For what I am doing here, see this.
It was not only in Ionia that serious and systematic thinking could be found. On the other side of the Greek world, in Elea, a Greek colony in Southern Italy, Parmenides began to think through the implications of change. Out of his conclusions grew what has come to be known as the Eleatic School. There were several major thinkers in this philosophical movement.
Parmenides: We have only one fragmentary poem from Parmenides, but it gives us some idea of what his philosophical views may have been. In the poem Parmenides has a vision of a goddess who tells him to distinguish the Way of Knowledge from the Way of Opinion. The Way of Knowledge concerns what exists, and what truly exists does not change. Reality (as we would call it) is immutable, complete, indivisible, and eternal. What exists cannot come from anything other than itself, because the only thing that is not what exists is what does not exist (nothing); and nothing comes from nothing. What exists cannot be known by sensory perception, but only by logos, reason: sensory perception only gives us what seems to exist, and what seems to exist is different from what exists. What seems to exist is mere appearance; it is not real.
The notable thing about this is that Parmenides has taken the discussion of change that we found in the Ionian philosophers and raised it to an entirely new level: we are not merely talking about changing things, but about existence itself. Also important is the fact that he raises as a serious and important issue the distinction between knowledge and opinion and tries to draw a precise line between the two. Both of these points will be very influential on later philosophers. The argument that what exists cannot change was particularly important for the later course of Greek philosophy; it is often called the Eleatic Challenge: how can what exists come from what does not exist, which seems to be what happens in every change?
Zeno of Elea: The most famous of Parmenides' followers is Zeno of Elea. In Parmenides (at least what survives of his work) we have a poetic expression of the Eleatic Challenge. Zeno took that Challenge and put it in rigorous analytical forms. Because of this Aristotle calls him the inventor of dialectic, the process of analyzing claims through careful use of arguments and counter-arguments. Some of these arguments are justly famous; although they come to very paradoxical conclusions, they are well designed and thought-provoking, and through their profound simplicity have stood the test of time.
Melissus of Samos: Like Zeno, Melissus took Parmenidean ideas and formulated arguments for them. One of his positions was that the universe is infinite in time and space; he also (according to Plutarch) claimed that we should not say anything about the gods because we do not know anything about them. The arguments attributed to him seem creative, and Melissus seems to have been influential in making people aware of Eleatic ideas, but the reception of his work does not seem to have been completely laudatory: Aristotle says that his thought was crude and his arguments were absurd, and clearly regards him as inferior to Parmenides and Zeno. It's difficult for us to say because we only have fragments of Melissus's work surviving in quotations found in other people's works.
The Eleatic Challenge plays an important role in the history of philosophy because it directs our attention to the problem of change. Any serious attempt to understand the world must come to grips with the question, "What is change?" It is a question that is not as easy to answer as it might at first seem. The emphasis Zeno and Melissus place on careful rational argument is of immense importance, too. With them we have gone beyond the Ionian attempt to find natural explanations; they show the value of trying to explain things in rational and well-argued ways.
