Abstract and Concrete
What do types, sets, universals, increases, theorems, species and governments have in common that distinguishes them from sticks, stones, mountains, molecules and cities? It's not that only the latter are causally efficacious: on many accounts (e.g. those of Lewis and Kim), events -- the paradigm examples of causal efficacy -- are sets; and why shouldn't one say that if a thing's being charged produces an effect, Charge (the universal) is just as much responsible for this as the thing itself? It's also not that only the latter are located in space or time: impure sets, species, Aristotelian universals and governments arguably are spatiotemporally located as well. And by the Helen Cartwright Theorem Theorem, theorems are sometimes written on blackboards. Indeed, I'm not sure whether anything at all clearly fails to be located in time, unless we require that something located in time must undergo intrinsic change or have a beginning or an end, which sounds ad hoc. Without such restrictions, I can't see a reason to deny that e.g. numbers exist at every time. (Oddly, I'm less inclined to say that numbers exist everywhere. But I might get used to it.)
Here is my suggestion: what distinguishes abstracta from concreta is that only the former are parts of the world (or perhaps, more generally, parts of some world). Every (actual) stick, stone, mountain, molecule and city is a part of our world. But no part of our world is the book type called "Philosophical Investigations" (what about the fusion of all tokens? I'd say that if this fusion is the book type, then the book type isn't abstract in any intuitive sense); no part of our world is a set (what about the empty set in Lewis' set theory? Again, it isn't clearly abstract); no part of our world is a universal, or an increase, or a theorem, or a species, or a government (unless these are fusions of individuals).
To make this work, the world has to contain all and only concrete things as parts. If I simply say that by 'the world' I mean the fusion of all concrete things, my account is empty. So I have to say something else. Perhaps something like: the world is the fusion of all microphysial things. At least for a reductive substance physicalist this yields the correct results. It might even work for substance dualists if they are prepared to call the irreducibly nonphysical things 'abstract'. It definitely won't work for people who believe that even sticks and stones and mountains are not ultimately fusions of microphysical things. Unfortunately there are a lot of those people. Perhaps I must rest content with explicating 'abstract' in my theory of everything.
On the present suggestion I could also just say that something is concrete iff it is a fusion of microphysical things. I prefer the roundabout way involving worlds because I want to leave room for some other way to characterize the world.
An interesting alternative (that might also work for worlds with completely different physics) is to define worlds somehow along Lewis' lines, as maximal fusions of spatiotemporally or quasi-spatiotemporally related things, where things are spatiotemporally related if they are located at some distance from each other. Unfortunately, I find this suggestion rather unclear. Moreover, if my singleton is located exactly where I am, and if the Helen Cartwright Theorem Theorem is located on the blackboard then they will end up being parts of the world.
Why do I care? I don't think the concepts 'abstract' and 'concrete' are particularly useful or important. But it seems to me that our classifications aren't too arbitrary, so conceptual analysis should be able to find some principles behind the classifications.