Matthieu Ricard, after saying this, as quoted in earlier post:
Is he speaking of a "stream of consciousness" or not?
No. He is speaking of the "conditioning of a stream of consciousness" :
"First of all, it’s important to understand that what’s called reincarnation in Buddhism has nothing to do with the transmigration of some “entity” like an autonomous “self”. It’s not a process of metempsychosis.
As long as one thinks in terms of entities rather than function and continuity of experience, it’s impossible to understand the Buddhist concept of rebirth. As it’s said, “There is no thread passing through the beads of the necklace of rebirths.”
Over successive rebirths, what is maintained is not the identity of a “person,” but the conditioning of a stream of consciousness. "goes on to say this:
"In conventional terms, we can talk about an “individual” consciousness, even if the individual doesn’t exist as an isolated entity. The fact that there’s no such discontinuous entity being transferred from one life to the next doesn’t mean that there can’t be a continuity of functioning and a particular history.
That the self has no true existence doesn’t prevent one particular stream of consciousness from having qualities that distinguish it from another stream. The fact that there’s no boat floating down the river doesn’t prevent the water from being full of mud, polluted by a paper factory, or clean and clear. The state of the river at any given moment is the result of its history. In the same way, an individual stream of consciousness is loaded with all the traces left on it by positive and negative thoughts, as well as by actions and words arising from those thoughts. What we’re trying to do by spiritual practice is to gradually purify the river.
The ultimate state of complete clarity is what we call spiritual realization. All the negative emotions, all the obscurations that render the underlying wisdom invisible, have then been dissolved. It’s not a question of annihilating the self, which has never really existed, but simply of uncovering its imposture. Indeed, if the self did have any intrinsic existence we’d never be able to bring it from existence into non-existence."