Mathematician explains how geometry helps us understand the world

*Originally published on May 11, 2022.

Give Jordan Ellenberg a number and he'll tell you its square root. It's a party trick he used to do in college. 

The mathematician is a child prodigy. He scored a perfect 800 on the math portion of his college entrance exams — the SAT's — at the age of 12. And now Ellenberg is a Guggenheim Fellow, a Harvard graduate, and a professor of mathematics at the University of Wisconsin-Madison. 

He's also the author of Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. It's not a book about the kind of geometry some people feel intimidated by. It's about geometry as a way of thinking. As a method of reasoning and argument. And as a system for making sense of the world.

Ellenberg spoke to IDEAS host Nahlah Ayed about how geometry is embedded in our daily life.

NA: You describe geometry not just as a branch of math, but as an actual way of thinking. What does that way of thinking entail? 

JE: I think people don't appreciate how much it's woven into our ordinary way of thinking that we do when we're not in the classroom working on problems with a pencil and paper. But to make that claim, I suppose I should say that my interpretation of what counts as geometry is very broad. 

You might not be surprised to hear. I mean, the word literally means the measurement of the world, right? Ok, I'm not going to try to speak Greek to you guys, but like geos and metros, earth and measure. And I would say whenever we talk about distance, whenever we talk about two things being close to each other, like a close relative. If you think about your family tree, when you talk about a close relative versus a distant relative, that's a geometric way of thinking about your family. 

When you describe it as a tree, when you draw a picture you're drawing a fundamentally geometric diagram that takes these familial relationships you have and geometries them —  sort of makes a picture a shape that your family forms.

I often say that geometry is the cilantro of math. People are not neutral about it.- Jordan Ellenberg

When you talk about something like two sides of an argument, you're sort of imposing a geometric metaphor. And I actually think being conscious of that is very useful because often you say to yourself, 'Wait, why does it have two sides? Like lots of figures have different numbers of sides, not just to it. If you sort of interrogate these metaphors a little bit and think more seriously about what they say I think that can be a very valuable practice. 

Walk me through that utility — help me as an interviewer. If I'm talking about a two-way argument, why is it useful to think of it as a geometric metaphor? 

Nobody ever asked me this before. Everybody always just accepted this face value. You're going to make me go through my paces, which is awesome. So let me improvise. What I would say is that if you sort of thoughtlessly say there are two sides to the argument, what are you doing?

First of all, as I said, you're saying there are only two sides. You may be missing something — an approach that you haven't thought of. But thing two is, I think if your metaphor is, 'Oh, it's like the two sides of a stop sign, the front and the back.' You're saying something very specific about the relationship of those two positions, that they are opposite of each other, that whatever is true of one is false of the other. And whatever is true of the other is false of the other. 

The most interesting social or political or intellectual or literary or philosophical arguments are not actually like that. Right. The two sides of the argument, if you like, may be in opposition but they are not literally mirror images of one another, which is what you might think of, which is that what the metaphor applied carelessly might lead you to think? 

In your book, you write that geometry is not just a way of measuring ourselves or measuring, but it's a form of honesty. What's so honest about geometry? 

That's another feature. I often say that geometry is the cilantro of math. People are not neutral about it. There are people who love it and people who hate it.

And if you talk to people about it, which, you know, going around talking about math, people come to me with their math stories. I'm like the therapist and they've been waiting 20 years to unload this trauma that they have. I mean that very seriously. But what I find that is part of the impetus of writing this particular subject is that there are two kinds of people: there are people who are like 'I loved math in high school except for geometry. Like what was up with that?' And there was this weird interlude where we just drew pictures, improved, obvious things like, why do we do that?'

Then there are other people, I would actually say maybe a little more numerous who are like, 'I hated math in high school except for geometry. Why wasn't math all like that? That was the part that made sense.' But people are not neutral. People recognize that it's different. 

I am coming to answer your question, by the way. You know, if the case I was making was [geometry] really organizes the way we think about the world, I can say it about algebra, too. I could say it about probability, too. But I've got to think about things probabilistically. We've got to think about things algebraically. Of course, I believe all those things. I'm a mathematician, but I do think that geometry has a special role that makes it very different from everything else we learn in school about math — and frankly, everything else we learn in school.

The honesty piece is this: geometry is where we prove things, where we write proofs of statements, and that is very, very special. It's the place in school where we can make our own knowledge. We're not reliant on the authority of the teacher. We're not reliant on the authority of a book. There are certain rules and we can construct a fact, even if it's a somewhat abstract fact about a triangle or a circle. We can make it from scratch and nobody can tell us that we're wrong. That's extremely powerful, and I think people respond to that. There's a certain electricity to it and people can respond to it positively or negatively, but I think that's what sets it apart.

*This episode was produced by Matthew Lazin-Ryder