given the ‘a priori’ nature of math, is there anything about which a researcher can be dishonest about such that it invalidates their results? I understand that plagiarism might be a thing, but that doesn’t invalidate the plagiarized results’ validity. Is there any equivalent to fabrication of data, misrepresentation of sources, etc?
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Well they do use generate large data sets. Even for a simple to state theory things like “4 colour maps” and Fermats postulates …
So the data could be “repaired..”
Back in the day, Mersenne ? gave a list of perfect primes. Then put “note, I didn’t verify the last half dozen”.. well some of those were not … (When someone proved one was not prime, but did not provide its offending divisors , another guy spent 3 years finding them…this was done mentally and on paper..no computer… This just shows much data and there could be in just finding out why a Mersenne near-prime wasn’t prime )
In mathematics you can get something wrong in good faith easily. You develop a proof by casebash (Aussie term for it, I believe case exhaustion and proof by exhaustion are also terms used?), but miss a hard to solve case. If that case isn’t obvious, your error might be missed.
But getting something wrong in bad faith is unlikely.
It is almost always far easier to verify a result than to come up with one, which means that anyone who can understand your paper can likely verify it.
Possible exception: You know there’s a missing case in your proof, and you collude with a peer reviewer who also ‘misses’ the same case. Your paper is thus published despite you knowing it is incorrect.
I remember on advanced algebra trig one of my classmates got a question wrong with the wrong formula. the teacher was quite shocked when he checked the person’s work and if they had done the math correctly the formula actually would have worked even though it wasn’t a normal formula. you can easily do something that may work for some cases and so I can find out that it doesn’t work in other cases therefore making your theory and valid because it only works in some cases
Many maths papers have very sloppy proofs. Often the theorem is still true, so it’s hard to argue they are outright “wrong”, but I’ve found at least 3 theorems in published papers which were just wrong. In each case they were reasonably easy to fix (missing a required condition).
Of course, the sloppier you are, the easier it is to get papers published. This mostly only happens with proofs no one cared that much about, as they get less scrutiny.
Writing the answer without showing your work.
Oh yes! Most definitely, one of the biggest frauds in science is actually a mathematician! And the worst thing of it all: people think he is legit. Check this out: https://www.youtube.com/watch?v=broyJbHweZI ; Abdon Atangana. He has several retractions so far for his fraud, but he is expected to get a lot more as most of his papers are nonsense. Just check his pubpeer record and you know already enough. More on him here as well: https://retractionwatch.com/2021/02/03/mathematician-ranked-as-clarivate-highly-cited-researcher-has-third-paper-retracted/