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I am currently writing a paper that I wish to publish in a mathematics journal. During the course of my research, I have discovered a result that is aesthetically pleasing, i.e., contains a form of symmetry in its definition that can be seen by some as "elegant", and moreover connects several distant theorems together. However, this result turns out to be useless for practical use, furthermore it adds nothing to the other proofs and theorems presented in the paper. Is it worth publishing/mentioning, even as a corollary?

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    If this is not the main result of the paper, why not let the reviewers decide? – henning Sep 14 '17 at 12:38
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    As a pure mathematician, I'm having trouble understanding what you mean. If I couldn't publish results that were aesthetically pleasing but had practical value, I wouldn't have any publications. – Alexander Woo Sep 14 '17 at 16:11
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    What is practical use? I'm serious here, I'm not sure what you mean by 'beautiful but useless'; aren't beautiful things ipso facto useful by definition? I understand that something could be beautiful without enabling the perception of beauty in certain aesthetic paradigms (i.e. the result has intrinsic beauty but it is impossible for any being to appreciate/observe it), but I seriously doubt that's what you meant. – the dark wanderer Sep 15 '17 at 1:06
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    @AlexanderWoo Did you mean "If I couldn't publish results that were aesthetically pleasing but had no practical value"? – Klangen Sep 15 '17 at 8:55
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    @Pickle: Yes. Typo. – Alexander Woo Sep 15 '17 at 15:05
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Yes, this is fine. Math papers very often contain results just because they are interesting or instructive, even if they do not seem to be "useful".

You could mention this when introducing the result, with something like "The following theorem may help to illustrate the connection between blah blah blah..."

Authors also sometimes signal this sort of thing by describing a result as "pleasant", "amusing", etc, though "elegant" is probably a little too egotistical.

If the referee feels it's too much of a digression, they might suggest you take it out. But I don't think this would be the difference between acceptance and rejection.

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    @GEdgar: I'm not a mathematician by any means, but why bother noting that "it's useless for practical purposes"? Today's useless knowledge is tomorrow's key to unbreakable cryptography (or whatnot). Shouldn't the reader be the judge of that? – tonysdg Sep 14 '17 at 15:30
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    @user3658307 Various neural network models were for a long time theoretically appealing but practically useless; only recently have we had enough data storage (for large data sets) and computational power to make them actually work. – Jeffrey Bosboom Sep 14 '17 at 21:55
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    I'm surprised nobody has brought up the old G.H. Hardy quote yet: "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years". He said that in 1941 of all things. So it's really hard to judge what's useful and what isn't. – Voo Sep 15 '17 at 13:57
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    @StellaBiderman GPS was developed specifically for the US military and was initially released for civilian purposes after a Korean plane accidentally flew into Soviet airspace and was shot down (at least that's the story..it's certainly possible that civilian use was planned before that). – Bryan Krause Sep 15 '17 at 15:28
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    @StellaBiderman Relativity is the concept behind nuclear weapons. The weaponeers said "Hey, if E=mc^2 then we can make a big boom!" Also I'm hard pressed to say that number theory wasn't the basis for things like the Enigma machine. – Shane Sep 15 '17 at 17:54
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To add to Nate Eldredge's correct (and useful!) answer and to Alexander Woo's sarcastic quip highlighting the same point, one should keep in mind that pure mathematics is, by its very definition, the part of mathematics that seeks to study mathematical structures for the sake of the pure intellectual and aesthetic value of the mathematical ideas one is trying to discover. Yes, it helps that a lot of pure mathematics has turned out to be useful beyond the wildest dreams of the people who discovered it -- a totally weird phenomenon that no one seems to understand -- but that is not the primary concern (or even the secondary or tertiary concern, usually) of the pure mathematician.

Lack of (caring about) usefulness is a feature, not a bug.

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If your result "moreover connects several distant theorems together." I'd like to know that. You may not find a useful application of that result, but knowing what you just stated may help me to come up with something useful.

So publish it.

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    Check "Langlands Program" - yes, connecting distant theorems is definitely worth publishing. – Captain Emacs Sep 15 '17 at 20:49
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Absolutely. Not just because, to many, the point of science and mathematics is understanding and appreciating the beauty of reality, but also because it might become practical in the future! I doubt the people who worked on number theory foresaw crytography, for instance, or the esoteric probability theory making its way into machine learning now.

Also, I think tying together distant theorems is a practical application in some sense. Or at least an educational one for practitioners who might be trying to understand something, and realize something useful based on your theorem tying it to something else.

As the other answer says, I think it makes sense as long as you make it coherent with the rest of the paper!

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Engineer here. I marvel at all those math papers that contain nothing but "hey this looks cool!" I really like those. Some of these are even easy enough for me to understand :-) And do not fret about applications. You are doing math. You are doing theory. It's the engineers job to find an application for it.

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(pats OP on the back)

Congratulations, you're now officially a Mathematician! Publish away.

On a slightly more serious note: Spend time working on a good introduction that communicates the pleasing elegance of your results (or rather the lack of pleasing elegance without them). If for some reason the journals/conferences think it's totally useless, they'll reject.

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This is perfectly fine. Academically, mathematical research is done for its own sake, not for its practical usefulness. Mathematicians are not concerned with how the information they have will be used anymore than engineers are concerned with how the information they used was discovered.

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