TITLE: Explanation of a Phrase from Prof. Ravi Vakil's Website
QUESTION [8 upvotes]: I was just browsing through Ravi Vakil's website when i found a nice article written on what he demands from his students. Here is the webpage. (For those interested!)

http://math.stanford.edu/~vakil/potentialstudents.html

I actually found a line which caught my attention. He says:

"Mathematics isn't just about answering questions; even more so, it is about asking the right questions, and that skill is a difficult one to master."

I would like somebody to explain the meaning of this statement. What does asking the right questions mean here?

REPLY [3 votes]: I would say that most of the advancements in Mathematics were made as a result of trying to resolve conjectures.
Good conjectures themselves arise from asking questions, picking the right ones and failing to answer them.
Even during problem solving, many times, one finds that you need to make assumptions (which are really questions in disguise) which help solve the problem, and then verify/prove that the assumptions are true. It is probably easier to verify/prove the assumptions than coming up with them, and without making the right assumptions the problem would probably be unsolved.
Without new questions, there would be very little advancement.