"The impasse of the Parmenides is that of establishing that both the one and the others [i.e. the many or multiple] do and do not possess all thinkable determinations, that they are totally everything and that they are not so. We are thus led to a general ruin of thought as such by the entire dialectic of the one." Alain Badiou Being and Event
One could assert that a major element in the history of philosophy stretches from Plato's dizzying Parmenides to a moment in Nietzsche's Twilight of the Idols. Although Nietzsche is not directly invoked, Badiou certainly sees his work as taking off from a realization that the thinking of the one and the many has been at the very heart of philosophy since its appearance in Plato as inspired by the Pre-Socratics and that, generally, we have not escaped from the manner in which Plato presented the problem in the course of his works. It is this escape that Badiou wants to stage drawing, fundamentally, upon a failure in the way that the Parmenides conceived of the problem.
The reason I suggest the history of the problem could be read as stretching from Plato to Nietzsche is because Nietzsche's unparalleled assault upon the two world dualism, which he sees as the heart of almost all metaphysics, can be read as a version of this same problem of the one and the many. Indeed, there is a way to read the conclusion of both Plato and Nietzsche's grappling with this problem as indicating, against each of their conscious wills, the way to jump out of the dialectic. It is, I would argue, precisely this point which Heidegger sees when he suggests that the challenge for philosophy now is to think the Nothing.
The Parmenides ends with an exhausting attempt to think out the speculative implications of positing the existence of The One and The Many. The conclusions, roughly, are that if The One exists it must be the only thing which can be thought to exist and all difference and plurality must be illusion. One of the main ideas here is that similarity and difference between things, such that we could compare The One to others, requires that parts or elements be discernible within The One and thus present or lacking in the others. However, if The One is truly one it has no parts and no elements, no characteristics which can be divided from its oneness. So, if The One exists we do away with similarity and difference, and thus with plurality. But, if this is the case, then oneness becomes meaningless and unthinkable.
"Thus if one is, the one is all things and is not even one, both in relation to itself and, likewise, in relation to the others." Plato's Parmenides
On the other hand, if The One does not exist we are left with an unthinkable multiplicity. This multiplicity is unthinkable because, again, similarity requires some level of oneness which is, in some sense, not total oneness. In other words, for the multiplicity to be a multiplicity of things we need a way to think the oneness of a thing and the oneness implied in sharing something with other things. The communion required for similarity itself requires oneness. This leads us to the ultimate conclusion of the dialogue:
"If one is not, nothing is."
It is possible to read the dialogue as a defense of the Parmenidean point that the ultimate truth of reality must be that change and difference are illusions. The one must be, but if the one is it must be all. However, as Badiou reads it for example, the dialogue actually demonstrates that the relation between the one and the many is the reef upon which the ship of thought is ruined. But, as our discussion should suggest, this reef is really the problem of similarity and it is this reef, I would claim, upon which Nietzsche built his entire life's work. (Think I mixed metaphors there? I rather think I didn't. If ever there was a thinker who willfully sought to build a perpetually changing and endangered edifice upon shifting submerged reefs of thought it was Nietzsche.)
If we look at some of Nietzche's early work we find a thread that continues throughout the course of the rest of his work. This thread is found, for example, in Truth and Lie in an Extramoral Sense when we are told that similarity and, eventually, truth are built out of the basic "lies" created through the limiting nature of our senses and the metaphoric nature of thought and language. All sameness, unity, stability and generality are constructed, artificially, through basic illusions overlaid on an actual flux of change and difference. Reality, then, is the unthinkable multiplicity without similarity or continuity and all else is a poetic creation. This same point is repeated throughout Nietzsche's work surfacing, for example, in The Gay Science and finally in Twilight of the Idols. If we look at the section "'Reason' in Philosophy" in Twilight of the Idols we see very clearly the presentation of reason as precisely this lie, or metaphor, constructing collection of instincts. However, it becomes clear that the drive to posit similarity goes beyond the construction of our everyday experience of the world. It eventually gives rise to the complete rejection of this world precisely because we STILL find, after all our processes of falsification of experience, too much change in the world. This leads to the positing of another world, the Real world, which is stable, unchanging and in comparison with which the world of experience and multiplicity must be rejected. We can see as well, here, the connection between this artificial origin of similarity/relation and the positing of a final, ultimate, singular perspective which would count as the Truth about any given subject. To achieve Truth is to see things from that perspective from which all difference and change disappears as illusory. From the rejection of this view we get Nietzsche's perspectivalism.
There is, however, a problem with the very chain of ideas I have just presented for Nietzsche. The problem is that the original insight that similarity is artificial arises from a Real World vs. Illusion dichotomy and, indeed, in Truth and Lies in an Extramoral Sense Nietzsche is at his most Schopenhauerian and metaphysical. For him, at that time, there sometimes seems to be a hidden ultimate reality, a thing in itself, which is something like chaos, flux or Will to Power. I believe Nietzsche ultimately does not hold to this commitment, but it is necessary in order to get into the viewpoint I have been presenting. The rejection of two world dualism, then, originates from a two world dualism.
It is precisely this point, in a lovely echo of Plato's Parmenides, that Nietzsche presents so clearly in the section of Twilight of the Idols entitled "How the 'Real World' at last Became a Myth". There we see how the two world dualism of Plato eventually leads to its own overcoming, just as the positing of either The One or The Many eventually lead to the collapse of both. The end of the story, likewise, is the same as the end of the Parmenides. Specifically, we end with a very troubling paradox.
"We have abolished the real world: what world is left? the apparent world perhaps? ... But no! with the real world we have also abolished the apparent world!"
This same paradox leads both Nietzsche and the characters in the Parmenides to the same conclusion:
"If one is not, nothing is." Plato
"The characteristics which have been assigned to the 'real being' of things are the characteristics of non-being, of nothingess -" Nietzsche
Although not seemingly the intended meaning, both moments in the history of metaphysics point us to the necessity of thinking the Nothing and its relation to being. What would it mean to reject the One, and thus the Many, and instead assert Nothingness? What does it mean to say that the characteristics of being are precisely those of the Nothing? This is the question which Heidegger believed could get us beyond nihilism and which Badiou, in a totally different manner from Heidegger, also seeks to answer.