From Not To Go on Four Legs:

INTERVIEWER: So what is it like to spend years of your life working on a mathematical problem of this magnitude?

PROF. BARRA: My students have asked me that, and I have a rather elaborate metaphor, if you don't mind. It's like a dream in which you go to climb a rock pinnace. You can't see the top of it, and while you can plan your ascent from the bottom to some degree--I'll put my hand there and my foot there, and then I'll be able to reach there, and so forth--you can't really know how you can climb until you begin. And of course you don't really know what's at the top.

I: But you have an inkling.

B: Yes. You know something of the shape of the rock, and something about how you're going to climb it, and something about what you'll find at the top. But not very much! And as you climb, you can spend as much time examining each hand hold as you like. In fact, since it's a dream you have that peculiar ability to focus on one object to the exclusion of everything else, and you can see every detail perfectly. That object can become your whole world, and it's easy to forget about the rest of the climb.

I: What about the climb?

B: Well, if you can keep a memory of where you're headed, you just keep finding these holds. They can be as tenuous as you want, as long as they'll support you. It's just a question of finding a new hold and moving a little bit further every time. Of course you can get stuck!

I: And then what?

B: Well, you can try to climb down a bit and find a new path. Or you can try to carve out a hand hold. But sometimes you fall. Of course the only thing that happens if you fall is that you wake up. Nobody has ever died of an unproved theorem.* But I'm sure you know how hard it can be get back to a dream after you've awoken....

*Does Archimedes count, do you think? μή μου τούς κύκλους τάραττε and all that.