An illusion is an experience of something that doesn't really happen. When you have a fever, you may experience a feeling of cold, but it is an illusion, you're actually very hot (it can be argued that it's not an illusion, the feeling of cold is just misinterpreted, but that's beside the point).

This experience of continuity is built from memory, I think I am the same person I was 2 minutes ago because I can remember what it was like to be that person. If I take a roofie or drink too much or sleep walk, I will feel that the things that I did happened to another person, because I don't remember them at all. Likewise, the experience of continuity is often lost - I generally experience sleep as "falling asleep" -> "waking up", even if an EEG would show that I am not even entirely losing consciousness for this time.

> How is everything different? On the contrary, I'd say that actually most things seem to change only marginally and as time periods approach the infinitesimal, so do the changes. And I don't understand what you mean by clear boundaries. Why would there be any? Isn't lacking clear boundaries what actually makes it continuous.

Yeah, that phrase was really bad, I didn't get my point across at all. What I wanted to say is that in physics you actually look at every moment in time individually, even if functions of time are continuous; and you look at every piece of every object individually - any object can be equivalently represented as multiple objects held together by strings (or springs). There is no need or use for object identity across time.

> Let me ask you, why do you care about your future 'self'? That is, why do you not just maximize _your_ (current) pleasure, rather than caring about the experience of someone that you seem to think isn't actually yourself. At the very least, it seems like from your point of view there is no reason to place more importance on your future self than any other future person.

Because in this view, a single experience is extremely fleeting, far too little to gain any great pleasure. Also, my future "self" is the person I can empathize most with in the world, so I feel it is natural to want to help him. Not to mention, for any desire I have, he is the most likely to actually see it through.

Also, we often sacrifice our happiness for other people, often ones that we have much less reason to empathize with than our future selves. Not to mention, of course I occasionally sacrifice my future self's happiness for my current one - such as eating an extra piece of cake, or drinking another glass. But there has to be some limit, especially when I have such an intimate idea of how much the future person will suffer.

> 'time-stable' identity doesn't mean that it must be forever frozen, but rather that it's continuous in the time dimension and happens slowly enough that the subject can experience this.

Yes, I think we mean the same thing by time-stable. It doesn't have to be frozen, but it must be well defined for more than one moment of time (it's pretty trivial to define an object at time T, even the Atlantic Ocean, but it's much harder to define an object that remains itself over some interval T0-T1).

> You can replace parts of the 'soul material', without identity being lost as long.

This is the part I don't understand. In the materialistic computational theory of the mind, it's pretty clear why the mind doesn't change just because cells die and are replaced, or just because every atom in a body is replaced. But if each atom carries a part of the soul, I really don't understand how this works, or at least what it buys you in terms of identity or mind.