7:46 AM
As an intermediate step, shall we update the tag wikis based on the classifications in the answers below? — barto 10 hours ago
@barto I honestly don't know what to do. I am somewhat hesitant to make changes in tag-wiki based on an answer with total score +3 in a matter which is relevant to large number of questions. On the other hand, this question has been around for almost a month and it will probably be away from the frontpage soon. So the probability that many other users will come here to discuss this matter or at least vote is not too high. — Martin Sleziak 25 secs ago
From the discussion on and .
20

The tags elementary-number-theory and number-theory have been recently mentioned in this question: There are 1,732 questions tagged both elementary-number-theory and number-theory However, in that question these two tags serve only as an illustration of a more general issues. In this post I'd lik...

3 hours later…
10:45 AM
Since there is no clear support for an "artificial" solution, why not just go with what happens in practice and is self-explanatory, viz. number-theory is for Number Theory? — quid 35 mins ago
@quid I think that what happens in practice is that most user use (number-theory) for advanced topics. Just have a look, for example, at recent Andres' retags. (As an example of a user who is quite long on this site, so he has experience with the tagging system and his actions clearly indicate that he understands division between the two tags in this way. So do many other users, but it would take some time to collect examples of edits corroborating this.) — Martin Sleziak 36 secs ago

1 hour later…
12:04 PM
@MartinSleziak this is somebody trying to enforce what does not happen in practice. The idea to have a restriction on an extremly common tag is in my opinion ill-advised. The effort would be much better spent adding more specific number-theoretic subtags. — quid 1 hour ago

1 hour later…
1:27 PM
@quid That's an interesting point of view. We can't verify if every existing number theory question is tagged according to the new 'rules'. Introducing more specific subtags is never a bad idea. But then what should we do with the existing tags? Merge them? — barto 1 hour ago
@barto it is in fact very easy to verify that it is not the case, one just has to browse the list. My solution would be to say: The tag number-theory is for questions in Number Theory; since this is a very broad field consider to add a more specific tag too: [list of common subtags]. I will try to elaborate on this later. — quid 43 mins ago
Sounds good. You can post this as an answer I think. It would be good to start making that list of common subtags, so we would know exactly what it would be like if the community supports this idea. — barto 23 mins ago
I definitely disagree with the claim that is used in the way as a general tag and that is a subtag.
I remember seeing many retags by experienced users which removed tag from low-level questions.
I pointed out Andres' edits simply because they were quite recent and I remembered them.
Typically, if was used for such a question, it was done by relatively new users.
Even the tag-excerpt for number-theory explicitly says that: Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.
If needed, I can try to dig up examples from such retags.
But I think that I am definitely not the only user who understood these two tags in this way.
@quid It seems that discussion about how these tags were (are) used could be quite long. I would suggest to move it to chat so that we do not fill this with too many comments. — Martin Sleziak 14 secs ago
1

The tag number-theory should be an all encompassing tag for questions in Number Theory. It thus would naturally contain elementary-number-theory, analytic-number-theory, algebraic-number-theory and so on. Granted, this is a broad tag but still it conveys some relevant information. More specific ...

I will quote Qiaochu's anwer from 2011: Some tags do have sophisticated and unsophisticated versions, such as [number-theory] vs. [elementary-number-theory].
So at least at the time these tags were view in this way.

1:52 PM
quid's proposal to merge number theory with elementary number theory seems good to me, provided that there are enough subtags to make the distinction between them. This way the distinction between number theory and elementary number theory would not be made based on only one tag, but one should make this distinction for oneself based on the topic.
It seems a conceptually very good idea, but it's important to decide if there should be introduced more tags (not just tags like, say, primitive-roots, but also subtopic tags such as algebraic-number-theory, p-adic-number-theory etc) and if so, which.

I am not saying that this proposal is bad.
I am simply saying that this is not how the tags have been used so far.

I was not my intention to post that as a reaction to yours, apologies if this offended you.

I don't see anything offensive there. (Sorry if I sounded offended.)
To get some kind of sample I went through all 50 questions which were shown to me on the 50th-page of recently active questions in .
(Of course, the content of that page is changing constantly.
Those were questions which were last active around 2014/08/06.
Most of them are tagged only (elementary-number-theory).
I found 5 questions where was removed at some point: 1, 2, 3, 4, 5.
I found two question tagged with both tags: 1, 2.
Sorry, I cannot use chat at the moment, but then I also should not continue this debate right now. I will come back to this later. Only one clarification: I know very well that this distinction does not originate here with you and is encoded in the current tag-wiki. It is still not a good idea in my opinion and should be discontinued/modified. — quid 4 mins ago
In that case it seems just a misunderstanding on my side. I thought that you are aiming to say that the tags are (and always have been) used in the way that (elementary-number-theory) is subtag of (number-theory).
My impression was that most experienced users used them differently (at least it was the case maybe a year ago). And many inexperienced users used both tag, but most of such questions were quickly retagged.
Of course, it can be that it is only my impression and it is completely wrong.

Diophantine equations are another problem; if we keep the distinction between number-theory and elementary-number-theory there's no objective way to decide which diophantine equation belongs where.

So I simply misunderstood what you were saying.
I know very well that this distinction does not originate here with you and is encoded in the current tag-wiki. I certainly agree with this.
It is still not a good idea in my opinion and should be discontinued/modified. I am not sure this is the case, but if this will be the consensus from this discussion, I will certainly do my best to retag new questions (and also older questions, which will be bumped for some other reasons) in accordance with the general agreement.
I have some doubts whether it will be easy to reach some kind of agreement.
@barto About Diophantine equations: Certainly, it is not clear immediately whether $x^3+y^3=z^3$ requires only elementary methods or something more advanced where you first see this equation. This one is well known, but if I modify it to, say, $x^3+y^3=z^3+1$, who knows - maybe the new one is easier and can be solved by elementary methods.
But it was agreed that it is o.k. to retag question after they are answered, if the answers indicate that question is related to some other areas (some other tags).

2:22 PM
I agree that we can retag questions after they are answered, but when we are retagging a diophantine equation from, for example, elementary-n-t to n-t, is this judging more advanced diophantines as 'non-elementary'?
If for example a diophantine is tagged as n-t, receives an answer involving advanced techniques but one month later someone finds an elementary approach. Is it ok to retag it as elementary-n-t?

I would say it can be tagged ent, if the equation can be solved by method like: Check possible remainders modulo 7.
If a solution uses elliptic curves, I would tag it n-t.
If the solution uses unique factorization in Gaussian integers, I would tend to tag it as n-t. (Maybe some users will disagree with this classification...)
I don't see having questions tagged by both tags as a problem.
For example something which can be easily solved if you know some advanced stuff; but it can also be solved using elementary methods, even though the solution is quite lengthy.
For example let us assume that there is a question about Fermat's theorem on sums of two squares. And among the answers we have Zagier's one sentence proof, proof using Minkowski theorem and also some of the proofs which do not use anything advanced.
In that case it seems as the most reasonable thing to use both tags.
(Assuming we divide number theory questions into e-n-t and n-t in the way suggested in my answer. Which, I think, is the way these tags have been used for a long time.)

3:14 PM
I agree, but unfortunately one user has forced the decision the other way by having gone on a massive retagging spree during the past month. I pointed him to this thread and asked hm to wait for a community consensus, but he continued. I also notified a moderator (RobJohn) but received no reply. So much for democracy. — Bill Dubuque 36 mins ago
@BillDubuque We will see whether that user will respond here, but as he posted some comments in this thread, he is certainly aware of this question. I want just point out that many other users made retags of the same type in the past. Here is a random sample. — Martin Sleziak 33 mins ago
@Martin Well if one user is free to remove them, then another is free to add them back. Maybe I'll start doing that. If you knew this was occuring, then why didn't you say something long ago? It was very wrong that this massive campaign was hidden from this discussion on the matter. — Bill Dubuque 8 mins ago
@BillDubuque (and quid): I noticed those retags too, but I did not look at them in detail. (I assumed that those are retags of new questions.) Since this thread doesn't give any consensus at the moment, it is difficult to say what would be the best course of action. Maybe a separate post (separate answer), asking users not to retag questions in these two tags until a consensus is reached? Such post would also be much better place to discuss this. So far the comments posted to this answer do not have much in common with the proposal given in quid's post. — Martin Sleziak 44 secs ago
@Martin When did you first become aware of this massive behind-the-scenes campaign? — Bill Dubuque 2 mins ago
@BillDubuque I don't recall exactly. As I already said I think that this is not directly linked to quid's proposal. As I do not want to add more off-topic comments, I'd suggest either making a separate post for discussing this or moving the discussion to chat. And I should also say that the best person to give some explanation about actions of a particular user is that user himself. — Martin Sleziak 46 secs ago
@Martin I am quite disappointed given that you are the OP, and you knew this was going on for some time, but did not udate this thread so the community could become aware of this. It took other users (quuid and I) to stumble upon this to realize what was happening - just like what happened with the massive deletion campaign, Forcing decisions like this sets very bad precendents. — Bill Dubuque 29 secs ago
@MartinSleziak it seems to me that mainly dually tagged questions are retagged, also quite old ones. (While it might not seem like it given the many comments I write, I am a bit short on time ATM so I did not check carefully.) On the one hand, it is not unusual that somebody does this now as the issue came to attention, on the other hand if somebody was seemingly asked to postpone the activity and does not it would not be so nice. — quid 43 secs ago

3:59 PM
@barto Maybe in the light of the above comments, we should stop doing retags removing from elementary questions until the issue is settled in some way.
I probably did a few such retags during last months. (It seemed that the discussion about these tags is going nowhere and neither of the two opinions was gaining strong support. So I thought it was fair game to make such retags.)

4:26 PM
Yes, I agree. I must say I did some retags too last weeks

4:57 PM
Would be too localised?

I am not sure.
We will see whether someone will respond to your question here.
If not, there are two possible ways to find out.
You can either create the tag. If some other user doesn't like the new tag, they will either remove it or make a post on meta to discuss the new tag.
Or you can ask on meta before creating the tag.
To me it seems a bit too specific. But probably not enough to go ahead and remove the tag, if somebody else created it @barto.

1 hour later…
6:32 PM
To digress a bit from the heated discussion about number theory-related tags:
What about . What is it supposed to be for?
1

If $B \ne C$, prove that the perpendicular distance from $A$ to the line through $B$ and $C$ is $$\dfrac {|| (A-B)\times(C-B)||}{||B-C||}$$ where $\times$ means the vector cross product. Attempt: Let the perpendicular from point $A$ meet the vector joining $B$ and $C$ at the point $P$. Let ...

6:53 PM
0

When I see a recently active discrete-mathematics question about e.g. number theory being poorly tagged, I tend not only to add the relevant, more specified, tags but also I remove the tag discrete-mathematics. I thought it would be ok to do this, because many number-theory questions do not have ...

1 hour later…
8:22 PM
Only recently I have noticed the tag. It does not contain that many questions. But it probably can be useful, so it might be worth keeping.

1 hour later…
9:42 PM
Should and be synonyms?
The questions about ceiling function and question about floor function definitely have a lot in common.

@MartinSleziak yes. In fact I think they could be subsumed into something like "rounding" or "rounding-functions" I was somewhat astonished that I found no "rounding" tag. But perhaps it is just called differently.

Maybe.

Now, for number theory: while I understand there is some tradition of keeping el NT and NT separate I do not really understand the rational.
The fact that there are 1000+ questions tagged with both shows this does not work.
The benefit if any from keeping them separate seems marginal to me. The downsite is permanent "fight" against mis-tags.

Well, perhaps my main objection to the change is that this was the way they were tagged already before my time.
As I have said a few times, if some consensus is reached, I will try to help with "correct" tagging.
"Correct" meaning whichever way the consensus will go.
The way I see it, he benefit from keeping would be that questions on very easy stuff (gcd, divisibility) can be kept separately. So users such as Matt E could easily find question closer to their level, easy stuff would be left to mere mortals to answer.
BTW sorry if the exchange seemed a bit heated, that was definitely not my intention. And also it seems that it was simply misunderstanding.
One thing that bother me about the whole issue is that I have some doubts whether some consensus will be found. And while the discussion is not settled, we are in a kind of limbo.

9:59 PM
No worries.
I agree the most unfortunate would be if it were unclear.

@quid I have to admit that I am less active in retagging questions that I used to be. (And also I am more interested in other tags.) It might be that I simply did not notice when the situation got worse. But my impression is that the separation used to be quite clear.

On Matt E, he also answered elementary NT and left MO for this site to find more basic things (at least so my understanding).

Wow! If there were 1732 questions with both tags at the time of Najib's post and there are 1236 of them now, some users must have been busy either deleting or retagging.
BTW this is definitely not work of only one user.
Well, Matt E was just an example. But I can understand that some people are more interested in seeing more advanced question.

Check the activity of the user mention by BD, pages over pages of tag edits. 20 pages over the last two weeks.

Well, yes, I went through sorted by recent activity and I saw there two users popping up quite frequently. But I do not want to turn this to witchhunt.
As I said, I have noticed this. (I think it is hard not to nice it for anybody regularly visiting the site.)
But the number of retags I saw seemed not that big.
I did not see any reason to investigate actions of an experienced user. And I considered it fair game to retag new questions "in old way" until the discussion is settled.

10:09 PM
It is not a witchhunt. Not few might consider it as a service. Anyway these are some 400 edits since mid january.. Mainly tagedits in NT it seems. So this accounts pretty much for the dfference.

It seems that Andres, me, maybe some other users understand that current usage of these tags is such that is for more advanced questions.
I have tried to go through randomly picked posts from mid last year to see which way they have been retagged if they had two tags.

Everybody that reads the tag-wiki excerpt should understand it in this way. This includes me. I want this changed;
Also, I do not think you have any particular responsibility. That user is "old enough" to know what they are doing.

But as you rightly say, 1k+ posts in both tags suggest that large part of posts is not tagged in accordance with the tag-wiki.
As the question was bumped today, it got a few more views and some users voted on some of the answers.
We will see whether with time a few more users will vote.

@MartinSleziak My main issue is that also advanced number theory is extremly mixed.

@quid I certainly agree with this.
I posted the question with the hope that we (as a community) will be able to somehow clarify the usage of those two tags.

10:20 PM
And I do not mean it is mixed because of tagging not being acurate but because it is so diverse as a field.

And since the list of topics seemed to work find in and , I have posted some suggestion of list of topics here.
The tag-wiki for those two set-theoretical tags are certainly detailed enough to help with most of the posts.
For the number-theoretic tags we only had something like this:
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 — Jyrki Lahtonen ♦ Jan 1 at 14:16
I understand your arguments about common tag for all of . They seem quite reasonable.

I think for set-theory the situation might be a bit more clearcut. Everybody will be exposed to some "naive" set theory that perhaps is not even really set theory. And then there is actual set theory as a field. But this is not like this in NT. (Maybe it is also not like this in set theory, but since the main active users in that tag seem to be happy they do as they like. No reason for me to get invovled.)

Maybe some kind of consensus will be reached, but it will definitely take some time.
Well, it's the only other tag with little elementary brother. So it is natural to compare the two.
And maybe situation is not that clear even there. For example, I'd guess that some people would consider ordinals and transfinite induction an advanced topic, which should not be in . Yet, it is listed in the tag-wiki.
But if there are two set-theoretic tags, the division has to be made somewhere.
The same is true if we are going to have division between elementary and more advanced topics in number theory. There will be some gray areas, but at least to some extent it should be agree what belongs where.

I'd guess in practice for, say, cardinal arithmetic it depends a bit what exactly is asked.

Well, certainly.
There is difference between the question about $\mathfrak c^{\mathfrak c}$.

10:30 PM
One issue for number theory is for example today was asked for which $n$ is $2^n - 7$ a perfect square.

And the question about possible values of $\aleph_\omega^{\aleph_0}$.
Yes, I saw that question. I think that someone was able to find some older copy of the same question.

I'd guess so, knowing how it is called should make this easy.

Sawarnik's question is here.

What I mean is it is not a priori clear what type of number theory this even is.

Yes. I would tag this as . (Based on the criterion that Diophantine equations belong to this tag.)
But if it is clear from the answer that more advanced methods are needed, tag can be added.

10:35 PM
But the common proof is algebraic number theory.

Of course, what I wrote above is the way I understand the tagging system the way it is at the moment.

And then take Catalan's equation. Also elementary NT? But then for a long time it was mainly part of analytic number theory (linear form in logs).
And then it was proved algebraically.

It seems I would have to use at least 4 tags in the current system on the Ramanujan–Nagell question: e-n-t, n-t, alg-n-t, diophantine-equations.
We will probably not solve this right now.
Let us wait and see how people will vote on that question.
Although it seems that voting and democracy sometimes do not work too well on meta.
(I have used all my daily votes second day in a row. I am procrastinating big time.)
I am not sure what time zone you're in @quid, but it's getting late here. So maybe I'll go to sleep in a while.
I don't think there's much more to be said here, anyway.

Yes, let us continue another time @MartinSleziak (same timezone here).

So good night then!