How does nominalism differ from logical realism?

In the philosophy of logic, the distinction between nominalism and logical realism centers on the question of ontological commitment—specifically, what entities a theory must acknowledge as existing.1

Nominalism is the doctrine that an ontology consisting solely of individuals (particulars) is sufficient. Modern nominalists reject the existence of abstract entities, such as universals, sets, numbers, and properties, often invoking Ockham’s razor to argue against multiplying entities beyond necessity. Because these abstracta lack spatiotemporal or causal relations to humans, nominalists face the challenge of explaining how we can refer to or know truths about them without admitting them into our ontology.12

Logical realism is the opposing view, which posits that a theory is committed to the existence of the entities that serve as the values of its bound variables. While nominalism restricts this range to individuals, realism accepts that higher-order logic and mathematical theories require a commitment to abstract entities like sets, properties, and relations. Realists argue that these entities are indispensable for accounting for the facts of mathematics and scientific theories.12

Ultimately, the debate persists because while nominalism is ontologically economical, it struggles to provide a foundation for mathematics that avoids reference to abstracta.21