“Carnap” and “the Polish logician”

InThe Many Faces of Realism and elsewhere, Hilary Putnam has presented an argument for the conclusion that there is no fact of the matter as to how many objects there are. In brief: “Carnap” says that a certain imaginary world contains three objects, ×1, ×2, and ×3. The “Polish logician” says that this same world must contain four other objects (×1 + ×2, ×1 + ×2 + ×3, etc.). Putnam maintains that there can be no fact of the matter as to whether the imaginary world contains three or seven objects. I examine Putnam’s argument and find it, at bottom, unintelligible.