How can a mathematician handle the pressure to discover something new?
This is ancient history, and considering my age, I may have told this story here before. I started at Harvard graduate school in 1957, the same year that Hironaka arrived there to work with Zariski. He was already an accomplished mathematician, even if he didn’t yet have a PhD. Early that year, I must have said to him that I couldn’t imagine ever doing research, and he said, in essence, Oh, you learn about some subject, think about it in depth, and before you know it, you’re proving Theorems. I thought, This guy must be in Cloud Cuckoo Land, I’ll never do that. But of course, that’s exactly what happens.
The theoretical physicist Richard Feynman was in a similar state of mind, he referred to it as a "burn-out" feeling: Now that he had landed the University professorship he had strived for, he felt the obligation to do something "important", but he had lost the joy of doing science just "for fun". In his autobiography he describes how he recovered:
Then I had another thought: Physics disgusts me a little bit now, but I used to enjoy doing physics. Why did I enjoy it? I used to play with it. I used to do whatever I felt like doing - it didn't have to do with whether it was important for the development of nuclear physics, but whether it was interesting and amusing for me to play with. So I got this new attitude. Now that I am burned out and I'll never accomplish anything, I've got this nice position at the university teaching classes which I rather enjoy, and just like I read the Arabian Nights for pleasure, I'm going to play with physics, whenever I want to, without worrying about any importance whatsoever.
Within a week I was in the cafeteria and some guy, fooling around, throws a plate in the air. As the plate went up in the air I saw it wobble, and I noticed the red medallion of Cornell on the plate going around. It was pretty obvious to me that the medallion went around faster than the wobbling. I had nothing to do, so I start figuring out the motion of the rotating plate. [...] Then I thought about how the electron orbits start to move in relativity. Then there's the Dirac equation in electrodynamics. And then quantum electrodynamics. [...] The whole business that I got the Nobel prize for came from that fiddling around with the wobbling plate.
A friend of mine once told me quite brightly about mathematics that "when you search, you find". As naive as it sounds, the more the times passes, the more I believe it.
Moreover, mathematics is not only about finding. Part of the job is also made of acquiring a deeper understanding of concepts, relearning your field, sharing your knowledge and interests, and many "new" ideas come from this crossed points of view and interactive community. Take any topic, spend time into it as a whole, and you will get used enough to it to naturally understand what is missing, what could be asked, what could be tried, etc. In this spirit, "failure" (not finding as easily and what expected) is also part of the job: failing to do something is underlining what is missing, being able to share it, and this will provoke ideas, will lead you to ask specific researchers for the missing steps, and will plant seeds that will eventually blossom.
What is not easy to handle is that it takes time, and hence it requires confidence. This is an important psychological aspect of doing mathematics. When you share with other researchers, you realize that they are just as human as you, and not the apparently perfect spirits they seem behind their articles :) I believe it is necessary to mourn the will to be perfect and embrace everything, we are all doing part of the effort, this is essential to understand and believe.
Maybe could I suggest reading Netz's article on "deuteronomic texts". It is a famous work on history of mathematics that shows how much of the advances and efforts in (greek) mathematics have been made after the one who we credit with the discovery, by "digestion": treating examples, spreading the ideas, rewriting, unifying, choosing brighter notations, etc. I always thought it gives a good taste of how a community should work. And mathematicians are a community, it always worked through sharing, attempts and failures (look even at Gauss' or Serre's letters).