The Parmenides is Plato’s account of a meeting between the Italian-Greek philosopher Parmenides, then a venerable sage of sixty-five, and a young Socrates. Unlike earlier dialogues, which tended to focus on a particular topic, usually some aspect of virtue, it is not completely clear what Plato was trying to accomplish in the Parmenides. It is clear that Plato held Parmenides himself in high esteem, even though he would have been too young to have met him personally. A simple wish to memorialize him would not have been enough reason for him to write a dialog, however–particularly one as technical as this one. It is possible that he meant it as some sort of teaching aid for the dialectal method, but this also seems unlikely since there are so many good examples of dialectic in his earlier works. In addition, this is not the best written or most dramatic of Plato’s works. It flows awkwardly and the characters are poorly developed. It is easy to see why this has traditionally been one of the less popular Platonic dialogues with readers. So is this just an example of mid-career slump, or is there a deeper message to be gained?The dialog begins in a somewhat convoluted manner as a recollection by Cephalus of a recollection by Adeimantus (Plato’s older half-brother) of the meeting. Parmenides and Zeno are in town and Socrates and his friends have gone to see Zeno recite one of his own dialogues. Afterward, Socrates begins asking questions and, in the course of the conversation, starts arguing for an early version of his (Plato’s?) theory of ideas. Parmenides breaks in and offers several objections which Socrates is unable to answer. He then advises the young philosopher to be more rigorous in exploring all the implications of his hypotheses to their ultimate conclusion. Socrates then convinces Parmenides to provide a demonstration of his dialectal methods which he does, with Adeimantus as interlocutor.
Zeno has been speaking about “the One” (as opposed to “the Many”) so Parmenides chooses to examine the null hypothesis “The One does not exist”. There follows a rather long and tortuous dialog, the main structure of which are summarized in Jowett’s preface to his translation of the dialog):
1. One is.
2. One is not.
If one is, it is nothing.
If one is not, it is everything.
But is and is not may be taken in two senses:
Either one is one,
Or, one has being,
from which opposite consequences are deduced,
1.a. If one is one, it is nothing.
1.b. If one has being, it is all things.
To which are appended two subordinate consequences:
1.aa. If one has being, all other things are.
1.bb. If one is one, all other things are not.
The same distinction is then applied to the negative hypothesis:
2.a. If one is not one, it is all things.
2.b. If one has not being, it is nothing.
Involving two parallel consequences respecting the other or remainder:
2.aa. If one is not one, other things are all.
2.bb. If one has not being, other things are not.
This is barely more easy to follow than the dialog itself. In the end, though, Parmenides proves that the one must exist:
Parmenides: Then may we not sum up the argument in a word and say truly: If one is
not, then nothing is?
Parmenides: Let thus much be said; and further let us affirm what seems to be the truth, that, whether one is or is not, one and the others in relation to themselves and one another, all of them, in every way, are and are not, and appear to be and appear not to be.
Adeimantus: Most true.
We are told that after this Adeimantus gave up philosophy and focused on training horses.
The existence of “The One” or Unity is certainly an important part of number theory and (much later) abstract algebra, but I don’t think Plato had math on his mind when he decided to write this dialog. We don’t know as much about Parmenides as we would like, but it seems from what we do know that his primary interest was cosmology. “The One” has a central place in Platonic and neo-Platonic cosmology as the First Hypostasis of the godhead: The One, The First Existant, The Unknowable, Infinite Unity which Christian theologians would eventually equate with The Father.
The first half of the dialog might have been a chance to memorialize a respected philosopher, but I think that Plato, who was himself turning more to metaphysics and cosmology in his later career, deliberately used the second half of the dialog to record an important proof that he knew would be useful in later work by himself and his students.