Things add up.
Tag Archives: Poincaré’s Conjecture
December 14, 2010Posted by on
Today I read the famous article on the proof of Poincaré’s conjecture, Manifold Destiny on The New Yorker. A brilliant and colorful story and you can find an informative Wiki-page on it, the Controversy section is very interesting.
Personally, I found it quite curious why so many people got interested in this story of a battle for credit. Just as Pereman himself put it,
Everybody understood that if the proof is correct then no other recognition is needed.
However, one quote of a Russian mathematician, Burago, caught my eyes. When describing Pereman, he said something like,
He was not fast. Speed means nothing. Math doesn’t depend on speed. It is about deep.
This reminded me of one of my professors (he was once an associate professor at Berkley), who told me that he was surprised by the harsh time constraint of exams at my university. Back in Berkley, he mentioned, they always give students much-more-than-needed time. Actually, he conducted some two-hour exams on Algebra this semester and many students finished within 15 minutes.
This suggests a contrast between mathematics education in the U.S., where deep and independent thoughts are encouraged, and in China, where more emphasize is put on mastering the techniques (Well, to do things fast, you have to be quite familiar with all the well-established results and too often, you are left with no time to develop your independent thoughts or work on what you find intriguing). Personally, I have a preference for the former. After all, mathematics is not just a bunch of techniques but about imagination, which is what really distinguishes great ones and the ordinary. However, your imagination and thoughts sometimes become a burden if you have to struggle with lots of exercises.
Moreover, it is not those who solve problems that push mathematics forward but those who raise good problems (take Fermat and Poincaré as examples, whose problems actually engines the development of mathematics for centuries). So it is hard to justify the emphasize on techniques but not imagination and creative thoughts.
Maybe this also explains why many Chinese work so hard in the science field but few contributes originative ideas.