Structural Nominalism

This paper defends Structural Nominalism, the view that abstract objects do not exist as mind-independent substances, but that real structural regularities instantiated in the physical world are sufficient to ground mathematical and logical necessity without appeal to a Platonic realm, a divine mind, or any transcendent ontological category. The paper argues, first, that immaterial objects fail a basic causal criterion of existence (Quine, 1948; Armstrong, 1978); second, that abstractions are functional shadows, real as states of physical cognitive systems, unreal as substances; third, that the apparent necessity of mathematics is explained by the stable structural regularities of physical reality, accessed through embodied cognitive simulation rather than rational intuition of a separate realm; and fourth, that this position transcends both classical Platonism and bare nominalism by integrating structuralism into a fully naturalist ontology (Shapiro, 1997; Resnik, 1997). The paper closes by arguing that Plato’s Cave, properly understood, is precisely inverted: the abstract realm, not the physical world, is the shadow.