Is ‘How would you know to do the next step?’ always a bad question?
We have a user who keeps posting questions of the form, ‘How would you [tortured synonym for “know”] to [do the next step in a proof]?’ Leaving aside the various other reasons that these posts are bad[1], my question is whether the question is intrinsically bad based on its form alone. (If so, I would assume that the policy should be to close any such question, as no amount of editing to solve the other problems would save it.)
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I don't want those other issues to be a distraction from the central question here. I'm only mentioning them so that people can consider what these posts might look like if they were otherwise written to be exemplary questions: if they showed understanding of or engagement with the surrounding context (well-researched), if they rewrote the central concepts in their own words instead of (or perhaps in addition to) posting large screenshots of the source material, if they were written using a vocabulary that wasn't needlessly obtuse and distracting (good English), and if they were questions about mathematical concepts instead of questions about understanding the non-mathematical parts of what an author is communicating (basic reading comprehension). ↩︎
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I believe that yes, such questions are intrinsically bad.
There are two ways to attempt to answer such a question. First, the answerer could try to get into the head of the author of the particular proof being examined, and figure out how they made the necessary leap. If the proof is well written, the proof itself should include motivation for any tricky steps, either explicitly or because subsequent steps make it clear why this step is needed. But in that case, the asker should also have been able to extract that information from the proof. If the asker can't understand some aspect of the proof because of some unfamiliar mathematical term, or because the argument is missing a step that the asker can't fill in, those are different questions (and better ones, not least because they show some attempt at engaging with the material instead of posting as soon as something confusing is encountered). If the asker simply doesn't understand the written word well enough to read what is laid out before them, they are beyond hope. On the other hand, if the proof is poorly written, an answerer is in trouble if they attempt to read the proof author's mind. Any such answer would be a subjective guess, and not itself of high quality.
The other approach is to forget about reading the proof author's mind, and instead try to answer the question literally (‘How would you know to...’) and coach the asker on the art of proof writing as if they were a student. The problem with these answers is that they are all the same. Writing proofs is an art, not a mechanical process; one might as well ask, ‘How would you know to write the next sentence in a novel?’ Figuring out a next step (not ‘the’ next step; there usually isn't one and only one way to make progress) is always a process of trial and error, guided by intuition, sometimes alternating between working forward from givens and backward from goals. There are a small number of general tricks, like proof by contradiction, but for the most part it isn't the case that whenever you see X, you know you'll need a proof of type Y. We don't need dozens of questions of this form if they all have the same answer.
Coaching an art, such as proof writing, is not a good fit for a question-and-answer site, because it needs to be an interactive process, and it depends not only on the specifics of the task at hand (which could conceivably be encountered by other people searching for related terms and thus would be a useful public resource) but also on the specifics of the asker: their aesthetic, their intuition, and their current level of ability, all of which are difficult if not impossible to search for. Narrower questions related to the art are of course a different story. But surely a question like ‘How would you know to write this specific sentence?’ would be unwelcome in the Writing Codidact, and likewise I think these sorts of questions should be unwelcome here.
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Absolutely not. They're the kind of questions as https://math.stackexchange.com/questions/tagged/motivation?tab=Votes and https://math.stackexchange.com/questions/tagged/intuition?tab=Votes.
So the most time mathematicians are working, they're concerned with much more than proofs, they're concerned with ideas, understanding why this is true, what leads where, possible links. You play around in your mind with a whole host of ill-defined things.
I can't remember which book. If you know, edit my post. Field Medallist Michael Atiyah wrote somewhere.
Is there one big question that has always guided you?
I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion.
People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.
And somewhere else, Atiyah wrote ...
So the most time mathematicians are working, they're concerned with much more than proofs, they're concerned with ideas, understanding why this is true, what leads where, possible links. You play around in your mind with a whole host of ill-defined things.
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I have few opinions on it. I am in a homework helping site (where people says to solve their homework (show their works)).
bad English
It may not be your fault you don't know English well, but it's not my fault either. ~ Olin Lathrop
When I joined that homework helping site, I asked a question with no work (But, I had did some work I just didn't show my attempt) then mod had deleted that post. Then, I asked the same question again with solution attempt. Then, no one directly answered. They said, try it "this way, that way". They never answers directly. They just suggest what to do next. That way the author (I) could learn those things. If those people had directly answered $a=b$ than I couldn't learn why $a$ is equal to $b$.
When an user is keep asking lot of questions without any attempt
Tell him that he must show his attempt what he had tried (say it in few posts if he keeps asking without any attempt and keep ignoring your comment then I will say to start deleting those posts and don't undelete until he improve his posts. If he still asking new questions without any attempt then warn him. And everyone knows what to do after warning (ban him)).
The user was showing attempt earlier but not anymore
Answer his few questions if you can. And if you see that he have asked more than 10 questions without any attempt then tell him to show attempt then, do as I said above.
The user sometimes show attempt sometimes doesn't
I will say this kind of behavior is acceptable. I will say to give him hint for this kind of problems (when he don't show attempt). For this kind of behaviour, we can be sure that the user have idea what the topic actually is. But, when a user keep posting without any attempt that sounds like the user don't have any idea of that he just wants us to solve his problems (Simply called them backbencher).
How would you know to do the next step?
Telling from my experience, we sometimes don't do next step cause we think that our answer will be wrong. While Math CD wasn't active. I was asking lots of questions there. A person had told me,"always try to do next step it's OK if that's wrong". But, sometimes I can't understand what to do next because of "I don't know lot of stuffs" (Since, I don't follow academic books). For that reason, I will say (to author of those kind of question),"when you can't understand next step then just request us to give hint instead of requesting to give the whole answer".
But, if you have any idea how to do that (It's OK if that's wrong) then just do cause that could be correct also. Sometimes, we think that our answer is wrong but it may not be. So, never scare of doing next step.
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