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I'm an undergraduate math major and I really enjoy pure math. I don't really have a specific area of interest yet, but I'm looking. That being said, I often ask myself the question "what's the point of pure math?" and "what is it good for?" and struggle to find answers to these questions. I'm able to understand and appreciate the beauty in pure math (and that's very rewarding), but to me this doesn't seem like enough of a reason to dedicate myself to pure math. Do any people in math community (including people who do math/research for a living) have any views on how I can look at this question, or any interesting answers?

Again, I'm not really looking for something along the lines of "because math is so beautiful" or "math in itself is an art" - I completely agree with that. But why should I (or anyone) pursue it? Is there nothing else to it other than furthering its own results/solving its own problems?

PS - I'm also not looking for answers along the lines of "but what's the point of sports that way?" -- the point of sports is to entertain/involve people in an activity (you can disagree with this, but that's not important to my question). Tldr - I still can't properly answer the following question to myself: what's the point of pure math?

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    $\begingroup$ Pure math has built over the years tons of abstractions and ways of thinking about things that can be very helpful to other disciplines such as computer science and physics. But also, not only do the ideas and ways of thinking of pure maths permeate through disciplines over the years, its results actually get there too. For instance at first, noneuclidean geometry was utterly inapplicable, only about a hundred years later did it start to be interesting to physicists studying space-time. You can say something similar about Galois theory and cryptography for instance. $\endgroup$ – Max May 2 '19 at 18:50
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    $\begingroup$ Vanity, vanity, all is vanity. $\endgroup$ – B. Goddard May 2 '19 at 18:50
  • $\begingroup$ Another, more recent, example to build upon Max's point: en.wikipedia.org/wiki/Mirror_symmetry_(string_theory) $\endgroup$ – bounceback May 2 '19 at 18:51
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    $\begingroup$ That way you can ask why should people study anything. And by the way, people who think that pure math is useless are very mistaken. (I know you didn't say that, I mean in general. I met people who actually think pure math is useless) $\endgroup$ – Mark May 2 '19 at 18:53
  • $\begingroup$ Even "pure" math often has applications. If you are looking for monetary reward - it is not uncommon for math PhD's to earn six figure salaries after graduating. $\endgroup$ – Jair Taylor Jun 4 '19 at 5:20
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Many reasons.

  1. Because it's interesting to you. There are a lot of mathematicians who study it simply because it's interesting. This is sort of the point of most of human endeavour, to be honest. Why make a faster smart phone, or learn to scuba dive? Realistically none of these things influence our survival, but that doesn't mean they are worthless.

  2. Because what is pure math now may not be pure math later. For a huge example, Fermat's Last Theorem, which on its own is totally useless to us, led to the deep connection of number theory and elliptic curves that has applications in cryptography, which I think we can all agree is "useful". If you're looking for a utilitarian reason, then it's simply that we cannot predict what will be needed in the future to solve a problem.

  3. Because it can simplify things and illuminate problems. This is similar to the lines of number 2 above. People might have real, practical problems that they are trying to solve but are doing it the wrong way. For example, people used to try to find a general equation for finding roots of polynomial equations. There are LOTS of reasons why this is not just a pure math problem, and root finding has huge applications in many fields. The solution to the problem of finding a general root finding algorithm (you can't) was found through the creation and application of new types of "pure math".

  4. Because there is no such thing as "pure math". We LOVE making distinctions (like pure and applied math), but that's not fair. Certainly some math is much more obviously applicable to "real" problems than other math. I can't argue that. But that doesn't mean that "pure math" is not also "applied", or that results in a pure math field can't be used to simplify or expand or solve problems in an applied field. The distinction is false and propagated by society and academia, not mathematics itself.

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  • $\begingroup$ I'd argue that we can and do predict (if imperfectly) what will be need in the future to solve a problem, and that's part of how we decide what research topics are important, even in pure math. We look for problems that are simple to state but hard to solve, and there's a good chance that solving those problems will lead to insights later on. $\endgroup$ – Misha Lavrov May 2 '19 at 20:05
  • $\begingroup$ @MishaLavrov agreed. What I meant more is that we can't predict now what pure math concepts will have a "real" application, partly because we may not know what is applicable until the applied math problem arises, and partly because what "applied" means is vague anyway $\endgroup$ – Michael Stachowsky May 2 '19 at 20:11
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Here's a good (and long) article written in early 1900 by Indian mathematician about study of mathematics. Mainly aimed to high school-undergraduate students.

https://www.docdroid.net/xDZ8FIL/in-woods-of-god-realization-vol-9-pages-58-92.pdf

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