Here's a good book to get you to the next level with the Presocratics: THE PRESOCRATIC PHILOSOPHERS, by Kirk, Raven, and Schofield.

Remember we are in a geocentric universe and science and philosophy is breaking away from older traditions of literary myth-making. Take a look at the first pages of Hesiod's THEOGONY and you may recognize some of the influence on the language and ideas of the Presocratics. You may also want to visit Book 18 of Homer's ILIAD (478-608), where the poet describes the shield that Hephaestus fashions for Achilles. Homer, in general, is full of illustrations of the relationship between divinity and morals; and he also gives major accounts in THE ODYSSEY (see Book 10, for instance) of the shades of once-living mortals in the underworld. This includes ideas of how these shades interact with the world "above"; how they submit to justice for the lives they have lived; what kind of memory and knowledge they possess as shades; and what the nature of their non-corporeal substance is. These issues all tie in very importantly with the ideas of the Presocratics.

The major tool for Greek science is geometry, a branch of learning that will be codified in Euclid's ELEMENTS. Astronomers, working in the Ptolomaic system, will employ geometry to calculate the velocity of celestial bodies in motion. Below, I've posted some representations of Ptolomy's universe:

Here's the basic way the system works -- it's circles stacked on circles to describe the uneven motion of a given planet observed from earth:

This is a more complex model, which features an "ad hoc" invention -- an imaginary second center, which forms the basis of the calculations of planetary acceleration. Of course, if they only knew the equation for an ellipse, they would have had an easier time:

This shows the "outer regions" of the system, with the sphere of the fixed stars:

Zeno of Elea (Southern Italy), early 5th Century.

Zeno is, of course, aware of this -- that scientists use the concept of plotting points on a line to calculate the time a body with a constant velocity will arrive at a given destination. (Acceleration -- difficult to measure until Newton -- was treated as a kind of compound of constant velocities. But the acceleration of a body couldn't be too complex or else the math would become inaccurate.) What kinds of consequences does the geometric concept of time and space have for the phenomenon of motion in "reality"? How might some of the ideas of our philosophers reflect the rift between discrete and continuous models of motion?

Here's a mathematical illustration of one of Zeno's paradoxes, famously referred to as "Achilles and the tortise":

Permenides of Elea (Southern Italy), early 5th Century.

Parmenides frames the biggest problematic in metaphysics and what will later be termed "epistemology"—that of the relationship between being and becoming, and between truth and appearance. (See Curd’s introduction, p.p. 43-4.) His discourse takes the form of an epic poem. Why do you think he chooses this form? What advantages would it give a philosopher of “being”? Maybe you could consider the details of the opening passage of the poem (Curd, 45). Keep the character of the wisdom-giving goddess in mind when we get to Boethius.

Some key generalities from the reading:

--Parmenides seems to be denying the validity of all so-called "knowledge" (actually only doxa, or "opinions") based on SENSE PERCEPTION.

--He seems to be claiming that we can only know things that ARE, but that it is impossible that the things that ARE could have come to be.

--Why is this? Because BEING has all these characteristics of PERMANENCE.

--So, for Parmenides, does change even exist? If NOT, this is a HUGE conclusion, tantamount to Zeno's paradoxes drawn from mathematics except that it applies to the relationship between the physical and metaphysical, between becoming and being. Do you accept the elements of his argument? If so, how can we recuperate the physical, material world as a "real" domain? What things are there that are "real" and "knowable"?

Passage 1, Curd, 45: Here is a requote of the end of the passage, with a few key words highlighted:

There is a need for you to learn all things [panta]—

both the unshaken heart of persuasive Truth [Aletheia]

and the opinions [doxa] of mortals, in which there is no true reliance.

But nevertheless you will learn these too—that the things which appear [ta dokounta: “what is believed”]

Must genuinely be, being always, indeed, all things [dia pantos panta peronta: “being all things through all”].

From Kirk, Raven & Schofield: “Parmenides seeks to leave the familiar world of ordinary experience where night and day alternate, an alternation governed—as Anaximander would have agreed [Curd, 12]—by law or ‘justice’. He makes instead for a path of thought (‘a highway’) which leads to a transcendent comprehension both of changeless truth and of mortal opinion. No less important is his message about the obstacle to achievement of this goal: the barrier to escape from mortal opinion is formidable, but it yields to ‘gentle argument’.

          "The motifs of the gates of Day and Night and of divine revelation, modeled on materials in Hesiod’s Theogony, are well chosen to convey both the immense gulf which in Parmenides’ view separates rational enquiry from common human understanding and the unexpectedness of what his own reason has disclosed to him (cf. for both these points Heraclitus, e.g. [frag. 25, Curd, 32; frag 85, Curd, 38; frag. 36, Curd, 33]."

Passage 2, Curd, 45. Here is a tough question: The wording seems to indicate that there are two mutually exclusive “ways” to go about any subject of inquiry (the it of “it is…”). Thus we have the grammatical construction “the one […] the other […]”. But critics have bigger problems determining the sense of the Greek word for “is” (estin). The main debate is whether the “is” should be read as existential (“exists”) or predicative (“is [something or other])”. What do you think? Can examining other passages from the poem help to illuminate the sense of “is”?

Passage 8, Curd 47:

From KRS, 248: “If we must avoid the way ‘is not’, our only hope as enquirers lies in pursuit of the way ‘is’. At first sight it would appear that if we embrace that alternative, there open for us limitless possibilities of exploration: the requirement that any subject we investigate must exist seems to impose scarcely any restriction on what we might be able to discover about it; and the argument that what is available to be thought of must exist [frag. 6, line 1, Curd, 46] makes it look as though the range of possible subjects of investigation is enormous, including centaurs and chimeras as well as rats and restaurants. But in the course of a mere 49 lines Parmenides succeeds in reducing this infinity of possibilities to exactly one. For the ‘signs’ programmatically listed in [frag. 8, line 2, Curd, 47] in fact constitute further formal requirements which any subject of enquiry must satisfy; and they impose formidable constraints (note the metaphor of chains in [the passage]) on the interpretation of what is compatible with saying of something that it exists. The upshot of Parmenides’ subsequent argument for these requirements is a form of monism: it certainly transpires that everything there is must have one and the same character; and it is doubtful whether in fact anything could have that character except reality as a whole.

          “[Lines 5-21, concluding with ‘In this way, coming to be has been extinguished and destruction is unheard of’] are designed to prove that what is can neither come to be or perish. Parmenides is content to marshal explicit arguments only against coming into being, taking it as obvious that a parallel case against perishing could be constructed by parity of reasoning. He advances two principal considerations corresponding to the dual interrogative: ‘How and whence did it grow?’ (line 7). He assumes that the only reasonable answer to ‘whence?’ could be: ‘from not existing’, which he rejects as already excluded by his argument against ‘is not’ (lines 7-9). In his treatment of ‘how?’ he appeals to the Principle of Sufficient Reason. He assumes that anything which comes to be must contain within it some principle of development (‘need’, chreos) sufficient to explain its generation. But if something does not exist, how can it contain any such principle?”