I found another proof of my theorem. Can I present two different proofs in the paper?

Question

My area is mathematics. I have just found another proof of the main theorem in my paper. It is slightly more elegant than the previous proof that I wrote, but the idea is very different. When submitting this paper, can I include two proof in the paper, or must I select only one? Personally, I don’t want to delete either of them, but I don’t know how the editor will deal with this situation.

Answer 1

If the two proofs are sufficiently different, then it's entirely reasonable to include both of them in your paper. You might, however, want to think first about whether the proofs might actually lead (easily) to different generalizations of your theorem. If they do, then you get two benefits: It's then clear that the two proofs are genuinely different. And you have some more general results.

Answer 2

There are three alternatives here:

1) Include the original proof only, perhaps with a remark that you later found a different proof.

2) Include the more recent proof only, perhaps with a remark that you had earlier found a different proof.

3) Include both proofs.

All three of these are certainly acceptable, and depending upon the circumstances any one of them may be best. There is a lot to say here about the nuances of various situations: here is an incomplete discussion.

When might 1) be best? Suppose you have already written the paper and that you have been planning to submit it at a certain near date (and perhaps you have received funding for this work). Suppose that both proofs are relatively long and that including both of them would make the paper close to twice as long. (In fact, the second proof uses material which is not yet present in the paper, including both could more than double the length of the paper.) Suppose that the second proof, although "slightly more elegant", does not have any other specific advantage to it, e.g. no further applications that you can see. Or suppose that it suggests to you a new possible approach or avenue, but you haven't developed it much and would like to spend more time on this than you can spare in delaying the submission of your paper.

Then it is plausible to just submit your paper as is, possibly with the remark about the second proof. Math papers have a temporality to them. By that I mean that although it might be intellectually more satisfying to work out every aspect of a single problem before you submit a paper on it, in practice this is almost never possible, because (i) mathematicians, like other professionals, need to show evidence of work done and put out product in a reasonably regular manner, and (ii) mathematics is potentially infinite: there may well never be a natural stopping point. Andrew Wiles settled for a special case of the Taniyma-Shimura Conjecture when he must have suspected that a few more years of work could yield the whole thing. Pierre Deligne wrote a paper solving the Weil Conjectures and then another solving them in a better way (though the first was good enough!) six years later. What chance do the rest of us have?

When might 2) be best? Suppose you've already written up the second proof and/or you know it would be acceptably easy and fast to do so. Suppose the second proof significantly shortens the length of the paper. Then, in confluence with the rest of the circumstances above, it is plausible to submit the paper with the second proof only (possibly with a remark...).

In general, space in strong math journals is at quite a premium. Journals like papers which have "no fat", and they especially like short papers which get in, prove a strong result, and get out. My best publication is five pages long (for the non-mathematicians: this is really short for a contemporary math paper) and it was accepted as is with absolutely no revisions requested or critical comments made. They often don't like papers which have "too much exposition" or "too little content for their length". I hope you can hear my eyes rolling as I type out these sentiments, but I'm just telling it like I think it is. If you write a paper which is "twice as long with the same content", then you risk a journal liking it less.

You can always try to publish the other proof later: one point in favor of withholding the better proof is that it will be easier to publish it later. It will be hard to publish a "less elegant proof that I found earlier" in a journal of the same stature as your original publication, but there are other venues for mathematical content. For instance you could put a longer version of the paper on your own website where you include both proofs -- or, if it seems preferable, a "supplement" to the paper containing the first proof. How important is this work to you and to the community? Maybe some day you will be teaching an advanced course and/or writing a book: that's a great place to include both proofs. Or maybe this work is just one stop for you on the road of mathematical research, and it happens that you were perceptive enough to find two proofs and don't feel the need to publicize both of them. (But please read the second bullet point.) There are quite a lot of "less elegant proofs" that the rest of us do not see.

When might 3) be best? If both proofs are relatively short, easy and fast to write up, and do not add substantially to the length of the paper. Especially, if the proofs really do look different from each other and/or when it seems like each may have its own applications. Or if the difference between the two proofs is itself interesting or is something you'd like to receive feedback on.

Let me end with two (relatively!) quick comments:

• I have recently been in a somewhat similar situation. About six weeks ago I submitted a paper -- relatively quickly after starting work in a certain subfield, for certain reasons that I needn't go into -- for publication. About three weeks ago I found a different proof of the main theorem, say A, of this paper, which all in all I like better: it leads relatively easily to a stronger result B. I am not completely decided on what to do -- certainly it depends on what happens with the submitted paper -- but I am leaning towards the first option. One reason for this is one I didn't mention above because it didn't seem to apply to the OP: there are coauthors involved, and that makes the prospect of slowing down / jeopardizing the acceptance of the paper less appealing. Another reason is that the second proof opens up the possibility of further improvements on Y: in fact, about two weeks ago, after doing some further reading suggested by one of my coauthors, I was able to get an improvement C. I am pretty confident that adding an extra ingredient (which I have not yet mastered) would lead to a further improvement D, and there are still further improvements E that I aspire to but do not yet know whether they are possible. Well, how many times should I rewrite one paper? I also freely admit that researchwise I do A and then B,C, probably D and possibly E, then I am thinking in terms of multiple publications rather than just one. And conversely: B and C are pretty good, but I would like to have the chance to see whether I can get to D and E before I publish B and C. These are not easy decisions: perhaps I'll change my mind.

• Publications aside, the experience of discovering a second proof of a theorem is a really important and positive one for a mathematician, more so than senior mathematicians seem to successfully communicate to our students and junior colleagues. I have been thinking recently about the "bipartite structure of theorems and proofs": roughly, proofs are viewed as being secondary objects to theorems, but I think that rather both are basic, and the incidence relation between them is a key one in the traversal of the mathematical landscape. If you two different proofs of a theorem, try to figure out whether they are actually the same. If they are, then you'll have made a new (to you, at least) connection between two things you already knew: again, this may not sound so sexy but I claim that it really is. Or if they aren't, then (well, it doesn't follow from graph theory, but I claim that's far more likely than not) you've actually proved at least one further theorem, which you should carefully write down and then look for alternate proofs of. And so forth! (Of course, now that you've heard my thoughts about this you know how hard my eyes roll when journals say that they are interested in new results, not "merely" new proofs.)

Answer 3

I very much understand your personal satisfaction here - it is a pleasure for a mathematician to "cross-proof" a result by two different methods. But what is best for your readers? Are both of the proofs equally aesthetically pleasing and easy to comprehend? Are the ideas behind them really significantly different?

I would recommend you to discuss your proofs with a supervisor or senior colleagues, probably on a local seminar in your group. Collect their opinions on which proof is better and why. Then think again and maybe it will be easier to see which one is preferable.

Answer 4

As another strategy, I suggest that first explain one of the proofs which has more consistency with other contents of your paper and then briefly outline your other proof (maybe in a remark after your theorem). In this way, the reader easily reads one proof and has the option to consider another proof too.