Frodo as Paradox
Frodo's character development is one piece of evidence many Tolkienites use to support the statement that in LotR Tolkien wasn't just imitating older forms of literature but was, in fact, writing a very modern book. If we pause to look at it, Frodo's psychological complexity is at least as strong as any character in a "modern" novel. The fact that this complexity is so built into the character that we might not notice it if we don't pause to look at it, speaks of Tolkien's skill as a writer along with, I believe, his understanding of human nature. I think the only kind of author who could have made Frodo real would have been an author who knew that human beings are living mysteries, because Frodo is a walking paradox. But we accept his paradox as true because we've probably experienced it within ourselves.
At this point, I'm looking beyond the idea of Gollum as Frodo's shadow side, and even beyond Shippey's insightful statement that, for Tolkien, the line between good and evil "runs through the human heart," although that applies vividly to Frodo. We don't have to look for evil in Frodo to see the seeming contradictions he lives with especially, but not only, during the Quest. They're similar to those of most people who try to follow the right course although they're not always sure what the right course is, who rely on God's help although they're not always confident it will come, and pray even when they can't say with certainty that there's anyone to hear them, who cry out, "I do believe. Help my lack of faith." That is, people who are a lot like us.
This became more clear to me as I reflected on the "Did Frodo Fail?" question. And it's vital as we look at Frodo's spiritual journey. Even "the best hobbit in the Shire" isn't perfect. He may be St. Frodo, but with few exceptions saints have to struggle with their faith as much - or even more - than those who don't travel as far along the Road to spiritual perfection in this life.
There's one aspect of the Quest that can be so difficult to deal with (at least from anything other than a faith-based approach) that both animated and live-action movies have completely ignored it. That's the fact that, from the very beginning, Frodo knows he can't destroy the Ring; so do Gandalf and Elrond. Tolkien shows us early on that Frodo can't perform the necessary action even in his own parlor, and also told a reader that Frodo accepts the Quest with love and humility, "knowing himself to be completely inadequate to the task." Of course, anyone would have been inadequate to the task; one of Frodo's strengths is that he recognizes and admits it. He accepts the role of Ring-bearer because he knows that he's been called to it, not because he thinks he can accomplish something that no one else can.
If we keep this fact in our minds throughout the Quest, especially after Frodo and Sam leave the rest of the Fellowship, it becomes an amazing - almost incredible - statement of faith. All the while Frodo is looking for a way out of the Emyn Muil, forcing himself through the Dead Marshes, planning to enter the Black Gate and then accepting Gollum's promise of another route, climbing the stairs, encountering Shelob, following Sam from Cirith Ungol, crossing Mordor while struggling against not only physical thirst and exhaustion but also the mental torment caused by the Ring, and crawling up the side of Mount Doom, he knows he can't destroy the Ring! In fact, he becomes more and more aware of that as time goes on and the Ring begins to overpower him: 'And if you tried to take it from me, I should go mad.' Even Sam begins to understand this, although he doesn't want to listen to himself about it: 'He won't be able to do anything for himself' is a pretty loaded statement under the circumstances.
Despite that, Frodo keeps going, trusting not in himself but in Whoever it was who called him to the task. It's been said that Frodo's courage is in putting one foot in front of the other; if we look at that in the context of his knowledge of his own inability to complete the task, it's a statement not only of courage but also of radical faith in that Caller whose existence he knows only from the fact that he's been called.
But (a paradox always needs a "but") if Frodo accepts the Quest in all humility, knowing himself to be unequal to the task, why does he feel guilty about not being able to complete it? Tolkien calls it "unreasoning guilt" - it's not rational or logical. But I think it's very, very human: so human, in fact, that it might not seem like a paradox to us at all.
This same paradox exists even during the Quest, as when Frodo tells Faramir, 'I must find the Mountain of Fire and cast the thing into the gulf of Doom.' He says that, though, before the part of the Quest that's primarily responsible for breaking him down so that he can be made into something quite different. And that process is never simple, easy, rational, or without paradox. Later, when deciding what path to take in Mordor, Frodo tells Sam that he has to '...do the best I can. At present that is to avoid being captured as long as possible.'
True humility is seeing yourself as you really are. Pride can be a barrier to holiness, but so can the false humility that might cause us to see ourselves as less than we are. Tolkien says that in order to be healed in the Undying Lands, Frodo would need to understand himself "in his littleness and in his greatness," which I think is a marvelous description of true humility. Frodo didn't singlehandedly save Middle-earth, but yet, as Elrond says at the Council, his place is among the mighty elf-friends of old. This isn't a contradiction, but post-Quest Frodo likely saw it as one, perhaps even feeling that he'd lost that place by his "failure" at the Sammath Naur. His healing would need to include the reconciliation of these two sides of what he saw as a contradiction, or, at least, his acceptance of it as a paradox that he couldn't fully grasp, part of the mystery of himself as an infinitely knowable person.