Is our impression of continuity the source of continuity or is it the point of departure for an analysis of self-identity? What I mean by self-identity here is the feeling of the continuance of a self and not the empty expression implying that there is a self that is identical to itself.
First of all the analysis of personal identity into synchronic and diachronic may be unsustainable as it is obvious that traits and abilities require time for their expression. A closed and suspicious person will require less time to be disabused of a new acquaintance than an open and trusting one but both traits will require time to unfold. It is like the moving point of the cone of memory a la Bergson. Everything is focused at that point of entry into the plane of history but because everything that happens is poured into the cone of memory and alters its contents, it is clearly the case that we never step into the same river twice. There is then constant alteration but there is at the same time a style of alteration. To offer a planetary metaphor this style or interaction between the elements of the personality is like the gravity that keeps the ‘falling’ planets in the same relation to each other. So by the anatman doctrine of the Buddhists and the ‘bundle’ theory of Hume the self is a vacuous concept.
The wily advaitins expand the vocabulary of the problem and refer to that disputed entity as the jiva or individual person and reserve the appellation ‘self’ for the consciousness or awareness that takes different mental shapes but remains fundamentally the same through all its manifestations. The metaphor offered is that of ‘clay’ and various vessels that are made of clay.
This is a position that you cannot logically think your way into, the force of it must be felt through meditation or to put it into Platonic terms, it is a formal apprehension.
All in All
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