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I am quite tempted to ask this question because I have committed this a few times recently (e.g. Is this a useful concept in trying to solve the Riemann Hypothesis? Or am I still to naïve? , Why isn't this a bijection from N -> R? or Formulating a philosophical observation in math ...)

On one hand, those are people who have gross misconceptions about basic notions in maths, which means that the questions they ask are in most part nonsense; on the other hand, they are not necessarily deluded people and it is hard not to notice their genuine interest in the subject.

So, is it better to engage with them, even if the discussion may stray a long way and end up mostly useless for anyone else (possibly also useless for the OP and for the person providing the answer - sometimes it feels like talking to a wall), but with a potential benefit that they may learn something new, and they may feel that their interest has been appreciated?

Or, close down such questions quickly without engaging, which doesn't waste anyone's time but may breed resentment and come across as rude - as it is hard to close the question with a reason meaningful to the OP without engaging the OP at least to some extent?

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  • $\begingroup$ I once taught an OEIS contributor basic modular arithmetic from learning elementary set theory. It all depends on how they look at things. $\endgroup$ – user645636 Jan 31 '20 at 21:16
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    $\begingroup$ From our Help Center on What type of questions should I avoid asking: If your motivation for asking the question is “I would like to participate in a discussion about ______”, then you should not be asking here. However, if your motivation is “I would like others to explain ______ to me”, then you are probably OK. I think this is a good general guideline. Not easy to judge the motives of an asker, but keeping this rule in mind helps. $\endgroup$ – Jyrki Lahtonen Feb 1 '20 at 6:08
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Allow me to weigh in, in part as a long time contributor, in part as a set theorist who gets exposed to these type of questions perhaps more than the average mathematician, and in part as a moderator.

There is a simple test here: does the person want to learn? Do they want to understand their mistake, or are they here to preach?

If the question is phrased as "Cantor was wrong", ignore it, close it, downvote it to oblivion, and send the author into a question block.

But if the question is "What am I missing?", then that means the author accepts that they might be missing something. That they are willing to learn. In this case, engage with the author, at least until they prove to be of the first type (if at all, sometimes they really just want to understand their mistake, and that's the best possible outcome).

Some years ago there was a person contributing here who was very much on the borderline of crank, believing that Peano arithmetic is inconsistent. But they tried to prove it rigourously, and they asked serious mathematical questions and admitted mistakes when these were pointed to them. That, to me, was the best possible discussion one can have with a layperson (or at least a laylogician, I guess).

On the other hand, other people occasionally come to the site to post a few soapboxing questions just to engage in a heated argument about how everyone were wrong for the last 150 years, and Cantor's proof is in fact mistaken. Well, it's useless to explain things to a wall...

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    $\begingroup$ When I was a young PhD student I found myself on a mailing list of some crank alongside very famous mathematicians, some set theorists and some not. By chance, I ran into one of them the following week at a conference and I mentioned that. He said that he has a rule, when someone emails him some crankery email, he replies the first time pointing the mistakes. If the person insists, he adds them to the spam list. $\endgroup$ – Asaf Karagila♦ Jan 31 '20 at 23:41
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    $\begingroup$ Asaf has articulated my approach as well. Reply (at most) once. If they do not want to learn, ignore future communications. $\endgroup$ – GEdgar Feb 1 '20 at 0:57
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    $\begingroup$ I was going to say "Please vote them down" but I love Asaf's standard -- it is compassionate, considerate, and sensible. Just be aware that if you decide to engage you're probably setting yourself up for spending more time than with the average question. $\endgroup$ – JonathanZ supports MonicaC Feb 1 '20 at 15:11
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    $\begingroup$ There are some real risks in engaging with extreme cases. For example, John Conway and I (and others) tried to politely explain on sci.math why Archimedes Plutonium's attempted proofs of the Twin Prime Conjecture etc were wrong. Eventually this led to Archie Pu posting uncountable threads over many years about all of our "errors" and how we can't do proofs or teach math, etc (they show high in google search results on our names because he put our names in the title). He also called our universities (Princeton & MIT) and tried to stir up trouble. $\endgroup$ – Bill Dubuque Feb 1 '20 at 23:57
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    $\begingroup$ So be careful, esp. if you are younger and/or just embarking on your career and need to worry about impact on your evolving online rep when job searching. The HR folks etc aren't always sharp enough to filter out such nonsense when evaluating job applicants. $\endgroup$ – Bill Dubuque Feb 1 '20 at 23:57
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Inherently, I don't think that "fringe" is a useful category to judge questions - as always, a good test to apply to a question is to consider if it does all it can to support a good answer. Some specific guidelines that seem frequently relevant when dealing with fringe questions are the following:

  1. Is there some clear question? Sometimes, fringe questions are really just rants in disguise - which fail to pose a question. "Why is this wrong?" or "Does this work?" are fine questions, whereas "Why are mathematicians IGNORING this proof?" is not really a question (and neither is a rant about how everything and everyone in math must be wrong). Similarly, the text of the question ought to be devoted to a (single) coherent question - it's common that fringe questions are more of a rambling journey with questions thrown in here and there, which makes writing good answers basically impossible.

  2. Is the question about math? I've seen plenty of fringe questions which are really about philosophical concerns that don't really draw upon mathematical expertise - or which, perhaps, regard the presentation of math in culture or in teaching, but not actually the content of that presentation.

  3. Is the OP's background sufficiently clear? Sometimes a fringe question is truly unanswerable because the OP has made up all sorts of vocabulary and refused to explain what it means - or because they seem to have redefined something fundamental without really explaining. Conversely, even if the question is really odd and is phrased in odd terms, I've sometimes felt, as an answerer, that the OP's perspective is clear enough that I can write an answer starting from their perspective.

  4. Is the OP acting within the expectations of the site? Fringe questions have a higher than usual tendency to come with moving goalposts, to get reposted over and over with inconsequential modification, or to result in endless dissatisfied comments from the OP on every answer they receive. These are moderation issues that are reasonable to respond to by disengaging, downvoting, close voting, or, for issues beyond the reach of such measures, by flagging.

It's worth being explicit about these more general aspects of quality since fringe questions often miss the mark of quality required by this site, but sometimes are close enough that the OP can fix them with a bit of guidance from comments judging the question against the site's expectation rather than against mathematical rigor. Importantly, I don't think that it is appropriate to use down votes or close votes simply because a question has misunderstood mathematics (even though I certainly understand the impulse to do so) - the purpose of questions is to provide a good foundation for an answer, not to communicate in mathematical rigor. Indeed, I think several of my favorite answers on this site come from questions which contain really bad mathematics, but which contain everything needed to write an answer.

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  • $\begingroup$ I agree that these criteria suffice for most cases, but there is one case in which Asaf's approach is better in my opinion. Consider a Collatz crank who posts a 20 page poof of the conjecture after a header question asking "I can't possibly have solved an open problem so easily, so where is the mistake in my proof?" And suppose that all 20 pages comprise nothing but complicated case whacking, and many seem correct. This crank can repeat the process; even if someone spends a few hours to find the single error, he would just replace that bit with a couple more pages and post again... $\endgroup$ – user21820 Feb 12 '20 at 3:14
  • $\begingroup$ Maybe you included the reposting behaviour in your point (4), but even without reposting, I don't think such kind of posts have any value. $\endgroup$ – user21820 Feb 12 '20 at 3:17
  • $\begingroup$ @user21820 I would still think about why we wouldn't like such a question in terms of quality, not crankery though - even such a question could be improved if the OP narrowed their question to be more along the lines of "Would a proof with this structure work?" (not asking about equations specifically and not requiring them to be reproduced for the question) or "Is this specific step of algebraic manipulation valid?" (focussing on a single small detail). The question you posit would surely fall under the "Needs more focus" close reason without such revision. $\endgroup$ – Milo Brandt Feb 12 '20 at 3:44
  • $\begingroup$ I don't disagree. Though we don't even know whether the Collatz conjecture can be proven by analysis of 10000 cases (just like the 4-colour theorem), so I'm not sure we can answer "can a proof with this structure work?"... The issue is that if such a crank posts it as a proof-verification question, it is not currently classified as off-topic per se. $\endgroup$ – user21820 Feb 12 '20 at 8:49
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In broad strokes I am in agreement with Asaf Karagila’s answer. However I wish to extend on their response in a small way. The audience for a question is not just the person who write ask the question but other people who view this site.

People who engage in psydomath are often operating under a profound misunderstanding of how mathematics operates. While these people often have an ideological commitment to their incorrect ideas[1] I think there are cases where demonstrating how a mathematician examines a mathematical argument has value. Of cause answers like this have diminishing returns and there is no value in doing this for the nth minor variation on a bijection between reals and naturals.

One last thing I have to say is that I often gain much personal value from some of the fringe questions when researching and thinking through the answer leads me to interesting mathematics. For example a series of fring questions about spherical geometry motivated me to have a better understanding of the geodesic.

[1] For want of a better term.

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