Philosophy in the Age of Distraction

I am struggling to define some relationships between Homer, Heraclitus and Protagoras based upon my limited knowledge of Greek poetry and philosophy. I suppose one might use the lengthy quality of Homeric epic poetry versus the short aphorisms of the latter two philosophers as a basis of comparison. I have also considered Heraclitus and Protagoras as thinkers working within different frameworks of "philosophy" as we might define the term today. Philosophy is in my opinion a formalistic inquiry into how the world exists and how reality is defined. This includes how humans might conduct themselves and also how they might construct a system of justice or define political life. We can see these lines of inquiry emerging in Protagoras and Heraclitis. As these kinds of philosophical considerations are embedded in Homer's oral epic poetry I feel that they are less systematically described.

Additionally, philosophizing in the time of the Presocratics appears to be open to debate, and debate does not seem to be part of the manner by which Homer constructs his narrative. Although the characters within the Iliad may debate each other, Homer's epic was not recited nor performed to elicit debate; in other words, I gather that Homer was not thinking about "philosophy as rhetoric" per se in his performances. He was not concerned with crafting an art of speech to defend one's rights and property like Protagoras who, according to Aristotle, even justified his right to take money to help people "better" themselves in the art of rhetoric.

Snell argues that the "Homer conceived of the thing which we call intellect in a different manner, and that in a sense the intellect existed for him, though not qua intellect." I am not quite sure what Snell means by intellect. Does Snell mean the construction of human subjectivity when he uses the term "intellect"? Another Snell question might be important here: "What did the Greeks at any given time know about themselves, and what did they not yet know?" Does Snell mean "self-consciousness", which could mean anything from having consciousness about one's world to theorizing about it? Aristotle writes about (and this is taken from a fragment in the Protagoras reading) how Protagoras struggled with geometry and mathematics as a legitimate subject of inquiry ("As Protagoras says of mathematics, the subject-matter is unknowable, and the terminology distasteful.") I cite this as an example of something that some "Greeks" did not yet know, or at least something that they could not agree on as an object of inquiry.

Finally, a comment on methodology: I can't even imagine how difficult it must be to construct a coherent philosophy of thinkers like Heraclitus and Protagoras from fragments of mostly secondhand sources.