I was once in a presentation for a math PhD thesis. During the thesis, the evaluator of the thesis noticed a flaw in their proof. The student understood and then asked “What now?” The evaluator prof simply shrugged.
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I personally know a story in this vein with a (sort of) happy ending.
A PhD student discovered that a result of his professor would imply the solution to a big conjecture in another field. The people in that field then analyzed the prof's result and found that the proof was flawed. The student was still allowed to graduate based on this since the finding of the connection between fields was brilliant. Then he quit academia (not because of this story, he had planned it before). Then a year later the prof figured out how to fix the flaw in his proof and published a paper with his former student, thus solving the conjecture. The two are still on good terms, writing papers together.
> The two are still on good terms, writing papers together.
Why are you framing this like this? Do mathematician take these kind of things personally?
Because I think the median reader would expect this sort of thing to come up at some point in the story.
That sounds like the worst "exam nightmare" scenario imaginable, but did the student get the PhD in the end?
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...
> 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.
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.
"Negative results" in the sense of replication failures and trial pre-registration are incredibly important. Lean is in some senses a mathematics response to the maths replication problem -- fields are so specialised, and proof checking so onerous, that many errors will go uncorrected.
Arguably mathematics should have pre-registration, so you can see who has tried what before, rather than just by knowing everyone in your field. As LLMs progress, we might get pre-registration of LLM-aided research. "We plan to spend $10K on Fable tokens to look at conjecture X is algebraic co-homology."
> "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.
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.
I first heard that kind of story about a thesis defense at Princeton. The twist was that they had been short one person to judge it, so they roped in a professor who was available at that moment ... and he found a counterexample on the fly.
The professor was John Milnor.
A recording of a car crash: discovering on live radio/podcast that the central tenet of your book is wrong, and amateurishly so.
Naomi Wolf 'death recorded' on BBC[1], skip to 5:51. After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies.
This is astonishing. How could a person go to the lengths of writing an entire book without ever looking up this sort of thing?
The term she mistook for meaning a death/execution had been recorded was "death recorded". So it's a mistake that could be made similarly to how one might confidently assume that a "public school" is a school which is a school that is either government funded or open to anybody in the local area.
More generally I think people are inappropriately confident in general. When you go through history just about every century prior people believed many things that the next century would come to see as plainly false. People in modern times seemed to have stopped believing this was true, yet it's likely that people during every given century also stopped believing such was true, because what you believe "now" is always assumed to be correct. Nobody wants to believe they hold false beliefs, even though there's a practically 100% certainty that all of us do, and in quite healthy quantities.
...what is public school then?
A fee charging privately operated institution that does not receive public funding. [1] Brits have a way with words.
[1] - https://en.wikipedia.org/wiki/Public_school_(United_Kingdom)
The US still somewhat has a similar distinction with public and private colleges. Private vs public funding rather than private vs public admission.
A similar distinction with opposite polarity, however.
A public school in Britain does not receive public funding.
A public school in the US is entirely publicly funded.
When rich people started having their kids educated with children of other rich people instead of at home by tutors.
You have to be confident by default or you'll never get anything done. I always find myself stuck in analysis paralysis and then I look back at something like the history of Minecraft and realise he wrote the whole game (minus content additions) from scratch in only about two years, and all the updates I remember not liking were just another Tuesday for him as there had been similar updates every few weeks during that entire time period.
Like, I bought the game just before the Nether was added, and I thought that was not a very good feature, but just a few weeks before that, he'd added monster spawners, and just a few weeks before that, he'd added redstone, and a few weeks before that, minecarts, and a few weeks before that, signs, and a few weeks before that, the world wasn't infinite yet. Just another few months back, and the game didn't have an inventory. That guy had the opposite mindset of me and he got shit done.
> You have to be confident by default or you'll never get anything done.
Crippling doubt can happen. Blind confidence has its own hazards. You only need to be confident that you can proceed, in spite of having provisional, incomplete knowledge. "But if you grip to your beliefs too tightly, you will not recognise The Truth when it comes knocking on your door." (forget who said that)
The most astonishing thing is that she still holds the PhD whose thesis was the basis of the book. A stain on Oxford reputation.
When I read your comment I completely give myself over to the understanding that you mean she wrote a whole and continuous book, without ever researching whether you meant a book about uncastrated male animals. Imagine my surprise when I go the radio to discuss your comment and a zoologist calls up to correct my understanding of a common English phrase.
Sorry, I don't follow. Did I make a dumb mistake and fail to notice? Am I a total hypocrite?
Is the video available not through a proprietary player?
> After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies.
That's quite an extreme shift considering she had previously been a leading figure in third-wave feminism and an OWS activist.
The other Naomi, Naomi Klein, wrote an entire book that uses Wolf's "transformation" as a structural scaffold to explore the rise of the new right/MAGA etc. etc. It's called "Doppelganger".
Which electronics youtuber Naomi got disappeared in China for showing bad parts of china? Was that a third Naomi?
Naomi Wu aka Sexy Cyborg - she did a lot of electronics work, and showcasing tech
I couldn’t get the video to work. Here’s the relevant clip I found on youtube.
BBC iPlayer is not available outside the UK.
It worked fine for me in the US
Some people are just contrarians who want their ideas to be the most outlandish.
This is a real problem if you're one of the people involved in something like OWS who wants to see effective change, not just making a circus about yourself.
Why? She went from crazy (that you sympathize with) to crazy (that you don't sympathize with). She was always crazy.
I think the crazy magnitude increased massively. If you subscribe to the horseshoe model, the change in bearing wasn't that large.
You underestimate the craziness of Occupy Wall Street and other leftist causes.
Laura Loomer is another crazy person who went from extremely crazy to just... untrustworthy but not totally crazy on Ukraine. I still think she is crazy underneath, though.
It doesn't seem too egregious an error to assume "death recorded" means they were executed, rather than the opposite where death was recorded as the verdict but not actually executed.
Not checking newspaper reports of the period is lazy though and exactly the kind of thing I'd expect a journalist to do and therefore be the one to discover the flaw.
If you're not a historian it wouldn't be an egregious error.
I wasn't sure if I would be able to listen again - it's so hard to hear.
I just listened to it. I think she handled it remarkably well.
She had poise in the moment.