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" "Draw pictures, draw pictures! As a guideline, when I do research in any area (fluid mechanics, geometry, dynamical systems, topology, combinatorics, representation theory), I always draw pictures when I'm doing research. I'm drawing one picture every few minutes, so by the end of the day, after maybe six or seven hours, I have maybe 30, 40 pictures, if not more. So in a week that's hundreds. You should be drawing lots and lots of pictures, trying lots of pictures. Some of these pictures can be in your head, but you should start by drawing lots of real pictures, on paper. But not a few pictures. Not tens of pictures. Hundreds of pictures, please, hundreds. Because that's how we can, eventually, listen to Mozart's music - in mathematics!
(Japanese: 時枝正; born 1968) is a Japanese mathematician, working in mathematical physics. He is a professor of mathematics at Stanford University; previously he was a fellow and Director of Studies of Mathematics at Trinity Hall, Cambridge. He is also very active in inventing, collecting, and studying toys that uniquely reveal and explore real-world surprises of mathematics and physics. In comparison with most mathematicians, he had an unusual path in life: he started as a painter, and then became a classical philologist, before switching to mathematics.
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[T]here are lots of things that one does which are essential, indespensable for survival and which is foundational for everything else, about which people never ask... "What's exciting about it?" What's exciting about breathing for example. ...[I]f you stop breathing, you are no longer. ...You're aware of breathing sometimes. It's not that you're completely unconcsiously invisible, but you don't ask that question. What's exciting about... living itself? Of course there are ups and downs. There are dramas in life, but people don't live because it's exciting. People live because it's natural for them and because that's what they want to do, despite everything sometimes, or in some lucky cases, because of some things. ...But people live because it's a basic and natural way of existing as humans, as indeed, biological creatures... [S]cientists, when they are unhampered and unencumbered by those dictates of sociology... where you have to publish in certain ways because you want to enhance your career, because you want to achieve some status, because you want to... ensure you have a certain standard of living and so on. If they are doing science where they do science because, almost, they have to, because that's their existence... If I lost my job... I have to be able to live somehow, but let's assume that I have some kind of income, and I have to move to and live in isolation. I think after... the initial period of being really depressed... "Why am I stuck here?" and so on, I think I'd end up doing science, because that's... who I am.
[E]very human life... is unique, especially seen from the inside. ...You might look like the boring doldrums and the standard... path to somebody else, but for each individual... that person is living only once, and unique experiences... are not... replaceable by anything else... I'm not sure that my experience is qualitatively different from other people's... [P]eople struggle through various difficulties and have moments of joy and... discovery and sometimes... get fed up and... want to leave... [T]hen sometimes they come back and so on... I don't think that it's that different. ...People should realize that their experience is unique and it's interesting, if you make it interesting. ...[I]f you decide that, "Oh, I'm a boring person..." of course you become, , a boring person... [O]ther people will not help you out. They'll say you're boring, but... you live only once, and... I'm sure there's lot's going on in your brain that the rest of the world cannot see, naturally... [Y]ou should cherish it...
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I'm fortunate enough to be friends and personally acquainted with some of the leading mathematicians of our age, but... I have never met a genius. They are all understandable. They are wonderful, wonderful people, and they really love what they are doing, and their creations... open up a whole world for you... but I don't think... I ever met a genius. ...In practical terms, it's much more useful to ...focus on other things. So that's why I'm skirting around your question on innate ability.