Inspirational interview by Michael Atiyah, winner of both Fields Medal and Abel Prize, currently age 86!
Excerpt from the interview:
Is there one big question that has always guided you?
I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion.
People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.
Interesting comment that “A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works.”. Sometimes, intuitive understanding is needed, along with formal proof.
One example in high school mathematics is proving . It is possible to prove it by induction without actually understanding how the formula comes about!