Can you commit fraud in a mathematics publication? Or is this a privilege of empirical sciences?

If a mathematician jumps from one bit of information to another that does not follow logically, from more basic principles, then, that's not fraud, but a logical fallacy, akin to saying 1+1=3.

If an empirical scientist gets a results of 3.3341, but claims it was 3.7341 that's fraud.

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    A professor friend of mine just happened to mention to me today that when reviewing mathematics papers written by non-native English speakers, it is surprisingly easy to spot plagiarizing. The plagiarized passages are the ones where the grammar and spelling suddenly has no errors. – Eric Lippert Apr 13 '16 at 18:49
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    @EricLippert That's the same way in which, frequently, you can detect plagiarism with students, even when they're writing in their own language. – Massimo Ortolano Apr 13 '16 at 19:53
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    As Stephen's answer points out, your example of a mathematician making an unjustified logical leap can be fraud, if the mathematician knows it is unjustified and intentionally does it anyway. But it would be hard to prove they knew it was unjustified, and barring a "smoking gun" (like a letter to a collaborator saying "this has a huge gap but let's publish anyway and hope nobody notices"), they'd likely get the benefit of the doubt. – Nate Eldredge Apr 13 '16 at 22:01
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    Fermat's proof of Fermat's last theorem. – Count Iblis Apr 13 '16 at 22:23
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    There is one practice I didn't see mentioned here which is not quite a fraud, but certainly dishonest and hurts the community. Namely, some authors know their theorem can be proved by simpler means, but deliberately use complex techniques and fancy machinery in hopes of publishing at a better conference or journal. – dtldarek Apr 13 '16 at 22:23

Wikipedia defines fraud as "deliberate deception". A couple of mathematical frauds I could think of:

  • Passing off someone else's result as one's own; plagiarism.
  • Using a result in a proof although one knows full well that its preconditions are not met.
  • Making other claims one knows are false, e.g., "it is easy to see that" or "by a tedious computation we see that".

Fraud is certainly possible in mathematics. It's probably harder to distinguish fraud from bona fide errors in math than in other sciences. Who is going to prove that you knew your "simple but tedious enumeration" would not work? Conversely, reusing graphics ostensibly stemming from very different experiments is very hard to explain as a simple error.

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    The last point is what we call proof by intimidation. – Matthew Leingang Apr 13 '16 at 17:16
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    I would add "deliberate obfuscation" to the list. Making proofs look deeper is rumored to help with getting papers into snobby journals. Naturally, it is impossible to prove. – Boris Bukh Apr 13 '16 at 20:54
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    Third bullet point reminds me of some (at the time) unproved theorem written on the margin of a book... – Mindwin Apr 13 '16 at 20:56
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    In other words, the distinction between "error" and "fraud" is one of intent, of accident versus deliberate action. And your third bullet is analogous to the example in the question. The empirical scientist committed fraud by reporting a value of 3.7341 because they knew they didn't have evidence that it was accurate. Likewise, a mathematician commits fraud by claiming to have proved something even though they know they have not. – Nate Eldredge Apr 13 '16 at 21:58
  • But the latter two seem easy, relative to a posteriori fields, to detect. So mostly it's just the first one? – BCLC Apr 17 '16 at 10:33

Sure. It is fraud if you copy-paste someone else's paper or preprint into a paper of your own and try to get it published. It is also fraud if you develop an algorithm, prove that it converges, and illustrate its practical convergence properties using made-up numbers.

  • Having proposed the existence of this type of fraud, can you illustrate it with a real-world instance? :) (seriously, curious) – Alejandro Erickson Apr 13 '16 at 23:19
  • Yes, I have reviewed papers and proposals that were copied from other people's papers. As for made-up data points -- that's hard to prove, but I'm pretty sure I've seen graphs that look too good to be true, or where outlying data points have conveniently been omitted. – Wolfgang Bangerth Apr 14 '16 at 23:16

I have seen this in mathematics.

This would be 30 years ago or more. An entire paper, translated from Chinese to English by a young mathematician, then published in an Eastern European math journal as his own work.

This was back in the days of paper publications stored in libraries. I was looking for a paper in that journal, and found a one-page notice published the journal about that fraud that had taken place in the past.

A colleague recently mentioned a story that happened to him many years ago. As a referee for a paper, he saw how to generalize the authors' results considerably, so he told the editor that the paper was not acceptable in its current form, but that he would be happy to join the authors as a coauthor so that he could write the more general arguments. The editor passed on this offer to the original authors, who accepted, and the paper eventually appeared in that journal. Then, a couple of years later my colleague received the original manuscript to referee for a different journal, with no mention of the revised paper that considerably generalized the results! He informed the editor of the story and it was rejected.

I think this is a clear case of fraud, basically trying to get the same paper published twice. If they had referenced the revised version, this might be defensible, but they were pretending it didn't exist.

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    We do not know the details of the situation, but in this case even the action of your colleague is highly questionable. The reviewer is in a position of power, and the authors have no chance getting past the reviewer without him blocking the paper. There is no question that he contributed an addition to the paper, but the question and the first version of the proof was formulated by the authors. The paper should have been let through and a separate addition later published by the reviewer (if he then wants to switch sides, even together with the authors). – Captain Emacs Apr 13 '16 at 15:32
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    The authors definitively should not have resubmitted the original, weaker paper, but perhaps this was prompted by the highly questionable - I am tempted to even say unethical - behaviour of the reviewer/editor couple which encroached on their legitimate expectation of protection of their authorship. I have seen multiple cases of very significant contributions by reviewers without named attribution. If at all, such suggestions should have come from the editor, not the reviewer. Thank you for this interesting and at the same time unsavoury case, demonstrating breaches of code at multiple levels. – Captain Emacs Apr 13 '16 at 15:39
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    @CaptainEmacs: agree with you completely. – Jim Conant Apr 13 '16 at 16:16
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    @CaptainEmacs: I should add that my colleague felt the paper was too weak to publish as is, and only became suitable for the journal after his additions were made. That said, I still have an uneasy feeling about how he (and the editor) handled the situation. – Jim Conant Apr 14 '16 at 13:31
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    As an editor, this kind of behavior from a reviewer would have raised all sorts of red flags with me! It is certainly borderline unethical. I would have probably withheld the review and asked someone else to review the paper. – Wolfgang Bangerth Apr 14 '16 at 23:18

A quick browse of Retraction Watch suggest that many retractions in math are due to plagarism, which is a form of fraud.

Another way fraud, of sorts, is committed is with automatic paper generators, like SCigen. This has been successful in mathematics and other disciplines.

In today's search of retraction watch I also found peer review rackets in mathematics, where one professor reviewed a lot of their own papers for a special issue of the journal. The post on RW said that they were an author of 11/13 papers in the issue!

What I did not find, and have never heard of, is someone accused of faking data or deliberately misconstruing something false as being true. This could be because it is usually hard to distinguish this from genuine errors, but also because the claims in math papers are typically verifiable, either by hand or by computer. If a referee doesn't believe the claims then they may recommend rejection of the paper or ask for a revision the fraudster cannot make.

Writing math papers that seem strong but have false claims and convincing referees/editors at good journals that the claims are true seems quite challenging to me!

  • Maybe if you do some kind of optimization with lots of local minima, you do multistart optimization or genetic algorithm or something, and you think it should converge on X, and it only came close, so you say it actually reached X. I know I had a paper where I wanted an optimization routine to reach a certain value, and it never did. Later I found a way to analytically prove what the global minimum was, and published it that way. :) But it would have been very easy to lie about the results of the numerical optimization I had done, almost like lying about an experiment. – neuronet Apr 13 '16 at 23:27
  • I see what you mean, but the difference I see is that the opportunity for fraud was serendipitous, rather than deliberately created. Further to that though, you'd have an "end justifies the means" situation, where you bet something is correct and can't prove it. If it does turn out to be correct, someone will just publish on that and if anything, you'll be credited for proposing it first. Again, can't tell the difference between error and fraud. Maybe it happens all the time! – Alejandro Erickson Apr 13 '16 at 23:36
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    I wouldn't describe the auto-generated papers as fraud, because there is no claim that they are real research. The acceptance of a Mathgen paper by a journal could be described as fraud by the journal. – Jessica B Sep 1 '17 at 6:35

tldr: I see good reasons it is harder to produce fraudulent results in mathematics than other fields, but I would not be hubristic without seeing a sociology-of-science study demonstrating this.

Of course there is plagiarism and other forms of fraud. You seem to be asking about false results making it into publication. My view is this: it is a bit cheaper in mathematics to reproduce an experiment than in other sciences. It usually means working through the logic of a proof and convincing oneself of its validity. A highly important result will be reviewed very thoroughly. This is not true for costly experiments, whether requiring the LHC or the study of 500 college students interviewed after lunchtime.

That being said, this answer is incomplete. I would be highly interested in empirical data on how often false results make it into mathematics journals, how important these results are (even using a crude metric like # of citations), and how this compares to other fields (I'm aware of recent pessimistic studies on how frighteningly often reproduction fails). If recent results show scientists have had too much hubris, mathematicians should not respond to this with even more hubris.

Well, a way to fraud in the strict sense would be if you know that your logic is wrong, you know where the problem is, but you actively construct your proof in a way that makes the error harder to spot. One could, for example, move the error into a passage that seems either very hard, very dull, or very easy. Playing more into the psychology of the reviewer than anything else (hard => some may give the benefit of the doubt; dull => reviewer my fall asleep and not notice; easy => might just skip it as "obviously correct").

Of course, a good fraud, if detected then would need to pass as a simple error, to avoid repercussions.

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    Wasn't there this guy who proved a lot of theorems for finite skew-symmetric fields, only for him to see proven that finite skew-symmetric fields do not exist? I cannot imagine a more devastating outcome for a mathematician to find out that the field you worked in basically crashes under your feet. Now doing this from the outset would require a seriously sick mind. But perhaps Gödel found that there is actually a contradiction in mathematics and waits for the punchline of his joke to sink in in the community? ;-) – Captain Emacs Apr 17 '16 at 0:08

As an example, a review by Almgren of a book by Fomenko comes quite close to an accusation of what, with some stretch, could be considered a fraud:

The reviewer has known Fomenko personally for more than two decades and still is at a loss to understand why he is not more responsible in his mathematical claims. The following are two particular examples of concern.

The book cover states "In this volume, the solution of the Plateau problem in the class of all manifolds with fixed boundary is given in detail ... " Fomenko made a similar claim in a lecture at and in the proceedings of the 1974 International Congress in Vancouver, in the introduction to a major paper (in Russian), and in an interview published in the Mathematical Intelligencer. His preface in the volume under review is ambiguous about this issue. In any case, the claim is not proved, as he acknowledges privately <...> The only significant contributions to this representation problem are due to B. White.

Borrowing ideas from Literature (specifically I have Jorge Luis Borges in mind who has written "reviews" of and discussed non-existing books, without revealing that they were non-existent of course, as a sui-generis art form), a really interesting fraud would be to back mathematical claims citing non-existing papers (in say, intermediate parts of a proof).

Coming up with convincing such citations, in terms of the claim made but also regarding the journal/cited author chosen, and ideally not easy to find/verify, would be no easy task and the artist, excuse me, fraudulent scholar, would have to spend a visible amount of time and intellectual energy to the task... proving first and foremost that he is a hustler at heart, since he could spend said resources in actually proving something.

In the age of internet and digitized archives, I guess this has become harder to achieve...

I am not claiming originality of this fraud-idea, I just don't know if it has been spotted already in the mathematical (or other) scientific literature.

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