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(Prompted by this question)

As you can see here, there are (as of January 2020) 5,931 questions tagged and . (For the record, there are respectively 28,327 and 31,874 questions in these tags, so that's not insignificant).

It's always been my impression that these sort of tags are mutually exclusive: vs , vs , vs , etc. In any case it makes zero sense for a question to have both tags, it's just redundant; they can always be tagged just .

I believe the differentiation between and works because they are names of actual classes in standard curriculums in the US, and there are very active users (user?) dealing with set theory questions. However, things didn't work out so well for probability theory, as we can see in the question I linked at the top, and number theory is going the same way.

Rather than dealing with specific cases, I'd like to ask general questions:

  • Is differentiating tags by level useful?
  • Is differentiating tags by level feasible? How?
  • How to decide when a tag needs to be split like this? How to decide what the threshold should?
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    $\begingroup$ Yeah, we dealt with this mess (although on a much smaller scale) with a large intersection of set theory and elementary set theory. $\endgroup$ – Asaf Karagila Jan 1 '15 at 12:44
  • $\begingroup$ In the case of set theory the tag info does the good job. If you look at the tag-info for elementary-set-theory and set-theory you will see that a list of topics which are covered by the tag. Of course, the lists are not exhaustive, but they cover a lot of questions and help to decide which tag to use. $\endgroup$ – Martin Sleziak Jan 1 '15 at 13:55
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    $\begingroup$ I have noticed this, too. I occasionally hunt these down, and edit the tags. My conservative estimate (no actual data I'm afraid) is that in 90% of the cases I either removed the NT tag, or replaced it with ENT. The rule of thumb is that if I can answer the question without rubbing the three grey cells together real hard it is ENT. Not foolproof, and the algorithm is still evolving $\endgroup$ – Jyrki Lahtonen Jan 1 '15 at 14:16
  • $\begingroup$ @Martin I doubt it's the tag wiki. Look at the case of probability: it's explained right in the tag excerpt what tag covers what, people still confuse the two. $\endgroup$ – Najib Idrissi Jan 1 '15 at 14:16
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    $\begingroup$ New users frequently tag their question NT, because they learned about the material in a course titled (Intro to) NT. If we could make them actually read the tag excerpts, the problem would disappear. Not holding my breath. $\endgroup$ – Jyrki Lahtonen Jan 1 '15 at 14:18
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    $\begingroup$ Two unrelated points: .) If elementary-number-theory is intended only as "low level number theory" than elementary-number-theory is not a good name. .) I do not see the issue with redundancy in itself. Is is also a problem if something is tagged analytic-number-theory and number-theory? This is inherent to the tagging system. $\endgroup$ – quid Jan 1 '15 at 14:19
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    $\begingroup$ @quid Regarding redundancy, I guess I could have said that better: number-theory is explicitly for higher level than elementary-number-theory. I assume users ask elementary questions then add both tags because they both seem to apply, but they don't. You're right that redundancy is not the right word, what I mean is that if elementary-number-theory can apply (and OP thought so by tagging the question that way), number-theory cannot apply. It's different from, say, tagging a question homotopy-theory and algebraic-topology. $\endgroup$ – Najib Idrissi Jan 1 '15 at 14:25
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    $\begingroup$ @NajibIdrissi What I meant is that in the case of the two set theory tags, the tag-info does a good job in explaining what belongs to which tag. (The other question is whether the users actually apply the tags correctly.) In the case of number theory, the tag-wikis for elementary-number-theory and number-theory are more-or-less identical. So they do not help much when deciding which tag to choose. $\endgroup$ – Martin Sleziak Jan 1 '15 at 14:26
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    $\begingroup$ I will just mention that also elementary-general-topology tag was discussed. This older post is also to some extent related to the general questions asked at the end of your post. $\endgroup$ – Martin Sleziak Jan 1 '15 at 15:31
  • $\begingroup$ As @Martin mentions, such issues have been raised before. Tagging questions with "sophistication" leads to various problems, e.g. in the thread Martin cites see this answer and its comments. As in the example linked there, questions often receive answers of various levels of sophistication - which is pedagogically desirable since it often motivates students and plants in mind seeds of important generalizations that they will later need. It would help to have the ability to tag the level of the answer. $\endgroup$ – Bill Dubuque Jan 1 '15 at 16:24
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    $\begingroup$ [cont] I dislike the idea of meta "sophistication" tags (or meta tags at all) And I don't believe I'm talking about sophistication in this thread: this seems to apply to answers, not questions. Even if it's possible to answer an elementary question with a great amount of sophistication using very advanced notions, it still remains an elementary question. $\endgroup$ – Najib Idrissi Jan 1 '15 at 16:58
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    $\begingroup$ @MartinSleziak Yes of course, you're right, that's not what I'm saying. Fixing the tag wiki is important. But I believe that it would be great if the need for retagging questions was alleviated by having users tagging them correctly in the first place, and it seems to me that the tag name is more important in that respect than the tag wiki. $\endgroup$ – Najib Idrissi Jan 1 '15 at 19:19
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    $\begingroup$ @Najib I was under the impression that "elementary number theory" is number theory other than "algebraic number theory" or "analytic number theory." So no imaginary numbers, no algebraic integers, no asymptotics or heuristics. $\endgroup$ – Robert Soupe Jan 4 '15 at 19:20
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    $\begingroup$ @RobertSoupe it's true that Elementary Number Theory is actually a technical term which pertains to specific topics, which is different from the case with, say, set theory (at least as far as I understand set theory). Certainly elementary number theory is an area of active research, which should indicate the tag is being misapplied more than anything. A more ideal solution would be to allow both tags or--if users just cannot get past using "elementary" for easy things--deleting the elementary tag altogether. $\endgroup$ – Adam Hughes Jan 9 '15 at 10:11
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    $\begingroup$ @URL Please don't make useless edits like this. $\endgroup$ – Najib Idrissi Jan 10 at 9:15
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I believe that, although most of the times only one of the tags should be used, there are situations where both are appropriate. Consider, for example, this highly upvoted question. OP asks a question involving only prime numbers and elementary arithmetic:

Definition: A positive integer $n$ is called interesting if it has a prime factorisation $n=pq$ with $p\ne q$ such that the prime factorisation of $n+1$ is $p'q'$ where $p'$ is the prime after $p$ and $q'$ the prime before $q$.

Are there other interesting numbers [other than $14$ and $21$]?

However, the only answer makes use of the highly non-trivial and non-elementary result that, for $n\geq25$, there's always a prime between $n$ and $\frac65n$. Given the fact that nobody else bothered answering, it doesn't seem like the problem can be solved in any substantially simpler way.

I believe that, if a question (and its answers) might make use of both elementary and advanced concepts, using both tags should be appropriate.

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In the past week or so, I edited the tag-wiki for ENT to make it far less wishy-washy that primitive roots and quadratic residues were NT and everything before that in a typical class or textbook are ENT. (It's not quite easy vs. non-easy. For instance, Pythagorean triples are the solutions to a non-linear Diophantine equation and therefore fall under NT. But that has its own tag so it probably doesn't come up too often.)

Personally, I retag questions as I encounter them, but don't feel inclined to drain the backlog of questions that are surely overwhelmingly mistagged. (In theory, it could be made worth my while... coughtagbadgeprogresscough) It's not an ideal situation, but at the same time I'm not sure that much of anyone is harmed by the current state of affairs.

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  • $\begingroup$ I will mention that the division between these two tags was also discussed here: How to differentiate between (elementary-number-theory) and (number-theory). But one cannot say that some consensus was reached. (In fact, some users suggest that (elementary-number-theory should be a subset of elementary-number-theory). $\endgroup$ – Martin Sleziak Jan 9 at 23:22
  • $\begingroup$ I revised your edit - the topics you considered "advanced" are actually part of ENT by most definitions. Please be more careful before making such edits. $\endgroup$ – Bill Dubuque Jan 10 at 1:27
  • $\begingroup$ Thank you for your vigilance, @BillDubuque, but please note that the statement that quadratic residue and primitive root is NT has been an explicit part of the NT tag since August 2015. That was the source of care that I took when I made my edit to have the two tags both explicitly state that they are mutually exclusive. If you don't think that those topics should be NT, you probably should fix that tag as well. $\endgroup$ – Matthew Daly Jan 10 at 5:54
  • $\begingroup$ Thanks for pointing that out. I updated that too. $\endgroup$ – Bill Dubuque Jan 10 at 6:28

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