Take the concept of equality. We must admit that we have no experience, among sensible objects, of exact equality; we see only approximate equality. How, then, do we arrive at the idea of absolute equality? Or do we, perhaps, have no such idea?
Russell clearly falls into the "we have no idea of absolute equality camp."
Plato's answer, on the other hand, is that knowledge is recollection, so even if we had no experience in this life of absolute equality, we did in some past existence.
This is a plank in his argument for immortality, which in Russell's presentation is not persuasive. For one, why must a priori knowledge come from past existence--why not, I don't know, from some form of pure (non-empirical) reason? Surely the Plato of the world of ideas must entertain that such abstract ratiocination is possible. But even if a priori knowledge is from a past life, must past existence guarantee future existence? Russell really rips the immortality argument apart.
At the same time, even if Plato's observation doesn't get him where he wants it to go, the observation that we possess a priori knowledge (from somewhere!) is nonetheless sound. Not to Russell:
Let us take a concrete case. The metre is defined as the length of a certain rod in Paris at a certain termperature. What should we mean if we said, of some other rod, that its length was exactly one metre? I don't think we should mean anything. We could say: The most accurate processes of measusement known to science at the present day fail to show that our rod is either longer or shorter than the standard metre in Paris... But this is still an empirical statement, in the sense that empirical evidence may at any moment disprove it. I do not think we really possess the idea of absolute equality that Plato supposes us to possess.
Quine spent some time hammering at the synthetic-analytic distinction. I feel this is surely related to Russell's criticism of the a priori/a posteriori divide, but I regret my current inability to explain how.
Either way, this whole explanation by Russell doesn't do a thing for me. Now, I agree with Plato, we'll never see absolute equality in nature. Nonetheless, I have complete faith, and I really don't see how Russell cannot, that I could recognize a rod exactly one meter should I ever come across it. At present, I can also accurately recognize a rod that is not exactly one meter. (By meter, in all this, I use Russell's definition: "The length of that rod in Paris"). Moreover, not only could I recognize such equality, I can even imagine it. It takes virtually no effort to dream up two absolutely equal rods--how do I do this if I have not the knowledge Russell denies?
This is all wishful thinking on Russell's part. People don't like the irksome problem of "Where do we get our premises from?" (Russell even admits this puzzle earlier). Grounding them as deductive won't do, since deduction can only produce conclusions from premises thrown in, and empirical knowledge produces the same endless line of justification (how do we know we know we know?) So we either deny such premises exist or distract ourselves by harping on the empirical side of the divide, where our starting points are more universal and thus, more apt to escape notice.
Incidentally, Russell has a better criticism of Plato's argument, so far as immortality goes. Even, says he, if we grant that we have a notion of absolute equality, it would seem to only be possessed by fully reasoning adults, certainly not newborns. Where are these supposed memories of the immortal child before he learns to properly reason? Held in escrow somewhere?