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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?

Klangen
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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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    Could you add this as an appendix or maybe as a "final note" directly before the conclusion of the paper? – tonysdg Sep 14 '17 at 13:53
  • @tonysdg I could, but it has better "flow" to put it directly after a related theorem. – Klangen Sep 14 '17 at 14:03
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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. – Please stop being evil Sep 15 '17 at 01:06
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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 08:55
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    Your description makes the paper sound normal for a mathematics journal. Make your contribution (esp. connections between distant theorems) clear and let the reviewers decide if that's enough for publication. Also maybe pick a journal that publishes more theoretical work than something like a (fictional) Journal of Practical Mathematics. – WBT Sep 15 '17 at 13:58
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    @Pickle: Yes. Typo. – Alexander Woo Sep 15 '17 at 15:05
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    @Pickle - Remember that you are already making a huge assumption: my paper is useless. Just because YOU don't see the value doesn't mean another might not. Look at gravity. Newton: "Hey, stuff falls down, probably for a reason. I mean, pretty obvious, no real use, but I'll write it down." Later Galileo: "Hey, that gravity stuff is cool! It affects everything the same way, regardless of weight!" You don't know when, or where, how, or if your work will be built off of, you do know you have a paper that you can publish. – EvSunWoodard Sep 15 '17 at 15:10
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    The aesthetic pleasure, the symmetry, the elegance, the connection between several distant theorems — that is the practical value. – ShreevatsaR Sep 15 '17 at 16:02
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    @EvSunWoodard while I am decidedly on the side of people who think lack of usefulness is no reason to not study or publish something, a pet peeve of mine is that people often bend over backwards (as I feel you are doing by invoking Newton and Galileo) to argue that seemingly useless research should be published because it may prove useful someday. The conclusion is correct, but the reasoning is wrong, and harmful IMO. Mathematicians should not be constantly apologizing for doing "useless" research - to do so is missing the point of why we do what we do and how we think it benefits the world. – Dan Romik Sep 15 '17 at 19:13
  • Unless you have some way of proving that it's useless, do not underestimate it's utility. Someone, sometime, may find a use for it. If nothing else, if it gives readers a different way of thinking (e.g., by connecting disparate theorems together as you say), this can enhance the reader's understanding and may lead to new discoveries or innovations. – brianmearns Sep 15 '17 at 20:52
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    Connections between distant theorems is one of the most practical things! – TT_ stands with Russia Sep 15 '17 at 22:11
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    @TT_: You should turn that into an answer so that I could make 30 fake accounts and upvote it 30 times. – Tobia Tesan Sep 17 '17 at 09:52
  • @AlexanderWoo: As a pure mathematician, you should now that a missing "no" can make a pretty big difference in a sentence ;) – Eric Duminil Sep 18 '17 at 07:06
  • If it connects distant theorems together, it can be useful since proceedings in one field then can automatically have impact in other fields. – willeM_ Van Onsem Sep 18 '17 at 10:38

7 Answers7

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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.

Nate Eldredge
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    +1 ... of course include it. (And note that it is useless for practical purposes if you wish.) – GEdgar Sep 14 '17 at 14:30
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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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    @tonysdg Lol, just saw your comment. It seems like crypto is sort of the "go-to example" for seemingly useless math having practical implications... we need more examples of this I think... – user3658307 Sep 14 '17 at 15:43
  • @user3658307: Doesn't help that I'm a computer engineer, so my mind automatically goes in that direction ;) – tonysdg Sep 14 '17 at 16:15
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    @tonysdg ... Pickle was wondering whether to include it, because it is useless. I say, include it. And if Pickle is worried about it being useless, say so. Or don't say so, and let the reader decide how useful it is. – GEdgar Sep 14 '17 at 18:17
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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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    In biology we just like to hedge for the future, often generically. "Could be useful for xxxxx in the future" roughly translates to "We found something interesting/unexpected and hopefully someone can make some sense of it someday." Popular press sometimes takes this as "CURE IMMINENT FOR (insert what ails you)," but I'd like to think this approach could make sense in a math paper, too. – Bryan Krause Sep 14 '17 at 22:42
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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
  • @Voo I stand by the assertion that relativity and number theory haven't been used for warlike purposes, to this very day. I guess GPS is the closest relativity has come to being martially relevant, but that's a lot more than just a system of war. – Stella Biderman Sep 15 '17 at 14:46
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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
  • @BryanKrause That's really interesting! I had no idea about that history :) – Stella Biderman Sep 15 '17 at 15:31
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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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    @Stella Number theory is at the base of all modern encryption. And encryption is one of THE most important parts of warfare ever. It's estimated that the cracking of Enigma shortened the second world war by two years - how much more applicable to war can you get? – Voo Sep 17 '17 at 18:12
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    @Shane: Relativity is not the concept behind nuclear explosives any more than it is the concept behind chemical explosives; the reactions in either a chemical explosive or a nuclear one cause a loss of mass, but it's more noticeable in nuclear reactions because so much more energy is released. – user2357112 Sep 18 '17 at 08:10
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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.

Dan Romik
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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.

Tom Au
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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.

Attila Kinali
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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!

user3658307
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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.

einpoklum
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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.

kingfrito_5005
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