One way to look at Nietzsche's "overturning of Platonism" is by seeing what Platonism's Idea entails as a concept. That is, what it allows us to think.
Simply put, broadly speaking, the Idea is and does two things:
1): The Idea establishes a teaching of Truth. For example, the Idea Beauty is entirely itself, entirely beauty, self-identical. And, it is through this Beauty that we can know particular instances of Beauty: this man, this work of art, and so on. In this way, the Idea gives us the mark of truth. What does truth look like? Perfect self-identity. And it allows us to recognize truth.
2): The Idea allows for a criterium of judgment. On the basis of the Idea of Beauty, established in 1, we can judge things as to their beauty or ugliness by determining to what degree they approach the Idea of the Beautiful, to what degree they share in the Idea of Beauty. This is then what determines value: proximity to an Idea. The more you share in the Idea of Courage, the more courageous you are. As such, the Idea allows us to determine differences between people and things. And it allows us to ground these differences.
In Platonism, 2 is secondary to 1. It is on the basis of 1 that 2 is possible. Nietzsche’s “overturning” can be seen as a reversal of this relation. He tries to show how the judgment concerning the beautiful precedes the knowledge and/or truth of Beauty as an Idea, and how in Platonism’s Idea of say Beauty, it is really the judgment that precedes the truth in the argument.
“This is beautiful” is not made possible by the Idea of Beauty. Rather, the judgment that this is beautiful makes possible the Idea of Beauty.
“Pathos of distance.” Things are different, not in relation to a common criterium (some Idea), but totally and radically from within themselves and their natural genesis.
Someone’s beauty or ugliness is not determined by someone’s proximity to Beauty, but by one’s own beauty, the ground of which is not a ‘Beauty’ separate from the person, but the concrete way in which this person has come to be beautiful or ugly; his genealogy, his upbringing, his particular biological and spiritual dispositions, his mode of life, his environment, and so on.
In fact, we can now say that the Idea was created to allow for judgment. How do we distinguish the real philosopher from the impostor? The Idea allows us to do this. Judgment precedes truth.
And so Nietzsche’s task: to think judgment and hierarchy without recourse to transcendent criteria. This means the “overturning of Platonism.” We do not establish hierarchies on the basis of Truths. No, Truths are established on the basis of natural hierarchies. But how to think natural difference and hierarchy now that we are so bound up in a thinking that we can only think such differences and hierarchies by recourse to transcendent criteria, be it Ideas or anything else?
It is also related to how the question is asked. When Socrates asks his fellow Athenians: "What is beauty? What is justice? What is courage?" They often point to particular instances. “Well, I don’t know, this man is beautiful, and this one seems courageous, and this one just. Do you not have eyes to see my dear Socrates? Maybe you are getting old, or maybe you are too ugly yourself to recognize beauty even as it stands before you?”
Of course, these are false answers to the question. But who says that the question itself is not false to begin with? That is, maybe if we want to know what beauty is, the question “Who is beautiful?” is a more useful and productive question than the question “What is beauty?” In the former case, the answer will give us a concrete example and genealogy of how beauty comes to be. The latter question only gives abstract answers.
Nietzsche says against Socrates: "Who is it?", this is the question that sets us up for much more results than this question “What is it?” This question after the 'essence' of beauty, it leaves beauty undetermined and tells us nothing of the genesis of beauty, what forces make it so that someone becomes beautiful or ugly. Forces that can only be seen by looking at the life of the beautiful person. And so, in the name of truth and reality, Nietzsche turns against the question for Truth conceived of as the “being” of something. We want to think reality, but to what degree does this question ‘what is?’ not make this impossible?
“Genealogy” shows itself as a superior method for determining the beautiful, the courageous, and so on. The Greeks knew this in the sense of ‘paideia’, breeding and education in all of its senses, ‘culture.’
Which is in its own way, just like the Idea, a method for knowing what is, and for judgment. "Who is the real Greek? who is the Barbarian?" Well, simple, look at where they come from, and look at their real genealogy, both of blood and training, that has made them who they are. They are who they are, and what they are is the result of their becoming.
What the Socratic/Platonic represents is the creation of a new method of knowing and judgment. One's value is now determined by one's proximity to the Idea, which also becomes the ground from which knowledge occurs. We no longer know what ‘is’ by looking at how things become. No, we know the becoming of things by looking at what is.
What also occurs is a duplication of being. A beautiful person appears. This is how it is, the brutality of fact. How is it that this person is how he or she is? Because of their becoming, says the old answer. And so there are two elements: that which is, and the becoming that allowed the thing to be how it is.
But with Plato, the fact is explained by reference to a different being, the being of the idea. And so now there are three elements; that which is, the becoming that allowed the thing to be how it is, and a second Being that is said to be the real determining factor that will replace the explanation by becoming.
And so, the simple appearance of beauty in which at first being had its stay must now be deemed an inferior kind of being, mere appearance, as opposed to the real Being which resides in the Idea. And so eventually, the simple appearing of beauty or whatever, is no longer the domain of being, but only a shadow. And so, in short, the Idea, at first competing with the explanation by genealogy or becoming, ends up also competing with the appearance as such, with reality as such.
This is why Nietzsche has such hope for art, because the artist never sees appearance as an argument against reality, and only the artist can bring appearance back to its full reality. The artist sees appearance as higher than ‘Truth.’ That is, reality is revealed more fully and indubitably in the brutality of the fact of appearing, than it is in the Truth.
This is also why in Nietzsche, art and truth compete with each other. Because ‘truth’ has come to stand for that way of thinking that denies appearing any power. Whereas art is that way of thinking in which mere appearing is taken to the ends of its powers and shown to be creative, equally and even more creative than the philosopher of the Idea. In fact, it is seen as the true ground of ‘Truth’ even. Platonism starts his investigation with looking at particular instances of beauty and so on, and on the basis of this rises up to the Idea of Beauty through reason.
The “overturning” of Platonism does not mean an affirmation of some sensuous realm and a doing away with some super-sensuous realm, an affirmation of ‘life’ over ‘intellect’ or some other banality. Rather, it means a radical shaking up of the way in which Platonism has sectioned up reality into higher and lower in a specific way. It means also a full recognition of the creative act of Plato, and an understanding of it as such. Appearance only became ‘mere appearance’ after Plato’s creative act. And so a doing away with Platonism is a doing away with thinking the differences in the world by recourse to a transcendent unity. Why? So reality can be affirmed as such and no longer needs an explanation outside of itself. This is as much a continuation of Plato as it is an ‘overturning’. It is precisely of the Idea to allow us to think something that is self-justifying and self-explanatory. An original power that as such can not be thought but allows us to think. This force, this power that hides in the great notion of the Idea, it must apply to reality as such, to ‘mere appearance.’
For Nietzsche, the problem is not some replacement of “life” by logic and dialectic, it is that for him this Socratic dialectic is a second-grade logic incapable of grasping life. It does not allow us to think the real ways in which a certain type of life, higher or lower, comes to be what it is. The Idea of Beauty does not explain how the beautiful work of art comes to be. What would get us much closer is an investigation of the artist’s life. And so in this way, the question: “What type of life leads to the creation of beauty?” tells us much more about the reality of beauty than the question: “What is beauty?”
“In praxi, this means that moral judgments are torn from their conditionality, in which they have grown and alone possess any meaning, from their Greek and Greek-political ground and soil, to be denaturalized under the pretense of sublimation. The great concepts "good” and “just” are severed from the positions to which they belong and, as liberated “ideas,” become objects of dialectic. One looks for truth in them, one takes them for entities or signs of entities: one invents a world where they are at home, where they originate—” (Nietzsche, WTP, §430)
How did he become so courageous? “Because he is in close proximity to the Idea of Courage." This is really, even for Plato, a roundabout way of saying that someone is courageous because he is courageous. “The safest answer” to the question “what makes something beautiful?” is that something is beautiful because of the presence of the beautiful, as the Symposium puts it.
To be sure, in Platonism, the question of the truth of an Idea is never entirely separated from the phenomenological materiality and self-evidence of said Idea. It only takes a glance at Plotinus’ treatise on beauty, which starts with the statement that some things appear to us as beautiful and some as ugly. There is a natural judgment that is considered true and evident by grace of its phenomenological materiality only, and only in a second movement of thought do we rise up to the grand notion of the Idea of Beauty which will ground this experience. But what is revealed in the experience as such is never questioned, and that the experience of the beautiful reveals beauty to us is never questioned. Whatever else might be said about ‘beauty’, the fact is that it appears to us.
Most importantly, Nietzsche shows the Idea to be related to judgment, it is a procedure of selection. Why was such a procedure needed? Let us speculate.
Perhaps the event of democracy in which men can no longer be determined as different in value by nature and rearing, and so a new manner of judgment must be created.
The question "how did he become so courageous?" can no longer be asked when the answer is outlawed, the answer which necessarily consists in pointing to genealogy and natural differences. And so a new manner of judgment must be created that generally speaking preserves the same outcome, but through different means. “He is courageous because in him courage is present.” The same fact of reality, but a much safer answer.
Even with the new manner of judgment, it will still be the beautiful people that will be said to be closest to Beauty. The outcome is the same.
And this is part of what the "power of the false" and "truth as falsification" consists of and what constitutes Nietzsche's greatest respect for Plato. Even if the Idea falsifies the real way in which the beautiful or the courageous become beautiful, the judgment is still possible, the hierarchy is preserved. And so, natural difference survives the terrible equalization that is democracy. This is what the Idea allows and what gives it its tremendous power as a concept that still makes us think today. A falsification that allows us to think reality, just like art. And perhaps it is only in a "democratic" environment that this is possible.
What the Idea expresses in the most basic sense: not that which is thought, but that which allows us to think. That is, it allows us to judge, and it allows us to create.
And so, judgment precedes Truth. Or, "Truth" is a necessary falsification that allows us to judge, that allows us to think.
How can beauty, courage, and so on be preserved and thought in this new environment which carries with it the danger of total anarchy? The Idea, which allows us to preserve and explain the existence of beauty by pointing to the very fact of its existence, and it allows us to judge, to separate the beautiful from the ugly by pointing to the mere fact of something being beautiful or ugly.
And so, the Idea seeks to ground the appearance of beauty, courage, justice, and so on. Why would you seek to give such an absolute ground to something, if you did not care for its existence? Plato achieved the greatest thing one could possibly achieve; he was able to give an absolute ground to the self-evident appearing of beauty in life. So that even if a time would when beauty is nowhere to be seen, the souls of man could still be focussed on it. Focussed on it as an Idea, that is, as the ground and justification for action and thought, and as its end. The Idea of the Good allows us to be directed toward goodness, even if there is no goodness to be seen.
This is one reading. Some Nietzscheans would say however that Plato himself is the democratic danger entering thought, and not the great artist who gives life a new foundation when its previous conditions for existence were slowly slipping away.
Nietzsche writes:
“What dawns on philosophers last of all: they must no longer accept concepts as a gift, nor merely purify and polish them, but first make and create them, present them and make them convincing. Hitherto one has generally trusted one’s concepts as if they were a wonderful dowry from some sort of wonderland; but they are, after all, the inheritance from our most remote, most foolish as well as most intelligent ancestors.” (Nietzsche, WTP, §408)
An ‘idea’ is a construction, a crystallized form of a will to power, a judgment, a selection. Just like an artist carves out a piece of reality, and invests it with powers that can in turn empower life.
This is to say also that not all ‘truths’ are equal, in the same way that not all works of art are equal. Some truths are entirely uninteresting, others are of significance. Some ideas, like some works of art, express an entire world. Others express nothing at all.
Some ideas tell us nothing at all, other ideas allow us to think the entirety of reality, like the concepts of all the greatest philosophers.
But it is questionable if ‘truth’ as such is a criterium that allows us to judge on the value of an idea in philosophy. In science this might be true, but in philosophy it is different because philosophical concepts are also concerned with judgment and the positing of value, hierarchy, and so on. They do not just describe reality, they also partition it. They do not just allow us to see reality, they tell us how to look at reality. This is not to say that philosophical ideas should have no concern for what is, absolutely not.
The full value of a truth is determined by relating it to the powers of judgment that are invested into it, and by looking at the possibilities for thought and life that it makes possible. What does it allow us to think?
In this sense, there is no greater idea than that of the Idea. It at once allows us to think beauty, courage, and so much more, and in the same movement is able to posit these things as ends. In this light, Kant’s great theory of Ideas appears as much as a great tragedy as a great achievement; it keeps the essential form of the Idea and its abstract function, but it loses the noble modes of life for which these forms were invented.
One’s belief in the truth of ‘Ideas’ is irrelevant, it is a question of being able to see what the power of the Idea was and still is.
"And so eventually, the simple appearing of beauty is [...] but only a shadow.
[...] "And so, natural difference survives the terrible equalization that is democracy."
This is illustrated hilariously, gruesomely well in a recent film called 'Triangle of Sadness.' Highly recommended.. It appears very shallow to begin with but descends, and describes this idea magnificently by the end.
I could spoil it to make a clearer argument, but won't in case anyone needs a good film. I saw it over a month ago and still flash back to it every few days for its relevant insights.
"The Idea of Beauty does not explain how the beautiful work of art comes to be."
Ananda Coomaraswamy asserted artists were virtually a protected species in India once, by those with rasa or 'taste'; an artist would be sponsored by such a rasa'd man. The Idea of beauty existed in the mind (of the man with rasa/taste) who recognised beauty flowing through the artist from a transcendent principle. In a religious sense one could say the artist is being used as God's medium, and the artwork is the filtered result of the life of the artist. The man with rasa wasn't able to use the Idea as it stood within himself, but was able to see it channeled through the artist.
Perhaps the man with rasa could answer “What type of life leads to the creation of beauty?” by funding the lifestyle of the artist because an artist can't create if they are constantly battling to survive, which is also the most significant problem with the west, according to Coomaraswamy. What happened to great art? Survival.
Modern art is a form of money laundering these days...