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 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, are subsumed by and find their place in his more comprehensive and “scientific” system, notably

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 – 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, the text at times pauses to comment on its own incomprehensibility (e.g. the Incomprehensibility of the beginning or of matter being the unity of space and time PN§261b), much like the mystic Meister Eckhart did in his texts. Hegel himself acknowledges accomplishments of gnosticism and mysticism for philosophy, but indicates that he regards his system as more refined (EncPreface2ndEd). 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, 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.

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 A History of Western Philosophy, as being obfuscating and in fact nonsensical (e.g. Grassmann 1844, p. xv, 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 type theory 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.

Therefore:

nPOV

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, inspired by various suggestions by William Lawvere (an nPOV). The way this formalization dictionary works is summarized below in The formalization dictionary. A diagram showing the resulting process is in Overview of the formalization process.

(Ironically, the type theory which, this way, may be argued to provide intelligibility for Hegel’s Logic, goes back, ultimately, to Bertrand Russell’s theory of ramified types laid out in his Principia Mathematica.)

Contents

Formalization in Categorical logic / in Modal type theory

Formalization of Unity of Opposites

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

\flat \dashv \sharp

flat modality \dashv sharp modality

This gives unifying triples of the form

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

for any type XX.

Several examples of this appear below.

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

For instance for the points-to-pieces transform induced by the shape modality \dashv flat modality dichotomy ʃʃ \dashv \flat, we have, as discussed at differential cohomology hexagon, that the dualities generate for each type a hexagonal web of which the above unifying tripl is the bottom stage.

ʃ dRΩA dRΣA θ A ʃ dRΩA A ʃ dRΣA ʃθ A A ʃA, \array{ && ʃ_{dR} \Omega A && \longrightarrow && \flat_{dR}\Sigma A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \flat ʃ_{dR} \Omega A && && A && && ʃ \flat_{dR}\Sigma A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{ʃ \theta_A}} \\ && \flat A && \longrightarrow && ʃ A } \,,

This appears below in two stages, first in the Seinslogik at around §714, and then (“reflected”) in the Wesenslogik around §989.

Formalization dictionary

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

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

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

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

The formalization dictionary
Hegel’s logicmodal homotopy type theory
Subjektive Logic
Begriffslogiktype theory/natural deduction§1280, Law94b
Begriff, concept, notiontype§1280, MaLö73, Sale 77, Se94 LaPr14
UrteiljudgementMaLö96
Schlussnatural deductionGe35
Grund (unvermittelt)antecedent§1021
das Begründete (vermittelt)succedent/consequent§1021, §1035
entering into existenceterm introduction§1033, §1035
momentmodality, modal operator
unity of oppositesadjoint modality (moment \dashv co-moment)Law91, Law94, Law96
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
moment of two negations, somethingdouble negation modality ¬¬\not \not, more generally: bracket type/(-1)-truncation modality§210
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 ʃʃ§395
qualityadjoint modality (attraction \dashv repulsion) = (ʃʃ \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
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{ ʃ &\dashv& \flat \\ \bot && \bot \\ \flat &\dashv& \sharp }\right) §699, §708, §714, §725
vanishing of infinitesimalsreduction modality \Re§174, §404
being-for-oneinfinitesimal flat modality \Im§322
being-for-selfinfinitesimal shape modality &\& §305
ideality (inf. quality)unity of opposites (& \& \dashv \Im )§305, §322
Aufhebung of finiteness(& ʃ )\left(\array{\& &\dashv& \Im \\ \vee && \vee \\ ʃ &\dashv& \flat }\right)§304 , §305
realityadjoint modality (&\Re \dashv \&)§304 , §305
unity of ideality and realitydifferential cohesion adjoint triple adjoint modality ( & & )\left( \array{ \Re &\dashv& \& \\ \bot&\stackrel{}{}& \bot && \\ \& &\dashv& \Im} \right)§304, §324, EL§214, §1636
absolute indifferenceadjoint modality (ididid \dashv id)§803, §808, §812
Wesenslogikhomotopy type theory
Wesen, essencethe ambient category§803, §812
possibilitypossibility monad \lozenge = dependent sum followed by context extension
necessitynecessity comonad \Box = dependent product followed by context extension
absolute actuality/Wirklichkeit, unity of possibility and necessityadjoint pair of (co-)monads ()(\lozenge \dashv \Box), hence local cartesian closure§1160 (with §1159), §1190, §1192
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 §860
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;§973
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
Ideethe term model of the above modal type theory, in particular the true propositions§1630b, §1631, §1633, §1634
Naturmodel (representation) of the above modal type theoryPN§192, PN§193b, EL§244, §1782, §1817
Raum-Zeitétale stacks, being the models of infinitesimal shape &\&-modal typesPN§254a
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

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

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

Overview of the formalized process

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. 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 Materie ρ() Schwere ρ() ρ() ρ(&) Weichheit ρ(&) Elastizitaet ρ() Haerte Natur ρ(ʃ) Kohaesion ρ() Dichtigkeit ρ() ρ() ρ() ρ(*) Entaeusserung Idee Substanz id id Existenz Form ʃ Ding Inhalt abs.Grund AufhebungdesWiderspruchs falsch abs.Widerspruch * wahr Wesen Erscheinung best.Reflexionen setzendeReflexion Type 0Type 1Type 2 auessereReflexion An&Fuersichsein id id ObjektiveLogik AufhebungderDifferenzen VerschwindenderInfinitesimalen Realitaet & Unendlichkeit Fuersichsein & inf.QualitaetIdealitaet/ Fuereinessein DieKategorien AufhebungderEndlichkeit Ansichsein Attraktion ʃ QualitaetEtwas Repulsion SeinfuerAnderes Eichmass Endlichkeit Dasein Diskretion Quantitaet Kontinuitaet AufhebungdesWerdens reinesSein Nichts Werden * Sein Sein Moeglichkeit abs.Wirklichkeit Notwendigkeit 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{Materie}{} & \bot \\ && && && && \rho(\rightsquigarrow) & \stackrel{Schwere}{\dashv} & \rho(\,\,) \\ && && && && \vee & & \vee \\ && && && && \rho(\Re) & \dashv & \rho(\&) \\ && && && && \bot & \stackrel{}{} & \bot \\ && && && &\stackrel{Weichheit}{}& \rho(\&) &\stackrel{Elastizitaet}{\dashv} & \rho(\Im) & \stackrel{Haerte}{} \\ && && && && \vee && \vee \\ Natur && && && & & \rho(&#643;) &\stackrel{Kohaesion}{\dashv}& \rho(\flat) & \\ && && && && \bot && \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{Existenz}{} &\stackrel{Form}{} & '&#643;' & \stackrel{Ding}{\dashv}& '\flat' & \stackrel{Inhalt}{} \\ && && && && \bot && \bot \\ && && && && '\flat' &\dashv& '\sharp' & \stackrel{abs.\,Grund}{} \\ && && &&&& \vee &\stackrel{Aufhebung \atop {des\;Widerspruchs}}{}& \vee \\ && &&&&\stackrel{}{}&\stackrel{falsch}{}& '\emptyset' &\stackrel{abs.\,Widerspruch}{\dashv}& '\ast' & \stackrel{wahr}{} \\ \\ && && && Wesen \\ && && && && & Erscheinung \\ &&\stackrel{best.\; Reflexionen}{}&& && &\stackrel{setzende \atop Reflexion}{} & &\stackrel{\stackrel{\stackrel{\vdots}{Type_2}}{Type_1}}{Type_0}& & \stackrel{auessere \atop Reflexion}{} \\ \\ && && &&\stackrel{An\& Fuersichsein}{}&& id &\stackrel{}{\dashv}& id \\ {Objektive \atop Logik}{} && && &&&& \vee &\stackrel{{Aufhebung \atop {der\;Differenzen}}}{}& \vee \\ && && &&& \stackrel{{Verschwinden\;der}\atop Infinitesimalen}{}& \Re &\stackrel{Realitaet}{\dashv}& \& \\ && && \stackrel{Unendlichkeit}{} &&&& \bot &\stackrel{}{}& \bot \\ && && &&\stackrel{Fuersichsein}{} && \& &\stackrel{Idealitaet /}{\stackrel{inf.\,Qualitaet}{\dashv}}& \Im &&\stackrel{Fuer-eines-sein}{} \\ && \stackrel{Die\;Kategorien}{}&& &&&& \vee &\stackrel{{Aufhebung \atop {der\;Endlichkeit}}}{}& \vee \\ && && &&\stackrel{Ansichsein}{}&\stackrel{Attraktion}{}& &#643; &\stackrel{Etwas}{\stackrel{Qualitaet}{\dashv}}& \flat & \stackrel{Repulsion}{} & \stackrel{Sein-fuer-Anderes}{} \\ && && &&&& \bot &\stackrel{Eichmass}{}& \bot \\ && && \stackrel{Endlichkeit}{} &&\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}{} \\ && && && Sein \\ && && && &\stackrel{Moeglichkeit}{}& \lozenge &\stackrel{abs. Wirklichkeit}{\dashv}& \Box & \stackrel{Notwendigkeit}{} \\ {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.

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 above we identify this becoming (following Lawvere 91) as the universal factorization

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

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

Indeed, later in §174 it says:

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

Also, below it says

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

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, type, something

§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.

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

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

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

§221 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 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 &\& and infinitesimal flat modality \Im.

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 & \& \dashv \Im

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

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

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 &\& .

This happens to make good sense: the modal types of &\& 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 \&

makes good sense.

Notice that – and this is of course precisely what the second-order duality with ideality:&ideality \colon \& \dashv \Im 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}& \& \\ && \bot && \bot \\ & & \& &\stackrel{ideality/ \atop {inf.\, quality} }{\dashv}& \Im \\ && \vee && \vee \\ && &#643; &\stackrel{quality}{\dashv}& \flat }

Moreover, here the bottom step upward is an Aufhebung in the mathematical sense that ʃʃ\Im \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}& \& \\ && \bot &\stackrel{Idee}{}& \bot \\ & & \& &\stackrel{ideality/ \atop {inf.\, quality} }{\dashv}& \Im \\ && \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{ \Im \\ \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 & \& &\stackrel{Idealitaet}{\dashv}& \Im & 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 dahsv\dahsv 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

§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} }
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

A. Die Zahl
B. Extensives und Intensives Quantum

extensive and intensive quantity

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.

####### Anmerkung 1

####### Anmerkung 2

c. Die Unendlichkeit des Quantums

####### 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 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.

§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 itslef

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 §860 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

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

§834 Das Wesen ist Reflexion;

§834 Essence is reflection

See discussion at Wesen als Reflexion in Ihm Selbst

(1) Die setzende Reflexion

(2) Die äußere Reflexion

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

§853 Die bestimmende Reflexion ist überhaupt die Einheit der setzenden und der äußeren Reflexion. Dieß ist näher zu betrachten. Die äußere Reflexion fängt vom unmittelbaren Seyn all, die setzende vom Nichts.

§853 Determining reflection is in general the unity of positing and external reflection. This is to be considered in more detail. 1. External reflection starts from immediate being, positing reflection from nothing.

Chapter 2. Die Wesenheiten oder die Reflexions-Bestimmungen / The Essentialities or Determination of Reflection

§860 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.

§860 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

\;\; Remark A=AA = A

§863 Thus the essential category of identity is enunciated in the proposition: everything is identical with itself, A = A.

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

A Identity

§869 Essence is therefore simple identity with self.

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

Below this is called te First original law of thought.

\;\; Remark 1: Abstract identity
\;\; Remark 2: First original law of thought

§875 In this remark, I will consider in more detail identity as the law of identity which is usually adduced as the first law of thought.

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

The reflector term constructor in an identity type.

B Difference
\;\; (a) Absolute difference
\;\; (b) Diversity
\;\; Remark: The Law of Diversity

§903 All things are different; or: there are no two things like each other.

Reminiscent of identity types in intensional type theory.

C 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.

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 posits itself; 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.

§973 Das Wesen hat eine Form,

§973 Essence has a 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.

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 §973 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 §973 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.
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 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.

§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.

This is a unity of opposites:

Notice by §1159 that the Absolute is the absolute reality

(beware that the possibility monad and necessity comonad are not in general idempotent).

Also beware that by §1190, §1192 below there is also plain (not absolute) reality which goes along with possibility at the same level, so that maybe one should write something like

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, the absolute reality here translates to this ambient category being a locally Cartesian closed category.

§1161 Drittens die Einheit des Absoluten und seiner Reflexion ist das absolute Verhältniß, oder vielmehr das Absolute als Verhältniß zu sich selbst; Substanz.

Thirdly, the unity of the absolute and its reflection is the absolute relation, or rather the absolute as relation to itself — substance.

So by §1159 “the absolute (absolute reality)” here translates to the locally Cartesian closed ambient category and by §816 etc. the reflection here translates to the type of types, so the above would then translate to

But this is made more precise below in §1235 where it is explicit that the unity embodied by substance also includes all the determinations of being, hence and all the adjoint modalities. So substance is the full differential cohesive homotopy theory.

Das Absolute

Die Wirklichkeit

§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.

again: (Wirklichkeit-possibility)-necessity, see §1160 above.

A Zufälligkeit oder formelle Wirklichkeit, Möglichkeit und Nothwendigkeit/ 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.

again: Wirklichkeit \rightarrow possibility

B. Relative Nothwendigkeit oder reale Wirklichkeit, Möglichkeit und Nothwendigkeit / Relative Necessity, or Real Actuality, Possibility and Necessity
C. Absolute Nothwendigkeit / Absolute Necessity

Das absolute Verhältniß

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.

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.

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

Schluss

Der objektive Begriff / Objectivity

Mechanismus

Chemismus

Teleologie

§1599 However unsatisfactory, therefore, the discussion of the teleological principle is in respect of its essential point of view, nevertheless the position that Kant gives to it is worthy of note. In ascribing it to a reflective judgment, he makes it a connecting middle term between the universal of reason and the individual of intuition; further, he distinguishes this reflective judgment from the determining judgment, the latter merely subsuming the particular under the universal. Such a universal which merely subsumes, is an abstraction which only becomes concrete in something else, in the particular. End, on the contrary, is the concrete universal, which possesses in its own self the moment of particularity and externality and is therefore active and the urge to repel itself from itself. The Notion, as end, is of course an objective judgment in which one determination, the subject, namely the concrete Notion, is self-determined, while the other is not merely a predicate but external objectivity. But the end relation is not for that reason a reflective judging that considers external objects only according to a unity, as though an intelligence had given this unity for the convenience of our cognitive faculty; on the contrary it is the absolute truth that judges objectively and determines external objectivity absolutely. Thus the end relation is more than judgment; it is the syllogism of the self-subsistent free Notion that unites itself with itself through objectivity.

concrete general

Der subjektive Zweck
Das Mittel
Der ausgeführte Zweck

§1630a First we saw subjectivity, the Notion’s being-for-self, pass over; into its in-itself, objectivity, to be followed by the reappearance in the latter of the negativity of the Notion’s being-for-self; in that negativity the Notion has determined itself in such a manner that its particularity is an external objectivity, or it has determined itself as a simple concrete unity whose externality is its self-determination.

§1630b The movement of the end has now reached the stage where the moment of externality is not merely posited in the Notion, where the end is not merely an ought-to-be and a striving to realise itself, but as a concrete totality is identical with the immediate objectivity. This identity is on the one hand the simple Notion and the equally immediate objectivity, but on the other hand, it is just as essentially a mediation, and only through the latter as a self-sublating mediation is it that simple immediacy; the Notion is therefore essentially this: to be distinct as an explicit identity from its implicit objectivity, and thereby to possess externality, yet in this external totality to be the totality’s self-determining identity. As such, the Notion is now the Idea.

a concrete general as in §1323.

Die Idee / The Idea

§1631 Die Idee ist der adäquate Begriff, das objektive Wahre oder das Wahre als solches. Wenn irgend etwas Wahrheit hat, hat es sie durch seine Idee, oder etwas hat nur Wahrheit, insofern es Idee ist.

§1631 The Idea is the adequate Notion, that which is objectively true, or the true as such. When anything whatever possesses truth, it possesses it through its Idea, or, something possesses truth only in so far as it is Idea.

§1633 Indem nun der Ausdruck Idee für den objektiven oder realen Begriff zurückbehalten und von dem Begriff selbst

§1633 Reserving then the expression ‘Idea’ for the objective or real Notion and distinguishing it from the Notion itself

§1634 Indem sich aber das Resultat ergeben hat, daß die Idee die Einheit des Begriffs und der Objektivität, das Wahre ist, so ist sie nicht nur als ein Ziel zu betrachten, dem sich anzunähern sei, das aber selbst immer eine Art von Jenseits bleibe, sondern daß alles Wirkliche nur insofern ist, als es die Idee in sich hat und sie ausdrückt. Der Gegenstand, die objektive und subjektive Welt überhaupt sollen mit der Idee nicht bloß kongruieren, sondern sie sind selbst die Kongruenz des Begriffs und der Realität; diejenige Realität, welche dem Begriffe nicht entspricht, ist bloße Erscheinung, das Subjektive, Zufällige, Willkürliche, das nicht die Wahrheit ist.

§1634 But having reached the result that the Idea is the unity of the Notion and objectivity, is the true, it must not be regarded merely as a goal to which we have to approximate but which itself always remains a kind of beyond; on the contrary, we must recognise that everything actual is only in so far as it possesses the Idea and expresses it. It is not merely that the object, the objective and subjective world in general, ought to be congruous with the Idea, but they are themselves the congruence of Notion and reality; the reality that does not correspond to the Notion is mere Appearance, the subjective, contingent, capricious element that is not the truth.

With Notion/Begriff being type and true being (under propositions as types) inhabited types or rather particular terms inhabiting a type, the Idea is the system of definable terms that may be constructed (from the basic type term introduction rules and those coming with the above modal operators).

Moreover, by §1630b this is to have a concrete and external aspect to it. Hence both by the understanding of externalization as representation/model as well as by the understanding of concrete general as a category of models of a theory the Idea is to be thought of as the category of these definable terms. This is known as the term model of the given (modal type theory).

Specifically, under propositions as types this includes all witnesses of the truth of propositions, hence the Idea is the collection of all the “true concepts”, the actual facts that may be proven from the given modal type theory.

§1635 When it is said that no object is to be found in experience that is perfectly congruous with the Idea, one is opposing the Idea as a subjective standard to the actual; but what anything actual is supposed in truth to be, if its Notion is not in it and if its objectivity docs not correspond to its Notion at all, it is impossible to say; for it would be nothing. It is true that the mechanical and chemical object, like the nonspiritual subject and the spirit that is conscious only of the finite, not of its essence, do not, according to their various natures, have their Notion existent in them in its own free form.

But they can only be true at all in so far as they are the union of their Notion and reality, of their soul and their body. Wholes like the state and the church cease to exist when the unity of their Notion and their reality is dissolved; man, the living being, is dead when soul and body are parted in him; dead nature, the mechanical and chemical world — taking, that is, the dead world to mean the inorganic world, otherwise it would have no positive meaning at all — dead nature, then, if it is separated into its Notion and its reality, is nothing but the subjective abstraction of a thought form and a formless matter. Spirit that was not Idea, was not the unity of the Notion with its own self, or the Notion that did not have the Notion itself for its reality would be dead, spiritless spirit, a material object.

§1636 The Idea being the unity of Notion and reality, being has attained the significance of truth; therefore what now is is only what is Idea.

Der Gegenstand, die objektive und subjektive Welt überhaupt sollen mit der Idee nicht bloß kongruieren, sondern sie sind selbst die Kongruenz des Begriffs und der Realität;

Sein hat die Bedeutung der Wahrheit erreicht, indem die Idee die Einheit des Begriffs und der Realität ist; es ist also nunmehr nur das, was Idee ist.

Die Idee hat aber nicht nur den allgemeineren Sinn des wahrhaften Seins, der Einheit von Begriff und Realität, sondern den bestimmteren von subjektivem Begriffe und der Objektivität.

§1636 The Idea being the unity of Notion and reality, being has attained the significance of truth;

§1634 But having reached the result that the Idea is the unity of the Notion and objectivity,

§1636 Die Idee ist die Einheit des Begriffs und der Realität

§1636 The Idea being the unity of Notion and reality,

§1638 However, the Idea has not merely the more general meaning of the true being, of the unity of Notion and reality, but the more specific one of the unity of subjective Notion and objectivity.

So

  • Idee = Begriff & Realität

See also EL§214, which instead has

  • Idee = das Ideelle & das Reelle .

But of course the Notion is ideell, by §304 “wie auch weiterhin der Begriff als ein Ideelles und zwar als ein nur Ideelles” and §316 “wie noch mehr der Begriff, die Idee, der Geist, Ideelles zu nennen ist”.

(Notice that it is Realität §304, §305 which appears here, not Wirklichkeit ( actuality ) as in §1160.)

§1640 As this relation, the Idea is the process of sundering itself into individuality and its inorganic nature, and again of bringing this inorganic nature under the power of the subject and returning to the first simple universality. The identity of the Idea with itself is one with the process; the thought which liberates actuality from the illusory show of purposeless mutability and transfigures it into the Idea must not represent this truth of actuality as a dead repose, as a mere picture, lifeless, without impulse or movement, as a genius or number, or an abstract thought; by virtue of the freedom which the Notion attains in the Idea, the Idea possesses within itself also the most stubborn opposition; its repose consists in the security and certainty with which it eternally creates and eternally overcomes that opposition, in it meeting with itself

In the Lectures on the History of Philosophy it is clarified that The Idea here indeed relates to the doctrine of ideas of Plato:

HistoryOfPhilosophy – Plato – In this account of Philosophy, we at once see what the so much talked of Ideas of Plato are. The Idea is nothing else than that which is known to us more familiarly by the name of the Universal, regarded, however, not as the formal Universal, which is only a property of things, but as implicitly and explicitly existent, as reality, as that which alone is true. We translate eidoς first of all as species or kind; and the Idea is no doubt the species, but rather as it is apprehended by and exists for Thought. Of course when we understand by species nothing but the gathering together by our reflection, and for convenience sake, of the like characteristics of several individuals as indicating their distinguishing features, we have the universal in quite an external form. But the specific character of the animal is its being alive; this being alive is that which makes it what it is, and deprived of this. it ceases to exist. To Plato, accordingly, Philosophy is really the science of this implicitly universal, to which, as contrasted with the particular, he always continues to return. “When Plato spoke of tableness and cupness, Diogenes the Cynic said: ‘I see a table and a cup, to be sure, but not tableness and cupness.’ ‘Right,’ answered Plato; ‘for you have eyes wherewith to see the table and the cup, but mind, by which one sees tableness and cupness, you have not (noun ouk exeiς).” What Socrates began was carried out by Plato, who acknowledged only the Universal, the Idea, the Good, as that which has existence. Through the presentation of his Ideas. Plato opened up the intellectual world, which, however, is not beyond reality, in heaven, in another place, but is the real world. With Leucippus, too, the Ideal is brought closer to reality, and not — metaphysically — thrust away behind Nature. The essence of the doctrine of Ideas is thus the view that the True is not that which exists for the senses, but that only what has its determination in itself, the implicitly and explicitly Universal, truly exists in the world; the intellectual world is therefore the True, that which is worthy to be known — indeed, the Eternal, the implicitly and explicitly divine. The differences are not essential, but only transitory; yet the Absolute of Plato, as being the one in itself and identical with itself, is at the same time concrete in itself, in that it is a movement returning into itself, and is eternally at home with itself. But love for Ideas is that which Plato calls enthusiasm.

Now that the Idea has appeared in the Logic, nature springs out of it (as announced in §53). The shorter Logic ends with EL§244.

Leben

EL§216 Die unmittelbare Idee ist das Leben. Der Begriff ist als Seele in einem Leibe realisiert.

§1645 The Idea of Life is concerned with a subject matter so concrete, and if you will, so real, that with it we may seem to have overstepped the domain of logic as it is commonly conceived.

The living individual
The life-process
The genus

§1669 The living individual, at first disengaged from the universal Notion of life, is a presupposition that is not as yet authenticated by the living individual itself. Through its process with the simultaneously presupposed world, it has posited itself on its own account as the negative unity of its otherness, as the foundation of itself; as such it is the actuality of the Idea, in such a manner that now the individual brings itself forth out of actuality, whereas before it proceeded only from the Notion, and that its genesis which was an act of presupposing, now becomes its production.

§1670 But the further determination that it has attained by the sublation of the opposition is that of being the genus as identity of itself with its previously indifferent otherness. This Idea of the individual, since it is this essential identity, is essentially the particularisation of itself. This its diremption, in accordance with the totality from which it proceeds, is the duplication of the individual — a presupposing of an objectivity that is identical with it, and a relationship of the living being to itself as to another living being.

§1671 This universal is the third stage, the truth of life in so far as this is still confined within its sphere. This sphere is the self-related process of the individual, where externality is its immanent moment; secondly, this externality is itself, as a living totality, an objectivity that for the individual is its own self, an objectivity in which, not as sublated but as persisting, the individual has the certainty of itself.

§1672 Now because the relationship of the genus is the identity of individual self-feeling in what is at the same time another self-subsistent individual, it is contradiction; thus the living being is again an urge. Now the genus is indeed the consummation of the Idea of life, but at first it is still within the sphere of immediacy; this universality is therefore actual in an individual shape — the Notion, whose reality has the form of immediate objectivity. Consequently, though the individual is indeed in itself genus, it is not explicitly or for itself the genus; what is for it is as yet only another living individual; the Notion distinguished from itself has for object, with which it is identical, not itself as Notion but a Notion that as a living being has at the same time external objectivity for it, a form that is therefore immediately reciprocal.

§1673a The identity with the other individual, the individual’s universality, is thus as yet only internal or subjective; it therefore has the longing to posit this and to realise itself as a universal. But this urge of the genus can realise itself only by sublating the single individualities which are still particular relatively to one another.

§1673b In that first instance, in so far as it is these latter which, in themselves, universal, satisfy the tension of their longing and dissolve themselves into the universality of their genus, their realised end identity is the negative unity of the genus that is reflected into itself out of its disremption.

§1674 It is thus the individuality of life itself, generated no longer from its Notion, but from the actual Idea. In the first instance, it is itself only Notion that has yet to objectify itself, but it is the actual Notion — the germ of a living individual. The germ is visible evidence to ordinary perception of what the Notion is, and it demonstrates that the subjective Notion has external actuality. For the germ of the living being is the complete concretion of individuality, in which all its diverse aspects, properties and articulated differences are contained in their entire determinateness, and the initially immaterial, subjective totality is undeveloped, simple and non-sensuous; the germ is thus the entire living being in the inner form of the Notion.

§1675 The reflection of the genus into itself is from this side the means whereby it obtains actuality, the moment of negative unity and individuality being thereby posited in it — the propagation of the living species. The Idea, which as life, is still in the form of immediacy, thus falls back into actuality and this its reflection is only repetition and the infinite progress, in which it does not emerge from the finitude of its immediacy. But this return into its first Notion has also the higher side, that the Idea has not merely run through the mediation of its processes within its immediacy, but by this very act has sublated this immediacy and thereby raised itself to a higher form of its existence.

§1676 That is to say, the process of the genus, in which the single individuals sublate in one another their indifferent immediate existence and in this negative unity expire, has further for the other side of its product the realised genus, which has posited itself identical with the Notion. In the genus process, the separated individualities of individual life perish; the negative identity in which the genus returns into itself, while it is on the one hand the process of generating individuality, is on the other hand the sublating of it, and is thus the genus coming together with itself, the universality of the Idea in process of becoming for itself. In copulation the immediacy of the living individuality perishes; the death of this life is the procession of spirit. The Idea, which as genus is implicit, is now explicit, in that it has sublated its particularity which constituted the living species, and has thereby given itself a reality that is itself simple universality. As such it is the Idea that relates itself to itself as Idea, the universal that has universality for its determinateness and existence — the Idea of cognition.

See also PS§456b.

Erkennen

Absolute Idee

EL§244 Die Idee, welche für sich ist, nach dieser ihrer Einheit mit sich betrachtet, ist sie Anschauen, und die anschauende Idee Natur. Als Anschauen aber ist die Idee in einseitiger Bestimmung der Unmittelbarkeit oder Negation durch äußerliche Reflexion gesetzt. Die absolute Freiheit der Idee aber ist, daß sie nicht bloß ins Leben übergeht, noch als endliches Erkennen dasselbe in sich scheinen läßt, sondern in der absoluten Wahrheit ihrer selbst sich entschließt, das Moment ihrer Besonderheit oder des ersten Bestimmens und Andersseins, die unmittelbare Idee als ihren Widerschein, sich als Natur frei aus sich zu entlassen.

§1782 Sie ist der einzige Gegenstand und Inhalt der Philosophie Indem sie alle Bestimmtheit in sich enthält, und ihr Wesen dieß ist, durch ihre Selbstbestimmung oder Besonderung zu sich zurückzukehren, so hat sie verschiedene Gestaltungen, und das Geschäft der Philosophie ist, sie in diesen zu erkennen. Die Natur und der Geist sind überhaupt unterschiedene Weisen, ihr Daseyn darzustellen; Kunst und Religion ihre verschiedenen Weisen, sich zu erfassen und ein sich angemessenes Daseyn zu geben; die Philosophie hat mit Kunst und Religion denselben Inhalt und denselben Zweck; aber sie ist die höchste Weise, die absolute Idee zu erfassen, weil ihre Weise die höchste, der Begriff, ist. Sie faßt daher jene Gestaltungen der reellen und ideellen Endlichkeit, so wie der Unendlichkeit und Heiligkeit in sich, und begreift sie und sich selbst. Die Ableitung und Erkenntniß dieser besonderen Weisen ist nun das fernere Geschäft der besonderen philosophischen Wissenschaften.

§1782 It is the sole subject matter and content of philosophy. Since it contains all determinations within it, and its essential nature is to return to itself through its self-determination or particularisation, it has various shapes, and the business of philosophy is to cognise it in these. Nature and spirit are in general different modes of presenting its existence, art and religion its different modes of apprehending itself and giving itself an adequate existence. Philosophy has the same content and the same end as art and religion; but it is the highest mode of apprehending the absolute idea, because its mode is the highest mode, the Notion.

nature is representation of the (absolute) idea, but the idea is the term model, hence nature is a model of the theory (the substance). see also in PN§193b, PN§250

(Die dialektische Methode)

§1783a The logical aspect of the absolute Idea may also be called a mode of it; but whereas mode signifies a particular kind, a determinateness of form, the logical aspect, on the contrary, is the universal mode in which all particular modes are sublated and enfolded. The logical Idea os the Idea itself in its pure essence, the Idea enclosed in simple identity within its Notion prior to its immediate reflection in a form-determinateness.

§1783b Hence logic exhibits the self-movement of the absolute Idea only as original word, which is an outwardising or utterance, but an utterance that in being has immediately vanished again as something outer; the Idea is, therefore, only in this self-determination of apprehending itself; it is in pure thought, in which difference is not yet otherness, but is and remains perfectly transparent to itself. Thus the logical Idea has itself as the infinite form for its content — form which constitutes the opposite to content to this extent that the content is the form-determination withdrawn into itself and sublated in the identity in such a manner that this concrete identity stands opposed to the identity explicated as form; the content has the shape of an other and a datum as against the form which as such stands simply in relation, and its determinateness is at the same time posited as an illusory being. More exactly, the determination is its own completed totality, the pure Notion. Now the determinateness of the Idea and the entire course followed by this determinateness has constituted the subject matter of the science of logic, from which course the absolute Idea itself has issued into an existence of its own; but the nature of this existence has shown itself to be this, that determinateness does not have the shape of content, but exists wholly as form, and that accordingly the Idea is the absolutely universal Idea. Therefore what remains to be considered here is not a content as such, but the universal aspect of its form — that is, the method.

§1785a Accordingly, what is to be considered here as method is only the movement of the Notion itself, the nature of which movement has already been cognised; but first, there is now the added significance that the Notion is everything, and its movement is the universal absolute activity, the self-determining and self-realising movement. The method is therefore to be recognised as the unrestrictedly universal, internal and external mode; and as the absolutely infinite force, to which no object, presenting itself as something external, remote from and independent of reason, could offer resistance or be of a particular nature in opposition to it, or could not be penetrated by it. It is therefore soul and substance, and anything whatever is comprehended and known in its truth only when it is completely subjugated to the method; it is the method proper to every subject matter because its activity is the Notion. This is also the truer meaning if its universality; according to the universality of reflection it is regarded merely as the method for everything; but according to the universality of the Idea, it is both the manner peculiar to cognition, to the subjectively selfknowing Notion, and also the objective manner, or rather the substantiality, of things — that is of Notions, in so far as they appear primarily to representation and reflection as others. It is therefore not only the highest force, or rather the sole and absolute force of reason, but also its supreme and sole urge to find and cognise itself by means of itself in everything. Here, secondly, is indicated the difference of the method from the Notion as such, the particular aspect of the method. The Notion, when it was considered by itself, appeared in its immediacy; the reflection, or the Notion that considered it, fell within our knowing.

§1785b The method is this knowing itself, for which the Notion is not merely the subject matter, but knowing’s own subjective act, the instrument and means of the cognising activity, distinguished from that activity, but only as the activity’s own essentiality. In the cognition of enquiry, the method likewise occupies the position of an instrument, of a means standing on the subjective side by which this side relates itself to the object. In this syllogism the subject is one extreme and the object the other, and the former by means of its method unites with the latter, but in doing so it does not unite with itself. The extremes remain diverse because subject, method, and object are not posited as the one identical Notion; the syllogism is therefore still the formal syllogism; the premises in which the subject posits the form on its side as its method is an immediate determination, and therefore contains the determinations of form, as we have seen, of definition, division, and so forth, as facts found existing in the subject. In true cognition on the contrary, the method is not merely an aggregate of certain determinations, but the Notion that is determined in and for itself; and the Notion is the middle term only because it has equally the significance of the objective, and consequently in the conclusion the objective does not merely attain an external determinateness by means of the method, but is posited in its identity with the subjective Notion.

§1786a 1. Thus what constitute the method are the determinations of the Notion itself and their relations, which we have now to consider in their significance as determinations of the method. In doing so we must first begin with the beginning. Of the beginning we have already spoken at the beginning of the Logic itself, and also above, when dealing with subjective cognition, and we have shown that, if it is not made arbitrarily and with a categorical unconsciousness, it may indeed seem to involve a number of difficulties but nevertheless is of an extremely simple nature. Because it is the beginning, its content is an immediate, but an immediate that has the significance and form of abstract universality. Be it otherwise a content of being, or of essence, or of the Notion, it is as an immediate something assumed, found already in existence, assertorical. But first of all it is not an immediate of sensuous intuition or of representation, but of thinking, which on account of its immediacy may also be called a supersensuous inner intuition.

§1786b The immediate of sensuous intuition is a manifold and an individual. But cognition is thinking by means of notions, and therefore its beginning also is only in the element of thought — it is a simple and a universal. This form has already been discussed under definition. At the beginning of finite cognition universality is likewise recognised as an essential determination, but it is taken as a determination of thought and of Notion only in opposition to being. In point of fact this first universality is an immediate one, and for that reason has equally the significance of being; for being is precisely this abstract relation-to-self. Being requires no further derivation, as though it belonged to the abstract product of definition only because it is taken from sensuous intuition or elsewhere, and in so far as it is pointed out to us. This pointing out and derivation is a matter of mediation, which is more than a mere beginning, and is a mediation of a kind that does not belong to a comprehension by means of thinking, but is the elevation of ordinary thinking, of the empirical and ratiocinative consciousness, to the standpoint of thought. According to the current opposition of thought or concept and being it is regarded as an important truth that no being belongs as yet to the former, taken on its own, and that the latter has a ground of its own that is independent of thought. But the simple determination of being is in itself so meagre that, if only for that reason, there is no need to make much fuss about it; the universal is immediately itself this immediate, since as abstract it also is merely the abstract relation-to-self, which is being. As a matter of fact, the demand that being should be exhibited for us to see has a further, inner meaning involving more than this abstract determination; what is meant by it is in general the demand for the realisation of the Notion, which realisation does not lie in the beginning itself, but is rather the goal and the task of the entire further development of cognition. Further, since the content of the beginning is supposed to be justified and authenticated as something true or correct by its being pointed out in inner or outer perception, it is no longer the form of universality as such that is meant, but its determinateness, of which we shall need to speak presently. The authentication of the determinate content with which the beginning is made seems to lie behind it; but in fact it is to be considered as an advance, that is, if it belongs to philosophical cognition.

§1798 On this point, 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. Formal thinking does in fact think contradiction, only it at once looks away from it, and in saying that it is unthinkable it merely passes over from it into abstract negation.

§1799 Now the negativity just considered constitutes the turning point of the movement of the Notion. It is the simple point of the negative relation to self, the innermost source of all activity of all animate and spiritual self-movement, the dialectical soul that everything true possesses and through which alone it is true; for on this subjectivity alone rests the sublating of the opposition between the Notion and reality, and the unity that is truth. The second negative, the negative of the negative, at which we have arrived, is this sublating of the contradiction, but just as little as the contradiction is it an act of external reflection, but rather the innermost, most objective moment of life and spirit through which a subject, a person, a free being, exists.

§1805 It is here that the content of cognition as such first enters into the circle of consideration, since, as deduced, it now belongs to the method. The method itself by means of this moment expands itself into a system. At first the beginning had to be, for the method, wholly indeterminate in respect of content; to this extent it appears as the merely formal soul, for and by which the beginning was determined simply and solely in regard to its form, namely, as the immediate and the universal. Through the movement we have indicated, the subject matter has obtained for itself a determinateness that is a content, because the negativity that has withdrawn into simplicity is the sublated form, and as simple determinateness stands over against its development, and first of all over against its very opposition to universality.

§1812 It is in this manner that each step of the advance in the process of further determination, while getting further away from the indeterminate beginning is also getting back nearer to it, and that therefore, what at first sight may appear to be different, the retrogressive grounding of the beginning, and the progressive further determining of it, coincide and are the same. The method, which thus winds itself into a circle, cannot anticipate in a development in time that the beginning is, as such, already something derived; it is sufficient for the beginning in its immediacy that it is simple universality. In being that, it has its complete condition; and there is no need to deprecate the fact that it may only be accepted provisionally and hypothetically. Whatever objections to it might be raised — say, the limitations of human knowledge, the need to examine critically the instrument of cognition before starting to deal with the subject matter — are themselves presuppositions, which as concrete determinations involve the demand for their mediation and proof. Since therefore they possess no formal advantage over the beginning with the subject matter against which they protest, but on the contrary themselves require deduction on account of their more concrete content, their claim to prior consideration must be treated as an empty presumption. They have an untrue content, for they convert what we know to be finite and untrue into something incontestable and absolute, namely, a limited cognition determined as form and instrument relatively to its content; this untrue cognition is itself also the form, the process of seeking grounds, that is retrogressive.. The method of truth, too, knows the beginning to be incomplete, because it is a beginning; but at the same time it knows this incompleteness to be a necessity, because truth only comes to be itself through negativity of immediacy.

§1814 By virtue of the nature of the method just indicated, the science exhibits itself as a circle returning upon itself, the end being wound back into the beginning, the simple ground, by the mediation; this circle is moreover a circle of circles, for each individual member as ensouled by the method is reflected into itself, so that in returning into the beginning it is at the same time the beginning of a new member. Links of this chain are the individual sciences [of logic, nature and spirit], each of which has an antecedent and a successor — or, expressed more accurately, has only the antecedent and indicates its successor in its conclusion

§1815 Thus then logic, too, in the absolute Idea, has withdrawn into that same simple unity which its beginning is; the pure immediacy of being in which at first every determination appears to be extinguished or removed by abstraction, is the Idea that has reached through mediation, that is, through the sublation of mediation, a likeness correspondent to itself. The method is the Pure Notion that relates itself only to itself; it is therefore the simple self-relation that is being. But now it is also fulfilled being, the Notion that comprehends itself, being as the concrete and so absolutely intensive totality. In conclusion, there remains only this to be said about this Idea, that in it, first, the science of logic has grasped its own Notion.

(Übergang zur Natur)

§1816a In the sphere of being, the beginning of its content, its Notion appears as a knowing in a subjective reflection external to that content. But in the Idea of absolute cognition the Notion has become the Idea’s own content. The Idea is itself the pure Notion that has itself for subject matter and which, in running itself as subject matter through the totality of its determinations, develops itself into the whole of its reality, into the system of the science [of logic], and concludes by apprehending this process of comprehending itself, thereby superseding its standing as content and subject matter and cognising the Notion of the science.

§1816b Secondly, this Idea is still logical, it is enclosed within pure thought and is the science only of the divine Notion. True, the systematic exposition is itself a realisation of the Idea but confined within the same sphere. Because the pure Idea of cognition is so far confined within subjectivity, it is the urge to sublate this, and pure truth as the last result becomes also the beginning of another sphere and science. It only remains here to indicate this transition.

§1817 The Idea, namely, in positing itself as absolute unity of the pure Notion and its reality and thus contracting itself into the immediacy of being, is the totality in this form — nature. \,\,\,\,\,\,\, But this determination has not issued from a process of becoming, nor is it a transition, as when above, the subjective Notion in its totality becomes objectivity, and the subjective end becomes life. On the contrary, the pure Idea in which the determinateness or reality of the Notion is itself raised into Notion, is an absolute liberation for which there is no longer any immediate determination that is not equally posited and itself Notion; in this freedom, therefore, no transition takes place; the simple being to which the Idea determines itself remains perfectly transparent to it and is the Notion that, in its determination, abides with itself. The passage is therefore to be understood here rather in this manner, that the Idea freely releases itself in its absolute self-assurance and inner poise. By reason of this freedom, the form of its determinateness is also utterly free — the externality of space and time existing absolutely on its own account without the moment of subjectivity. In so far as this externality presents itself only in the abstract immediacy of being and is apprehended from the standpoint of consciousness, it exists as mere objectivity and external life; but in the Idea it remains essentially and actually [in and for itself] the totality of the Notion, and science in the relationship to nature of divine cognition. \,\,\,\,\,\,\, But in this next resolve of the pure Idea to determine itself as external Idea, it thereby only posits for itself the mediation out of which the Notion ascends as a free Existence that has withdrawn into itself from exteranlity, that completes its self-liberation in the science of spirit, and that finds the supreme Notion of itself in the science of logic as the self-comprehending pure Notion.

The theme of the idea expressing itself via the spirit in nature is summarized well here.

Die Philosophie der Natur / Philosophy of Nature

The next book in the system, after the Science of Logic, is the Philosophy of Nature. It starts continuing from EL§244 in PN§192.

(Notice however that “matter” and in fact “physical fields” were already in the Wesenslogik, around §1068, and “life” was already in the Begriffslogik above.)

Betrachtungsweisen der Natur

Begriff der Natur

PN§192. Die Natur hat sich als die Idee in der Form des Andersseins ergeben. Da die Idee so als das Negative ihrer selbst oder sich äußerlich ist, so ist die Natur nicht äußerlich nur relativ gegen diese Idee (und gegen die subjektive Existenz derselben, den Geist), sondern die Äußerlichkeit macht die Bestimmung aus, in welcher sie als Natur ist.

PN§192 Nature has come into being as the idea in the form of otherness“.

Hence Nature is the externalization of the Idea, and, by PN§193b is representation of the Idea. By the above, the Idea is the last stage in the system of modalities. Possible technical terminology for a model of the modal type theory is essentially verbatim “representation” and “externalization”.

PN§193a Die Natur zeigt daher in ihrem Dasein keine Freiheit, sondern Notwendigkeit und Zufälligkeit.

PN§193a Hence nature exhibits no freedom in its existence, but only necessity and contingency.

PN§193b Weil sie jedoch, [[]] Darstellung der Idee ist,

PN§193b [[]] nature is a representation of the idea,

Darstellung/representation: model of the theory. See also §1782 and PN§250

PN§195. Nature is, in itself a living whole. The movement of its idea through its sequence of stages is more precisely this: the idea posits itself as that which it is in itself; or, what is the same thing, it goes into itself out of that immediacy and externality which is death in order to go into itself; yet further, it suspends this determinacy of the idea, in which it is only life, and becomes spirit, which is its truth.

PN§202a Other mathematical determinations, such as infinity and its relationships, the infinitesimal, factors, powers, and so on, have their true concepts in philosophy itself. It is awkward to want to take and derive these from mathematics, where they are employed in a nonconceptual, often meaningless way; rather, they must await their justification and significance from philosophy.

Einteilung

Die Idee als Natur ist:

I. in der Bestimmung des Außereinander, der unendlichen Vereinzelung, außerhalb welcher die Einheit der Form, diese daher als eine ideelle, nur an sich seiende und daher nur gesuchte ist, die Materie und deren ideelles System, – Mechanik;

II. in der Bestimmung der Besonderheit, so daß die Realität mit immanenter Formbestimmtheit und an ihr existierender Differenz gesetzt ist, ein Reflexionsverhältnis, dessen Insichsein die natürliche Individualität ist, – Physik;

III. in der Bestimmung der Subjektivität, in welcher die realen Unterschiede der Form ebenso zur ideellen Einheit, die sich selbst gefunden und für sich ist, zurückgebracht sind, – Organik

On the importance of the discussion of Measure above in the passage to nature:

PN§202b 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.

PN§250 Der Widerspruch der Idee, indem sie als Natur sich selbst äußerlich ist

see also PN§193b

Die Mechanik

Mathematische Mechanik – Raum und Zeit

Der Raum

PN§254a Die erste oder unmittelbare Bestimmung der Natur ist die abstrakte Allgemeinheit ihres Außersichseins, – dessen vermittlungslose Gleichgültigkeit, der Raum. Er ist das ganz ideelle Nebeneinander, weil er das Außersichsein ist, und schlechthin kontinuierlich, weil dies Außereinander noch ganz. abstrakt ist und keinen bestimmten Unterschied in sich hat.

“ideelles Nebeneinander” étale stack via infinitesimal shape modality &\&

PN§254b Von Raumpunkten zu sprechen, als ob sie das positive Element des Raums ausmachten, ist unstatthaft,

PN§254b To speak of points of space, as if they constituted the positive element of space, is inadmissible

synthetic geometry, see also PN§256b

PN§256a Aber der Unterschied ist wesentlich bestimmter, qualitativer Unterschied. Als solcher ist er α\alpha) zunächst Negation des Raumes selbst, weil dieser das unmittelbar unterschiedlose Außersichsein ist, – der Punkt. β\beta) Die Negation ist aber Negation des Raumes, d. i. sie ist selbst räumlich; der Punkt als wesentlich diese Beziehung, d.i. als sich aufhebend, ist die Line, das erste anders-, d.i. Räumlich-sein des Punktes. γ\gamma) Die Wahrheit des Andersseins ist aber die Negation der Negation. Die Linie geht daher in die Fläche über, welche einerseits eine Bestimmtheit gegen Linie und Punkt, und so Fläche überhaupt, andererseits aber die aufgehobene Negation des Raumes ist, somit Wiederherstellung der räumlichen Totalität, welche nunmehr das negative Moment an ihr hat; – umschließende Oberfläche, die einen einzelnen ganzen Raum absondert.

the brane bouquet: form the (super-)point 0|N\mathbb{R}^{0|N} emanate the d|N\mathbb{R}^{d|N} by central extension via the (Γ)(-\Gamma-).

PN§256b Daß die Linie nicht aus Punkten, die Fläche nicht aus Linien besteht, geht aus ihrem Begriffe hervor, da die Linie vielmehr der Punkt als außer sich seiend, nämlich sich auf den Raum beziehend und sich aufhebend, die Fläche ebenso die aufgehobene, außer sich seiende Linie ist. –

PN§256b That the line does not consist of points, nor the plane of lines, follows from their concepts

synthetic geometry, see also PN§254b

Die Zeit
Einheit von Raum und Zeit
Ort und Bewegung

PN§260 Der Raum ist in sich selbst der Widerspruch des gleichgültigen Auseinanderseins und der unterschiedlosen Kontinuität, die reine Negativität seiner selbst und das Übergehen zunächst in die Zeit. Ebenso ist die Zeit, da deren in Eins zusammengehaltene entgegengesetzte Momente sich unmittelbar aufheben, das unmittelbare Zusammenfallen in die Indifferenz, in das ununterschiedene Außereinander oder den Raum. So ist an diesem die negative Bestimmung, der ausschließende Punkt, nicht mehr nur an sich dem Begriffe nach, sondern gesetzt und in sich concret durch die totale Negativität welche die Zeit ist; – der so konkrete Punkt ist der Ort.

modern terminology here would be: spacetime event

Materie und Bewegung

PN§261 Dies Vergehen und Sich-wiedererzeugen des Raumes in der Zeit und der Zeit im Raum, [[]] ist die Bewegung. Dies Werden ist aber selbst eben so sehr das in sich Zusammenfallen seines Widerspruchs, die unmittelbar identische daseiende Einheit beider, die Materie.

This disappearance and regeneration of space in time and of time in space is motion;– a becoming, which, however, is itself just as much immediately the identically existing unity of both, or matter.

So space and time transmute into each other, forming a unity. The appropriate modern word for this is clearly spacetime. That Hegel’s perspective fits well with modern general relativity has been argued in (Wandschneider82).

Moreover, this unity of space and time in spacetime is matter and by PN§26d it is matter with the forces acting on it included. This is very much the perspective of the modern concept of Kaluza-Klein compactification in which theories of pure (supergravity-)gravity give rise to matter and forces depending on how parts of spacetime take certain shape. For instance models such as the G2-MSSM are theories of pure gravity (in this case 11d supergravity) taking shape in the form of a KK-compactification on a G2-manifold fiber bundle, and all the matter (and force field) content of the standard model of particle physics arises from just this (super-)geometry.

A vaguely similar synthethis had been suggested in (Weyl 1919). In the 1960s John Wheeler highlighted this idea of producing matter from pure spacetime geometry, coining the term geometrodynamics for it. His slogan “mass without mass” referred to mass arising from pure spacetime geometry. At this level of detail, this is rather close to PN§261.

PN§261b Der Uebergang von der Idealität zur Realität, von der Abstraktion zum konkreten Dasein, hier von Raum und Zeit zu der Realität, welche als Materie erscheint, ist für den Verstand unbegreiflich

PN§261c Die geläufige Vorstellung ist, Raum und Zeit als leer, gleichgueltig gegen ihre Erfuellung, und doch immer als voll zu betrachten, sie als leer von außen her mit der Materie erfuellen zu lassen, und einerseits auf diese Weise die materiellen Dinge als gleichgueltig gegen Raum ind Zeit, und andererseits zugleich als wesentlich räumlich und zeitlich anzunehmen.

PN§26d Es gehlrt ferner zu dieser begrifflosen Reflexion, die sogenannten Kräfte als der Materie eingepflanzt, das ist, als ihr urpsrünglich äußerlich anzusehen, so daß eben diese Identität der Zeit und des Raumes, welche bei der Reflexionsbestimmung von Kraft vorschwebt und welche in Wahrheit das Wesen der Materie ausmacht, als etwas ihr fremdes und Zufälliges von außen in sie gebrachtes gesetzt ist.

Endliche Mechanik – Materie und Bewegung.

PN§262a Die Materie hält sich gegen ihre Identität mit sich, durch das Moment ihrer Negativität, ihrer Abstrakten Vereinzelung, auseinander; die Repulsion der Materie.

In modern terminology this intrinsic repulsion which keeps solid matter from collapsing to a condensate is the degeneracy pressure of fermion matter induced by the Pauli exclusion principle. See also PN§290.

PN§262b Ebenso wesentlich ist, weil diese Verschiedenen ein und dasselbe sind, die negative Einheit dieses außereinander-seienden Fürsichseins; die Materie is somit kontinuierlich – ihre Attraktion.

PN§262c Die Materie ist untrennbar beides, und negative Einheit dieser Momente, Einzelnheit, aber gegen das unmittelbare Außereinander der Materie noch unterschieden und darum selbst noch nicht als materiell gesetzte, idelle Einzelheit, Mittelpunkt – die Schwere.

PN§262d Die Schwere ist von der bloßen Attraktion wesentlich zu unterscheiden.

PN§262e Die Schwere macht die Substantialität der Materie aus, diese selbst ist das Streben nach dem – aber, – dies ist die andere Wesentliche Bestimmung, außer ihr fallenden Mittelpunkt.

See also PN§271.

PN§262 Matter in itself holds itself apart from itself through the moment of its negativity, diversity, or abstract separation into parts; it has repulsion. Its being apart from itself is just as essential, however, because these differences are one and the same: the negative unity of this existence apart from itself as being for itself, and thus continuous. Matter therefore has attraction. The unity of these moments is gravity.

So we have a unity of opposites:

AussereinanderSchwere Aussereinander \dashv Schwere

Below in PN§290 we see more in detail what Außereinander / asunderness is and we give the formalization of this opposition there.

Notice that before in the Seinslogik, the unity of attraction with repulsion was in §369, where we interpreted it as the unity of opposites ʃ&#643; \dashv \flat of the shape modality and the flat modality. By the discussion below §990, this unity indeed describes gauge fields (via the differential cohomology hexagon) among which one may count gravity (when thinking of it as a Cartan connection).

See also on Science of Logic the Comment on the Construction of Matter from the Forces.

Absolute Mechanik – Gravitation

Die allgemeine Gravitation

PN§269 Die Gravitation ist der wahrhafte und bestimmte Begriff der materiellen Körperlichkeit, der zur Idee realisiert ist

Die Kepplerschen Gesetze
Die Totalität des Sonnen-Systems

PN§271 Die Substanz der Materie, die Schwere, zur Totalität der Form entwickelt hat das außersichsein der Materie nicht mehr außer ihr. … Ihr abstraktes dumpfes In-sich-sein, als schwer ueberhaupt, ist zur Form entschlossen, sie ist die qualifizierte Materie; – Physik.

Die Physik

PN§218 Gravity, as the essence of matter existing in itself only inner identity, transforms, since its concept is the essential externality, into the manifestation of the essence. As such it is the totality of the determinations of reflection, but these as thrown apart from each other, so that each appears as particular, qualified matter which, not yet determined as individuality, is a formless element.

Die Physik der allgemeinen Individualität

Die freien physikalischen Körper
Das Licht

PN§275 Die erste qualifizierte Materie… existierende allgemeine Selbst der Materie ist das Licht.

PN§276 Als das abstrakte Selbst der Materie ist das Licht das absolute-leichte, und als Materie, und als Materie ist sie unendliches Außersichsein aber als reines Manifestieren, materielle Idealität untrennbares und einfaches Außersichsein

We may grant light here as a placeholder for the modern concept of bosons and relate “light” to the bosonic modality.

Die Körper des Gegensatzes

PN§279 Das Dunkle, zunächst das Negative des Lichts, ist der Gegensatz gegen dessen abstrakte-identische Idealität, – der Gegensatz an ihm selbst; er hat materielle Realität under zerfällt in sich in die Zweihet, α\alpha) der körperlichen Verschiedenheit, d.i. des materiellen Fürsichseins, der Starrheit, β\beta) der Entgegensetzung…

This material reality and rigidity opposing light which by PN§276 we identified with the bosonic modality is clearly captured well by its left adjoint, the fermionic modality e\e. Hence we have a unity of opposites

rigid matter \, fermionic modality e\e \dashv \rightsquigarrow bosonic modality \, light

Die Körper der Individualität
Die Elemente

Die Physik der besonderen Individualität

PN§290 Die vorher elementarischen Bestimmtheiten nun der individuellen Einheit unterworfen, so ist diese die immanente Form, welche für sich die Materie gegen ihre Schwere bestimmt. Die Schwere als suchen des Einheitspunktes tut dem Außereinander der Materie keinen Eintrag, d.i. der Raum und zwar nach einem Quantum ist das Maaß der Besonderungen der Unterschiede der schweren Materie, der Massen, (…) jetzt durch die gesetzte Individualität der Materie ist sie in ihrem Außereinander selbst ein Zentralisieren gegen dieß ihr Außereinander und gegen dessen suchen der Individualität, different gegen das ideelle Zentralisieren der Schwere, ein immanentes anderes Bestimmenn der materiellen Räumlichkeit als durch die Schwere nach der Richtung derselben. Dieser Teil der Physik ist die individualisierende Mechanik, indem die Materie durch die immanente Form und zwar nach dem Räumlichen bestimmt wird.

In modern terminology, this asunderness (see also PN§262c) of matter that opposes its gravitational collapse is the Pauli exclusion principle obeyed by fermions. Mathematically this originates in the odd grading involved in superalgebra/supergeometry. In terms of adjoint modalities/unity of opposites, this is embodied by the adjoint triple between plain and superalgebra, hence by the (fermionic modality e\e \dashv \rightsquigarrow bosonic modality)-adjunction on super smooth infinity-groupoids. This was identified above in PN§279 with the unity of opposites between rigid matter and light. Hence we find

rigid matter \, fermionic modality easunderness\e \stackrel{asunderness}{\dashv} \rightsquigarrow bosonic modality \, light

Since there is a further right adjoint (by the discussion at super smooth infinity-groupoid) we get a second order duality, and since by PN§262c this opposite of asunderness is Schwere, i.e. gravity we add to the Process) the level:

starreMaterie ρ(e) Aussereinander ρ() Licht ρ() Schwere ρ() \array{ \stackrel{starre \atop Materie}{} & \rho(\e) &\stackrel{Aussereinander}{\dashv}& \rho(\rightsquigarrow) & \stackrel{Licht}{} \\ \bot && \bot \\ & \rho(\rightsquigarrow) &\stackrel{Schwere}{\dashv}& \rho(\,) }

Notice that this fits well in 11-dimensional supergravity (which by PN§261 is indeed the case at hand), where the bosonic part of the physical fields is the graviton, which after KK-compactification becomes also light and the other bosonic force field, and the rest is the fermionic matter content resisting its gravitational collapse via the Pauli exclusion principle.

Die spezifische Schwere

PN§293 Die einfache, abstrakte Spezifikation ist die spezifische Schwere oder Dichtigkeit der Materie, ein Verhältnis des Gewichts der Masse zu dem Volumen, wodurch das Materielle als selbstisch sich von dem abstrakten Verhältnisse zum Zentralkörper, der allgemeinen Schwere, losreißt, aufhört, die gleichförmige Erfüllung des Raums zu sein, und dem abstrakten Außereinander ein spezifisches Insichsein entgegensetzt.

this being the first abstract determination, by comparison with the previous stages it should be at the level of \flat \dashv \sharp. In the Seinslogik this was the, the quantity/cardinality “of points in space”, which harmonizes with the suggestion of specific weight.

§294 Die Dichtigkeit ist nur erst einfache Bestimmtheit der schweren Materie; aber indem die Materie das wesentliche Außereinander bleibt, so ist die Formbestimmung weiter eine spezifische Weise der räumlichen Beziehung ihres Vielfachen aufeinander, - Kohäsion.

Die Kohäsion

PN§295 In der Kohäsion setzt die immanente Form eine andere Weise des räumlichen Nebeneinanderseins der materiellen Teile, als durch die Richtung der Schwere bestimmt ist. Diese somit spezifische Weise des Zusammenhalts des Materiellen ist erst am Verschiedenen überhaupt gesetzt, noch nicht zu in sich beschlossener Totalität (Gestalt) zurückgegangen; sie kommt somit nur gegen gleichfalls verschiedene, und kohärent verschiedene, Massen zur Erscheinung und zeigt sich daher als eine eigentümliche Weise des Widerstands im mechanischen Verhalten gegen andere Massen.

cohesion

PN§296 Die eigentliche qualitative Kohäsion ist ein Zusammenhalten der homogenen Massen durch immanente, eigentümliche Form oder Begrenzung, welche sich hier als die abstrakten Dimensionen des Raums expliziert. Die eigentümliche Gestaltung kann nämlich keine andere sein als eine Weise bestimmter Räumlichkeit, die der Körper an sich zeichnet. Denn die Kohärenz ist die Identität des Körpers in seinem Außereinander, die qualitative Kohärenz ist also eine bestimmte Weise des Außereinanderseins, d. h. eine Raumdetermination. Diese Einheit ist in der individuellen Materie selbst, als ein Zusammenhalten gegen die allgemeine Einheit, welche sie in der Schwere sucht. Die Materie erhält jetzt nach vielerlei Seiten eigentümliche Richtungen in sich selbst, die von der nur vertikalen Richtung der Schwere verschieden sind. Diese Kohäsion, obgleich Individualität, ist aber zugleich noch bedingte Individualität, weil sie nur durch das Einwirken von anderen Körpern zum Vorschein kommt; sie ist noch nicht die freie Individualität als Gestalt, d. h. noch nicht die Individualität als Totalität ihrer durch sie gesetzten Formen. Die to tale Gestalt nämlich ist da, mechanisch bestimmt, mit solchen Seiten und Winkeln. Hier aber ist der Charakter der Materie nur erst die innere Gestalt derselben, d. h. eben eine solche, die noch nicht in ihrer Bestimmtheit und Entwicklung da ist.

PN§297 γ) Das Körperliche, gegen dessen Gewalt ein Körperliches im Nachgeben zugleich seine Eigentümlichkeit behauptet, ist ein anderes Körperindividuum. Aber als kohärent ist der Körper auch an ihm selbst außereinanderseiende Materialität, deren Teile, indem das Ganze Gewalt leidet, gegeneinander Gewalt ausüben und nachgeben, aber als ebenso selbständig die erlittene Negation aufheben und sich herstellen. Das Nachgeben und darin die eigentümliche Selbsterhaltung nach außen ist daher unmittelbar verknüpft mit diesem inneren Nachgeben und Selbsterhalten gegen sich selbst, die Elastizität.

PN§297Zusatz Die Elastizität ist die Kohäsion, die sich in der Bewegung darstellt, das Ganze der Kohäsion.

elasticity

PN§298 Wenn hier und sonst von materiellen Teilen die Rede ist, so sind nicht Atome, noch Moleküle, d. h. nicht abgesondert für sich bestehende zu verstehen, sondern nur quantitativ oder zufällig unterschiedene, so daß ihre Kontinuität wesentlich von ihrer Unterschiedenheit nicht zu trennen ist; die Elastizität ist die Existenz der Dialektik dieser Momente selbst. Der Ort des Materiellen ist sein gleichgültiges bestimmtes Bestehen; die Idealität dieses Bestehens ist somit die als reelle Einheit gesetzte Kontinuität, d. i. daß zwei vorher außereinander bestehende materielle Teile, die also als in verschiedenen Orten befindlich vorzustellen sind, jetzt in einem und demselben Orte sich befinden. Es ist dies der Widerspruch, und er existiert hier materiell. Es ist derselbe Widerspruch, welcher der Zenonischen Dialektik der Bewegung zum Grunde liegt, nur daß er bei der Bewegung abstrakte Orte betrifft, hier aber materielle Orte, materielle Teile. In der Bewegung setzt sich der Raum zeitlich und die Zeit räumlich (§ 260); die Bewegung fällt in die Zenonische Antinomie, die unauflöslich ist, wenn die Orte als Raumpunkte und die Zeitmomente als Zeitpunkte isoliert werden, und die Auflösung der Antinomie, d. i. die Bewegung, ist nur so zu fassen, daß Raum und Zeit in sich kontinuierlich sind und der sich bewegende Körper in demselben Orte zugleich ist und nicht, d. i. zugleich in einem anderen ist, und ebenso derselbe Zeitpunkt zugleich ist und nicht, d. i. ein anderer zugleich ist.

Zeno's paradox of motion is what is resolved by modern analysis, by the concept of the infinitesimal, encoded by the reduction modality \dashv infinitesimal shape modality &\Im \dashv \&

Hence “elasticity is the existence of the dialetic of these moments themselves” points to a unity of opposites

ρ()Elastizitaetρ(&)\rho(\Im) \stackrel{Elastizitaet}{\dashv} \rho(\&).

Indem ein Körper sich im andern setzt und sie jetzt von einer gewissen Dichtigkeit sind, so wird erstens die spezifische Schwere dessen, in dem sich der andere setzt, verändert. Das zweite Moment ist das Widerstandleisten, das Negieren, das sich abstrakt Verhalten; das dritte ist, daß der Körper reagiert und den ersten von sich abstößt. Das sind die drei Momente, die als Weichheit, Härte und Elastizität bekannt sind.

hence we further label this unity of opposites as

Weichheit ρ() Elastizitaet ρ(&) Haerte \array{ Weichheit & \rho(\Im) & \stackrel{Elastizitaet}{\dashv} & \rho(\&) & Haerte }

PN§298b So ist in der Elastizität der materielle Teil, Atom, Molekül, zugleich als affirmativ seinen Raum einnehmend, bestehend gesetzt, und ebenso zugleich nicht bestehend, - als Quantum, in einem als extensive Größe und als nur intensive Größe.

extensive and intensive quantity

Der Klang

PN§300 Die spezifische Einfachheit der Bestimmtheit, welche der Körper in der Dichtigkeit und dem Prinzip seiner Kohäsion hat, diese zuerst innerliche Form, hindurchgegangen durch ihr Versenktsein in das materielle Außereinander, wird frei in der Negation des für sich Bestehens dieses seines Außereinanderseins. Es ist dies das Übergehen der materiellen Räumlichkeit in materielle Zeitlichkeit. Damit, daß diese Form so im Erzittern, d. i. durch die momentane ebenso Negation der Teile wie Negation dieser ihrer Negation, die aneinander gebunden eine durch die andere erweckt wird, und so, als ein Oszillieren des Bestehens und der Negation der spezifischen Schwere und Kohäsion, am Materiellen als dessen Idealität ist, ist die einfache Form für sich existierend und kommt als diese mechanische Seelenhaftigkeit zur Erscheinung.

Die Geburt des Klanges ist schwer zu fassen. Das spezifische Insichsein, von der Schwere geschieden, ist, als hervortretend, der Klang; er ist die Klage des Ideellen in dieser Gewalt des Anderen, ebenso aber auch sein Triumph über dieselbe, indem es sich in ihr erhält. Der Klang hat zweierlei Weisen seiner Hervorbringung: α) durch Reibung, β) durch eigentliches Schwingen, Elastizität des Insichseins. Bei der Reibung ist auch dieses vorhanden, daß während ihrer Dauer eine Mannigfaltigkeit in eins gesetzt wird, indem die verschiedenen außereinander seienden Teile momentan in Berührung gebracht werden. Die Stelle eines jeden, somit seine Materialität, wird aufgehoben; sie stellt sich aber ebenso wieder her. Diese Elastizität ist es eben, die sich durch den Klang kundgibt.

Following the above, the cohesion and elasticity here is not so much that of actual physical bodies, but more of mathematical space itself (matter is the union of space and time anyway PN§261). Viewed this way speaking of a “sound of space” here is precisely what modern mathematics does, in the discussion of hearing the shape of a drum, where it refers to the spectrum of Laplace operators on Riemannian manifolds. Now Riemannian geometry is an example of a Cartan geometry, and Cartan geometry is one of the hallmark concepts formalized in differential cohesion. At the same time, followin PN§298 we think of differential cohesion as the formalization of Hegel’s elasticity.

Summing this up. Hegel’s “sound” matches well with the structure of higher Cartan geometry formalized in differential cohesion.

Die Wärme

Die Physik der totalen Individualität

Die Gestalt
Die Besonderheit des individuellen Körpers
Der chemische Prozeß

Die Organik

A. Die geologische Natur

PN§338 The first organism is the body of Earth. PN§339 The process of nature that leaves Earth in its independency are its solar, lunar and cometary life. PN§341 This crystal of life is the subject of a meteorological process by which it is fertilized to live. Land and sea blossom in an infinitude of points of puntual temporary life, immesurable phosphorescent points of life in the sea.

B. Die vegetabilische Natur

C. Der tierische Organismus

The last paragraph is

In this way nature has passed over into its truth, into the subjectivity of the concept, whose objectivity is itself the suspended immediacy of individuality, the concrete generality, the concept which has the concept as its existence — into the spirit.

Die Philosophie des Geistes / Philosophy of Spirit

Einleitung

Begriff des Geistes

PG§381 Der Geist hat für uns die Natur zu seiner Voraussetzung, deren Wahrheit und damit deren absolut Erstes er ist. In dieser Wahrheit ist die Natur verschwunden, und der Geist hat sich als die zu ihrem Fürsichsein gelangte Idee ergeben, deren Objekt ebensowohl als das Subjekt der Begriff ist. Diese Identität ist absolute Negativität, weil in der Natur der Begriff seine vollkommene äußerliche Objektivität hat, diese seine Entäußerung aber aufgehoben und er in dieser identisch mit sich geworden ist. Er ist diese Identität somit zugleich nur als Zurückkommen aus der Natur.

PS§381 Spirit has for us nature as its presupposition, of which it is truth. In this truth, its concept, nature has disappeared; spirit has therefore produced itself as idea, of which the concept is both the object and the subject. This identity is absolute negativity, because in nature the concept has its completely external objectivity. But it has suspended its articulation, and in this it has become identical with itself. It is this identity only insofar as it is a return from nature.

The essence of the spirit is therefore freedom, the identity of the absolute negativity of the concept with itself. It can distance itself from everything external and from its own externality as well as from its being, and thus bear infinite pain, the negation of its individual immediacy; in other words, it can be identical for itself in this negativity. This possibility is its self-contained being in itself its simple concept, or absolute generality itself.

The theme of the idea expressing itself via the spirit in nature is summarized well here.

PG§381 Das Wesen des Geistes ist deswegen formell die Freiheit

§ 384 Das Offenbaren, welches als das Offenbaren der abstrakten Idee unmittelbarer Übergang, Werden der Natur ist, ist als Offenbaren des Geistes, der frei ist, Setzen der Natur als seiner Welt; ein Setzen, das als Reflexion zugleich Voraussetzen der Welt als selbständiger Natur ist. Das Offenbaren im Begriffe ist Erschaffen derselben als seines Seins, in welchem er die Affirmation und Wahrheit seiner Freiheit sich gibt.

Das Absolute ist der Geist, dies ist die höchste Definition des Absoluten. – Diese Definition zu finden und ihren Sinn und Inhalt zu begreifen, dies, kann man sagen, war die absolute Tendenz aller Bildung und Philosophie, auf diesen Punkt hat sich alle Religion und Wissenschaft gedrängt; aus diesem Drang allein ist die Weltgeschichte zu begreifen. – Das Wort und die Vorstellung des Geistes ist früh gefunden, und der Inhalt der christlichen Religion[29] ist, Gott als Geist zu erkennen zu geben. Dies, was hier der Vorstellung gegeben und was an sich das Wesen ist, in seinem eigenen Elemente, dem Begriffe, zu fassen, ist die Aufgabe der Philosophie, welche so lange nicht wahrhaft und immanent gelöst ist, als der Begriff und die Freiheit nicht ihr Gegenstand und ihre Seele ist.[30]

Erste Abteilung. Der subjektive Geist

A. Die Seele – Anthropologie

PS§308 Spirit came into being as the truth of nature which has translated and suspended itself. But spirit is, then, not merely true and primordial: its transition into the realm of the concept is not only reflection into others and reflection into itself but it is also free judgment. The becoming of spirit in this way indicates that nature suspends itself in itself as untruth, and that spirit no longer presupposes itself as immediacy self-externalised in physical individuality, but as general and as that immediacy, simple in its concreteness, in which it is soul.

a. Natürliche Seele

PS§311 Spirit as the abstract soul of nature is simple, sidereal, and terrestrial life. It is the nous of the ancients, the simple, unconscious thought, which (a) as this general essence is the inner idea and would have its reality in the underlying externality of nature. But since it, as soul, is immediate substance, its existence is the particularisation of its natural being, an immediate and natural determinacy, which has its presupposed reality in the individual earth.

b. Träumende Seele
c. Wirkliche Seele

PN§308 Spirit came into being as the truth of nature which has translated and suspended itself. But spirit is, then, not merely true and primordial: its transition into the realm of the concept is not only reflection into others and reflection into itself but it is also free judgment. The becoming of spirit in this way indicates that nature suspends itself in itself as untruth, and that spirit no longer presupposes itself as immediacy self-externalised in physical individuality, but as general and as that immediacy, simple in its concreteness, in which it is soul.

B. Bewußtsein – Phänomenologie

PdGPreface15 Hence the important thing for the student of science is to make himself undergo the strenuous toil of conceptual reflection, of thinking in the form of the notion. This demands concentrated attention on the notion as such, on simple and ultimate determinations like being-in-itself, being-for-itself, self-identity, and so on; for these are elemental, pure, self-determined functions of a kind we might call souls, were it not that their conceptual nature denotes something higher than that term contains. The interruption by conceptual thought of the habit of always thinking in figurative ideas (Vorstellungen) is as annoying and troublesome to this way of thinking as to that process of formal intelligence which in its reasoning rambles about with no real thoughts to reason with. The former, the habit, may be called materialised thinking, a fortuitous mental state, one that is absorbed in what is material, and hence finds it very distasteful at once to lift its self clear of this matter and be with itself alone. The latter, the process of raisonnement, is, on the other hand, detachment from all content, and conceited superiority to it. What is wanted here is the effort and struggle to give up this kind of freedom, and instead of being a merely arbitrary principle directing the content anyhow, this freedom should sink into and pervade the content, should let it be directed and controlled by its own proper nature, i.e. by the self as its own self, and should observe this process taking place. We must abstain from interrupting the immanent rhythm of the movement of conceptual thought; we must refrain from arbitrarily interfering with it, and introducing ideas and reflections that have been obtained elsewhere. Restraint of this sort is itself an essential condition of attending to and getting at the real nature of the notion.

a. Bewußtsein als solches
b. Selbstbewußtsein
c. Vernunft

B. Geist – Psychologie

a. Theoretischer Geist

PS§363 Spirit has shown itself as the unity of the soul and consciousness, — the former a simple immediate totality, and the latter is knowledge which is not limited by any object, and no longer stands in relation to it, but is knowledge of the simple, neither subjective nor objective totality. Spirit originates, therefore, only from its own being, and only relates itself to its own determinations.

PS§365 The progress of the spirit is development, because its existing phase, knowledge, involves consciousness in and for itself as the purpose or rationale. Thus the action of translating this purpose into reality is strictly only this formal transition into manifestation. Insofar as knowledge is infinite negativity, this translation in the concept is creativity in general. Insofar as knowledge is only abstract or formal, the spirit in it does not conform to its concept, and its purpose is to bring forth the absolute fulfilment and the absolute freedom of its knowledge.

PS§456b This common element is either any one particular side of the object raised to the form of universality, such as, for example, in the rose, the red colour; or the concrete universal, the genus, for example, in the rose, the plant;

concrete general, see also at The genus.

b. Praktischer Geist
α\alpha. Praktisches Gefühl
β\beta. Trieb
γ\gamma. Willkür und Glückseligkeit

Zweite Abteilung. Der objektive Geist

A. Das Recht

B. Die Moralität

C. Die Sittlichkeit

Dritte Abteilung. Der absolute Geist

PS§553 The concept of the spirit has its reality in the spirit. If this reality is in completed identity with that concept as the knowledge of the absolute idea, then the necessary aspect is that the implicitly free intelligence liberates itself for its concept, in order for it to be a shape worthy of it. The subjective and the objective spirit can therefore be seen as the path on which this side of reality or existence forms itself (§ 304). Conversely, this path also has the significance that the subjective spirit is seen as the first entity which exists in its immediacy without the concept, grasps its essence and forms itself from there, and thereby reaches its free identity with the concept, its absolute reality.

A. Die Kunst

B. Die geoffenbarte Religion

C. Die Philosophie

PG§574 Dieser Begriff der Philosophie ist die sich denkende Idee, die wissende Wahrheit (§ 236), das Logische mit der Bedeutung, daß es die im konkreten Inhalte als in seiner Wirklichkeit bewährte Allgemeinheit ist. Die Wissenschaft ist auf diese Weise in ihren Anfang zurückgegangen und das Logische so ihr Resultat als das Geistige, daß es aus dem voraussetzenden Urteilen, worin der Begriff nur an sich und der Anfang ein Unmittelbares war, hiermit aus der Erscheinung, die es darin an ihm hatte, in sein reines Prinzip zugleich als in sein Element sich erhoben hat.

return to the beginning

The second edition of the Enzyklopädie der philos. Wiss. from 1827 ends here. Later the following three paragraphs are added, concerning three kinds of Schluß.

PG§575 Es ist dieses Erscheinen, welches zunächst die weitere Entwicklung begründet. Die erste Erscheinung macht der Schluß aus, welcher das Logische zum Grunde als Ausgangspunkt und die Natur zur Mitte hat, die den Geist mit demselben zusammenschließt. Das Logische wird zur Natur und die Natur zum Geiste. Die Natur, die zwischen dem Geiste und seinem Wesen steht, trennt sie zwar nicht zu Extremen endlicher Abstraktion, noch sich von ihnen zu einem Selbständigen, das als Anderes nur Andere zusammenschlösse; denn der Schluß ist in der Idee und die Natur wesentlich nur als Durchgangspunkt und negatives Moment bestimmt und an sich die Idee; aber die Vermittlung des Begriffs hat die äußerliche Form des Übergehens und die Wissenschaft die des Ganges der Notwendigkeit, so daß nur in dem einen Extreme die Freiheit des Begriffs als sein Zusammenschließen mit sich selbst gesetzt ist.

LogikNaturGeist Logik \stackrel{Natur}{\longrightarrow} Geist

PG§576 Diese Erscheinung ist im zweiten Schlüsse insoweit aufgehoben, als dieser bereits der Standpunkt des Geistes selbst ist, welcher das Vermittelnde des Prozesses ist, die Natur voraussetzt und sie mit dem Logischen zusammenschließt. Es ist der Schluß der geistigen Reflexion in der Idee; die Wissenschaft erscheint als ein subjektives Erkennen, dessen Zwecks die Freiheit und es selbst der Weg ist, sich dieselbe hervorzubringen.

§577 Der dritte Schluß ist die Idee der Philosophie, welche die sich wissende Vernunft, das Absolut-Allgemeine zu ihrer Mitte hat, die sich in Geist und Natur entzweit, jenen zur Voraussetzung als den Prozeß der subjektiven Tätigkeit der Idee und diese zum allgemeinen Extreme macht, als den Prozeß der an sich, objektiv, seienden Idee. Das Sich-Urteilen der Idee in die beiden Erscheinungen (§ 575/6) bestimmt dieselben als ihre (der sich wissenden Vernunft) Manifestationen, und es vereinigt sich in ihr, daß die Natur der Sache, der Begriff, es ist, die sich fortbewegt und entwickelt, und diese Bewegung ebensosehr die Tätigkeit des Erkennens ist, die ewige an und für sich seiende Idee sich ewig als absoluter Geist betätigt, erzeugt und genießt.[394]

References

Texts with background, introduction and survey include

  • John McTaggart, A Commentary on Hegel’s Logic Cambridge University Press, 1910 (web)

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

  • Stephen Houlgate, The opening of Hegel’s Logic, Purdue University Press, 2006 (pdf)

  • John Grier Hibben, Eric v.d. Luft, Hegel’s Shorter Logic: An Introduction and Commentary

  • David Carlson, Hegel’s theory of measure (web)

  • David Carlson, Hegel and what is actual (pdf)

  • Richard Dien Winfield, Lecture Course in Hegel’s Science of Logic (web)

A quick survey of the theme of the idea expressing itself via the spirit in nature is in

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

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

Further comments on Hegel’s text include

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

  • Inwood, Hegel, 1983

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

Comments on the general aim of a fundamental logic based on dialectic are in

  • Dieter Wandschneider, Dialektik als Letztbegründung der Logik, in Koreanische Hegelgesellschaft (ed.), Festschrift für Sok-Zin Lim Seoul 1999, 255–278 (pdf)

Texts amplifying the hermeticism, gnosticism and mysticism of Hegel’s system include

Comments on the similarity of Hegel’s physics to aspects of general relativity are in

and the idea of matter emerging from space and time has apparently influenced

Related discussion in view of infinitesimals is in

  • Hermann Cohen, Das Prinzip der Infinitesimal-Methode und seine Geschichte , Berlin 1883. (html)

Proposals for formalizing some of Hegel’s thoughts in terms of algebra may be identified in

and, apparently following this, according to

  • William Lawvere, A new branch of mathematics, “The Ausdehnungslehre of 1844,” and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis, Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106 (publisher)

proposals for formalization of Hegel’s objective logic in categorical logic have been put forward by William Lawvere in several places, for instance in

Related commentary is in

  • Andrei Rodin, section 5.8 Categorical Logic and Hegelian Dialectics of Axiomatic Method and Category Theory (arXiv:1210.1478), Springer 1914

Discussion of formalization of the topics of the “subjective logic” within type theory include

  • Per Martin-Löf, On the Meanings of the Logical Constants and the Justifications of the Logical Laws, Nordic Journal of Philosophical Logic, 1(1): 11–60, 1996, (pdf)

which gives a detailed discussion of the term judgement as used in the history philosophy and then in type theory.

Texts on foundations of mathematics which comment on the circularity (or not) of foundations include

The formalization of Schluss by natural deduction originates in

The idea that the types in type theory are (mathematical) concepts or notions (i.e. “Begriffe”, as in the formalization dictionary above ) is arguably implicit in the original

  • Per Martin-Löf, An intuitionistic theory of types: predicative part, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), 73-118. (web)

where it says in section 1.1:

Every mathematical object is of a certain type [] a type is well defined if we understand (or grasp to use a word favoured by Kreisel 1970) what it means to be an object of that type.

The explicit statement “a type is a concept” appears for instance (referring both to types of mathematical objects as well as to data types in computer science) in:

  • Arhtur Sale, Primitive data types, The Australian Computer Journal, Vol. 9, No. 2, July 1977 (pdf)

The explicit statement “a type is a ‘mathematical’ concept” appears on p. 6 of

  • Francis Sergeraert, The computability problem in algebraic topology, Advances in mathematics 104, 1-29, 1994 pdf

An explicit suggestion that “type” in homotopy type theory should be read as a “concept” is in

Revised on March 4, 2015 22:00:20 by Urs Schreiber (195.113.30.252)