Terence Tao – Kepler, Newton, and the true nature of mathematical discovery

**Dwarkesh Patel** (0:00)
Okay, today, I'm chatting with Terence Tao, who needs an 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.

**Terence Tao** (0:15)
Okay. Yeah. So I've always had an amateur interest in astronomy, and so I've loved stories of how the early astronomers worked out the nature of the universe. So Kepler was building on the work of Copernicus, who was himself building on the work of Aristarchus. So Copernicus very famously proposed the heliocentric model that instead of the planets and the sun going around the earth, that the sun was at the center of the solar system and the other planets were going around the sun. And Copernicus proposed that the orbits of the planets were perfect circles. And his theory kind of fit the observations that the Greeks and the Arabs and the Indians had worked out over centuries.
I think, Kepler got interested, he 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. I think he started proposing that, you know, if you take, say, the orbit of, say, the earth and you enclose it in, I think, maybe a cube, the outer sphere that encloses the cube almost match perfectly the orbit of Mars. And so forth. And there were six planets known at the time, five gaps between them, and there were five perfect platonic solids, the cube, the tetrahedron, the isochrydron, octahedron and dodecahedron. And so he had this theory which he thought was absolutely beautiful, that he could inscribe these platonic solids between the spheres of the planets. And it seemed to fit, and it seemed to be to him like, you know, God's design of the planets was matching this mathematical perfection of the platonic solids. So he needed data to confirm this theory. And at the time, there was only one really high quality data set almost in existence, which was the... So Taika Brahe, this Danish astronomer, very wealthy, eccentric astronomer, had managed to convince the Danish government to fund this extremely expensive observatory, this, in fact, an entire island, where he had taken decades of observations of all the planets, Mars, Jupiter, every night, at least every night for which the weather was clear, with the naked eye, actually. He was the last of the naked-eye astronomers. And so he had all this data which Kepler could use to confirm his theory. And so Kepler started working with Taika, but Taika was very jealous of the data. He only gave little bits of it at a time. And I think Kepler eventually just stole the data. He copied it and had to have a fight with Brahe's descendants. But he did work out, he did get the data, and then he worked out to kind of his disappointment that his beautiful theory didn't quite work. Like the data was sort of off from his platonic solid theory by about 10% or something. And he tried all kinds of fudges, moving the circles around and things. 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. And that was an incredibly clever, genius amount of data analysis. And then he eventually worked out that the ellipses, not circles, which was shocking to him. And then he worked out the two laws of planetary motion, the ellipses, also equal areas, sweep out equal times. And then 10 years later, after collecting a lot of data, the furthest planets like Saturn and Jupiter were the hardest for him to work out. But then he finally worked out this third law also, that the orbits, the time it takes for a planet to complete its orbit was proportional to some power of the distance to the sun. And these are the three famous laws of motion.
And he had no explanation for them. It was just all driven by experiment. And it took Newton a century later to give a theory that explained all three laws at once.

**Dwarkesh Patel** (4:09)
The take I want to try on you is that Kepler was a high temperature alum.
Where Newton comes up with this explanation of why the three laws of planetary motion must be true. And 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 then, through his career, he's just trying random relationships. And in fact, in the book in which he writes down the third law of planetary motion, it's sort of on the side, on the harmonics of the world, which is this book about 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 mipha mi, that's the note of Earth. And so 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, which is, as you're detailing, if you add that to Newton's F equals ma and then 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 I feel like LLMs can do the kind of thing of like 20 years, let's try random relationships, some of which make no sense. As long as there's a verifiable data bank like Brahe's data set where, okay, I'm going to try out random things about like musical notes, I'm going to try out random things about platonic objects, I'm going to all these different geometries have this bias that if there's some important thing about the geometry of these orbits and then one thing works. And as long as you can verify it, it can then draw these empirical regularities can then drive actual deep scientific progress.