Here are two translations of Brahma-Sutra 2.2.25:
And (a permanent soul has to be admitted) because of the fact of remembrance (ie., memory).
(Translated by Swami Gambhirananda here)
And on account of memory (the permanency of the experiencer has to be recognized).
(Translated by Swami Vireshwarananda here)
Commentary by Swami Vireshwarananda:
A further refutation of the momentariness of things is given here. If everything is momentary, the experiencer or enjoyer of something must also be momentary. But that the enjoyer is not momentary and abides longer is realized from the fact that people have the memory of past experiences. Memory is possible only in a person who has previously experienced it, for what is experienced by one man is not remembered by another. So the agent of the experience and the remembrance being the same, he is connected with at least two moments—which refutes the doctrine of momentariness.
Continuity of memory is used as an argument for the evidence of a permanent soul.
What happens to a person who loses his memory due to amnesia, dementia or brain injury? Does that mean that he has lost his permanent soul? If not, how can such a condition be justified?
An electronic device such as a smartphone can remember info and keep memory, without having a permanent soul. So, how can this argument be justified in this case? Does proof of memory mean that the smartphone has a permanent soul?