121–130 of 265 posts
That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.
1) I recommend the wonderful 1976 book Proofs and Refutations by Imre Lakatos.
2) There is a considerable list of books dedicated to counterexamples, e.g. in topology, probability, analysis, etc.
That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.
proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions. and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.
proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions. and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
Couldn't people in principle continue to study a problem that's only been shown to break at one point? Prove something adjacent, or slightly weaker, or elaborate the counter example into a powerful explanatory framework?
> The Jacobian Conjecture Interestingly, Yitang Zhang of the twin-prime-conjecture fame spent 7 years working on the Jacobian conjecture under the advisor Tzuong-Tsieng Moh at Purdue. A key step in his thesis used a corollary of Moh's. It turned out that the corollary was incorrect. As a result, Moh refused to write any recommendation letter for Zhang, and Zhang couldn't find any teaching or research job and ended up…
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.
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.
When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute. On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It…
This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly
I suppose it will fall to AI as well to compose the mathematical equivalent of The Ballad of John Henry. Who will be the human champion, the last great hero who can deliver proofs "from the book" that a machine cannot outperform? [1] https://en.wikipedia.org/wiki/John_Henry_(folklore) [2] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK
There's no structural reason to expect acceleration any more or less than an asymptotic behavior (if even that, acceleration is probably the bigger ask). Different problems yield to a new solvent, maybe that's also more, but it could go either way and we definitionally don't know yet because we don't understand the convexity of AI capability, we cannot directly access it interiority, we don't know if it's sandbagging (other than that it does sometimes, it can). It's an emergent phenomemon that might actively resist measurement. Or it might be as predictable as a clock in a few years.
No one knows, or if they do, they aren't talking. The loud people don't know anything.
Earlier quoted context omitted.
[flagged]
No it's not that they don't check facts, it's that facts don't matter. She found a community where poor scholarship and vibes was celebrated.
Alice is a football stats wiz. Alice is a WR Jones fan because he has an incredible 40-yd dash time and leads the league in YAC.
Bob is a WR Smith fan. Bob knows the stats are all cooked. Bob knows that if the stats weren't cooked, WR Smith would lead the league in all categories. Therefore, Bob doesn't bother with stats.
Earlier quoted context omitted.
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. [1] https://www.bbc.com/news/av/world-us-canada-48639663
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.
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.
Earlier quoted context omitted.
Proof fix-ups are quite common, but if it was not possible, then no, not for that topic.
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…
It would have to show something new and interesting.