This is Part 8 of an ongoing series on mathematical culture.
I think that my pure math research is bottlenecked by the lack of dynamic media I need to invent in order to broaden my understanding of particular mathematical structures.
Here are some quotes and rough notes:
From Mathematica (David Bessis):
Grothendieck did it differently. He knew that it was worthless to gather information about things that you can’t yet see. Instead, he allowed himself to imagine the things right away, without waiting, even when he was well aware that it might not work and his mental images would likely be terribly wrong. He wasn’t afraid of failure. He was even certain that he would be wrong, and that’s exactly what he was looking for.
Interrogating things, listening to the voice of things, means trying to imagine them, examining the mental images that form within you, seeking to solidify these images and make them clearer, working at unveiling more and more details, as when you try to recall a dream.
Students generally don’t feel like they have the right to imagine mathematical objects that they don’t yet understand. They feel they need to know more before daring to picture them. In the meantime, they’re content to try to decipher word by word, symbol by symbol. They may not understand what they’re reading, it might give them a headache, but they tell themselves that if they keep on trying they’ll get to the point where they feel confident enough to finally imagine what’s behind the words. But this approach almost never works.
From A Tenacious Explorer of Abstract Surfaces:
“[Maryam Mirzakhani] had a sort of daring imagination,” recalled McMullen, a 1998 Fields medalist. “She would formulate in her mind an imaginary picture of what must be going on, then come to my office and describe it. At the end, she would turn to me and say, ‘Is it right?’ I was always very flattered that she thought I would know.”
From a conversation with David Deutsch:
So brilliant engineers, brilliant racing drivers, brilliant physicists, all have in common they're kind of very different personalities, very different in the way they lay their lives into the field. But they have in common that they are fearless about conjecturing a novelty.
It is difficult to understand Universal Hyperbolic Geometry without exploring it through semi-dynamic pictures. Watching lectures is insufficient.
This is sort of a continuation of my previous essay How Mathematical Frames Emerge.