Meinong's object theory suggests the possibility of making progress in a third alternative with respect to the long-standing apparently intractable collision in the metaphysics of Platonic realism versus nominalism. Meinong's own views on the existence of such abstract mathematical entities as numbers and geometrical figures are considered, and the possibility of treating relations in particular as nonexistent Meinongian intended objects is developed at length. Russell's argument that relations must exist as universals, even if qualities at first are not assumed to be existent, in order to do justice to the truth conditions of our predications of ordinarily properties to multiple ordinary objects, is also considered in this context as potentially undermining the Meinongian interpretation of relations as nonexistent. The effect of making relations nonexistent despite Russell's argument is to achieve a considerable ontic reduction in a logic's referential semantic domain, in accord with the methodological principle of Ockham's razor not to multiply entities beyond explanatory necessity.