Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries | Lex Fridman Podcast #190

The following is a conversation with Jordan Ellenberg, a mathematician at University of Wisconsin and an author who masterfully reveals the beauty and power of mathematics in his 2014 book How Not To Be Wrong, and his new book just released recently called SHAPE: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. Quick mention of our sponsors: Secret Sauce, ExpressVPN, Blinkist, and Indeed. Check them out in the description to support this podcast. As a side note, let me say that geometry is what made me fall in love with mathematics when I was young. It first showed me that something definitive could be stated about this world through intuitive, visual proofs. Somehow that convinced me that math is not just abstract numbers devoid of life, but a part of life, part of this world, part of our search for meaning. This is the Lex Fridman podcast, and here is my conversation with Jordan Ellenberg. If the brain is a cake-

It is? (laughs)

Okay. (laughs) Well, let's just, let's just go with me on this, okay?

Okay. We'll pause it.

So for Noam Chomsky, um, language, the universal grammar, the framework from which language springs i- is like most of the cake, the delicious chocolate center, and then the rest of cognition that we think of is built on top, extra layers, maybe the icing on the cake, maybe just, um, maybe consciousness is just like a cherry on top. Uh, f- where do you put in this cake, uh, mathematical thinking? Is it as fundamental as language in the Chomsky view? Is it more fundamental than language? Is it echoes of the same kind of abstract framework does he- he's thinking about in terms of language that they're all, like, really tightly interconnected?

That's a really interesting question. You're getting me to reflect on this question of whether the feeling of producing mathematical output, if you want, is like the process of, you know, uttering language or producing linguistic output. I think it feels something like that, and it's certainly the case... Let me put it this way. It's hard to imagine doing mathematics in a completely non-linguistic way. It's hard to imagine doing mathematics without talking about mathematics and sort of thinking in propositions. But, you know, maybe it's just because that's the way I do mathematics (laughs) and maybe I can't imagine it any other way, right? It's a...

Well, what about visualizing shapes, visualizing concepts to which language is not obviously attachable?

Ah, that's a really interesting question. And, you know, one thing it reminds me of is one thing I talk about, uh, in the book, is dissection proofs, these very beautiful proofs of geometric propositions. Um, there's a very famous one by Bhaskara of the, the Pythagorean theorem. Um, proofs which are purely visual, proofs where you show that, uh, two quantities are the same by taking the same pieces and putting them together one way, uh, and making one shape, and putting them together another way and making a different shape, and then observing those two shapes must have the same area because they were built out of the same pieces. Um, you know, there's a, there's a famous story, and it's a little bit disputed about how accurate this is, but that in Bhaskara's manuscript he sort of gives this proof, just gives the diagram, and then the, uh, the entire, uh, verbal content of the proof is he just writes under it, "Behold."