I don't think we really can. The categories of concrete and abstract objects are supposed to carve reality at its joins: I see a chair, I prove a theorem. You can't really do this sort of analysis without reference to the chairs and the theorems, and if you do make those references, you must have already settled the question of whether a chair is concrete, and a fortiori whether concrete objects exist. The alternative, studying concepts that were originally intended to carve reality at its joins without intending to do so yourself, has historically been unproductive, except to some extent in math.