Why does mathematical nominalism reject the existence of numbers and sets?

Mathematical nominalism rejects the existence of numbers and sets primarily because it denies the existence of all abstract entities. Nominalists argue that because such objects are nonspatiotemporal and noncausal, they cannot interact with human beings, making it impossible to explain how we could refer to or acquire knowledge about them.1

Nominalists often invoke Ockham’s razor to support their position, arguing that we should not multiply entities beyond necessity. By rejecting abstracta, they aim for greater ontological economy.1

Because mathematical theories appear indispensable to science, nominalists must account for how these theories are used without committing to the existence of abstract objects. They approach this through various strategies:

• Paraphrase nominalism: Argues that mathematical sentences do not actually refer to objects but can be rephrased (e.g., as conditional statements) to avoid such claims.2
• Fictionalism: Maintains that mathematical sentences are not literally true but are useful "fictions" or stories.2
• Neo-Meinongianism: Claims that mathematical sentences can be literally true even if the objects they describe do not exist.2

Ultimately, nominalism stands in opposition to Platonism, which asserts that mathematical objects are real, objective, and independent of human thought.34