“Of Personal Identity” is the final chapter of Book I of Hume’s seminal A Treatise of Human Understanding. In this chapter he critically examines the idea of ‘self’, especially the Cartesian notion of an indubitable and necessarily existing self. Hume applies the criteria he has been using throughout the work so far, in his critical analysis of ideas, that any idea must have somewhere in it a reference to a past sensory impression. But, our idea of self is not based on one, constant, continual impression, but is an idea of something that all of our impressions and ideas instead relate to. Accordingly, Hume casts the idea away as senseless. There is not one, stable, continuing self, but merely a bundle of sensations, impressions, memories, and so on.
Hume’s emphasis on the positive empirical contents pushes him into a view of the subject as flux, out of which emerges a fiction of self – in a manner akin to how the idea of causality was revealed to be mere convention. Strictly speaking, there are no selves, identical to themselves over time, or even identical to themselves at a single time. This radical idea was to have huge import long after Hume’s time.
However, this analysis being done, in the Appendix of the entire volume Hume suddenly realizes a problem. It seems he is unable to explain how a single ‘bundle’ of impressions and memories should hang together, and coordinate to create such a fiction. However, he is defiant against the Cartesians to the end: he is sure that some further principle could bring this all together, without having to return to the idea of some ineffable yet stable mental substance in which perceptions could inhere.
Read more about Hume’s identity woes here.