与陶哲轩的精彩对话

Here is the audio, video, and transcript. Here is part of the episode summary: Tyler opens by asking Terence what it feels like to contemplate Euler’s identity, quantum mechanics, the Riemann hypothesis, and Gödel’s incompleteness theorem, before moving into why mathematical history gets stripped out of classrooms, whether abstract math is a lottery ticket or a unified whole, string theory’s beauty versus its lack of confirmation, why real analysis arrived so late, why zero took until the 7th century to be invented, game theory, the Black-Scholes model, Bostrom’s simulation argument, and Newcomb’s paradox, math education and the case for teaching more statistics, Tao’s view that AI has pushed mathematics from an era of “proof scarcity” into “proof abundance,” what that means for the research paper, why video talks may become more valuable than papers themselves, Tao’s favorite Hong Kong action movies, his unconventional path through an Australian university at age 13, his most unusual work habit, his verdict on the Collatz conjecture, and more. Excerpt: COWEN: When you look at mathematical economics over the last two decades, is it interesting to you as math, or is it just a bunch of people doing something different that, to you, is low level? TAO: Oh, yes, I was always fascinated by economics. I love game theory, and actually often there are pure math problems where it’s actually good to think in game theory terms. In analysis, you have all these complicated statements, like for every ε, there exists a δ, blah, blah, blah. It looks very intimidating, but if you phrase it as a game, your opponent is giving you an ε, you need to counter that ε with a δ of your own. If you view it as a game, then actually you can talk about strategies and who’s ahead, who’s behind. It actually activates really good modes of thinking. I’ve used economic thinking in my own work. We talk about cost functions and minimizing loss. There are lots of concepts. Ricardo’s law of comparative advantage, I use that to run my life. There are so many things I can do. What do I choose to spend my time on? It’s actually a very useful principle: comparative advantage. Economics, I find it’s a very good arena to understand the difference between models and reality. Physics, we’re spoiled because we’ve had these great physicists who’ve created these very accurate models of the real world, classical mechanics, quantum mechanics, and they’re very well tested and very accurate. You can just go to town exploring their consequences and still predict reality. Economics is a much more mixed situation where you have models, and sometimes they predict the real world very well, but sometimes they don’t. In addition to doing the math, you also have to do the modeling, and you have to understand the limits of when a model starts breaking down and what you need to do to adjust it. I find that fascinating. It’s somewhere between the hard sciences and the social sciences. There’s enough hard models that you can do math, but there’s also enough real-world messiness that you have to also grapple with the subjectivity. As an outsider, I find it a really fascinating subject. COWEN: When you looked at the Black-Scholes model, were you shocked when the probability doesn’t matter? Was that beautiful to you? TAO: Yes, yes, yes. There is a special feature of Gaussian fluctuations that you can price options just by knowing risk. There is one algebraic miracle that makes this work. Normally, predicting future costs of anything is just an intractable problem, and there just happens to be this miracle. If you have Gaussian fluctuations, then it is beautiful. This is actually a really good example of when you need to know when the models work and when they don’t work. If you don’t have Gaussian fluctuations, then actually Black-Scholes is somewhat dangerous to use, especially if you compound it into derivatives over and over again. Again, that’s an old story. You know what happened in 2009. Self-recommending, and of interest throughout. There is some on AI, but he has spoken so much on that topic lately I decided not to emphasize it. And Terry does have a new book out, namely Six Math Essentials. The post My excellent Conversation with Terence Tao appeared first on Marginal REVOLUTION.