On what did these two base their philosophies? What did they see as the authority for what they had to present and on which their arguments rested?
This is a question that is not as easy as it seems. It depends first what you mean by authority. Take Christianity for instance. We might think that it was based on the ultimate authority of the life and teaching of the Lord Jesus and on the ideas He taught. Simple, is it not? No, it is not at all simple. There are plenty of organizations which claim to speak for Him. They all claim to be “God's organization” or “the True Church” or something similar. Then there are the problems of Bible versions, each one, or rather its publisher claiming that their's is the most accurate or closer to the original text. It is something of a joke that a man like Rupert Murdoch, the pornographer in chief, of popular tabloids, has the privilege of publishing and distributing at least one version of the Bible.
So then, if we say that Plato's authority is “The Good” we are immediately faced with the problem of what it is that “The Good” is. Well what is it? The simple answer to that is, “I do not know,” nor it seems did Plato. If he did he is remarkable coy about defining it even though Socrates often asks his stooges for definitions. When, in The Republic, Socrates sets about finding what "justice" is, all he can give us is the example of an ideal state which we are meant to believe is a "just" society. It is "just" apparently because it is governed by philosophers who have had a vision of “The Good” and, therefore know what justice is and how to rule accordingly.
In a series of analogies Plato, through the mouthpiece of Socrates, gives us the idea of what it is and how it is found. The analogy of the “Cave” is one. People sit in darkness and look at shadows of things cast by a light behind them on the wall in front of them. If someone were to escape and to get out into the sunlight he would at first be dazzled, but then, as he grew accustomed, he would be able to see things as they really are. The climax of this is that the gifted person would, at last be able to see the sun itself, the giver of light, and this is the same as first seeing good things and then “the Good” itself. It is somewhat like a concept, but writ large.
The “divided line” is another analogy, though it is hardly that, it is just a list of levels of perception and understanding with a division between the things in the world of sense and in the heavenlies, where only philosophers venture. It is in the latter that one perceives things as they really are, and not their changing improperly perceived state as by folk in this world. This is why Plato, in the famous picture, is shown pointing upwards and Aristotle is pointing downwards.
In our world of sense, lit by the sun, we see, in the dark of the cave only reflections and shadows. Moving upward we see, with increasing clarity, animals, trees and manufactured things. The first state is one of conjecture, the second of belief. Next, in the real world, the world of ideas, as in the heavenlies, lie thought, images, ideas. Here comes the study of abstracts, it is the area of mind. On the lower part of the line was the world of sight only. On the second part of the line comes mathematical thought, then finally, dialectical thought with the pure abstractions of justice, goodness, beauty and such.
In order to bring these out Plato insisted on a general education, including military service then a rigorous study of mathematics and music before going on to a diet of the "hard stuff" as Bertrand Russell has put it. This lasted up to thirty years of age and the successful graduate became a Guardian of the "just" state.
(BookVI,508E-511E and BookVII,514A-519C)
In another way it might be said that while “The Good” is an absolute, like God, yet it requires the education of reason for the superior human being to attain to “the Good.” So we might say that it is "reason" or rationality by means of dialectical argument that is also authority for Plato, in that reason is necessary if we are to attain to “The Good”.
The “founding lie,” is more than just a way of getting round difficulties. On one hand it is the perennial attempt of the elite to keep knowledge from the lower classes. Plato says, “while God moulded you, He mingled gold in the generation of some, and those are the ones fit to rule, who are therefore the most precious. He mingled silver in the assistants(auxiliaries), and iron and brass in farmers and the other craftsmen.” (Book III 414A-415E) We see that Plato had in mind a caste system and deemed it necessary to have the common folk believe the lie/myth or fable that inequalities were designed by God.
