Terence Tao – Wie der weltbeste Mathematiker KI einsetzt

Today, I'm chatting with Terence Tao, who needs no introduction. Terence, I want to begin by having you retell the story of how Kepler discovered the laws of planetary motion because I think this will be a great jumping off point to talk about AI for math. I've always had an amateur interest in astronomy. I've loved stories of how the early astronomers worked out the nature of the universe. Kepler was building on the work of Copernicus, who was himself building on the work of Aristarchus. Copernicus very famously proposed the heliocentric model, that instead of the planets and the Sun going around the Earth, the Sun was at the center of the solar system and the other planets were going around the Sun. Copernicus proposed that the orbits of the planets were perfect circles. His theory fit the observations that the Greeks, the Arabs, and the Indians had worked out over centuries. Kepler learned about these theories in his studies, and he made this observation that the ratios of the size of the orbits that Copernicus predicted seemed to have some geometric meaning. He started proposing that if you take the orbit of the Earth and you enclose it in a cube, the outer sphere that encloses the cube almost perfectly matched the orbit of Mars, and so forth. There were six planets known at the time and five gaps between them, and there were five perfect Platonic solids: the cube, the tetrahedron, icosahedron, octahedron, and dodecahedron. So he had this theory, which he thought was absolutely beautiful, that you could inscribe these Platonic solids between the spheres of the planets. It seemed to fit, and it seemed to him that God's design of the planets was matching this mathematical perfection of the Platonic solids. He needed data to confirm this theory. At the time, there was only one really high-quality dataset in existence. Tycho Brahe, this very wealthy, eccentric Danish astronomer, had managed to convince the Danish government to fund this extremely expensive observatory. In fact, it was an entire island where he had taken decades of observations of all the planets, like Mars and Jupiter, at least every night for which the weather was clear, with the naked eye. He was the last of the naked-eye astronomers. He had all this data which Kepler could use to confirm his theory. Kepler started working with Tycho, but Tycho was very jealous of the data. He only gave him little bits of it at a time. Kepler eventually just stole the data. He copied it and had to have a fight with Brahe's descendants. He did get the data, and then he worked out, to his disappointment, that his beautiful theory didn't quite work. The data was off from his Platonic solid theory by 10% or something. He tried all kinds of fudges, moving the circles around, and it didn't quite work. But he worked on this problem for years and years, and eventually, he figured out how to use the data to work out the actual orbits of the planets. That was an incredibly clever, genius amount of data analysis. And then he worked out that the orbits were actually ellipses, not circles, which was shocking for him. So he worked out the two laws of planetary motion: the ellipses, and also that equal areas sweep out equal times. Then ten years later, after collecting a lot of data—the furthest planets like Saturn and Jupiter were the hardest for him to work out—he finally worked out this third law, that the time it takes for a planet to complete its orbit was proportional to some power of the distance to the Sun. These are the three famous Kepler's laws of motion. He had no explanation for them. It was all driven by experiment, and it took Newton a century later to give a theory that explained all three laws at once. The take I want to try on you is that Kepler was a high-temperature LLM. Newton comes up with this explanation of why the three laws of planetary motion must be true. Of course, the way that Kepler discovers the laws of planetary motion, or figures out the relative orbits of the different planets, is as you say a work of genius. But through his career, he's just trying random relationships. In fact, in the book in which he writes down the third law of planetary motion, it's an aside on The Harmonics of the World, which is just a book about how all these different planets have these different harmonies. And the reason there's so much famine and misery on Earth is because the Earth is mi-fa-mi, that's the note of Earth. It's all this random astrology, but in there is the cube-square law, which tells you what relationship the period has to a planet's distance from the Sun. As you were detailing, if you add that to Newton's F=ma and the equation for centripetal acceleration, you get the inverse-square law. And so Newton works that out. But the reason I think this is an interesting story is that I feel LLMs can do the kind of thing of trying random relationships for twenty years, some of which make no sense, as long as there's a verifiable data bank like Brahe's dataset. "Ok, I'm going to try out random things about musical notes, Platonic objects, or different geometries, I have this bias that there's some important thing about the geometry of these orbits." Then one thing works. As long as you can verify it, these empirical regularities can then drive actual deep scientific progress. Traditionally, when we talk about the history of science, idea generation has always been the prestige part of science. A scientific problem comes with many steps. You have to identify a problem, and then you have to identify a good, fruitful problem to work on. Then you need to collect data, figure out a strategy to analyze the data, and make a hypothesis. At this point, you need to propose a good hypothesis, and then you need to validate. Then you need to write things up and explain. There are a dozen different components. The ones we celebrate are these eureka genius moments of idea generation. Kepler certainly had to cycle through many ideas, several of which didn't work. I bet there were many that he didn't even publish at all because they just didn't fit. That's an important part of the process, trying all kinds of random things and seeing if they worked. But as you say, it has to be matched by an equal amount of verification, otherwise it's slop. We celebrate Kepler, but we should also celebrate Brahe for his assiduous data collection, which was ten times more precise than any previous observation. That extra decimal point of accuracy was essential for Kepler to get his results. He was using Euclidean geometry and the most advanced mathematics he could use at the time to match his models with the data. All aspects had to be in play: the data, the theory, and the hypothesis generation. I'm not sure nowadays that hypothesis generation is the bottleneck anymore. Science has changed in the century since. Classically, the two big paradigms for science were theory and experiment. Then in the 20th century, numerical simulation came along, so you can do computer simulations to test theories. Finally, in the late 20th century, we had big data. We had the era of data analysis. A lot of new progress is actually driven now by analyzing massive datasets first. You collect large datasets and then draw patterns from them to deduce thoughts. This is a little bit different from how science used to work, where you make a few observations or have one out-of-the-blue idea, and then collect data to test your idea. That's the classic scientific method. Now it's almost reversed. You collect big data first, and then you try to get hypotheses from it. Kepler was maybe one of the first early data scientists, but even he didn't start with Tycho's dataset and then analyze it. He had some preconceived theories first. It seems like this is less and less the way we make progress, just because the data is so much more massive and useful. Oh, interesting. I feel like the 20th-century science that you're describing actually very well describes what happened with Kepler. He did have these ideas—1595 and '96 is where he comes up with the polygons and then the Platonic objects theory—but they were wrong. Then a few years later, he gets Brahe's data, and it's only after twenty years of trying random things that he gets this empirical regularity. It actually feels a bit closer to Brahe's data being analogous to some massive data bank of simulations, and now that you've got the data, you can keep trying random things. If it wasn't for that, Kepler would be out there just writing books about harmonics and Platonic objects, and there would be nothing to actually verify against. The data was extremely important. The distinction I was trying to make was that traditionally, you make a hypothesis and then you test it against data. But now with machine learning, data analysis, and statistics, you can start with data and through statistics work out laws that were not present before. Kepler's third law is a little bit like this, except that instead of having the thousand data points that Brahe had, Kepler had six data points. For every planet, he knew the length of the orbit and the distance to the Sun. There were five or six data points, and he did what we would now call regression. He fit a curve to these six data points and got a square-cube law, which was amazing. But he was quite lucky that these six data points gave him the right conclusion. That's not enough data to be really reliable. There was a later astronomer, Johann Bode, who took the same data—the distances to the planets—and inspired by Kepler, he had a prediction that the distances to the planets formed a shifted geometric progression. He also fit a curve, except there was one point missing. There was a big gap between Mars and Jupiter. His law predicted that there was a missing planet. It was kind of a crank theory, except when Uranus was discovered by Herschel, the distance to Uranus fit exactly this pattern. Then Ceres was discovered in the asteroid belt, and it also fit the pattern. People got really excited that Bode had discovered this amazing new law of nature.