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My opinion is that if you are going to ask a question, then it is in the best interest of everybody, with no exception, to include as much information as you (strictly) need in order to make yourself understood.

According to this philosophy, I would recommend that it is not only a very good idea, but should be mandatory to make the effort to explain (in a few words, that's all) what your notation is.

I made a comment on a post recently (which no longer exists, it was deleted as a duplicate question) that the notations $\Bbb F_2$ and $\langle {x^3 + x^2 + 1}\rangle$ (or some such) needed to be explained in the body of the question. However, I received a rebuke in the comments that it is not necessary to define them, because if you don't immediately know what they are from knowledge of the context in which the question is being asked (that context, in this case, being apparent from the fact that the word "field" appeared in the title), then you won't understand what they mean even if this information was, and that it is sufficient for them to be explained in a comment.

In short, if you don't understand what the post is about, you don't get to be told.

Fair enough, you need to know your knowledge domain before you can answer it, but I would have thought that MSE was more than just a question-and-answer forum. Ideally it would be a place where less capable mathematicians and students of mathematics can come to learn something new, and to be able to read an existing question and answer and pick up on an area of maths which they aren't familiar with.

What do others think the "purpose" of MSE actually is? To answer questions posed as pithily and efficiently as possible, or to contribute towards the spread of knowledge and skill in general?

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    $\begingroup$ I agree with the spirit of your Question, but not the first paragraph; you simply cannot include a textbook into your Question, and once you do, the Question becomes something I would vote to close for lack of clarity. I think giving the definitions is good, but perhaps don't bother to define the terms in the definition. Another solution is to add links to e.g. wikipedia. $\endgroup$
    – Calvin Khor
    Commented Jun 29, 2021 at 12:14
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    $\begingroup$ @CalvinKhor I'm not saying include a textbook. All I'm suggesting is that you include a few short words: "... where $\Bbb D$ denotes an arbitrary ordered integral domain" or whatever. It can then be up to the reader to go and look up what an ordered integral domain is, which would be impossible to do unless you throw the poor reader a bone by stating that it's an ordered integral domain in the first place. $\endgroup$
    – Prime Mover
    Commented Jun 29, 2021 at 12:20
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    $\begingroup$ Right, which is precisely what I meant; give the definition, and not the definitions used by the definition. This is not quite the same as "include as much information as you can". Sorry for the rather tangential point, it was a reflex in response to such a strongly worded paragraph ("with no exception") $\endgroup$
    – Calvin Khor
    Commented Jun 29, 2021 at 12:22
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    $\begingroup$ @CalvinKhor +1, but the question is specifically about notation. I read it to mean that, for example, $\mathbb{F}_2$ should be stated as denoting the field with two elements, but there is no need to define a field. $\endgroup$
    – user1729
    Commented Jun 29, 2021 at 12:22
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    $\begingroup$ @CalvinKhor Amended the relevant sentence to say "need". Apologies, I get where you're coming from now. $\endgroup$
    – Prime Mover
    Commented Jun 29, 2021 at 12:31
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    $\begingroup$ To address the specific post you asked about, if you had had $\mathbb F_2$ and $\langle x^2 + x+1\rangle$ explained to you, do you think you would have been able to answer the question? Do you have experience with finite fields and ideals in rings? $\endgroup$
    – JonathanZ
    Commented Jun 29, 2021 at 13:02
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    $\begingroup$ @JonathanZsupportsMonicaC Personally, I was taught $(x^2+x+1)$ denotes the ideal generated by the element $x^2+x+1$ (I also found that Brilliant.org uses this notation, and other people+places too - including the question Arctic Char just linked to, below). So I assume that $\langle x^2+x+1\rangle$ means the same thing, and could probably solve the question under this assumption. But I shouldn't have the make this assumption. $\endgroup$
    – user1729
    Commented Jun 29, 2021 at 13:26
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    $\begingroup$ @JonathanZsupportsMonicaC Yes I like to think I would. Whether I would actually do so or not is beside the point: this for me would be revision of stuff that I'd done long years ago but since forgot. But whether I did or not is completely irrelevant. $\endgroup$
    – Prime Mover
    Commented Jun 29, 2021 at 13:30
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    $\begingroup$ @ArcticChar No it wasn't, but it was in the same, er, field. :-) $\endgroup$
    – Prime Mover
    Commented Jun 29, 2021 at 13:32
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    $\begingroup$ It would take almost no extra work to write "$\mathbb{F}_2$ (the field with two elements)" and "$\langle x^2+x+1\rangle$ (the ideal generated by the polynomial $x^2+x+1$". These brief explanations of notation are helpful, and should be included in the question. $\endgroup$
    – Xander Henderson Mod
    Commented Jun 29, 2021 at 13:48
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    $\begingroup$ I strongly disagree with this requirement. Starting to include explanation for each bit of notation, each small term, each so and so, that sounds like too much work. If I were a serious student who wanted to ask a serious question here, knowing that I have to essentially practice-run the preliminaries section of my thesis on the site before each question would be a great reason to never ask any serious question. What this question is saying, essentially, is that any question that cannot be understood by a freshman is not giving enough context, and with that I strongly disagree. $\endgroup$
    – Asaf Karagila Mod
    Commented Jun 29, 2021 at 14:16
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    $\begingroup$ I don't see why someone who is asking a question about forcing needs to explain the notation $M\models p\Vdash\dot x\in\dot y$ to a level that someone who is not a set theorist could understand it. I'm not saying it can't be done, but for me, as someone who might answer this question, it is a complete waste of my time, since it means that a question that was previously two or three paragraphs and well-written for me, is now a few screen-lengths with stuff that I've known since a decade, and nobody is expected to learn from MSE anyway. $\endgroup$
    – Asaf Karagila Mod
    Commented Jun 29, 2021 at 14:18
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    $\begingroup$ @AsafKaragila You don't need to explain it, you need to define it. "where the above means M is a model for p therefore the upper closure of x is in the upper closure of y". Or whatever it means. How certain are you that everybody uses the same notation as you? $\endgroup$
    – Prime Mover
    Commented Jun 29, 2021 at 15:00
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    $\begingroup$ @PrimeMover: In this case? Very. $\endgroup$
    – Asaf Karagila Mod
    Commented Jun 29, 2021 at 15:17
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    $\begingroup$ Is $0\in\mathbb{N}$? $\endgroup$
    – JRN
    Commented Jul 22, 2021 at 14:53

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