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In all likelihood, Anaximander was the student of Thales. He was born around 610 B.C., also in Miletus. Like his teacher, his main interests were in natural philosophy, geometry, and astronomy. Also like his teacher, he was apparently a very busy figure; reports tell us that he was the first man to construct a map of the known world, the first to build a sundial, and the first to build a celestial globe with a chart of the stars.
Anaximander was a material monist, but the physis he posits is much more conceptually sophisticated than that of Thales. The evidence we have for him provides a slightly more thorough picture of his thought. Among the many impressive bits and pieces, we can find the first use of the famous principle of sufficient reason and what appears to be the first statement of evolutionary theory.
In Thales' system water was supposed to be the source out of which everything arose. In Anaximander's system the Unbounded (or apeiron) is both the source out of which everything derives and also the unifier within nature. That is to say, in Anaximander's worldview, everything in the universe is in fact just a variation on the Unbounded.
As a physis, the Unbounded has many theoretical advantages over water. As Anaximander no doubt reasoned, the water theory leads to a real problem for the existence of fire. If everything derived from water (and especially if everything still is water in some sense), fire could not possibly survive. In order to allow for the existence of all of the opposites, Anaximander decided to make his physis indefinite, having no particular qualities of its own. Because the Unbounded is entirely neutral between opposites, it does not pose a threat to any of them.
This is clearly a major conceptual step forward. Anaximander has separated his explanatory entity from the entities that need explaining—always a good first step in any explanatory enterprise. He has posited, in fact, the first theoretical entity—an entity that we cannot observe, but that whose existence we infer because of its explanatory role.
In addition to being indefinite, the Unbounded is also limitless or infinite, both temporally and spatially. The Unbounded must be limitless because it must be inexhaustible in order to give rise to everything else in the universe. It must have unlimited potentialities.
The line "is this a dagger which I see before me" is from Macbeth, not Hamlet. C'mon, Sparknotes! I expect better from you.
Is it possible that Parmenides was referring to the object orientation of our thoughts with his famous saying that "what is is and what is not is not"? Consider that when separating an object from its background, we can conceive of the object as something but can not conceive the background as a thing. The object is "what is", while the background is "what is not". This interpretation fits well with several ideas of the time, for example that opposites had a special position in our thought, that the universe is one (Zeno's paradoxes
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