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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

#291

Post by data_maan »

gavagai691 wrote: post #5

Two additional notes: 1. Zhang posted an attempt at solving this problem in 2007 that he later more or less admitted was flawed: https://mathoverflow.net/questions/131221/yitang-zhangs-2007... . But speaking with mathematicians who are intimately familiar with Zhang's previous work, there seems to be good reason to be optimistic nevertheless. First, the idea behind Zhang's proof is similar to the zero-repulsion ideas…

Raoul Bott is another good example of someone switching from (electrical engineering I believe?) to math in his 40s.

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

#292

Post by dagw »

skyde wrote: post #155
danuker wrote: post #115

Earlier quoted context omitted.

I suspect computers can also help us get deeper. Stuff like computer algebra systems. Maybe some CAS-assisted work gets us into feedback loops allowing us to go indefinitely, as in a technological singularity. But the "shallow" part is also quite wide. You can teach people what you've learned forever, for instance.

do you have experience with CAS? i would love to learn how to use CAS to write proof more effectively

One very common and simple use case is looking for counter examples. If your theory states for example that "all Matrices with property X also have property Y" then it is quick and easy to generate a whole bunch of matrices with property X and check that they all have property Y. Of course that doesn't actually prove anything, but it can be used to disprove a statement and save you a bunch of time chasing down a dead end.

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

#293

Post by hoseja »

maegul wrote: post #169
pa7x1 wrote: post #78

Math used to be a young person game, 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…

> 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…

It's also a major failure in didactics. It feels like very little of the new knowledge since twentieth century has been truly digested for easy teaching. Why isn't general relativity taught in elementary school? It should be possible.

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

#294

Post by data_maan »

gavagai691 wrote: post #8
cshimmin wrote: post #6

Wow, from the 2015 article: [A journal reviewer of his famous paper says]: "you should be careful. This guy posted a paper once, and it was wrong. He never published it, but he didn’t take it down, either.’ ” The reader meant a paper that Zhang posted on the Web site arxiv.org, where mathematicians often post results before submitting them to a journal, in order to have them seen quickly. Zhang posted a paper in 2007…

If the proof is right, Zhang is in contention for greatest living mathematician with seven papers total, and on the basis of two of them (and should win all the major prizes he has not won yet and is still eligible for: Abel, Wolf, etc.). Would truly live up to Gauss' motto: "pauca, sed matura!"

I think the word eligible is problematic, since it is actually aluding to age discrimination, which is what the Fields medal is doing and what should not be allowed.

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

#295

Post by fredley »

zimpenfish wrote: post #260

Brings to mind Gould's quote - "I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops." https://www.goodreads.com/quotes/99345-i-am-somehow-less-int...

I think about this often. Having studied mathematics, it's a field where there is a huge range of talents, even at 18 (and younger). There are genuinely gifted people who stride ahead, but the sad reality is that even at top institutions most of the undergraduate cohort will be merely averagely clever people who have received great (often private) tuition.

In a field where finding every genius really matters because of the difficulty of expanding the frontiers it's heartbreaking to realise most just never get the opportunity of good mathematical education in the first place.

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

#296

Post by [deleted] »

maegul wrote: post #169
pa7x1 wrote: post #78

Math used to be a young person game, 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…

> 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…

[deleted]

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

#297

Post by dheera »

pa7x1 wrote: post #78

Math used to be a young person game, 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…

> we may reach a point where a mathematician's intellectually productive life is not sufficient to contribute Arguably the same could have been said in the 1700s when people lived till a mean life expectancy of about 40 and they would have perhaps been concerned that at some point in their future nobody would be able to make scientific advances in their lifespan. Yet here we are, living till 80s and beyond, making accomplishments in our 50s and 60s.

So we could solve this a few ways:

- Continue to extend the length of human life. Living to 150 will be the norm in a few generations.

- Create AGI that are smarter than us and make the advances for us

- Genetically engineer humans to be twice as smart, and learn as much as an average 30-year-old in 15 years

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

#298

Post by germandiago »

wheels wrote: post #239
pa7x1 wrote: post #78

Math used to be a young person game, 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…

> [...] 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. That's not really the case here. Zhang spent 10 years out of school working in fields in the cultural revolution, and didn't start college until he was 23. After his PhD, he couldn't get an academic job for 8 yea…

A self-made man :)

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

#299

Post by maegul »

hoseja wrote: post #293
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…

It's also a major failure in didactics. It feels like very little of the new knowledge since twentieth century has been truly digested for easy teaching. Why isn't general relativity taught in elementary school? It should be possible.

Interesting! As other comments ts have pointed out, didactics as an important aspect of how humanity responds to such a dynamic.

Otherwise, I’d say our two thoughts are connected. With increasing difficulty in understanding new progress, there could be an inertial tendency to over emphasise the importance of old knowledge because it’s comforting/easier/pragmatic for teachers and parents.

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

#300

Post by germandiago »

hoseja wrote: post #293
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…

It's also a major failure in didactics. It feels like very little of the new knowledge since twentieth century has been truly digested for easy teaching. Why isn't general relativity taught in elementary school? It should be possible.

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, why they were invented, made a point about its primarily practical origins.

To explain things, he could most of the time come with real-life instances of application and there were much more often intuitive or geometric interpretations of the techniques used much more often even before starting the explanation itself to have an intuitive idea and visualization of what you were achieving.

After that, I noticed that to learn math, the first thing is to develop an intuitive, non-mathy idea of what you are doing and later formalize it.

At school and high school they just taught it as almost-memorize tables, apply formulas.

Talking about Spain, btw.

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