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(I'm by no means in the situation described below, I'm just hypothesizing/daydreaming.)

Say that a professional mathematician thinks that he can get some interesting results by applying a certain method he found. Being a seasoned mathematician, he is confident in his attempt, only to find out that the results yielded by the method were trivial. Is it still meaningful to publish the result, as to tell other mathematicians that come up with the same method "Hey, I've already checked this cave, its empty!"? In particular, is this frequently done?

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closed as off-topic by Tom Oldfield, Start wearing purple, Hakim, achille hui, user88595 Jun 2 '14 at 22:12

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is not about mathematics, within the scope defined in the help center." – Tom Oldfield, Start wearing purple, Hakim, achille hui, user88595
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Can the close-voter please provide a reason? $\endgroup$ – Andrew Thompson Jun 2 '14 at 21:38
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    $\begingroup$ Sure, I voted to close since this question seems to be both primarily opinion based, and off topic for the site. Whilst it may be related to the field of mathematics, it isn't mathematical in nature. $\endgroup$ – Tom Oldfield Jun 2 '14 at 21:49
  • $\begingroup$ I have edited the question (i.e added one sentence) that should erase the "primarily opinion based" part, in addition to fit the scope defined in the help center. $\endgroup$ – Andrew Thompson Jun 2 '14 at 22:25
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This paper is great: How not to prove the Poincare Conjecture by John Stallings (now deceased unfortunately.) A great mathematician with a great sense of humor.

Bibliographic info:

Stallings, John, How not to prove the Poincaré conjecture. Topology Seminar, Wisconsin, 1965, 83–88, Ann. of Math. Stud., 60, Princeton Univ. Press, Princeton, NJ, 1966.

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  • $\begingroup$ Haha, funny title! But he never published the attempt before the conjecture was proven by Perelman, correct? $\endgroup$ – Andrew Thompson Jun 2 '14 at 21:37
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    $\begingroup$ "The second point is that I was unable to find flaws in my 'proof' for quite a while, even though the error was obvious. It was a psychological problem, a blindness, an excitement, an inhibition of reasoning by an underlying fear of being wrong. Techniques leading to the abandonment of such inhibitions should be cultivated by every honest mathematician." $\endgroup$ – Cheerful Parsnip Jun 2 '14 at 21:37
  • $\begingroup$ I always thought he had published it, but it looks more like he simply posted it on his webpage. $\endgroup$ – Cheerful Parsnip Jun 2 '14 at 21:37
  • $\begingroup$ He did publish it. I will update my answer to include the info. $\endgroup$ – Cheerful Parsnip Jun 2 '14 at 21:42
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    $\begingroup$ It would be amusing if everyone who unsuccessfully tried to prove a major conjecture were able to publish their methods $\endgroup$ – Zarrax Jun 2 '14 at 21:45
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Taking the viewpoint that "attempts" are "experiments" which may or may not yield information, a "failure" may yield considerable information even while not achieving a decisive conclusion... and, therefore, be interesting.

Oppositely, some experiments seem fairly pointless "in advance", and/or prove unsurprisingly boring and uninformative, ... so are not interesting.

A possibly useful distinction is that a relatively expert mathematician may be able to relatively easily anticipate that a given experiment will yield no (interesting) information, while a relatively inexperienced person might not see the pointlessness of going down a particular road. At the same time, an inexperienced person may themselves acquire to-them-valuable information by performing to-expert-pointless experiments, to see first-hand (and viscerally?) how things work. Yet, as privately informative as this might be, it would fail by the stronger criterion of "informative to experienced experts".

Altogether hard-to-say, subjective.

Also, "it depends" on what one means by "publication": trying to get something "peer-reviewed" to score professional points is quite different from literal publication, e.g., on the internet, to inform any interested parties.

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In most cases, no, negative results are very hard to publish. Stallings's situation was an exception; recall he had published major work on this subject.

This is a general nuisance in science, not only in math. If you look at your favorite math journals they rarely if ever will they have papers showing a possible method doesn't work.

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