Additions to 'Hegel and modern topology'

The something and the other. In the Science of Logic Hegel thinks of the idea of two things bearing the same relation to each other and indistinguishable in any other way. Now a good illustration of this situation in found in the two possible orientations of a vector space. Each bears the same relation to the other and there is absolutely no way to uniquely identify one of them in distinction to the other.  In the same place Hegel talks about the meaning of proper names and states that they have none !

Hegel's strategy. All of Hegel is based on the dynamics and structure of consciousness as it reveals itself to itself. But equally important it is all based on positing this structure of consciousness  to be essentially objective and not subjective.  Thus 'being a subject' is seen as a stage in the development of the object while mere  'subjectivity' is seen as a failure and partiality of objective consciousness in not living up to its full objectivity or actuality (or knowledge and realization thereof).

Thus ideality and infinity are the key structure of an object which has developed to a point of being a subject. 

Finitude and limit:  a closed set.  Yet the boundary also defines what the set is not (cf. discussion on Heyting algebra of open sets, etc.). Thus the boundary contains implicitly its own negation.

Limitation : an open set (also analytic continuation). Limit outside itself.

The ought... manifestation of the germ of a space through a given open set representation of the equivalence class. Any particular one is insufficient and can be replaced by another which is also insufficient.

False infinity: taking simply the set (or diagram) of such  open set representatives.  True infinity taking the limit (equivalence class) or limit (in the categorical sense)..

Degrees of interpenetration and transparency between the individual and the universal (and between self-consciousness and essence) in the Phenomenology of Spirit.  From the rudimentary form in the ethical life to: the universal is in the individual, the individual in the universal and the universal is in the relation between individuals and the relation between individuals is in the universal (mutual confession and forgiveness). 

Milne's notes on Class Field Theory has a quote of Chevalley (1940) which seems couched in Hegelian terms: the Abelian extensions of a number field are determined by the number field itself, the number field  'contains within itself the elements of its own transcendence'.

And yet : and yet it is really basic to use ontology and logic inspired by (or abstracted from) biology and psychology  - that is, a kind of general systems theory and universal logic based fundamentally on biology and consciousness. The fact that mathematics exhibits philosophical analogy with biology suggests that biological ontology and logic is fundamental. René Thom already did this by suggesting that concepts have analogies with living organisms.

'Pure' category theory is very poor in results, like Hegel's poor, empty, abstract universal.  But once we consider the particular, like Abelian or Monoidal categories, things become very rich.