I am a professional mathematician and number theorist, who has studied work closely related to Zhang's.
I had never heard before of an older (40+) mathematician, who has done essentially no meaningful work in the subject before, and has virtually no publication record, coming out of seemingly nowhere and proving such a big theorem.
This is in no way a slight against older mathematicians, indeed many spectacular results are proved by people over 40, but they typically accumulate an excellent track record on the way.
Indeed, I fully assumed that this guy was full of shit. However, I have heard that well-known experts in the area have closely read Zhang's paper and found it to be correct.
This is enough to put a smile on my face. There is something wonderful about skepticism and cynicism being proven wrong, especially when the skepticism is my own.
The most dramatic example I know of someone who is older and also not a well known who found a major result is Kurt Heegner. He was a 59 year old private tutor and radio engineer with no previous published results when he proved the Class Number 1 problem. His proof was initially rejected by the mathematical establishment and he died before learning that the proof was a major advance and essentially correct.
> This is enough to put a smile on my face. There is something wonderful about skepticism and cynicism being proven wrong, especially when the skepticism is my own.
Mathematicians have always amused me with their weird way of turning their craft upon themselves. There's an astonishing amount of work done (I assume by procrastinating PhD candidates) on the statistics of performance in mathematics. Mathematics is supposed to be such a "pure" science, but this work seems to be motivated by insecurity and a base of other negative emotions. The skepticism it breeds isn't at all useful or beneficial for the field; if anything it's destructive, but yet it seems to persist.
Malcolm Gladwell wrote a great piece for The New Yorker  on late vs early bloomers in the art world. Gladwell writes that early bloomers are often driven by a sort of internal energy, and since they haven't taken time to refine their process they tend to be more abstract or conceptual. Late bloomers, he suggests, tend to take years or decades honing their craft. They're extreme perfectionists who, instead of working on building a piece, tend to work on refining the skills they need to build a piece.
In Gladwell's model for genius, Zhang is obviously the latter. I think as a society we'd benefit from celebrating the successes (and even the failures) of "late-bloomers" like Zhang a lot more. Maybe it would promote the kinds of intrinsic motivation which would encourage more brilliant people to continue their struggle.
All of that said, the thing that brings a smile to my face is the fact that Zhang didn't come from one of the MITs, Harvards, Stanfords, or Cornells of the world. He came from a "lowly" state school. Here's hoping he stays there.
> I think as a society we'd benefit from celebrating the successes (and even the failures) of "late-bloomers" like Zhang a lot more.
What do you mean? It seems like Zhang's success is being fully recognized and celebrated, by mathematicians and the public alike. Do you think it deserves still more press than it is getting?
> Mathematicians have always amused me with their weird way of turning their craft upon themselves
> this work seems to be motivated by insecurity and a base of other negative emotions.
On what basis do you make these assertions? How many mathematicians do you personally know?
Mathematicians are overall a positive and cheerful group who are positive, welcoming, and as optimistic as circumstances allow. It's one reason I love working in this subject. Our portrayal in the media and Hollywood is, I think, a bit misleading.
When the editors of the Annals of Mathematics received Zhang's manuscript, from someone in his fifties and entirely unknown, what was their response? To forward the paper to experts, and ask for an evaluation. And, indeed, the paper was evaluated on its merits.
> The skepticism it breeds isn't at all useful or beneficial for the field; if anything it's destructive, but yet it seems to persist.
Since I don't understand what you're talking about, could I please ask you to translate your criticism into advice, which I might then consider following?
Ah, I did not realize he had not really published before. I suppose that is very surprising. Also, the way your post is written seems to suggest that most results out of nowhere are "crankish". Are there really people who spend their time working on these things that aren't professional mathematicians and who submit flawed papers to math journals? I would think that would be an unproductive use of one's time...
Consider that a lot of problems seems "obvious" to beginners who does not know the full literature on the subject, pretty much regardless of subject.
I studied computer science, and during our data structures and algorithms course must have "discovered" a dozen or more algorithms that I'd then find treated or discredited a chapter or two later in our course material.
Sometimes someone out of nowhere benefits from not having been explained why a specific idea "can't" work, but much more often they end up wasting their time on it. And some of them go on to think they have valid, significant results.
E.g., there are regularly people that are sure they have solutions to problems that are reducible to solving the halting problem, but that don't have the theoretical background to realize it is reducible to the halting problem or why that means their "solution" doesn't work.
> E.g., there are regularly people that are sure they have
> solutions to problems that are reducible to solving the
> halting problem, but that don't have the theoretical
> background to realize it is reducible to the halting
> problem or why that means their "solution" doesn't work.
I think you might have gotten this the wrong way: All decidable problems are trivially reducible to the halting problem (just decide them and then map to a halting/non-halting Turing machine). On the other hand, if you can reduce the halting problem to a problem, it means it's undecidable.
I would be curious though which "natural" undecidable problems people claim to solve regularly.
You misunderstand what I wrote. I should perhaps have been more precise, as I can see that "solutions to problems" might be misunderstood. "Problems" in this context was referring to what people might have tried to prove, not the process they were trying to apply it to. The point being that there are plenty of processes where to prove specific outcomes is a problem that can be reformulated as finding a generic algorithm for solving the halting problem.
The most common variations are simply obfuscated variations where it is not clear that a sub-part of the process is turing complete, and where that sub-part can determine whether or not a specific state is reached.
Quite a lot of software have subsystems that turn out to be Turing complete, whether through the explicit inclusion of scripting support, or through more convoluted means.
I'm not sure, but I think I did understand what you were saying. I was genuinely interested in undecidable problems people might run into and that are easily seen to be at least as powerful as the halting problem. Thanks for the example!
> Are there really people who spend their time working on these things that aren't professional mathematicians and who submit flawed papers to math journals?
Yes, as well as people who are professional mathematicians (usually not well known) who also do so.
There is not a flood of such papers, but my thesis advisor, in his capacity as editor of the Proceedings of the American Mathematical Society, gets at least a dozen or so such papers each year. I refereed some of them, and had to explain to these authors why their proofs were mistaken.
Attempted proofs of Fermat's Last theorem by cranks used to be so common that Edmund Landau, a German mathematician, had a form letter for them: “Dear Sir/Madam: Your proof of Fermat’s Last Theorem has been received. The first mistake is on page _____, line _____.”
I have sometimes seen notes on the pages of prominent mathematicians that they don't have time to examine unsolicited attempts at major problems by amateurs so there must still be a good number. I also have known someone whose sibling is a software engineer and whose hobby is trying to resolve P ? NP.
I heard a talk by one of the main vetters of the P?NP problem (which itself was underwhelming; clearly presentations were not his strong suit) and he said there were vast numbers of attempted proofs, only a small fraction of which are even treated seriously, and none of which are considered for long enough to be considered "close" to complete. He seemed to believe that the only significant hope for the problem was in the hands of Ketan Mulmuley at UChicago (http://www.cs.uchicago.edu/people/mulmuley), but realistically a solution would not be found in our lifetimes.
This is not only interesting for that specific problem, but it was revealing in how many amateurs (including those with doctorates) who are completely over their head at the edge of our knowledge—the people who understand it well enough to evaluate an approach adequately don't submit because they understand the difficulty in ways most don't.
Back in the 19th century, non-Euclidean geometry was considered the work of the devil and many amateurs and cranks attempted to "prove" the parallel postulate (which had already been proven to be independent of the other postulates).