Mathematical Proof of the ABC Conjecture Will Be Published (nature.com) 39
AmiMoJo shares a report from Nature: After an eight-year struggle, embattled Japanese mathematician Shinichi Mochizuki has finally received some validation. His 600-page proof of the abc conjecture, one of the biggest open problems in number theory, has been accepted for publication. Acceptance of the work in Publications of the Research Institute for Mathematical Sciences (RIMS) is the latest development in a long and acrimonious controversy over the mathematicians' proof. Mochizuki is chief editor but was not involved in the review.
Eight years ago, Mochizuki posted four massive papers online, claiming to have solved the abc conjecture. The work baffled mathematicians, who spent years trying to understand it. Then, in 2018, two highly respected mathematicians said they were confident that they had found a flaw in Mochizuki's proof -- something many saw as death blow to his claims. The "abc conjecture," the problem Mochizuki claims to have solved, expresses a profound link between the addition and multiplication of integer numbers. Any integer can be factored into prime numbers, its 'divisors': for example, 60 = 5 x 3 x 2 x 2. The conjecture roughly states that if a lot of small primes divide two numbers a and b, then only a few, large ones divide their sum, c. A proof, if confirmed, could change the face of number theory, by, for example, providing a novel approach to proving Fermat's last theorem, the legendary problem formulated by Pierre de Fermat in 1637 and solved only in 1995. Some experts say Mochizuki failed to fix the fatal flaw in the solution. "I think it is safe to say that there has not been much change in the community opinion since 2018," says Kiran Kedlaya, a number theorist at the University of California, San Diego.
Another mathematician, Edward Frenkel of the University of California, Berkeley, says, "I will withhold my judgment on the publication of this work until it actually happens, as new information might emerge."
Eight years ago, Mochizuki posted four massive papers online, claiming to have solved the abc conjecture. The work baffled mathematicians, who spent years trying to understand it. Then, in 2018, two highly respected mathematicians said they were confident that they had found a flaw in Mochizuki's proof -- something many saw as death blow to his claims. The "abc conjecture," the problem Mochizuki claims to have solved, expresses a profound link between the addition and multiplication of integer numbers. Any integer can be factored into prime numbers, its 'divisors': for example, 60 = 5 x 3 x 2 x 2. The conjecture roughly states that if a lot of small primes divide two numbers a and b, then only a few, large ones divide their sum, c. A proof, if confirmed, could change the face of number theory, by, for example, providing a novel approach to proving Fermat's last theorem, the legendary problem formulated by Pierre de Fermat in 1637 and solved only in 1995. Some experts say Mochizuki failed to fix the fatal flaw in the solution. "I think it is safe to say that there has not been much change in the community opinion since 2018," says Kiran Kedlaya, a number theorist at the University of California, San Diego.
Another mathematician, Edward Frenkel of the University of California, Berkeley, says, "I will withhold my judgment on the publication of this work until it actually happens, as new information might emerge."
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Half the time I mod something and it does the opposite of what its supposed to.
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Half the time I mod something and it does the opposite of what its supposed to.
Looking at your user name... It probably doesn't help that you mod everything, "Meathead". :-)
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Honestly, why is anyone doing anything that doesn't involve studying the coronavirus to find a vaccine?
how far are you ? I am getting closer by the day.
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Remember, dear publishers: (Score:2)
If you don't see failure as useful information too, you are not a scientist, and missed the point of science. ... dear mathematicians. ;)
(Apart from that it must have useful predictions
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If you don't see failure as useful information too, you are not a scientist
If you run a science experiment, and you get a negative result, that is useful information. A great example is the Michelson-Morley experiment, which failed to detect the ether and led to a revolution in physics.
Likewise, a mathematical proof that proves a conjecture false is very useful. Godel didn't prove that math was complete and consistent. He proved it wasn't.
But if an experiment fails because of faulty experiment design, or defective equipment, or fudged data from a lazy postdoc, that isn't so use
Re:Remember, dear publishers: (Score:5, Interesting)
"Complex, opaque, not clear that it's of any use to anyone" also applied to most of Euler's work at the time. The only way to distinguish between "blazing a trail with revolutionary new methods" and "on crack" is to review the work. Seems like the right thing was done here, as just a couple of mathematicians dove int this work in case he was on to something, instead of just on something. Sometimes even if the proof is flawed, if the proof uses a different approach that approach could turn out to be useful. Not so much here it seems, but someone had to get their head around it to be sure.
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"Complex, opaque, not clear that it's of any use to anyone" also applied to most of Euler's work at the time.
Euler was widely recognized as a genius during his own lifetime.
Many of his methods were complex and opaque, but his results were not.
e^pi+1=0 took genius to derive, but is breathtakingly simple.
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Sure, but most of what he wrote simply could not be understood in his time. Most of 20th century mathematics was a race to re-discover something Euler wrote about before the "translators" published it.
Euler made up his own notation and wrote all his notes using such notation. A lot of what is fundamental in modern math notation comes from a textbook Euler wrote, including "f(x)" for "a function of x". His notation, outside of what he published, was very opaque and takes an expert in his notation to under
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i --- you dropped this.
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> e^pi+1=0 took genius to derive, but is breathtakingly simple.
For clarity that should be:
e^(i*pi) + 1 = 0.
I know what you mean but not everyone does. (It may not be obvious that you meant p and i when you used pi -- and maybe misleading for the layman.)
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Mathematics is not science. It doesn't make predictions - certainly not about the real world. Scientists use mathematics to make predictions, but it is not the same thing.
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Mathematics is that part of science that doesn't require a universe.
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Huh? (Score:1)
The conjecture roughly states that if a lot of small primes divide two numbers a and b, then only a few, large ones divide their sum, c.
What defines the terms "a lot," "a few," "small," and "large?" This seems too vague to be useful at all.
Re: Huh? (Score:3, Informative)
Basically if you add a ton of primes together the result is only divisible by a few large numbers.
Intuitively this seems to make sense given the nature of primes, but proving or disproving this is really hard.
This guy seems to have invented a whole new field of maths to prove it. If he's right, he's basically a modern day Newton but at this point nobody seems to understand the paper he wrote - like Newton but on steroids.
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Why? Did he invent a new language, too?
Re: Huh? (Score:5, Interesting)
Yes.
‘INTER-UNIVERSAL TEICHMÜLLER THEORY I:CONSTRUCTION OF HODGE THEATERS’
‘The goal [...] is to establish anarithmetic version of Teichmüller theory for number fields equipped with an elliptic curve—which we refer to as “inter-universal Teichmüller theory”— by applying the theory of semi-graphs of *anabelioids, Frobenioids, the étale theta function, and log-shells* developed in earlier papers by the author. ’
The majority of terminology and constructs that follow is totally lost on the others working in the same field.
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What defines the terms "a lot," "a few," "small," and "large?" This seems too vague to be useful at all.
It's call a layman's summary. For the purposes of summarizing an unfamiliar topic for everyday readers, that level of detail is actually optimal. I'm sure there are more precise definitions of those terms, but the general reader doesn't need to know those. They would clutter the summary, hence be *less* than useful to include. If you ask someone "summarize your work in 5 minutes" they have to keep it vague so that they give you the gist of it: going into details on every point would waste the 5 minutes.
How do they come up with these conjectures? (Score:3)
What amazes me more than the eventual proof is how these guys come up with these conjectures... for example, Fermat. Guy lives in the 1600s, no electricity or running water, and just theorizes that x^n +y^n = z^n has no integral solutions for n>2. How? Did he compute a bunch of values by hand? Did he have a legion of assistants who would `crunch the numbers' to ensure that the proof is checked for at least small values of n?
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Back in that era, there was no Internet, no television, no radio, no newspapers.
People took whatever time was required for their work because there wasn't much else to do, unlike in 2020 where you should be doing something more productive than reading this post.
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So, it has come to this (Score:4, Funny)
I'm so used to read about anti-science events happening in the United States that I read the title as "Mathematical proof of the ABC conjecture will be punished".
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Primes have a reputation for being spooky and mysterious; if mathematicians go around proving theorems about their structure, then that's harmful to the brand. Big prime has a lot riding on making sure that there's a lot of FUD around alleged proofs of these things.
In a related news.. (Score:5, Funny)
Research Institute for Mathematical Sciences (Score:2)
After 8 years of people messing with him and calling his work a joke as he tried to get this recognized by a prestigious organization, you could say when it ended he turned the joke around on them and got a RIM Shot.
I'm waiting for the proof of the (Score:2)
LMNOP conjecture.