Say person A and person B are collaborating on a mathematics paper. It goes like this: person A repeatedly suggests proofs, and person B repeatedly finds flaws in them. Over time, this process culminates in a correct proof.

Does person B deserve to be an author on this paper? One might argue that the final paper would not have existed without B, so they deserve authorship. On the other hand, they did not actually contribute anything in the final work - each successive proof was generated by A alone.

Wikipedia says that the development of RSA went something like this: "Rivest and Shamir, as computer scientists, proposed many potential functions while Adleman, as a mathematician, was responsible for finding their weaknesses," until Rivest hit on the final answer. But that might be overly reductive, and I don't know of any other examples.

(As a final note, I'm not A or B in this scenario - I'm just curious.)

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    To directly answer the question in bold: hell yes. That is: if A is not capable of finding flaws in his or her own "proofs", then A does not deserve sole credit, especially when this is a repeated process. I take the view (not universally shared) that coming up with ideas for proofs is the easy part, most of the time, just as having a wishlist is much easier than actually having an achievable plan of action – Yemon Choi Jul 29 '16 at 13:36
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    If B keeps finding "real flaws", then why is his contribution negative? Is not it positive since he is actually improving the state of the paper? – The Guy Jul 29 '16 at 13:37
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    @Barbot Pointing out that something is wrong - and why - is making an improvement to the work – Landric Jul 29 '16 at 13:42
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    I understand. But as @Landric said " Pointing out that something is wrong - and why - is making an improvement to the work". The issue on hand is that contributing does not only mean to find "solutions". It is an umbrella that covers many aspects. In some cases, one can say that B need only to be acknowledged and not be a co-author. However, the way I see it is B can be a co-author (maybe even deserves to be one) since s/he went over many proofs (it was not a one time thing). – The Guy Jul 29 '16 at 13:49
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    "[Adleman] told Ron, 'Take my name off the paper. It's your work'. But Rivest insisted and eventually prevailed upon him. I thought, 'Well, it's going to be the least important paper I've ever been on...'." -- cstheory.stackexchange.com/a/17917 – user2768 Jul 29 '16 at 15:44
up vote 100 down vote accepted

To moderate Yemon Choi's comment: yes. Your first assertion is that A and B are collaborating, which means they should be co-authors, unless one actively backs out.

For a mathematical project, it's easy to have lots of ideas but not enough time to pursue them all to see which (if any) work. If someone can shoot down ideas and tell you they definitely (or with high likelihood) won't work, this can help put you on the right track. So in your situation I would say B was instrumental in finding a correct solution.

(In a somewhat different abstract scenario where B dismisses some approaches to a problem that A suggests, and B does not otherwise actively work on the project, it may depends on the situation and they should have a discussion about whether B is a co-author or not. And some people will have different opinions about the same situation, e.g. RS versus A in the RSA example.)

In general in a mathematical collaboration, if there's one key idea it's unlikely that both collaborators arrive at it together. Maybe through discussion they enhance each other's understanding of the problem, and then one will have the key idea and the other will encourage/validate it. That doesn't mean was the other person was unnecessary, even if you can't pinpoint parts of the final paper as being "their contribution."

(And if collaborations became competitive to the point of dropping co-authors just because the didn't see the final solution first, who would want to collaborate?)

  • Does that mean a graduate student's advisor should always get to be an author on the student's paper, even if the student came up with the problem and most of the ideas? By the nature of the relationship, the student is going to share a lot of the work with the advisor, and likely the advisor will give some direction. But co-authorship could diminish the student's contributions in this case, if the advisor has a "big" name. – Forever Mozart Jul 30 '16 at 1:01
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    @ForeverMozart: In mathematics, the usual rules for coauthorship are frequently partially suspended for work that is part of a doctoral dissertation - in many cases advisors are not authors on papers (but instead listed in acknowledgements) when, under other circumstances, their contributions would merit coauthorship. This is a known community standard. – Alexander Woo Jul 30 '16 at 1:59
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    @ForeverMozart And in my field (not mathematics), the advisor is always the last name on the paper and no grad student ever publishes solo. It's just understood that the head of the lab is the final author and readers have a general idea of what that kind of contribution means. – tpg2114 Jul 30 '16 at 16:03
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    @ForeverMozart Also, more specific to math, this MathOverflow question (also linked to from JeffE's links): mathoverflow.net/q/57337/6518 – Kimball Aug 1 '16 at 1:26

Pointing out errors is certainly a contribution to the paper, especially if done repeatedly. Indeed, it's possible that B's contribution exceeds A's. If A is getting lots of information of the form "this won't work" and "that won't work", the possibilities for something that does work can get narrowed down to the point where it's rather easy to find a proof (especially if the "won't work" information comes with an indication of why things won't work).

Rather than talking about two people collaborating, think about what happens when you write a solo paper. If your experience is like mine, it may well happen that discovering that early attempts won't work and understanding why they won't work is a bigger part of the job than finding the proof that finally does work.

Pointing flaws isn't something negative in any way. Does the paper/theory worth anything if it has several flaws ? In my opinion he is doing an excellent job. Find every possible flaw in a paper, and fixing it, is what makes it consistent and reliable.

Get an example for something else external to mathematics, like the engineering process of a car. Those who test the car for safety, aren't doing a purely negative contribution, is a step that is mandatory, that is to check for any flaws. The car resulting from a process where no tests were made, will be a choice for you ? Will be accepted by the market ? The same goes for the paper, if B didn't work finding flaws, someone else will, it happens all the time, even with more than one person focusing on finding flaws, happens to someone else, external to the group, to find something.

Your question is quite interesting. However, I don't think the case of RSA is a good example for your question.

Suppose a scenario: X finds a proof, and submits it to a journal, where the reviewer finds the flaw and rejects it. He then improves the proof and submitted it again. Does X need to include this reviewer as a co-author? I think not.

What if X repeats this process until he has a correct proof? Does he need to add all reviewers as co-authors, or he only needs to do so when all reviewers happen to be the same person?


The case of RSA is very different. Rivest and Shamir proposed many potential one-way functions. These were not proof, they could be viewed as conjectures as best. By finding the weaknesses of those functions, Adleman actually provided the proofs (by counter-example???) that those functions were not one-way. Saying he just falsified the proof trivializes his contribution.

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