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

Okay. Today, I'm chatting with Terence Tao, who needs no introduction. Terence, I wanna 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.

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