Nominalism and Abstract Objects

Core Idea

Nominalism denies the existence of abstract objects such as numbers, properties, and propositions, maintaining that only concrete physical particulars exist. This position offers an ontologically parsimonious metaphysics but faces challenges explaining how we refer to and reason about abstract entities. Nominalists develop various strategies to reconstruct our discourse about abstractions without positing abstract objects themselves.

How It's Best Learned

Study concrete nominalist proposals (field nominalism, structural nominalism about mathematics) and evaluate their success in handling paradigm abstract entities like numbers and mathematical objects.

Common Misconceptions

Confusing nominalism with strict empiricism or phenomenalism. Thinking nominalism requires denying all entities except observable concrete objects.

Explainer

From your background on abstract objects, you're already acquainted with entities like numbers, properties, sets, and propositions — things that seem not to exist at any place or time, don't enter causal relations, and cannot be perceived, yet seem to be what mathematical and logical claims are about. Platonism (or realism about abstract objects) holds that these entities genuinely exist, mind-independently, in an abstract realm. Nominalism is the denial of this: only concrete, particular, spatiotemporally located things exist. Abstract objects, on the nominalist view, are a philosophical illusion — a category mistake generated by taking certain ways of talking too literally.

The nominalist's primary motivation is ontological parsimony — the principle that we should not multiply entities beyond necessity (Ockham's razor applied to ontology). If we can give a complete account of everything that needs explaining — including mathematics, science, and ordinary language — without positing an abstract realm, we should not posit one. But the challenge is immediate: mathematical discourse seems to work. "7 is prime" appears to be true. "There are infinitely many primes" is demonstrably correct. What makes these statements true, if not numbers that actually exist? The Platonist has a clean answer. The nominalist must do more work.

Various nominalist strategies have been developed. Fictionalism (associated with Hartry Field) says mathematical claims are like claims within a fiction: "7 is prime" is true-in-the-mathematical-fiction, as "Sherlock Holmes lives in Baker Street" is true-in-Conan-Doyle's-fiction, without there being a real Baker Street resident. Mathematical language is indispensable for science as a useful representational tool, but we need not believe in the abstract objects it seems to describe. Structuralism shifts attention from objects to structures: mathematics is about structural patterns, not specific entities — what matters is the relational form, not whether any particular things play the roles. Nominalization programs (Field's most ambitious project) attempt to paraphrase scientific theories so that apparent reference to abstract objects disappears, leaving only claims about concrete physical things.

Each strategy trades one set of problems for another. Fictionalism must explain why a fiction is reliable enough to do scientific work. Structuralism struggles to say what structures are without reintroducing abstract objects. Nominalization programs have proved difficult to execute fully. But nominalism also poses a challenge back to Platonism: if numbers exist outside space and time, causally inert and imperceptible, how do we come to know truths about them? This is the epistemological objection to Platonism (Benacerraf's challenge), and it is part of what keeps the debate live. Nominalism's appeal is not merely parsimony but a commitment to keeping our ontology answerable to our causal and epistemic situation in the world.