Empedocles tried to meet the Eleatic challenge by positing four elements that were themselves Parmenidean Reals, out of which the rest of the world arose. In this way he meant to account for apparent generation, destruction, and change by arguing that these phenomena are, in fact, just the mixing and separating out of the eternal, unchanging elements. Anaxagoras follows this model, but with some modifications. In order to better account for the full diversity of objects that populate our world, he posits an infinite number of Parmenidean Reals, called the homeomeric substances. Any substance without differentiated parts, on this view, such as flesh, blood, earth, or fire, counts as a Parmenidean Real.
Anaxagoras was born in Clazomenae in Ionia (the land of the Milesians) around 500 B.C. Like his Milesian predecessors, he was a busy public figure. For thirty years he lived in Athens, where he was the first philosopher to become a well-known teacher in the city that would soon become the hotbed of philosophy. Among his students were the dramatist Euripides and the famous Athenian politician Pericles. His association with Pericles ended up getting him in trouble; In 450 B.C. (or 430 B.C., some sources vary) he was prosecuted for impiety by the Athenian state (like Socrates and Aristotle after him), an event that was probably orchestrated by political enemies of Pericles. Unfortunately, popular outcry against Anaxagoras was heated, fueled in large part by his declaration that the sun was not a god but a hot mass of molten rock, larger than the Peloponese. He was convicted of atheism and exiled to the northern Ionian city of Lampascus, near Troy. He died there in 428 B.C.
A common belief among ancient Greeks was the idea that like generates like. In other words, ancient Greeks believed that X comes from X. In part, this belief was fueled by the Parmenidean-inspired conviction that X cannot possibly come from not X. This principle, commonly referred to as the "like-like" principle, is in part what led Empedocles to his theory of the four elements, but it is put to even better use by Anaxagoras. Though the four elements cover some of the most basic natural kinds in the observable world, they clearly do not exhaust all worldly existents. There are many qualities that seem to have nothing to do with these four elements. Where, Anaxagoras asks, do these qualities come from?
In place of the four elements, he posits an infinite number of Parmenidean Reals, or basic substances of existence, out of which everything else arises. In particular, he posits as the basic constituents of reality all substances without differentiated parts (homeomeric substances). A substance without differentiated parts is a substance that will remain the same substance no matter how small of large a piece of it you have. So, for instance, flesh is a homeomeric substance because a piece of flesh is just a piece of flesh no matter what size. A human body, however, is not a homeomeric substance because a piece of a human body is not the same as a whole human body. In a human body, the part is different from the whole. When it comes to bone, flesh, and marrow, on the other hand, whole and part are the same.
Like the four elements, the homeomeric substances possess some of the crucial properties of the Parmenidean Real: They cannot be generated or destroyed, nor can they change qualitatively. Also like the four elements, the homeomeric substances can be viewed as Anaxogoras's solution to the problem of physis, both in the sense of the original material out of which everything arose and in the sense of the unifiers within nature, of which everything else is a variation.
Being post-Eleatic, Anaxagoras acknowledges the fact that nothing comes into being from non-being, so he adds to his metaphysical theory the stipulation that all things are mixed in with all things. In other words, nothing comes into being from not being because there is a little of everything mixed in everything else. So, for instance, when a baby is born bald and then grows hair, that hair is not coming into being from not-being, rather what is happening is that formerly tiny, imperceptible hair parts mixed in with the scalp, are growing larger.