Identity Conditions
Sometimes we wonder whether some thing A is identical with some thing B: Is the man in the brown hat (the same as) my neighbour? Is the table in the mirror over there (the same as) the one here in front of me? Is the square root of 841 (equal to) 29?
What determines whether A really is identical with B? According to a view I find very irritating it's the identity conditions of A and B. The idea is that all things fall under kinds, and every kind comes with an associated identity condition. Different kinds may be associated with the same identity condition, but there is never more than one identity condition for a kind. So to find out whether A = B, we first have to find the relevant identity conditions of A and B. A good way to find those is to find the relevant kinds and look for their associated identity conditions. If the identity conditions differ, A is not identical with B. If they are the same, they tell us under which conditions A and B is identical, and we only have to find out whether these conditions obtain.
(Bob Hale and Crispin Wright base their solution to the Julius Caesar problem on this view: Hume's Principle gives the identity conditions for cardinals, but not for humans; therefore no cardinal is identical with any human.)
I would have thought that there are exactly two possibilities: either A = B or not. If A = B, we have only one thing, not two, and this one thing is identical to itself. Moreover, no special conditions have to obtain for anything to be identical with itself. In particular, the self-identity is not due to further facts about whatever kind the thing falls under. If, on the other hand, A is not identical with B then we have two things, and again no special conditions need to obtain or not to obtain for two things to be non-identical.
Of course it is not always trivial to find out whether A = B. And sometimes sortals and their associated identity conditions do play an important role in this process. But their role is not to determine whether the given objects A and B are identical. Rather, they are used to determine which objects we talk about in the first place. Roughly, the identity condition associated with a kind specifies the spatiotemporal and modal extension of instances of the kind. For instance, if I don't know what 'person' means, and you ask me, pointing at the man with the brown hat, whether this person is the same as the one we saw here yesterday, I can't answer your question because I don't know what things (A and B) you're talking about: Do you mean A = that man and B = the man we saw here yesterday? Or do you mean A = that timeslice of a man, and B = the timeslice we saw yesterday? Or do you mean A = the fusion of that man with all men larger than 1.80m, and B = the fusion of the man we saw yestarday with all men larger than 1.80m?
By modal extension I mean the fusion of all a thing's counterparts. I need to know the modal extension of things falling under 'person' to determine whether that person is the same as some possible person in another possible world. These questions are usually not put in terms of identity but in terms of a thing's nature or essence.
I would like to know if the irritating view about identity conditions is a consequence of endurantism and literal trans-world-identity. Certainly the alternative view is much easier to describe using perdurantism and counterpart theory, and those who defend the strange view, like Wiggins and Lowe, usually also defend endurantism. But that's not really conclusive evidence.
There's no way to divide a mereological atom. Hence 'the mereological atom covering that point in spacetime and no point in any other spacetime' precisely determines the spatiotemporal and modal extension of what you referred to (assuming you've managed to pick out a determinate point by 'that point' and that point is not empty and not covered by more than one mereological atom). Let's assume the atom in question is spatiotemporally extended over, say, 10 m^3 and 20 years. Still we might reasonably ask whether that large mereological atom here is the same as the one we saw yesterday. That's more or less exactly what van Inwagen thinks is going on when we ask if this man is the same as the man we saw yesterday. (Endurantists believe that men are temporal atoms, van Inwagen believes that they are also spatial atoms.) I don't see how it forces upon us the strange view about identity conditions mentioned above.
Incidentally, I would (still) like to know more about the question of extended mereological atoms. For example: can one plausibly accept spatiotemporally extended atoms but reject trans-world atoms? I don't think so.