Engage with people from the fringes - or not?

Question

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?

Answer 1

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).

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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...

Answer 2

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.

Answer 3

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.