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