Quora, a sort of wikipedia of question-and-answer sessions, has an excellent discussion about what it’s like to work in advanced mathematics. You should read it all here. A few tidbits* to whet your appetite:
. Mathematicians don’t really care about “the answer” to any particular question; even the most sought-after theorems, like Fermat’s Last Theorem, are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing.
. When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights. (The trivial version of this – i.e., math at my level – is to substitute 0 or 1 for X in the equation you’re trying to solve and see what happens.)
. You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion
*titbits – a term that brings out the 12-year-old boy in me – if you’re British