Parmenides is one of the dialogues of Plato. It is perhaps Plato's most challenging dialogue.
The Parmenides purports to be an account of a meeting between the two great philosophers of the Eleatic school, Parmenides and Zeno of Elea, and a very young Socrates. The occasion of the meeting was the reading by Zeno of his treatise defending Parmenidean monism against those partisans of plurality who asserted that Parmenides' supposition that there is a one gives rise to intolerable absurdities and contradictions.
Employing his customary method of attack, the reductio ad absurdum, Zeno has argued that if as the pluralists say things are many, then they will be both like and unlike; but this is an impossible situation, for unlike things cannot be like, nor like things unlike. But this difficulty vanishes, says Socrates, if we are prepared to make the distinction between sensibles on the one hand and Forms, in which sensible participate, on the other. Thus one and the same thing can be both like and unlike, or one and many, by participating in the Forms of Likeness and Unlikeness, of Unity and Plurality; I am one man, and as such partake of the Form of Unity, but I also have many parts and in this respect I partake of the Form of Plurality. There is no problem in demonstrating that sensible things may have opposite attributes; what would cause consternation, and earn the admiration of Socrates, would be if someone were to show that the Forms themselves were capable of admitting contrary predicates.
At this point Parmenides takes over as Socrates' interlocutor and indeed dominates the remainder of the dialogue. After establishing that Socrates himself has made the distinction between Forms and sensibles, Parmenides asks him what sorts of Form he is prepared to recognize. Socrates relies that he has no doubt about the existence of mathematical, ethical and aesthetic Forms (e.g. Unity, Plurality, Goodness, Beauty), but is unsure of Forms of Man, Fire and Water; he is almost certain, though admits to some reservations, that undignified objects like hair, mud and dirt do not have Forms. Parmenides suggests that when he is older and more committed to philosophy, he will not be so deferent to public opinion and will admit Forms of even the lowest objects.
For the remainder of the first part of the dialogue, Parmenides draws Socrates out on certain aspects of the Theory of Forms and in the process brings to bear five arguments against the theory.
Argument 1. If particular things come to partake of the Form of Beauty or Likeness or Largeness they thereby become beautiful or like or large. Now each particular thing must receive either the whole of the Form of which it partakes, or a part of that Form. Either way, however, the Form becomes many - in the first case by multiplication, in the second by division - and thus will not still be one.
Argument 2. Socrates' reason for believing in the existence of a single Form in each case is that when he views a number of large (for example) things, there appears to be a single character which they all share, viz the character of Largeness. But consider the series of large things x,y,z, Largeness Itself. If all members of this series partake of a single Form, it must be the Form Largeness Two. Similarly, x,y,z, Largeness and Largeness Two must all partake of a further Form, Largeness Three, and so on ad infinitum. Hence, instead of there being one Form in every case we are confronted with an indefinite number. This is the famous Third Man Argument. (TMA)
Argument 3. To the suggestion that each Form is a thought existing in a soul, thus maintaining the unity of the Form, Parmenides replies that a thought must be a thought of something that is a Form. Thus we still have to explain the participation relation. Further, if things share in Forms which are no more that thoughts, then either things consist of thoughts and think, or else they are thoughts, yet do not think.
Argument 4. Socrates then suggests that the relationship between Forms and particulars may be one of imaging. Parmenides then shows that this account is susceptible to the TMA, for if x,y and z are large because they resemble Largeness, then Largeness must resemble x,y and z, resemblance being a two-way relationship. But if x, y, z and Largeness all resemble each other they must have some common character, viz a resemblance to a further Form, Largeness Two, and again a regress is generated.
Argument 5. The gravest difficulty with the theory of Forms arises as a consequence of the assertion of the separate existence of the Forms. Forms do not exist in our world but have their being with reference to one another in their own world. Similarly, things of our world are related among themselves, but not to Forms. Just as Mastership has its being relative to Slavery, so mastership in our world has its being relative to slavery in our world. No terrestrial master is master of Slave itself, and no terrestrial master-slave relation has any relationship to the ideal Master-Slave relation. And so it is with knowledge. All our knowledge is such with respect to our world, not to the world of the Forms, while ideal Knowledge is knowledge of the things not of our world but of the world of the Forms. Hence, we cannot know the Forms. What is more, the gods who dwell in the divine world, can have no knowledge of us, and nor can their ideal mastership rule us.