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I asked this question a week ago, and it quickly drew critical comments which were upvoted. I felt that the concern expressed in those comments had been addressed in the body of the question, and tried to elaborate on that, but there was no direct response to what I said and the question has not received any answers.

While no-one has publicly voted to close, that could just be because no-one with the power to do so has noticed its existence. [Edit: my mistake, see comment below by Xander Henderson.] On the other hand, there have been a couple of upvotes.

Normally, when there've been no answers after a week, I would post the question on MathOverflow. But the critical comments and their upvotes make me think that perhaps this was just not a good question. I'd really appreciate any advice.

Further comments:

I agree the presentation is rambling. I struggle with this but am working on it.

"Interesting" was an unhelpful word; it was meant to save space. I'll give a hopefully less-subjective definition here, in case it doesn't improve things. Please note, I wrote the question with the interesting number paradox in mind; the template below, with integers instead of PDEs, would just give all of them.

Let $P$ be a property of PDEs; I'll characterize what I'm looking for in $P$ (there's probably multiple viable candidates, hence the soft-question tag). The idea is: $P$ should be broad enough to include the "small sliver" of PDEs which are actually studied, but also have a short mathematical characterization.

(The analogy with Reverse Mathematics is: there are weak subtheories of second-order arithmetic which are broad enough to prove the mathematical results actually used in science and engineering, but also have short characterizations; they turn out to be "natural" in the sense that many alternative candidate theories turn out to be equivalent.)

So: "actually used by scientists, engineers etc." is sufficient (but not necessary) for $P$ to hold. $P$ should also be closed under the sorts of operations used in practice to derive new PDEs from old ones, such as: placing constraints on certain variables / parameters, or the relationships between them; taking limits as they approach particular values; changing co-ordinates. Taking the closure under such operations will make $P$ not only broad but also, I hope, simple to characterize. At the same time, I would still expect it to exclude pathological examples like $u^π_{xxxx}=u^2_ye^{u_z}$.

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    $\begingroup$ Try editing it, so that users can find it in the home page or consider to start a bounty. $\endgroup$
    – 19aksh
    Aug 16, 2019 at 14:32
  • $\begingroup$ RE the particular Questoin (I'm one of the commenters on your "soft" Question): I have read through it a few times, and on its face you ask for "necessary conditions" for some mathematical object (PDEs in this instance) to be "interesting". The presentation is somewhat rambling, e.g. your wish to apply a "closure operation" seems more applicable to sufficient conditions than to necessary conditions, and the recent comments to "motivate" the Question by analogy to Gödel's Incompleteness Theorem do not have any resonance with me. $\endgroup$
    – hardmath
    Aug 16, 2019 at 17:30
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    $\begingroup$ See for context the interesting number paradox, which many analyses say hinges on the subjective characterization of what is interesting. $\endgroup$
    – hardmath
    Aug 16, 2019 at 21:28
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    $\begingroup$ You say that no one publicly voted to close the question. However, I voted to close it on the 8th of August (and therefore cannot vote to close it again right now), and a it went through review at the same time---it garnered one more close vote, but three "leave open" votes. Given that it currently has no votes-to-close, I suspect the votes cast last week have expired. $\endgroup$
    – Xander Henderson Mod
    Aug 16, 2019 at 23:39
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    $\begingroup$ Regarding the justification of voting-to-close, I hope that I made myself clear in my comments: the question relies on an opinion about what is "interesting". Questions whose answers are a opinion-based are explicitly off-topic on MSE. $\endgroup$
    – Xander Henderson Mod
    Aug 16, 2019 at 23:41
  • $\begingroup$ Thanks to everyone for your feedback so far. I've added some responses above. $\endgroup$
    – Robin Saunders
    Aug 17, 2019 at 12:46
  • $\begingroup$ I've now integrated most of the additions into the original question. I'd be very grateful for any feedback, here or there. $\endgroup$
    – Robin Saunders
    Aug 20, 2019 at 18:09
  • $\begingroup$ In general, it's a good practice to keep questions shorter than one screen, particularly when there aren't any equations or bullet points making whitespace. This isn't a rule, of course, but it may help your question get an answer. In its current form, it's too long, and doesn't get to the point quickly enough. I stopped reading before the end of the yellow box. $\endgroup$
    – Alexander Gruber Mod
    Aug 21, 2019 at 5:39
  • $\begingroup$ @AlexanderGruber I take your point, but the shorter version was consistently misinterpreted! I could really use some help here. $\endgroup$
    – Robin Saunders
    Aug 21, 2019 at 12:15

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