I have come across a couple of questions which have asked about the "Real World" applications of things (specifically, of the Fibonacci series and of Groups, Rings and Fields). It seems to me that these questions have attracted rather passive-aggressive* comments, along the lines of "Who cares about applications, we are pure mathematicians!". This seems to be a relatively recent phenomenon (for example, compare the reception of the recent question on the application of the Fibonacci sequences with an older, but identical, question.). Examples of comments include the following.

"Not all mathematics has to have an application..."

"Would it be so terrible to have learned something beautiful that lacked practical applications? I hope not."

"Groups, rings and fields are everywhere in mathematics. I don't much care about their real-life applications."

I find such comments unhelpful. They don't answer the question and they are a tired, unoriginal opinion. However, my issue lies deeper than this. Although it is true that pure mathematicians are not motivated by applications, it is not unhelpful to keep one eye on possible applications. Indeed, the UK funding agency in maths, EPSRC, makes it very clear that you need to keep an eye on possible applications. Therefore, if you want money (aka you want a job as a pure mathematician) you cannot have this insular attitude. But more than this, applications of pure maths can be exciting! For example, Ricci flows are being used to diagnose colonic cancer, while there is a big push in group theory to apply the plethora of decision problems in this setting to cryptography (as current encryption protocols are susceptible to quantum computers).

In summary, and to borrow a comment from another user in the Rings and things question,

"can we not do the whole "I don't need real life (sneer) applications" routine? Good for you if you don't need them, but the applications to other academic fields have always been a huge source of inspiration for mathematics and play a large part in making it so wonderfully rich. Acting like "I don't need them" $\Rightarrow$ "we should all ignore them" is just as ignorant as the reverse view"

I am posting this in order to bring attention to these comments, but one can ask a question: Are these comments an issue? If so, what is the appropriate course of action? Should these be flagged? Or just ignored?

If I wanted to ask vaguely controversial and mildly philosophical question, I would ask the following:

Is this a commentary on the decline of quality within this site?

This is motivated by the difference in the two Fibonacci sequence questions. In 2010, we see multiple answers which give genuine, real-world applications - people know the answer! In 2013, noone gives a genuine real-world application, and the question is closed (I mean no disrespect to those who answered - this is meant as a commentary on the breadth of knowledge within members the site, not of certain individuals).

*I say "passive aggressive" because they are implying that the OP should try and give a reason for wanting to know about applications, as if being curious was not enough! Probably passive aggressive is not the correct phrase. Feel free to edit this phrase and put words into my mouth.

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    $\begingroup$ I don't think it's fair to characterize those two questions as "identical" when the more recent one contains this as a subquestion: "If there are no applications, why do mathematicians examine mathematical constructs which have no use in the real world? This does not make sense." It looks to me like most of the comments you're reading as passive-aggressive are attempts to respond to that part of the question (with varying degrees of success, admittedly). $\endgroup$
    – Micah
    Jul 22, 2013 at 20:17
  • $\begingroup$ On the application of group theory to cryptography, it would be more accurate to say that some current encryption primitives are susceptible to quantum computers (and specifically to the application of the quantum Fourier transform to the hidden subgroup problem). There are alternatives. $\endgroup$ Jul 22, 2013 at 20:25
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    $\begingroup$ @1729, Maybe you mean "aggressive", "snide", "combative", or something. Passive-aggressive is attacking the question or its asker in deceptively indirect fashion that ostensibly talks about something else, or hints at a problem instead of stating "[thing] is a problem!". $\endgroup$
    – zyx
    Jul 23, 2013 at 6:00
  • $\begingroup$ Why is this question tagged (comment-replies)? If I look at the tag wiki for these tags, this seems to be about something completely different. $\endgroup$ Jul 23, 2013 at 7:21
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    $\begingroup$ @Micah You make a good point about the sub-question. However, I am reminded of the old joke about the programmer who was sent to the shop with his wife's orders ringing in his ears: "Buy a loaf of bread. If you find eggs, get a dozen." He returned home with 12 loafs of bread. $\endgroup$
    – user1729
    Jul 23, 2013 at 8:50
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    $\begingroup$ @user1729: How is that even related? $\endgroup$
    – tomasz
    Jul 23, 2013 at 13:48
  • $\begingroup$ @tomasz "If there are no applications..." $\endgroup$
    – user1729
    Jul 23, 2013 at 13:55
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    $\begingroup$ As a quoted commenter, I should say (reinforcing Micah's point) that the two questions are different. Such comments were made (at least in my case) because the OP suggested, in his second question, that there was no point to study if it had no applications - this seemed unnecessarily antagonistic, although may only have been due to ignorance or lack of forethought in writing the question. As such, it was met with with civil, if somewhat terse, comments suggesting a differing opinion. As to the closure, the OP asks three different questions; two of which could be considered big-list candidates. $\endgroup$
    – Dan Rust
    Jul 24, 2013 at 0:39
  • $\begingroup$ @DanielRust big-lists are discouraged, but not illegal. $\endgroup$
    – user1729
    Jul 24, 2013 at 9:52
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    $\begingroup$ I think you should take the comments less seriously. $\endgroup$
    – Pedro Mod
    Jul 24, 2013 at 10:02
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    $\begingroup$ @user1729 Of course big-lists are fine, but when three questions are asked, two of which could be considered inviting big lists, I think closing as too broad is perfectly reasonable. If the OP wishes to split the questions up in to separate questions, then I see no reason why the question could not be re-opened. $\endgroup$
    – Dan Rust
    Jul 24, 2013 at 10:24
  • $\begingroup$ @DanielRust Then you should tell the OP this! (Although question (1) is a duplication while the if in question (2) makes it redundant. So Question (3) is the only one worth asking...but then we could just look at Wikipedia...) $\endgroup$
    – user1729
    Jul 24, 2013 at 10:30
  • $\begingroup$ Since I'm quoted in this post ("Would it be so terrible...") and my quote is described as passive-aggressive, I'd like to respond briefly. Applications are important, and I am not advocating that math should be useless. My quote meant exactly as it reads: even if, hypothetically, pure math lacked applications, it would still be worth learning for its beauty alone. To reiterate: My hope is that the OP would not consider it a waste of time to learn something beautiful (like art or literature) that lacked practical applications. $\endgroup$ Aug 3, 2013 at 16:51
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    $\begingroup$ I admit that I didn't care much about applications myself when I was a student. And pure math is indeed beautiful. However, my experiences since are that trying to apply math forces you to explore unknown pathways that are not only rewarding but ultimately lead to... more beautiful mathematics! I mostly interpret disdain for applications as a lack of experience (or maturity). But I also fear that operating in an exclusively academic context can prevent growth in this respect. In general, I would welcome more friendly and open minded conversations on math.se. $\endgroup$
    – WimC
    Aug 4, 2013 at 19:10

4 Answers 4


I hesitated to do so at first, but if everyone will remember that we're all in this together, I will give an example of what I was talking about in my comments. I had in mind this question, in which a PhD in computer science admits that he's never seen any applications of groups, rings or fields in his cs work and wonders whether such applications exist. I couldn't find anything in the statement of the question to indicate that the OP was in doubt that abstract algebra is interesting, that its study is intrinsically rewarding, and so forth. Rather he is asking whether these are the only reasons for computer scientists to study abstract algebra. At one point the OP comments: "Absolutely NOT! Math is beautiful! I wanted to know if it's only the beauty I need to appreciate or am I being superficial and not seeing the 'hidden' beauty of its application(s). Hence the question PS: I study math topics just 'cause they are beautiful and don't care so much about their practical applications. But engineering schools rarely teach you something for beauty and thus was curious."

Here are some comments on this question (most of them highly upvoted):

1) It made you smarter.

$ $

2) You didn't learn about them because you found it fun and interesting?

$ $

3) Would it be so terrible to have learned something beautiful that lacked practical applications? I hope not.

$ $

4) Groups, rings and fields are everywhere in mathematics. I don't much care about their real-life applications.

$ $

5) Ignorant question, there is no need to emphasize the importance of groups, it being obvious. apart from that 'one rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.'

Here are my comments on these comments. Before you read them, please remember that I have devoted much of my adult life to thinking about groups, rings and fields: I like them about as much as anyone I have ever met.

1) This is at best totally irrelevant. Studying mathematics is probably good for the mind, as is studying lots of other things. There is a suggestion here that a computer scientist who studies abstract algebra will become smarter than a computer scientist who doesn't: this is questionable and even slightly obnoxious.

2) This comment seems to totally ignore the fact that there are completely legitimate reasons to study abstract algebra beyond finding it fun and interesting, and that if abstract algebra is a required course for a student of something other than mathematics, it is almost certainly required for reasons other than being fun and interesting.

3) I included this comment for balance: it reminds the OP about the virtues of pure mathematics in a way which seems totally appropriate to me. (But it only had one upvote; the other ones quoted here have many more.)

4) If someone asks a question about $X$, then replying "I don't care about $X$" is at best irrelevant. But really it suggests that the OP's question and, by extension, applied mathematics, is somehow illegitimate.

5) I flagged this comment. Calling the OP's question "ignorant" is very rude. There follows a statement -- I think it's a quote -- which tries to convince the reader that pure mathematics is better than applied mathematics. The OP got this quote by asking about applications? Is it appropriate for someone to suffer insult and proselytization just for asking a question?

There are also currently three votes to close this question. One of the votes says that the question is "off-topic" which I find absurd and dangerous: how can a question asking about the applications of mathematics be off-topic on a math Q&A site?? Two of the votes say "too broad". I can see that it is broad enough to be made community wiki, but it is much less broad than many other questions which encourage the creation of infinitely long lists of things for reasons unexplained or without clear virtue. The inherent virtue of explaining to people in what way various parts of mathematics can be applied seems clear. And it is clear that a good answer would be an application of abstract algebra to do something which is especially important or valued by computer scientists, or to which a large number of people are currently working on, and so forth. Since the number of people who are truly informed about current applications of abstract algebra to cs must be rather small, this is a question which, although broad, is potentially very useful and helpful.

  • $\begingroup$ Sorry, I didn't realize this was the question you were referring to in your earlier comment to me; as the author of comment 2, I do know this question. P.S. I am the one current upvote on comment 3, and I did not vote to close. Of course, I'm glad that the OP seems to already be appreciative of math for its own sake, so that these comments turned out to be unnecessary (the "absolutely NOT" comment you refer to comes after all 5 of these). However, I personally see the title "Why did I learn..." as providing plenty of reason to have initially thought that the OP did not find abstract algebra $\endgroup$ Jul 24, 2013 at 7:57
  • $\begingroup$ interesting or rewarding - otherwise these would have been answers to that question (and I was aiming in this general direction with my comment, though not ideally expressed or fully thought through, since it is possible to be required to take a course). Now it is clear that the intended question is more like "Why was I required to learn abstract algebra in the course of my study of computer science". $\endgroup$ Jul 24, 2013 at 7:59
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    $\begingroup$ Comment 5) quotes Hardy, A mathematician's apology first paragraph on page 41. $\endgroup$
    – Martin
    Jul 24, 2013 at 8:26
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    $\begingroup$ @Martin: thanks for that; it certainly sounded familiar. Although I have read and enjoyed Hardy's book many times, it certainly contains some sentiments and passages which can be construed as rude/obnoxious, especially when taken out of context (but sometimes even when taken in context!). And of course a lot of what Hardy says is simply no longer true, e.g. when he takes refuge in number theory as having no warlike applications. When he speaks of applied mathematics and how it is taught, he is speaking of how things were more than 70 years ago. $\endgroup$ Jul 24, 2013 at 8:40
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    $\begingroup$ @Zev: I think the logic and intent of the question is clear: (i) the OP took abstract algebra as an undergraduate under the claim that it would be useful in cs; (ii) he went on to get a PhD in cs without using abstract algebra, so (iii) he wonders what happened: are there in fact applications of abstract algebra to cs of which he is unaware? Whether the OP has a positive or negative attitude towards pure mathematics seems somewhere between peripheral and completely irrelevant: the question that he has asked is meaningful either way, and the answers will not depend on it. $\endgroup$ Jul 24, 2013 at 8:47
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    $\begingroup$ @Zev: Looking back at the question, it seems to me that a lot of the commenters fixated on the title of the question ("Why did I learn groups, rings and fields?") Your comment is relevant to that, but (in my opinion of course) not to what is said in the body. Maybe one lesson here is that a bad title can cause a lot of trouble. Just now I have edited the title to something which I hope will make for a more productive commentary while staying faithful to the OP's question. $\endgroup$ Jul 24, 2013 at 8:56
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    $\begingroup$ @Pete: I definitely agree with your assessment that lot of people's negative reaction was due to the original title; thanks for taking the lead and editing it. I fully agree that the OP's attitude towards pure mathematics is irrelevant to their actual question about applications. But in general, if anyone on the site (intentionally or accidentally) makes it appear like they are "opposed" to either pure or applied math, it seems certain that people will want to address that regardless of its relevance to the question, and comments seem like the least-bad place for that discussion. $\endgroup$ Jul 24, 2013 at 10:08
  • $\begingroup$ As usual, this is nice, balanced, and well thought-out. Thank you. Something I am not sure I understand, though: "One of the votes says that the question is "off-topic" which I find absurd and dangerous". Could you please elaborate what you mean here? In what sense, and why dangerous? $\endgroup$ Jul 24, 2013 at 22:15
  • $\begingroup$ @PeteL.Clark My comment was a comment exactly because it didn't answer the question. $\endgroup$
    – Git Gud
    Jul 25, 2013 at 0:46
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    $\begingroup$ @Andres: The applications of mathematics, especially to something like CS, are a core part of mathematics. So I find it absurd that a question about applied mathematics could be off-topic for an all-purpose math Q&A site. The danger is making a site which will be off-putting and unwelcoming to students and practitioners of applied mathematics. This already seems to be true, to an extent. $\endgroup$ Jul 25, 2013 at 2:59
  • $\begingroup$ Ah, I see, thanks. (And I agree.) $\endgroup$ Jul 25, 2013 at 3:10
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    $\begingroup$ Comment (4) is merely an observation giving the commenter’s own view of the matter. The claim that ‘really it suggests that the OP's question and, by extension, applied mathematics, is somehow illegitimate’ is merely a subjective reaction to the observation, not a statement of fact. That comment is harmless, it is relevant to the topic, albeit not to the specific question, and above all, it is merely a comment: it does not purport to be an answer. $\endgroup$ Jul 27, 2013 at 20:47
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    $\begingroup$ @Brian: Sure, what I'm saying is a subjective reaction to the observation. My whole answer consists of subjective reactions; how could it be otherwise? "That comment is harmless" You know, that's merely a subjective reaction, not a statement of fact. I disagree: expressing lack of interest or contempt for things can cause harm. "it is relevant to the topic, albeit not to the specific question" Is it too much to ask for comments on a question to be relevant to the specific question? I would say no. $\endgroup$ Jul 30, 2013 at 1:27
  • $\begingroup$ You’re moving the goalposts: expressing lack of interest and expressing contempt are two very different things. I agree that the latter can cause harm; if the former does so, I think that the problem is more with the reader than with the commenter. At most the former suggests that it’s all right not to have an interest — which is perfectly true. $\endgroup$ Jul 30, 2013 at 4:43
  • $\begingroup$ @Brian: A totally unsolicited statement of lack of interest seems tantamount to a mild form of contempt. You don't think that there is possible harm when public figures express a total lack of interest in one intellectual endeavor or another? If the most charitable interpretation of such a comment is that it's a purely personal statement of opinion not meant to influence anyone, then what is the benefit of such comments anyway? $\endgroup$ Jul 30, 2013 at 5:22

In answer to your first question about what to do with these comments and are they an issue:

I don't think these should be flagged for deletion. (Of course, if someone is obnoxious about it--for example, saying "only someone who is stupid would want all math to be applied"--go ahead and flag.) I think the proper response is to simply reply to question with a comment like "@(whoever left the "bad" comment) Yes, it is true that not all math need have an application. But, it is often interesting to see real-world manifestations of math, especially in things that would seem to not have any application at all."

I think that most people posting the comments referred to in this thread aren't really thinking about how they can be perceived; a comment like the one I suggested above is intended to help them realize that maybe the OP isn't being as antagonistic towards pure math as originally thought.

In response to your second question about a decline of the site:

I don't think this represents a decline in the quality of the site, but rather a shift in audience. In the early days of the site (I'm saying all this based on the Area 51 proposal--I wasn't actually here), it appears that the site was very much more tolerant towards "newbie" questions (e.g. What's so special about e? or "Is there any practical situation where $0^0 \ne 1$?").

Also, there appeared to be more applied questions--the people here appeared to be migrants from Stack Overflow with a programming background, who happened to have questions about math. This group would be much more tolerant of the "how is this practical" sort of questions.

Fast forward several years. Now we have (what appears to be) a majority of professors or math students who love math just for the sake of math. Thus, when someone comes along with a "how is this practical?" sort of question, this group is offended and squawks loudly.

Is this a decline in quality? I'd say this is like comparing apples to oranges. We have more mathematicians who love math just because it is, rather than people who love math because of what it does.

  • $\begingroup$ That is an interesting thought about the shift in audience. Thanks. (Although it doesn't quite explain why 3 years ago people knew the (an) answer, but seemingly not now.) $\endgroup$
    – user1729
    Jul 22, 2013 at 15:59
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    $\begingroup$ Of course this is a diverse site. But why must one group completely disparage another? I tend to shy away from pure abstraction, probably because my mind is not geared toward that. But you will never see me pop into a discussion on, say, rings and demand that there be some application lest it be worthless. Applied mathematics is quite legitimate, let's leave it at that. $\endgroup$
    – Ron Gordon
    Jul 22, 2013 at 17:17
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    $\begingroup$ @RonGordon And I totally agree. I hope I'm not making it sound like I disrespect applied math--I love to see how math is applied. :) $\endgroup$
    – apnorton
    Jul 22, 2013 at 21:25
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    $\begingroup$ @anorton: thanks, man. And vice-versa, I believe that math is worth studying for its own sake, but I am also taken with its applications. $\endgroup$
    – Ron Gordon
    Jul 23, 2013 at 13:57
  • $\begingroup$ I strongly agree that mathematics is worth studying for its own sake, as an art form, for its beauty, and as an intellectual exercise. What I do not support is paying people to engage in this idyllic ethereal activity. Sounds like the UK funding agency ESPRC shares this view. Good. $\endgroup$
    – bubba
    Jul 28, 2013 at 9:58

The fact that many others have shared an opinion makes it no less valid or appropriate to express (is yours a tired, unoriginal opinion also?).

If someone asks a question in a manner that suggests they believe only applied mathematics is worthwhile, it would be valuable to them to hear an opposing viewpoint: that math is also beautiful and can be done for its own sake. Of course, given the limited size of comments and human nature in general, it is far easier to express the extreme opposite view than a nuanced middle ground. But regardless, I think these comments are an appropriate method of opposing (perceived) bias against or ignorance of pure mathematics in a post, and therefore should not be flagged or removed. Of course, these comments should remain within the bounds of civility.

I don't see how the presence of such comments would be a decline at all.

P.S. Please don't get me started on the detrimental effects (and politics) of "requiring" that research have practical application.

  • $\begingroup$ I certainly wasn't meaning to say that being required to find applications of mathematics is a good thing. It gives me a headache! But rather, I am trying to say that asking about applications in an informal situation such as this should be encouraged (or at least not discouraged/sneered at). $\endgroup$
    – user1729
    Jul 22, 2013 at 16:14
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    $\begingroup$ Re: "Don't you like using <everyday item that needs it>?" .. makes a valid point and hopefully convinces them to moderate their views." Does not sound either convincing or a valid point. I use welded products every day, and see no need to ever learn welding. $\endgroup$
    – 40 votes
    Jul 22, 2013 at 20:49
  • $\begingroup$ @40votes: Yes, well that's what I get for trying to make up a hypothetical argument in the opposite direction I'm used to. Feel free to put in whatever sort of imagined exchange you want about why someone should learn applied mathematics. $\endgroup$ Jul 22, 2013 at 21:17
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    $\begingroup$ But Zev: while asking about practical applications of X piece of mathematics is not in any way a slight against pure mathematics, replying "Who cares about practical applications of any piece of mathematics?" is a huge slight against applied mathematics. I find the former question constructive and the latter kind of response purely negative. I don't understand why such negativity towards mathematics should be encouraged on a math Q&A site. $\endgroup$ Jul 23, 2013 at 0:49
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    $\begingroup$ I agree that if the lack of value of pure mathematics is part of the premise of the question, then the question is based on a faulty premise, and it's fair game to talk about that. But I don't think that's what uer1729 is talking about.... $\endgroup$ Jul 23, 2013 at 0:51
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    $\begingroup$ ...There was one recent question from a PhD in cs, asking "What are the applications of abstract algebra? Because I have never seen any in my work." And then several people jumped on the OP for being such a philistine as to suggest that one should inquire about the applications of abstract algebra. They talked about how wonderful abstract algebra is as a branch of pure mathematics. How is that an answer to the question asked?? (It seems to suggest that there are not significant applications of algebra in cs, which is of course completely wrong.) $\endgroup$ Jul 23, 2013 at 0:53
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    $\begingroup$ @PeteL.Clark: Comments are not answers. If they answered a question without putting any effort into explaining that algebra does, in fact, have applications in cs (and frankly, I find it rather baffling that a PhD in cs could not know that), but if the question was strongly biased in the other direction, I think comments in that direction are valid to a degree, as Zev said. $\endgroup$
    – tomasz
    Jul 23, 2013 at 14:00
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    $\begingroup$ @Pete: I don't know anything about your specific example, but such questions can be, and often are, phrased in a way as to imply something more: "What are the applications of abstract algebra? Because I have never seen any (and I don't think there are any, and I think that reflects badly on abstract algebra)". Even for someone who has no animosity towards applied math, a natural response to such a post would be to argue (in the comments, since it is not an answer) that applications are not necessary to have value. Just telling the OP applications doesn't address the more important issue. $\endgroup$ Jul 23, 2013 at 21:00
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    $\begingroup$ @Pete: Of course, I agree that saying "Who cares about any practical applications" is a very negative thing to say about applied math. I don't think that's what's been said here though, at least not as far as I've seen. I think people are so used to application questions having the hidden meaning I mentioned that they are simply eager to remind OPs that enjoyment / beauty are also valid reasons to do things. $\endgroup$ Jul 24, 2013 at 1:29
  • $\begingroup$ Perhaps I might go so far as to say that, in some situations (depending on the area of math asked about and the phrasing used in the question), there is an onus on the OP to clarify that while they are asking for applications, they understand that applications are not necessary to be worthwhile. $\endgroup$ Jul 24, 2013 at 1:35

I remember the question you are referring to, I even gave an example of "real-world" application there - the Fibonacci retracement used in finance for technical analysis. However, the question was closed so quickly that I decided to delete my anwer.

I am pretty sure the question could have many decent responses if it were not closed. And this is a more general problem, which I think has nothing to do with the perception of applications by really pure mathematicians. Another part of that problem is e.g. that wrong or minor edits are very quickly approved even when a post clearly contains more important issues to be corrected.

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    $\begingroup$ You make a good point about it being closed. I remember a question on notation which was closed simply because noone had come across the notation before, that is to say, because noone knew the answer. Which is a rubbish close reason. So maybe over-closing is a problem. Hmm....(As a side point - the question is a duplicate. You should re-post your answer in the old question, for the sake of completeness.) $\endgroup$
    – user1729
    Jul 22, 2013 at 19:04
  • $\begingroup$ I've already put a question here raising the last issue you mentioned: meta.math.stackexchange.com/q/9794/30222 . $\endgroup$
    – tomasz
    Jul 23, 2013 at 13:50

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