Including alternative proofs

Question

Suppose I have found two or even more proofs of a theorem and I prepare a paper on it. Is it considered to be a good practice to write down all of the proofs? Or is it considered to be my job as an author to select the best one (which can be a tricky task!) and omit the other ones? What about a compromise, such as including a remark which sketches the alternative proofs? If your answer depends on additional information, what are relevant criteria which decide if the alternative proofs are included? For example, is it OK to include an alternative proof even if the proof follows a similar idea, albeit doesn't use as much machinery and hence is accessible for more readers, even though it is not as elegant? In my opinion, all proofs should be included because they offer a different perspective on the same thing, which I think is always a good thing, but I don't know if there is a standard common practice. Does this perhaps even depend on the journal where the paper is published?

Edit: Please post your answers as answers, not in the comment section. ;)

Answer 1

This is a great question. Since the OP explicitly asked for answers as answers rather than comments, let me try to get it off the unanswered queue. As far as I'm aware, there are three options:

1. Choose one proof (whichever matches best with the flow of the article) to put in the main text, and include the second proof in an appendix.

2. Only include one proof, and either put the second in the arxiv version as an appendix, or on your personal webpage, or just never publish it.

3. Include the second proof as a stand-alone paper.

My personal opinion largely agrees with that of LSpice, in favor of option (1). The downside is that it makes your paper slightly longer, which can make it take longer to publish (e.g., more time waiting for a referee report), can slightly raise the bar for publication (in the mind of the editor), and can rule out some journals (like Proceedings of the AMS) that have a page limit. Still, having two different proofs of the same result should in general make the paper stronger, especially if you sell that as a strength in the introduction, e.g., how proof A is more suitable to generalization in direction X while proof B is suitable to generalization in direction Y.

I don't like option (2), because I feel like the arxiv paper should match the published paper if possible, personal webpages won't last forever, and it's good to disseminate math instead of sitting on it.

Option (3) works if it's a big and important result. There are plenty of publications like "a new proof of X's theorem." I assume the OP knew this already and has already determined that this proof is not sufficient for a stand-alone paper, so I think option (1) is best.

One last note (certainly known to the OP but perhaps useful for other readers): it's essential to avoid the impression that you're providing two proofs because one (or both) might not be airtight. When my students write their first research papers, sometimes they put in two bad proofs instead of one good proof (often, with weasel words). This would raise an immediate red flag in the mind of any good referee. But I think the strategy I suggested above, of remarking on why it's nice to have two different proofs, would take away any such concern.