfache Unmittelbarkeit und Gleichgültigkeit. Das Sein hat sich im Wesen erhalten, insofern dieses an seiner unendlichen Negativität diese Gleichheit mit sich selbst hat; hierdurch ist das Wesen selbst das Sein. Die Unmittelbarkeit, welche die Bestimmtheit am Scheine gegen das Wesen hat, ist daher nichts anderes als die eigene Unmittelbarkeit des Wesens; aber nicht die seiende Unmittelbarkeit, sondern die schlechthin vermittelte oder reflektierte Unmittelbarkeit, welche der Schein ist, - das Sein nicht als Sein, sondern nur als die Bestimmtheit des Seins, gegen die Vermittlung; das Sein als Moment.

§827 It is the immediacy of non-being that constitutes illusory being; but this non-being is nothing else but the negativity of essence present within it. In essence, being is non-being. Its intrinsic nothingness is the negative nature of essence itself. But the immediacy or indifference which this non-being contains is essence’s own absolute being-in-itself. The negativity of essence is its equality with itself or its simple immediacy and indifference. Being has preserved itself in essence in so far as the latter in its infinite negativity has this equality with itself; it is through this that essence itself is being. The immediacy of the determinateness in illusory being over against essence is consequently nothing other than essence’s own immediacy; but the immediacy is not simply affirmative [seiend], but is the purely mediated or reflected immediacy that is illusory being-being, not as being, but only as the determinateness of being as opposed to mediation; being as a moment.

§828 Diese beiden Momente, die Nichtigkeit, aber als Bestehen, und das Sein, aber als Moment, oder die an sich seiende Negativität und die reflektierte Unmittelbarkeit, welche die Momente des Scheins ausmachen, sind somit die Momente des Wesens selbst: es ist nicht ein Schein des Seins am Wesen oder ein Schein des Wesens am Sein vorhanden; der Schein im Wesen ist nicht der Schein eines Anderen, sondern er ist der Schein an sich, der Schein des Wesens selbst.

§828 These two moments, namely the nothingness which yet is and the being which is only a moment, or the implicit negativity and the reflected immediacy that constitute the moments of illusory being, are thus the moments of essence itself. What we have here is not an illusory show of being in essence, or an illusory show of essence in being; the illusory being in essence is not the illusory being of an other, but is illusory being per se, the illusory being of essence itself. What we have here is not an illusory show of being in essence, or an illusory show of essence in being; the illusory being in essence is not the illusory being of an other, but is illusory being per se, the illusory being of essence itself.

Reading Nichtigkeit here via §823 is ididid \dashv id and reflektierte Unmittelbarkeit as the self-reflection of the \infty-topos in its type universe via §833, then §828 says that the two moments of the Essence are

(idid),Type (id \dashv id) \;\;\;\;, \;\;\;\; Type

which in turn is the ambient category and its type universe. Hence this gives again that Essence is the ambient \infty-topos.

C Die Reflexion / Reflection

§833 Der Schein ist dasselbe was die Reflexion ist, aber er ist die Reflexion als unmittlebare.

§833 Illusory being is the same thing as reflection; but it is reflection as immediate.

type universe/object classifier

§834 Das Wesen ist Reflexion;…

§834 Essence is reflection…

See discussion at Wesen als Reflexion in Ihm Selbst

§834 Das Wesen ist Reflexion; die Bewegung des Werdens und Übergehens, das in sich selbst bleibt, worin das Unterschiedene schlechthin nur als das an sich Negative, als Schein bestimmt ist. - In dem Werden des Seins liegt der Bestimmtheit das Sein zugrunde, und sie ist Beziehung auf Anderes. Die reflektierende Bewegung hingegen ist das Andere als die Negation an sich, die nur als sich auf sich beziehende Negation ein Sein hat. Oder indem diese Beziehung auf sich eben dies Negieren der Negation ist, so ist die Negation als Negation vorhanden, als ein solches, das sein Sein in seinem Negiertsein hat, als Schein. Das Andere ist hier also nicht das Sein mit der Negation oder Grenze, sondern die Negation mit der Negation. Das Erste aber gegen dies Andere, das Umittelbare oder Sein, ist nur diese Gleichheit selbst der Negation mit sich, die negierte Negation, die absolute Negativität. Diese Gleichheit mit sich oder Unmittelbarkeit ist daher nicht ein Erstes, von dem angefangen wird und das in seine Negation überginge, noch ist es ein seiendes Substrat, das sich durch die Reflexion hindurch bewegte; sondern die Unmittelbarkeit ist nur diese Bewegung selbst.

§834 Essence is reflection, the movement of becoming and transition that remains internal to it, in which the differentiated moment is determined simply as that which in itself is only negative, as illusory being. At the base of becoming in the sphere of being, there lies the determinateness of being, and this is relation to other. The movement of reflection, on the other hand, is the other as the negation in itself, which has a being only as self-related negation. Or, since the self-relation is precisely this negating of negation, the negation as negation is present in such wise that it has its being in its negatedness, as illusory being. Here, therefore, the other is not being with a negation, or limit, but negation with the negation. But the first, over against this other, the immediate or being, is only this very equality of the negation with itself, the negated negation, absolute negativity. This equality with itself, or immediacy, is consequently not a first from which the beginning was made and which passed over into its negation; nor is it an affirmatively present substrate that moves through reflection; on the contrary, immediacy is only this movement itself.

§835 Das Werden im Wesen, seine reflektierende Bewegung, ist daher die Bewegung von Nichts zu Nichts und dadurch zu sich selbst zurück.

§835 Consequently, becoming is essence, its reflective movement, is the movement of nothing to nothing, and so back to itself.

in view of §823 we read this as the ultimate opposition

ididid \dashv id

see also §839

(1) Die setzende Reflexion / Positing reflection

§839 Zunächst ist die Reflexion die Bewegung des Nichts zu Nichts, somit die mit sich selbst zusammengehende Negation. Dieses Zusammengehen mit sich ist überhaupt einfache Gleichheit mit sich, die Unmittelbarkeit. Aber dies Zusammenfallen ist nicht Übergehen der Negation in die Gleichheit mit sich als in ihr Anderssein, sondern die Reflexion ist Übergehen als Aufheben des Übergehens; denn sie ist unmittelbares Zusammenfallen des Negativen mit sich selbst; so ist dies Zusammengehen erstlich Gleichheit mit sich oder Unmittelbarkeit; aber zweitens ist diese Unmittelbarkeit die Gleichheit des Negativen mit sich, somit die sich selbst negierende Gleichheit; die Unmittelbarkeit, die an sich das Negative, das Negative ihrer selbst ist, dies zu sein, was sie nicht ist.

§839 In the first place, reflection is the movement of nothing to nothing and is the negation that coincides with itself. This coincidence with itself is, in general, simple equality-with-self, immediacy. But this coincidence is not a transition of the negation into equality-with-self as into its otherness: on the contrary, reflection is transition as sublating of the transition; for reflection is immediate coincidence of the negative with itself. This coincidence is thus first, equality-with-self or immediacy; but secondly, this immediacy is the equality of the negative with itself, hence self-negating equality, the immediacy that is in itself the negative, the negative of itself, that consists in being that which it is not.

This amplifies nicely on §835

“die Bewegung des Nichts zu Nichts … ist unmittelbares Zusammenfallen des Negativen mit sich selbst”

hence coincidence of the opposite with itself hence the trivial unity of opposites

ididid \dashv id

see also §813

For the following paragraphs it may be helpful to keep in mind the outer part of the movement, remark 6:

id id =*¯ * =id¯ \array{ id &\dashv& id & = \overline{\ast} \\ \vee && \vee \\ \vdots && \vdots \\ \vee && \vee \\ \emptyset &\dashv& \ast & = \overline{id} }

with its interpretation as the initial opposition of being *\ast and nothing \emptyset in becoming, finding its ultimate Aufhebung in the self-opposition of identity, which is at the same time the negation of pure being, and vice versa.

§840 Die Beziehung des Negativen auf sich selbst ist also seine Rückkehr in sich; sie ist Unmittelbarkeit als das Aufheben des Negativen; aber Unmittelbarkeit schlechthin nur als diese Beziehung oder als Rückkehr aus einem, somit sich selbst aufhebende Unmittelbarkeit. - Dies ist das Gesetztsein, die Unmittelbarkeit rein nur als Bestimmtheit oder als sich reflektierend. Diese Unmittelbarkeit, die nur als Rückkehr des Negativen in sich ist, ist jene Unmittelbarkeit, welche die Bestimmtheit des Scheins ausmacht und von der vorhin die reflektierende Bewegung anzufangen schien. Statt von dieser Unmittelbarkeit anfangen zu können, ist diese vielmehr erst als die Rückkehr oder als die Reflexion selbst. Die Reflexion ist also die Bewegung, die, indem sie die Rückkehr ist, erst darin das ist, das anfängt oder das zurückkehrt.

§840 The self-relation of the negative is, therefore, its return into itself; it is immediacy as the sublating of the negative; but immediacy simply and solely as this relation or as return from a negative, and hence a self-sublating immediacy. This is posited being or positedness, immediacy purely and simply as determinateness or as self-reflecting. This immediacy which is only as return of the negative into itself, is that immediacy which constitutes the determinateness of illusory being and which previously seemed to be the starting point of the reflective moment. But this immediacy, instead of being able to form the starting point is, on the contrary, immediacy only as the return or as reflection itself. Reflection therefore is the movement that starts or returns only in so far as the negative has already returned into itself.

§841 Sie ist Setzen, insofern sie die Unmittelbarkeit als ein Rückkehren ist; es ist nämlich nicht ein Anderes vorhanden, weder ein solches, aus dem sie, noch in das sie zurückkehrte; sie ist also nur als Rückkehren oder als das Negative ihrer selbst. Aber ferner ist diese Unmittelbarkeit die aufgehobene Negation und die aufgehobene Rückkehr in sich. Die Reflexion ist als Aufheben des Negativen Aufheben ihres Anderen, der Unmittelbarkeit. Indem sie also die Unmittelbarkeit als ein Rückkehren, Zusammengehen des Negativen mit sich selbst ist, so ist sie ebenso Negation des Negativen als des Negativen. So ist sie Voraussetzen. - Oder die Unmittelbarkeit ist als Rückkehren nur das Negative ihrer selbst, nur dies, nicht Unmittelbarkeit zu sein; aber die Reflexion ist das Aufheben des Negativen seiner selbst, sie ist Zusammengehen mit sich; sie hebt also ihr Setzen auf, und indem sie das Aufheben des Setzens in ihrem Setzen ist, ist sie Voraussetzen. - In dem Voraussetzen bestimmt die Reflexion die Rückkehr in sich als das Negative ihrer selbst, als dasjenige, dessen Aufheben das Wesen ist. Es ist sein Verhalten zu sich selbst, aber zu sich als dem Negativen seiner; nur so ist es die insichbleibende, sich auf sich beziehende Negativität. Die Unmittelbarkeit kommt überhaupt nur als Rückkehr hervor und ist dasjenige Negative, welches der Schein des Anfangs ist, der durch die Rückkehr negiert wird. Die Rückkehr des Wesens ist somit sein Sich-Abstoßen von sich selbst. Oder die Reflexion-in-sich ist wesentlich das Voraussetzen dessen, aus dem sie die Rückkehr ist.

§841 It is a positing in so far as it is immediacy as a returning movement; for there is no other on hand, either an other from which or into which immediacy returns; it is, therefore, only as a returning movement, or as the negative of itself. Furthermore, this immediacy is the sublated negation and the sublated return-into-self. Reflection, as sublating the negative, is a sublating of its other, of immediacy. Since, therefore, it is immediacy as a returning movement, as a coincidence of the negative with itself, it is equally a negative of the negative as negative. Thus it is a presupposing. Or immediacy, as a returning movement, is only the negative of itself, only this, to be not immediacy; but reflection is the sublating of the negative of itself, it is a coincidence with itself; it therefore sublates its positing, and since in its positing it sublates its positing, it is a presupposing. Reflection, in its presupposing, determines the return-into-self as the negative of itself, as that, the sublating of which is essence. The presupposing is the manner in which it relates itself to itself, but to itself as the negative of itself; only thus is it the self-relating negativity that remains internal to itself. Immediacy presents itself simply and solely as a return and is that negative which is the illusory being of the beginning, the illusory being which is negated by the return. Accordingly, the return of essence is its self-repulsion. In other words, reflection-into-self is essentially the presupposing of that from which it is the return.

§842 Es ist das Aufheben seiner Gleichheit mit sich, wodurch das Wesen erst die Gleichheit mit sich ist. Es setzt sich selbst voraus, und das Aufheben dieser Voraussetzung ist es selbst; umgekehrt ist dies Aufheben seiner Voraussetzung die Voraussetzung selbst. - Die Reflexion also findet ein Unmittelbares vor, über das sie hinausgeht und aus dem sie die Rückkehr ist. Aber diese Rückkehr ist erst das Voraussetzen des Vorgefundenen. Dies Vorgefundene wird nur darin, daß es verlassen wird; seine Unmittelbarkeit ist die aufgehobene Unmittelbarkeit. - Die aufgehobene Unmittelbarkeit umgekehrt ist die Rückkehr in sich, das Ankommen des Wesens bei sich, das einfache sich selbst gleiche Sein. Damit ist dieses Ankommen bei sich das Aufheben seiner und die von sich selbst abstoßende, voraussetzende Reflexion, und ihr Abstoßen von sich ist das Ankommen bei sich selbst.

§842 It is only when essence has sublated its equality-with-self that it is equality-with-self. It presupposes itself and the sublating of this presupposition is essence itself; conversely, this sublating of its presupposition is the presupposition itself. Reflection therefore finds before it an immediate which it transcends and from which it is the return. But this return is only the presupposing of what reflection finds before it. What is thus found only comes to be through being left behind; its immediacy is sublated immediacy. Conversely, the sublated immediacy is the return-into-self, the coming-to-itself of essence, simple, self-equal being. Hence this coming-to-itself is the sublating of itself and is the self-repelling, presupposing reflection, and its self-repelling is the coming-to-itself of reflection.

§843 Die reflektierende Bewegung ist somit nach dem Betrachteten als absoluter Gegenstoß in sich selbst zu nehmen. Denn die Voraussetzung der Rückkehr in sich - das, woraus das Wesen herkommt und erst als dieses Zurückkommen ist -, ist nur in der Rückkehr selbst. Das Hinausgehen über das Unmittelbare, von dem die Reflexion anfängt, ist vielmehr erst durch dies Hinausgehen; und das Hinausgehen über das Unmittelbare ist das Ankommen bei demselben. Die Bewegung wendet sich als Fortgehen unmittelbar in ihr selbst um und ist nur so Selbstbewegung - Bewegung, die aus sich kommt, insofern die setzende Reflexion voraussetzende, aber als voraussetzende Reflexion schlechthin setzende ist.

§843 It follows, therefore, from the foregoing considerations that the reflective movement is to be taken as an absolute recoil upon itself. For the presupposition of the return-into-self-that from which essence comes, and is only as this return-is only in the return itself. The transcending of the immediate from which reflection starts is rather the outcome of this transcending; and the transcending of the immediate is the arrival at it. The movement, as an advance, immediately turns round upon itself and only so is self-movement — a movement which comes from itself in so far as positing reflection is presupposing, but, as presupposing reflection, is simply positing reflection.

§844 So ist die Reflexion sie selbst und ihr Nichtsein, und ist nur sie selbst, indem sie das Negative ihrer ist, denn nur so ist das aufheben des Negativen zugleich als ein Zusammengehen mit sich.

§844 Thus reflection is itself and its non-being, and is only itself, in that it is the negative of itself, for only thus is the sublating of the negative at the same time a coincidence with itself.

(2) Die äußere Reflexion

§848 Diese äußere Reflexion ist der Schluß, in welchem die beiden Extreme, das Unmittelbare und die Reflexion-in-sich, sind; die Mitte desselben ist die Beziehung beider, das bestimmte Unmittelbare, so daß der eine Teil derselben, die Unmittelbarkeit, nur dem einen Extreme, die andere, die Bestimmtheit oder Negation, nur dem anderen Extreme zukommt.

§848 This external reflection is the syllogism in which are the two extremes, the immediate and reflection-into-self; the middle term of the syllogism is the connection of the two, the determinate immediate, so that one part of the middle term, immediacy, belongs only to one of the extremes, the other, determinateness or negation, belongs only to the other extreme.

Anmerkung

§850 Es ist aber hier nicht, weder von der Reflexion des Bewußtseyns, noch von der bestimmteren Reflexion des Verstandes, die das Besondere und Allgemeine zu ihren Bestimmungen hat, sondern von der Reflexion überhaupt die Rede. Jene Reflexion, der Kant das Aufsuchen des Allgemeinen zum gegebenen Besondern zuschreibt, ist, wie erhellt, gleichfalls nur die äußere Reflexion, die sich auf das Unmittelbare als auf ein gegebenes bezieht.

Reflection is usually taken in a subjective sense as the movement of the faculty of judgement that goes beyond a given immediate conception and seeks universal determinations for it or compares such determinations with it. Kant opposes reflective judgement to determining judgement. He defines the faculty of judgement in general as the ability to think the particular as subsumed under the universal.

(3) Bestimmende Reflexion

§853a Die bestimmende Reflexion ist überhaupt die Einheit der setzenden und der äußeren Reflexion. Dieß ist näher zu betrachten.

§853a Determining reflection is in general the unity of positing and external reflection. This is to be considered in more detail.

§853b 1. Die äußere Reflexion fängt vom unmittelbaren Seyn an, die setzende vom Nichts. Die äußere Reflexion, die bestimmend wird, setzt ein Anderes, aber das Wesen, an die Stelle des aufgehobenen Seins;

§853b 1. External reflection starts from immediate being, positing reflection from nothing. External reflection, when it determines, posits an other-but this is essence-in the place of the sublated being;

id id =*¯ * =id¯ \array{ id &\dashv& id & = \overline{\ast} \\ \vee && \vee \\ \vdots &\uparrow& \vdots \\ \vee && \vee \\ \emptyset &\dashv& \ast & = \overline{id} }

§854a Das Gesetzte ist daher ein Anderes, aber so, daß die Gleichheit der Reflexion mit sich schlechthin erhalten ist; denn das Gesetzte ist nur als Aufgehobenes, als Beziehung auf die Rückkehr in sich selbst. - In der Sphäre des Seins war das Dasein das Sein, das die Negation an ihm hatte, und das Sein der unmittelbare Boden und Element dieser Negation, die daher selbst die unmittelbare war. Dem Dasein entspricht in der Sphäre des Wesens das Gesetztsein. Es ist gleichfalls ein Dasein, aber sein Boden ist das Sein als Wesen oder als reine Negativität; es ist eine Bestimmtheit oder Negation nicht als seiend, sondern unmittelbar als aufgehoben.

§854a What is posited is consequently an other, but in such a manner that the equality of reflection with itself is completely preserved; for what is posited is only as sublated, as a relation to the returninto-self. In the sphere of being, determinate being was the being in which negation was present, and being was the immediate base and element of this negation, which consequently was itself immediate. In the sphere of essence, positedness corresponds to determinate being. It is likewise a determinate being but its base is being as essence or as pure negativity; it is a determinateness or negation, not as affirmatively present but immediately as sublated.

§854b Das Dasein ist nur Gesetztsein; dies ist der Satz des Wesens vom Dasein.

§854b Determinate being is merely posited being or positedness; this is the proposition of essence about determinate being.

§854c Das Gesetztsein steht einerseits dem Dasein, andererseits dem Wesen gegenüber und ist als die Mitte zu betrachten, welche das Dasein mit dem Wesen und umgekehrt das Wesen mit dem Dasein zusammenschließt. - Wenn man sagt, eine Bestimmung ist nur ein Gesetztsein, so kann dies daher den doppelten Sinn haben; sie ist dies im Gegensatze gegen das Dasein oder gegen das Wesen. In jenem Sinne wird das Dasein für etwas Höheres genommen als das Gesetztsein und dieses der äußeren Reflexion, dem Subjektiven zugeschrieben. In der Tat aber ist das Gesetztsein das Höhere; denn als Gesetztsein ist das Dasein als das, was es an sich ist, als Negatives, ein schlechthin nur auf die Rückkehr in sich bezogenes. Deswegen ist das Gesetztsein nur ein Gesetztsein in Rücksicht auf das Wesen, als die Negation des Zurückgekehrtseins in sich selbst.

§854c Positedness stands opposed, on the one hand, to determinate being, and on the other, to essence, and is to be considered as the middle term which unites determinate being with essence, and conversely, essence with determinate being. Accordingly, when it is said that a determination is only a positedness, this can have a twofold meaning; it is a positedness as opposed to determinate being or as opposed to essence. In the former meaning, determinate being is taken to be superior to positedness and the latter is ascribed to external reflection, to the subjective side. But in fact positedness is the superior; for as positedness, determinate being is that which it is in itself, a negative, something that is simply and solely related to the return-into-self. It is for this reason that positedness is only a positedness with respect to essence, as the negation of the accomplished return-into-self.

Chapter 2. Die Wesenheiten oder die Reflexions-Bestimmungen / The Essentialities or Determination of Reflection

§860a Die Reflexion ist bestimmte Reflexion; somit ist das Wesen bestimmtes Wesen, oder es ist Wesenheit.

§860a Reflection is determinate reflection; hence essence is determinate essence, or it is an essentiality.

§860b Die Reflexion ist das Scheinen des Wesens in sich selbst. Das Wesen als unendliche Rückkehr in sich ist nicht unmittelbare, sondern negative Einfachheit; es ist eine Bewegung durch unterschiedene Momente, absolute Vermittelung mit sich. Aber es scheint in diese seine Momente; sie sind daher selbst in sich reflektirte Bestimmungen.

§860b Reflection is the showing of the illusory being of essence within essence itself. Essence, as infinite return-into-self, is not immediate but negative simplicity; it is a movement through distinct moments, absolute self-mediation. But it reflects itself into these its moments which consequently are themselves determinations reflected into themselves.

Type 1Type 2Type 3Type 4 Type_1 \subset Type_2 \subset Type_3 \subset Type_4 \subset \cdots

See above at §816

§863 So wird die wesentliche Bestimmung der Identität in dem Satze ausgesprochen: Alles ist sich selbst gleich; A=AA = A.

§863 Thus the essential category of identity is enunciated in the proposition: everything is identical with itself, A=AA = A.

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

§864 Die Kategorie ist ihrer Etymologie und der Definition des Aristoteles nach, dasjenige, was von dem Seyenden gesagt, behauptet wird.

§864 According to its etymology and Aristotle’s definition, category is what is predicated or asserted of the existent.

category (philosophy)

§865 Die Reflexions-Bestimmungen dagegen sind nicht von qualitativer Art.

§865 The determinations of reflection, on the contrary, are not of a qualitative kind.

A Identity

§869 Essence is therefore simple identity with self.

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

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 Der Unterschied / Difference
(a) Der absolute Unterschied / Absolute difference

§886 Darin, drückt man sich aus, sind zwei Dinge unterschieden, daß sie u.s.f.— Darin, das heißt, in einer und derselben Rücksicht, in demselben Bestimmungsgrunde. Er ist der Unterschied der Reflexion, nicht das Andersseyn des Daseyns. Ein Daseyn und ein anderes Daseyn sind gesetzt als außereinanderfallend, jedes der gegen einander bestimmten Daseyn hat ein unmittelbares Seyn für sich. Das Andre des Wesens dagegen ist das Andre an und für sich, nicht das Andre als eines andern außer ihm Befindlichen; die einfache Bestimmtheit an sich. Auch in der Sphäre des Daseyns erwies sich das Andersseyn und die Bestimmtheit von dieser Natur, einfache Bestimmtheit, identischer Gegensatz zu seyn; aber diese Identität zeigte sich nur als das Übergehen einer Bestimmtheit in die andere. Hier in der Sphäre der Reflexion tritt der Unterschied als reflektirter auf, der so gesetzt ist, wie er an sich ist.

§886 1. Two things are different, it is said, in that they, etc. ‘In that’ is, in one and the same respect, in the same ground of determination. It is the difference of reflection, not the otherness of determinate being. One determinate being and another determinate being are posited as falling apart, each of them, as determined against the other, has an immediate being for itself. The other of essence, on the contrary, is the other in and for itself, not the other as other of an other, existing outside it but simple determinateness in itself. In the sphere of determinate being, too, otherness ‘and determinateness proved to be of this nature, to be simple determinateness, identical opposition; but this identity revealed itself only as the transition of one determinateness into the other. Here, in the sphere of reflection, difference appears as reflected difference, which is thus posited as it is in itself.

Given a moment \bigcirc and two types XX and YY that are not equivalent, then if also X\bigcirc X is not equivalent to Y\bigcirc Y we may say that XX and YY are different in that their \bigcirc-moments are (already) different.

If hower these moments are equivalent, then XX and YY are similar in this respect. similarity.

(b) Die Verschiedenheit / Diversity

§897 An der sich entfremdeten Reflexion kommen also die Gleichheit und Ungleichheit als gegen einander selbst unbezogene hervor, und sie trennt sie, indem sie sie auf ein und dasselbe bezieht, durch die Insoferns, Seiten und Rücksichten. Die Verschiedenen, die das eine und dasselbe sind, worauf beide, die Gleichheit und Ungleichheit, bezogen werden, sind also nach der einen Seite einander gleich, nach der andern Seite aber ungleich, und insofern sie gleich sind, insofern sind sie nicht ungleich. Die Gleichheit bezieht sich nur auf sich, und die Ungleichheit ist ebenso nur Ungleichheit.

§897 In the self-alienated reflection, therefore, likeness and unlikeness appear as mutually unrelated, and in relating them to one and the same thing, it separates them by the introduction of ‘in so far’, of sides and respects. The diverse, which are one and the same, to which both likeness and unlikeness are related, are therefore, from one side like one another, but from another side are unlike, and in so far as they are like, they are not unlike. Likeness is related only to itself, and similarly unlikeness is only unlikeness.

§898 Durch diese ihre Trennung von einander aber heben sie sich nur auf. Gerade, was den Widerspruch und die Auflösung von ihnen abhalten soll, daß nämlich Etwas einem Andern in einer Rücksicht gleich, in einer andern aber ungleich sey;—dieß Auseinanderhalten der Gleichheit und Ungleichheit ist ihre Zerstörung. Denn beide sind Bestimmungen des Unterschiedes; sie sind Beziehungen aufeinander, das eine, zu seyn, was das andere nicht ist; gleich ist nicht ungleich, und ungleich ist nicht gleich; und beide haben wesentlich diese Beziehung, und außer ihr keine Bedeutung; als Bestimmungen des Unterschiedes ist jedes das was es ist, als unterschieden von seinem andern. Durch ihre Gleichgültigkeit aber gegen einander, ist die Gleichheit nur bezogen auf sich, die Ungleichheit ist ebenso eine eigene Rücksicht und Reflexion für sich; jede ist somit sich selbst gleich; der Unterschied ist verschwunden, da sie keine Bestimmtheit gegen einander haben; oder jede ist hiermit nur Gleichheit.

§898 But by this separation of one from the other they merely sublate themselves. The very thing that was supposed to hold off contradiction and dissolution from them, namely, that something is like something else in one respect, but is unlike it in another - this holding apart of likeness and unlikeness is their destruction. For both are determinations of difference; they are relations to one another, the one being what the other is not; like is not unlike and unlike is not like; and both essentially have this relation and have no meaning apart from it; as determinations of difference, each is what it is as distinct from its other. But through this mutual indifference, likeness is only self-referred, and unlikeness similarly is self-referred and a reflective determination on its own; each, therefore, is like itself; the difference has vanished, since they cannot have any determinateness over against one another; in other words, each therefore is only likeness.

“The diverse [[]] are related [[]] from one side like one another, but from another side are unlike [[]] that something is like something else in one respect, but is unlike it in another ”

similarity

Bemerkung 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.

§906 Zwei Dinge sind nicht vollkommen gleich; so sind sie gleich und ungleich zugleich; gleich schon darin, daß sie Dinge oder zwei überhaupt sind, denn jedes ist ein Ding und ein Eins so gut als das andere,jedes also dasselbe, was das andere; ungleich aber sind sie durch die Annahme.

§906 Two things are not perfectly alike; so they are at once alike and unlike; alike, simply because they are things, or just two, without further qualification — for each is a thing and a one, no less than the other — but they are unlike ex hypothesi.

Via the formalization of similarity this “alike, simply because they are things” is similarity with respec to the *\ast-modality, example 4.

(c) Der Gegensatz / Opposition

§908 Im Gegensatze ist die bestimmte Reflexion, der Unterschied vollendet. Er ist die Einheit der Identität und der Verschiedenheit;

§908 In opposition, the determinate rejection, difference, finds its completion. It is the unity of identity and difference; its moments are different in one identity and thus are opposites.

§911 Diese in sich reflektirte Gleichheit mit sich, die in ihr selbst die Beziehung auf die Ungleichheit enthält, ist das Positive; so die Ungleichheit die in ihr selbst die Beziehung auf ihr Nichtseyn, die Gleichheit enthält, ist das Negative.—Oder beide sind das Gesetztseyn; insofern nun die unterschiedene Bestimmtheit als unterschiedene bestimmte Beziehung des Gesetztseyns auf sich genommen wird, so ist der Gegensatz eines Theils das Gesetztseyn in seine Gleichheit mit sich reflektirt; andern Theils dasselbe in seine Ungleichheit mit sich reflektirt; das Positive und Negative.—Das Positive ist das Gesetztseyn als in die Gleichheit mit sich reflektirt; aber das reflektirte ist das Gesetztseyn, das ist, die Negation als Negation, so hat diese Reflexion in sich die Beziehung auf das Andere zu ihrer Bestimmung. Das Negative ist das Gesetztseyn als in die Ungleichheit reflektirt; aber das Gesetztseyn ist die Ungleichheit selbst, so ist diese Reflexion somit die Identität der Ungleichheit mit sich selbst und absolute Beziehung auf sich.—Beide also, das in die Gleichheit mit sich reflektirte Gesetztseyn hat die Ungleichheit, und das in die Ungleichheit mit sich reflektirte Gesetztseyn hat auch die Gleichheit an ihm.

§911 This self-likeness reflected into itself that contains within itself the reference to unlikeness, is the positive; and the unlikeness that contains within itself the reference to its non-being, to likeness, is the negative. Or, both are a positedness; now in so far as the differentiated determinateness is taken as a differentiated determinate self-reference of positedness, the opposition is, on the one hand, positedness reflected into its likeness to itself and on the other hand, positedness reflected into its unlikeness to itself — the positive and the negative. The positive is positedness as reflected into self-likeness; but what is reflected is positedness, that is, the negation as negation, and so this reflection-into-self has reference-to-other for its determination. The negative is positedness as reflected into unlikeness; but the positedness is unlikeness itself, and this reflection is therefore the identity of unlikeness with itself and absolute self-reference. Each is the whole; the positedness reflected into likeness-to-self contains unlikeness, and the positedness reflected into unlikeness-to-self also contains likeness.

§912 Das Positive und das Negative sind so die selbstständig gewordenen Seiten des Gegensatzes. Sie sind selbstständig, indem sie die Reflexion des Ganzen in sich sind, und sie gehören dem Gegensatze an, insofern es die Bestimmtheit ist, die als Ganzes in sich reflektirt ist. Um ihrer Selbstständigkeit willen machen sie den an sich bestimmten Gegensatz aus. Jedes ist es selbst und sein Anderes, dadurch hat jedes seine Bestimmtheit nicht an einem andern, sondern an ihm selbst.—Jedes bezieht sich auf sich selbst, nur als sich beziehend auf sein Anderes. Dieß hat die doppelte Seite; jedes ist Beziehung auf sein Nichtseyn als Aufheben dieses Andersseyns in sich; so ist sein Nichtseyn nur ein Moment in ihm. Aber andern Theils ist hier das Gesetztseyn ein Seyn, ein gleichgültiges Bestehen geworden; das andre seiner, das jedes enthält, ist daher auch das Nichtseyn dessen, in welchem es nur als Moment enthalten seyn soll. Jedes ist daher nur, insofern sein Nichtseyn ist, und zwar in einer identischen Beziehung.

§912 The positive and the negative are thus the sides of the opposition that have become self-subsistent. They are self-subsistent in that they are the reflection of the whole into themselves, and they belong to the opposition in so far as this is the determinateness which, as a whole, is reflected into itself. On account of their selfsubsistence, they constitute the implicitly determined opposition. Each is itself and its other; consequently, each has its determinateness not in an other, but in its own self. Each is self-referred, and the reference to its other is only a self-reference. This has a twofold aspect: each is a reference to its non-being as a sublating of this otherness within it; thus its non-being is only a moment in it. But on the other hand positedness here has become a being, an indifferent ssistence; consequently, the other of itself which each contains is also the non-being of that in which it is supposed to be contained only as a moment. Each therefore is, only in so far as its non-being is, and is in an identical relationship with it.

§913 Die Bestimmungen, welche das Positive und Negative konstituiren, bestehen also darin, daß das Positive und das Negative erstens absolute Momente des Gegensatzes sind; ihr Bestehen ist untrennbar Eine Reflexion; es ist Eine Vermittelung, in welcher jedes durch das Nichtseyn seines Andern, damit durch sein Anderes oder sein eigenes Nichtseyn ist.—So sind sie Entgegengesetzte überhaupt; oder jedes ist nur das Entgegengesetzte des Andern; das eine ist noch nicht positiv, und das andre noch nicht negativ, sondern beide sind negativ gegen einander. Jedes ist so überhaupt erstens insofern das Andre ist; es ist durch das Andre, durch sein eignes Nichtseyn, das was es ist; es ist nur Gesetztseyn; zweitens es ist insofern das Andre nicht ist; es ist durch das Nichtseyn des Andern das was es ist; es ist Reflexion in sich.—Dieses beides ist aber die eine Vermittelung des Gegensatzes überhaupt, in der sie überhaupt nur Gesetzte sind.

§913 The determinations which constitute the positive and negative consist, therefore, in the fact that the positive and negative are, in the first place, absolute moments of the opposition; their subsistence is inseparably one reflection; it is a single mediation in which each is through the non-being of its other, and so is through its other or its own non-being. Thus they are simply opposites; in other words, each is only the opposite of the other, the one is not as yet positive, and the other is not as yet negative, but both are negative to one another. In the first place, then, each is, only in so far as the other is; it is what it is, through the other, through its own non-being; it is only a positedness; secondly, it is, in so far as the other is not; it is what it is, through the non-being of the other; it is reflection-into-self. But these two are the one mediation of the opposition as such, in which they are simply only posited moments

§914 Aber ferner dieß bloße Gesetztseyn ist in sich reflektirt überhaupt; das Positive und Negative ist nach diesem Momente der äußern Reflexion gleichgültig gegen jene erste Identität, worin sie nur Momente sind; oder indem jene erste Reflexion die eigne Reflexion des Positiven und Negativen in sich selbst, jedes sein Gesetztseyn an ihm selbst ist, so ist jedes gleichgültig gegen diese seine Reflexion in sein Nichtseyn, gegen sein eigenes Gesetztseyn. Die beiden Seiten sind so bloß verschiedene, und insofern ihre Bestimmtheit, positiv und negativ zu seyn, ihr Gesetztseyn gegen einander ausmacht, so ist jede nicht an ihr selbst so bestimmt, sondern ist nur Bestimmtheit überhaupt; jeder Seite kommt daher zwar eine der Bestimmtheiten von Positivem und Negativem zu; aber sie können verwechselt werden, und jede Seite ist von der Art, daß sie ebenso gut als positiv wie als negativ genommen werden kann.

§914 Further, however, this mere positedness is simply reflected into itself; in accordance with this moment of external reflection the positive and negative are indifferent to that first identity in which they are only moments; in other words, since that first reflection is the positive’s and negative’s own reflection into themselves, each is in its own self its positedness, so each is indifferent to this its reflection into its non-being, to its own positedness. The two sides are thus merely different, and in so far as their being determined as positive and negative constitutes their positedness in relation to one another, each is not in its own self so determined but is only determinateness in general. Therefore, although one of the determinatenesses of positive and negative belongs to each side, they can be changed round, and each side is of such a kind that it can be taken equally well as positive as negative.

§915 Aber das Positive und Negative ist drittens nicht nur ein Gesetztes, noch bloß ein Gleichgültiges, sondern ihr Gesetztseyn oder die Beziehung auf das andere in einer Einheit, die nicht sie selbst sind, ist in jedes zurückgenommen. Jedes ist an ihm selbst positiv und negativ; das Positive und Negative ist die Reflexionsbestimmung an und für sich; erst in dieser Reflexion des Entgegengesetzten in sich ist es positiv und negativ. Das Positive hat die Beziehung auf das Andere, in der die Bestimmtheit des Positiven ist, an ihm selbst; ebenso das Negative ist nicht Negatives als gegen ein anderes, sondern hat die Bestimmtheit, wodurch es negativ ist, gleichfalls in ihm selbst.

§915 But thirdly, the positive and negative are not only something posited, not merely an indifferent something, but their positedness, or the reference-to-other in a unity which they are not themselves, is taken back into each. Each is in its own self positive and negative; the positive and negative are the determination of reflection in and for itself; it is only in this reflection of opposites into themselves that they are positive and negative. The positive has within itself the reference-to-other in which the determinateness of the positive is; similarly, the negative is not a negative as contrasted with an other, but likewise possesses within itself the determinateness whereby it is negative.

C Der Widerspruch / Contradiction
1. Der Unterschied ueberhaupt

§931 Der Unterschied überhaupt enthält seine beiden Seiten als Momente; in der Verschiedenheit fallen sie gleichgültig auseinander; im Gegensatze als solchem sind sie Seiten des Unterschiedes, eines nur durchs andere bestimmt, somit nur Momente; aber sie sind ebenso sehr bestimmt an ihnen selbst, gleichgültig gegen einander und sich gegenseitig ausschließend; die selbstständigen Reflexions-Bestimmungen.

§931 Difference as such contains its two sides as moments; in diversity they fall indifferently apart; in opposition as such, they are sides of the difference, one being determined only by the other, and therefore only moments; but they are no less determined within themselves, mutually indifferent and mutually exclusive: the self-subsistent determinations of reflection.

§934 Der Unterschied überhaupt ist schon der Widerspruch an sich; denn er ist die Einheit von solchen, die nur sind, insofern sie nicht eins sind,—und die Trennung solcher, die nur sind als in derselben Beziehung getrennte. Das Positive und Negative aber sind der gesetzte Widerspruch, weil sie als negative Einheiten, selbst das Setzen ihrer, und darin jedes das Aufheben seiner und das Setzen seines Gegentheils ist.—Sie machen die bestimmende Reflexion als ausschließende aus; weil das Ausschließen Ein Unterscheiden, und jedes der Unterschiedenen als Ausschließendes selbst das ganze Ausschließen ist, so schließt jedes in ihm selbst sich aus.

§934 Difference as such is already implicitly contradiction; for it is the unity of sides which are, only in so far as they are not one-and it is the separation of sides which are, only as separated in the same relation. But the positive and negative are the posited contradiction because, as negative unities, they are themselves the positing of themselves, and in this positing each is the sublating of itself and the positing of its opposite. They constitute the determining reflection as exclusive; and because the excluding of the sides is a single act of distinguishing and each of the distinguished sides in excluding the other is itself the whole act of exclusion, each side in its own self excludes itself.

Here we need a unity of opposites that expresses “difference as such”. The obvious candidate is the opposition between false and true. And indeed, in type theory/categorical logic these are again given by empty type \emptyset and unit type *\ast which form an adjunction

Abs.Contradiction:false * true Abs.\,Contradiction \colon \array{ false & \emptyset &\dashv& \ast & true }

Technically this is the same adjunction as that between nothing and being as around §134 in the Seinslogik. Indeed that makes sense: the tower of determination of the Seinslokig should repeat in the Wesenslogik, but reflected, and hence with different meaning.

§938 Das Negative ist also die ganze, als Entgegensetzung auf sich beruhende Entgegensetzung, der absolute sich nicht auf Anderes beziehende Unterschied; er schließt als Entgegensetzung die Identität von sich aus; aber somit sich selbst, denn als Beziehung auf sich bestimmt er sich als die Identität selbst, die er ausschließt.

§938 The negative is, therefore, the whole opposition based, as opposition, on itself, absolute difference that is not related to an other; as opposition, it excludes identity from itself — but in doing so excludes itself; for as self-relation it is determined as the very identity that it excludes.

2. Der Widerspruch löst sich auf

§943 Nach dieser positiven Seite, daß die Selbstständigkeit im Gegensatze, als ausschließende Reflexion sich zum Gesetztseyn macht, und es ebenso sehr aufhebt, Gesetztseyn zu seyn, ist der Gegensatz nicht nur zu Grunde, sondern in seinen Grund zurückgegangen.— Die ausschließende Reflexion des selbstständigen Gegensatzes macht ihn zu einem Negativen, nur Gesetzten; sie setzt dadurch ihre zunächst selbstständigen Bestimmungen, das Positive und Negative, zu solchen herab, welche nur Bestimmungen sind; und indem so das Gesetztseyn zum Gesetztseyn gemacht wird, ist es überhaupt in seine Einheit mit sich zurückgekehrt; es ist das einfache Wesen, aber das Wesen als Grund. Durch das Aufheben der sich an sich selbst widersprechenden Bestimmungen des Wesens, ist dieses wiederhergestellt, jedoch mit der Bestimmung, ausschließende Reflexionseinheit zu seyn,—einfache Einheit, welche sich selbst als Negatives bestimmt, aber in diesem Gesetztseyn unmittelbar sich selbst gleich und mit sich zusammen-gegangen ist.

§ 943 According to this positive side, in which the self-subsistence in opposition, as the excluding reflection, converts itself into a positedness which it no less sublates, opposition is not only destroyed [zugrunde gegangen] but has withdrawn into its ground. The excluding reflection of the self-subsistent opposition converts this into a negative, into something posited; it thereby reduces its primarily self-subsistent determinations, the positive and negative, to the status of mere determinations; and the positedness, being thus made into a positedness, has simply returned into its unity with itself; it is simple essence, but essence as ground. Through the sublating of its inherently self-contradictory determinations, essence has been restored, but with this determination, that it is the excluding unity of reflection-a simple unity that determines itself as a negative, but in this positedness is immediately like itself and united with itself.

§944 Zunächst geht also der selbstständige Gegensatz durch seinen Widerspruch in den Grund zurück; jener ist das Erste, Unmittelbare, von dem angefangen wird, und der aufgehobene Gegensatz oder das aufgehobene Gesetztseyn ist selbst ein Gesetztseyn. Somit ist das Wesen als Grund ein Gesetztseyn, ein Gewordenes. Aber umgekehrt hat sich nur dieß gesetzt, daß der Gegensatz oder das Gesetztseyn ein Aufgehobenes, nur als Gesetztseyn ist. Das Wesen ist also als Grund so ausschließende Reflexion, daß es sich selbst zum Gesetztseyn macht, daß der Gegensatz, von dem vorhin der Anfang gemacht wurde und der das Unmittelbare war, die nur gesetzte, bestimmte Selbstständigkeit des Wesens ist, und daß er nur das sich an ihm selbst Aufhebende, das Wesen aber das in seiner Bestimmtheit in sich Reflektirte ist. Das Wesen schließt als Grund sich von sich selbst aus, es setzt sich; sein Gesetztseyn,—welches das Ausgeschlossene ist,—ist nur als Gesetztseyn, als Identität des Negativen mit sich selbst. Dieß Selbstständige ist das Negative, gesetzt als Negatives; ein sich selbst Widersprechendes, das daher unmittelbar im Wesen als seinem Grunde bleibt.

§944 In the first place, therefore, the self-subsistent opposition through its contradiction withdraws into ground; this opposition is the prius, the immediate, that forms the starting point, and the sublated opposition or the sublated positedness is itself a positedness. Thus essence as ground is a positedness, something that has become. But conversely, what has been posited is only this, that opposition or positedness is a sublated positedness, only is as positedness. Therefore essence as ground is the excluding reflection in such wise that it makes its own self into a positedness, that the opposition from which we started and which was the immediate, is the merely posited, determinate self-subsistence of essence, and that opposition is merely that which sublates itself within itself, whereas essence is that which, in its determinateness, is reflected into itself. Essence as ground excludes itself from itself, it posf its elf; its positedness — which is what is excluded — is only as positedness, as identity, of the negative with itself. This self-subsistent is the negative posited as negative; it is self-contradictory and therefore remains immediately in essence as-init ground.

§945 Der aufgelöste Widerspruch ist also der Grund,

§945 The resolved contradiction is therefore ground, essence as unity of the positive and negative.

By the discussion at §931 the contradiction in question is given by the adjunction between false=empty type and true=unit type. The Aufhebung of that proceeds via the sharp modality exactly as for becoming as discussed around §183 following the technical discussion at Aufhebung – over cohesive sites.

AufhebungdesWiderspruchs abs.Widerspruch * Wesen \array{ && \vdots && \vdots \\ && \bot && \bot \\ && \flat &\dashv& \sharp \\ &&\vee &\stackrel{Aufhebung \atop {des\;Widerspruchs}}{}& \vee \\ &&\emptyset &\stackrel{abs.\,Widerspruch}{\dashv}& \ast \\ Wesen }

The question then remains which part of this diagram is to carry the name “Grund”. From §945 Grund might be the Aufhebung itself, that would make it the analog in the Wesenslogik of the Dasein in the Seinslogik, which was the Aufhebung of becoming.

However, the analog of Dasein in the Wesen should be another determination of being, and “Grund” seems not to be the right word for a determination of being. It seems rather that Grund is to go along with “Existenz” which is a decent name for a determination of being.

So if Grund is not the process of the above Aufhebung, then maybe it is that wherein which we have Aufhebung. By the above these are the sharp-modal types.

This now has a certain charm to it, because these of course form the base topos/base infinity topos of the topos which is the Wesen, and for that the term “Grund” is rather fitting.

So in the Proceß we tentatively label the fragment as

Grund AufhebungdesWiderspruchs abs.Widerspruch * Wesen \array{ && \vdots && \vdots \\ && \bot && \bot \\ && \flat &\dashv& \sharp & \stackrel{Grund}{} \\ &&\vee &\stackrel{Aufhebung \atop {des\;Widerspruchs}}{}& \vee \\ &&\emptyset &\stackrel{abs.\;Widerspruch}{\dashv}& \ast \\ Wesen }

Der Grund

§964 Das Wesen bestimmt sich selbst als Grund.

Wie das Nichts zuerst mit dem Seyn in einfacher unmittelbarer Einheit, so ist auch hier zuerst die einfache Identität des Wesens mit seiner absoluten Negativität in unmittelbarer Einheit. Das Wesen ist nur diese seine Negativität, welche die reine Reflexion ist. Es ist diese reine Negativität als die Rückkehr des Seyns in sich; so ist es an sich oder für uns bestimmt, als der Grund, in dem sich das Seyn auflöst. Aber diese Bestimmtheit ist nicht durch es selbst gesetzt; oder es ist nicht Grund, eben insofern es diese seine Bestimmtheit nicht selbst gesetzt hat. Seine Reflexion aber besteht darin, sich als das, was es an sich ist, als Negatives zu setzen und sich zu bestimmen. Das Positive und Negative machen die wesenhafte Bestimmung aus, in die es als in seine Negation verloren ist. Diese selbstständigen Reflexions-Bestimmungen heben sich auf, und die zu Grunde gegangene Bestimmung ist die wahrhafte Bestimmung des Wesens.

§964 Essence determines itself as ground.

§964 Just as nothing is at first in simple immediate unity with being, so here too the simple identity of essence is at first in immediate unity with its absolute negativity. Essence is only this its negativity, which is pure reflection. It is this pure negativity as the return of being into itself; as such, it is determined in itself, or for us, as ground in which being is dissolved. But this determinateness is not posited by essence itself; in other words, essence is not ground except in so far as it has itself posited this its determinateness. Its reflection, however, consists in its positing and determining itself as that which it is in itself, as a negative. The positive and negative constitute that determination of essence in which essence is lost in its negation. These self-subsistent determinations of reflection sublate themselves, and the determination that has fallen to the ground [zugrunde gegangene] is the true determination of essence.

On this see the discussion around §945

§964 Essence determines itself as ground.

§968 Der Grund ist zuerst absoluter Grund, in dem das Wesen zunächst als Grundlage überhaupt für die Grundbeziehung ist; näher bestimmt er sich aber als Form und Materie, und giebt sich einen Inhalt.

Zweitens ist er bestimmter Grund, als Grund von einem bestimmten Inhalt; indem die Grundbeziehung sich in ihrer Realisirung überhaupt äußerlich wird, geht sie in die bedingende Vermittelung über.

Drittens, der Grund setzt eine Bedingung voraus; aber die Bedingung setzt ebenso sehr den Grund voraus; das Unbedingte ist ihre Einheit, die Sache an sich, die durch die Vermittelung der bedingenden Beziehung in die Existenz übergeht.

Ground is first, absolute ground, in which essence is, in the first instance, a substrate for the ground relation; but it further determines itself as form and matter and gives itself a content.

Secondly, it is a determinate ground as ground of a determinate content; in that the ground relation in its realisation as such becomes external to itself, it passes over into conditioning mediation.

Thirdly, ground presupposes a condition; but the condition no less presupposes the ground; the unconditioned is their unity, the fact in itself, which through the mediation of the conditioning relation passes over into Existence.

A. Der absolute Grund
a. Form und Wesen

§970 der Grund ist als das aufgehobene Bestimmtseyn nicht das Unbestimmte, sondern das durch sich selbst bestimmte Wesen, aber als unbestimmt oder als aufgehobenes Gesetztseyn Bestimmtes. Er ist das Wesen, das in seiner Negativität mit sich identisch ist.

§970 The determination of reflection, in so far as it withdraws into ground, is a first, an immediate determinate being in general, which forms the starting point. But determinate being still has only the meaning of positedness and essentially presupposes a ground-in the sense that it does not really posit a ground, that this positing is a sublating of itself, that really it is the immediate that is the posited, and ground the not-posited. As we have seen, this presupposing is positing that recoils on that which posits: ground, as the determination that has been sublated, is not indeterminate; it is essence determined through itself, but determined as undetermined, or as a sublated positedness. Ground is essence that in its negativity is identical with itself.

Der Form gehört überhaupt alles Bestimmte an; es ist Formbestimmung, insofern es ein Gesetztes, hiermit von einem solchen, dessen Form es ist, Unterschiedenes ist; die Bestimmtheit als Qualität ist eins mit ihrem Substrat, dem Seyn; das Seyn ist das unmittelbar Bestimmte, das von seiner Bestimmtheit noch nicht unterschieden,—oder das in ihr noch nicht in sich reflektirt, so wie diese daher eine seyende, noch nicht eine Gesetzte ist.—Die Formbestimmungen des Wesens sind ferner als die Reflexions-Bestimmtheiten, ihrer nähern Bestimmtheit nach, die oben betrachteten Momente der Reflexion. Die Identität, und der Unterschied, dieser Theils als Verschiedenheit, Theils als Gegensatz. Ferner aber gehört auch die Grundbeziehung dazu, insofern sie zwar die aufgehobene Reflexions-Bestimmung aber dadurch das Wesen zugleich als Gesetztes ist. Dagegen gehört zur Form nicht die Identität, welche der Grund in sich hat, nämlich daß das Gesetztseyn als aufgehobenes und das Gesetztseyn als solches,—der Grund und das Begründete,—Eine Reflexion ist, welche das Wesen als einfache Grundlage ausmacht, die das Bestehen der Form ist. Allein dieß Bestehen ist im Grunde gesetzt; oder dieß Wesen ist selbst wesentlich als bestimmtes; somit ist es auch wieder das Moment der Grundbeziehung und Form.—Dieß ist die absolute Wechselbeziehung der Form und des Wesens, daß dieses einfache Einheit des Grundes und des Begründeten, darin aber eben selbst bestimmt oder Negatives ist, und sich als Grundlage von der Form unterscheidet, aber so zugleich selbst Grund und Moment der Form wird.

§973a Das Wesen hat eine Form und Bestimmungen derselben.

§973a Essence has a form and determinations of the form.

Notice that “Form” also means shape. By the discussion at §812,§816, the Essence is formalized as the ambient topos. In view of this §973 translates to “The topos has a shape”, and indeed there is the concept of shape of an infinity-topos. This is just what is reflected by the shape modality ʃ&#643;.

Morover, given a type XX in the Wesen, hence in cohesive homotopy type theory, it makes good sense to refer to ʃX&#643; X as the shape of that homotopy type – which is in the established traditional sense of shape theory . This is indeed what the name “shape modality” is alluding to.

See the discussion below §989 for the dual moment.

§973b Erst als Grund hat es eine feste Unmittelbarkeit oder ist Substrat.

§973b It is only as ground that it has a fixed immediacy or is a substrate.

b. Form und Materie

§978 Das Wesen wird zur Materie, indem seine Reflexion sich bestimmt, zu demselben als zu dem formlosen Unbestimmten sich zu verhalten.

§978 Essence becomes matter in that its reflection is determined as relating itself to essence as to the formless indeterminate.

§980 Die Materie muß daher formirt werden, und die Form muß sich materialisiren,

§981 2. Die Form bestimmt daher die Materie, und die Materie wird von der Form bestimmt.

§981 2. Hence form determines matter, and matter is determined by form.

Notice by §1068 that here indeed Materie refers to the physical world (even if physical nature only appears much further down in PN§192) and explicitly refers also to physical fields.

c. Form und Inhalt.

§989 Die Form steht zuerst dem Wesen gegenüber; so ist sie Grundbeziehung überhaupt, und ihre Bestimmungen, der Grund und das Begründete. Alsdenn steht sie der Materie gegenüber; so ist sie bestimmende Reflexion und ihre Bestimmungen sind die Reflexionsbestimmung selbst und das Bestehen derselben. Endlich steht sie dem Inhalte gegenüber,

§989 At first, form stands opposed to essence; it is then the simple ground relation, and its determinations are the ground and the grounded. Secondly, it stands opposed to matter; it is then determining reflection, and its determinations are the reflected determination itself and the subsistence of the determination. Lastly, it stands opposed to content;

Above in the discussion at §973a we identified the moment of “form” with the shape modality at the reflected level of the Wesen, since for XX a type in cohesive homotopy type theory, then ʃX&#643; X is indeed naturally pronounced as the “shape of the homotopy type” in the traditional sense of shape theory.

In this vein, the collection of all global points *X\ast \to X in XX, hence its flat modality X\flat X is naturally pronounced as the “content” of that type. It is literally the collection of unit types ( ones §340)– that it contains .

Indeed, if one thinks of XX as a cohesive homotopy type, then X\flat X is the underlying bare homotopy type with its cohesion forgotten.

This terminology is motivated from and well adapted to the picture in chemistry (see at motivation for cohesive toposes): imagine a chunk of chemical substance, then its plain content of substance is the collection of all the separate molecules – quantified by the number of moles of the substance – whereas in remembering just this number all memory of the shape and cohesion of the substance has been forgotten. Indeed, by §1068, this reference to chemistry seems to be entirely intended.

Therefore it is natural to pronounce the flat modality \flat, as the content, Inhalt. And so then by §973a its adjunction with the shape modality yields a unity of opposites

form ʃ content \array{ form & &#643; & \dashv & \flat & content }

which is naturally identified with what the text in §989 alludes to.

It remains to find the name of this unity of opposites:

§990 Der Inhalt hat erstlich eine Form und eine Materie, die ihm angehören und wesentlich sind; er ist ihre Einheit.

§990 The content is, first, a form and a matter which belong to it and are essential; it is their unity

So form-content-matter form a unity of opposites. The exact form of §990 suggests to take (formcontentmatter)(form \stackrel{content}{\dashv} matter), which however seems a bit awkward. But by §989 there seems to be some flexibility in these three terms opposing each other, and if we appeal to that and declare that we should put

form ʃ matter content \array{ form & &#643; & \stackrel{matter}{\dashv} & \flat & content }

then it works out nicely: Notice by §1068 that “matter” here includes physical fields such as explicitly the electromagnetic field, hence gauge fields. Now this matches: by the discussion at differential cohomology hexagon the adjunction ʃ&#643; \dashv \flat is precisely what axiomatizes higher gauge fields in the form of cocycles in differential cohomology.

This fits indeed rather well, as it means that we recover at the stage of the Wesenslogik what we already had at the stage of the Seinslogik (where the appearance of gauge fields via the above adjunction is discussed around §714).

But we should maybe make notationally more explicit (than would have been possibl in 1812) that, by §1068, “matter” here means “physical fields and matter” and specifically gauge fields. Therefore in the Proceß we add the stage:

form ʃ (gauge)fields content. \array{ form & &#643; & \stackrel{(gauge)\,fields}{\dashv} & \flat & content } \,.

Notice that in the Naturphilosophie PN§204 this unity of opposites of attraction and repulsion becomes gravity.

B. Der bestimmte Grund
a. Der formelle Grund
b. Der reale Grund
c. Der vollstaendige Grund
C. Die Bedingung
a. Das relativ Unbedingte

§1021 Der Grund ist das Unmittelbare und das Begründete das Vermittelte.

§1021 Ground is the immediate, and the grounded the mediated.

Begriffslogiknatural deduction
unmittlebarGrundantecedent
vermitteltdas Begründetesuccedent/consequent
b. Das absolute Unbedingte
c. Hervorgang der Sache in die Existenz

§1033 Wenn alle Bedingungen einer Sache vorhanden sind, so tritt sie in die Existenz.

§1033 When all the conditions of a fact are present, it enters into Existence.

This is term introduction via the natural deduction from the antecedent of the term introduction rule. More of this in §1035

§1035 Die Sache geht aus dem Grunde hervor. Sie wird nicht durch ihn so begründet oder gesetzt, daß er noch unten bliebe, sondern das Setzen ist die Herausbewegung des Grundes zu sich selbst, und das einfache Verschwinden desselben. Er erhält durch die Vereinigung mit den Bedingungen die äußerliche Unmittelbarkeit und das Moment des Seyns.

§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.

Die Erscheinung

§1036 Das Wesen muß erscheinen.

§1036 Essence must appear

Notice that by §816 this “appear” is short for “appear in itself”.

§1037 So erscheint das Wesen. Die Reflexion ist das Scheinen des Wesens in ihm selbst.

Thus appears essence . Reflection is the appearance of essence within itself.

This nicely explicitly re-iterates §816. See the discussion there about translating this to “The ambient category appears reflected within itself”.

Die Existenz / Existence.

§1040 Just as the proposition of ground states that whatever is has a ground, or is something posited or mediated, so too we must formulate a proposition of Existence, and in these terms: whatever is, exists. The truth of being is to be, not a first immediate, but essence that has emerged into immediacy.

A. Das Ding und seine Eigenschaften.

§1048 Das Ding wird von seiner Existenz unterschieden, wie das Etwas von seinem Seyn unterschieden werden kann.

§1048 The thing is distinct from its Existence just as something can be distinguished from its being.

This means that the Ding repeats (reflected) the Etwas from the Seinslogik in §221:

momentunitycomoment
SeinslogikAnsichseynEtwasSein-fuer-Anderes
WesenslogikExistenzDing

Hence we locate these terms at matching positions in the Proceß.

a. Ding an sich und Existenz.
b. Die Eigenschaft.

§1056 Die Qualität ist die unmittelbare Bestimmtheit des Etwas; das Negative selbst, wodurch das Seyn Etwas ist.

§1056 Quality is the immediate determinateness of something, the negative itself through which being is something.

c. Die Wechselwirkung der Dinge.

§1064 Das Ding-an-sich existiert wesentlich; die äußerliche Unmittelbarkeit und die Bestimmtheit gehört zu seinem Ansichsein oder zu seiner Reflexion-in-sich. Das Ding-an-sich ist dadurch ein Ding, das Eigenschaften hat, und es sind dadurch mehrere Dinge, die nicht durch eine ihnen fremde Ruecksicht, sondern sich durch sich selbst voneinander unterscheiden.

§1064 The thing-in-itself essentially exists; the external immediacy and determinateness belongs to its in-itself or to its reflection-into-self. By virtue of this, the thing-in-itself is a thing which has properties, and hence there are a number of things which are distinguished from one another not in respect of something alien to them but through themselves.

§1065a Diese mehreren verschiedenen Dinge stehen in wesentlicher Wechselwirkung durch ihre Eigenschaften; die Eigenschaft ist diese Wechselbeziehung selbst, und das Ding ist nichts außer derselben;

§1065a These many different things stand in essential reciprocal action via their properties; the property is this reciprocal relation itself and apart from it the thing is nothing;

This is the concept of structuralism. As discussed there, at least to some extent this is captured by category theory, where the nature of objects is entirely determined and only determined by the morphisms relating objects to each other.

Hence we read thing here as object of the ambient topos.

§1065b die gegenseitige Bestimmung, die Mitte der Dinge-an-sich, die als Extreme gleichgültig gegen diese ihre Beziehung bleiben sollten, ist selbst die mit sich identische Reflexion und das Ding-an-sich, das jene Extreme sein sollten.

§1065b the reciprocal determination, the middle terms of the things-in-themselves, which, as extremes, are supposed to remain indifferent to this their relation, is itself the self-identical reflection and the thing-in-itself which these extremes are supposed to be. Thinghood is thus reduced to the form of indeterminate identity-with-self which has its essentiality only in its property. If, therefore, one is speaking of a thing or things in general without any determinate property, then their difference is merely indifferent, quantitative. What is considered as one thing can equally be made into or considered as several things; the separation or union of them is external. A book is a thing and each of its leaves is also a thing, and so too is each bit of its pages, and so on to infinity. The determinateness through which one thing is this thing only, lies solely in its properties. Through them it distinguishes itself from other things, because property is negative reflection and a distinguishing; the thing therefore contains the difference of itself from other things solely in its property. This is the difference reflected into itself, through which the thing, in its positedness, that is, in its relation to another, is at the same time indifferent to the other and to its relation to it. All that remains therefore to the thing without its properties is abstract being-in-self or initselfness, an unessential compass and external holding together. The true in-itself is the in-itself in its positedness: and this is property. With this, thinghood has passed over into property.

B. Das Bestehen des Dings aus Materien.

§1068 Der Übergang der Eigenschaft in eine Materie oder in einen selbstständigen Stoff ist der bekannte Übergang, den an der sinnlichen Materie die Chemie macht, indem sie die Eigenschaften der Farbe, des Geruchs, des Geschmacks u.s.f. als Lichtstoff, Färbestoff, Riechstoff, sauren, bittern u.s.f. Stoff darzustellen sucht oder andere wie den Wärmestoff, die elektrische, magnetische Materie geradezu nur annimmt, und damit die Eigenschaften in ihrer Wahrhaftigkeit zu handhaben überzeugt ist.—Ebenso geläufig ist der Ausdruck, daß die Dinge aus verschiedenen Materien oder Stoffen bestehen. Man hütet sich, diese Materien oder Stoffe Dinge zu nennen; ob man wohl auch einräumen wird, daß z.B. ein Pigment, ein Ding ist; ich weiß aber nicht, ob z.B. auch der Lichtstoff, der Wärmestoff, oder die elektrische Materie u.s.f. Dinge genannt werden. Man unterscheidet die Dinge und ihre Bestandtheile, ohne genau anzugeben, ob diese und in wie weit sie auch Dinge, oder etwa nur Halbdinge seyen; aber Existirende überhaupt sind sie wenigstens.

§1068 The transition of property into a matter or into a self-subsistent stuff is the familiar transition performed on sensible matter by chemistry when it seeks to represent the properties of colour, smell, taste and so on, as luminous matter, colouring matter, odorific matter, sour, bitter matter and so on, or merely straightway postulates others like heat matter or caloric, electrical and magnetic matter, in the conviction that it has got hold of properties in their truth. Equally current is the expression that things consist of various matters. One is careful not to call these matters things; although it would certainly be admitted that, e.g. a pigment is a thing; but I do not know whether e.g. luminous matter, heat matter or electrical matter and so on, are also called things. Things and their constituents are distinguished without it being exactly stated whether and to what extent the latter are also things or perhaps only half things; but they are at least existents in general.

Despite – and in fact via – this cautioning remark, this says that the word Materie as used before in Form und Materie and Form und Inhalt is indeed meant what the physical world is made of. Notice

  1. how the reference to chemical substances harmonizes with the chemical imagery going with cohesion in the discussion below §989;

  2. that not just genuine matter but also what in modern parlance is called physical fields, notably gauge fields (“Lichtstoff” = light, “elektrische Materie” =electromagnetic field), is explicitly included.

To highlight this we should maybe write Materiefelder or just Felder or even Eichfelder instead, which is what we do in the Proceß-diagram.

It is maybe noteworthy that some physics appears here in the Wesenslogik, even though nature will not appear before PN§192.

Die Nothwendigkeit, von den Eigenschaften zu Materien überzugehen, oder daß die Eigenschaften in Wahrheit Materien sind, hat sich daraus ergeben, daß sie das Wesentliche und damit das wahrhaft Selbstständige der Dinge sind.

C. Die Auflösung des Dinges.

Die Existenz hat in diesem Dinge ihre Vollständigkeit erreicht, nämlich in Einem an sich seyendes Seyn oder selbstständiges Bestehen, und unwesentliche Existenz zu seyn; die Wahrheit der Existenz ist daher, ihr Ansichseyn in der Unwesentlichkeit, oder ihr Bestehen in einem Andern und zwar dem absolut Andern, oder zu ihrer Grundlage ihre Nichtigkeit zu haben. Sie ist daher Erscheinung.

Die Erscheinung

§1084 Die Erscheinung ist daher Einheit des Scheins und der Existenz

§1084 Appearance is accordingly the unity of illusory being and Existence.

(Here the English translation does not really seem to work…)

Das wesentliche Verhältnis

Das Verhältnis des Ganzen und der Teile
Das Verhältnis der Kraft und ihrer Auesserung
Verhaeltnis des Inneren und Aeusseren

§1149 Das innere ist als die Form der reflektierten Unmittelbarkeit oder des Wesens, gegen das Aussere als die Form des Seins bestimmt, aber beide sind nur eine Identitaet.

§1149 The inner is determined as the form of reflected immediacy or of essence over against the outer as the form of being, but the two are only one identity.

The “form of being” is to refer to the category of being, hence to the ambient \infty-category. Opposed to that is the type universe which is the inner reflection of that inside itself, see also §1163b.

Their unity hence must be univalence. See also (§1159, §1187).

Die Wirklichkeit

§1158 Die Wirklichkeit ist die Einheit des Wesens und der Existenz;

§1158 Actuality is the unity of essence and Existence

§1159 Diese Einheit des Innern und Äußern ist die absolute Wirklichkeit. Diese Wirklichkeit aber ist zunächst das Absolute als solches;

§1159 This unity of inner and outer is absolute actuality. But this actuality is, in the first instance, the absolute as such — in so far as it is posited as a unity in which form has sublated itself and made itself into the empty or outer difference of an outer and inner.

If the inner is the type universe, being the inner reflection of the ambient category, hence of the outer (§1149) then the unity of the inner and the outer is univalence.

§1160 Zweitens die eigentliche Wirklichkeit. Wirklichkeit, Möglichkeit und Nothwendigkeit machen die formellen Momente des Absoluten, oder die Reflexion desselben aus.

§1160 Secondly, we have actuality proper. Actuality, possibility and necessity constitute the formal moments of the absolute, or its reflection.

(beware that the possibility monad and necessity comonad are not in general idempotent).

In any case, by the discussion at necessity and possibility – As modality in dependent type theory the adjunction (possibility \dashv necessity) characterizes locally cartesian closed category. So with the “Wesen” (“Essence”) translating to the ambient category by §812, actuality proper here translates to this ambient category being a locally Cartesian closed category.

For more on this see below §1191.

§1161 Drittens die Einheit des Absoluten und seiner Reflexion ist das absolute Verhältniß, oder vielmehr das Absolute als Verhältniß zu sich selbst; Substanz.

§1161 Thirdly, the unity of the absolute and its reflection is the absolute relation, or rather the absolute as relation to itself — substance.

This we come to in more detail below.

Das Absolute

A. Die Auslegung des Absoluten

§1163b die Beziehung von Seyn und Wesen hat sich bis zum Verhältnisse des Innern und Aeußern fortgebildet. Das Innere ist das Wesen aber als die Totalität, welche wesentlich die Bestimmung hat, auf das Seyn bezogen und unmittelbar Seyn zu seyn. Das Aeußere ist das Seyn, aber mit der wesentlichen Bestimmung, auf die Reflexion bezogen unmittelbar ebenso verhältnißlose Identität mit dem Wesen zu seyn.

§1163b the connection between being and essence has progressed to the relation of inner and outer. The inner is essence, but as totality, which essentially has the determination of connection with being and immediately to be being. The outer is being, but with the essential determination of being connected with reflection, and equally to be immediately a relationless identity with essence.

We found that being, the category of being, is the full ambient \infty-topos H\mathbf{H}, while essence, in its appearence in itself, is the internal type universe TypeHType \in \mathbf{H}. Hence the outer is the “outer type universe”, the \infty-topos H\mathbf{H}, regarded as the full category of being, and the inner is its essence, hence by the above its internal reflection, hence is its inner type universe TypeHType \in \mathbf{H}.

§1163c Das Absolute selbst ist die absolute Einheit beider; es ist dasjenige, was überhaupt den Grund des wesentlichen Verhältnisses ausmacht, das als Verhältniß nur noch nicht in diese seine Identität zurückgegangen, und dessen Grund noch nicht gesetzt ist.

§1163c The absolute itself is the absolute unity of both; it is that which constitutes in general the ground of the essential relation which, as relation, merely has not yet withdrawn into this its identity and whose ground is not yet posited.

§1169 Das Absolute ist nur das Absolute, weil es nicht die abstrakte Identität, sondern die Identität des Seyns und Wesens, oder die Identität des Innern und Aeußern ist. Es ist also selbst die absolute Form, welche es in sich scheinen macht, und es zum Attribut bestimmt.

§1169 The absolute is the absolute only because it is not abstract identity, but the identity of being and essence, or the identity of inner and outer; it is therefore itself the absolute form which makes it reflect itself into itself and determines it into attribute.

This unity is univalence, see §1149, §1159, §1187.

B. Das absolute Attribut
C. Der Modus des Absoluten

Die Wirklichkeit / Actuality

§1187 Das Absolute ist die Einheit des Innern und Aeußern als erste, ansichseyende Einheit.

§1187 The absolute is the unity of inner and outer as initial, implicit unity.

univalence, as in §1159

§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.

§1191 Aber zunächst, indem Wirkliches und Mögliches formelle Unterschiede sind, ist ihre Beziehung gleichfalls nur formell, und besteht nur darinn, daß das Eine wie das Andere ein Gesetztseyn ist, oder in der Zufälligkeit.

§1191 But first of all, since the actual and the possible are formal differences, their relation is likewise merely formal and consist only in the fact that the one like the other is a positedness, or in contingency.

These paragraphs as well as those of the following subsections below explore Wirklichkeit (reality, actuality) as expressed in a triple of modalities

  1. Möglichkeit (pssobility)

  2. Notwendigkeit (necessity)

  3. Zufälligkeit (randomness, contingency)

(Notice the original German of the third. While the existing English translations chose “contingency”, arguably “randomness” may be closer to the spirit of the orginal.)

Formalization in formal logic of the modalities of possibility and necessity via modal operators constitutes the field of modal logic, which has by now a long tradition and is well establlished. In the following we highlight how this traditional formalization refines faithfully and usefully to the type theory/categorical logic that we are using for formalization here (modal type theory), namely by identifying the necessity and possibility monads (as also discussed there) as one of the two adjoint pairs induced by the adjoint triple of base change into a context of a type of “possible worlds”. Then we discuss that the other adjoint pair induced by this adjoint triple indeed captures randomness, in that its monad may be thought of as producing space of random variables.

Let WW be a type (space) of possible worlds in some contex and consider the base change adjoint triple of dependent sum W\sum_W, context extension W *W^\ast and dependent product X\prod_X over XX

H /X XW * XH. \mathbf{H}_{/X} \stackrel{\overset{\sum_X}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_X}{\longrightarrow}}} \mathbf{H} \,.

Then, as discussed in some detail at necessity and possibility, we have that

  • the dependent sum X\sum_X expresses possibility: if PP is a proposition about elements of WW, then WP\sum_W P is the type which under propositions as types expresses the proposition that there is a wWw \in W such that P(w)P(w) holds, hence such that it is possible (in some world ww) that PP holds;

  • the dependent product X\prod_X expresses necessity: if PP is a proposition about elements of WW, then WP\prod_W P holds when PP holds for all wWw \in W, hence when PP holds necessarily, independently of which particular possible world is realized.

Or rather, to stay in the context of the possible worlds WW, we consider the possibility monad

WW * W \lozenge_W \coloneqq W^\ast \sum_W

and the necessity comonad

WW * W, \Box_W \coloneqq W^\ast \prod_W \,,

both operating on the slice H /W\mathbf{H}_{/W}.

But then there is also the induced monad operating on H\mathbf{H}, given by

WW *:HH, \prod_W W^\ast \colon \mathbf{H} \to \mathbf{H} \,,

often called the function monad. To see which meaning this has we need to consider its action on a type which is not a mere proposition (for a mere proposition PP in the absolute context is either true or false and hence so is WW *\prod_W W^\ast, there is nothing interesting happening in this case). For VHV\in \mathbf{H} any type, then

WW *V[W,V]=(WV) \prod_W W^\ast V \simeq [W,V] = (W\to V)

is the space of VV-valued functions on the space of possible worlds. A modern mathematical incarnation of spaces of “possible worlds” (possible configurations of a system under consideration) are probability spaces (e.g. Toronto-McCarthy 10b, slide 23). Now the space of (measurable) functions on a probability space WW is to be interpreted as its space of random variables (stochastic variables).

In the context of monads in computer science, the function monad [W,][W,-] is called the reader monad (notably in the Haskell programming language). This terminology reflects the intended usage where WW is a type of input that is read in from a device such as a keyboard. When used this way, then the terms of WW are indeed entirely random as far as the program that reads them in is concerned. Indeed, software prgrams that require genuine randomness (as opposed to pseudo-randomness) as input, such as programs the generate keys for cryptography, commonly ask the user to hit some keys on the keyboard or read other data from external input devices. When such software models input via a reader monad, then the terms of the relevant type [W,V][W,V] are precisely the kind of random variables in the physical universe that probability theory is designed to model.

In this vein, the Haskell-introduction Verdier 14 says:

The intuition behind the Reader monad, for a mathematician, is perhaps stochastic variables. A stochastic variable is a function from a probability space to some other space. So we see a stochastic variable as a monadic value.

In a more technical style the very same point may be found made in (Toronto-McCarthy 10b, slide 24) and then Toronto-McCarthy 10b, slide 35, where it says:

you could interpret this by regarding random variables as reader monad computations.

In the Toronto-McCarthy 10a, 2.2 the authors call the monad WW *=[W,]\prod_W W^\ast = [W,-] the random variable idiom.

Traditional Haskell tutorials model random states wWw \in W via the state monad

[W,W×()]= WW * WW *() [W, W \times (-)] = \prod_W W^\ast \sum_W W^\ast (-)

for W=StdGenW = StdGen a type of random numbers. This allows programs Y[W,W×Y]Y \to [W, W\times Y] which not only depend on the state wWw\in W but may also modify it. If one restricts attention to the case where the environment possible world ww is not to be changed by the program, then this construct may be reduced to the reader monad, as above.

From all this we may conclude that WW *\prod_W W^\ast expresses core aspects of Zufälligkeit, randomness, contingency much like W * WW^\ast \prod_W expresses necessity.

We may make this yet more concrete as follows. By the discussion at dependent linear type theory and Quantization via Linear homotopy types, the secondary integral transforms on linear types which formalize aspects of path integral quantization are all controled by the action of WW *\prod_W W^\ast on linear types. Of course an intrinsic randomness is the very hallmark of quantum mechanics, which is the actual reality, hence we see that in linear homotopy type theory the relation of WW *\prod_W W^\ast to aspects of randomness is stronger still.

Observe furthermore that for that discussion of secondary integral transforms one needs to choose fundamental classes in linear homotopy type theory,which are essentially choices of equivalences WW *() WW *()\prod_W W^\ast (-) \simeq \sum_W W^\ast(-) (see there for details). Hence in the context the fourth (co-)monad induced by the base change adjoint triple coincides or essentially coincides with the third one.

In summary we get that the base change adjoint triple of a locally Cartesian closed category

H /W WW * WH \mathbf{H}_{/W} \stackrel{\overset{\sum_W}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_W}{\longrightarrow}}} \mathbf{H}

induces an adjoint pair of (co-)monads on H /W\mathbf{H}_{/W} as well as to (co-)monads on H\mathbf{H} which in application to quantum physics are forced to coincide on their relevant input types, such as to yield a total of three operations which may be pronounced thus:

possibility necessity , randomness Moeglichkeit Notwendigkeit , Zufaelligkeit W * W W * W , WW *Wirklichkeit \underset{Wirklichkeit}{ \underbrace{ \array{ possibility &\dashv& necessity &\;\;\;\;,\;\;& randomness \\ Moeglichkeit &\dashv& Notwendigkeit &\;\;\;\;,\;\;& Zufaelligkeit \\ W^\ast\sum_W &\dashv& W^\ast \prod_W &\;\;\;\;,\;\;& \prod_W W^\ast } } }
A Zufälligkeit oder formelle Wirklichkeit, Möglichkeit und Nothwendigkeit/ Randomness or Formal Actuality, Possibility and Necessity

§1192 1. 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 1. 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.

see below §1191

B. Relative Nothwendigkeit oder reale Wirklichkeit, Möglichkeit und Nothwendigkeit / Relative Necessity, or Real Actuality, Possibility and Necessity

see below §1191

C. Absolute Nothwendigkeit / Absolute Necessity

Das absolute Verhältniß

§1234 Dieß Verhältniß in seinem unmittelbaren Begriff ist das Verhältniß der Substanz und der Accidenzen, das unmittelbare Verschwinden und Werden des absoluten Scheines in sich selbst. Indem die Substanz sich zum Fürsichseyn gegen ein Anderes bestimmt, oder das absolute Verhältniß als reales, ist das Verhältniß der Kausalität. Endlich indem dieses als sich auf sich Beziehendes in Wechselwirkung übergeht, so ist damit das absolute Verhältniß nach den Bestimmungen, welche es enthält, auch gesetzt; diese gesetzte Einheit seiner in seinen Bestimmungen, die als das Ganze selbst und damit ebenso sehr als Bestimmungen gesetzt sind, ist alsdann der Begriff.

§1234 This relation in its immediate Notion is the relation of substance and accidents, the immediate vanishing and becoming of the absolute illusory being within itself. When substance determines itself to being-for-self over against an other, or the absolute relation determines itself as real, then we have the relation of causality. Lastly, when this as self-relating passes over into reciprocity, the absolute relation is also posited in accordance with the determinations it contains; this posited unity of itself in its determinations which are posited as themselves the whole, but equally as determinations, is then the Notion.

substance and accidence?

Das Verhaeltniß der Sustantialitaet / The relation of substantiality

§1235 Absolute necessity is absolute relation because it is not being as such, but being that is because it is, being as absolute self-mediation. This being is substance; as the final unity of essence and being it is the being in all being; it is neither the unreflected immediate, nor an abstract being standing behind Existence and Appearance, but it is immediate actuality itself and this as absolute reflectedness-into-self, as a subsisting in and for itself. Substance as this unity of being and reflection is essentially the reflective movement [Scheinen] and positedness of itself. The reflective movement is the reflective movement that is self-related, and it is thus that it is; this being is substance as such. Conversely, this being is only the positedness that is identical with itself, and as such it is totality in the form of illusory being, accidentality.

This means that

  • substance is the unity of being and reflection

recall also the announcement of substance in §1161.

§1238 Substance, as this identity of the reflective movement, is the totality of the whole and embraces accidentality within it, and accidentality is the whole substance itself.

Since above we identified

  1. the logic of being with the system of adjoint modalities of differential cohesion;

  2. the logic of essence with the theory of the type universe, hence with the ambient homotopy type theory/(infinity,1)-topos;

the union of the two is the totality of the whole, the full differential cohesive homotopy type theory/cohesive (infinity,1)-topos.

Therefore we label in the Process the \infty-topos as its own maximal subcategory H\mathbf{H} as “substance”.

Notice here that every (higher) topos has two geometric interpretations (see at topos – Idea): on the one hand as a category of spaces and on the other hand as one single space (see there) itself, the latter at least if it is a ringed topos or more generally a structured (infinity,1)-topos. The latter is canonically the case for H\mathbf{H}, by the discussion at differential cohesion – structure sheaves. Under this interpretation, all the objects (types/concepts) inside H\mathbf{H} may naturally be interpreted as spaces modeled on the space H\mathbf{H}, hence as spaces built out of the substance H\mathbf{H}.

A modern imagery may be suggestive here: consider this differential cohesive homotopy type theory coded (as has partially been done already, see the cohesive HoTT – References) in a programming language such as Coq. Then it is a piece of software, the modern absolute substance. If one now sets out, as the aim is, to use this as a kind of computer algebra system for reasoning about fundamental physics, then this software is farily literally the substance on which the world, nature, is to be modeled.

This imagery also seems to be well suited to do away with the notorious issue with whether Spinoza's substance is “material”. No, it is not matter as in the nature which “runs” on it, still, it is in evident sense the substance out of which this nature is formed, but, if one wishes, a substance of a more idealistic form that plain matter.

Compare this to the statement of §1287 that this substance is the substance of Spinoza's system, which Spinoza introduces as

Per Substantiam Intelligo– By substance I understand what is in itself and is conceived through itself, i.e., that whose concept does not require the concept of another thing, from which it must be formed.

Back in the imagery of a computer algebra software for formalized fundamental physics, this is indeed the situation after the moment of booting the system and seeing the green prompt on an empty black screen. Nothing has been introduced yet on the basis of anything, all we have is the software kernel running, from which now everything is to be formed. In this imagery, Spinoza’s

Prop 15 Whatever is, is in the substance, and nothing can be or be conceived without the substance.

seems clear.

§1236 Die Bewegung der Accidentalität stellt daher an jedem ihrer Momente das Scheinen der Kategorien des Seyns und der Reflexions-Bestimmungen des Wesens in einander dar.

§1236 The movement of accidentality therefore exhibits in each of its moments the illusory showing in one another of the categories of being and of the reflective determinations of essence.

§1237 Diese Bewegung der Accidentalität ist die Aktuosität der Substanz, als ruhiges Hervorgehen ihrer selbst.

§1237 This movement of accidentality is the actuosity of substance as a tranquil coming forth of itself.

Where for instance the shape modality ʃ&#643; determines a category of being, as discussed above, we may ask if it happens to be, accidentally, exhibited by a type/object 𝔸\mathbb{A} in that ʃloc 𝔸&#643; \simeq loc_{\mathbb{A}} is the localization at 𝔸*\mathbb{A}\to\ast. If so, then this 𝔸\mathbb{A} has the interpretation of theing the continuum, e.g. the real line in smooth infinity-groupoids.

§1238 Die Substanz als diese Identität des Scheinens ist die Totalität des Ganzen, und begreift die Accidentalität in sich, und die Accidentalität ist die ganze Substanz selbst. Der Unterschied ihrer in die einfache Identität des Seyns, und in den Wechsel der Accidenzen an derselben ist eine Form ihres Scheins. Jenes ist die formlose Substanz des Vorstellens, dem der Schein sich nicht als Schein bestimmt hat, sondern das als an einem Absoluten an solcher unbestimmten Identität festhält, die keine Wahrheit hat, nur die Bestimmtheit der unmittelbaren Wirklichkeit oder ebenso des Ansichseyns oder der Möglichkeit ist;—Formbestimmungen, welche in die Accidentalität fallen.

§1238 Substance, as this identity of the reflective movement, is the totality of the whole and embraces accidentality within it, and accidentality is the whole substance itself. The differentiation of itself into the simple identity of being and the flux of accidents in it, is a form of its illusory being. The former is the formless substance of ordinary thinking for which illusory being has not determined itself as illusory being, but which clings to such an indeterminate identity as an absolute, an identity which has no truth and is only the determinateness of immediate actuality or equally of the in-itself or possibility — form determinations which fall into accidentality.

The relation of causality
Reciprocity

Die Lehre vom Begriff / The doctrine of the notion

Vom Begriff im Allgemeinen

§1280 Der Begriff ist von dieser Seite zunächst überhaupt als das Dritte zum Seyn und Wesen, zum Unmittelbaren und zur Reflexion anzusehen. Seyn und Wesen sind insofern die Momente seines Werdens; er aber ist ihre Grundlage und Wahrheit, als die Identität, in welcher sie untergegangen und enthalten sind. Sie sind in ihm, weil er ihr Resultat ist, enthalten, aber nicht mehr als Seyn und als Wesen; diese Bestimmung haben sie nur, insofern sie noch nicht in diese ihre Einheit zurückgegangen sind.

§1280 From this aspect the Notion is to be regarded in the first instance simply as the third to being and essence, to the immediate and to reflection. Being and essence are so far the moments of its becoming; but it is their foundation and truth as the identity in which they are submerged and contained.

This justifies, despite the order of the books and chapters, to order the Notion below Being and the Essence in the Process. We read concept/notion as type and so the doctrine of the notion as the ambient type theory, literally the foundation in which the determinations of being (the adjoint modalities) and the reflections of essence (the type of types) are formulated.

§1281 Objective logic therefore, which treats of being and essence constitutes properly the genetic exposition of the Notion. More precisely, substance is already real essence, or essence in so far as it is united with being and has entered into actuality. Consequently, the Notion has substance for its immediate presupposition; what is implicit in substance is manifested in the Notion. Thus the dialectical movement of substance through causality and reciprocity is the immediate genesis of the Notion, the exposition of the process of its becoming. But the significance of its becoming, as of every becoming is that it is the reflection of the transient into its ground and that the at first apparent other into which the former has passed constitutes its truth. Accordingly the Notion is the truth of substance; and since substance has necessity for its specific mode of relationship, freedom reveals itself as the truth of necessity and as the mode of relationship proper to the Notion.

§1286 Diese unendliche Reflexion in sich selbst, daß das An- und Fürsichseyn erst dadurch ist, daß es Gesetztseyn ist, ist die Vollendung der Substanz. Aber diese Vollendung ist nicht mehr die Substanz selbst, sondern ist ein Höheres, der Begriff das Subjekt. Der Uebergang des Substantialitäts-Verhältnisses geschieht durch seine eigene immanente Nothwendigkeit, und ist weiter nichts, als die Manifestation ihrer selbst, daß der Begriff ihre Wahrheit, und die Freiheit die Wahrheit der Nothwendigkeit ist.

§1286 This infinite reflection-into-self, namely, that being is in and for itself only in so far as it is posited, is the consummation of substance. But this consummation is no longer substance itself but something higher, the Notion, the subject. The transition of the relation of substantiality takes place through its own immanent necessity and is nothing more than the manifestation of itself, that the Notion is its truth, and that freedom is the truth of necessity.

§1287 Es ist schon früher im zweiten Buch der objektiven Logik S. 194 f. Anm. erinnert worden, daß die Philosophie, welche sich auf den Standpunkt der Substanz stellt und darauf stehen bleibt, das System des Spinoza ist.

§1287 I have already mentioned in the Second Book of the Objective Logic that the philosophy which adopts the standpoint of substance and stops there is the system of Spinoza.

Spinoza's system

§B160 Der Begriff ist das Freie, als die für sich seiende Macht der Substanz; – und als die Totalität dieser Negativität, in welcher jedes der Momente das Ganze ist, das er ist, und als ungetrennte Einheit mit ihm gesetzt ist, ist er in seiner Identität mit sich das an und für sich bestimmte.

§1291a The foregoing is to be regarded as the Notion of the Notion. It may seem to differ from what is elsewhere understood by ‘notion’ and in that case we might be asked to indicate how that which we have here found to be the Notion is contained in other conceptions or explanations. On the one hand, however, there can be no question of a confirmation based on the authority of the ordinary understanding of the term; in the science of the Notion its content and character can be guaranteed solely by the immanent deduction which contains its genesis and which already lies behind us. On the other hand, the Notion as here deduced must, of course, be recognisable in principle in what is elsewhere presented as the concept of the Notion. But it is not so easy to discover what others have said about the nature of the Notion. For in the main they do not concern themselves at all with the question, presupposing that everyone who uses the word automatically knows what it means. Latterly, one could have felt all the more relieved from any need to trouble about the Notion since, just as it was the fashion for a while to say everything bad about the imagination, and then the memory, so in philosophy it became the habit some time ago, a habit which in some measure still exists, to heap every kind of slander on the Notion, on what is supreme in thought, while the incomprehensible and non-comprehension are, on the contrary, regarded as the pinnacle of science and morality.

§1291b Ich beschränke mich hier auf eine Bemerkung, die für das Auffassen der hier entwickelten Begriffe dienen kann und es erleichtern mag, sich darein zu finden. Der Begriff, insofern er zu einer solchen Existenz gediehen ist, welche selbst frei ist, ist nichts anderes als Ich oder das reine Selbstbewußtsein. Ich habe wohl Begriffe, d.h. bestimmte Begriffe; aber Ich ist der reine Begriff selbst, der als Begriff zum Dasein gekommen ist. Wenn man daher an die Grundbestimmungen, welche die Natur des Ich ausmachen, erinnert, so darf man voraussetzen, daß an etwas Bekanntes, d. i. der Vorstellung Geläufiges erinnert wird. Ich aber ist erstlich diese reine sich auf sich beziehende Einheit, und dies nicht unmittelbar, sondern indem es von aller Bestimmtheit und Inhalt abstrahiert und in die Freiheit der schrankenlosen Gleichheit mit sich selbst zurückgeht. So ist es Allgemeinheit, Einheit, welche nur durch jenes negative Verhalten, welches als das Abstrahieren erscheint, Einheit mit sich ist und dadurch alles Bestimmtsein in sich aufgelöst enthält. Zweitens ist Ich ebenso unmittelbar als die sich auf sich selbst beziehende Negativität Einzelheit, absolutes Bestimmtsein, welches sich Anderem gegenüberstellt und es ausschließt; individuelle Persönlichkeit. Jene absolute Allgemeinheit, die ebenso unmittelbar absolute Vereinzelung ist, und ein Anundfürsichsein, welches schlechthin Gesetztsein und nur dies Anundfürsichsein durch die Einheit mit dem Gesetztsein ist, macht ebenso die Natur des Ich als des Begriffes aus; von dem einen und dem anderen ist nichts zu begreifen, wenn nicht die angegebenen beiden Momente zugleich in ihrer Abstraktion und zugleich in ihrer vollkommenen Einheit aufgefaßt werden

§1291b I will confine myself here to a remark which may help one to grasp the notions here developed and may make it easier to find one’s bearings in them. The Notion, when it has developed into a concrete existence that is itself free, is none other than the I or pure self-consciousness. True, I have notions, that is to say, determinate notions; but the I is the pure Notion itself which, as Notion, has come into existence. When, therefore, reference is made to the fundamental determinations which constitute the nature of the I, we may presuppose that the reference is to something familiar, that is, a commonplace of our ordinary thinking. But the I is, first, this pure self-related unity, and it is so not immediately but only as making abstraction from all determinateness and content and withdrawing into the freedom of unrestricted equality with itself. As such it is universality; a unity that is unity with itself only through its negative attitude, which appears as a process of abstraction, and that consequently contains all determinedness dissolved in it. Secondly, the I as self-related negativity is no less immediately individuality or is absolutely determined, opposing itself to all that is other and excluding it — individual personality. This absolute universality which is also immediately an absolute individualisation, and an absolutely determined being, which is a pure positedness and is this absolutely determined being it only through its unity with the positedness, this constitutes the nature of the I — as well as of the Notion; neither the one nor the other can be truly comprehended unless the two indicated moments are grasped at the same time both in their abstraction and also in their perfect unity.

Der subjektive Begriff / Subjectivity.

Begriff

§1322 Understanding is the term usually employed to express the faculty of notions; as so used, it is distinguished from the faculty of judgment and the faculty of syllogisms, of the formal reason But it is with reason that it is especially contrasted; in that case, however, it does not signify the faculty of the notion in general, but of determinate notions, and the idea prevails that the notion is only a determinate notion. When the understanding in this signification is distinguished from the formal faculty of judgment and from the formal reason, it is to be taken as the faculty of the single determinate notion. For the judgment and the syllogism or reason are, as formal, only a product of the understanding since they stand under the form of the abstract determinateness of the Notion. Here, however, the Notion emphatically does not rank as something merely abstractly determinate; consequently, the understanding is to be distinguished from reason only in the sense that the former is merely the faculty of the notion in general.

§1323 This universal Notion, which we have now to consider here, contains the three moments: universality, particularity and individuality. The difference and the determinations which the Notion gives itself in its distinguishing, constitute the side which was previously called positedness. As this is identical in the Notion with being-in-and-for-self, each of these moments is no less the whole Notion than it is a determinate Notion and a determination of the Notion.

abstract general, concrete general, concrete particular

see also EL§61

§1324a In the first instance, it is the pure Notion or the determination of universality. But the pure or universal Notion is also only a determinate or particular Notion, which takes its place alongside other Notions. Because the Notion is a totality, and therefore in its universality or pure identical self-relation is essentially a determining and a distinguishing, it therefore contains within itself the standard by which this form of its self-identity, in pervading and embracing all the moments, no less immediately determines itself to be only the universal over against the distinguishedness of the moments.

§1324b Secondly, the Notion is thereby posited as this particular or determinate Notion, distinct from others.

§1324c Thirdly, individuality is the Notion reflecting itself out of the difference into absolute negativity. This is, at the same time, the moment in which it has passed out of its identity into its otherness, and becomes the judgment.

Der allgemeine Begriff
Der besondere Begriff

§1337a Now determinateness, it is true, is the abstract, as against the other, determinateness; but this other is only universality itself which is, therefore, also abstract, and the determinateness of the Notion, or particularity, is again nothing more than a determinate universality. In this, the Notion is outside itself; since it is the Notion that is here outside itself, the abstract universal contains all the moments of the Notion. It is (a) universality, (b) determinateness, (c) the simple unity of both; but this unity is immediate, and therefore particularity is not present as totality. In itself it is also this totality and mediation; it is essentially an exclusive relation to an other, or sublation of the negation, namely, of the other determinateness – an other, however, that exists only in imagination, for it vanishes immediately and shows itself to be the same as its supposed other. Therefore, what makes this universality abstract is that the mediation is only a condition or is not posited in the universality itself. Because it is not posited, the unity of the abstract universality has the form of immediacy, and the content has the form of indifference to its universality, for the content is not present as the totality which is the universality of absolute negativity. Hence the abstract universal is, indeed, the Notion, yet it is without the Notion; it is the Notion that is not posited as such.

§1337b When people talk of the determinate Notion, what is usually meant is merely such an abstract universal. Even by notion as such, what is generally understood is only this notion that is no Notion, and the understanding denotes the faculty of such notions. Demonstration appertains to this understanding in so far as it progresses by notions, that is to say, merely by determinations. Such a progression by notions, therefore, does not get beyond finitude and necessity; for it, the highest is the negative infinite, the abstraction of the supreme being [des höchsten Wesen], which is itself the determinateness of indeterminateness. Absolute substance, too, though it is not this empty abstraction – from the point of view of its content it is rather the totality – is nevertheless abstract because it lacks the absolute form; its inmost truth is not constituted by the Notion; true, it is the identity of universality and particularity, or of thought and asunderness, yet this identity is not the determinateness of the Notion; on the contrary, outside substance there is an understanding – and just because it is outside it, a contingent understanding – in which and for which substance is present in various attributes and modes.

§1337c Moreover, abstraction is not empty as it is usually said to be; it is the determinate Notion and has some determinateness or other for its content. Even the supreme being, the pure abstraction, has, as already remarked, the determinateness of indeterminateness; but indeterminateness is a determinateness, because it is supposed to stand opposed to the determinate. But the enunciation of what it is, itself sublates what it is supposed to be; it is enunciated as one with determinateness, and in this way, out of the abstraction is established its truth and the Notion. But every determinate Notion is, of course, empty in so far as it does not contain the totality, but only a one-sided determinateness. Even when it has some other concrete content, for example man, the state, animal, etc., it still remains an empty Notion, since its determinateness is not the principle of its differences; a principle contains the beginning and the essential nature of its development and realization; any other determinateness of the notion, however, is sterile. To reproach the Notion generally with being empty, is to misjudge that absolute determinateness of the Notion which is the difference of the Notion and the only true content in the element of the Notion.

Das Einzelne

Urtheil / Judgement

Das Urteil des Daseins
Das Urteil der Reflexion
Das Urteil der Notwendigkeit
Das Urteil des Begriffs

Schluss / Syllogism

syllogism

see above

§1436 We have found the syllogism to be the restoration of the Notion in the judgment, and consequently the unity and truth of both. The Notion as such holds its moments sublated in unity; in the judgment this unity is internal or, what is the same thing, external; and the moments, although related, are posited as self-subsistent extremes. In the syllogism the Notion determinations are like the extremes of the judgment, and at the same time their determinate unity is posited.

§1437 Thus the syllogism is the completely posited Notion; it is therefore the rational. The understanding is regarded as the faculty of the determinate Notion which is held fast in isolation by abstraction and the form of universality. But in reason the determinate Notions are posited in their totality and unity. Therefore, not only is the syllogism rational, but everything rational is a syllogism. The syllogistic process has for a long time been ascribed to reason; yet on the other hand reason in and for itself, rational principles and laws, are spoken of in such a way that it is not clear what is the connection between the former reason which syllogises and the latter reason which is the source of laws and other eternal truths and absolute thoughts. If the former is supposed to be merely formal reason, while the latter is supposed to be creative of content, then according to this distinction it is precisely the form of reason, the syllogism, that must not be lacking in the latter. Nevertheless, to such a degree are the two commonly held apart, and not mentioned together, that it seems as though the reason of absolute thoughts was ashamed of the reason of the syllogism and as though it was only in deference to tradition that the syllogism was also adduced as an activity of reason. Yet it is obvious, as we have just remarked, that the logical reason, if it is regarded as formal reason, must essentially be recognisable also in the reason that is concerned with a content; the fact is that no content can be rational except through the rational form. In this matter we cannot look for any help in the common chatter about reason; for this refrains from stating what is to be understood by reason; this supposedly rational cognition is mostly so busy with its objects that it forgets to cognise reason itself and only distinguishes and characterises it by the objects that it possesses. If reason is supposed to be the cognition that knows about God, freedom, right and duty, the infinite, unconditioned, supersensuous, or even gives only ideas and feelings of these objects, then for one thing these latter are only negative objects, and for another thing the first question still remains, what it is in all these objects that makes them rational. It is this, that the infinitude of these objects is not the empty abstraction from the finite, not the universality that lacks content and determinateness, but the universality that is fulfilled or realised, the Notion that is determinate and possesses its determinateness in this true way, namely, that it differentiates itself within itself and is the unity of these fixed and determinate differences. It is only thus that reason rises above the finite, conditioned, sensuous, call it what you will, and in this negativity is essentially pregnant with content, for it is the unity of determinate extremes; as such, however, the rational is nothing but the syllogism.

§1438 Now the syllogism, like the judgment, is in the first instance immediate; hence its determinations are simple, abstract determinatenesses; in this form it is the syllogism of the understanding. If we stop short at this form of the syllogism, then the rationality in it, although undoubtedly present and posited, is not apparent. The essential feature of the syllogism is the unity of the extremes, the middle term which unites them, and the ground which supports them. Abstraction, in holding rigidly to the self-subsistence of the extremes, opposes this unity to them as a determinateness which likewise is fixed and self-subsistent, and in this way apprehends it rather as non-unity than as unity. The expression middle term is taken from spatial representation and contributes its share to the stopping short at the mutual externality of the terms. Now if the syllogism consists in the unity of the extremes being posited in it, and if, all the same, this unity is simply taken on the one hand as a particular on its own, and on the other hand as a merely external relation, and non-unity is made the essential relationship of the syllogism, then the reason which constitutes the syllogism contributes nothing to rationality.

§1439 First, the syllogism of existence in which the terms are thus immediately and abstractly determined, demonstrates in itself (since, like the judgment, it is their relation) that they are not in fact such abstract terms, but that each contains the relation to the other and that the middle term is not particularity as opposed to the determinations of the extremes but contains these terms posited in it.

§1440 Through this its dialectic it is converted into the syllogism of reflection, into the second syllogism. The terms of this are such that each essentially shows in, or is reflected into, the other; in other words they are posited as mediated, which they are supposed to be in accordance with the nature of the syllogism in general.

§1441 Thirdly, in that this reflecting or mediatedness of the extremes is reflected into itself, the syllogism is determined as the syllogism of necessity, in which the mediating element is the objective nature of the thing. As this syllogism determines the extremes of the Notion equally as totalities, the syllogism has attained to the correspondence of its Notion or the middle term, and its existence of the difference of its extremes; that is, it has attained to its truth and in doing so has passed out of subjectivity into objectivity.

Der Schluß des Daseins
Der Schluß der Reflexion
Der Schluß der Notwendigkeit

§1529a However, this determination of the Notion which has been considered as reality, is, conversely, equally a positedness. For it is not only in this result that the truth of the Notion has exhibited itself as the identity of its inwardness and externality; already in the judgment the moments of the Notion remain, even in their mutual indifference, determinations that have their significance only in their relation.

§1529b Der Schluß ist Vermittlung, der vollständige Begriff in seinem Gesetztsein. Seine Bewegung ist das Aufheben dieser Vermittlung, in welcher nichts an und für sich, sondern jedes nur vermittels eines Anderen ist. Das Resultat ist daher eine Unmittelbarkeit, die durch Aufheben der Vermittlung hervorgegangen, ein Sein, das ebensosehr identisch mit der Vermittlung und der Begriff ist, der aus und in seinem Anderssein sich selbst hergestellt hat. Dies Sein ist daher eine Sache, die an und für sich ist, – die Objektivität.

§1529b The syllogism is mediation, the complete Notion in its positedness. Its movement is the sublating of this mediation, in which nothing is in and for itself, but each term is only by means of an other. The result is therefore an immediacy which has issued from the sublating of the mediation, a being which is no less identical with the mediation, and which is the Notion that has

Der objektive Begriff / Objectivity

§1530 In Book One of the Objective Logic, abstract being was exhibited as passing over into determinate being, but equally as withdrawing into essence. In Book Two, essence reveals itself as determining itself into ground, thereby entering into Existence and realising itself as substance, but again withdrawing into the Notion. Of the Notion, now, we have shown to begin with that it determines itself into objectivity. It is self-evident that this latter transition is identical in character with what formerly appeared in metaphysics as the inference from the notion, namely, the notion of God, to his existence, or as the so-called ontological proof of the existence of God. It is equally well known that Descartes’ sublimest thought, that God is that whose notion includes within itself its being, after being degraded into the defective form of the formal syllogism, that is, into the form of the said proof, finally succumbed to the Critique of Reason and to the thought that existence cannot be extracted from the notion. Some points connected with this proof have already been elucidated. In Vol. 1, pp. 86 sqq., where being has vanished in its immediate opposite, non-being, and becoming has shown itself as the truth of both, attention was drawn to the confusion that arises when, in the case of a particular determinate being, what is fixed on is not the being of that determinate being but its determinate content; then, comparing this determinate content, for example a hundred dollars, with another determinate content, for example, with the context of my perception or the state of my finances, it is found that it makes a difference whether the former content is added to the latter or not – and it is imagined that what has been discussed is the difference between being and non-being, or even the difference between being and the Notion. Further, in the same Vol., p. 112 and Vol. II, p. 442 we elucidated a determination that occurs in the ontological proof, that of a sum-total of all realities. But the essential subject matter of that proof, the connection of the Notion and determinate being, is the concern of our consideration of the Notion just concluded, and the entire course through which the Notion determines itself into objectivity. The Notion, as absolutely self-identical negativity, is self-determining; we have remarked that the Notion, in determining itself into judgment in individuality, is already positing itself as something real, something that is; this still abstract reality completes itself in objectivity.

§1531 Now though it might seem that the transition from the Notion into objectivity is not the same thing as the transition from the Notion of God to his existence, it should be borne in mind on the one hand that the determinate content, God, makes no difference in the logical process, and the ontological proof is merely an application of this logical process to the said content. On the other hand however it is essential to bear in mind the remark made above that the subject only obtains determinateness and content in its predicate; until then, no matter what it may be for feeling, intuition and pictorial thinking, for rational cognition it is only a name; but in the predicate with its determinateness there begins, at the same time, realisation in general. The predicates, however, must be grasped as themselves still included within the Notion, hence as something subjective, which so far has not emerged into existence; to this extent we must admit on the one hand that the realisation of the Notion in the judgment is still not complete. On the other hand however the mere determination of an object by predicates, when that determination is not at the same time the realisation and objectifying of the Notion, also remains something so subjective that it is not even the genuine cognition and determination of the Notion of the object-subjective in the sense of abstract reflection and uncomprehended pictorial thinking. God, as the living God, and still more as absolute spirit, is known only in his activity; man was early instructed to recognise God in his works; only from these can proceed the determinations, which are called his properties, and in which, too, his being is contained. Thus the philosophical [begreifende] cognition of his activity, that is, of himself, grasps the Notion of God in his being and his being in his Notion. Being merely as such, or even determinate being, is such a meagre and restricted determination, that the difficulty of finding it in the Notion may well be the result of not having considered what being or determinate being itself is. Being as the wholly abstract, immediate relation to self, is nothing else than the abstract moment of the Notion, which moment is abstract universality. This universality also effects what one demands of being, namely, to be outside the Notion; for though this universality is moment of the Notion, it is equally the difference, or abstract judgment, of the Notion in which it opposes itself to itself.

The Notion, even as formal, already immediately contains being in a truer and richer form, in that, as self-related negativity, it is individuality.

§1532 But of course the difficulty of finding being in the Notion as such and equally in the Notion of God, becomes insuperable when the being is supposed to be that which obtains in the context of outer experience or in the form of sensuous perception, like the hundred dollars in my finances, something to be grasped with the hand, not with the mind, something visible essentially to the outer, not to the inner eye; in other words, when that being which things possess as sensuous, temporal and perishable, is given the name of reality or truth. A philosophising that in its view of being does not rise above sense, naturally stops short at merely abstract thought, too, in its view of the Notion; such thought stands opposed to being.

§1533 The custom of regarding the Notion merely as something one-sided, such as abstract thought is, will already hinder the acceptance of what was suggested above, namely, to regard the transition from the Notion of God to his being, as an application of the logical course of objectification of the Notion presented above. Yet if it is granted, as it commonly is, that the logical element as the formal element constitutes the form for the cognition of every determinate content, then the above relation must at least be conceded, unless in this opposition between Notion and objectivity, one stops short at the untrue Notion and an equally untrue reality, as something ultimate. But in the exposition of the pure Notion, it was further made clear that this is the absolute, divine Notion itself, so that in truth the relationship of our application would not obtain, and the logical process in question would in fact be the immediate exposition of God’s self-determination to being. But on this point it is to be remarked that if the Notion is to be presented as the Notion of God, it is to be apprehended as it is when taken up into the Idea. This pure Notion passes through the finite forms of the judgment and syllogism because it is not yet posited as in its own nature explicitly one with objectivity but is grasped only in process of becoming it. Similarly this objectivity, too, is not yet the divine existence, is not yet the reality that is reflected in the divine Idea. Yet objectivity is just that much richer and higher than the being or existence of the ontological proof, as the pure Notion is richer and higher than that metaphysical void of the sum total of all reality. But I reserve for another occasion the more detailed elucidation of the manifold misunderstanding that has been brought by logical formalism into the ontological, as well as the other, so-called proofs of God’s existence, as also the Kantian criticism of them, and by establishing their true significance, to restore the fundamental thoughts of these proofs to their worth and dignity.

§1534 As previously remarked, we have already met with several forms of immediacy, though in different determinations. In the sphere of being immediacy is being itself and determinate being; in the sphere of essence it is existence, and then actuality and substantiality; in the sphere of the Notion, besides immediacy as abstract universality, there is now objectivity. When the exactitude of philosophical distinctions of the Notion is not involved, these expressions may be used as synonymous; but the determinations mentioned have issued from the necessity of the Notion. Being is in general the first immediacy, and determinate being is the same plus the first determinateness. Existence, along with things, is the immediacy that issues from the ground-from the self-sublating mediation of the simple reflection of essence. But actuality and substantiality is the immediacy that has issued from the sublated difference of the still unessential Existence as Appearance and its essentiality. Finally, objectivity is the immediacy to which the Notion determines itself by the sublation of its abstraction and mediation. Philosophy has the right to select from the language of common life which is made for the world of pictorial thinking, such expressions as seem to approximate to the determinations of the Notion. There cannot be any question of demonstrating for a word selected from the language of common life that in common life, too, one associates with it the same Notion for which philosophy employs it; for common life has no Notions, but only pictorial thoughts and general ideas, and to recognise the Notion in what is else a mere general idea is philosophy itself. It must suffice therefore if pictorial thinking, in the use of its expressions that are employed for philosophical determinations, has before it some vague idea of their distinctive meaning; just as it may be the case that in these expressions one recognises nuances of pictorial thought that are more closely related to the corresponding Notions. One will be less ready, perhaps, to admit that something can be without existing; but at least, one will hardly use ‘being’ as copula of the judgment as interchangeable with the expression ‘to exist’ and say, ‘this article exists dear, suitable, etc.’, ‘gold exists, a metal or metallic’, instead of ‘this article is dear, suitable, etc.’, ‘gold is a metal or metallic’.

§1535 And surely it is usual to distinguish between being and appearing, appearance and actuality, as well as to distinguish mere being from actuality, and still more all these expressions from objectivity. However, even should they be employed synonymously, philosophy will in any case be free to utilise such empty superfluity of language for its distinctions.

§1536 When treating of the apodeictic judgment — the consummation of the judgment — where the subject loses its determinateness as against the predicate, we referred to the twofold meaning of subjectivity originating therefrom, namely, the subjectivity of the Notion, and equally of the externality and contingency opposed to the Notion. A similar twofold meaning also appears for objectivity which stands opposed to the self-subsistent Notion, yet is also the being that is in-and-for-itself. In the former sense, the object stands opposed to the I = I which in subjective idealism is enunciated as the absolutely true; in that case it is the manifold world in its immediate existence with which the ego or the Notion only engages in never-ending struggle, in order, by the negation of the intrinsic nullity of this other, to give to the first certainty of self the actual truth of its equality with itself. In a less specific sense it denotes an object in general for any interest or activity of the subject.

§1537 But in the opposite sense, objectivity signifies that which is in and for itself, and free from limitation and opposition. Rational principles, perfect works of art, etc., are called objective in so far as they are free and above all contingency. Although rational, theoretical or ethical principles belong only to subjectivity, to consciousness, yet that element in the latter that is in and for itself is called objective; the cognition of truth is placed in cognising the object as object, free from anything added by subjective reflection, and right conduct in the obedience to objective laws that are not subjective in origin and admit no caprice and no treatment that might overthrow their necessity.

§1538 At the present standpoint of our exposition objectivity signifies, in the first instance, the absolute being of the Notion, that is, the Notion that has sublated the mediation posited in its self-determination and converted it into immediate relation-to-self. Consequently this immediacy is itself immediately and wholly pervaded by the Notion, just as the Notion’s totality is immediately identical with its being. But since, further, the Notion has equally to restore the free being-for-self of its subjectivity, there arises a relationship between the Notion as end and objectivity. In this relationship the immediacy of the objectivity becomes the negative element over against the end, an element to be determined by the activity of the end; this immediacy thus acquires the other significance, that of being in and for itself null in so far as it stands opposed to the Notion.

§1539 First, then, objectivity is an immediacy whose moments, by virtue of the totality of all the moments, exist in a self-subsistent indifference as objects outside one another, and in their relationship possess the subjective unity of the Notion only as an inner or an outer unity. This is Mechanism.

§1540 But secondly, this unity reveals itself as the immanent law of the objects themselves, and thus their relationship becomes their peculiar specific difference founded on their law; it becomes a relation in which their determinate self-subsistence sublates itself. This is Chemism.

§1541 Thirdly, this essential unity of the objects is thereby posited as distinct from their self-subsistence; it is the subjective Notion, but posited as in and for itself related to objectivity, as end. This is Teleology.

§1542 Since the end is the Notion that is posited as in its own self relating itself to objectivity and as sublating by its own act its defect of being subjective, the purposiveness which is at first external becomes, through the realisation of the end, internal and the Idea.

Mechanismus

To gauge the terminology here, see also the discussion of (LectHistPhil-Descartes) of the mechannism Descartes via deterministic (§1549) laws of nature (§1572, §1575). This is what today is called classical mechanics.

§1543 Da die Objektivität die in ihre Einheit zurückgegangene Totalität des Begriffes ist, so ist damit ein Unmittelbares gesetzt, das an und für sich jene Totalität und auch als solche gesetzt ist, in der aber die negative Einheit des Begriffs sich noch nicht von der Unmittelbarkeit dieser Totalität abgeschieden hat; – oder die Objektivität ist noch nicht als Urteil gesetzt. Insofern sie den Begriff immanent in sich hat, so ist der Unterschied desselben an ihr vorhanden; aber um der objektiven Totalität willen sind die Unterschiedenen vollständige und selbständige Objekte, die sich daher auch in ihrer Beziehung nur als selbständige zueinander verhalten und sich in jeder Verbindung äußerlich bleiben. – Dies macht den Charakter des Mechanismus aus, daß, welche Beziehung zwischen den Verbundenen stattfindet, diese Beziehung ihnen eine fremde ist, welche ihre Natur[409] nichts angeht und, wenn sie auch mit dem Schein eines Eins verknüpft ist, nichts weiter als Zusammensetzung, Vermischung, Haufen usf. bleibt. Wie der materielle Mechanismus, so besteht auch der geistige darin, daß die im Geiste Bezogenen sich einander und ihm selbst äußerlich bleiben. Eine mechanische Vorstellungsweise, ein mechanisches Gedächtnis, die Gewohnheit, eine mechanische Handlungsweise bedeuten, daß die eigentümliche Durchdringung und Gegenwart des Geistes bei demjenigen fehlt, was er auffaßt oder tut. Obzwar sein theoretischer oder praktischer Mechanismus nicht ohne seine Selbsttätigkeit, einen Trieb und Bewußtsein stattfinden kann, so fehlt darin doch die Freiheit der Individualität, und weil sie nicht darin erscheint, erscheint solches Tun als ein bloß äußerliches.

§1543 As objectivity is the totality of the Notion withdrawn into its unity, an immediate is thereby posited that is in and for itself this totality, and is also posited as such, although in it the negative unity of the Notion has not as yet detached itself from the immediacy of this totality; in other words, objectivity is not yet posited as judgment. In so far as it has the Notion immanent in it, it contains the difference of the Notion, but on account of the objective totality, the differentiated moments are complete and self-subsistent objects which consequently, even in their relation, stand to one another only as self-subsistent things and remain external to one another in every combination. This is what constitutes the character of mechanism, namely, that whatever relation obtains between the things combined, this relation is one extraneous to them that does not concern their nature at all, and even if it is accompanied by a semblance of unity it remains nothing more than composition, mixture, aggregation and the like. Spiritual mechanism also, like material, consists in this, that the things related in the spirit remain external to one another and to spirit itself. A mechanical style of thinking, a mechanical memory, habit, a mechanical way of acting, signify that the peculiar pervasion and presence of spirit is lacking in what spirit apprehends or does. Although its theoretical or practical mechanism cannot take place without its self-activity, without an impulse and consciousness, yet there is lacking in it the freedom of individuality, and because this freedom is not manifest in it such action appears as a merely external one.

Das mechanische Objekt

§1544 Das Objekt ist, wie sich ergeben hat, der Schluß, dessen Vermittlung ausgeglichen und daher unmittelbare Identität geworden ist. Es ist daher an und für sich Allgemeines; die Allgemeinheit nicht im Sinne einer Gemeinschaftlichkeit von Eigenschaften, sondern welche die Besonderheit durchdringt und in ihr unmittelbare Einzelheit ist.

§1544 The object is, as we have seen, the syllogism, whose mediation has been sublated [ausgeglichen] and has therefore become an immediate identity. It is therefore in and for itself a universal — universality not in the sense of a community of properties, but a universality that pervades the particularity and in it is immediate individuality.

proof term

§1549 Indem also das Objekt in seiner Bestimmtheit ebenso gleichgültig gegen sie ist, weist es durch sich selbst für sein Bestimmtsein außer sich hinaus, wieder zu