It's hard to explain the experience of expertise. That's why one of the first things they teach you in journalism school is to avoid questions like, "What's it like to be a mathematician?" It's hard for your interview subject to know how to respond and you seldom get a useful answer.
But not never.
On Quora, someone* asks, "What is it like to have an understanding of very advanced mathematics?" And the responses are surprisingly interesting. Especially the first, wherein an anonymous mathematician lays out a detailed account of how advanced mathematics have altered his/her view of the world and of being a mathematician.
• You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It's not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
• 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.
These are only two bullets on a multi-bullet post. You really should read the whole thing.
Great find, noggin!
*I couldn't tell who had asked the question. Maybe I'm just not familiar enough with Quora. If you can see a name for the thread's original author, let me know.