"He was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. I tried to imitate him. But I've realized that it's very difficult to make good mistakes."
- Goro Shimura's profound description of how Yutaka Taniyama approached mathematics.
I never understood the amount of creativity and ingenuity that goes into good math until I learned some number theory and abstract algebra back in my college days, and read about the many math greats throughout history.
Thanks for the link to the documentary. This was well worth the time.
There's a striking moment at about 41:00 where Wiles tearfully describes his ultimate breakthrough. Mathematicians and others in the hard sciences are often perceived as cold and joyless, but this is a remarkable counterexample.
Exactly. Proving the theorem was his dream and kick started his passion for maths. This is a man who has achieved his childhood dream. That's no different from a singer who's achieved their childhood dream.
For many, I imagine maths is cold and joyless so they associate that with those who enjoy it. Nobody can see the internal experience, which probably is just as warm and joyful as the singer or writer who achieved their dreams.
I had the great fortune of being shown this documentary during highschool calculus class. Even though I didn't understand the details at all, it was my first glimpse into what "real math" (so to speak) was about.
https://vimeo.com/18216532
It even gives a high level description of the methods used for those who are interested.