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Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

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Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#371

Post by RugnirViking »

rcarr wrote: post #368
germandiago wrote: post #300

Earlier quoted context omitted.

FWIW I notice how remarkably bad I was taught math at school. When I started to go to university I really noticed how bad it was. At university the jump forward was really noticeable. For example, at school they would show you a couple of simple explanations about derivative math or integrals, briefly and start with all the formulas. At university I used to have a teacher that started with: history of mathematics, wh…

This! Identical situation in the UK from my experience. Why maths is taught completely separated from its history and its purpose is baffling. Well actually it’s not: it’s because it serves the teachers. You hammer the formulas and patterns into the kids so they can pass the exams. Your school gets good grades and the principal is happy because he can now market his school as successful and the government inspectors…

> because it serves the teachers

I don't think this is the case. I think it's because the teachers themselves don't have a good grasp on "the story of maths and how it benefitted society".

There is a silly meme about asking high school maths teachers "how will we use this in life", and imo it's not because there isn't a good response, but rather that it requires a good understanding of the ways math is actually used. Few high school teachers have actually themselves used the math they teach for anything other than academic exercise. Someone trained in control theory or using physics equations can make things that appear almost magical using maths, and if they are talented, they can find a way to explain it to laypeople. However, people with that combination of talents are desired by just about everybody, from universities to companies, and high schools simply have no way to compete (not least because teaching high schoolers is a soul-crushing job for bureaucratic reasons)

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#372

Post by LudwigNagasena »

woodruffw wrote: post #356
LudwigNagasena wrote: post #307

Earlier quoted context omitted.

I don’t see any direct origins. Karl Popper was a small child when Einstein came up with relativity. Hobbes was just a reactionary apologist of absolute monarchy. And Frege’s philosophical contributions were world-changing as long as we can pretend that propositional calculus is philosophy.

I don’t think this is going to be productive, but to be clear: the claim isn’t that science didn’t exist before Popper, or that Popper somehow gets credit for all of science. It’s that Popper has produced the best description of science’s epistemic underpinnings thus far, allowing science to progress without imploding under the demands of positivism. In other words: when we “do” science, the epistemic model we use is…

> It’s that Popper has produced the best description of science’s epistemic underpinnings thus far, allowing science to progress without imploding under the demands of positivism

I am not sure what that claim means precisely. Popper saved science from… a certain group of philosophers?

> when we “do” science, the epistemic model we use is less than a century old

The epistemic model is whatever the scientist in question uses. One can be a Popperian, another can claim they are guided by God, and someone else would just say “shut up and calculate.”

Popper didn’t like Bohr and his Copenhagen interpretation from the philosophical standpoint yet he didn’t deny that Bohr was a good physicist.

> The claim that propositional calculus is in the domain of philosophy and mathematics is not controversial in either domain. Nor is it unusual: the history of philosophy is the history of discharging subjects once they develop a field of their own.

The context of this discussion is that philosophy progresses over time. You cannot say that it progresses just because we put a label “philosophy” on every new thing that isn’t formalized yet. In that trivial sense, of course, philosophy had great progress; especially in its subfield of natural philosophy also known nowadays as science. But did Frege really stand on the shoulders of Hegel, Spinoza and Kant and improved on their ideas?

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#373

Post by FabHK »

toyg wrote: post #343
ebjaas_2022 wrote: post #334

Earlier quoted context omitted.

> rather than on the finding and promotion of undiscovered geniuses. There's a catch in it as well, which is that you cannot really identify a genius up-front. A genius is, by definition, someone who thinks differently. A lot of people think differently. But the thing with the genius is that he thinks differently and correctly. The latter part won't be obvious until after the fact. When Alex Ferguson started out as a…

Alex Ferguson might have been a good football manager, but he wasn't a genius - and I say that as a Man United supporter. He didn't advance football in any significant way, and even towards the end of his career he was somewhat naive from a tactical perspective. He was just a good man-manager in what is a sea of managerial mediocrity - football management is largely restricted to ex-pro-footballers who can't do anyth…

On a similar note, Page and Brin were obviously smart and driven and all, but applying linear algebra to the search problem was arguably one of the things that was "in the air". It is hard to judge, because what seems obvious ex post was (obviously) not obvious ex ante. But, again, eigenvalue decomposition/SVD (and linear algebra more generally) - you throw it at the Netflix problem, you throw it at image compression, you throw it at anything really, something's gonna stick.

It's an interesting counterfactual: without Page, when would Page Rank have come around? The idea that the stationary distribution of a Markov Chain (under certain conditions) is given by the eigenvector to the (largest) eigenvalue 1 is certainly decades old, if not a century.

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#374

Post by FabHK »

andrepd wrote: post #347
wikfwikf wrote: post #303

Earlier quoted context omitted.

I don't think this is the whole story. Of course, it is a tragedy when people who had undeniable raw talent are neglected or passed over and don't get to achieve as they might. It's also sad and disappointing when we observe structural injustices in our society. However, isn't it possible that giving great teaching resources to people who aren't already at the top level in terms of raw talent, allows them to succeed…

> If so, it's not a zero sum game - the 'averagely clever people who have received great tuition' and who go on to succeed might be a sign of the education system working, not its failure. The failure is that mediocre students which come from an environment which helps them (good school, private tutors, stable home life) can be successful, while "naturally gifted" yet disadvantaged students don't. The basic fact that…

> parental income is still the best predictor of educational achievement

Is that controlled for genetic influence though? Suppose smart people 1) have high income, 2) have smart kids.

Disentangling the various causal factors is substantially harder than your quote above suggests.

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#375

Post by n4r9 »

robertlagrant wrote: post #314
germandiago wrote: post #300

Earlier quoted context omitted.

FWIW I notice how remarkably bad I was taught math at school. When I started to go to university I really noticed how bad it was. At university the jump forward was really noticeable. For example, at school they would show you a couple of simple explanations about derivative math or integrals, briefly and start with all the formulas. At university I used to have a teacher that started with: history of mathematics, wh…

When I came across the (obvious, in retrospect) visual explanation for why (a + b)(a + b) == a^2 + 2ab + b^2 I was blown away by how simple it was, and by why on earth I had to just memorise that formula in school.

By "visual explanation" do you mean imagining a square whose sides are length (a + b), and breaking it up into smaller squares and rectangles?

Or do you mean mulitplying out the terms:

  (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2

?

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#376

Post by m_nyongesa »

rutgersnj22 wrote: post #226
m_nyongesa wrote: post #49

As an older person currently working on a PhD, this guy was and is a something of a hero to me. He has an interesting life story. He was very into math at an early age, so he's different from people like me who got interested in it later in life, but he's also different in that his family was sent down to the countryside in China. I remember reading a lot about him a few years ago and relating to some of the professi…

you got to read this article from his sister: https://zhishifenzi-com.translate.goog/depth/character/480.h...

Thanks for that link. Wish I could read Mandarin Chinese so I could read the original too.

Clearly a flawed person. Not sure why he blew off his family so hardcore for so many years. Too bad to hear he was arrogant as a kid too, although a lot of smart kids are. Some of them turn out to be Peter Thiel, luckily this guy just wanted to work on math.

Anyway, I wish he had been better to his parents. On the other hand, he needed that big breakthrough to save his life as a mathematician: until that point, he was just an adjunct lecturer with no stability at all. Life is weird and complicated and we don't have full control of the choices we make, some choices can seem really hard to some people. I'm not excusing his behavior toward his family but I would be interested to know why he made that choice.

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#377

Post by FabHK »

kjellsbells wrote: post #235
maegul wrote: post #169

Earlier quoted context omitted.

> but it now requires so much knowledge just to get to the frontiers of human knowledge, not to speak of making a dent into uncharted territory, that results are being obtained later and later in life. When mathematicians have had time to accrue sufficient knowledge while still being sharp enough to make the intellectual leap. The sad part is that as the trend continues we may reach a point where a mathematician's in…

That's a very big claim, perhaps true only if human endeavor only ever builds linearly. But I dont think that is true. Certainly not in the arts or music, e.g. the Beatles did not need to ingest the entire corpus of Mozart or even Scott Joplin to be highly productive. Although, it certainly helped that they were extremely open minded to all types of music. A second example is that of recent advances in virtualization…

> only if human endeavor only ever builds linearly.

A lot of mathematics is cumulative, though.

And even if you can go further by going thinner (specialising more), maybe breakthroughs require lateral thinking and connections that are predicated on not being too specialised, but having breadth also. If that is the case, sooner or later we might be in trouble.

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#378

Post by robertlagrant »

n4r9 wrote: post #375
robertlagrant wrote: post #314

Earlier quoted context omitted.

When I came across the (obvious, in retrospect) visual explanation for why (a + b)(a + b) == a^2 + 2ab + b^2 I was blown away by how simple it was, and by why on earth I had to just memorise that formula in school.

By "visual explanation" do you mean imagining a square whose sides are length (a + b), and breaking it up into smaller squares and rectangles? Or do you mean mulitplying out the terms: (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2 ?

The former, yes.

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#379

Post by FabHK »

woodruffw wrote: post #356
LudwigNagasena wrote: post #307

Earlier quoted context omitted.

I don’t see any direct origins. Karl Popper was a small child when Einstein came up with relativity. Hobbes was just a reactionary apologist of absolute monarchy. And Frege’s philosophical contributions were world-changing as long as we can pretend that propositional calculus is philosophy.

I don’t think this is going to be productive, but to be clear: the claim isn’t that science didn’t exist before Popper, or that Popper somehow gets credit for all of science. It’s that Popper has produced the best description of science’s epistemic underpinnings thus far, allowing science to progress without imploding under the demands of positivism. In other words: when we “do” science, the epistemic model we use is…

> Popper has produced the best description of science’s epistemic underpinnings thus far

Ok, sure.

> allowing science to progress without imploding under the demands of positivism

How is giving a description of something necessary for its continued progress? Were people not having sex before evolutionary psychologists and biologists elucidated how and why?

Re: Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang

#380

Post by snovv_crash »

tsimionescu wrote: post #280
snovv_crash wrote: post #262

Earlier quoted context omitted.

Think of it like Rust eventually supplanting C++ once the ecosystem and libraries are complete enough for people to make the change. The cognitive burden reduces via the new foundations not requiring deep understanding of the old ones to be able to make new steps forward.

This is not a good example, as Rust does not build on C++, it simply replaces it as the foundation. Of course, the development of Rust is built upon lessons learned from C++. But there is no universal law that says that the foundations of a field must be simple enough for one human to understand them in 70 years. It may well be that the simplest possible statement of a field of knowledge is still too complex for a si…

> This is not a good example, as Rust does not build on C++, it simply replaces it as the foundation.

No, that's exactly the point. You can thereby drop the cognitive load of C++ and build greenfield on the Rust foundations.

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