Publishing independent proofs of a theorem in the same journal?

Question

In the comments here, a user suggested that when two proofs of the same theorem are obtained independently, it is a common practice (at least in physics) to submit them to the same journal.

I was wondering what the benefits of this approach are and how widespread it is. (I'm particularly interested in mathematics, but insight from other disciplines is also appreciated.)

Edit: Just to clarify, I'm not asking why it's worth publishing independent proofs of the same theorem - that's something that can be taken for granted, at least in mathematics. What I'm curious about is the practice of specifically publishing the two proofs in the same journal (as opposed to different journals).

Answer 1

I can only speak for mathematics. However, when you really understand the nature of mathematics you start to realize that the proofs of the theorems are often as or more important than the statements of the theorems themselves.

The proof, in showing why something is true, gives a roadmap to truth. Sometimes a technique, if it is not standard but is interesting, can be the most important aspect of a math paper because that same roadmap might just show a way to prove other things, some similar, and some not.

My own dissertation had interesting theorems, to be sure, but it was most useful for the proof of one of the theorems. The proof was unexpected and gave new ways to approach some problems in Analysis.

So, a journal that shows two independent proofs of the same thing can be especially interesting since the similarities and differences between the proofs can give hints of other things that might be shown.

I doubt that it is especially common, though in popular areas of research it must happen. Parallel research is pretty widespread, though if offset in time by only a bit, it won't be possible to have such things in the same issue of a journal.

On the other hand, getting beaten to a result may not be devastating if different approaches are taken. Other mathematicians can learn from that.

From the journal's standpoint such situations are especially satisfying as two different but interesting proof methodologies may represent the merging of two separate trains of thought. That in itself is interesting to a mathematician and may lead to consolidation or to further advances.

Answer 2

Generally, in such a situation, the journal in fact takes care to make sure the two papers are published in the same issue. This has a few effects:

1) There is official acknowledgement that the papers are simultaneous, independent work, and both sets of authors deserve credit for original "discovery" of the theorem.

2) There is official acknowledgement that some expert(s) consider both proofs of the theorem to be roughly equally good.

3) Presumably, the same editor has handled both papers, and this means that the editor has to spend less time understanding the basic contents of the papers (at least the statement of the theorem) than they might otherwise.

4) If the two proofs are similar enough that this is plausible, it's common for both papers to be assigned to the same referee, so that only one person has to spend time on the material rather than two different people having to spend time on it, and also so that the referee can make any necessary comparative judgements (which might be none). (The journal might decide, probably in addition to one referee who looks at both papers, to have someone from each group referee the other paper!)

Answer 3

Both Buffy Alexander Woo have provided good answers already. However, since I am the user who wrote that comment, perhaps I should weigh in on this too.

First of all, let me note that I wasn't thinking only about theorems, but more generally about results. Here is perhaps the clearest dividing line between mathematics (for the sake of this discussion broadly defined to include theoretical computer science, parts of mathematical physics, and similar fields that largely deal with theorems) and the natural sciences. If you have a valid proof of a theorem, reproducibility is guaranteed. In the natural sciences you rarely have that guarantee. Accordingly, multiple groups finding consistent results is kind of a big deal - whether the two groups use the same methods or not (think different samples, measurement techniques, numerical methods/codes, formalisms, approximations, etc.). If two groups can independently obtain consistent results, that raises both readers' and the journal's confidence in the papers. If a journal accepts this cultural attitude in general, there's no reason not to also apply it to theorems.

A second significant difference between mathematics and most subfields of physics is that authorship order matters. Combining two preprints into one joint paper could lead to a more impactful paper, but it would very likely also lead to some unpleasant conversations and compromises. The number of authors per paper is also larger on average, raising potential concerns about dilution of credit. Hence, such joint papers should be expected to be less common.

Having established that having two independent papers may be a good thing, what are the advantages of publishing them in the same journal (and likely issue)?

• From a reader's point of view I think it's mostly a wash. There might be a convenience factor to finding both papers next to each other, particularly in a print journal. On the other hand, if both papers are published in a journal you don't track closely, maybe you would have preferred to see them spread out. Of course, in arXiv-heavy disciplines you might already have found both anyway...

• From the authors' point of view it's nice to have what's essentially a rubber stamp saying that both works were independent, simultaneous, and of similar quality. It also helps provide equal visibility, and credit. Compare it to the case where one of the papers is published in a top journal, and the other in an obscure journal. It's not hard to guess which one is most likely to be overlooked. (Of course history provides a number of examples where the obscure paper came years before the other, so it's not an unfounded worry either.)

• From the journal's point of view, they're looking to publish interesting things. If both papers are interesting, then why not publish both? By handling two papers reaching the same result, the editor (who's likely not an expert) can also be more confident in the results and minimize the risks of having to publish errata in the future - or, in extreme cases, having to retract the paper. As Alexander Woo points out, there's also a factor of editorial efficiency. Finally, especially if it's an important result, a journal/publisher in today's impact factor-centric world would probably be incentivized not to share citations with a competitor.