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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?

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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.)


  1. 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 think you're referring me? No offense, but your post appears inequitable. "[tortured synonym for “k... (4 comments)
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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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2 comment threads

Emphasis on "the" (1 comment)
Writing a proof is not like writing a novel (5 comments)
Writing a proof is not like writing a novel
Derek Elkins‭ wrote over 2 years ago

There is actually quite a bit that can be mechanized in proof finding. Up to and including completely deriving a proof mechanically. And I don't just mean that automatic theorem provers exist. A decent amount of a proof is highly constrained. In my experience, it is not uncommon for people to struggle with even these extremely constrained proof steps. While writing a "good" proof (i.e. one which is concise yet enlightening) might be a bit of an art, writing a proof is much less so. There isn't a concept of a "valid novel" or a "valid next sentence" in writing. There certainly is for proof writing. Writing a proof is closer to designing a bridge. There are better and worse designs for bridge that nevertheless function (e.g. a worse design might be more expensive, labor intensive, and/or aesthetically unpleasing but still work), and there are designs that don't work at all.

Derek Elkins‭ wrote over 2 years ago

To give an example of the kind of thing I'm talking about, consider the following more positive rendition of the infinitude of primes: For any finite set of primes, there exists a prime not in that set. I can produce the following proof skeleton completely mechanically from the form of the proposition even if I didn't know what "finite", "set", or "prime" meant. Proof skeleton: "Assume we're given a finite set $S$. Let $p(S)$ be (...). $p(S)$ is prime because (...). Assuming $p(S)$ is in $S$ leads to the contradiction (...). Q.E.D." Now it still requires some cleverness to fill in the blanks, mostly the one defining $p(S)$ because that step is largely unconstrained. Nevertheless, at a low level in contrast to "for the most part it isn't the case that whenever you see X, you know you'll need a proof of type Y", it actually is the case that there are only a few relevant rules at any step of a proof, though some, like cut, have an unbounded number of largely unguided instantiations.

Derek Elkins‭ wrote over 2 years ago

It is also the case that the need for insight is often oversold. Many results about sums of binomial coefficients and closely related combinatorial identities have clever proofs involving insights and analogies. However, virtually all "textbook" identities of this form fall within the purview of a decision procedure which can not only immediately tell you whether the identity holds but also produce a certificate letting you easily verify it. Being aware of such decision procedures, i.e. when insight is not required, is also very useful. Indeed, the fact that there are two commonly known decision procedures for deciding the equality of polynomials (over $\mathbb Q$, say), namely 1) writing both in the normal form $\sum_{k=0}^n a_k x^k$, or 2) comparing the polynomials at a number of points greater than their degree, means we usually don't see spelled out proofs of polynomial equality.

r~~‭ wrote over 2 years ago · edited over 2 years ago

Derek Elkins‭, I agree that there are some ways that writing a proof is more like designing a bridge than writing a novel (and some ways that it is like neither). And yes, there are skeletons that get reused and are good to know. The sorts of questions I'm objecting to are not questions like, ‘What types of proof structures could be used to prove the infinitude of primes?’ but rather, ‘How did this author know to use $p(S) = 1 + \prod S$?’ followed by, ‘I'm not interested in an explanation of why $p(S)$ is prime; what I want to know is how someone got the idea to use that definition at all!’ Are you making the case that there's a better answer to that question than ‘experience, persistence, and luck’? It's great that in some circumstances, decision procedures can substitute for insight; but even in those circumstances, in the context of reviewing a proof in a textbook, unless that decision procedure is what's being taught, do you think that's a useful answer to the question?

Derek Elkins‭ wrote over 2 years ago

I haven't downvoted (or upvoted) this meta answer, but I haven't yet completely decided how I feel. On the one hand, I think in practice there are indeed a lot of bad questions of the form discussed. On the other hand, I am making the case that there are often better answers than "experience, persistence, and luck" especially across the whole class of questions of this type. As I've stated, I've often seen questions asked about the parts of a proof that are completely mechanical, e.g. many people learning proofs would struggle making the proof skeleton I illustrated while having no problem filling it in. That is, they don't understand what they need to do to have a proof. A bit more softly, a common general hang-up I've seen for people starting out in proofs is that they don't use definitions. Of course, there is also advice specific to fields, e.g. using a generating function or manipulating the problem into a form suitable for a decision procedure.