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Prominent Mathematicians Rebuke Recent Riemann Hypothesis Proof
Posted by
ScuttleMonkey
on Fri Jul 04, 2008 01:29 PM
from the here-is-your-peer-review dept.
from the here-is-your-peer-review dept.
Bryan writes "Xian-Jin Li's purported proof of the Riemann Hypothesis (reported on recently) has been rebuked by Fields Medalist Terence Tao. Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts." Relatedly, jim.shilliday writes "The proof cites and appears to be based in part on the work of the leading French theorist Alain Connes. A few hours ago, Connes posted a comment on his blog stating that the purported proof is so badly flawed that he stopped reading it."
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Claimed Proof of Riemann Hypothesis 345 comments
An anonymous reader writes "Xian-Jin Li claims to have proven the Riemann hypothesis in this preprint on the arXiv." We've mentioned recent advances in the search for a proof but if true, I'm told this is important stuff. Me, I use math to write dirty words on my calculator.
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Why "fortunately"? (Score:5, Interesting)
From the summary:
Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts
Why? I'm probably missing something obvious, I'm not even a mathematician to start with, but...
I mean, we (the world) do want to prove it right (or wrong) one day or another, don't we?
Re:Why "fortunately"? (Score:5, Informative)
I guess they mean that there's no shame in having failed, since many other respectable attempts also failed.
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Re:Why "fortunately"? (Score:5, Insightful)
They're just being polite by pointing out there's no shame in failing to prove the Riemann Hypothesis, since it has frustrated the attempts of many a prominent mathematician so far.
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Re:Why "fortunately"? (Score:5, Insightful)
Having said that, Li is no crank. I had not heard of him, but that's no surprise since I'm not a number theorist. But he has published several refereed papers in this area, has a position at BYU, and really ought to have known better than to explode on the scene like this.
I've gotten communications from genuine crackpots, wanting my comments on their work. Early in my career, I wrote back, gently pointing out the mistake. To my horror, friends then received slightly modified but still absurd drafts, listing me as a collaborator! Li is a real mathematician, probably with poor social skills, and a bad proof.
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Re:Why "fortunately"? (Score:4, Informative)
Are you sure about that? Getting a paper onto arxiv.org doesn't seem to be that hard, and there's lots of ways to find out about it (RSS feed, etc.). He may not have had any reason to believe that he'd get this sort of attention, as he may have thought everyone involved would simply assume that it wasn't worth much, not having been peer reviewed.
While I love the free and open flow of information that arxiv represents, this is hardly the first time that something has been posted on there and subsequently blown out of proportion. The Internet at large doesn't seem to really understand arxiv.org, that just because someone's got a fancy LaTeX paper up claiming some wild thing doesn't mean it's credible. A paper on arxiv.org shouldn't even be understood as being endorsed by the author, let alone "science". I always love when somebody backs up their argument about physics with a link to arxiv.org, it's like a red flag that it's time to just pack it in, you're not going to get through to this person, because they only understand the trappings of science, not the actual process.
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Re:Why "fortunately"? (Score:4, Interesting)
...with a link to arxiv.org, it's like a red flag...
An even redder flag is a link to New Scientist as if its some peer reviewed source. NS references arxiv.org heavily no matter how stupid the claims (aka Zero Point Energy).
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Re:Why "fortunately"? (Score:5, Informative)
I believe you're mixing this up with another hard problem that hasn't been proven yet. You're thinking about the NP = P [wikipedia.org] problem. The difference is that here we don't know what will be the outcome, whereas for the RH most people assume it's true. Having a proof for this wouldn't really change anything (apart from validating large parts of mathematics that assume it is true)
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I am a mathematician, and there's no reference for this claim, but RH is a problem in analytic number theory and none of the credible work on it (meaning not by random crackpots) uses anything involving factoring. Why would an algorithm to factor numbers have any use at all, especially since this isn't something that can be proven computationally anyway?
The best we've done algorithmically by assuming the Riemann hypothesis is come up with faster algorithms to test primality (like an unconditional Miller-Ra
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For the record, most people don't know what P or NP is ;)
Re:Why "fortunately"? (Score:5, Funny)
[..] lead to the undermining of current encryption methods, which depend on the difficulty of factoring large prime numbers.
That's a trivial problem.
All prime numbers have two factors: 1 and itself.
Goodbye encryption :-)
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Re:Why "fortunately"? (Score:5, Informative)
One possible explanation for your understanding (which in my understanding, is wrong), is the Miller-Rabin primality test algorithm.
The primality problem (telling whether a number is prime), although hard, was never proved to be NP-complete.
The Miller-Rabin primality test is a (actually, the 1st and possibly the only) polynomial deterministic algorithm that is based on the Riemann hypothesis (polinomial deterministic meaning "fast and accurate"). Proving RH would prove that Miller-Rabin is exact and therefore shown that primality testing is in P.
http://en.wikipedia.org/wiki/Miller-Rabin_primality_test [wikipedia.org]
Unfortunately, algorithm freaks were faster than math freaks (well, the algorithm freaks involved were math freaks too) and a new algorithm called AKS was developed that did everything Miller-Rabin did without relying on the Riemann Hypothesis.
http://en.wikipedia.org/wiki/AKS_primality_test [wikipedia.org]
So, to this day, we know primality testing is polynomial. The _real_ problem in cryptography is prime *factoring* (if it's not prime, then find 2 numbers that when multiplied produce the original number). Although it is not know whether that problem is P or NP-complete or both, it is believed to be outside NP because it is much harder than plain primality testing.
http://en.wikipedia.org/wiki/Integer_factorization [wikipedia.org]
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what does it all mean, Basil? (Score:2, Funny)
Re:what does it all mean, Basil? (Score:5, Funny)
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Re:what does it all mean, Basil? (Score:5, Informative)
There are a lot of results based on assuming the conjecture is true, including a variety of factoring and root finding algorithms that are computationally very useful.
Until it is proven you really don't know if these algorithms are giving correct answers.
This is why it is so important and has a big prize associated to it.
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Re: (Score:3, Insightful)
Unless it says the number is prime (you have to trust there are no factors) or gives factors that aren't primes.
Re:what does it all mean, Basil? (Score:4, Interesting)
Since the work based on the assumption that the hypothesis is true is in itself valuable, it will still be used.
It's just that a proof, if found, will elevate who-ever finds it to the status of mathematical superstar.
Consider this, we are still finding proof of various of Einstein's theories, but work based on his has been of real value for decades.
Here's another example that makes me sound all clever because I know it.
Newtons equations, and his entire body of work, completely failed to explain how it is that the moon can orbit the earth while the earth orbits the sun, and we *still* don't have the equation to explain that bugger.
There are specific n-body solutions, I've written one myself, but a solution for the general case? Nope, never been done.
Louis Pasteur spent most of his life on that particular problem, as have many other prominent scientists, all to no avail. We found a use for Newtons work regardless, and Einstein extended it successfully, even with that glaring hole.
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Re:what does it all mean, Basil? (Score:5, Insightful)
You are mixing the basic tenants of physics and mathematics, not a good thing to do. Science is a mix of inductive and deductive logic, math has a higher standard and doesn't admit inductive proofs.
Three guys were vacationing in Scotland. One was an astronomer, one a physicist and one a mathematician.
In their travels they chance on a black sheep grazing in a field.
Astronomer: All sheep in Scotland are black
Physicist: Some sheep in Scotland are black
Mathematician: There is one sheep in a field in Scotland that is black on at least one side.
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Re: (Score:3, Insightful)
Mathematician: There is one sheep in a field in Scotland that is black on at least one side.
I thought he was going to say all sheep in Scotland are grey?
Anyhow, what's with there's no inductive proof in Mathematics? There are many many inductive proofs, even at high-school you write "proof by induction" quite a lot. Google it, all the top hits are mathematical.
http://en.wikipedia.org/wiki/Mathematical_induction [wikipedia.org]
Re: (Score:3, Informative)
Inductive is a philosophical term, the inference of new facts based on previously known ones. In Physics, this means using experimental data in order to make general assumptions about the universe.
In mathematics, we use the term tongue-in-cheek, to refer to a particular and useful consequence of the least-element axiom. It resembles inductive reasoning, but it is indeed quite more rigorous.
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Addressing the latter first, Newton's equations describe to a very high degree of accuracy (perfectly, in the limit of ignoring relativistic and other high-order corrections) the interaction of any arbitrarily large number of bodies. The fact that we cannot solve these equations is in no way
Re:what does it all mean, Basil? (Score:4, Informative)
Nope. We can do calculations that involve n-bodies, of which obviously 3-body is part, but they involve using the 2-body solution of Newton for all unique pairs in a simulation.
A separate general three body solution probably does exist, but no-ones found it.
If found, it would quite possibly revolutionise n-body modelling, and prove useful to space science (if, and only if, it sped up calculations), but I doubt astronomers would care much.
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I had to go look it up after you asked... apparently to mathematicians, there are plenty of practical applications. See http://en.wikipedia.org/wiki/Riemann_hypothesis [wikipedia.org] for a few examples. Neither of them made sense to me yet, and I've already had coffee. If they are looking for one example where this theory is not true, and offering a million bucks, someone is sure to put a couple yellow dog games console clusters together and find out soon enough. (either that or prove Doom is written by zombies who don't
I don't know about you all... (Score:5, Funny)
The "proof" is that of Theorem 7.3 page 29 in Li's paper, but I stopped reading it when I saw that he is extending the test function h from ideles to adeles by 0 outside ideles and then using Fourier transform (see page 31). This cannot work and ideles form a set of measure 0 inside adeles (unlike what happens when one only deals with finitely many places).
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The "proof" is that of Theorem 7.3 page 29 in Li's paper, but I stopped reading it when I saw that he is extending the test function h from ideles to adeles by 0 outside ideles and then using Fourier transform (see page 31). This cannot work and ideles form a set of measure 0 inside adeles (unlike what happens when one only deals with finitely many places).
Re:I don't know about you all... (Score:5, Insightful)
It's called "Proof by Intimidation":
using the formula:
[ some formula ]
it is trivial to see that:
[ some other formula out of nowhere ]
therefore, combining the above, we can arrive at the easily obtained answer:
[ some MATLAB result ]
Don't forget, it works both ways; the people marking your assignment don't want to admit that they can't see the so-called "trivial" derivation.
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Re:I don't know about you all... (Score:5, Insightful)
Use the "Star Trek" filter:
he is extending the test function h from [ tech ] to [ tech2 ] by [ tech 3 ] and then using Fourier transform ... This cannot work and [ tech ] form a set of measure 0 [ tech 4 ] (unlike what happens when one only deals with finitely many places).
When he moved from one set to another and did the Fourier transform, he forgot that he ended up with an empty set instead of a finite number of points because that's apparently a property of whatever the hell he was talking about.
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Re:I don't know about you all... (Score:4, Informative)
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Re: (Score:3, Interesting)
This is not all that bad.
Probably many slashdotters are familiar with the discrete Fourier transform (used in JPEG encoding, incidentally). The DFT for sequences of length n fits together nicely with the DFT for longer sequences whose length is a multiple of n. If you try to put all these DFTs for sequences of different length together in a certain way and combine them with the continuous Fourier transform, you end up with something called the adelic Fourier transform. (That's a little bit different
Ow my head (Score:5, Funny)
The proof, and the rebuke, only proved my theory that there is a distinct surge in advil usage when something like this is posted on /. or digged.
Re: (Score:3, Funny)
I stopped reading when I saw that he is using Advil, this cannot work when it is well established that Excedrin is the preferred migraine reliever.
Preprint, not a reviewed paper (Score:5, Insightful)
Well duh this is what we have been saying - this is a preprint and is likely to have errors. Whether or not they can be repaired is open to question.
Wiles' proof of Fermat's last theorem took a long time to go through the review and repair process. And there was at least one pretty hard problem that had to be fixed.
Slashdot's "journalistic" process really suxors when it comes to this sort of stuff.
Re:Preprint, not a reviewed paper (Score:5, Funny)
Slashdot's "journalistic" process really suxors when it comes to this sort of stuff.
Wel of course it does. Slashdot is journalology, not journalonomy.
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Re: (Score:3, Interesting)
The comments made by Tao and Connes are the sort of comments one would make if the paper was irrevocably flawed. For instance, Tao notes that "the decomposition claimed in equation (6.9) ... is, in fact, impossible; it would endow the function h ... with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry ..."
In more simple terms: Partway into the paper the author proved something that is definitely false; he then relied on this false
Preprints are not ideal. (Score:3, Insightful)
Yeah, this is becoming a real problem with the preprint journals. Media groups like New Scientist will run a hyped-up story on some "ground-breaking new development" which will have propagated through the blog echo-chamber before other scientists have even had a chance to review it. It's not enough for the media to completely butcher the science they do present, now they have to present results which haven't even had cursory review. It's no wonder the public doesn't trust science considering what is is bein
Lazy title selection (Score:5, Funny)
"Renowned Researchers Rebuke Recent Riemann Reasoning"
What a shock... (Score:5, Interesting)
Re:What a shock... (Score:5, Interesting)
I work part-time for a couple of mathematics research journals and we do get the occasional crank submission. There's one guy who's been sending us, on average, a 'paper' every week or so for the past few years: typically a single, badly-written page of gibberish (we're talking Time Cube [wikipedia.org] standard lunacy here) which is clearly not the work of someone who's ever seen a real mathematics paper. We've never responded to him, or even acknowledged any of his submissions (helpfully he prints his return address on the back of the envelope, so these days they go straight in the bin, unopened and unread) and yet he still keeps sending them in.
The arXiv also tends to get its fair share of crank submissions, usually elementary attempted (but trivially broken) proofs of things like the Goldbach Conjecture, Fermat's Last Theorem and the like - I'm assuming that the really mad stuff is filtered out by the moderators.
In contrast, at a quick glance to my nonspecialist eyes (I'm a knot theorist) Xian-Jin Li's preprint looks like a genuine (if flawed) attempt by a serious, qualified mathematician who specialises in the relevant area. Fair play to him for trying, though. I'm also not sure I'd characterise Terence Tao or Alain Connes' refutations as 'rebukes' - they looked more like dispassionate analyses of the paper's flaws to me, the sort of discussion you'd expect from the peer-refereeing process.
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See if she weighs the same as a duck?
Prof Connes also a Fields medalist (Score:5, Informative)
Just wanted to point out that Professor Connes is also a Fields medalist (1982) [wikipedia.org].
I guess it is a testament to Xian-Jin Li excellent reputation and the importance of the topic that these two mathematical superstars took the time to look at his proof.
there was no rebuke (Score:5, Insightful)
The proof was a legitimate effort by a non-crackpot, but the ideas in it were well known to specialists in the field and were generally understood to not be powerful enough to crack the problem. So the errors were found fairly quickly. Scott Aaronson's post Ten Signs that a claimed mathematical breakthrough is wrong [scottaaronson.com] item #10 may be helpful in understanding what happened.
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Re:Not Making Yourself Look Good Here (Score:5, Insightful)
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I'm not sure, but the way I read that is the quote is saying that the mathematician whose work was the basis for the proof started to read it and he (the mathematician) stopped reading.
From the link pointing to Connes blog ...
Re:Not Making Yourself Look Good Here (Score:5, Insightful)
Yes, why don't you tell the Fields medalist how to make himself look good? I'm sure he needs your help desperately. Jeebus, you know that a Fields medal is objectively harder to get than a damned Nobel prize, right?
He did critique the 'proof' objectively. The claim was that by looking at the function on a certain domain ("ideles" whatever those are), one could look out from there and see how it would have to behave elsewhere ("adeles"). However, the "ideles" aren't big enough to give a good viewpoint of what's going on (i.e. the function at the ideles is not necessarily representative of the rest of the function). If you only look at multiples of 2pi, you could "prove" that sin(x)==0. Just because you or I couldn't notice the obvious problem in the RH proof, doesn't mean that it doesn't merit quick dismissal. Sometimes obvious mistakes are made in math (some would say that only obvious mistakes are made - but they are only obvious once they are pointed out).
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Re:Not Making Yourself Look Good Here (Score:5, Insightful)
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