This whole article was good, even this objection, though I strongly disagree with it:
Still, despite its clean lines and long history, Platonism [about numbers and universals- ed.] cannot be right either. Since the time of Plato himself, nominalists have been urging very convincing objections. Here’s one: if abstracta float somewhere outside our own universe of space and time, it’s hard to imagine how can we see them or have any other perceptual contact with them. So how do we know they’re there?
-Plato himself gives a clear answer to this in the Phaedo, sc. because we saw them before we had bodies. This might, however, be taken as proving the point – if our epistemology calls for pre-existence, it will certainly have a hard time convincing almost everybody nowadays.
-That said, Plato proves the pre-existence of soul from our intuition of abstract objects. The intuition itself is manifested from other sources. As odd as it seems to talk about “manifesting an intuition” (since intuitions must be, if anything, seen in themselves) it nevertheless has a clear sense in Plato. The sensible world is a cause of remembrance, but it is still a necessary cause. Plato’s theory of knowledge requires that knowledge begin with sensation, but this sensation is a principle of recalling, not of abstracting.
-In a word: Plato denies that we now know universals and ideas by direct intuition of another world. We now depend on sensation, and know by sensation first. Plato does not have a theory of intuition of universals, but of recollecting them.
-So if Plato, just like the Nominalists, agree that the sensible world is the given starting point, how do we decide between abstraction and recollection? Plato’s answer in Phaedo seems to be that abstraction requires that the thing we abstract the form from be like the form abstracted, but we do not always form ideas from things that are like the idea. We sometimes get a very clear idea of justice by witnessing at a flagrant injustice, a clear idea of the infinite from the finite, or of eternal things from contingent things. For that matter, we get an idea of the abstract from the concrete.
-Again, abstraction is, at the same time, an end result of a group of particulars and the governing concept that causes them to be grouped together in the first place. So is the universal initially just formed by chance? Is it implicit in sense knowledge alone? Even if the latter is true (and even animals have some rudimentary universals from sense alone) is it always so implicit?
-Plato’s argument, considered formally, does not require the pre-existence of the soul but any transempirical intuition (though this intuition, as we now exist, is knowable only by recollection). It is extraordinarily hard to avoid these all such intuitions. At the bare minimum, it’s hard to avoid them with respect our knowledge of our own selves, and if of ourselves, we have some knowledge of all that belongs essentially to the self- being, goodness, freedom, contingency, etc.
-Even if we pre-existed, this pre-existence does not account for transempirical intuition. The sense is that, without a body, mind would know in the way proper to it, as opposed to having to recollect.
-”Recollection” for Plato is just when, seeing one thing (whether alike or similar to another) we are caused to know that other. The parenthetical makes the difference between recollection and abstraction.
-I try to correct my students when they speak of “abstractions”. This is a theoretical account of the genesis of ideas or universals.