@BillDubuque: one issue with "confusing students into thinking the answer may be incorrect" is that, if they were to submit answers with no prose and unusual formatting for their homework assignments, they probably would be marked incorrect. To the extent that we are trying to teach students, we should teach them to write mathematics in the way it is usually written by professional mathematicians, which is to say using adequate prose and avoiding unusual abbreviations.

For many people who do teach students, a significant challenge is to get them to stop writing their work in entirely symbolic form, and to transition to using sentences.

@CarlMummert But no one is proposing that students should submit hints or sketches given here. We've had this exact discussion in the past. Did you forget?

But it was not this that was being debated above. Rather, it was matters of formatting style (inlined vs. displayed equations).

@CarlMummert PS, I don't know what fields you read professionally, but in my area (number theory and algebra) it is not at all unusual to see arguments written very concisely - as in some of my answers. If students don't gradually gain the skills to reason in mathematical (vs. natural) language then they will face great obstacles when reading research literature.

@BillDubuque: I didn't forget, but it wasn't clear to me if you did, since you were still implying that the answers were "perfectly correct", while the overall correctness of the answers is debatable

@CarlMummert If you read more carefully you will see that I am speaking about mathematical correctness - not "correct according to Carl Mummert's standard of mathematical style".

Again, the reader is at fault for failing to understand. :\

@XanderHenderson No, for failing to read more carefully

To-may-to, to-mah-to.

@Xander Henderson: I'm not arguing that, though, just that the "correctness" of a post is judged by the readers, in the end, not by the author.

@CarlMummert I was agreeing with you. I was grumbling that @BillDubuque was, again, blaming the readers for not understanding his words.

@XanderHenderson Yes, if you purposely twist my words I will object since that is not a constructive method of discussion.

@BillDubuque Whatever you say, sir.

Anyway, for the record, though I think that many answers are so verbose that they greatly obfuscate the essence of the matter, I don't downvote them into oblivion. I respect that others have different pedagogical styles and goals, and something of use may come from those answers. I hope that others will extend the same professional courtesy.

@CarlMummert I see there are many examples in your own posts that are far worse than anything I've written, e.g. i.sstatic.net/KXntQ.png To which the questioner replied "Thank you for your answer. While I lack the background in formal logic to make full use of it, I'm sure it can help others with similar questions." Kettle ... pot ...

@BillDubuque: indeed, I see that the last post is not at all in sentence form.

@BillDubuque could you give any recent and moderately mainstream examples for research literature in this style? (Bonus points for them not being widely considered as good mathematics written-up poorly.)

@CarlMummert I can't make any sense of your prior comment. Please give some context.

@BillDubuque: you linked to this screenshot i.sstatic.net/KXntQ.png . While it is true that it uses logical notation, the post is overall structured in sentence form, and moreover is virtually identical (modulo not actually using LaTex) to the way it would appear in a journal article.

@quid As soon as Carl provides his manual of style - so that we can avoid being downvoted for volating it

@BillDubuque I only made this explicit later, but I meant research literature.

^^ examples for this.

@CarlMummert But this site is not a journal. Did you seriously think that would be helpful to the OP. It looks like the output of an automated theorem proving program! If that's what you consider good style, then I'm proud that I don't follow your style guidelines.

@quid Pick almost any publication in algebra and number theory, e.g. looking at the paper currently on top of my pile (Hendrik Lenstra's Lectures on Euclidean Rings) one will find many examples of inlined equations more complex than those I used in that trivial proof that XH preferred displayed vs. inline.

@BillDubuque: it is interesting, you chose a post about the Tarski-Kuratowski computation, the purpose of which is specifically to write a formula as it would appear in a formal language (ie. like an automated theorem prover) and count the number of quantifiers in its prenex form. In any case, would you care to post a link to one of your papers so we can see how your ordinary style is, if you think that the style you use here is one that would be professionally recognized?

Indeed, looking at the whole post, math.stackexchange.com/questions/2801681/… , there is much more than just the three displayed formulas.

@CarlMummert Unlike you, I don't view this site as a journal, so I write much more informally because I believe that doing so allows one more latitude to focus on the essence of the matter.

I also write more informally here, but without disregarding the usual standards for mathematical exposition.

@CarlMummert fyi: informal mathematical exposition is rarely constrained to journal rules of style. That would be extremely cumbersome.

@CarlMummert You didn't answer the question: do you think that your answer was helpful to the OP? Given the OP's comment it seems not. That is the pertinent issue here - not whether it meets the style standards of your favorite journal.

@Bill Dubuque: I don't think that most people on this channel need any info about the usual standards for mathematical exposition.

As for that answer, there is an unfortunate pattern for people who are seeing facts in computability theory in a CS class to post that kind of exercise, without really having the background to answer it, because they have not seen enough techniques in their class. So, if the OP had been interested in actually learning computability theory, the post would have been helpful. If they just wanted to answer their homework problem and move on, more power to them

Looking over the entire post, I don't see anything calling out for revision. In particular, unlike the other answer, mine identifies that the set is $\Pi^0_2$ complete.

i don't think there's anything wrong about your writing style, @Bill, but if people find it hard to read, they're gonna downvote you. at least they're telling you why

@BillDubuque thanks. I don't think the issue is about the absolute complexity of the formula. Is this available online somewhere? Not sure if lectures notes from 1974 (that are not even typeset) count as recent [...] research literature.

if i started writing all of my answers in pink-on-black with all mathfrak variables they'd still be right but if people didn't like it i'd understand

Google books gives me snippets, which is why I know it's not typeset. I did not see overly big formulas so far; but I could not see much.

@AlexanderGruber The problem is that some folks look for any random excuse to downvote posts of those who have opposing political views, and this is one of those random reasons.

@BillDubuque: that argument would be stronger if they also downvoted posts of your that use proper mathematical writing, such as math.stackexchange.com/questions/2981235/…

@quid There were no "big formulas" in my answer either, so I'm not sure what you're looking for.

@AlexanderGruber Pink on purple! I insist!

@BillDubuque In any case, the style of the notes looks much more "verbal" to me than some of your answers.

@BillDubuque I think it is a mistake to dismiss feedback from peers like this.

@quid The discussion was about a specific answer, and about inlined vs. displayed equations.

@quid You're certainly entitled to your opinion.

@BillDubuque to pull up a non-typeset example from the typewriter times is a bit odd. But I'll concede that there is research literature that does not contain many displayed formulas. Overall it still does not at all resemble the style of many of your answers. (I'm even astonished you insist on this; I thought you considered your style as innovative and not the usual mainstream stuff.)

@quid My posts do contain many displayed formulas. It's only that one two-line post that received such a critique. CM doesn't like that I abbreviate words and God knows what else (probably color too)

@BillDubuque I'd say the issue is that some of your answers read a bit like pseudo-code. But, arguably that's fine for a network that started around a site for programming. But it's just not what most mathematicians are used to.

@quid I don't know your background so I can't comment on that. My more concise posts (hints and sketches) are exactly what I've seen written by many of the best teachers I've had - some quite eminent mathematicians (e.g. Artin, Rota). That someone prefers to use much more formal style here matters little to me.

@BillDubuque I am honestly very confused about how you seem to be interpreting my critique of your answer. I said: (1) I can understand people downvoting answers that are formatted in a way that hinders clarity, even if the mathematics is correct, (2) the particular answer you provided is one that could be made more clear with the addition of some whitespace, but (3) I don't think that it is worth downvoting that answer (and there is another answer in that thread which is a worse offender).

My critique was not meant to justify downvotes, was meant to point out only a small issue, and was meant to be constructive. I am sorry that you seem to have heard it otherwise. I will endeavor to be more clear in the future.

@BillDubuque it is not clear how my background is relevant to my point. They might have written like this, alright, but where? On a blackboard maybe or on a sketch-sheet? The medium matters though.

@XanderHenderson Thanks for clarifying that. I am glad to hear that you did not mean to justify such downvotes.

@quid Blackboards, problem set solutions, notes, (e)mail, back-of-envelope stuff, etc

@BillDubuque I see. As said, in my opinion, not the right style for the current medium.

We might look at how eminent mathematicians that are on SE write here.

@quid Alas, most of them have left due to poor site management.

@quid: or professional mathematicians - eminent mathematicians may or may not represent the norm of the profession, and we do have a large number of professional mathematicians who we could look at

@BillDubuque orthogonal.

@CarlMummert yes, that too.

@quid looking at number theory, we have "top users" such as Robert Israel and Barry Cipra,

I am a very informal person and that is reflected in my style of exposition in informal settings. CM appear to be much more formal than I (e.g. requesting in his profile that people address him using the proper spacing in his name rather than the SE-generated name). So it comes as no surprise that we may disagree on the level of formality to use in posts here. But I did find it surprising that he thinks such differences are valid reasons for downvoting posts.

@Bill Dubuque: I see few of your posts, and so without thinking about you in particular, I do think that the quality and clarity of writing as judged by each voter is a valid reason for them to vote up or down, as are many other factors. I think it is very debatable whether posts written in a nonstandard manner are "useful" either pedagogically for the OP or for long-term storage as a reference.

@CarlMummert In case you may not know, some users have written in comments that they have learned number theory from my posts. I also receive countless emails thanking me. This is the feedback that matters to me - not silly comments about style differences.

@BillDubuque it's interesting. I think we have a different meaning of "formal" but that's likely besides the point.

@quid Yes, there are a couple senses of "formal" being used above, but they are not unrelated.

Could this discussion be continued elsewhere, other than a room dedicated to requests for Close/Reopen/Undelete/Delete/Edits of questions and answers on Math SE? This conversation seems to have digressed.

@amWhy Yes, I agree, and apologies for the extreme tangent.

Perhaps math.meta chat? Or whatever, quid would be a great option, too, particularly given the welcome!

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@BillDubuque That is fantastic, and I would expect that those users have upvoted your answers.

However, I learned long ago that what works for one student may be toxic to another.

No matter how great you are as a writer or instructor, you should expect there to be a subset of your readers / students who don't find your approach useful, and who might even actively hate you (hyperbole aside).

Those readers / students are, with a certain probability, going to downvote some of your answers on the grounds that those answers are not useful to them. For whatever reason.

Such is life.

@XanderHenderson I don't recall every expressing any concern about downvotes from students. Rather my concern has to do with organized downvoting from crude due to perceived problems with the question.

In case it wasn't clear from my prior posts: I could not care less about SE "rep". What I care about is votes that mislead students into believing that the answer may be incorrect.

@BillDubuque Your perception of organized downvoting from CRUDE might be more to the point.

No such organized downvoting exists, so far as I know.

@XanderHenderson I'm not interesting in arguing about what is clear as day.

crude is not a user. Check usernames. There is not username "crude". Each voter on any post is an independent user: a student, professional, or both, or neither. Stop attributing to "crude" valid independent votes from users, whose votes are as valid as any you cast.

@amWhy CRUDE is well-named.

@BillDubuque You clearly are interested in arguing it, because you keep brining it up...

@XanderHenderson You pinged me.

@BillDubuque Stop changing the subject every time anyone makes a valid point you know not how to answer or address.

I pointed out that your style might not be a good fit for some students / readers (i.e. users of MSE), and that those users might downvote your answers for that reason. Your response is that you weren't talking about those users, but about organized voting from "CRUDE". You brought up CRUDE, not I.

@amWhy No change. The organized voting and punsihing in that room is both CRUDE and highly unethical in my opinion.

Stop moving the goal posts.

@BillDubuque Again, your PERCEPTION of organized voting.

@XanderHenderson As I said, it is clear as day. At least have the courage to accept responsibility for your actions.

Can you please point out what specific action that I have taken that I could/should accept responsibility for?

@BillDubuque That comment is better a self-meditation for you, @BillDubuque. Projecting your own shortcomings onto others is rather unbecoming.

@XanderHenderson I think it is quite clear to any active participant of that room how unethical and unfair some of the voting is againt some users (e.g. MR). I don't necessarily agree with MR's views, but that doesn't mean that I will behave unethically toward him. I've been collecting stats on the users targeted by CRUDE.

@BillDubuque Please be specific. What is MR?

@XanderHenderson The user MR - one of the most highly mentioned and targeted users inthat room

I don't know that I have ever interacted with a user with the handle "MR," thus I must admit that I have no idea what you are talking about.

And, for the record, while I might experience frustration with certain users from time to time, I work very hard not to mention users in CRUDE, nor to vote on questions/answers on the basis of who posted them.

@XanderHenderson I never said that you interacted with him. I find it highly implausible that any active member of that room doesn't know who that is

@BillDubuque Okay. If you say so.

@XanderHenderson Of course you don't have to, since someone else is organizing the lists of things "deserving" deletion and closure.

Be specific.

@BillDubuque How convenient: you always have a vague, ambiguous accusation, to deflect any responsibility on your part for maligning others.

Can you point out any specific post that has been mentioned for closure or deletion in CRUDE that you feel should be kept?

If so, did you mention it at the time?

@XanderHenderson Don't be ridiculous, you know damn well what I am talking about. Since you are not being sincere I have no interest in further wasting my time.

@BillDubuque I honestly have no idea what you are talking about.

Right, and I'm the pope. Bye

Okay, whatever.

@amWhy, do you know which user "MR" is?

I don't see a user with that handle on the users page...

@XanderHenderson Mr. Roberts? There was a movie by that name. I see MR and think "Mister"... Someone is playing with us.

Mister Roberts. Of course, there was also Mister Rogers (As in Mister Rogers Neighborhood. But neither are Math.SE users.

@XanderHenderson See the screenshot below for proof. Both of you surely know damn well who MR since there are many such discussion in CRUDE. Don't waste your time trying to explain your way out of that. It is by now clear to me that we are on completely different planets when it comes to matters of ethics. Best of luck with yours. i.sstatic.net/FLOEC.png

So, in other words, a screenshot in which MR is referenced after I comment is to be taken as evidence that I know exactly what you are talking about?

Why not just link to the place in chat where that conversation took place?

I.e. provide me with the context to understand what you are talking about?

@XanderHenderson Comments targeted AT you. As I said, don't waste your time. You have proven quite clearly that you are not sincere.

I don't remember that conversation, @BillDubuque. You could very easily act in good faith and provide a link to it.

I don't know why you are choosing to be intentionally obscure.

@XanderHenderson Please have the courtesy never to ping me again

Indeed, if @amWhy and I are referencing the same user, it looks like my take on the matter is "I though one thing about said user, but was convinced by someone else that i was wrong."

BUT I DON'T REMEMBER THAT INTERACTION, so the context to know which user is being referenced is lacking.

@BillDubuque For the record, I have found the context you are quoting.

The user to which I am referring is gimusi.

Is gimusi the same as MR?

The context, by the way, is "I noticed an odd behaviour, what might explain it?"

My initial interpretation was that it was nefarious; I was convinced otherwise by the folk who were active in CRUDE at the time.

@XanderHenderson PLEASE don't ping me any further.

Okay, take your ball and go home, @BillDubuque.

@quid The Johnny Johnny Yes Papa thing blew up because of how disturbing the video is to anyone not 3 years old. It was so bad that I think they deleted one of their videos at some point. In any case, its become quite the meme.

@XanderHenderson I think MR is Michael Rosenberg? I do remember you railing against said user.

@BillDubuque While I don't consider @XanderHenderson to be an angel, you seem to be quite incapable of taking criticism. Sure, you may be older than most of us, but that doesn't invalidate feedback.

@RushabhMehta I have no clue what you are talking about, nor am I interested in it.