How Grigori Perelman solved one of math's greatest mysteries
medium.com
medium.com
The key quote can be found in the New Yorker article
http://www.newyorker.com/magazine/2006/08/28/manifold-destin...
(someone else already posted it):
As for Yau, Perelman said, “I can’t say I’m outraged.
Other people do worse. Of course, there are many
mathematicians who are more or less honest. But
almost all of them are conformists. They are more
or less honest, but they tolerate those who are not
honest.”
It's interesting to see the political implications behind breakthroughs like this.
In Soviet Russia, the fat of the land lives off you.
The man actually lives within a walk distance from me (or used to at the moment of hype), you can both pay for your water / electricity / etc, and not starve by any means with a monthly income of $600, roughly.
From the newyorker article linked by deuill:
The prospect of being awarded a Fields Medal had forced
him to make a complete break with his profession. “As
long as I was not conspicuous, I had a choice,” Perelman
explained. “Either to make some ugly thing”—a fuss about
the math community’s lack of integrity—“or, if I didn’t do
this kind of thing, to be treated as a pet. Now, when I
become a very conspicuous person, I cannot stay a pet and
say nothing. That is why I had to quit.” We asked
Perelman whether, by refusing the Fields and withdrawing
from his profession, he was eliminating any possibility
of influencing the discipline. “I am not a politician!”
he replied, angrily.
It should be "Steklov Institute" because there's no х to be found in "Стеклов".
Specifically: the apostrophe doesn't mean "LOOK OUT HERE COMES AN `S'!".
As we know, the process of submitting to a scientific journal has, besides the diffusion of one’s results to the community, the aim of verifying those results. Here, such an approach was made impossible by Perelman, so some independent groups of scholars set at the highly difficult task to understand, complete, verify, and explain his work.
Peer review does not "verify results"; peer review is there to make sure there are no serious and obvious flaws. Duplication of studies and collection of additional data / use of other techniques is what verifies results.
It is possible Perelman's papers received a more rigorous review because they were not peer reviewed – giving people incentive to dig into the details, perhaps more than they would have if the papers had appeared in a journal. But, given the signficance of the problem he was attacking, I suspect the papers not being in a peer-reviewed journal made little difference, in terms of how much effort was expended to check his proofs.
What you're saying is true of Science than Math. There's a fundamental difference between Math research and Science research. Math research doesn't involve hypothesis and verification through experimentation. Perelman's paper is purely logical it starts with axioms and derives its conclusions from them. For research like that peer review is actually where you verify results.
> Mathematicians disagree about the amount of detail checking that has to be done by the referees. While some (few) mathematicians think that checking the correctness of the proofs is the main task of the referee, others disagree with this and consider mathematical correctness the problem of the author rather than that of the referee.
It later quotes an editor:
> There are situations where almost nothing needs be checked (e.g., the results come from a seminar where the results were checked, or I see the paper is not too good and then it is useless to check details, or the author is well-known and it is his concern to submit a correct paper). There are situations when I insist to check all the procedures (e.g., when it concerns good results from a less known author).
See how it's assumed that the author's reputation is used as a proxy for the quality of the paper?
It also quotes an opinion piece by Nathanson:
> Many great and important theorems don't actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community. But the community itself is tiny. In most fields of mathematics there are few experts. [. . .] In every field, there are `bosses' who proclaim the correctness or incorrectness of a new result, and its importance or unimportance. Sometimes they disagree, like gang leaders fighting over turf. In any case, there is a web of semi-proved theorems throughout mathematics.
and it reports questionnaire results sent to math journal editors asking if they require verification of all of the proof, or only partial verification: "six editors thought that the referee should check all proofs in detail; five thought that the referee should check some proofs in detail", and one of he six actually commented "but to be reasonable, I am happy when I find a referee doing [the latter]."
These comments seem to contradict the statement that "peer review is actually where you verify results".
Also, not all non-math/science papers involve "verification through experimentation." It's hard to verify through experiment a report of the neutrino interactions observed from supernova 1987A.
My comments on his papers potentially receiving more peer review was a distinct commment, which does not rely on my qubble with the author's definition of peer review for a "scientific journal". I should have made that more clear in my original comment.