This in the continuation of an earlier attempt to assign the major characters of the Republic places on the Divided Line:
Thrasymachus (dianoia as skeptical apistia) — Dianoia is the power of grasping the insufficiency of opinion as such. Opinion must be ultimately grounded in a higher order reality or it will wither under skeptical dianoia. Thrasymachus takes the cynical point of view that justice is an arbitrary invention of those in power. In doing so, he grasps the conventional/contingent/malleable nature of opinion as such, which is a defining feature of dianoia. Mapping him on the Cave Allegory, Thrasymachus sees that the puppets are not the realities the credulous masses believe them to be and wants to be a puppeteer. He even thinks Socrates’ “method” is exactly that, and consists of tricks to mold and generate self-serving beliefs in his interlocutors. (338d) Thrasymachus is faithless, i.e. suffers from apistia, and does not see the beliefs as defective images of some higher level reality. His dianoia is downward looking, a means of critiquing (others’) beliefs in order to undermine them and replace them with self-serving substitutes. Yet, his notion of “advantage” must be based on a criterion other than mere opinion in order to provide the infallibility his notion of justice seems to demand (341a). Advantage cannot itself be a product of arbitrary will, if will is to be well-guided by appeal to advantage.
Both Adeimantus (Dianoia trying to secure a better pistis) and Glaucon (Dianoia leading upward to noesis), brothers of Plato, accept that Thrasymachus’ cynical account of justice has some sort of force. They understand and raise questions about the contingent genesis of belief, even their own beliefs. But, unlike Thrasymachus, neither is willing to abandon their belief about justice. The two essentially beg Socrates to help them secure their belief on a securer basis against Thrasymachus’ threat. Both trust that justice must be more than an arbitrary product for the self-service of those in power. Their willingness to expose their own beliefs to testing is evidence of the dianoietic virtue in them. But there is an important difference between the two brothers. At one point in the dialogue while discoursing about The Good, Adeimantus essentially gives up looking up toward noesis and requests an adequate opinion from Socrates:
Adeimantus: But, Socrates, you must tell us whether you consider the good to be knowledge or pleasure or something else altogether.
Socrates: What a man! It’s been clear for some time that other people’s opinions about these matters won’t satisfy you.
Adeimantus: Well, Socrates, it doesn’t seem right to for you to be willing to state other people’s convictions and not your own, especially when you’ve spent so much time occupied with these matters.
Socrates: What? Do you think it’s right to talk about things one doesn’t know as if one does know them?
Adeimantus: Not as if one knows them…but one ought to be willing to state one’s opinions as such.
Socrates: What? Haven’t you noticed that opinions without knowledge are shameful and ugly things? The best of them are blind — or do you not think that those who express a true opinion without understanding are any different from blind people who happen to travel the right road?
Adeimantus: They’re no different. (Grube/Reeve, 506b-d — I have added the character names before each line.)
At this point Adeimantus drops out of active participation until Book VIII, while it is Glaucon who participates in the very heights of the dialogue: the Sun Allegory, the Divided Line and Cave Allegory. Adeimantus is ultimately downward-looking, using dianoia to hone and sharpen belief but never advancing beyond this honing. Glaucon, on the other hand, is upward-looking and never abandons the upward quest towards noetic truth.
Socrates (Noesis) — I am the least confident here, since Socrates continually refuses to own any claim to knowledge and knowledge is located at the noetic stage. But Socrates continually maintains noetic openness to his lack of other knowledge. He is never satisfied with mere opinion as such and never confuses opinion with knowledge. I think that Socrates, in denying any claim to knowledge is really denying any ability to adequately express his knowledge in a way understandable to others. Any logos of knowledge will be only a ready-to-hand opinion, however true. But I am unsure whether Socrates is just the highest stage of dianoia or a full participant in noesis. The Republic, after all, never advances beyond the dianoietic level. It’s claims are all hypothetical, defectively pointing toward realities that it is unable to express directly.