Most Platonists today are disappointed Platonists--people with Platonic expectations that are unfulfilled, because they accept Darwinian evolution as true, and therefore since all living forms have evolved, they cannot be eternal as conforming to Plato's intelligible realm of eternal Ideas. Moreover, if everything has evolved, this must include moral and political order, and thus there is no eternally unchanging Idea of the Good by which we can see absolute standards of right and wrong. Consequently, there are no moral absolutes, and we must accept moral relativism or nihilism. Darwinism is "true but deadly" (as Nietzsche said). And thus these disappointed Platonists become nihilists.
But if you do not have Platonic expectations, you will not be disappointed by the Darwinian conclusion that everything has evolved, and therefore human beings have evolved. Without the Platonic assumption that morality must be grounded in a moral cosmology, you will be satisfied with a Darwinian explanation of morality as grounded in a moral anthropology. Even if morality has no eternal grounding in a cosmic God, a cosmic Nature, or a cosmic Reason, human morality can still have an evolutionary grounding in human nature, human culture, and human judgment. And thus in contrast to the disappointed Platonists, the satisfied Darwinians are not nihilists.
Socrates teaches us that to know what is truly good, we must transcend the visible realm of Becoming in ascending to the invisible realm of Being, and finally ascending to the Idea of the Good. But it's not clear that this answer is fully convincing or satisfying. Socrates concludes the Republic with the myth of Er--a mythic story about eternal rewards and punishment in the afterlife. The problem with this myth is that it's only a myth, and so it's not clear why we should believe it. But the earlier answer--the Idea of the Good--is so incomprehensible that it's not clear why we should believe it either.
Socrates became Socrates (as opposed to one more philosophical conventionalist) when he realized that the cosmos can be comprehensible if and only if the principles that characterize it are, at least roughly, commensurate with the principles of our own common sense understanding of things. That is why Socrates returned to the city and (initially at least) abandoned direct investigations into natural phenomena. The keys to understanding nature lie in the opinions and especially in the contradictions between the opinions of ordinary people.