"The prize was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed."
He later abandoned mathematics and returned to obscurity.
It's unlikely that he's given up mathematics. Publishing, yes. Mathematics, no. If he'd finished his proof a few years later, I suspect Perelman would have followed Satoshi's lead and published as an anon. He just dumped dumped it on the arxiv, on the random.
In the documentary, Gromov seemed a little miffed at Perelman. As Gromov sees it, other mathematicians spent a lot of energy helping Perelman progress and he kind of "owes" it to the community to interact and mentor.
On the flip side, we should probably ask ourselves why someone like Perelman would rather be a recluse than participate in the community. The politics around his proof were particularly nasty, but I have to wonder if he sees deeper problems.
Proving the Poincaré conjecture wasn't enough? Why would he owe more than that? It's significantly more return on the investment of energy than anyone expected.
Of course, this whole debacle could have been avoided if Princeton had given Perelman tenure after he proved the Soul Conjecture. To an extent, I agree with Perelman’s opinions of the mathematics community, I just felt Gromov’s opinion was a bit more defensible than you were giving it credit for.
Useful for what? A lot of people see mathematics as and end unto itself.
What I mean by mathematics being an end unto itself is that mathematical results have intrinsic value. That is, they have value in and of themselves, regardless of how they may or may not be used.
Perhaps you thought I meant that mathematical results could be used to discover more mathematical results? No, that is not what I meant.
The Poincaré conjecture is interesting because it was a simple statement that ended up being very hard to prove. How someone proved it, and understanding why such an innocuous statement is so difficult to prove, is far more interesting than knowing that the obviously-true-sounding statement is true.
That's an interesting choice of language. For you to make that point at all indicates you feel it has some validity. But you didn't choose a compelling form of words to advocate your interest to other people. Why not?
Perhaps you are a mathematician, and perhaps this is some expression of the difference between maths and other endeavours.
The beauty of maths is that it exists independent of all other considerations. This seems to be where Perelman comes from. Perhaps Perelman goes further. It's hard to say without knowing more of his story, which Perelman doesn't give us.
You, on the other hand, find the human dimension of Perelman's story interesting - so much so that you dare to conjecture what he might think. Some emotional force drove you to post about his story, presumably because you desired other people to share your interest and commune with you in a way that might resonate and drive your enjoyment further. But you didn't push it. You only just barely said it at all. But you did suggest it.
I'm curious why you didn't choose a more strident form of advocacy to grab people by the lapel and drag them towards Perelman's story, where they might learn something and be entertained and enlightened.
Perhaps because advocacy isn't maths?
Perhaps because you have too much humility and respect to foist your own interests onto others?
What might Poincare make of this curiosity?
Okay, so why publish at all? Why not just sit in mathematical nirvana in a cave, having discovered the math-God?
So, I’d just like to point out that this is a very naive perspective on the philosophy of mathematics. Math is an intrinsically social activity, because you need to lead other mathematicians through your proofs/arguments.
I thought it was a good question and the wording was compelling enough.
It's the style of the remark that struck me:
> we should probably ask ourselves why
Or perhaps they are the same person?
There's a citation in the article to "Gardner, 1984 p. 9–10". But there's no footnote. Would that be the late Martin Gardner, formerly of Mathematical Recreations? He also had a talent for expressing hard subjects clearly, in plain words.
In August 2006, Perelman was offered the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo."
Some people are built differently, what a legend.
The Poincaré-Perelman theorem seems more accurate now.
Perelman proved a more general result: the geometrization conjecture that was initially stated by William Thurston: