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.
Anaxagoras's theory of mixture represents a totally new move in the plurality question. Whereas the other Presocratics (Parmenides and his followers excluded) asked how a plurality could arise out of a unity, Anaxagoras bypasses the whole issue by claiming that there is a full plurality to begin with. There never really was a unity at all, since in every piece of matter, no matter how small, pieces of every other homeomeric substance is mixed in (to make this view possible, incidentally, Anaxagoras needs to hold to his principle of infinite divisibility, for which he is indebted to Zeno of Elea).
Given that all things are in all things, it might seem that the process of identifying objects should get rather confusing, but in fact Anaxagoras has a neat solution to this worry. We identify things by the proportional amount of homeomeric substance in the mixture. Just like on Empedocles' view, objects were identified by the ratio of the four elements in their recipe, here too objects are identified by the dominant homeomeric substances in their recipe.
Like his Milesian forebears, Anaxagoras uses his metaphysical theory as the basis of a cosmogony (or theory of the origins of the world). He paints a picture in which originally what existed was a primordial mixture containing all things. This original mixture was then set into motion, and the different parts were separated off. These then recombined with each other to produce the world as we perceive it.
What Anaxagoras means by claiming that "all things" were in this original mixture is unclear. He probably means that all homeomeric substances were originally in this mixture, and that they were what then separated off when the motion began. This seems like the most plausible interpretation, especially given the fact that the homeomeric substances, as Parmenidean Reals, could not have been generated. Some commentators, however, have read this passage differently. Some believe that Anaxagoras meant to claim that all possible objects of the world were in this mixture, but there seems to be no reason for him to have made this strong claim. Others believe that it was only the opposites that were in this mixture, but this would then make it seem as if the homeomeric substances were generated.
Because of the Eleatic attack on change, Anaxagoras could not make a weak appeal to eternal motion when explaining how his original mixture got moving. To what, then, does he appeal? The confused mass gets started in the process of differentiation by the one thing not mixed in with anything else: mind or nous.
Anaxagoras's mind is somewhat akin to Heraclitus' logos, insofar as it is the rationality that controls nature. Mind is infinite and self-ruled, and like the logos, it is very much a part of the physical world. It also, however, takes on the role of Empedocles' love and strife as the motive force in the world.
In order to arrive at the notion of mind, Anaxagoras looks to the human analogy, drawing once again on the microcosm/macrocosm principle. In the human sphere, he reasons, when things are in confusion, it is by an activity of mind that they are set in order. The human mind sorts out the confusion, distinguishing one thing from another and putting them all in their proper place. So in the cosmos it must be a mind, which is somehow active in nature, that orders and controls. As we would expect from this analogy, Anaxagoras's mind is characterized primarily by the power to distinguish and separate one thing from another and thereby to create order in the universe.
Anaxagoras's idea of mind probably served as the inspiration for Aristotle's notion of a final cause or telos. Aristotle bases his entire scientific enterprise on the idea that there is a purpose within nature, and that all motion can be explained in terms of the striving of each individual object to fulfill that purpose. The idea of a controlling rationality within nature, that acts as the source of all motion, first found its voice in Anaxagoras.