Someone posed this question on Quora:
I’m interested to hear what very talented mathematicians and physicists have to say about “what it’s like” to have an internalized sense of very advanced mathematical concepts… I’ve always been curious about this. Does it feel analogous to having mastery of another language like in programming or linguistics?
And an anonymous writer posted a very detailed and interesting answer. A few snippets:
- You are often confident that something is true long before you have an airtight proof for it.
- 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.
- You go up in abstraction, “higher and higher”. The main object of study yesterday becomes just an example or a tiny part of what you are considering today.
- You move easily between multiple seemingly very different ways of representing a problem.
- Understanding something abstract or proving that something is true becomes a task a lot like building something.
- You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer.