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There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:

  • Organizing known results into a coherent narrative in the form of lecture notes or a textbook
  • Contributing code to open-source mathematical software

What are some other ways to make auxiliary contributions to mathematics?

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    $\begingroup$ Teaching is also important. $\endgroup$ – Henno Brandsma Jan 2 '15 at 11:02
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    $\begingroup$ This should probably be community wiki. $\endgroup$ – Dan Rust Jan 2 '15 at 11:02
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    $\begingroup$ It is not all about proofs; making conjectures is another way of contributing to mathematics, because it can lead attention of the mathematical community towards an unexplored challenge. $\endgroup$ – Felipe Gavilan Jan 2 '15 at 12:36
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    $\begingroup$ Apart from proving new theorems, one can also prove already well-known results using some novel techniques, giving new insights (e.g. use of ergodic theory to prove results like Szemeredi's theorem). $\endgroup$ – Wojowu Jan 2 '15 at 12:47
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    $\begingroup$ Don't forget disproving old theorems. :P $\endgroup$ – darij grinberg Jan 2 '15 at 15:33
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You can create new jobs for mathematicians, e.g. by funding institutes like Jim Simons. Arguably this does much more for mathematics than actually doing mathematics due to replaceability: the marginal effect of becoming a mathematician is that you do marginally better mathematics than the next best candidate for your job, which is a much smaller effect than creating a new mathematician job where there wasn't one before.

You can also work on tools for mathematicians to use like arXiv (or MathOverflow!). Arguably this also does much more for mathematics than actually doing mathematics. Incidentally, arXiv was developed by a physicist, Paul Ginsparg, and almost none of the mathematics graduate students I've talked to about this know his name.

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  • $\begingroup$ The "marginal effect" is much greater than you state, as you are considering only the first term of the series. The next term comes from the job that does get done by that next best candidate, which would otherwise have to be done by the next-next-best candidate, and so on. The series telescopes to the intuitive value: Your "marginal" contribution is indeed the total amount of math that you contribute. $\endgroup$ – Matt Jan 2 '15 at 20:39
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    $\begingroup$ @Matt: I don't agree. Let's say that there are a fixed number $N$ of mathematician jobs in the world, and that after you get one of them, the otherwise $N^{th}$ worst mathematician has to take a job outside of mathematics to which they're less well-suited. The marginal effect of doing this on mathematics is (the math you do) - (the math the $N^{th}$ worst mathematician would have done), plus other terms all involving jobs outside of mathematics (and presumably these are small by default). $\endgroup$ – Qiaochu Yuan Jan 2 '15 at 21:25
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    $\begingroup$ It sounds to me like the first part of the answer is "yes, you can pay others to prove theorems", which isn't really a satisfying answer. $\endgroup$ – Pål GD Jan 4 '15 at 12:01
  • $\begingroup$ @QiaochuYuan Did you mean to write “$N^{th}$ best mathematician” (as all of the remaining $N-1$ mathematician jobs would be filled by the next $N-1$ best mathematicians)? Even with that, I don’t follow your argument. (It doesn’t matter much, though, because I would have agreed with your assertion anyway.) $\endgroup$ – k.stm Jan 4 '15 at 15:44
  • $\begingroup$ @Qiaochu: Yes, I was assuming that as N goes high, the output goes to zero before we run out of math jobs. For example, there are many more math jobs in higher education than there are publishing mathematicians. But even if you don't agree with me, you also don't agree with the formula in your answer, which referenced the next best candidate for your job, not the worst hired mathematician. And you should point out that by this same reasoning, creating a new job would only add the Nth best mathematician to the overall pool. (I like many arguments for math jobs, but not ones with math mistakes!) $\endgroup$ – Matt Jan 24 '15 at 9:37
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Even people with no mathematical background at all can contribute to mathematics.

One obvious way is by running software such as the GIMPS client for finding Mersenne primes, though the value of such primes to theoretical mathematics is debatable.

Another, vastly more important one is to contribute anything, literally anything at all to human civilization. In reality, it is that complex civilization which makes mathematics possible in the first place. Because they do not have to scramble for food in the dirt at the risk of their lives, people have the time to explore "non-productive" subjects such as mathematics. The number of great minds who could have become Eulers or Euclids but did not because they had to work as swineherds or died of a flu at age nine surely outnumbers those who actually did ever do any mathematical work. By keeping civil order, hospitals and agriculture going, you keep mathematics going as well.

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    $\begingroup$ In no way I would count offering computation time or doing something vaguely productive as contributing to mathematics. Thus, I downvoted. (Actually, this is in my view the answer with the least relation to the question so far.) $\endgroup$ – k.stm Jan 3 '15 at 21:08
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    $\begingroup$ @k.stm I didn't downvote, but I agree that the answer isn't addressing the question being asked. While technically correct, it's a vacuous statement. By that argument, every medical doctor in the history of the world has contributed to every field and industry of the arts and sciences... $\endgroup$ – rationalis Jan 4 '15 at 5:21
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    $\begingroup$ @k.stm, rationalis: Does a mathematics professor's secretary contribute to mathematics? What about the cleaning crew that maintains the mathematical institute's building? Where does the chain end? It doesn't end. What is vacuous IMO are claims of the type "doctors save lives" – of course they do, but so do all those that work for them, and the suppliers of medical equipment, and indeed mathematicians. If one finds that one's talents are insufficient to contribute to mathematics directly, the next best bet is to keep the "system as a whole" running. $\endgroup$ – user139000 Jan 4 '15 at 7:32
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    $\begingroup$ It ends even before the cleaning crew, I think. And I also think you are confusing obvious with vacuous statements. $\endgroup$ – k.stm Jan 4 '15 at 8:03
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    $\begingroup$ @k.stm: About the drugs analogy, the thin line between socially accepted and intolerable is not written in stone (I live in the Netherlands ;). And neither is the point where the chain ends. OP mentioning auxiliary makes it even more open for broad interpretation. Pew made his point; where is yours? Your arguments so far were about semantics; please share your ideas where the chain ends. Though I must warn you: it can only lead to quibbling. $\endgroup$ – Ruud Helderman Jan 4 '15 at 11:27
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There is a lot of work done towards formalizing existing proofs into logic software and proof wikis. The formalization of a proof can be very helpful economically. It can lead to more confidence in proofs and making searching for results more feasible.

(Mario here:) I am an undergraduate who works with the program and library Metamath for doing formal proofs. There is a TON of work that still needs to be done in formalizing even just the standard undergraduate curriculum, and it's all easily accessible to any reasonably bright undergraduate. It helps to be good at programming or at least thinking like a programmer, since the work you do looks a lot like programming to the uninitiated and formal math is just as unforgiving to typos as any compiler. But even though there is a lot of general problem solving involved, the path is all more or less written out by other mathematicians (as "hand proofs"), so the way forward is always clearly delineated and it might even be ungraciously called "transcription".

I'm going to try to keep it up in graduate school, but even just as a pastime it's good for the brain and gives you a great sense of accomplishment, in addition to making you understand the corresponding hand proof on a deeper level than pretty much any kind of study out there. It's not for the faint of heart, but for precision-minded amateur or professional mathematicians with a programming bent I could not recommend it enough.

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  • $\begingroup$ Interesting. Is this proof formalization an area in which a motivated undergraduate could make a significant original contribution? $\endgroup$ – David Zhang Jan 2 '15 at 16:57
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    $\begingroup$ @DavidZhang If you look around, possibly. If you are interested I would recommend contacting the authors of a proof wiki and see if there is anything that they would suggest. As far as "significant" goes...I'd say yes probably significant enough to put on a resume, but probably not significant enough to expect the girls at the bar to know your name before you walk in. $\endgroup$ – DanielV Jan 2 '15 at 17:02
  • $\begingroup$ @DanielV But I mean, how many mathematicians are significant enough to expect the girls at the bar to know their name? I can't think of a single (pun intended) living academic who could walk into an average bar (well, where I live anyways) and expect to be recognized on the merit of their contributions to their field. $\endgroup$ – rationalis Jan 4 '15 at 5:29
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    $\begingroup$ @DavidZhang As someone who does exactly this on a regular basis, yes I would absolutely say that you can make a significant original contribution. My first large project was proving Bertrand's postulate in Metamath, and since then I've been invited to a conference, wrote a paper, and I feel like my mathematical career is finally starting to take off. All while still in undergrad. $\endgroup$ – Mario Carneiro Jan 7 '15 at 5:25
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    $\begingroup$ @DavidZhang By the way, for anyone who's interested I maintain a blog chronicling my quest to prove those 100 theorems in Metamath. (40 down, 60 to go!) $\endgroup$ – Mario Carneiro Jan 7 '15 at 14:07
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I would say that though obtaining results is crucial, introducing new concepts, new connections, or even new perspectives looking at classical mathematics sometimes are more important.

Gauss, for example, introduced the concept of congruence in number theory and, arguably, number theory has then been developed systematically. The introduction of irrational numbers also solves old problems such as squaring a circle.

Einstein, for instance, found the connection between gravitation and curvature, so that differential geometry has been involved with physics. Kolmogorov, for another example, noted the connection between probability and measure theory, so that we have the modern theory of probability.

Klein, for example, proposed to view the geometries from the group-theoretic point of view and contributed a lot to modern geometries. Hilbert, for another instance, looked through the nature of mathematics and rigorously treated mathematics axiomatically.

Personally, I believe that these contributions to mathematics are of higher form of contributions.

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As @Hennobrandsma said, you can teach mathematics well.

In addition to developing tools which aid mathematics research (as you mention), you can also work to develop tools which aid teaching mathematics.

Without coming up with "new mathematics", you can apply mathematics in novel ways to other fields. I think this is especially true if you can find natural and really useful applications for some piece of mathematics which has not been applied yet. This is the best kind of "advertisement" mathematics can get.

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Another way to contribute that hasn't been mentioned yet is advocacy. It is important to raise awareness on the importance of mathematics and scientific literacy in general. For instance, here are a few concepts that it would be useful to disseminate to the general public:

  • that some knowledge of mathematics, statistics and science is important for the average person, too.
  • that there is still active research in mathematics; theorems weren't all discovered 300 years ago.
  • that mathematics (and STEM in general) is a viable career path, and people interested in it shouldn't be laughed at.
  • that research, even basic research, is an important endeavour and needs funding.
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    $\begingroup$ I don't know how many people think that higher math (even, say, linear algebra or calculus, which most non-STEM undergraduates don't have to take) are applicable to their lives, but high school algebra/geometry/statistics doesn't take much convincing. Also, I'm pretty sure that every adult in a "developed" country is well aware that STEM is a viable career path, though I think there is a general stereotype that it is only for very gifted, passionate individuals. As for what does and doesn't deserve funding, that is oftentimes a political issue more than an educational one. $\endgroup$ – rationalis Jan 4 '15 at 5:34
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    $\begingroup$ @rationalis Some people have trouble believing that math research has the need for funding, even before deciding whether it deserves it or not. Has no one ever replied to you "Funding? What for? You basically need just a blackboard and that's all!". $\endgroup$ – Federico Poloni Jan 4 '15 at 9:23
  • $\begingroup$ @Federico My comment in another question:I have a feeling that that wasn't the intended question. I think maybe the person meant to ask if there are research costs besides paying the researchers for their time and effort e.g. laboratories, equipment, etc...? – BCLC Aug 23 '14 at 22:13 academia.stackexchange.com/questions/27377/… $\endgroup$ – BCLC Jan 14 '15 at 6:58
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It is useful to contribute to databases of mathematical objects such as Sloane's Encyclopedia of Integer Sequences, FindStat and others. These resources have become invaluable for mathematicians searching for references to objects that they don't know the name of. Using these databases, mathematicians are able to find connections between seemingly distant fields if both make use of a certain integer sequence or other object in a database. For more information, see Billey and Tenner's manifesto here.

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Find problems to solve. These can be open mathematical questions but (more important) new fields of application that may also need new mathematical concepts. Calculus e.g. was founded because it was needed for physics. So any scientist who dares to show his problems to mathematicians helps mathematics as well.

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Many problems of theoretical physics (e.g., string theory) are related to the development of some new math. This severely requires the services of able mathematicians. In the past, Riemann, Grassman, Hilbert, Poincare, Elie Cartan, deed so for the discovery of Einstein's equation in GR. Physicist Witten was awarded Field Award for his services in Math research.

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