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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!)

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?

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+1. Good question :-) Should this be CW though? – Aryabhata Feb 3 '11 at 10:26
Probably one of the greatest examples of "asking the right questions" is that of Hilbert's twenty-three problems. – Rahul Feb 3 '11 at 10:45
As a side comment: that is not a sentiment that is original or limited to Ravil Vakil. Almost every senior mathematician I've met said that (in some form) at one point or another, and it is something that I also plan on telling my (future) students. One version is due to Georg Cantor: "To ask the right question is harder than to answer it." (Which I saw quoted around the preface to Arnold's Problems by VI Arnold.) It probably goes back even further. – Willie Wong Feb 3 '11 at 12:01
up vote 8 down vote accepted

Here's my take; I'm sure there are other, better answers. (community wiki?)

On the one hand, you could ask a question that isn't very meaningful (e.g. why is 2 the only even prime), while on the other hand you could ask a question that in all likelihood you'll never be able to answer (e.g. describe explicitly the group of diffeomorphisms of a given manifold). The trick is to shoot somewhere in between, to find fruit hanging high enough that it's interesting and deep but low enough that you actually have a fighting chance.

Another (related) interpretation is that there are certain questions that have motivated a whole lot of fascinating research and discovery (why isn't there a quintic equation, what are the maps between spheres up to homotopy, are there or aren't there an infinite number of twin primes, etc.), and it takes a good understanding of the meta-structure of the field of mathematics and the way it progresses (or perhaps occasionally a blatant disregard for that meta-structure!) to hit upon such questions. Anyone can plug away through book after book, but it's as least as important to try to come to grips with the bigger picture.

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Once you move on to original research, there will no longer be exercise questions on some sheet of papert. There might be some open problems in your area of interest. But in general, it will have to be you yourself who finds an interesting problem, formulates a (new) question about this problem and answers it. A set of such question-answer pairs might then lead to a solution of the problem.

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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.

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