Is there any "good" definition for what constitutes "applied mathematics"?

Wikipedia lists stuff such as statistics, optimization. However, e.g these have certainly "pure mathematical" aspects to them. So what makes them "applied mathematics"?

Or what would make "pure mathematics" not "applied mathematics"? Is the distinction arbitrary (e.g. because even if something initially seems like it doesn't have applications, then it could be, that we just haven't seem them yet)?

Based on this I'd say the distinction is arbitrary. Rather, there's just mathematics and applications of mathematics.

  • $\begingroup$ Applied math tries to be useful for something outside of math, such as engineering or science, whereas pure math is done only because it's beautiful and because people enjoy exploring this realm of ideas. $\endgroup$ – littleO Dec 17 '18 at 3:40
  • $\begingroup$ @littleO Arbitrary? I think a lot of computer science is very aesthetic. Yet it seems that most of it is related to application in computing. One could even perhaps argue that since the foundations of all mathematics are so "general", then there may not exist distinction between applied and pure, since they "stand" on the same basic ideas? Or i.e., they must be epistemologically similar, they must refer to the same kinds of things. $\endgroup$ – mavavilj Dec 17 '18 at 3:47
  • $\begingroup$ Perhaps if there's mathematics that one can say is "irrealizable in the real-world (e.g. one cannot build anything that resembles it)", then it could be "pure"? But is there such? Computers seem to stir this quite a bit, because they can express a lot of mathematics that would not be realizable in the real-world. $\endgroup$ – mavavilj Dec 17 '18 at 3:50
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    $\begingroup$ The distinction between pure and applied math is a little fuzzy, but I think they are still useful concepts. If someone tells you whether they focus on pure or applied math, you immediately get a much better idea of what kinds of things they are likely to know and what their motivations might be. $\endgroup$ – littleO Dec 17 '18 at 3:52
  • $\begingroup$ It might be worth noting (though this is not an answer by itself) that Vladimir Arnold took the stance that mathematics is actually a part of physics, "where experiments are cheap". This philosophy is somewhat extreme, and as far as I know it is not popular, but the more I think about it the more truth I see in it. (For the record, I have no opinion on his comments on French teaching, except to say that some of what he says sounds, for lack of a better word, rude; but this is presumably intentional.) Overall I agree with littleO. $\endgroup$ – Will R Dec 17 '18 at 3:53

My school had different course grouping numbers for pure and applied mathematics, but we were explicitly told that the grouping was meaningless and was only there because it was there historically. Pure math was $640$ and applied math was $642$.

I mostly took $640$ courses, but I did take one $642$ course and it could hardly have been called "applied" in the sense of it being useful in science or industry. It was just an ordinary graduate level combinatorics course.

That said, there can be a meaningful distinction between the two, and using the term "arbitrary" suggests to me that the distinction is not meaningful. My research is not applied math, and mathematical physics is. Between them, one can use discretion, but those are two things I know.

  • $\begingroup$ mathoverflow.net/questions/257529/… $\endgroup$ – mavavilj Dec 17 '18 at 3:19
  • $\begingroup$ I really wonder if one can display that some part of mathematics is "absolutely non-applied". $\endgroup$ – mavavilj Dec 17 '18 at 3:20
  • $\begingroup$ @mav I'm aware of this extremely distant relation to physics. Trust me it cannot be called an application of Schubert calculus, at least not the modern areas of study. $\endgroup$ – Matt Samuel Dec 17 '18 at 3:21
  • $\begingroup$ @mava I narrowed it down to my research only. I don't think they'll be needing my dissertation or any of the problems I'm trying to solve for applications any time in the next 500 years. Note I'm not necessarily proud of this, I don't hold any sort of disdain for applied math, just being realistic. $\endgroup$ – Matt Samuel Dec 17 '18 at 3:25
  • $\begingroup$ What is considered "applied mathematics" or "pure mathematics" at a given place and time may be a matter of history, and sometimes academic territoriality. $\endgroup$ – Robert Israel Dec 17 '18 at 4:18

I don't think the distinction between pure math and applied math lies in whether the concepts can be applied in other disciplines. Instead, I think the distinction lies in the goal when people do them. People who do pure maths are interested in solving problems in fields of math which arise only from our imagination, and they do so because it's fun or interesting to them. By contrast, people who do applied math couldn't care less about "imaginary" concepts, and will only pay any attention to them if they somehow help to solve a physical or statistical problem; i.e. a problem in the "real world". For example, you won't see many applied mathematicians studying the theory of algebraic geometry, because it's not interesting to them at all--there are essentially no applications to statistics and the like. But you will find many pure mathematicians doing so, because it explores a "beautiful" link between geometry and algebra. Another example is partial differential equations; comparatively little pure mathematicians study PDEs as compared to applied mathematicians, because PDEs are much more relevant to fields such as physics and describing the evolution of physical systems, than anything in pure maths.


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