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As a senior grad student, junior researcher and an introspective individual, I always face the question of how math ought to be studied. Among the many successful researchers whom I've had the pleasure of making acquaintance, only a very small portion had research works that would actually contribute to real-life problems. When the question of "why math" resurfaces I'd think well it is something I enjoy doing, and working out every single problem makes me feel accomplished, and I think I'm good at it. Do you think merely "enjoying something" and that "maybe it'll find its applications one day" are good enough justifications for a life-long math career even if we never see an immediate by-product of our efforts in real life problems?

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closed as primarily opinion-based by Matthew Towers, José Carlos Santos, ahulpke, JonMark Perry, Namaste Dec 25 '17 at 6:25

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I certainly do. As a personal philosophy, I don't find anything to have inherent value or meaning. We, as conscious beings, decide what is of value to ourselves. Pure math is somewhat of an art form to me, and I see no reason why its ability (or inability) to apply to a 'real' application would make it any more or less worthwhile. $\endgroup$ – infinitylord Dec 24 '17 at 1:49
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    $\begingroup$ When you say math career, by whom will you be employed ? $\endgroup$ – Rene Schipperus Dec 24 '17 at 1:49
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    $\begingroup$ I think justifying high level pure math research through potential applications is a fool's game. The reason mathematicians do it is because it is intellectually satisfying. It's a form of exploration which involves extreme creativity and rigor. Once you finally crack a proof you've been working on for months or years, the sense of accomplishment is fleeting but intense. To me the reason for pure math research is similar to the reason for art. $\endgroup$ – Cheerful Parsnip Dec 24 '17 at 1:57
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    $\begingroup$ You can do mathematics because it's useful in science and engineering, or you can do mathematics because it's a beautiful art form. People ought to be paid for the first activity, obviously. For the second activity, I think mathematicians ought to be rewarded the same way painters, poets, and musicians are rewarded -- by people who enjoy their work. But the prospective audience for mathematics-as-art is tiny, and society's sponsorship of the activity is small and shrinking, so don't expect to be able to feed your family this way. $\endgroup$ – bubba Dec 24 '17 at 2:29
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    $\begingroup$ In my day job, I do mathematics that solves current problems in engineering and manufacturing. I get paid a lot of money for this. In my play time, I do mathematics because it's fun and I find beauty in it. I don't care whether the play stuff is useful or not, but don't expect to get paid for it, either. $\endgroup$ – bubba Dec 24 '17 at 3:19
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Actually, this is frequently the way it has gone. Some theory/system will be constructed which is interesting but has no actual applications. Then years/centuries down the line someone comes across it and says "hrm- this would actually be a really good way to model/solve this problem I'm looking at".

That being said with how quickly information is being discovered/communicated these days, the wait is frequently shorter. I absolutely believe that anyone going into mathematics with the intent of only producing results that are practical and useful can have a very fulfilling and meaningful career.

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    $\begingroup$ You can point to pure mathematics developed in past centuries that found applications in modern times, but this is the exception, rather than the rule, I think. And my impression is that mathematics is much more abstract today than it was in the past, so applications are even less likely. Of the papers published in pure mathematics in the past year, what percentage do you think will be useful (outside of pure mathematics) in the next thousand years? Not a rhetorical/confrontational question -- I'm genuinely interested in the answer. $\endgroup$ – bubba Dec 24 '17 at 3:26
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    $\begingroup$ Yes, I'm aware of many of the examples, though it's always nice to have more. But, still, I think these are the exceptions -- a tiny fraction of pure mathematics. $\endgroup$ – bubba Dec 24 '17 at 4:49
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    $\begingroup$ Defining "pure mathematics" sounds like a job for someone wiser than I am. If I say "algebra" is part of pure mathematics, I expect some pedant will ask me to define algebra. That road doesn't lead anywhere interesting. Presumably AMS or the Math Reviews folks have definitions. As a representative sample, maybe we could take anything published in the Bulletin of the AMS in the last 10 years or so. $\endgroup$ – bubba Dec 24 '17 at 5:01
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    $\begingroup$ Well, as a mathematician, I'm sure you can see the difference between the statements: (1) all science benefits from mathematics, and (2) all mathematics is of benefit to science. Also, I would not consider numerical linear algebra to be part of pure mathematics. $\endgroup$ – bubba Dec 24 '17 at 5:24

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