Given that pure mathematics is, by definition, not concerned with applications, how does one decide that one problem is more valuable than another? Is it just a matter of certain topics becoming popular among the community of research mathematicians? Is it just a matter of doing something very difficult? Is it about solving something that people previously failed to solve? Is it about doing something that will get you a better job, and if so, how do employers decide what is "important?"

I know from experience that when one is working on an interesting problem, this question tends to matter less. The math is interesting for its own sake. But that doesn't mean it is going to be deemed valuable by the greater community. I also feel that much of mathematics, especially at the beginning of one's career, is motivated by trying to impress people who are more established. But how do they decide which problems matter and which ones don't? Is it just a matter of taste?

I expect that this question might be closed or voted as off-topic, etc., but I feel strongly that this is a valuable question for a mathematician to have an answer to. It's also something one is always confronted by as a teacher -- one needs to have an answer when students inevitably ask "Why do we need to learn this?" Also, being someone who has completed my PhD and has a (so far) successful research career, the fact that I don't have a totally solid answer to the question evidences its non-triviality.

  • $\begingroup$ Who told you that in pure mathematics, applications weren't important? $\endgroup$ – Thomas Andrews Nov 15 '17 at 3:52
  • $\begingroup$ The OP probably means practical applications. $\endgroup$ – Qudit Nov 15 '17 at 4:00
  • $\begingroup$ The 'importance' of maths comes down to the individual. When answering a question, some people are able to see through the working, and visualize the beauty of maths behind it. That's what fuels people to continue to do maths. In my regard, all maths questions are important (except for statistics), because they all contribute to a seemingly infinite pool of ideas that we call mathematics. $\endgroup$ – Landuros Nov 15 '17 at 4:00
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    $\begingroup$ I think this is a great question & it would be illuminating to hear answers from working mathematicians. Readers should note that OP is not a naive troll but holds a PhD himself. It is a serious question and could inspire thoughtful answers. My hunch is that working practitioners will say the "most important" problems are the ones with "deep" consequences: the result ties together seemingly unrelated branches or ideas, or the method used to get the result isn't merely ad hoc but has powerful applications in other contexts. Do all sufficiently "hard" problems have this property? I don't know. $\endgroup$ – symplectomorphic Nov 15 '17 at 4:10
  • $\begingroup$ @ThomasAndrews I'm not saying they aren't important, but they are, by definition, not the priority. If you'd like to make a case that applications are what decides that one problem is more important than another, I'd be very interested to read that. $\endgroup$ – j0equ1nn Nov 15 '17 at 4:29

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