4 ms·

Without intending to take anything away at all from the work of the researchers, I think “breakthrough” might just be overselling it. They have improved a boun

by jl6 2y ago

Without intending to take anything away at all from the work of the researchers, I think “breakthrough” might just be overselling it. They have improved a bound but I can’t tell that their method will open up a plausible path to a full proof. It feels like we’ve managed to increase the top speed of a car from 200mph to 227mph, but are no closer to understanding how the engine works.

magnio 2y ago
Change that to CPU performance in 2023 and you got a breakthrough as well.
math_dandy 2y ago
It may take many breakthroughs to yield a proof of the Riemann Hypothesis. Progress on a notoriously hard problem through introduction of innovative new ideas can certainly qualify as a breakthrough, imo. Naturally, the importance of the breakthrough will be judged by what can be built upon it.
- fishtockos 2y ago
  This is clever application of existing ideas, not the sort of new idea which is needed. No one can really imagine what sort of machinery will need to be built to prove the RH. A proof is completely outside the realm of speculation at this point.
btilly 2y ago
When you've been stuck for 80 years, any improvement at all seems like a big deal. But more importantly, a new idea like this often points the way to variations that may also create improvement. Therefore as other researchers pile on, we may see this bound come down further. So yes, it is like a wall in a maze was breached, and there is the opportunity to go further. And it is fair to call that a breakthrough. Of course you're still charging into a maze, and further progress is far from guaranteed.
gus_massa 2y ago
The author of the post is Terence Tao that is the best live mathematician. If he says it's a "breakthrough", it's a breakthrough. (Anyway, I agree that if you are not in this research area, it's still an internal result that you can safetly ignore.) Also, in the last years he was leading a few project to crowsource improvements after breakthroughs in other conjetures. I'm too far from this area to be sure, but if he sees there is a chance to improve this, I expect him to launch a new one. From the article: > but they do a number of clever and unexpected maneuvers If he post something like that about me, I'd frame it an hang it in the wall.
- jlarcombe 2y ago
  Well, the two authors of the paper he's talking about are also amongst the foremost mathematicians in the world today. James Maynard won the Fields medal a couple of years ago. So they've probably got used to approbation from the likes of Terence Tao. Nonetheless, exciting!
- nybsjytm 2y ago
  > The author of the post is Terence Tao that is the best live mathematician. If he says it's a "breakthrough", it's a breakthrough. I think it's pretty silly to say that Tao is the best living mathematician, but even if he were I don't think that this would be a useful way to think about things.
  - deleted 2y ago
    [deleted]
  - kadoban 2y ago
    Yeah, random people on HN are much better judges of importance of mathematical results.
  - gus_massa 2y ago
    > I think it's pretty silly to say that Tao is the best living mathematician I agree. I was going to write "one of the best" or "probably the best" or something like that. It's like discussing if Messi or Maradona were the best football players. [1]. Anyway, all three of them are quite good. > but even if he were I don't think that this would be a useful way to think about things. I also agree. It's just and argument by authority. For a decition that would change the lives of millons of persons and has a lot of subtle tradeoff and unknown unknowns, I'd ask for a better justification and evidence. But deciding if this is a breakthrough or not, may only change the lives of a few graduate students and a few grown up mathematicians, so I'm happy to take the word of Tao. [1] Hi from Argentina!
abetusk 2y ago
I'm no expert and I don't like engaging in arguments to authority but Terrence Tao is one of the leading mathematicians of our age, akin to Erdos. When he's excited about a result, that should be a high signal indicator that it's worth paying attention. In terms of the result itself, as Tao explains: > ... making the first substantial improvement to a classical 1940 ... Meaning, it's been 80 years since any progress has been made on this front (bounds of zeros of the Riemann zeta function). I would make the argument that it's not so much making a top speed car from 200mph to 227mph but discovering an engine that uses gasoline instead of steam. Presumably the methods used in the paper might be able to be extended or refined to get better results.
QuesnayJr 2y ago
It probably doesn't open up a path to a full proof, simply because Guth and Maynard are excellent mathematicians, and they wouldn't have given away their approach yet if they thought they could prove the full thing. It's still important because there hasn't been much new evidence in years whether the conjecture is true or false. Now we have some new evidence that it's true.
- JadeNB 2y ago
  I don't think that's really conclusive; surely they've taken it as far as they think they can in a reasonable amount of time, but (1) they've no real need to be secretive--why not share it as soon as there's a concrete result?, and (2) sometimes the inventor of a technique can be too used to the old way of thinking to use their new technique to its full potential, and it takes less experienced eyes on it.
- lupire 2y ago
  That's not how math community generally works. Everyone knows what the important idea of a proof is. Publishing the work that grates a new field of mathematics that leads to a proof is not worse than toiling away privately (for longer!). Gregory Perelman, the only Millennium Prize winner, declined the prize because he refused to accept credit for Richard S Hamilton's foundational work.
deleted 2y ago
[deleted]

by jl6 2y ago

magnio 2y ago

Change that to CPU performance in 2023 and you got a breakthrough as well.

math_dandy 2y ago

It may take many breakthroughs to yield a proof of the Riemann Hypothesis. Progress on a notoriously hard problem through introduction of innovative new ideas can certainly qualify as a breakthrough, imo. Naturally, the importance of the breakthrough will be judged by what can be built upon it.

fishtockos 2y ago
This is clever application of existing ideas, not the sort of new idea which is needed. No one can really imagine what sort of machinery will need to be built to prove the RH. A proof is completely outside the realm of speculation at this point.

btilly 2y ago

When you've been stuck for 80 years, any improvement at all seems like a big deal. But more importantly, a new idea like this often points the way to variations that may also create improvement. Therefore as other researchers pile on, we may see this bound come down further. So yes, it is like a wall in a maze was breached, and there is the opportunity to go further. And it is fair to call that a breakthrough. Of course you're still charging into a maze, and further progress is far from guaranteed.

gus_massa 2y ago

The author of the post is Terence Tao that is the best live mathematician. If he says it's a "breakthrough", it's a breakthrough. (Anyway, I agree that if you are not in this research area, it's still an internal result that you can safetly ignore.) Also, in the last years he was leading a few project to crowsource improvements after breakthroughs in other conjetures. I'm too far from this area to be sure, but if he sees there is a chance to improve this, I expect him to launch a new one. From the article: > but they do a number of clever and unexpected maneuvers If he post something like that about me, I'd frame it an hang it in the wall.

jlarcombe 2y ago
Well, the two authors of the paper he's talking about are also amongst the foremost mathematicians in the world today. James Maynard won the Fields medal a couple of years ago. So they've probably got used to approbation from the likes of Terence Tao. Nonetheless, exciting!
nybsjytm 2y ago
> The author of the post is Terence Tao that is the best live mathematician. If he says it's a "breakthrough", it's a breakthrough. I think it's pretty silly to say that Tao is the best living mathematician, but even if he were I don't think that this would be a useful way to think about things.
- deleted 2y ago
  [deleted]
- kadoban 2y ago
  Yeah, random people on HN are much better judges of importance of mathematical results.
- gus_massa 2y ago
  > I think it's pretty silly to say that Tao is the best living mathematician I agree. I was going to write "one of the best" or "probably the best" or something like that. It's like discussing if Messi or Maradona were the best football players. [1]. Anyway, all three of them are quite good. > but even if he were I don't think that this would be a useful way to think about things. I also agree. It's just and argument by authority. For a decition that would change the lives of millons of persons and has a lot of subtle tradeoff and unknown unknowns, I'd ask for a better justification and evidence. But deciding if this is a breakthrough or not, may only change the lives of a few graduate students and a few grown up mathematicians, so I'm happy to take the word of Tao. [1] Hi from Argentina!

abetusk 2y ago

I'm no expert and I don't like engaging in arguments to authority but Terrence Tao is one of the leading mathematicians of our age, akin to Erdos. When he's excited about a result, that should be a high signal indicator that it's worth paying attention. In terms of the result itself, as Tao explains: > ... making the first substantial improvement to a classical 1940 ... Meaning, it's been 80 years since any progress has been made on this front (bounds of zeros of the Riemann zeta function). I would make the argument that it's not so much making a top speed car from 200mph to 227mph but discovering an engine that uses gasoline instead of steam. Presumably the methods used in the paper might be able to be extended or refined to get better results.

QuesnayJr 2y ago

It probably doesn't open up a path to a full proof, simply because Guth and Maynard are excellent mathematicians, and they wouldn't have given away their approach yet if they thought they could prove the full thing. It's still important because there hasn't been much new evidence in years whether the conjecture is true or false. Now we have some new evidence that it's true.

JadeNB 2y ago
I don't think that's really conclusive; surely they've taken it as far as they think they can in a reasonable amount of time, but (1) they've no real need to be secretive--why not share it as soon as there's a concrete result?, and (2) sometimes the inventor of a technique can be too used to the old way of thinking to use their new technique to its full potential, and it takes less experienced eyes on it.
lupire 2y ago
That's not how math community generally works. Everyone knows what the important idea of a proof is. Publishing the work that grates a new field of mathematics that leads to a proof is not worse than toiling away privately (for longer!). Gregory Perelman, the only Millennium Prize winner, declined the prize because he refused to accept credit for Richard S Hamilton's foundational work.

deleted 2y ago

[deleted]