Mistakes in proofs are relatively common, but such a mistake doesn't automatically mean that the proof is entirely wrong. Many times, the mistake is just in the exposition, and can be fixed easily. Other times, the mistake is fixable and the fix is apparent. Perhaps the author forget to treat a relatively trivial edge case. In the first two cases, the student would likely just pass with minor corrections to be submitted soon.
Sometimes, of course, the proof is just wrong. That is the dangerous case, which will cause either major corrections or failure.
A few months after Andrew Wiles presented the proof for Fermat's last theorem, some mistakes were found that subsequently took him over a year to patch over. Imagine being 12 months into trying to put the pieces back together, after all the years of work and announcing your finding publicly.
I can recommend without reservation the book by Simon Singh, at least for a lay audience (don't know how an expert would experience it).
It's understandably the natural place to get the dramatic tension from what at least purports to be a sober account with some momentum. It's reasonably consistent with the way it's treated on the lectures I've seen from Sir Andrew Wiles on YouTube.
It sounds like a nightmare. He had intentionally set the proof up in a high stakes way, working in private, very quiet on what he was really doing. I gather this is not the done thing, eccentric at best kind of vibes, and I think by that point it was almost crackpot adjacent to even try seriously. His advisor forcefully pushed him off even trying earlier. So it's compound on the stakes. The beg reveal is intentionally as dramatic as possible. And then the questions start to land, most are minor clarifications, notation deficits, but one is sticky, one won't go away.
A Ph.D. is usually a collection of papers/chapters, so even if a fatal flaw is found in one of the chapters, it does not normally result in "failing the Ph.D." It just means that that particular paper isn't publishable. The bar for getting a Ph.D. is actually quite low; what is truly difficult is clearing the bar in terms of publications for academic positions and later tenure.
You would have to go back and do more work though, and if you couldn't fix it you would have to do another defence. But yes, it is a different situation to actually getting a good post doc.
I guess there's no tradition for publishing "negative results" in mathematics? By that I mean not proof of something negative, but rather, "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either".
Probably there are understandable reasons for that... But I think "negative science" is really important, and soft results like "we tried that for a long time and it wasn't very fruitful" are actually very important for progress even if they're poorly attested in the written record. I guess in mathematics they come informally from your thesis advisor...
> "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either"
There is a related tradition of making conjectures about things that you can not prove, and being known for having made such conjectures. Conjectures along with definitions and problem statements are incredibly important in mathematics. But usually they are introduced in the context of some other publishable work.
> I guess there's no tradition for publishing "negative results" in mathematics? By that I mean not proof of something negative, but rather, "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either".
Indeed there isn't such tradition. I have one or two results like that -- proofs that some proof strategies cannot work because some object does not exist, but since that object would not be interesting for anyone not trying that particular proof strategy for that particular (already obscure) problem, one cannot publish.
There are mathematical statements that have been shown to be undecidable (no proof for or against is possible) and such a demonstration is a huge deal.
Mistakes in proofs are relatively common, but such a mistake doesn't automatically mean that the proof is entirely wrong. Many times, the mistake is just in the exposition, and can be fixed easily. Other times, the mistake is fixable and the fix is apparent. Perhaps the author forget to treat a relatively trivial edge case. In the first two cases, the student would likely just pass with minor corrections to be submitted soon.
Sometimes, of course, the proof is just wrong. That is the dangerous case, which will cause either major corrections or failure.
A few months after Andrew Wiles presented the proof for Fermat's last theorem, some mistakes were found that subsequently took him over a year to patch over. Imagine being 12 months into trying to put the pieces back together, after all the years of work and announcing your finding publicly.
I can recommend without reservation the book by Simon Singh, at least for a lay audience (don't know how an expert would experience it).
It's understandably the natural place to get the dramatic tension from what at least purports to be a sober account with some momentum. It's reasonably consistent with the way it's treated on the lectures I've seen from Sir Andrew Wiles on YouTube.
It sounds like a nightmare. He had intentionally set the proof up in a high stakes way, working in private, very quiet on what he was really doing. I gather this is not the done thing, eccentric at best kind of vibes, and I think by that point it was almost crackpot adjacent to even try seriously. His advisor forcefully pushed him off even trying earlier. So it's compound on the stakes. The beg reveal is intentionally as dramatic as possible. And then the questions start to land, most are minor clarifications, notation deficits, but one is sticky, one won't go away.
Kept me up late reading about it.
Proof fix-ups are quite common, but if it was not possible, then no, not for that topic.
A Ph.D. is usually a collection of papers/chapters, so even if a fatal flaw is found in one of the chapters, it does not normally result in "failing the Ph.D." It just means that that particular paper isn't publishable. The bar for getting a Ph.D. is actually quite low; what is truly difficult is clearing the bar in terms of publications for academic positions and later tenure.
You would have to go back and do more work though, and if you couldn't fix it you would have to do another defence. But yes, it is a different situation to actually getting a good post doc.
I guess there's no tradition for publishing "negative results" in mathematics? By that I mean not proof of something negative, but rather, "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either".
Probably there are understandable reasons for that... But I think "negative science" is really important, and soft results like "we tried that for a long time and it wasn't very fruitful" are actually very important for progress even if they're poorly attested in the written record. I guess in mathematics they come informally from your thesis advisor...
> "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either"
There is a related tradition of making conjectures about things that you can not prove, and being known for having made such conjectures. Conjectures along with definitions and problem statements are incredibly important in mathematics. But usually they are introduced in the context of some other publishable work.
You would have to show what you did try.
Typically, you can describe that in the form of a partial result (see for example https://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Partia...) or as another conjecture.
> I guess there's no tradition for publishing "negative results" in mathematics? By that I mean not proof of something negative, but rather, "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either".
Indeed there isn't such tradition. I have one or two results like that -- proofs that some proof strategies cannot work because some object does not exist, but since that object would not be interesting for anyone not trying that particular proof strategy for that particular (already obscure) problem, one cannot publish.
There are mathematical statements that have been shown to be undecidable (no proof for or against is possible) and such a demonstration is a huge deal.
I mean it would be ridiculously easy to make up any number of incorrect proofs.
It would have to show something new and interesting.