#190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries
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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, in 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. As usual, I'll do a few minutes of ads now. I try to make these interesting, but I give you timestamps, so if you skip, please still check out the sponsors by clicking the links in the description. It's the best way to support this podcast. I don't do ads in the middle. I think for me at least, they get in the way of the conversation. I'm fortunate to be able to be very selective with the sponsors we take on, so hopefully if you buy their stuff, you'll find value in it just as I have. This show is sponsored by Wandery's series called Secret Sauce, hosted by John Fry and Sam Donner, where they explore the stories and successes behind some of the most inspiring businesses, creative innovators and intrepid entrepreneurs. And at the top of the list is Johnny Ive. Probably one of my favorite humans ever. The intricate, the fascinating push and pull, the complementary relationship between Johnny Ive and Steve Jobs created some of the most, I would say, amazing products in the history of human civilization. The gentleness of Johnny and then the harshness and the brutal drive of Steve Jobs. I think those two things combined beautifully. The artistry and the pragmatism created a fascinating dance of genius. And secret sauce covers just this relationship. Listen to Secret Sauce on Apple Podcasts, Amazon Music, or you can listen one week early and ad free by joining Wandery plus in the Wandery app. The tagline is wandery. Feel the story. This show is also sponsored by ExpressVPN. They protect your privacy and earn your and my trust by doing a bunch of things like using a trusted server that makes it impossible for them to store your data. I think companies that operate, at least in part online have a responsibility to be stewards of your data. 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There's a lot of amazing books on there, like Sapiens and Homo Deus by Yuval Noah Harari. So I read both of these books in their entirety, but I went to Blinkist before I read them and after before to see if I want to read them, and after to review some of the main ideas. I think that's a great way to use Blinkist is basically first to decide whether you want to read the book, and second to review the book. Also, it's a great way to get a sense of the key ideas in the book if you just don't have the time to read that particular book. We only have a limited time on this earth, but there's a bunch of interesting books that people discuss, so you at least want to get a sense of the key ideas in the book in order to participate in the conversation. Go to blinkist.com Lex to start your free seven day trial and get 25% off a Blinkist premium membership. That's blinkist.com Flex spelled Blinkist blinkist.com Flex this episode is brought to you by, indeed a hiring website. I've used them as part of many hiring efforts I've done in the past for the teams I've led. They have tools like indeed instant match that gives you quality candidates whose resumes indeed fit your job description immediately. I think all of the stages in the hiring process are difficult. The first one, when you have a giant pool of people and you want to narrow it down to a set of strong potential candidates, that's really difficult. The next stage is doing the initial interviewing to narrow down the field of candidates, all of whom are pretty good but you're looking for fit. And then maybe finally, is to grill the ones that are left to figure out whether they're going to be great members of the team. They're going to stand up to the pressure. They have the right level of passion, whether they align with your vision. They have that kind of fire in their eyes that would make you excited to show up to work every single day. So all of those are difficult, I think, indeed really helps with that initial stage of getting a good set of candidates and narrowing down that set of candidates. But then one on one interviewing, that's a whole nother ballgame. That's an art form, and that's on you or on me if I'm hiring anyway right now, get a free $75 sponsored job credit to upgrade your job post@indeed.com. Lex. Get it@indeed.com. Lex. Terms and conditions apply. Offer valid through June 30. Indeed, Lex. This is the Lex Friedman podcast, and here is my conversation with Jordan Ellenberg. If the brain is a cake, it is. Let's just go with me on this.
Okay, we'll pause it.
So for Noam Chomsky, language, the universal grammar, the framework from which language springs, 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 maybe, consciousness is just like a cherry on top. Where do you put in this cake mathematical thinking? Is it as fundamental as language, in the Chomsky view? Is it more fundamental in language? Is it echoes of the same kind of abstract framework that he's thinking about in terms of language, that they're all 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 uttering language, of 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 and propositions. But maybe it's just because that's the way I do mathematics, and maybe I can't imagine it any other way. Right.
Well, what about visualizing shapes, visualizing concepts to which language is not obviously attachable?
That's a really interesting question. And one thing it reminds me of is one thing I talk about in the book is dissection proofs, these very beautiful proofs of geometric propositions. There's a very famous one by Bhaskara of the pythagorean theorem proofs, which are purely visual proofs, where you show that two quantities are the same by taking the same pieces and putting them together one way and making one shape and putting them together another way and making a different shape, and then observing that those two shapes must have the same area because they were built out of the same. Know, there's a famous story, and it's a little bit disputed about how accurate this is, but that in Bascara's manuscript, he sort of gives his proof, just gives the diagram, and then the entire verbal content of the proof is. He just writes under it. There's some dispute about exactly how accurate that is. So then there's an interesting question. If your proof is a diagram, if your proof is a picture, or even if your proof is like a movie of the same pieces, like coming together in two different formations to make two different things. Is that language? I'm not sure I have a good answer. What do you think?
I think it is. I think the process of manipulating the visual elements is the same as the process of manipulating the elements of language. And I think probably the manipulating, the aggregation, the stitching stuff together is the important part. It's not the actual specific elements. It's more like, to me, language is a process, and math is a process. It's not just specific symbols, it's inaction.
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