On December18th I posted here a discussion of Galen Strawson's 2011 book, THE EVIDENT CONNEXION: HUME ON PERSONAL IDENTITY. I promised then that I would come back to the subject.
The very short version is this. Hume understands personal identity to depend on the mind. It can't depend on the brain, or the body generally, because for Hume our knowledge of enduring physical objects is quite limited -- we know bundles of sensations that act together with regularity.
So personal identity depends on the mind, and Humean philosophy needs an active (to some extent an enduring) mind to do the work of generating "fictions" (in some sense) about enduring physical objects and causal relations among them.
But when Hume tries to analyze the mind and what we know of it, it too rather crumbles in his hands. He finds that he could make sense of the knowing mind only if he believed either in a soul or in a strong knowable connection of cause and effect. Hume rejects them both. The second of those possibilities suggests a succession of mind states in which each would be the cause of the next one and the effect of the one before, in a strong metaphysical sense of "causation" -- again something Hume regards as unknowable. So he has no resources left for this project and admits defeat.
This gets us to William James. Despite James' use of the phrase "stream of consciousness," which everyone remembers largely because of its later adoption by literary critics, there was a sense of discontinuity to James' account of the mind. He saw consciousness, or thought, as a series of pulses, with each pulse remembering earlier pulses and claiming them as its own. So in each specious moment the self is creating itself anew by remembering certain earlier selves from the inside.
This could put James in much the same predicament as Hume. James believes in a self that does a lot of creative things, and these things require endurance over time (many spacious presents).
Strawson includes a compare-and-contrast exercise involving David Hume and William James. Can that self still exist if we think of it as a collection of ontologically separate pulses?
Well, yes. And James can say so with a consistency not available for Hume, because James has a resource Hume doesn't have. The brain. James begins his two-volume work with a detailed discussion of what was known in his day of the brain. Chapter 1 is merely a brief comment on the scope of psychology as a science. Chapters 2 and 3, each lengthy, each have the word "brain" in their title. With this start, James doesn't have Hume's difficulty with continuity because he assumes this particular physical fact as a datum.
The brain is the place where successive experiences, thoughts that are also the thinker, occur or reside. As Strawson puts it, "It's enough for him that short-lived selves, numerically distinct, arise successively from the same brain (from brain conditions that have considerable similarity from moment to moment, even as they change.)" Strawson can find "no grounds for an objection to the substance of [James'] proposal" regarding personal identity.
Later in life, James became more of a philosophical radical himself, and adopted, or at the least seriously entertained, a sort of neutral monism where everything might ontologically be "pure experience." That was a more Humean position, and might indeed be susceptible to the objections that Strawson says Hume leveled at himself.