Comments on Is ‘How would you know to do the next step?’ always a bad question?
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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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