Parmenides' great work, On Nature is in the form of an epic poem, written in the hexameters of Homer and Hesiod. And although he is best known for his somewhat mysterious (perhaps Pythagorean?) claim that what exists is eternal and unchanging, the third part of his poem contains reflections on the Ionian "physiologoi" who replaced mythical gods with natural explanations for phenomena.

Parmenides was the source of Plato's claim that Parmenidean Being is more "real" than Heraclitean Becoming, which may only be an "illusion."

For Plato, his forms or "ideas" are prior to any instance of an object with a given form. The forms exist in another "realm" that is more "real" than the everyday physical world of material objects. The forms are properly outside of time, like Immanuel Kant's noumenal world. Aristotle challenged Plato's idea and argued that the forms are merely "perfect" and "idealized" abstractions from the many "imperfect" examples found in the world.

In mathematics, the ideal circle consists of an infinite number of infinitesimal points that satisfy an equation. Such an infinity is never realized in the empirical world, in which objects are composed of a finite number of material particles, for example, atoms. Arguably, an ideal circle has an unchanging, eternal nature. It will be the same for any thinking entity, now in the real world, and forever in any possible world.

Plato thus set up the fundamental dualism of philosophy, the distinction between idealism and materialism, between abstract eternal essences and concrete ephemeral existences, between In his Timaeus 27d, Plato asked "What is Being always, but has no Becoming (origin or genesis), and what is Becoming always, and never Being?"

τί τὸ ὂν ἀεί͵ γένεσιν δὲ οὐκ ἔχον͵ καὶ τί τὸ γιγνόμενον μὲν ἀεί͵ ὂν δὲ οὐδέποτε;

In Plato's Parmenides, there is much talk of "Being" as "the One," but it is not clear whether Plato accepts the One completely, as the Socratic dialectic avoids coming to any conclusion. The dialogue is full of dazzling wordplay about infinite regresses, as well as claims that many things both are and are not, in various respects. The "One" is both like and unlike itself. (147c ff) The "One" both touches, and does not touch, both itself and others. (149d) The "One" is alike equal to, greater than, and less than, both itself and others. (151b)

Since the existent has not-being and the nonexistent has being, the "One" also, since it does not exist, must have being in order to be nonexistent. Thus it appears that the "One" has "being," if it is nonexistent, and also, since it is not existent, has not-being. (162b) The nonexistent "One" both comes to be and ceases to be, and also does not come to be or cease to be. (163b)

καὶ τὸ ἓν ἄρα μὴ ὂν ἀλλοιούμενον μὲν γίγνεταί τε καὶ ἀπόλλυται, μὴ ἀλλοιούμενον δὲ οὔτε γίγνεται οὔτε ἀπόλλυται: καὶ οὕτω τὸ ἓν μὴ ὂν γίγνεταί τε καὶ ἀπόλλυται, καὶ οὔτε γίγνεται οὔτ᾽ ἀπόλλυται. (163b)

Despite the empty verbal debates, the principal goal for Parmenides is to show that some one thing cannot be many things. In particular, it cannot be like another thing (in the sense of having a property) and yet not like that thing, that is have one property and yet not have that property.

Socrates demolishes Parmenides by arguing that properties are relative. One can have the property of being tall and not tall. Simmias is tall because heis taller than Socrates. But he also is short, shorter than Phaedo (Phaedo 102b). SImmias is both tall (with respect to Socrates) and not tall (with respect to Phaedo).

Socrates dispenses with Parmenides' claim that one cannot be many. He is one of the many philosophers and yet consists of many parts - head, hands, etc. Socrates then turns to a suggestion that the "Forms" are just "Thoughts," the ideas in some mind. Parmenides is well known for claiming that "Being is Thinking." Parmenides objects that if a form is a thought, that then any object with a form is a thinking thing. This "panpsychism" is unacceptable to both Parmenides and Socrates.

Socrates then suggests that forms are merely "patterns" in nature. This is the essence of information philosophy. When a form/pattern in an object is isomorphic to the form/pattern in a mind, when some part of the information in a structure is the same information stored in a mind, we can say the the thinker has some knowledge of the object. But one object can contain many different "patterns" or properties.

Much in the Parmenides has the character of Heraclitus's thought. He concluded rather dialectically that we both "step and do not step into the river, that we are and are not," sounding obscurely like the modern obscurant Hegel.

ποταμοῖσ τοῖσ αὐτοῖσ ἐμβαίνομεν τε και οὐκ ἐμβαίνομεν, εἶμεν τε και οὐκεἶμεν.
(Diels-Kranz B49a, "Homeric Questions.")

It concludes on a difficult note,

Thus, in sum, we may conclude, If there is no one, there is nothing at all.

To this we may add the conclusion. It seems that, whether there is or is not a one, both that one and the others alike are and are not, and appear and do not appear to be, all manner of things in all manner of ways, with respect to themselves and to one another.

Most true.