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Old 09-29-2018, 03:15 AM
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Has the Riemann Hypothesis been proven?

I use SDMB as my primary news source but see no mention of the fact that Sir Michael Atiyah has claimed to have proved RH. Google News doesn't mention it in its headlines, so my non-mathematical friend had to inform me! (I even lost a bottle of beer to him saying "Nah, SDMB would have told me if there were any serious claim to a proof of RH; I'll bet a beer you're misled!")

For those who don't follow mathematics RH (Riemann Hypothesis) is the most famous of the math problems for which $1,000,000 prizes have been offered.

I wasn't sure whether to post this in GD or GQ. If it were in IMHO I'd have to mention my own uninformed opinion, which is that Atiyah's paper reads just like rambling amateurish crackpottery. (The paper is about deriving physics' FSC as a fundamental math constant! It doesn't mention RH which supposedly is a simple corollary of the theory he developed for FSC.)

There are two different YouTubes of his recent lecture, which goes into less technical detail than the paper, itself already lacking in detail. Click here to get his voice and the slides or click here to watch him speak but with slides too small to be legible.

As I say: It seems like crackpottery and Google suggests that other mathematicians think this is an old old man who has finally published one paper too many.

On the other hand, Atiyah has been called the world's greatest living mathematician.
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Old 09-29-2018, 04:03 AM
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If it were in IMHO I'd have to mention my own uninformed opinion, which is that Atiyah's paper reads just like rambling amateurish crackpottery.
Sure sounds the same way to me. But I'm only here to mention that Sir Arthur Eddington once claimed that the Fine Structure Constant was exactly 1/136. Somehow this was related to there being precisely 1362256 electrons in the universe. Later, it was found that the FSC was closer to 1/137. Eddington changed his proof to show that the FSC was exactly 1/137. For that, he earned the name "Sir Arthur Adding-One".
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Old 09-29-2018, 05:52 AM
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In the linked paper, Sir Michael writes:
The number Ж is a mathematical candidate for 1/α. It satisfies the criteria of both Feynman and Good, restores the reputation of Eddington and ...
...
Ironically, Eddington was later laughed out of court, when 137 was found to need a long string of corrections. In fact these corrections just arise from the iterative process that defines Ж, so Eddington’s two mistakes cancel each other out.
IIUC the fact that he has found a simple math expression that evaluates to empirical FSC (α) with 9 digits accuracy is itself rather amazing.

Last edited by septimus; 09-29-2018 at 05:55 AM. Reason: 9 digits not 8
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Old 09-29-2018, 07:05 AM
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His first relation is:
Ч/γ = Ж/π

He then computes Ч via the limit of a sum of an integral. I tried punching it into Wolfram Alpha, but it timed out .

Well. Surely we can compute it to a few hundred places, and then see if it matches reality once a few more digits are measured. I'm sure it'll only take a few million Feynman diagrams this time...
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Old 09-29-2018, 08:39 AM
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His first relation is:
Ч/γ = Ж/π

He then computes Ч via the limit of a sum of an integral. I tried punching it into Wolfram Alpha, but it timed out .

Well. Surely we can compute it to a few hundred places, and then see if it matches reality once a few more digits are measured. I'm sure it'll only take a few million Feynman diagrams this time...
Equations like " Ч/γ = Ж/π " which introduces, redundantly, two constants — in Cyrillic alphabet no less — make me wonder if this is all some elaborate practical joke.

The FSC = 137.03599917 is known empirically to ten digits: that final '7' is '4'. Atiyah claims to have computed Ж to 9 digits with an exact match to empirical FSC — too close to be coincidence. Has anyone checked Atiyah's arithmetic?

[off-topic] Koide's formula is another unexplained numeric coincidence involving physical constants. A relationship among the masses of electron, muon, and tau gives 2/3 = 0.66666, exact within the uncertainty in those mass measurements. But that 5-digit match might be by chance. Atiyah claims a 9-digit match with FSC.

Last edited by septimus; 09-29-2018 at 08:42 AM.
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Old 09-29-2018, 03:14 PM
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There is also the lovely result that from the von Neumann hyperfinite factor A of type II and the Hirzebruch Formalism, we can renormalize the usual Euler formula e2πi to e2Жw. It merely requires introducing the new constant w = πi/Ж.

Last edited by Dr. Strangelove; 09-29-2018 at 03:14 PM.
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Old 09-29-2018, 05:12 PM
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This is the first I have heard of the RH having been proved. Sir Michael Atiyah is a great mathematician, arguably among the top five of the 20th century, but his specialty was differential geometry, not number theory and I suspect that the top number theorists of my department would be talking of little else if it were true. On this basis, I have to doubt it.

I have heard that the abc has been claimed to have been proved (a technical conjecture in number theory that would readily prove the Fermat conjecture and much more) and the proof has been claimed to be wrong and the parties have not come to a resolution. You might thing this sort of thing doesn't happen in math, but it does. The first purported "proof" of the 4 color theorem stood for 10 years before the error was found.

But I will ask around and see what people are saying.
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Old 09-29-2018, 05:22 PM
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He's not some crank with a theory, he is a well respected mathematician with quite a bit of credit to his reputation.

That said, it is a complicated proof, and it may take quite a while to find an error in it, and quite a bit longer to fix the error.

Fermat's last theorem went through the same thing, with people thinking that they had solved it, only to find a small flaw in their hundreds of pages. that did eventually get proved, but it took a bit longer.

Also, my understanding is that it is a proof by contradiction, which some mathematicians don't consider to be a "real" proof. It means that if you assume the Riemann hypothesis to *not* be true, then you get logical contradictions that he shows with his work.
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Old 09-29-2018, 05:57 PM
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Also, my understanding is that it is a proof by contradiction, which some mathematicians don't consider to be a "real" proof. It means that if you assume the Riemann hypothesis to *not* be true, then you get logical contradictions that he shows with his work.
Why in the world would a proof by contradiction not be a real proof. Cantor's diagonal proof is a proof by contradiction, and I think mathematicians consider it to be a real proof.
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Old 09-29-2018, 06:22 PM
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Why in the world would a proof by contradiction not be a real proof. Cantor's diagonal proof is a proof by contradiction, and I think mathematicians consider it to be a real proof.
I am trying to find the quote, but it was something that Ayitah said in his announcement, I assume somewhat tongue in cheek.

But, while it is a proof, it is not as satisfying. You can say that it must be true, but you are not really saying "why".

There have been proofs that have been updated to have a "better" proof, whether it be simpler or uses fewer assumptions.

This could be one where a new proof could be as "exiting" as a straightforward proof that does say "why".
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Old 09-29-2018, 06:36 PM
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Could someone please summarize what the fine-structure constant has to do with the Riemann hypothesis? Especially if it's not in the paper. Does it have to do with Hilbert's old idea of realizing the zeros of the ζ function as eigenvalues of some self-adjoint operator?
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Old 09-29-2018, 06:46 PM
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He's not some crank with a theory, he is a well respected mathematician with quite a bit of credit to his reputation.
He is a well respected mathematician but not a physicist and a lot of that manuscript is making claims about solving random physics, music etc....without looking at a single property of the Riemann zeta function .

The lack of a real proof and the kitchen sink nature of that paper should be a red flag but here are a few hit on the Physics Crackpot Index too.

  • 5 points for each mention of "Einstien", "Hawkins" or "Feynmann".
  • 10 points for arguing that while a current well-established theory predicts phenomena correctly, it doesn't explain "why" they occur, or fails to provide a "mechanism".
  • 20 points for talking about how great your theory is, but never actually explaining it.
  • 40 points for claiming that when your theory is finally appreciated, present-day science will be seen for the sham it truly is. (30 more points for fantasizing about show trials in which scientists who mocked your theories will be forced to recant.)
  • 50 points for claiming you have a revolutionary theory but giving no concrete testable predictions.


The overwhelming consensus of qualified mathematicians is that it proves nothing.

Don't take the fact that I linked to something called the crackpot index to tarnish his earlier work. Unfortunately it appears that he is suffering from what we all will suffer from one day, mortality.

Perhaps we are just missing the brilliance but the odds are not pretty remote.
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Old 09-29-2018, 06:46 PM
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Why in the world would a proof by contradiction not be a real proof. Cantor's diagonal proof is a proof by contradiction, and I think mathematicians consider it to be a real proof.
I don't think it's that popular, but adherents of intuitionist logic reject the law of the excluded middle, and so don't get much mileage out of proof by contradiction.
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Old 09-29-2018, 06:56 PM
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Could someone please summarize what the fine-structure constant has to do with the Riemann hypothesis? Especially if it's not in the paper. Does it have to do with Hilbert's old idea of realizing the zeros of the ζ function as eigenvalues of some self-adjoint operator?
Nothing that anyone can tell. Unfortunately it seems to be embarrassing chapter of an aging mathematician. The lack of public denouncement by others in the field seems to be due to respect for the man.

Apparently he started mentioning his work on a grand unifying theory shortly after his beloved wife passed.

If you watch the presentation this may be a bit more clear.

https://www.youtube.com/watch?v=jXugkzFW5qY

I wish him the best.
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Old 09-30-2018, 03:26 PM
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The Riemann Hypothesis is probably well known enough that when it is solved... well and truly solved... it will be broadcast far and wide enough so that millions of people who have never heard of the RH will quickly know it's been solved.
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Old 09-30-2018, 03:32 PM
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The Riemann Hypothesis is probably well known enough that when it is solved... well and truly solved... it will be broadcast far and wide enough so that millions of people who have never heard of the RH will quickly know it's been solved.
Ask a random person if Fermat's last theorem has been solved, and I doubt they will even know what you are talking about.
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Old 09-30-2018, 03:55 PM
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The "millions" who find out won't be random, they'll be partially self-selected by being newspaper subscribers, journalists, bloggers, googlers, members of forums
cryptographers, students, etc who will be exposed to the news via their own sources.

But, yeah, like Fermat, the millions who learn will be outweighed by the billions who won't.

Last edited by JohnT; 09-30-2018 at 03:56 PM.
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Old 09-30-2018, 06:02 PM
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On the other hand, Atiyah has been called the world's greatest living mathematician.
That's probably Serre, but Atiyah is ridiculously brilliant and accomplished. (I'm admittedly biased, since one of the areas I'm in, topological K-theory, is one in which Atiyah is a major figure. He's also a great author.)

That having been said, this paper honestly does look like something a crackpot would come up with. It reads like a half-remembered summary of another paper the author read somewhere. Math papers (and these look like notes rather than a real paper) methodically go through points in order and detail; this looks like notes a student would take at someone else's lecture. I can't follow what's going on in it, primarily because there is nothing going on. It's just gibberish.

It also pings nearly every crackpot index: lots of discussion of philosophy and grandoise claism without anything to back them up; lots of discussion of math, but few (if any) actual equations and results; analogies to historical results (Archimedes' formula for the area of a circle) that have nothing to do with the topic at hand; and lots of promises of awe-inspiring results with with nothing to back them up. Weird, considering that Atiyah is a legimate genius; it would be hard to come up with someone with a better reputation among living mathematicians.

Last edited by Itself; 09-30-2018 at 06:02 PM.
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Old 09-30-2018, 06:09 PM
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I should clarify above that I was referring to the notes in the OP about the fine structure constant. I did run across a similar 5-page blurb from Atiyah about his purported proof of the RH, and it's bizarre. At one part, for example, he constructs a complex function F, notes that F(s) = 2 F(s), and then concludes that "Since C is not of characteristic 2, it folows that F(s) is identically 0." That's insanely obvious and not something you need to explicitly state, and it's true even in characteristic 2 (where the equation above literally becomes F(s) = 0); just subtract F from both sides. He goes on to conclude that a certain polynomial T "is not the zero polynomial and so it is invertible in the field of meromorphic functions." I mean...sure, anyone reading this paper knows what 'meromorphic' means, and that's patently obvious. Weird. It's also odd in that crackpots tend to have the same combination of obsessive, pedantic detail about completely obvious things and hand-waving carelessness about nontrivial, important things. Atiyah's not a crackpot, but these notes are just as bad as any internet crank's.

He's an amazing guy (and, indeed, everything I've seen so far on various social media have addressed the matter much more kindly than they would if it were some random nobody), but it unfortunately looks like he's lost it.

Last edited by Itself; 09-30-2018 at 06:12 PM.
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Old 09-30-2018, 09:00 PM
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I don't think it's that popular, but adherents of intuitionist logic reject the law of the excluded middle, and so don't get much mileage out of proof by contradiction.
True, entirely true, but the intuitionists reject all of modern mathematics. I cannot take them seriously.

The last few posts have convinced me that it is all likely early stages of dementia. A shame, because he was a great mathematician.
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Old 09-30-2018, 09:46 PM
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True, entirely true, but the intuitionists reject all of modern mathematics. I cannot take them seriously.
I'm not sure they take themselves 100% seriously... but I don't know. Still--I think it's a nice effort to prove things using the weakest possible set of axioms. Plus, constructive arguments do tend to be more satisfying than non-constructive ones. Like the axiom of choice--use it if you need to, but it's nicer if you don't.
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Old 09-30-2018, 09:55 PM
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Nothing that anyone can tell. Unfortunately it seems to be embarrassing chapter of an aging mathematician. The lack of public denouncement by others in the field seems to be due to respect for the man.

Apparently he started mentioning his work on a grand unifying theory shortly after his beloved wife passed.

If you watch the presentation this may be a bit more clear.

https://www.youtube.com/watch?v=jXugkzFW5qY

I wish him the best.
I listened. The truly noteworthy fact is his claim that it came out of his study of the fine structure constant. Now, the fine structure constant is an observable quantity, related to Planck's constant and not a mathematical quantity at all. In a different universe it could, as far as anyone knows, have a different value. It seems inconceivable to me that any study of the FSC could bear on the RH.
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Old 09-30-2018, 10:03 PM
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Why in the world would a proof by contradiction not be a real proof. Cantor's diagonal proof is a proof by contradiction, and I think mathematicians consider it to be a real proof.
We had a discussion of proof by contradiction a while back, by the way https://boards.straightdope.com/sdmb...d.php?t=689130
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Old 09-30-2018, 10:20 PM
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I listened. The truly noteworthy fact is his claim that it came out of his study of the fine structure constant. Now, the fine structure constant is an observable quantity, related to Planck's constant and not a mathematical quantity at all. In a different universe it could, as far as anyone knows, have a different value. It seems inconceivable to me that any study of the FSC could bear on the RH.
While I hesitate to do so out of respect, look into his claimed 12-page Feit Thompson proof and paper on 6-spheres before you dedicate too much time.

I started to itemize issues or required points of clarification with this claim but stopped as it felt like I was stepping on puppies.

I'll leave it to others.
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Old 09-30-2018, 10:43 PM
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It looks like his purported 12-page proof of Feit-Thompson isn't available anywhere online, but he has a paper about complex structures on S^6 on the arxiv and another in a book called "Foundations of Mathematics and Physics One Century After Hilbert." The former is just nonsense. I can only find a preview of the first two pages of the latter, but they're just elementary, unrelated stuff about Minkowski space, and the entire thing is only four pages long. Oh well.
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Old 01-12-2019, 04:36 PM
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Sir Michael Atiyah just passed away:
https://www.bbc.com/news/science-environment-46850763

More extensive obituary:
https://www.nytimes.com/2019/01/11/o...iyah-dead.html
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Old 01-12-2019, 06:53 PM
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Sorry to hear that. There is a line in the NYT obituary that says his proof of RH did not hold up.
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Old 01-12-2019, 07:16 PM
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As an aside, I've long suspected that the fine-structure constant can, in fact, be mathematically calculated. It's not like it'd be the first time that a dimensionless constant of fundamental physics has been discovered to be calculable. But at the same time, any claim of a calculation should be treated with skepticism, especially if it doesn't come from a coherent physical model which makes other testable predictions. I'm not convinced by arguments like "this simple formula produces α to ten decimal places, and that can't be coincidence!", because even "simple" formulae generally contain more information than they seem to, and coincidences happen, especially when you have the whole universe of numbers to look for.
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