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If you're working in a field of mathematics that is not driven by real world problems (or the abstraction of real world problems), then how do you gauge the value of your work? How do you decide what is important to work on? Is it purely on a whim?

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closed as not a real question by rschwieb, Pedro Tamaroff, lhf, GEdgar, Sasha Sep 28 '12 at 4:18

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As with all things related to "value," it really depends on your values. – Thomas Andrews Sep 27 '12 at 18:00
A related question on MathOverflow:… – Jonas Meyer Sep 27 '12 at 18:01
The beginning answers to this can be found in G.H. Hardy's "A Mathematician's Apology." – Thomas Andrews Sep 27 '12 at 18:02
I never cared about such judgements; I simply did what interested me. I always felt that my real work was teaching anyway. – Brian M. Scott Sep 27 '12 at 20:03

Warning: My experience is limited, but I will try to offer a perspective.

What I end up doing, without really having to concentrate on doing this, is following what lots of other people in the area are doing - the assumption being that if a lot of people are interested in something, somebody at some point must have felt there was an important reason for doing it. This is probably not guaranteed to always work, but I can't think of a counterexample.

Partly this is because I'm a graduate student, so I'm being led (or at least nudged) in certain directions by my supervisor, but it seems to be true in more generality. Even if you're an expert in a field, if a lot of people suddenly become interested in something, it's worth checking to see if it has any useful application to your own work.

My feeling (guess?) is that even the most abstract areas of mathematics are ultimately driven by real-world problems, sometimes to a surprising extent. Most people would describe the sorts of things I study as being very abstract pure mathematics (and that's also how I think about it), but there are a lot of particle physicists interested in the theory as well, to the point where I often end up trying to read their papers.

As Thomas Andrews points out in the comments, it does depend on what you mean by value. If I'm being perfectly honest, I study the things I do because I find them intrinsically interesting, and it wouldn't really bother me if they didn't have other applications. However, I don't think this is the only sense in which they are valuable.

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I am not a working mathematician, but it seems like there are a lot of things that would make particular math work "valuable."

  1. Does it interest me? Is it valuable to me in it's own right?
  2. Does answering questions in this area get me any notice, fame, job offers, etc? Conversely, would I be working in a mathematical ghetto with nobody to talk to about the problems? This is value as defined, essentially, by your peers. Failure to consider this value can affect your ability to find employment, amongst other things.
  3. Does answering a particular question yield understanding beyond this question - does it give even a hint at something underlying, or a technique that might be used to tackle similar problems? (This is actually an extension of (1) and (2) - I am interested in "big" questions, and they also tend to get bigger notice. But the unanswered "bigger" questions also tend to be harder, because, by their nature, more people have tried to answer them and failed.)
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Time has shown that in returns of their endeavor, mathematicians (not necessarily individually but collectively) will be considered worthy of honor and recognition, somehow more profoundly than the intellectuals of other enterprises, like who would beat Libnitz in the impact of his ideas and contribution to the humanity, given that he sowed the seeds of computer and so much more to come in engineering. But, at the desk of a modern philosopher-mathematician, the true goal of the modern mathematics, is closer to its platonic views more than ever, where the emphasis and pay off is in Beauty. Nietzsche would have said that, mathematics is humanistic, alas all too humanistic.

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