There’s bad news on the math front. Penny Smith has *withdrawn* her Navier Stokes paper, because of the discovery of a serious error.
But to be optimistic for a moment, this doesn’t mean that there’s nothing there. Remember that when Andrew Wiles first showed his proof of Fermat’s last theorem, he discovered a very serious error. After that, it took him a couple of years, and some help from a colleague, but he *did* eventually fix the problem and complete the proof.
Whatever develops, it remains true that Professor Smith has made *huge* strides in her work on Navier-Stokes, and if she hasn’t found the solution yet, she has at least helped pave to road to it. Here’s hoping that she finishes it!
Good luck, Professor Smith! We’re pulling for you.
Good Math/Bad Math
The beauty of math; the humor of stupidity.
The way the academia works, someone else will probably scoop her and claim credit for providing the entire proof.
Could you share the title of the paper or the URL? Thanks.
I’m hoping that this work provides some assumptions or simplifications that make solving the NS equations more trivial but, alas, fluid mechanics is rather dynamic.
I’m glad math works the way it does. Would St. Augustine ever have “withdrawn” City of God if anyone had claimed to spot a “serious error” in its logic? Would Thomas Aquinas withdraw the Summa Theologica?
I’m sure this is a stressful time for Dr. Smith, and as we speak I’m throwing my condolences into the Internet. (Buck up, curse the hype and try again! I sure couldn’t have come even close!) But if we take a slightly broader perspective, it’s trivially obvious to the most casual observer that mathematics cannot progress without error correction.
We can make progress, and criticism is the only known antidote to error.
It’s great to see a REAL researcher showing the scientific frauds how it’s supposed to be done …
As a guy writing a Java 3D Viscous Finite Volume solver, I am eager to see what comes of this. I truly hope the error is minor and can be overcome.
DV8 2XL:
True, and I wish I had mentioned that in the original post. Publicly acknowledging an error the way Prof. Smith did is a sign of true professionalism. None of the pseudo-scientific crackpots out there – whether creationist, velikovskian, HIV denialist, woo-peddler, or whatever – none of them would have the courage or professionalism to admit to a mistake. Real scientists and real mathematicians make mistakes, admit them, and move on. Crackpots endlessly babble and deflect criticism, refusing to admit to any mistake, no matter how obvious.
There’s a dramatic record of events at the comments over at Peter Woit’s blog. A commenter named Euler pointed out that the paper’s assumptions includes both Euler’s and Burger’s equations, which both have shock wave solutions. Reductio ad absurdam. Shortly thereafter, Smith identified the error and honorably withdrew her paper.
This unsought publicity really sucks for her, as she says herself.
That which does not kill us … She deserves support for both the boldness to attempt a problem this challenging and the professionalism of handling a mistake.
I kinda actually wonder if the immense pressure surrounding Clay Prize level problems can actually inhibit the academic process of finding solutions. I can’t imagine it’s easy to collaborate on or work on publishing solutions to a problem like this when success can have a million dollar prize hanging over it and short-term errors in the proof process can result in embarrassment in headlines across the world.
Ashwin:
The paper was withdrawn because of the error, so it’s no longer online. On another blog, Penny Smith did say that while the error gets rid of the immortal solution, there are still finite time solutions that she proves possible using a different approach, and she’ll be submitting a revised paper with the weaker, but correct, conclusion.
And hopefully, in time, she’ll be back with the complete solution.
Mark: Just to clarify, Smith’s work *was* a non-constructive existence theorum, without immediate practical implications.
I’d venture a hypothesis that the system for obtaining tenure inhibits the process the most, at least for the younger crowd. Untenured mathematicians would be considered crazy to attack such a problem, as there is a high probability of such work being fruitless and resulting in a long period without publications. (Though once tenured, there’s less reason not to go for it, but there’s also less reason to do any work at all.)
For the most part, though, it doesn’t seem like very many mathematicians give serious consideration to (directly) attacking a Clay-level problem. But I don’t think there’s much of the kind of pressure you’re talking about.
Withdrawing the paper doesn’t mean there’s nothing there, but in fact in this case there really is nothing there. Any “huge strides” made in this paper were in the wrong direction.
I’m actually pretty angry about this whole mess. Smith was absolutely unprofessional to claim publicly that she had a solution without the slightest corroboration. If you think you’ve solved the most famous problem in the field with a month’s work, by finding a simple reduction to a special case of a previous theorem of yours published in a so-so journal, then you should be very suspicious. Going public with such a claim before you manage to convince even _one_ person who will vouch for your proof (a friend, a colleague, a mentor, a collaborator – Christina Sormani doesn’t count because she never claimed to understand the proof or to have checked it) is ridiculous. It gets worse in this case, because Smith’s methods simply didn’t address the real difficulties in the problem. It was clear there was an error when people realized that not only did Navier-Stokes satisfy the hypotheses of her older theorem, but so did some equations that were well known not to have immortal solutions. When you solve a famous problem, the first thing you should do is to ask yourself why you succeeded when everyone else failed. If you can’t pinpoint some key idea, and instead it just looks like the difficulties magically vanished, then you’ve really got to think twice. It’s not a guarantee that you’re wrong, but it’s the next best thing to a guarantee.
I suppose I’m sorry for her, given her public embarrassment, but she could easily have avoided it by being less hasty.
In any case, maybe Smith will solve the problem someday, maybe someone else will, but right now the best thing is just to forget about this incident. It’s patronizing to praise her for admitting her mistake (“Wow, she’s neither incompetent nor dishonest!”), and she doesn’t benefit from exaggerated praise by people who know nothing about the field or her work in it.
P.S. For the person who wanted to see her paper: she has withdrawn it from the arXiv, but due to the arXiv’s policy of openness all previous versions of papers remain available. The last revision prior to withdrawal is here:
http://arxiv.org/abs/math.DG/0609740v4
anon:
Based on what I’ve heard from people who understand her work more than I do, there *are* significant contributions in it. She may have blown the big point, which is the existence of the immortal solution, but she found a new approach to make elipticals, if I recall correctly, 3-countable.
I don’t think that Professor Smith was unprofessional. She produced an exciting result, and told a few people about it. She didn’t got trumpeting it to newspapers, holding press releases about her discovery, or anything over-the-top like that. She submitted the paper for publication, and made the pre-print available via arxiv. *We*, meaning the bloggers and net-journalists are the ones who blew it out of proportion.
And I *strongly* disagree that she shouldn’t be praised for publicly admitting her error. There are plenty of people out there who *don’t* admit to mistakes. Professor Smith listened respectfully to people who saw a problem with her proof, and gracefully admitted her mistake. That’s what a good scientist does. It doesn’t make her public hero number 1 or anything – but it is a sign of her honesty, integrity, and professionalism as a scientist and mathematician.
Nod to Mark. The only publicizing that Smith did was literally publishing her paper on arXiv. Everything else was a lot of people getting excited on their own about a solution to a Clay problem. She did not seek that attention; it came to her.
And publicly admitting failure even after something like this happened, after your result exploded on the blogosphere? That *does* take guts. It *is* praiseworthy. Look at all the douchebags that Mark takes on. Have any of them ever admitted failure and withdrawn their comments? It *does* take a good measure of integrity and dedication to science to do it right.
The most I can fault her for is your “How did I overcome what no one else could?” critique. From a casual spectator’s perspective, it does appear that not enough questioning of the result went on. However, she didn’t go crazy over it and she withdrew gracefully when it went crazy by itself, so that is tolerable.
As Wikipedia says, Be Bold.
so what’s the mistake? if she didn’t reveal that, then she’s in no way better than that Thomas Aquinas people above like to poit at.
Alexei:
She did describe what the mistake was; it’s just well
beyond the scope of this blog. From one of her comments about it over at Not Even Wrong: