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