Plato's answer to the one and the many is that many beautiful things share in one Form of Beauty. The many are here and changing; the one is elsewhere and does not change.

Everything then depends on the word share. What is the relation between a beautiful face and Beauty itself?

Plato uses several words for it — methexis, participation; mimesis, imitation; parousia, presence — and in the Phaedo he is unusually candid, saying he does not insist on any particular account, only that it is by the beautiful that beautiful things are beautiful. That is a statement of the problem rather than a solution, and he knew it.

The objections Plato made himself

The Parmenides contains the most severe criticism of the theory of Forms ever written, and Plato wrote it.

Is the Form divided or whole in each? If each beautiful thing has a part of Beauty, then Beauty has parts and each thing has only a fragment of it, which is not what was wanted. If the whole Form is in each, then one thing is entirely present in many separate places at once.

The third man. If a beautiful thing and Beauty are both beautiful, then by the reasoning that produced the Form in the first place there must be a further Form in virtue of which both are beautiful. And then another. The explanation regresses without end, and it does so by its own logic.

This is the objection resemblance cannot survive. If participation is likeness, then likeness is symmetric — the Form is like the particular exactly as much as the particular is like the Form — and the regress starts immediately.

Forms of undignified things. Is there a Form of hair, or mud, or dirt? Young Socrates in the dialogue says no and is told he is not yet philosopher enough to be indifferent to such distinctions. The theory does not get to pick which things are explained.

Aristotle's verdict

Aristotle took the objections as fatal and said the relation was empty. To say that things participate in Forms is to speak in metaphors and poetic figures, and a metaphor is not a mechanism.

His alternative removes the distance the relation had to cross. The form of a horse is not in another realm; it is in the horse, as the organizing principle of that matter — hylomorphism, with act and potency to explain how the matter comes to have it. There is no relation between realms to account for because there is one realm.

That is the cleaner theory. What it gives up is what the Forms were for: an account of how the objects of knowledge can be stable when everything we observe is not, which is why Platonism never went away despite the objections.

Where it went instead

The Neoplatonists kept participation and changed its shape. Instead of a relation between two realms holding across a gap, Plotinus makes it a procession — emanation, where the lower comes from the higher and remains dependent on it, the way light remains dependent on its source. A thing participates in the One by being an effect of it that has not left it.

Christian thought absorbed this wholesale. Aquinas's account of creatures participating in the divine being is participation-language doing theological work: God is being, and creatures have being, received and limited. That distinction between being something and having it is the whole content of the claim, and it is also why the analogy of being is required — the word is cannot mean quite the same thing in both cases.

The idea keeps returning because the alternative is unattractive. Drop participation and either the standard is just another instance, which explains nothing, or there is no standard, which is nominalism. Neither is comfortable, and a relation nobody can specify has proved easier to live with than either.