Identity is the concept of being exactly the same thing. If you create two objects and fill them with the same information, they’re not identical, because they’re still two separate objects.

Equivalence is the concept of meaning or containing the same thing. Those two objects are equivalent because they contain the same information, even though they aren’t identical. OTOH, if you change the contents of an object, its “past self” is still identical to its “future self”, but not equivalent. In C#, the Equals method tests equivalence, while the ReferenceEquals method (or, usually, the == operator) tests identity. (Pascal extends this to types: two records or pointers that have the same definition are equivalent, but not identical, unless one of them is defined as an alias of the other. Some operations only require type equivalence, but some require type identity.)

Now, how does this apply to your theory?

If I put off a task until tomorrow because I expect I’ll be more willing to do it tomorrow than I am today, that doesn’t mean I think my identity will change. I’ll still be the same person tomorrow that I am today; as a sane individual, I know that I’ve always been the same person and always will be. But who I am today is not equivalent to who I’ll be tomorrow. In fact, who I am now isn’t equivalent to who I was five minutes ago. Everyone is always changing because of their experiences, and I don’t think that should be news to anyone.

I’ve gotten to the point where I often assume that when I go to sleep someone else is going to wake up and continue in my place. And when I wake up in the morning I wonder whether my memories from the past day were implanted or real.

So why do I plan for the future at all? Only because it’s enjoyable to me in the present. Working towards a goal has always been one of the more delightful feelings for me. Whether or not I’m actually there to finish it is often irrelevant to me. As long as I can rest assured it’ll eventually get finished by someone.

Admittedly, our language is based around the notion of identity continuity, that a snapshot of me now is at its core the same as me at some other point in time. So it’s sometimes confusing for me to switch back and forth between my internal language and the society’s external language. But I think there’s an element of truth in both views… ie, I share a lot more in common with my past or future selves than I do with a random other person… but we don’t share everything. And so some of the time it makes sense to call us one and other times it makes more sense to call us many.