5:59 AM
And the downside of creating a new tag is...? — Mehrdad Apr 19 at 21:49
It is rather difficult to answer this within a space of one comment (600 characters limit) considering that there were many discussions about creating tags on meta.
Asaf's reacted by linking to a previous discussion on this:
Although reading his answer is probably sufficient answer to your question, I'll try to write a bit more about this and perhaps illustrate this on some examples.
Let me first try to mention here some past discussions and after that I will also try to summarize my thoughts.
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Past discussions on meta. I will list some previous discussion related to creation of tags, and occasionally point out some parts of them.
These posts probably better express what is the community consensus - what I post below are just my personal opinions.
Although there are still differences between the users about some tag-related issues and about specific tags, it seems that there are several things which are generally agreed.
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If one searches the word -theore in Tags one find some tags with the name of well known theorems as stokes-theorem or bayes-theorem. Also if one type lemm one finds only three tags, that are borel-cantelli-lemmas, pumping-lemma and hensels-lemma. I understand that is good have tags for special ...

Asaf's answer raises some reasonable points:
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Tags, as I see them, have three major roles to play, under ideal settings: They hint the reader what sort of topic the question will be, before opening it. Thus, using correct tags will help increasing the exposure of the question to its target audience. They help partition the database into mo...

It is probably worth reading in full, but here is a short quote: "Having too many tags on the site causes too much fragmentation, and may lead to questions having less exposure than ideally. So having specialized tags for specialized topics should be introduced when they help compensate for an actual problem."
There are many other related posts on meta.
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When coming to post a question, one looks for the most fitting tags. After a while, it may be apparent that a new tag is in order. Good examples are the elementay-number-theory and elementary-set-theory and so on. However, sometimes when the topics are of higher level, it might be as apparent (...

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This is a meta meta question on the principles by which we should decide which tags to create. The motivation for the question stems from this meta discussion, in which a majority thought that there shouldn't be an errata tag, whereas a significant minority of $5$ upvoted my answer saying there s...

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I have got the impression that a lot of new tags have been introduced lately. Is it intentional? Frankly, I cannot see why ultra-specific tags such as "simpsons-rule", "boundedness", "dot-product" or "product-space" could be useful.

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In several discussions here on meta, one user expressed the opinion, that every new tag (or at least most of them) should be discussed on meta before creating. For example here There is a saying that it's easier to apologize later than get permission first. I disagree with that when it comes...

As you can see, some users were frustrated with creation of various bad tags to suggest that more-or-less any new tag should first be discussed on meta. (However, this is a bit extreme position and I only remember one user explicitly saying something like that.)
Here is a discussion in chat which started from comments to the above question:

### Some thoughts on what makes tag useful

Oct 9 '14 at 11:44, 20 hours 44 minutes total – 33 messages, 2 users, 2 stars

Bookmarked Oct 22 '14 at 12:11 by Martin Sleziak

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My thoughts on this. Now let me get back to your question and summarize my personal views on tag creations.
And the downside of creating a new tag is...? — Mehrdad Apr 19 at 21:49
I definitely do not claim that creating a new tag is always a bad things. However, I'd guess that for a user somewhat experienced with this site and with usage of tags here, it should be self-evident that at least some thought should go into deciding when and how create a new tag.
For example, having several tags for the same thing is not good. Or using a tag name which has many different meanings and might therefore cause confusion when using tags.
What are tags useful for?
If we want to discuss how the tags should be created and used, we perhaps first should think about why do we need tags and what we want to use them for.
1. Searching. Tags might help with searching - you can restrict search by choosing the tags which are likely to be used in the question you are looking for. For some frequently asked question a very fast way to find them is look at frequent tab of the corresponding tag.
2. Lists of created questions. This is somewhat related to searching. AFAIK the lists of related questions (both the list shown when posting a question and the list generated after the question is posted and shown in the sidebar) are partly based on tags.
3. Exposure of questions.
4. Tags help keeping the site better organized. Which, in turn, helps users to use the site more efficiently.
The points 3 and 4 are closely related to each other. Considering how big the traffic on the site currently is, no user really reads all the questions. Many users probably mostly look at questions in the tags related to the areas they are familiar with.
I would not be surprised if there were some users which mostly view questions in their favorite tags. (For example, filtering questions by using intags:mine, see here or here for more details.)
When new tags can cause problems?
So if we agree on what tags are actually for, we should try to create new tags in such way that it helps these purposes.
Tags should help when searching the questions. This only helps if the questions which are about some topic also have that tag.
If we create too many very specific tags, it might be impossible to tag some questions which belong there, because a question can have at most 5 tags.
An example from elementary number theory: the questions about and are very likely to be closely related. Maybe it would be better to have one common tag as suggested here. (The proposal on meta has score +7 at the moment.)
There were several discussions on meta related to gcd. For some time it used to be synonym of , the synonym was removed as a result of discussions on meta: On (gcd) and (greatest-common-divisor) and other similar tags
For some time there used to be separate and tags - so wa had two tags for the same thing, which is definitely not ideal. This was resolved after this meta discussion: highest-common-factor $\cong$ greatest-common-divisor
Another thing is that tag do not help that much unless they are really used.
Take as an example. It has been created very recently and at the moment there are only two questions in that tag.
So if somebody is interested in a question about box product/box topology, then restricting the search using this tag does not help much.
If this tag stays and is eventually added to many of the questions about box topology which already are on this site, then it eventually becomes useful. Of course, this needs a lot retagging of existing questions.
And you could make basically the same argument about any new tag.
If a new tag is introduced, adding it to existing question will take quite a long time. (Bumping too many old questions just because of this is not a good idea.)
So it will take some time before a new tag really starts to be useful.
Let us compare the situation between some older tag and a relatively new tag related to the same topic.
I'll try to illustrate this on a topic I am at least a bit familiar with. Stone-Cech compactification is an important topic in general topology.
In fact I suggested some time ago that perhaps it might even have separate tag.
As a side note, I think that this is one of the situations when tags might help with searching. Namely, some questions might not explicitly mention the words Stone and Cech, but simply write down (the relatively common) notation $\beta X$, $\beta\omega$, $\beta\mathbb N$, $\beta D$, etc.
I have listed a few examples of posts which are about Stone-Cech compactification, but the words Stone and Cech do not appear in the text here.
The consensus from the discussion I linked was that the questions about Stone-Cech compactification should be tagged by the tag and the tag-wiki was edited to reflect that.
Last year the tag has been created. The discussion whether it should stay as a separate tag is still unresolved.
We might use this as a comparison to which extent tags are useful for finding questions.
We would expect the questions about Stone-Cech compactification to be under the tag if the convention agreed upon in the discussion of 2012 is followed.
However, if the tag stays, it is also a suitable tag for such questions. Since this tag exists since last August, perhaps it was enough time so that some questions about this topic get the tag.
Let us simply compare the numbers.
Searching for stone cech is:q returns 146 questions
If we restrict this to compactness tag we get 74 questions.
If we try compactification tag instead we get only 9 questions.
The above two examples perhaps show that new tag brings a lot of work in the sense that many old questions need retagging. They also might bring a lot of work if they are removed.
I can mention tag a recent example.
Feb 24 at 16:40, by Martin Sleziak
I'd say it is good to leave some time for other users to notice the new tag, which also gives them opportunity to post on meta, if they think that the tag should be removed. (And there are not that many posts to be removed, if the tag is not added too quickly. Not that long ago about 70 posts were bumped by adding and removing the tag when tag was created and the removed: http://chat.stackexchange.com/transcript/3740/2017/1/4 )
So a new tag was created, approximately 70 questions have been bumped when it was added, and approximately 70 question have been bumped when it was removed again.
Choice of name for a new tag matters. Especially if it contain a word which has several meanings or which is commonly used in non-mathematical meaning.
A good example of this is tag, which was originally created for divisors in algebraic geometry. Not surprisingly, many users start using it for divisors in elementary number theory.
This issue is not resolved yet. The relevant discussion on meta can be found here.
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I hope that the above thoughts and examples address at least partially the question from your comment.
@Mehrdad The answer Asaf has linked to addresses your question. And I have also tried to write something about this here. — Martin Sleziak 12 secs ago
Let me also mention that I tried to explain also in my post why I am asking about those tags: "In both cases the tags start growing relatively fast. Therefore I considered asking about community opinion as a reasonable thing to do. If the community consensus is that those tags should be removed, it is better to find out before the tags contain too many questions and removing the tags is a lot of work. "
I did not want to see a repeat of the situation similar to the situation with (straight-lines) tag mentioned above. (Many questions bumped by adding new tag, then many questions bumped by removing new tag.)

1 hour later…
7:25 AM
Holy cow, that's a long answer to such a short question. =P I wasn't trying to ask what the downside is in general; I was trying to ask what the downside is for the inequalities you mention. "Inequalities" is hardly appropriate for something about AM-GM, Cauchy-Schwartz, or Jensen's inequality, and something like "metric spaces" or "convex analysis" would be pretty overkill (though perhaps not entirely inappropriate). My view is that they're all used commonly enough that they would pretty obviously flourish on their own. — Mehrdad 2 mins ago
@Mehrdad So I misunderstood your question. But the general answer also answer the specific question. Too many too specific tags cause problems. So it is good to discuss which inequalities should have their own tag, so that we do not end up with many tags which are not really used. (BTW I think that (inequalities) tag is definitely the most suitable for the questions about AM-GM etc. The question is whether to create additional tags.) — Martin Sleziak 2 mins ago
@Mehrdad Perhaps we could say that the question in your comment is very similar to what I am asking in the question. Anyway, if you think that for example the three equations you mention deserve to have a separate tag, you can post that as an answer - this is exactly what I am asking and from reaction of other users to your answer we would see whether they agree with you.) Also, if you wish, we can continue this discussion (general or specific to inequalities) in chat. — Martin Sleziak 34 secs ago
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I think all of those that you mentioned (AM-GM, Jensen, Cauchy-Schwartz) are just fine for tags. I would even say Chebyshev's and Markov's inequalities are fine for tagging, though not the other two you mentioned. My rule of thumb would be that an inequality deserves a tag if: Mathematically...