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As those of you who have been participants in the debates over the last few days (see here, for example) may know, we have been essentially debating over the nature of the content we expect here at Math.SE.

To summarize, there are users (whom I will not name, and the one in the link is not the only one) who post answers free from context, demonstration, proof, origin, or reference. One user told me that (s)he will just continue to post Maple output because Maple clearly can do everything a human can do, but better. Another user posts beautiful formulae as answers to requests to show how this or that integral/sum/complicated expression may be evaluated in closed form, but the formulae only, without comment.

(Actually, that latter user chose to respond to requests for context by issuing the teenage-wiseass claim that he was merely stating axioms that do not require proof. Hardee-har-har.)

What to make of this? Technically, in a Q & A site, if the OP asks, "Is there a closed-form expression for this integral?" and a poster just says "Yes," then the question has been answered. If the OP asks further, "what is that value," and the poster then plops down a formula and nothing else, then that should also work. The fact that there are any upvotes means that the answer is useful to someone.

But could one of those people explain to me how on earth such an answer is useful?

Look, if all I care about is an answer, then, yeah, Maple and/or Mathematica will do in many cases. And if they can't do it, then I can evaluate things numerically. These black boxes have the whole thing covered. In fact, in a TED talk, Conrad Wolfram stated that we should change the way we teach mathematics to schoolkids because nobody is really interested in the mechanics behind computation anymore.

A personal story you may skip if you are tiring of me by now. In my previous life, I used to write software that simulates complex lithographic processes. Users of the software became trained to just trust the output of the software. Thus, when their models did not conform with their experiments, they spent a lot of valuable time in the fab checking experimental conditions, interrogating tool vendors, etc. Nobody thought to police the software vendors; this eventually became my job, because I was the only one who would design cases that could be solved analytically and demonstrate the inaccuracies in other vendors' software. Without skills needed in evaluating integrals, sums, etc., who will "watch the watchers"?

[And by the way, ponder this: how many of you can do long division on paper, quickly and accurately?]

So, I come to my point: is this a Mathematics site, where we expect folks to provide methodology and reason along with a result, or is this a Q & A site where we are OK with results plopped down from the heavens? My "not an answer" flags have been declined because some people find such answers useful. But why? How on earth is a contextless number or formula useful to anyone?

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    $\begingroup$ Just for clarification: your flag was declined because evidently some people find such answers useful. The evidence is in that those answers accrued some non-trivial number of up-votes. The moderators who participated in the discussion all agreed that the answers are not useful to us. But I hope you will agree that we moderators should not be given the blanket mandate to delete answers and questions that we don't find useful. (But just imagine how much smaller, and manageable, the site will be... :-p) $\endgroup$ – Willie Wong Nov 20 '13 at 15:41
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    $\begingroup$ @WillieWong: I totally get that, and thought I communicated that in my post. I think my question here regards why those who upvote such answers find them useful. $\endgroup$ – Ron Gordon Nov 20 '13 at 16:03
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    $\begingroup$ I think the upvotes are rather signs of admiration for skill rather than being a sign of someone generally being helped. $\endgroup$ – Michael Greinecker Nov 20 '13 at 16:04
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    $\begingroup$ @MichaelGreinecker: Perhaps, but that is then misguided a bit, isn't it? $\endgroup$ – Ron Gordon Nov 20 '13 at 16:06
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    $\begingroup$ @RonGordon I think so. $\endgroup$ – Michael Greinecker Nov 20 '13 at 16:07
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    $\begingroup$ @WillieWong: Oh my! I didn't think to wonder if that question was ambiguous. The apology should come from me - I apologize for the unclear language. $\endgroup$ – Ron Gordon Nov 20 '13 at 16:11
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    $\begingroup$ @MichaelGreinecker: I wonder how many of the people who upvote such answers spend the time verifying them. (Not an admiration of skill, as such, but an admiration of apparent (or perhaps purported) skill.) $\endgroup$ – user642796 Nov 20 '13 at 17:12
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    $\begingroup$ Welcome to the club of forumites who are perplexed by the motives of upvoting. The membership desk is kinda busy today. Please take a number. The line extends to the left. $\endgroup$ – Jyrki Lahtonen Nov 20 '13 at 17:32
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    $\begingroup$ And a $\Large{+1}$ for sharing your personal story. $\endgroup$ – Jyrki Lahtonen Nov 20 '13 at 17:34
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    $\begingroup$ @RonGordon, I think the displays of skill (such as your answers) are much more important for these hard integration problems, than the amount of help to the OP. First of all, this is not designed exactly as a help site, and OPs are not paying. More important, keeping the classical tradition alive and archiving that know-how in an accessible medium, is a huge service, like compiling a supplement to Whittaker and Watson and placing it on an open searchable web site. Highly developed integration skills are fairly rare, even in mathematicians, and documentation of the techniques is great. $\endgroup$ – zyx Nov 21 '13 at 1:58
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    $\begingroup$ @zyx: Thanks. I completely agree, and have always maintained that I do what I do here because I enjoy it. That said, one has a duty to address the OP's question or concern, but the reason why we do so is because we like what we do. I am grateful when I am told that they've learned something from what I have posted, and I hope I have been able to add to the brilliant content of this site. $\endgroup$ – Ron Gordon Nov 21 '13 at 2:40
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    $\begingroup$ @julien It seems unlikely because in many cases OPs awarded bounties for their questions, and they were awarded not to Cleo's answer, but to somebody else's answer that actually contained a proof. $\endgroup$ – Vladimir Reshetnikov Nov 21 '13 at 20:41
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    $\begingroup$ @JyrkiLahtonen: Nothing wrong with blowing one's trumpet, so long as there's a good tune in there. $\endgroup$ – Ron Gordon Nov 22 '13 at 1:14
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    $\begingroup$ @julien: as usual, you are very perceptive. Sigh...I think I've lost my sense of humor about this and am becoming a crank, or an old curmudgeon. Time for a beer. $\endgroup$ – Ron Gordon Nov 22 '13 at 6:21
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    $\begingroup$ @RonGordon If you want to save time for your family, you can just post crude answers with no explanations. Oops, I made a joke again... $\endgroup$ – Julien Nov 22 '13 at 14:06
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If someone asks a long, detailed combinatorics question, and I answer with just "25", that is not a particularly useful answer, even if it is "correct" in the trivial numerical sense. I would be free to post that answer, but other people would be completely justified in downvoting it.

The best answers convey a perspective from the answerer: this could be an explanation of a solution method, a pointer to a related topic, an explanation of the general concept, a generalization of the problem, or it could take another form. But purely numeric answers are not really better than just "Yes, I can solve this", and fall short of what I look for when I evaluate answers.

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    $\begingroup$ I would have to downvote your (hypothetical) answer as incorrect, since the correct answer would obviously be 42. $\endgroup$ – cardinal Nov 20 '13 at 21:40
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    $\begingroup$ @cardinal: Only for those unfortunates who never attended Pomona College. The rest of us know that the correct answer is 47. $\endgroup$ – Brian M. Scott Nov 21 '13 at 22:11
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    $\begingroup$ You can see lots of results without proofs in integral tables, e.g. Prudnikov, Brychkov, Marichev, and hardly anyone would argue they are not useful. $\endgroup$ – Vladimir Reshetnikov Feb 22 '15 at 21:50
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    $\begingroup$ @Vladimir Reshetnikov: the difference, in my mind, is that this site is intended for discussion of mathematics, rather than being intended as an online integral table. $\endgroup$ – Carl Mummert Feb 22 '15 at 22:42
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    $\begingroup$ @Carl Mummert Not necessary for discussion. It's intended to be a place where people can ask questions and get answers about mathematics. When I need an answer, sometimes I post a question at M.SE, and sometimes I look up in a table of integrals, and get a required answer there. $\endgroup$ – Vladimir Reshetnikov Feb 23 '15 at 0:27
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First of all, I completely agree that an answer containing a proof (based on commonly accepted axioms, not axioms tailored for a specific answer) are always more useful. But I still consider result-only answers as useful in some cases:

  • They challenge other users to look for a proof, that they might not even try otherwise because of a hopelessly-looking integral. Sometimes the shape of an answer can be a guide to where to look for a proof.
  • Some users of this forum are actually interested in a physics or another applied science, rather than in pure mathematics, and they post here difficult integrals occuring in their calculations. Believe me, in some cases they do not actually care much about the proof, they just need a result.
  • Eventually, a usefulness of an answer is determined by a community through the mechanism of up- and downvotes. If the statistics shows that many users consider an answer as useful, we should not create any rules to ban such answers (even moderators). If you consider an answer as not useful for you, just cast your personal downvote (remember that a serial downvoting is automatically reversed).
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    $\begingroup$ +1 When I was an undergrad a classmate in theoretical physics was better with integrals than I was. Partly because he faced more of these. Partly because I looked up things from Spiegel's (Schaum) Handbook, and he had had to buy a copy of Gradshtein & Ryzhik. $\endgroup$ – Jyrki Lahtonen Nov 21 '13 at 21:33
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    $\begingroup$ Voting does not determine the usefulness of an answer. In the very long run it may provide a decent proxy, though I’m not even sure of that. $\endgroup$ – Brian M. Scott Nov 21 '13 at 22:15
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    $\begingroup$ My personal impression of usefulness determines my voting. The are tooltips at up- and down-arrows explaining their meaning: "This answer is useful", "This answer is not useful". $\endgroup$ – Vladimir Reshetnikov Nov 21 '13 at 22:29
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    $\begingroup$ I disagree with point 2, in the sense that if you are a sceintist trying to publish a paper and the referee aks "why is that integral equal to $\pi/2$". The answer of "because I asked on Math.SE and that is the answer someone gave" is not going to cut the mustard. Saying "here is my calculation" or even "here is my Maple code" is a much better answer. $\endgroup$ – user1729 Nov 22 '13 at 9:24
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No. ${ }$

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    $\begingroup$ I cannot in good conscience upvote this answer. $\endgroup$ – user642796 Nov 20 '13 at 17:04
  • $\begingroup$ @ArthurFischer Obviously, you didn't see the joke. I'm just kidding. :P $\endgroup$ – Ahaan S. Rungta Nov 20 '13 at 20:28
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    $\begingroup$ Using lots of open and close braces in MathJax greatly slows down rendering (especially on some devices). You can also use   as effective filler. $\endgroup$ – cardinal Nov 23 '13 at 5:59
  • $\begingroup$ @cardinal Thanks, I didn't know that. $\endgroup$ – user1729 Nov 24 '13 at 16:16
  • $\begingroup$ No problem. I hope you didn't mind me going ahead with an edit. I'm not sure, but I suspect the cause is that, for each new open and close brace, a new variable scope must be created and evaluated in MathJax, and the speed with which that can be done depends on the efficiency of the browser's JavaScript implementation, among other factors. :-) $\endgroup$ – cardinal Nov 24 '13 at 16:20
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    $\begingroup$ Actually this answer is a self-proving theorem, whose proof is a proof by construction. $\endgroup$ – user21820 Dec 4 '14 at 6:28
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In some cases, I think it is useful to give a hint to help the OP do some work on his own. In one or two cases, that hint was just an answer. With the hope that the OP would use it to help in doing additional work, maybe even working out a solution. If the OP still cannot do the problem, the hope is that he will come back and ask for more explanation. Which I will give (unless that request was posted mere minutes later...)

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  • $\begingroup$ +1 for "that hint was just an answer." It reminds me of East Asian mathematics education. $\endgroup$ – Joel Reyes Noche Nov 22 '13 at 1:02
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IF this is a not a troll posting pre-cooked or deliberately wrong answers, the question seems to be about the merit of numeric prophecy as a style for posting answers that are generally correct.

As long as the rate of correctness of the prophesies is similar to that of the answers derived later by other means, there is no harm done, it can shorten the time to find complete answers, and it can draw more interest to the question.

It might be exasperating when people withhold information, but most of the answers on the site do not completely explain the inner thought processes and background knowledge leading to the solution. On meta, quite a few people advocate withholding information (for the good of the children), only posting hints and pseudo-answers lest the top secret be leaked. Or making OP's pay for drips and drops of wisdom by demonstrating some unspecified amount of (what passes for) effort. In such a context, I don't see the prophesies as any worse or different than a variety of other processes happening on the site, or praised on the meta. The only difference is that a prophet is more transparent about the refusal to share knowledge, and makes more explicit the assertion of superiority that is often involved in publicly acting out a refusal.

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    $\begingroup$ There is a difference between giving hints and stating an answer. For example: $$\text{Question: What is the inverse of }\:8\mod 13?$$ Are you suggesting that $$5$$ is a better answer than $$\text{Hint: Use the Euclidean algorithm to compute }m\text{ and }n\text{ such that } 8m+13n=1.$$? $\endgroup$ – user1729 Nov 21 '13 at 9:38
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    $\begingroup$ That has nothing to do with what I wrote, which was about information withholding done as public performance. What you call a hint is a full answer modulo looking up "Euclidean algorithm", since nearly every exposition of the algorithm explains its relation to solving $ax+by=1$. You also changed the question silently between the answer-example and the hint-example, in a way that favors the latter. $5$ is a fine answer to the question you wrote, and its analogue for the question you implicitly associated with the hint, is computer code for the Euclidean algorithm (or whatever other method). $\endgroup$ – zyx Nov 21 '13 at 15:50
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    $\begingroup$ To quote the sig of a famous poster from sci.math You can lead a horse's ass to knowledge, but you can't make it think. I am quite happy to withold a bit of knowledge from someone who doesn't want to think. In the case of elementary questions I am here to teach them to think. Been trying to do the same in real life for 20 some years. For the most part it is a singularly thankless job. The aspect of witholding information comes to the fore with more advanced questions, when the asker has already learned to think for themselves, but hasn't been exposed to a critical bit. $\endgroup$ – Jyrki Lahtonen Nov 21 '13 at 17:21
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    $\begingroup$ Cat and mouse challenge/response games and demonstrative withholding may have their uses, and can be initiated by some of the nicest people, but they are not elements of a free conversation between equals. Some OPs are willing to play the subordinate side of that game. A few complain and more walk away. Online forums like MSE can strive to replicate negative aspects of formal education, but a more appealing possibility is to cultivate something sui generis that does not place anyone in a role of teacher, student, or Socratic guru. @JyrkiLahtonen $\endgroup$ – zyx Nov 23 '13 at 6:43
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    $\begingroup$ I'd be very curious to know the reason behind the vote to delete. $\endgroup$ – user642796 Nov 23 '13 at 13:29
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I think there is a difference between "answer in MSE" and "answer in everyday life". A "result only" I not consider to an answer, rather comment, a useful comment. I would like to propose the possibility to move the result only answer into a comment. For example, if at least 5 people vote it. (A result only earned +100 recently.) What is your opinion?

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    $\begingroup$ In the SE model, it de-stabilizes things and causes bad feelings when other people take control over a user's posted answer, and especially when the control is used to perform some negative action. That includes deletion, conversion to comment or CW, and unauthorized large edits. Separate from that, I think there should be not any minimum requirement for posting an answer whether it concerns the length, level of detail, probability of correctness, probability OP will understand, or any other metric. $\endgroup$ – zyx Mar 5 '14 at 17:28

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