Nominalism denies that universals exist. There are red things; there is no redness in addition to them. General words are devices for grouping, not names of anything.

The motivation

Parsimony. If you can explain why we classify things together without positing an extra category of entity, the extra entity is unearned. William of Ockham pressed exactly this: do not multiply entities beyond necessity.

There is also a locational problem for the alternative. If redness is a real thing, it is wholly present in every red object at once, which is not how anything else behaves.

The versions

Predicate nominalism. Things are red because "red" applies to them. The obvious objection is that this reverses the explanation — the word applies because they are red, not the other way round.

Resemblance nominalism. Things are red because they resemble each other. The objection, pressed since antiquity, is that resemblance looks like a universal itself, and one that every pair of resembling things shares.

Trope theory. There are no shared universals, but properties are particular: this tomato has its own redness, distinct from that one's, and the two resemble without being identical. This preserves properties while keeping everything particular, which is why it is currently the most defended version.

Abelard's middle route

Peter Abelard argued that what individuals share is a status rather than a thing — the way they are, which the mind grasps as a concept. It is neither a free-standing entity nor an arbitrary noise, and it dominated twelfth-century discussion.

What hangs on it

Whether scientific laws describe real structure or summarize how particulars have behaved. On a strict nominalism a law is a very good summary, which brings the problem of induction directly into view. See realism.