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A soft question.

I am a PhD student, at early stages of my academic career; and have personally experienced the following many times. Sometimes you come up with a result, that you are not quite happy with (for whatever reason, maybe you think is not novel, or is simple, or has no applications, etc.)? How often do you put your research into trash?

Yet another question, I have seen people publishing sometimes papers that they don't feel quite happy about publishing, but every now and then, these papers start getting lots of citations, and being recognized. What do you think about this? Do you think unless the quality is really low you'd go ahead an post, at the very least on arXiv; or do you stick to perfectionism? To some degree, I feel like perfect is the enemy of the good!

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    $\begingroup$ Some people's junk is other people's treasure. My suggestion is to save what you proved in a TeX file with a detailed proof. Maybe in the future you will use this as a technical result in another paper. $\endgroup$ Mar 5, 2020 at 17:55
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    $\begingroup$ Never. All of it is practice for the results that you do share/publish. Even if you mess up the proof, you can still have the idea around for someone else to modify. As long as you qualify each attempt with your level of confidence in it and the amount you have checked it, you should be able to share it with those who respect such efforts. Gerhard "This Isn't About Journal Publishing" Paseman, 2020.03.05. $\endgroup$ Mar 5, 2020 at 20:54
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    $\begingroup$ Also, very small observations can be published on a blog or similar. Or given as a starting point for a masters thesis, where the student can try to extend the theorem, or apply it. $\endgroup$ Mar 6, 2020 at 7:27
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    $\begingroup$ If it is actually a result, then one can always write it out. If it is a small result, then you can write a small note and send it to Arxiv if you don't want to publish on a low-rank journal. If the result is so small that you don't want to finalize the detailed proofs, you could post the proof ideas on a forum or blog as mentioned above. $\endgroup$
    – dodo
    Nov 14, 2020 at 19:13
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    $\begingroup$ I like the following anecdote (it is perhaps a joke): A journalist did interview the cleaning man at Princeton and asked him how John von Neuman works. Reply: "He does not work at all. He writes the whole day and when he leaves he throws everything in the trash bin." $\endgroup$ Sep 2, 2021 at 16:05

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I have some strong opinions about this:

  1. Math is hard and interesting. Don't be snobbish about anyone's math, not least yourself. I have found fruitful research ideas from obscure papers that no-one read, from naive questions by undergraduate students or even my children. Some vague thoughts I had years ago when revisited now become full papers in decent journals since I have more tools and insight to develop them.

  2. Anything can be turned into a useful experience. That's what I told my kids before any tennis session. If they have to hit with someone weaker, focus on placing the ball perfectly for that person so they can hit it back. That helps them improve their skills and the other kids too. Same for your "weak" results. Try to extend them. Put them in bigger contexts. Write a good introduction. Don't be shy to speculate and ask open questions (but make sure you put some efforts in thinking about them first). These are crucial skills for "bigger" papers too. In this day and age you can always find a non-predatory journal whose quality your note can improve, for instance those published by math societies from countries with little history of math research. Send your note there. And you have done a good thing.

  3. You can always put them in a file called "Answers to future MO questions"! (-:

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