>There's only one: Two entities are equal if nothing can distinguish them. This is the “identity of indiscernibles”.
This doesn't exist. If you can distinguish past and present, you can distinguish anything by time. An object in one moment is not identical to an object in another moment because the moment changed? No. You have to decide on what invariants you care about to have a consistent notion of identity. Yours is inadequate to do anything, as you demand the whole universe be invariant in all aspects; such a thing is trivial and vacuous. And it surely is not the structure encoded in any usage of the term "identity", as that word has non-trivial structure.
> If you can distinguish past and present, you can distinguish anything by time.
That's the thing: Objects exist in time. Values don't. Values exist in the language's semantics, which is a timeless mathematical object. Does it make sense to ask when the number 2 suddenly came into existence?
> An object in one moment is not identical to an object in another moment because the moment changed? No. You have to decide on what invariants you care about to have a consistent notion of identity.
You're confusing “identity of indiscernibles” with “indiscernibility of identicals”.
And, obviously, you can't use the temporal properties of objects to determine whether their atemporal identities are equal.
> Yours is inadequate to do anything, as you demand the whole universe be invariant in all aspects; such a thing is trivial and vacuous.
Nope. It just requires you to distinguish between things that exist in time and things that exist independently of time.
This doesn't exist. If you can distinguish past and present, you can distinguish anything by time. An object in one moment is not identical to an object in another moment because the moment changed? No. You have to decide on what invariants you care about to have a consistent notion of identity. Yours is inadequate to do anything, as you demand the whole universe be invariant in all aspects; such a thing is trivial and vacuous. And it surely is not the structure encoded in any usage of the term "identity", as that word has non-trivial structure.