‘How to Think Clearly’

Watch Eugenia Cheng’s YouTube videos and you’ll start to believe she can do anything. She can, for example, play a beautiful rendition of a Schubert piano sonata. She can also bake the perfect mince pie, and, if need be, she can carve a Möbius strip out of a bagel. Plus she’s funny.

She proved it a couple of years ago when she appeared on The Late Show With Stephen Colbert. She more than held her own, including some impromptu fencing with rolling pins (go watch the clip).

Oh, and she’s a mathematician, too. Right now she’s the scientist in residence at the School of the Art Institute of Chicago, and, before that, she won tenure in pure mathematics at the University of Sheffield. Her research emphasis is category theory, a branch of mathematics that involves monoids and groupoids, pullbacks and pushouts. There are a lot of arrows used as well.

That would make more sense if she had explained it.

She’s good at explaining complicated things. In her new book, she takes on infinity, a complicated thing indeed. Beyond Infinity: An Expedition to the Outer Limits of Mathematics (Basic Books) is meant to persuade algebra-phobes that mathematics can be engaging, mysterious, and playful. She spoke with The Chronicle recently about her book, her love of music, teaching art majors to care about numbers, and the two new words she invented.

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In your new book, there are a number of food examples: baby carrots, sandwiches, and so on. Your previous book, How to Bake Pi (Basic Books, 2015), was rooted in recipes. Is there something about food that resonates?

It resonates for me because I love food so much. But, really, over the course of years of teaching, I’ve learned that the food stories make kids perk up. Those are always the most attractive stories that I’ve told.

I have a lot of musician friends and writer friends, and I knew that none of them would read the popular math books that I’d seen on the shelves, so I wanted something that would quickly say that this was for them.

You’re an accomplished pianist. Is the pleasure you get from composing and performing music different than the pleasure you get from thinking through a complex mathematical problem?

It’s different, but it’s related. One of the reasons mathematical research is frustrating is that you can spend hours and hours and get really, actually nowhere. I wanted to have another activity where I knew I could achieve something.

But then there’s the performance aspect, which is all about communicating. Whether it’s math or music, I’m trying to share my love of something with other people. I used to joke that when I’m playing music, I’m giving someone pleasure, and when I’m teaching math, I’m giving them pain, but I like to think that’s not so true now that I’m teaching math the way I think it should be taught.

You teach art students math. I imagine they might wonder why they would need math. Do you encounter that? Or are they open to the idea that it might be important to their work?

If they think math is what they did in high school, this is going to be quite different. They do arrive sometimes a bit skeptically. I love hearing their skepticism at the beginning, because my whole aim is to change their minds.

When you were young, you found math in school boring. Why did you find it boring — considering that you clearly have an aptitude for it — and did that early experience give you some empathy for students who feel similarly?

The reason math classes are so boring is because of the pressure teachers are under to bring students to a certain level. The only way they know to get students to pass those tests is to drill and drill and drill. And that’s not interesting for the people struggling, and it’s not interesting for the people who can already do it. It doesn’t encourage understanding and exploration, because all those things take too long. I was very lucky because my mother showed me things that were fantastic about math outside of class. I was inoculated against boring math classes by things that my mother showed me, and that led me to believe deep down that there was going to be something better waiting for me at the end of it.

A number of colleges have dropped their math requirement in recent years. I assume you think that’s not a good development?

It depends. We need to go back to the drawing board to think about why we are teaching math. There are three broad reasons to think math is important. One is direct use, which consists of things like understanding statistics so that people can’t be fooled by media misrepresentations. Then there’s math so people are equipped to go on to science and engineering careers — and that’s important to those people.

But there’s something else, and that’s just how to think clearly. And a liberal-arts system should be about becoming a human being who can think clearly.

A book published last year called The Math Myth argued that we should dispense with algebra and replace it with practical statistics, so people can understand news reports and mortgages and the math they need to get by in their daily lives. Are you sympathetic to that idea?

I am sympathetic to some part of that, but that’s not the view I take. That’s the first aspect of why math is important. I do think it’s one of the most boring ones. I don’t think we persuade young people to like something by telling them it’s useful. It’s not interesting to learn how mortgages work until it’s time to get a mortgage, and then it becomes rather important.

And, of course, the math involved in mortgages is very different from the sorts of questions you ask in the book about infinity. (For instance, how can a hotel with an unlimited number of rooms be full?)

It’s become accepted that for most people, physical fitness is a good thing. We know that good core strength helps with the rest of our strength. It’s the same with math. Yes, you can just work on your biceps or run up stairs a lot. With math, you can train yourself to understand how compound interest works, but if the core of your brain works — well, I truly believe I can learn any other part of math or any part of science if I really felt like it, because I understand how to think. It’s less direct but much more widely relevant.

I was scrolling through your Twitter, and this tweet caught my attention: “I don’t notice it’s a whole audience of men when I’m giving a research talk. I just notice before and after.” Care to expand on that?

When I’m teaching math majors, it’s quite male-dominated, but it is improving. Now that I’m teaching art students, I’m less accustomed to that, because it’s female-dominated. Most of the public talks, there are many women in the audience. When I’m suddenly in a room dominated by men, I think I’m noticing it more now.

That said — and this is what I observed — when I started talking about math, I didn’t notice it anymore, because we’d all gone into the abstract world of mathematics, and we’re just thinking about that. I looked around the room when I started and I thought, “This is all men.” Then I forgot about it. Then, when it was over, we started talking about going out for dinner, and I noticed it again.

In a lecture, I heard you use a couple of words that I think you made up: “ingressive” and “congressive.” Could you explain what they mean and why you think they’re useful?

When I started to be a mathematician, I tried to hide all aspects of my femininity, but I got frustrated with that, so I tried to bring in aspects of my femininity. Then I realized if I had new words, we could talk about the characteristic without saying which characteristic goes with which gender. So I came up with “ingressive” and “congressive.” Ingressive behavior is about taking risks and not caring about losing. Congressive behavior is about bringing people together and being kind.

It doesn’t make any sense to say that a woman is behaving in a masculine way, because if I’m a woman, then anything I do is feminine because it’s all pertaining to a woman. It’s like when people say to me, “You don’t look like a mathematician,” and I say, “I do look like a mathematician, because I am a mathematician.”

This interview has been edited for length and clarity.