What is the nature of 'dialectics' according to Plato, Plotinus and (in particular in the context of the Parmenides) ? How did it relate to other forms of ancient logic such as Stoic logic and Aristotle's Topics ? There is also the following interpretation of some aspects of dialectics. Given a logical system L we can study different axiomatic theories in L and how they relate to each other (for instance, are they mutually consistent) and whether they are in themselves consistent or incomplete. A major paradigm is starting from a hypothesis H and arriving at a contradiction or staring from the negation of H and arriving at a contradiction and taking this to be (as in classical logic) a proof of H.
And we can study different logical systems and their relationship as well as the relationship of their theories. However all such logical systems use and epistemically presuppose recursion theory and arithmetic - and along with deduction exhibit some of the order-characteristic of temporality and also it seems cyclic temporality. Also the different theories can be projected outwards in the form of concrete models, specially geometric models. Such models in turn can lead to other discoveries. And models can be reflected in other formal systems (see our theory of reflection) and concepts such as categoricity come into play, which must not be misunderstood in some kind of absolutist sense. And from a Fregean point of view we can consider the theory of the informal elucidation of the primitive terms and axioms. It is not at all clear how considering different theories (hypothesis) in logical systems can lead to disclosure of the primitive terms - but we must consider first of all the problem of the meaning of meaning, proposition, truth, of logical connectives and quantifiers as well as the concept of deduction and inference (this is already a self-reflection of logic), etc. The quest for primitive terms must involve the theory of definition. All these primitive terms, definitions, axioms in logical systems and theories concern the foundations of all possible knowledge and thought.
Mathematical logic and in particular formal theories of arithmetic and recursion must be seen as a reflection-into-self of logic, recursion theory and arithmetic itself. Gödel's incompleteness theorems are a unique example of reflection-into-self followed by reversion. Arithmetic projects itself outwards, reflects on the insufficiency of this projection and at the same time mediated by the projection cognizes a truth that leads it back to itself, the fact that the undecidable sentence is in fact true. Gödel's famous result gives us noetic knowledge.
Thus the projection into formal systems bound up absolutely with recursion theory and arithmetic (and hence combinatorics and graph theory and finitary set theory) is part of the cyclic process of investigation of the primitive concepts of thought (which appear to be known and clear but actually are not), a process which unfolds through formal projection and clash and comparison with other projections and hopefully leads to self-reversion.
What is the relationship between more purely 'logical' primitive terms and others which seem to relate more to ontology, metaphysics, philosophy of mind, physics, etc. ? In what sense the logical more fundamental (the old question of psychologism, etc) ? These concepts must be treated in the same way as logical and mathematical ones (see the quote from Leibniz under the blog header).
We can suspect a term is primitive if it does not seem to be easily definable. Can we define the logical connective 'and' ? We could group it together with other connectives and specify it by its truth-value properties in inferences, but in doing so we are already making use of it. For example, saying that 'A and B' is true iff A is true and B is true.
The biggest error of western philosophy was abandoning the neoplatonic (and augustinian) concept of the soul as an autonomous immaterial substance with potentially unlimited epistemic and ontological capabilities, and of taking the 'self' to be merely peripheral and mixed aspects of ordinary somatically and sensually conditioned psychological experience (this is the target of the original buddhist theory of anatta) or having a 'depth psychology' and elaborating a theory of the 'unsconscious' or 'subonscious' based merely on inferior aspects of the soul while totally ignoring the true spiritual depth which is both 'within and above' oneself.
The neoplatonic philosophy of mind and consciousness (through its theory of analogy and projection and reversion) allows us to reconcile logicism, realism and 'psychologism' and both species relativism and absolutism and both subjective idealism and natural science. Note that category theory besides being a rather specialized theory of relations is at the same time an interesting example of a theory of analogy and this is how it arose in the first place.
With regards to mathematics: how are we to understand why and how complex analysis and complex analytic geometry (for instance developed by the great geniuses Abel and Riemann) became so central in 19th century mathematics and beyond ? How is it connected to problems in number theory and physics (and the significance of the work of Grassmann and Clifford is yet to be fully explored) ? Hyperbolic geometry seems to be of immense philosophical interest, it perhaps represents the geometry of the soul or nous as opposed to the geometry of nature. The Beltrami surface gives an image of the 'inverted sphere' the return-to-self which is also the projection to infinity of the soul. Hyperbolic geometry expresses the consistency of infinite different possibilities (a point outside a given line has infinitely many lines going through it which do not intersect the given line).
The cause of the descent of the soul must be some kind of internal disorder and forgetfulness which, by means of the descent, is projected and given external manifestation intimately correlated with the soul's own inner activity, the goal being that the soul will recognize through the world and through this correlation the very internal disorder and forgetfulness it started with, but now known clearly as such and by this insight be lead to a spontaneous and total 'reversion'. So the descent of the soul is a fundamental 'mistake' and a 'fall' which at the same time is necessary to cure the internal 'mistake' that the soul was carrying within herself before the descent.
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