Comp and the capsule theory of memory (and "Memento") suggest that a "person" is a series of person-moments, each of which is considered to last somewhere around 1/10th of a second (it could be longer or shorter and the idea would still hold) and assumed be the same person due to being linked by memories. Generally this is considered to be because of physical continuity, but comp-style thought experiments, at least, can deconstruct that idea. The question is whether physical continuity has some bearing on identity, or is just incidental (i.e. nature hasn't found any other way to do it). The usual argument against the importance of physical continuity is that we replace our cells - even our brain cells, apparently - every few hours/days/years/whatever. And more specifically, the atoms involved in a thought might go on to do something else - take part in a different thought, form a memory, pay a visit to the big toe... they're constantly being moved around, even without being lost from the system.
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