Of Heraclitus we have about 140 fragments, some of dubious authenticity, all of them seemingly obscure and open to endless interpretation. He affirms change and becoming when he says, "Other and other waters touch those who go into the same river," a typical saying of Heraclitus in that a logical thought is couched in physical, everyday terms--water, a river, a body. Change and becoming are thought in terms of life and death: being is overcome by not-being and viceversa, a quality is conquered by its opposite, and so the cosmos, always becoming, is a field of constant war: "War is the father and king of all things, it shows some as gods, some as men, it makes some freemen and others slaves." The becoming of the cosmos is explained in physical terms: fire dies and is changed into air, air dies and becomes water, water dies to become earth, and so on, an interchanging of life and death between different elements which, however, would be wrong to call "a process," because "process" means something moving only one way--forward--whereas Heraclitus insists, in fragment after fragment, that this sequence of transformations in the cosmos goes both ways. In his own words, "The way up and the way down are one and the same." This is his deepest insight: all becoming is circular (notice that, indeed, in a circle the way up and the way down are one and the same). The same is true of human life: "In us the living and the dead, wakefulness and sleep, youth and old age, are one and the same: for the ones are changed into the others, and reciprocally." Heraclitus apparently believed in the cyclic recurrence of all things, including our lives. The German philosopher Nietzsche tried to revive that doctrine at the end of the 19th century. The circular nature of becoming was to mark deeply another famous German philosopher, Hegel (beginnings of 19th century).
Parmenides, on the other hand, has left us long fragments of a poem written in the same meter as the Homeric epics; although there is no lack in it of goddesses and mystical symbols, the main thrust is austerely logical. The poem has two parts: the first is "the way of Truth," the second, "the way of Opinion." Parmenides' main truth is: We cannot think nor say not-being. Thus, he rejects outright the possibility of what I called the horribly difficult thought of not-being. Let me explain how he does it. Suppose I say, "Dragons are not (i.e. they don't exist)." Parmenides would reply: either there are dragons out there, in which case you are uttering a lie, or there are not, in which case your word "dragon" (and your thought) are about nothing. But a thought or a word cannot be about nothing, words and thoughts are like arrows, or like wasp stings: they must hit a target. If you say, "But my word `dragon' hits an idea of dragon I have in my mind," he would reply, "Then you're changing the subject: your word means an idea, not an object out there, and in that case, when you say that dragons are not, you're uttering a lie, for you say that the idea is in your mind." Similarly, if you tell Parmenides that elephants are not flying animals, he will reply the following: "You're saying that flying elephants are not, but as I told you before..." In summary, we cannot say that something is not, nor can we say that A is not B. Remember that Aristotle's principle of contradiction states that you cannot say at once that A is B AND A is not B; but Parmenides was far stricter: he stated that we cannot say that A is not B, period. The consequences of this strict logic are stunning. Change and becoming are stopped in their tracks, and differences between things are erased, for saying A is different from B is tantamount to saying A is not B. For Parmenides the truth about our universe is that it is timeless, eternal, motionless, perfectly uniform, the same all throughout. Being no fool, he knew that's not the way we experience it with our senses, so he allowed for the way of opinion (dóxa). The word dóxa meant not only opinion but appearance, prestige, fame, and many other things. What Parmenides was after, then, is the truth behind appearances, and what he was saying is that becoming and change are merely appearances; true being is changeless. I said before that philosophy struggles with the difficult thought of not-being, and also that all philosophy is paradoxical. I may add now that at the time of the Greeks as well as in our own, the number of people who care to think those difficult thoughts is very small: philosophy is by nature elitist.
Heraclitus and Parmenides seem to be on opposite sides: one affirms becoming and change, the other denies them. Heidegger, however, has written that those two philosophers say the same thing. We cannot trust Heidegger at his word, for in the same book, An Introduction to Metaphysics (published in 1935) he wrote that the US and the Soviet Union "are metaphysically the same"; Heidegger was a Nazi, a thoroughly unsavory character. What he said about Heraclitus and Parmenides (who, for all we know, may have been pretty unsavory characters themselves), makes good sense, though. Remember that being means appearing, emerging and enduring. What's enduring, true being for Heraclitus is not endless becoming but its circular path: things change, being turns into not-being, life turns into death, but change itself is cyclical, repeated for ever, eternal: it truly is. For Parmenides, true being is whatever is changeless behind the appearance of change. In their differing ways, both philosophers struggled to rescue eternal being from the flux of appearance and change. Both, heroically, tried to stamp becoming with the seal of being, which is the intellectual way of abolishing death. Remember the bully Gilgamesh, who was no intellectual: he tried to become immortal, and failed. Philosophers try it in a different way, by thinking immortal thoughts.