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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

#311

Post by pa7x1 »

gavagai691 wrote: post #110
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…

"The sad part is that as the trend continues we may reach a point where a mathematician's intellectually productive life is not sufficient to contribute anything novel, statistically speaking." People talk about this a lot. While I think it could happen for certain subdisciplines (it already takes essentially an entirely PhD's worth of time to learn all the necessary background to be an algebraic geometer, so most al…

Isn't this evidence that this process has started to occur? The difficulty to make progress in certain areas of math pushes mathematicians to the shallower areas where it's easier to make a contribution. Progress in the first ones will stall and at some point the shallower areas will become less shallow and same phenomenon will occur.

Maybe we will keep forever discovering new shallow areas but I suspect this is not the case. In any case this is a phenomenon that I think will play out in the next few hundred years, not sufficiently impactful in the next few decades but more and more noticeable.

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

#312

Post by LeanderK »

earthboundkid wrote: post #227

The “lack of progress” in philosophy comes down to this. By the time of Socrates or Confucius, there had already been a lifetime’s worth of philosophical thought done that you needed to grapple with. Everyone since then has necessarily had to let something slip through in order to move forward at all.

which lack of progress? The few encounters with logic (in a philosophical sense, although it feels more "mathy") I had recently were all more recent works, certainly more recent than a large part of math I use!

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

#313

Post by jaapz »

TheOtherHobbes wrote: post #309
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…

I think the Beatles included three inherently gifted songwriters by sheer coincidence. No doubt the apprenticeship helped, but Harrison was writing songs like Taxman and experimenting with Indian classical music on Love You Too in 1966.

It's not either/or, it's both. They were gifted, but they also had the time, money and motivation to work on their skills and hone their craft. It's not by coincidence that in literally every community that talks about how to level up their skills, the first response is always "practice practice practice".

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

#314

Post by robertlagrant »

germandiago wrote: post #300
hoseja wrote: post #293

Earlier quoted context omitted.

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, 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.

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

#315

Post by pa7x1 »

abeppu wrote: post #263
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…

I can't pretend to be anywhere near the frontiers of math but aren't some contributions about providing a unifying framework or simplifying re-expression of something that was already known in a way that makes it more comprehensible? Is there not always hope that the right new perspective or innovation in one generation will allow subsequent generations to get up to speed faster?

This effect clearly exists. It suffices to check how physics and math were expressed in the papers that discovered them and how they are currently represented to notice that sometimes organizing knowledge in the right framework and simple notational changes can be sufficient to power up our ability to make new advances.

Question is, are we dedicating enough effort to this endeavors? Is rewriting known math under a different guise sufficient to survive the publish or perish attrition? And will these efforts keep returning results?

I suspect the answer is no to the first two questions, and I don't know the answer to the third.

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

#316

Post by romwell »

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…

>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 [...] The sad part is that as the trend continues we may reach a point where a mathematician's intellectually productive life is not sufficient to contribute anything novel, statistically speaking

Nonsense.

First, the frontier of mathematics is far closer than you think. Number theory is one of the oldest branches of mathematics, going back thousands of years. There are many more branches that have originated in the latter half of the 20th century. And new ones are coming into existence all the time.

Second, we don't need to focus and specialize — we need to do the opposite. Mathematics is about seeing connections and patterns. We need to teach philosophy and critical thinking, and we need to give people exposure to the vast universe of unexplored — and fun — world of mathematics so that they can go towards the frontiers and push them, rather than spend half a lifetime going in the well-trodden direction.

Finally, undergraduates are still producing new results, every year, in numerous REU programs around the US alone. What gives?

The problem isn't that math is so well-studied that you need so much education to do it. The problem is that we aren't teaching people to do math, we teach them about math that they may or may not use elsewhere.

We don't encourage (or give space) for them to play, experiment, explore, wonder, ask questions, venture into the unknown, and be surprised by what they see (outside the aforementioned REUs).

So many people simply don't get to even start doing mathematics until their third year of graduate school, simply because that's when the structure of our education allows them to.

None of that is necessary. We do it that way because professors are underfunded and mentorship is not rewarded (publish or perish), among other things.

The situation is due to structural problems in academia, not mathematics, humanity, or the advances we made.

Signed, —your neighborhood mathematics PhD

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

#317

Post by germandiago »

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.

This is exactly the kind of intuitions I am talking about. Much easier to explain like that.

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

#318

Post by romwell »

joe_the_user wrote: post #237
mlyle wrote: post #191

Earlier quoted context omitted.

Physics is limited by having to represent phenomena in our physical world simply. Mathematics is not just a small integer multiple larger than this.

This is almost correct. It's not that math is vastly larger (or more sophisticated) as a field. Both fields are infinitely large in many senses. Rather, the number of respectable starting points where you can do interesting things is much larger, orders larger in math.

There are plenty of unexplored things in physics too; the key word is "respectable".

Looking from the outside, physics suffers a lot from fashion/hot trend tendencies, where you need to be doing the "hot" thing to make the jumps necessary to the coveted Tenure Track — and otherwise, you get kicked out.

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

#319

Post by wslh »

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…

I would add that now we have new tools to enrich mathematicians works: new software tools and computing power. It is not about AI, AI is only a special case or a name for tools that can solve or ASSIST humans in cognitive tasks.

Age is (almost) not a limitation with the right tools, less in the near future.

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

#320

Post by Cthulhu_ »

dudeinjapan wrote: post #302
robin_reala wrote: post #289

Earlier quoted context omitted.

That’s US law though (no idea if Germany had it any different at the time, but it’s a different jurisdiction).

Germany outlawed Einstein.

Interesting phrasing to use for wanting to genocide him and all the other jews. Einstein got out on time, six million others didn't.
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