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Comment by vintermann | Hacker News Reader

I guess there's no tradition for publishing "negative results" in mathematics? By that I mean not proof of something negative, but rather, "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either".

Probably there are understandable reasons for that... But I think "negative science" is really important, and soft results like "we tried that for a long time and it wasn't very fruitful" are actually very important for progress even if they're poorly attested in the written record. I guess in mathematics they come informally from your thesis advisor...

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> I guess there's no tradition for publishing "negative results" in mathematics? By that I mean not proof of something negative, but rather, "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either".

Indeed there isn't such tradition. I have one or two results like that -- proofs that some proof strategies cannot work because some object does not exist, but since that object would not be interesting for anyone not trying that particular proof strategy for that particular (already obscure) problem, one cannot publish.

You would have to show what you did try.

Typically, you can describe that in the form of a partial result (see for example https://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Partia...) or as another conjecture.

"Negative results" in the sense of replication failures and trial pre-registration are incredibly important. Lean is in some senses a mathematics response to the maths replication problem -- fields are so specialised, and proof checking so onerous, that many errors will go uncorrected.

Arguably mathematics should have pre-registration, so you can see who has tried what before, rather than just by knowing everyone in your field. As LLMs progress, we might get pre-registration of LLM-aided research. "We plan to spend $10K on Fable tokens to look at conjecture X is algebraic co-homology."

> "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either"

There is a related tradition of making conjectures about things that you can not prove, and being known for having made such conjectures. Conjectures along with definitions and problem statements are incredibly important in mathematics. But usually they are introduced in the context of some other publishable work.

There are mathematical statements that have been shown to be undecidable (no proof for or against is possible) and such a demonstration is a huge deal.

I mean it would be ridiculously easy to make up any number of incorrect proofs.

It would have to show something new and interesting.

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