Is it possible that small bugs or assumptions in a root paper could cascade through referencing papers leading to wildly inaccurate outcomes 5 or 6 papers down the line?
The generalised quantum Stein's lemma [1][2] is (or was) a very powerful result that was used for over 10 years to prove things in this subfield. However last year it was noticed that there was an error in the proof of this lemma, and a whole bunch of results based on it weren't valid [3][4]. One of the authors of the paper where they wrote about the error gave a talk at the conference QIP 2023 this year, and there is a video of that talk available here [5]. Bartosz is a good speaker and I recommend watching the talk if you're interested, if you go to about 10 minutes, 30 seconds in the talk he discusses the consequences of this result now being not known to be true.
[1] Published paper https://link.springer.com/article/10.1007/s00220-010-1005-z
[2] Arvix version: https://arxiv.org/abs/0904.0281
[3] Published version: https://quantum-journal.org/papers/q-2023-09-07-1103/
[4] Arxiv version: https://arxiv.org/abs/2205.02813
[5] Youtube link: https://www.youtube.com/watch?app=desktop&v=2Xyodvh6DSY
Multiple papers on calculus claimed results about continuity and derivatives, but we’re using subtly different definitions.
The conflict between those results, and the counter-examples to demonstrate the difference, led to mathematicians building the modern machinery around proofs.
> The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness.
I wonder if they are good, or if it's selection/confirmation/other biases that lead is to think say.
Also, aren't surprising results the most interesting!?
Note that this is not true in general, and depends on the type of theorem. The idea is that while it’s easy to show why a particular proof is incorrect, it’s much more difficult to show that every proof is incorrect.
Formally, this idea is captured by CoNP, which is believed to be different from NP and hence a strict superset of P.
“a strict superset of P” doesn’t really follow from that. P is also believed to be different from NP, and P is certainly not a strict superset of P.
Of course, I assume you just misspoke slightly, and that whatever it is that you actually meant to say, is correct.
E.g. maybe you meant to say “coNP is believed to both be strict superset of P (as is also believed about NP), and distinct from NP.”
I think it is believed that the intersection of NP and coNP is also a strict superset of P, but that this is believed with less confidence than that NP and coNP are distinct?
I imagine I’m not telling you anything you don’t already know, but for some reason I wrote the following parenthetical, and I’d rather leave it in this comment than delete it.
(If P=NP, then, as P=coP, then NP=P=coP=coNP , but this is considered unlikely.
It is also possible (in the sense of “no-one has yet found a way to prove otherwise” that coNP = NP without them being equal to P.
As a separate alternative, it is also possible (in the same sense) that they are distinct, but that their intersection is equal to P.
So, the possibilities: P=NP=coNP, P≠cocapNP=NP=coNP, P=cocapNP≠NP,coNP, All 4 are different, (cocapNP is the intersection of NP and coNP)
Where the last of these is I think considered most likely? )
By the way it is not clear to me, if the theorem was false or if only proof was wrong.
https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...