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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

#241

Post by npunt »

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…

The Beatles themselves didn't need to ingest the entire corpus of past musicians to be productive & inventive, but they did stand on the shoulders of giants musically (vs being born in some prior century), and they were in the right time & place to be part of a growing scene whose smarts exceeded that of any of it's individual participants. This is the nature of 'scenius' and of golden ages - a lot of ideas are already teed up in the collective consciousness.

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

#242

Post by nephanth »

__MatrixMan__ wrote: post #157
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…

Another approach to the finite-mathematician-lifespan problem might be to develop new foundations that are closer to the edge. I expect there's a logical universe in which sets are bizarre and hard to construct but objects which would take a modern mathematician years to grok are convenient and simple to work with.

Homotopy type theory?

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

#243

Post by ves »

lupire wrote: post #129
gavagai691 wrote: post #110

Earlier quoted context omitted.

"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…

> If one part of math gets too deep, you can always go somewhere else, where the water is still "shallow." Yes, but the shallow areas aren't very interesting, which is why people work in the deep areas.

Most of the now-deep, now-interesting areas were once shallow and uninteresting.

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

#244

Post by adalacelove »

woodruffw wrote: post #234
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.

I think we sometimes understate philosophy's progress: it's certainly not measurable in decades, but the world of 2022 looks very different (in terms of philosophical priors and their consequences) than the world of Socrates or Confucius. A handful of examples that come to mind: * (Nearly) everybody on our planet lives under a government whose fundamental structure and right to power comes from modern (meaning 17th c…

We measure philosophy success relative to finding the meaning of life. On the other hand everybody agrees that computers have advanced enormously without achieving yet AGI.

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

#245

Post by LudwigNagasena »

woodruffw wrote: post #234
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.

I think we sometimes understate philosophy's progress: it's certainly not measurable in decades, but the world of 2022 looks very different (in terms of philosophical priors and their consequences) than the world of Socrates or Confucius. A handful of examples that come to mind: * (Nearly) everybody on our planet lives under a government whose fundamental structure and right to power comes from modern (meaning 17th c…

Did those things appear because philosophers talked about them or did philosophers talk about them because they appeared?

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

#246

Post by anonymous_i »

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…

I disagree with Math had to be learned when you are young.

Intelligent people will end up learning something profound when they are young.If they find something else interesting enough at a later stage in their life, they apply some transformation learning.

Leibniz did not start his training in Math until he was ~30

> Thus Leibniz went to Paris in 1672. Soon after arriving, he met Dutch physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was patchy. With Huygens as his mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including discovering his version of the differential and integral calculus.

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

#247

Post by cycomanic »

petesergeant wrote: post #240
fnordpiglet wrote: post #53

Earlier quoted context omitted.

Good luck finding a movie about Isaac Newton or Einstein, or literally anyone whose value to the world is more than pretending to be someone else or lying a lot, let alone a very interesting mathematician no one recognizes. (Ok I know there’s a few movies about folks like Turing and Nash, but it’s pretty slim pickings).

There's an excellent series in which Newton plays a large part, called The Baroque Cycle. I'd love a TV adaptation of it, although you'd have to gut much of academic stuff in it and just focus on the (excellent) plot

Yes Neil Stephenson is such a versatile writer. The Baroque Cycle is definitely recommended although not his best work.

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

#248

Post by SkyMarshal »

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…

Regarding #2, I think Andrew Wiles already disproved that conjecture, solving Fermat at 41 or thereabouts, but Zhang is certainly another nail in its coffin.

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

#249

Post by SkyMarshal »

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…

> Perhaps we could then rely on computer assisted theorem provers.

Or AI, particularly if we figure out AGI.

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

#250

Post by pencilguin »

Ar-Curunir wrote: post #124
pencilguin wrote: post #117

Earlier quoted context omitted.

RSA depends on large prime factors being hard to recover from their product.

Yes, that is what "factoring a composite number" means, and that's what I said in the last sentence.

Crucially, the factors involved have to be very large.
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