An anonymous Quora commenter has written an exhaustive and fascinating response to the question, "What is it like to understand advanced mathematics?"
While some of the answers focus on the feelings that go with a mathematical mindset, the most interesting parts describe the mindset itself: its preferential areas of focus ("You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles") and the things it shies away from ("Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable").
To me, the biggest misconception that non-mathematicians have about how mathematicians work is that there is some mysterious mental faculty that is used to crack a research problem all at once. It's true that sometimes you can solve a problem by pattern-matching, where you see the standard tool that will work; the first bullet above is about that phenomenon. This is nice, but not fundamentally more impressive than other confluences of memory and intuition that occur in normal life, as when you remember a trick to use for hanging a picture frame or notice that you once saw a painting of the street you're now looking at.
In any case, by the time a problem gets to be a research problem, it's almost guaranteed that simple pattern matching won't finish it. So in one's professional work, the process is piecemeal: you think a few moves ahead, trying out possible attacks from your arsenal on simple examples relating to the problem, trying to establish partial results, or looking to make analogies with other ideas you understand. This is the same way that you solve difficult problems in your first real maths courses in university and in competitions. What happens as you get more advanced is simply that the arsenal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try. Sometimes, during this process, a sudden insight comes, but it would not be possible without the painstaking groundwork [ http://terrytao.wordpress.com/ca… ].
Indeed, most of the bullet points here summarize feelings familiar to many serious students of mathematics who are in the middle of their undergraduate careers; as you learn more mathematics, these experiences apply to "bigger" things but have the same fundamental flavor.
What is it like to understand advanced mathematics?
[Anonymous/Quora]
(Thanks, Fipi Lele!)
(Image: A mathematics lecture, Tungsten, public domain)
