On Hegel's Logic

Addendum: one must not be mislead by the apparent negative attitude to mathematics expressed in certain passages Hegel's published works. The following extract from his letters (see Wallace's translation of the Encyclopedia Logic xiv-xv) is very revealing:

'I am a schoolmaster who has to teach philosophy, who, possibly for that reason, believes that philosophy like geometry is teachable, and must no less than geometry have a regular structure. But again, a knowledge of the facts in geometry and philosophy is one thing, and the mathematical or philosophical talent which procreates and discovers is another : my province is to discover that scientific form, or to aid in the formation of it'.
' You know that I have had too much to do not merely with ancient literature, but even with mathematics, latterly with the higher analysis, differential calculus, chemistry, to let myself be taken in by the humbug of Naturphilosophie, philosophising without knowledge of fact and by mere force of imagination, and treating mere fancies, even imbecile fancies, as Ideas.'

It is easy to see Hegel's logic as a kind of romantic epic panlogist retelling of Kant's critical idealism with a half-veiled current of mystical enthusiasm characteristic of his times. Arguably, there is little in Hegel's logic that cannot be found by a careful and attentive reason of Kant's Kritik der Reinen Vernunft (and this if we ignore the non-negligeable input for Proclus).  However, we would like to argue that:

1. There are actually strong connections between Hegel's project and the subsequent logicism (and anti-psychologism) of Lotze and Frege (cf. how Hegel stresses the Begriff is not psychological).  Somebody noticed the Hegel-Gödel-Günther connection.

2. That Hegel's logic can be seen as an anticipation and quite thorough alternative realization of Husserl's phenomenological project and also as asking contemporarily pertinent questions in the philosophy of logic, specially with regards to relevance logic, substructural logic, etc.

3. Hegel's logic as very interesting connections to modern category theory and categorical logic and its interdisciplinary applications. His critique of formalism is also fertile from this perspective.

4. There are many interesting connections to be explored with ancient philosophy, with such diametrically opposite philosophers as Sextus Empiricus and Proclus.

5. Together with Leibniz, Hegel seems to have anticipated the idea of computation in nature. 

The section on Essence in Hegel's logic details classical metaphysics, Kantian metaphysics and finally the idealism of his contemporaries (Substance) whose shortcomings are finally overcome with the transition to Notion. We should compare the treatment of  Thing there to that in the Phenomenology of Spirit.

In the section on Being, in the parts involving the one and the other, limit, the infinite, the one, and the successive transitions involved - all this suggests a connection to a universal construction in mathematics: that of a completion. We have a finite structure which is limited, incomplete in some determinate way. For instance a field may not be algebraically closed. Something is missing, something extrinsic that needs to become intrinsic (cf. the process of finite algebraic extensions, adjoining roots, etc.).  Yet the finite completion, the finite adjoining of the missing aspects generates in turn its own incompleteness at the next level, and so on. This situation is only overcome by assimilation, by incorporation of this extension process, this outflow, into the structure itself - now invariant under this finite 'passing the limit'. This 'incorporation' is itself a kind of  higher 'limit', a limit of limits, so to speak (which has a universal property, or a minimality property). Embodiments of this process and structure are not only found in algebra but in essential ways also in measure theory (the Borel $\sigma$-algebra, regular and complete measures) and analysis (completion and approximation theorems). Recall how Hegel states in the Encyclopedia that the determinations of Being are exterior to each other, their process is the passing into another.

The passage to quantity: we compare this to the emergence of measures. Consider the proof of Urysohn's lemma. The main ingredient is the fact that in a locally compact Hausdorff space $X$ if we have a compact set $K$ contained in an open set $U$ then we can find an open set $V$ such that $K \subset V \subset \overline{V} \subset U$. This expresses $K$'s overcoming of its limitation relative to $U$. In the proof of Urysohn's lemma this process of overcoming the limit is multiplied to countably infinite sets indexed by the rationals and then completed into a continuous function, into continuous quantity. Somehow this relationship to the real numbers is implicit in the abstract concept of a locally compact Hausdorff space.  We should study situations in which valuations, measures, uniparametric semigroups of automorphisms, emerge.  What is the meaning of continuous functions with compact support ? And of Borel measures emerging from bounded positive linear functionals on spaces of such functions ?  Also investigate the passage of quantity again into quality.

Interesting concepts are those of extension (of a functional), restriction,  Riesz representation via a measure and the maximum modulus type result (maximum only at the boundary).

Note the distinction between pure quantity and determinate quantity.  Pure quantity or indifferent quantity (apeiron) is clearly topological, corresponding to what is implicitly determined by a variable, a sequence, a net.  See our paper on the modern incarnations of the Aristotelian concepts of topos and continuity for a detailed discussion. Attraction is aggregation, cohesion, expressed by the open sets of a topology.  Repulsion is discreteness . This has been treated in our note on higher topos theory (in connections to Lawvere and Schreiber). The duality and mutual transition between the continuous and discrete is of utmost interest (geometric structures give rise to algebraic structures, these in turn have geometric realizations).  Intensive and extensive quality is nothing but the concept of sheaf. The treatment of measure recalls Morse theory and singularity theory, moduli spaces; also the imposition of growth conditions in analysis.  But in the initial section of the Greater Logic on measure there is a suggestion that measure is a kind of conservation law; that which remains invariant under actual or possible or conceivable change and variation. Measure is clearly related to mathematical physics; Hegel states that a measure now is not pure quantity but quantity in relationship to something exterior to itself. Hegel also raises the question of natural vs. artificial units of measure. He introduces gradualness (i.e. continuous change in quantity with surprising jumps in quality) and mentions the Sorites paradox. Hegel's interpretation is perhaps not at all incompatible with our own interpretation in terms of locally constant sheaves.  Local homogeneity and uniformity is reconciled with global non-triviality. Hegel's knotted line = constructible sheaf, the 'knots' correspond to the stratification (cf. singular points of a variety, bifurcation set, etc.).

The section on measure seems to echo some of our ideas on general systems theory concerning how systems combine into larger systems or split again, etc. Hegel uses chemistry and thermodynamics. It also suggests the idea of program or function which can be integrated with different inputs producing different behaviours yet still maintaining its essence. Note also that combination and mixture features in Aristotle's Topics (much of the discussion in Measure should be studied in conjunction with Aristotle's Physics).

The section of Essence is related to fundamental dualities in mathematics (equivalence between distinct categories) as we have seen before. Essence is captured by adjunctions between categories.  From one category we are lead naturally to ask about its correspondent in the other (cf. the inverse Galois problem). In algebraic topology this illustrates the mutual dependency and subsumption of quality (topological space) and quantity (for instance, abelian groups). 

But first of all, the homotopy type theoretic interpretation of an identity type $t =_A t$ as a closed path expresses admirably reflection into self by reflection into another. The emergence of difference is expressed by the monodromy of a closed path.  Hegel's observations on the law of the excluded middle suggests connections to intuitionism and the types as spaces paradigm (and the topological nature of genera). Grund is to be seen as the general idea of a deductive system.  In a category the determinations of an object spring from its relationship to others (the Thing).  For instance, being a terminal object.

It might be interesting to interpret the section on Appearance (phenomenon) in relationship to the duality between theory and model (in the sense of logic).  But Hegel focused on 'force',  on the interior vs. exterior relation.  Effective Reality might be expressed in Goguen's institutions, adjunctions between theories and models (or as in Goguen's approach to algebraic theories), that is the unification of both points of view and the incorporation of their mutual determination.  Cf. the duality and circle between specification and verification. Further development of Effective Reality, reciprocal action, can be seen as involving concurrency (in the sense of theoretical computer science) and considerations on modality (cf. Hegel's the truth of necessity is freedom).

Notion is related to group actions (or model theoretic generalizations). The category of group representations represents the manifold (in the sense of plural) but united realizations of the notion.  Principle bundle $PG$ (the universal), a cocycle (the particular), the associated bundle (individual). But of course a central place in the Notion will be played by categorical logic. The transitions in subjective notion to objective notion are reflected in the progress of internalization of proof as we outlined in a previous draft. Also at the level of objective notion systems theory comes in as well as a the duality between theory and model.

Notion is not a theory of mind but the necessary template for a theory of mind, upon which a theory of mind must depend.  Our post on 'Internalizing Tarski' may express some of the final higher stages of the unfolding of Notion, related to Knowledge in the aspect of limitation vs. infinite and reflection-into-self.

Hegel's Logic is to be viewed as the realization of the project of Proclus' metaphysics, psychology and epistemology: the unveiling of the  pure  universal essences implicit in mathematics but which in themselves transcend mathematics: the logoi.

What is Hegel's view on the operations on quantity, and their abstract properties, in particular distributivity and linearity ? In the Encyclopedia some considerations are given on arithmetical operations and the logical progression from succession, counting, addition, multiplication to exponentiation.  For instance multiplication is seen as a counting of multiplicities as units.  It would be interesting to study from a Hegelian perspective the passage from an ordinary category to an additive category to an abelian category which explodes in a richness of properties. The abelian group structure on morphisms is the introduction of quantity or measure in a previous abstract discrete set.  But the $\mathbf{0}$ object is interesting. In general for pointed categories the zero object is a synthesis between initial and terminal object. Indeed this follows from the general setting of requiring the hom-sets to be pointed sets. This entails the weakest form of coherence: there is a least one morphism connected any two objects in the category.  This can follow from postulating the existence of a morphism $\mathbf{1} \rightarrow \mathbf{0}$.  Note the automorphic simplicity of terminal and initial objects: $hom(\mathbf{0}, \mathbf{0}) = \{id_{\mathbf{0}}\}$, etc. Thus if there is a morphism of the initial object into the terminal object then the two are isomorphic. We must examine carefully the construction of quotient categories, in particular via multiplicative systems. For this is a abstraction of the definition of rational numbers. Find parallels between Hegel and the transition between an abelian category  $\mathcal{A}$ to the category of complexes $\mathbf{C}(\mathcal{A})$ to the homotopy category $\mathbf{K}(\mathcal{A})$ to the derived category $\mathbf{D}(\mathcal{A})$ where with the infinite regress of a complex is in a way overcome through a return to self via the triangulated category structure.