数学

"Some recognition is necessary and important, but yeah, it’s also important to not let these things take over your life and only be concerned about getting the next big award or whatever. So again, you see these people try to only solve really big math problems and not work on things that are less sexy, if you wish, but actually still interesting and instructive. As you say, the way the human mind works, we understand things better when they’re attached to humans, and also if they’re attached to a small number of humans. The way our human mind is wired, we can comprehend the relationships between 10 or 20 people. But once you get beyond like 100 people, there’s a limit, I think there’s a name for it, beyond which it just becomes the other."

"Yeah, that’s another difficult question. I suppose it has to do with… My mathematical style, my style as a mathematician, is that I don’t really like difficult mathematics. What I love is simple, clear, easy-to-understand arguments that prove a surprising result. That’s my favorite situation. And actually, the question of whether it’s a new result or not is somehow less important to me. And so that has to do with this question of the greats and so on, whoever does it first. Because I think, for example, if you prove a new result with a bad argument or a complicated argument, that’s great because you proved something new. But I still want to see the beautiful, simple, because that’s what I can understand."

"Well, the mathematical Platonic realm is… I’m not sure I would say it’s more real, but I’m saying we understand the reality of it in a much deeper and more— …a more convincing way. I don’t think we understand the nature of physical reality very well at all, and I think most people aren’t even scratching the surface of the question as I intend to be asking it. So, you know, obviously we understand physical reality. I mean, I knock on the table— …and so on, and we know all about what it’s like to, you know, have a birthday party or to drink a martini or whatever. And so we have a deep understanding of existing in the physical world. But maybe understanding is the wrong word. We have an experience of living in the world—"

"And so for small values of inputs, then it doesn’t make sense to talk about this polynomial time feasibility with respect to, say, the range of problem inputs that we will ever give it in our lifetime or in the span of human civilization or whatever. I mean, because it’s an asymptotic property, it’s really in the limit as the size of the inputs goes to infinity, that’s the only time that polynomial or NP becomes relevant. And so maybe it’s important to keep that in mind when… Sometimes you find kind of overblown remarks made about, you know, if P equals NP, then this will be incredibly important for human civilization because it means that we’ll have feasible algorithms for solving these incredibly important…"

"Yeah, I think so, because I guess part of the point of structuralism is that it doesn’t make sense to consider mathematical objects or individuals in isolation. What’s interesting and important about mathematical objects is how they interact with each other and how they behave in a system, and so maybe one wants to think about the structural role that the objects play in a larger system, a larger structure. There’s a famous question that Frege had asked actually when he was looking into the nature of numbers, because in his logicist program, he was trying to reduce all of mathematics to logic."