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Is tag for

  1. applications of mathematics to mathematics itself?

  2. applications of mathematics to other fields?

  3. applications of mathematics to any field?

  4. job applications? (e.g., Recommendations from supervisor to postdocs in mathematics)

I ask this question because tag has no usage guidance at the moment.

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    $\begingroup$ I think it should be for applications to any field $\endgroup$
    – vidyarthi
    Mar 9 at 11:04
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    $\begingroup$ Definitely not 4 (did you intend it as a joke?). Any of 2 and 3, and even 1, should go under this tag. $\endgroup$
    – Alex M.
    Mar 9 at 11:31
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    $\begingroup$ From the looks of things, taggers have interpreted "applications" in each of the above sense (1,2,3,4) at different times. I would suggest that we shouldn't have a tag called "applications" precisely because it means such different things to different people. We should split it into several tags. $\endgroup$
    – Tim Campion Mod
    Mar 9 at 13:36
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    $\begingroup$ Tentatively, I might also suggest that a tag "internal-to-math-applications" which is meant to signify applications from one area of math to another, or within a mathematical field from one theoretical apparatus to another -- would be a fairly useless tag. In fact, almost any question could be viewed as asking "what is an application of X to Y" in one form or another. So I think I'd prefer to see this tag reserved for applications of mathematics to other fields, or applications of other fields to mathematics. $\endgroup$
    – Tim Campion Mod
    Mar 9 at 14:08
  • $\begingroup$ @AlexM. Look at the tags on the question I linked to. $\endgroup$ Mar 9 at 14:59
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    $\begingroup$ I took the liberty to add a tag wiki excerpt (it's pending review). $\endgroup$
    – gmvh
    Mar 10 at 7:38
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    $\begingroup$ While I appreciated @gmvh's taking the initiative and making the edit, I voted against it in review because I agree with @‍TimCampion that having {applications} mean sense (1) makes it so broad that it applies to every question. What MO question doesn't involve applying mathematics to mathematics? $\endgroup$
    – LSpice
    Mar 10 at 14:43
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    $\begingroup$ @LSpice To backtrack a little: there is a reasonably well-defined class of questions of the form "What are some applications of mathematical theory X to mathematical theory Y?" (where Y is not "contained" in X -- such a question may also quantify over such Y), which is substantially smaller than the class of all MO questions. So it was too sweeping of me to claim that there's no distinction to be made at all about OP's (1). Now the question is: does it make sense to have a tag which signifies that a question is of this class? I think not, but I could imagine being persuaded otherwise. $\endgroup$
    – Tim Campion Mod
    Mar 10 at 17:41
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    $\begingroup$ @TimCampion If the tag is to be used for applications to non-mathematical problems, then perhaps it should be a synonym for tag applied-mathematics. $\endgroup$ Mar 10 at 17:43
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    $\begingroup$ Looking at the SE guidance about tags, I think that a tag of the form of OP's (1) is not useful for the primary purpose of tagging -- i.e. the purpose of connecting answerers to questions. It could perhaps be useful for the secondary purpose of tags -- viz. it might be fun from time to time to browse through questions tagged under "applications" in sense (1) just for the fun of it. But I think the same could be said of a lot of tags whose utility is marginal. $\endgroup$
    – Tim Campion Mod
    Mar 10 at 17:47
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    $\begingroup$ @RodrigodeAzevedo I think synonymity with the "applied mathematics" tag is a fair suggestion, with the following caveat. My impression is that "applied mathematics" is itself a distinct branch (or family of branches) of mathematics as a discipline, whereas I see a space for an "applications" tag which asks about applying tools which are not generally viewed as "applied math tools" to fields outside of mathematics.... but I could be persuaded that this space is not big enough to warrant its own tag. Probably as soon as something is applied this way, it becomes a tool of "applied math"! $\endgroup$
    – Tim Campion Mod
    Mar 10 at 17:50
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    $\begingroup$ Haha! The "applied mathematics" tag also has no usage guidance currently! $\endgroup$
    – Tim Campion Mod
    Mar 10 at 17:51
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Okay, I think maybe there's a bit of a consensus forming. Regarding senses (1,2,3,4) from the OP:

  1. There may be a distinction to be made here, but it seems that such a distinction isn't really the sort of distinction which merits having a tag of its own.

  2. As per Rodrigo de Azevedo's suggestion, this sense is basically a synonym for the applied-mathematics tag. Personally, I could imagine asking, say a question about applications of set theory to the real world. In such a case, I might think to add an "applications" tag while it might not occur to me to add an "applied-mathematics" tag (even though it really should). But if I understand the tag synonym system correctly, it serves exactly this purpose -- if I input "applications", it will be replaced with "applied-mathematics".

We should really have some tag guidance for the "applied-mathematics" tag too.

  1. This seems to be just the union of (1,2).

  2. To the extent that this is on-topic at all, it should be a different tag.

So following Rodrigo de Azevedo's suggestion, I'd propose that the thing to do is:

  • Add some tag guidance for "applied-mathematics".

  • Make "applications" a synonym of "applied-mathematics".

  • Whenever an old question comes up which is now tagged "applied-mathematics" because it was originally tagged "applications" in sense (1) or (4), remove the tag. But probably we don't have to go do a mass tag removal all at once.

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    $\begingroup$ There's also a sense (5) which comes up in some of these tags: applications of fields outside of mathematics to mathematics. Under this proposal, this sense of "applications" wouldn't really be covered by any tag. But I think this interpretation (5) of the tag was a bit tongue-in-cheek to begin with. Probably such questions don't need to be tagged together with applications of mathematics to other fields. $\endgroup$
    – Tim Campion Mod
    Mar 10 at 18:13

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