I have been active on MSE for around 3-4 days. I have noticed the fact that people either ignore or downvote intuitive/graphical/visual analogies of a problem. However I feel that intuition is one of the most important things that one should walk hand-in-hand with while learning a subject like mathematics. I have always been posting graphical ways of thinking/visualising a given problem. I had to delete a correct answer due to too many downvotes, just because my answer was graphical and not mathematically-rigorous. And the accepted answer was the one which couldn't be solved by hand and used WolframAlpha to find the correct answer. Shall I stop giving intuitive answers?
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3$\begingroup$ Maybe you can be more specific and show us the answer? This answer of yours for instance I think is good and greatly helped by the graphics. But I would also understand if a heavily graphical answer recieves downvotes in a question asking for a proof, especially if a proof is not also supplied. I would guess that the downvotes are not because your answer was graphical, but purely because it was "not mathematically-rigorous" $\endgroup$– Calvin KhorCommented Sep 14, 2020 at 5:38
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$\begingroup$ @CalvinKhor I have deleted the answer... Shall I undelete it? $\endgroup$– Soumyadwip ChandaCommented Sep 14, 2020 at 6:09
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$\begingroup$ That is one possibility, another is to provide the link to the question or answer. Users with over 10000 reputation can see deleted answers (despite being called "deleted") $\endgroup$– Calvin KhorCommented Sep 14, 2020 at 6:12
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$\begingroup$ math.stackexchange.com/questions/3825253/… $\endgroup$– Soumyadwip ChandaCommented Sep 14, 2020 at 6:16
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2$\begingroup$ In my opinion, this question is asking for a proof. Your answer gives 4 solutions but doesn't prove that there are no more, and the graphs are not very indicative of how the non-graphical proof can be made. I note though that the slick observation of Ross's comment on the question can turn your answer into an essentially rigourous answer, since a graph is essentially a pretty excel sheet (and its not so hard to trust a graph at integer points) $\endgroup$– Calvin KhorCommented Sep 14, 2020 at 6:28
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$\begingroup$ @CalvinKhor So what did we conclude? You may check my other answers. I have mostly relied on visualisations to answer questions. $\endgroup$– Soumyadwip ChandaCommented Sep 14, 2020 at 6:36
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$\begingroup$ I have checked a couple others, like here, I don't think the graphics add anything, and are only 'beautiful' when they are depicting an ideal case; e.g. if $-f(x)<g(x) < f(x)$ and $f(x)\to 0$ then $g(x)\to 0$, but this could have also been proven by using $2f$ instead of $f$, and then the picture is less nice. Also, the inequality only needs to hold in a neighbourhood of the point, but in your examples they are true for all $x$. So, I think if a graph can provide insight that is hard to say in words, like the first example, it is a good idea; $\endgroup$– Calvin KhorCommented Sep 14, 2020 at 6:48
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$\begingroup$ If the graph is added as a remark, it might take up the screen space of a reader who doesn't really care, and now just goes to a different page because they can't be bothered to decipher the image. Also, in contrast to memes, a good graphic requires the reader to really fight with it to understand it. This is a lot of effort that not many people would be willing to put in, which may partially explain the "ignoring" of these answers. And importantly, questions in Math.SE are often principally looking for proofs, and e.g. graphs might be sometimes seen as "missing the point" of the question $\endgroup$– Calvin KhorCommented Sep 14, 2020 at 6:48
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$\begingroup$ @CalvinKhor Thank you for your views. You may write it as an answer. I would accept that $\endgroup$– Soumyadwip ChandaCommented Sep 14, 2020 at 6:50
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1$\begingroup$ I'll see if I remember to consolidate the above into an answer later :) Or someone else can also provide their views in an answer. PS I have also extensively tried to use graphics in my answers. I do agree that the effort does not feel rewarded at times $\endgroup$– Calvin KhorCommented Sep 14, 2020 at 6:55
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2$\begingroup$ Looking at the timeline, it seems like the answer you deleted was the first answer given. I think people are more likely to react negatively to a "visualisation" or "intuition" answer, such as yours, if it is posted as the first answer unless it does actually answer the question. I know I would react negatively if I came across such an answer (one concrete reason for this reaction is that questions with answers are less visible). $\endgroup$– user1729Commented Sep 14, 2020 at 9:46
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$\begingroup$ @user1729 didn't that actually answer the question? $\endgroup$– Soumyadwip ChandaCommented Sep 14, 2020 at 9:47
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$\begingroup$ @SoumyadwipChanda Not really. As pointed out in the comments there, there is no justification, or even suggestion that justification is needed, that the solutions are $1, 17, 18, 19$, rather than "close to" these numbers. In my opinion, the graphic tells me that there are at most four solutions (maybe five, as graphics can be missleading and I'm not sure if it touches the line around $1$ twice). Of course, it is easy to verify that these four numbers are the solutions, but if you wrote this answer in my exam then you would miss marks for not verifying these solutions. $\endgroup$– user1729Commented Sep 14, 2020 at 10:04
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$\begingroup$ @user1729 I finally wrote there that the values of x can be obtained by putting [x]=1,19,17 and 18. I left the rest of the workup to the OP bcz I was discouraged to post the complete solution of problems on MSE $\endgroup$– Soumyadwip ChandaCommented Sep 14, 2020 at 10:13
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1$\begingroup$ I really don't think one can find intersections of curves with other curves or lines via a graph. The graph may be useful to figure out some inequalities related to the functions once the points of intersections of their graphs is found algebraically. $\endgroup$– Paramanand Singh ModCommented Sep 16, 2020 at 5:11
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1 Answer
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I think graphics and visuals can be great for giving hints, intuition or suggesting what the solution should be, but they frequently fall short of giving a rigorous proof (for instance, the missing square puzzle). If the OP is looking for a rigorous proof, then a rigorous proof should be given - it can always be accompanied by a visual or graphic to give a better sense of what's going on, if that's appropriate to the question at hand.
You can also use the visual as a hint and put the full solution implied by the visual under a set of spoiler tags, like this answer, which was well-received after some initial downvotes for not being rigorous (for users not able to view vote counts, it is at +12/-5 at the time of writing, which is more upvotes than the currently accepted answer, at +10). This was discussed here on meta and here on chat in the past.
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$\begingroup$ Thanks for your answer. Still, how shall I know that what the OP is looking for unless they mention it exclusively? If they would really mention that graphical proofs/solutions not accepted, then I would never bother to even touch that problem. $\endgroup$ Commented Sep 15, 2020 at 1:32
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$\begingroup$ My purpose of coming to MSE was only to show that intuitions and visualisations can make the topic more approachable. Once you have the intuition, you may go for the rigorous part. Almost similar to 3Blue1Brown $\endgroup$ Commented Sep 15, 2020 at 1:34
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2$\begingroup$ The OP should ideally make their intent clear to you via their post. If they haven't done that, then you can either ask them in the comments ("Are you looking for a full solution or intuition or just a hint to get started?") or use context clues (some tags/problem types/users/etc usually expect more rigor than others). I think there's space for providing hints, and I engage in this myself, but there are times when a complete and rigorous solution is the appropriate response. $\endgroup$– KReiserCommented Sep 15, 2020 at 2:32