What to do when a result is essentially proven but not stated in an existing publication?

Question

I am writing a paper in mathematics that uses some technical results from a classic paper in the field, but not exactly in the way these results were stated at the time. Basically, I found that interesting, slightly more general versions of the statements are also true.

I can't say that these new more general versions are entirely obvious, a few hours of staring at the original statements passed before I even began to consider generalizing them, and then I had to work through the proofs themselves to make sure that the original proof basically goes through verbatim in this new setting. Or in some cases the proof of some equivalence actually breaks down, but one of the implication still holds, and that's enough for my purposes.

These new versions of existing lemmas and theorems are not the core contribution of my work, and I don't think they are interesting in their own right, but I do use them to derive some (at least seemingly) novel theorems. And here is my dilemma: If I present the proofs, I am essentially copying multiple pages of proofs written by my predecessors, which doesn't seem perfectly ethical to me. If I don't, I feel like I am doing the reader a disservice: I am basically asking them to believe me when I say "If you find a copy of this book, and change all instances of the word macguffin to weak macguffin on pages 438-442, you will find that the statements and proofs remain correct".

EDIT: I asked some colleagues and found out that the original versions of the statement are not really common knowledge, since they are more akin to lemmas than theorems. With that in mind, I guess a reader that sees a statement or its proof for the first time would not care too much if it's not that different from something already published elsewhere (and of course, I will be explicitly mentioning the source).

Answer 1

I found myself in the position when I need some fact but the literature search produced something "close enough but not quite what I want" rather often. Moreover, I bet that this is a fairly usual experience for any mathematician who wants to do something a bit off the beaten path. My choice in such cases has always been to give a full reference and attribution to all those sources but to redo the whole thing in an appendix (not in the main body) in the shortest way that would get me exactly to the point at which I need to be with all relevant details. Why?

1. I find it much easier to trace a full formally presented argument than to reconstruct the details from the vague sentences like "slightly modifying the proof of Lemma 15 in [3]" or "Arguing similarly to the second half of [10]".

2. It makes me sure that I haven't missed any subtle errors in my modifications and generalizations. The proof is generally not reliable until it is fully written down and checked line by line by several people a few times after that. Granted, there do exist people that never make mistakes or make only "mistakes of a genius", but 99% of us, as far as I can tell, fall into neither category.

3. I do not want to take part in generating a long tree of references like "Applying modifications from [1] of the generalization from [2] of the main result of [3] to the setup similar to that in [4], we conclude that...", which would inevitably result from applying the alternative practice several times in a row and that would make the text nearly incomprehensible and extremely hard to verify.

4. Sometimes, when writing this appendix, I find a shorter or clearer approach even to the "well-known" results insofar as my needs are concerned and occasionally this may result in an overhaul of the main text as well.

Answer 2

It is acceptable to write "If you find a copy of this book, and change all instances of the word macguffin to weak macguffin on pages 438-442, you will find that the statements and proofs remain correct."

Except of course you should cite the book, rather than writing "find a copy of this book".

Also, if there are portions of the proof where the generalisation from the macguffin setting to the weak macguffin setting is not as obvious, it may be helpful for the reader to work that portion in the proof explicitly.

Answer 3

This is a judgement call. I've seen (and applied) different approaches, each of which has advantages and disadvantages.

1. You can say that the result follows by a modification of a proof from [insert citation]. This has the advantage that it saves you the time and work of writing the argument, and it saves the reader the need to read the argument if they're willing to trust you. This has the disadvantage of making it harder for the reader to follow the argument, and it makes it easier for you to miss a crucial technicality. With this approach, I feel like you're putting your reputation on the line: I would not blame a colleague for writing a complicated argument and missing a subtle detail which invalidates the reasoning, but I would slightly decrease my estimation of a colleague when they take a shortcut and end up being wrong. In my opinion, this approach is best suited to situation where you're very confident, especially if you have the reputation to back it up.

2. You can reproduce the argument from scratch. It is not in the least unethical to reproduce reasoning from a different paper almost verbatim as long as you're honest and open about what you are doing. It goes without saying that the source paper should be cited. In addition to that, it is helpful to add some remarks on precisely how closely the two arguments are related, and to add references linking specific points in the two argument (e.g. "equation (13) in our argument is the analogue of equation (43) in [insert citation]). As others have pointed out, if the argument ends up being long enough to interrupt the flow of your paper, it's best to put it in the appendix. This has advantages and disadvantages that mirror those of the previous option: It costs you more time and effort, but minimizes the chance of error. I find this to be the more pro-social option: it produces references that are more accessible. In my opinion, this approach is best suited to situations where you have some lingering doubt, wish to go out of your way to make other people's lives marginally easier, or simply feel like writing out the details of the argument for practice.

3. Finally, you can do a weighted combination of 1. and 2. This would include saying that most of the argument follows along the same lines, but then writing out the details of whichever steps you consider most interesting/trecherous/etc. The tricky part is to decide which steps need this treatment: depending on the implementation, this can end up being the best of both worlds, or the worst.

Answer 4

This is what appendices are for; and also, there are always ways in which the exposition of the proof can be streamlined and adapted to your particular purposes -- I am always baffled by students who believe that the only way in which the argument has already been developed by someone else is ipso facto the only way to state matters.

Answer 5

Let me see if I can sum up your current situation:

• The existing publication proved a lemma, but did not deem it worth of calling it a lemma and left it silently in the proof of another theorem.
• You noticed that this lemma was useful to you in a seemingly different context.
• You are planning to leave this lemma silent once more and just include it as part of the proof of another theorem in your paper.

Your insight that the unnamed lemma from an existing publication in a given context was applicable to your theorem in a different context is probably worth more than you think! This kind of cross-topic insight is extremely valuable in all branches of mathematics.

Please do consider giving a name to the lemma and proving it formally rather than leaving it only as a small part of a proof of another theorem.