In the empirical sciences, there are a number of journals that publish 'negative' results. Negative or null results occur when researchers are unable to confirm the findings obtained from earlier published reports. In the applied sciences, they may also come about when a scientist aims to to show that a particular technology (e.g., CRISPR) could alleviate a problem (e.g., a particular virus that kills that kills a specific type of plant), only to find out that it does quite the opposite (e.g. the technology led to the evolution of viruses that were more resistant to CRISPR).
In the formal sciences, including mathematics and logic, experiments like these aren't conducted*. However, it does happen that mathematicians develop machinery to tackle a particular thorny problem, only to find out it doesn't work. A good example is John R. Stallings' false proof of the Poincaré Conjecture.
Publications like these are few and far between. It seems to me that one of the reasons this is the case, is that there aren't any journals that are specifically geared to these types of papers. They are predominantly focussed on publishing articles that obtain 'positive' results, i.e. actually prove theorems or refute conjectures.
Yet it also appears to me that papers like these can be very useful to researchers in mathematics, for the following reasons:
A. They may inspire someone to slightly tweak the failed approach, in order to make it work and actually prove the theorem(s);
B. They may allow someone to see what has already been tried, and what types of avenues of research are probably not worth pursuing;
C. They may provide a platform for approaches to tackling difficult problems in mathematics, even if the methods don't work so far. Thus, they provide a place to share ideas, rather than throwing away months of work.
My question is twofold:
- Are there already any journals that are devoted to papers containing negative results in the above sense?
- Would it be worthwhile to set up such a journal, from your perspective?
(*) I am aware that experimental mathematics is a thing. The focal point of this question isn't really the experimental nature of the mathematics research, but it's about offering a venue to the failed approaches to solving problems developed through research – formal, experimental or otherwise.
Best Answer
I don't really know what an answer is for a question like "what about ...?" but I have some thoughts.
In fact, way back around 2006-2007 (according to the dates on the ArXiv, see Multiplying Modular Forms, if you want). What happened was I had written what I thought was a really nice paper explaining how to multiply modular forms whose associated representations (of a real Lie group) belonged to the discrete series. It clarified (to me) some ideas of others, and seemed to extend to all sorts of other kinds of modular forms like those on the exceptional group $G_2$.
Well... I was all happy about this and about to speak at a conference. But the day before, Gordan Savin told me about a mistake in the paper. I spent a long evening kicking around the paper and then kicking myself about it. It was a really subtle thing to me -- a difference between $K$-fixed vectors and $K \times K$-fixed vectors -- but a "well-known" issue to experts, ultimately involving the failure of discrete decomposability.
Anyways, the next day at the conference (AMS Special Session, Jan 8, 2007), I didn't really know what to say, but I suggested that someone start the "Journal of Doomed Proofs". And I wasn't kidding. It would be refereed and everything. Criteria for acceptance would be the following:
The paper contains an plausible approach to a problem of interest to the mathematical community.
The approach is sufficiently motivated that many other people might try it.
The approach is doomed, though not obviously from the beginning.
The paper explains why the approach is doomed, identifying the obstacles which really stop things from working. Or at least have to be worked around in the future.
I still think this is a good idea, and not just for the usual "science should publish negative results" reason. Mathematicians have a sort of secret oral tradition of "well-known" things (doomed ideas, silly apocrypha, the largest rank of an elliptic curve over Q, etc.). But those of us who teach in redwood forests don't really have access to this tradition any more. And some were never granted access in the first place. A journal might go a little way to correct this.
If anyone knows how to pitch a new journal, count me in. If it's JDP (Journal of Doomed Proofs) or JNR (Journal of Negative Results) or whatever, it's fine with me. But not with Elsevier please.