What It Is Like to Have an Understanding of Very Advanced Mathematics

1. You can answer many seemingly difficult questions quickly.

2. You are often confident that something is true long before you have an airtight proof for it.

3. You are comfortable with feeling like you have no deep understanding of the problem you are studying.

4. Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled.

5. 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.

6. 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.

7. The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques.

8. The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques.

9. 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.

10. You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles.

11. Understanding something abstract or proving that something is true becomes a task a lot like building something.

12. You are good at generating your own questions and your own clues in thinking about some new kind of abstraction.

13. You are easily annoyed by imprecision in talking about the quantitative or logical.

14. On the other hand, you are very comfortable with intentional imprecision or "hand waving" in areas you know, because you know how to fill in the details.

15. 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.