In Hegel's philosophy, «spirit» is the key concept, which relates to universal knowledge. It is the objective spirit. Under the subjective spirit is to be understood the individual consciousness.
For knowledge there must be both a knower and a known. The knower is the subject, the known the object.
A is the subject; Not-A is the object.
Spirit A itself knows itself (for this and because of this, it exists, materializing the object, seeing the world). Spirit opposes to itself matter, truth, Not-A in order to know itself. Spirit is free, therefore it does what it wants. To know itself, it is necessary for Spirit to create the object, truth, Not-A, that is, Not-Spirit. Since the goal of Spirit is to know itself, it should return from the matter to itself.
In this way, to find itself it must negate the matter, that is, it must find Not-Not-A.
Not-Not-A (Not-Not-Spirit) is itself A, but A having known itself, having found itself, and having found itself in its having negated itself. This is the foundational, albeit a bit unexpected, phrase in Hegel's system — the law of the negation of negation (Not-Not-A).
Simplifying: I cannot see myself without that, which would show me myself. I take the mirror. I negate myself in the mirror (yes, the mirror is a Not-I), but the mirror negates itself (since the Not-I is the face in the mirror, it is the Not-mirror). In this way, I see myself.
Spirit knows itself through people (history, art, politics), since people are both nature and Spirit. When a person (in the given instance, Hegel) knows this law (negation of negation), Spirit itself knows itself through this law.
Spirit knows itself through history. The varieties of culture, historical processes are that in which Spirit, roughly speaking, «tries itself out».
Perestroika was the negation of the Soviet Union, (A is the Soviet Union, but now Not-A has been built up), but then everything returns in a different form, one gets Not-Not-A. Or the great French revolution and Napoleon's coup of 18 Brumaire.
A person lives in Oslo, but he has to see another world and return, so as not to die of depression. Therefore the person flies to Melbourne (Not-Oslo), returns, and looks at everything differently, although it is still the same Oslo. Leaving Melbourne, the person negates the city (Not-Not-Oslo). The person returns to Oslo; but it is not the same Oslo, in which he had lived before his trip.
Maybe it's for this reason that friendship after a fight is so strong. Or that teenagers, rejecting their parents, will grow up and become like them.
The problem is that with this schema one can explain everything one wants, and to such schemes always there arises suspicion.