Seeing What One Sees
I was reading an article* in the New York Times in which the author is trying to describe the excitement among mathematicians over the apparent proof of the "Poincaré conjecture"# by Russian mathematician Grigory Perelman. There's human interst and mystery, too, since there's a million-dollar award on offer to the one whose proof sustains three years of professional scrutiny, but Dr. Perelman has not been seen since initially publicizing his proof. If I had more time I'd use it as a plot to a crime novel.
Anyway, in the midst of this article came a stunning statement by a mathematician named William Thurston of Cornell (for those familiar with my biography, that would be the other Cornell). Since I am an admirer of the movie "Buckaroo Banzai: Across the 8th Dimension", I felt that I was particularly receptive to Dr. Thurston's message:
“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”
Despite my having a bit of fun with Dr. Thurston's manifestly true observation, I'll say that the article I've referred to is actually rather informative, giving us the human-interest story about the elusive Dr. Perelman but also getting across some of the flavor of the mathematical work and community without doing violence to the accuracy of the ideas. Now, that's a rare treat.
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*Dennis Overbye, "Elusive Proof, Elusive Prover: A New Mathematical Mystery", New York Times, 15 August 2006.
#I'm sorry to say that I know nothing more about the Poincaré conjecture than I get from these NYT articles — which isn't much, except that it seems to be a result that is now considered part of the mathematical subfield of topology. However, Mr. Overbye gives it a go with the following, which sounds quite sensible and probably captures the essence rather nicely (and he does expand some on this later in the same piece):
Depending on who is talking, Poincaré’s conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.
The real trick here, applying Mr. Thurston's understanding of understanding, is to see that this rather simple idea is of fundamental import, and then to see that that notion implies that it could be quite difficult to prove it mathematically even though it seems a rather straightforward idea.
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on Wednesday, 23 August 2006 at 19.41
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I'm the antithesis of mathematician. So, it probably comes as no surprise that I'm left behind by the notion that reducing the loop to a point does anything to the space the loop resides in. Well, anything besides leave the space with a point somewhere in or on it. Alas, the whole thing evokes no intrigue in me.
However, after reading Thurston's statement above, I think it would be hilarious to sit him and Don Rumsfeld down for a conversation.
Thurston could hold forth on when you see what you see, etc. Rumsfeld could weigh in about not knowing what you don't know, etc.
The trick, of course, would be to keep Rumsfeld from completely ignoring Thurston in favor of asking himself questions and then answering his own questions.