Hegel claims to have logically derived his philosophy;
replaced a quote that was out of context with this one
<< Hegel's Science of Logic, introduction §62
"In the Phenomenology of Mind I have expounded an example of this method in application to a more concrete object, namely to consciousness. Here we are dealing with forms of consciousness each of which in realising itself at the same time resolves itself, has for its result its own negation — and so passes into a higher form . 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."
He continues to say in §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.">>
Hegelian philosophy has been partially formalized in mathematics, particularly in adjoint-modal-type-theory where one finds deep correspondence between the two. In my perspective, it receives further justification from formalizations of math in-itself in texts like type theory in type theory (just serious math there), which can be interpreted as the self-construing method Hegel talks about, in that math doesn't need mathematicians to operate( "circular" foundations). Since Hegel worked before the existence of advanced math and considered math as a subordinate science, am interested in knowing whether anyone truly understands his dialectic enough to independently apply it (if so, how?). Preferably, a first-person point-of-view will suffice.
EDIT: To be more specific, am asking for the way Hegel understood dialectics and utilised it consistently because I believe it provides an alternative "pre-mathematical" way of doing math, without need for random ideas or symbols. Not how it corresponds to math.