Stephen has something to prove in this chapter. In a way, this is hardest on the reader, because it means he has all of his genius going at full blast, and his argument can become quite involved and difficult to follow. Most immediately, he is trying to show off for the renowned Irish critic John Eglinton and the well-known mystical poet George William Russell (A.E.). Stephen refuses to align himself with either of the men. He wants no part of their role in the Irish Literary Revival, and does not think much of Russell's mystical approach to writing. At the very start of the chapter, he argues for the importance of Aristotle, who thought art should be grounded in the material world, against that of Plato, who thought that art needed to capture formless spiritual essences. Despite his defiance, however, Stephen very much wants to impress both of them. He is particularly hurt when they make no motion to invite him to George Moore's reading that evening, and when he discovers that Russell is not going to include him in his collection of young Irish poets.
But there's another figure that Stephen is struggling with in this chapter, one not actually present in the room. The figure is William Shakespeare. As Stephen elaborates his theory, we see that it is quite far from a traditional literary argument. It's wildly clever and often stated quite eloquently, but what becomes clear as the argument moves on is that Stephen is attempting to figure out his own artistic purpose through Shakespeare. Stephen argues for an art that can be grounded in everyday life (as Joyce did) by elaborating ways that Shakespeare pulled material from his family life for his plays.
Occasionally, we see Stephen's thoughts turn to his own experience even as he is discussing Shakespeare. For instance, when he is talking about what a raucous time Shakespeare must have had in London, he remembers his own conversation with a prostitute while he was in Paris. We here get a glimpse of the intensity of Stephen's literary ambition: he views himself as being on par with Shakespeare.
You'll notice that Stephen's argument becomes particularly pointed when he argues for the irrelevance of fathers. Stephen says, "a father is a necessary evil," and argues that "fatherhood, in the sense of conscious begetting, is unknown to man. It is a mystical estate, an apostolic succession, from only begetter to only begotten" (9.301). On the simplest level, what he is saying is that the process of childbirth is almost completely relegated to women. If a mother doubts that she is going to create a new human life, then she is reminded of it day in and day out for nine months, which culminates in the pain of childbirth. For men, however, the process of creating a new human life is tied to a relatively brief sexual act. Beyond that, they must struggle to imagine what it means to create another human being. This is the reason that fatherhood "is unknown to man." The theme will come up again in "Oxen of the Sun" when we see the process of literary creation compared to childbirth. In short, women give birth, and men, to compensate, write books.
It's not just the comparison of literary creation to childbirth that has Stephen so impassioned. We have seen evidence in the first eight episodes that Stephen feels estranged from his father, Simon, who is more of a drunkard and a town jokester than he is a father. There is little to indicate that Stephen's father takes much interest in his son or understands his ambitions. The lack of a father figure is one of the major reasons that Stephen seems so isolated in the novel, and it will become increasingly apparent as the novel develops that Bloom can become a sort of surrogate father figure for Stephen. For the moment, though, that surrogate father is busy looking up the skirts of Greek statues in the Library lobby.
Now, in Book 12 of the Odyssey, Odysseus and his men must find a passage between Scylla and Charybdis. Scylla is a six-headed monster that lives on a sharp mountain peak, and Charybdis is a giant whirlpool. Circe has already told Odysseus that to pass through he must steer closer to the rock of Scylla than to the yawning mouth of Charybdis. The correlation to the two figures is clearest in the opening of Stephen's literary argument. The way that Aristotle grounds art in material reality is comparable to the hard rock of Scylla, whereas the way that Plato pushes for art as a revelation of the ideal, of formless spiritual essences, is more like the whirlpool. Both Eglinton and Russell are more sympathetic to Plato whereas Stephen stays closer to Scylla (Aristotle).
In the Odyssey, Odysseus forgets Circe's advice and tries to engage Scylla in combat. As we see the details of Stephen's biographical argument become more and more complex, we might draw another comparison. Stephen briefly forgets to steer his course and instead gets too wrapped up in his own argument. Ultimately, it is Eglinton who has to remind him that the truth lies midway between the two different stances.
Stephen's ability to get too involved in his own argument becomes especially apparent toward the end of the chapter. Even as he and Mulligan are walking away, he is still summoning details to support his point. He has to silently coax himself to stop thinking about Shakespeare. In "Proteus" Stephen's intellectual ramblings seem very free and loose. Here, we get a sense of how one's own mind can seem like a prison when one is as intelligent as Stephen is. In short, he can't stop thinking. We see this when Stephen wraps up his argument: "he laughed to free his mind from his mind's bondage" (9.365). Laughter, for Stephen, is a way of calming down his mind, a way of not taking himself too seriously.
Stephen's inability to stop intellectualizing again becomes apparent when Eglinton asks him if he believes his own theory. He promptly says that he does not. Why not? Well, to believe in something is not the same as to know something. Belief requires that, at some point, one suspend one's questioning and intellectualizing and just accept that which one feels to be true. At this point in the novel, Stephen is incapable of controlling his thoughts. As much as he wants to, he doesn't yet believe in anything.