Eugenia Cheng is Scientist in Residence at the School of the Art Institute of Chicago and honorary fellow in the School of Mathematics and Statistics at the University of Sheffield. Her research is in category theory, which she aptly describes as the mathematics of mathematics. Her popular mathematics books How to Bake Pi and Beyond Infinity received wide acclaim. Eugenia has a knack for explaining complex mathematical ideas in a simple and accessible way to a wide audience. For that, she’s been compared to science communicator Neil deGrasse Tyson. She’s the only mathematician I know of who appeared on the Colbert Show.
Eugenia is an amazing spokesperson for mathematics and she does so with humor and insight. I jumped right into asking about math phobia and her answer is essential reading.
AB: Why is math phobia so prevalent, and what can we do as mathematicians to fix it?
EC: Math phobia has many causes, and one is because of how it is taught. I don’t want to sound like I am blaming teachers—I think I am blaming the entire education system along with the curriculum, standards, and the way standardized testing is imposed. I hope standardized testing was done in good faith to raise standards, but I think teachers feel under pressure to teach to those tests. As a result, math has become little more than a bunch of tests that you have to do well in
If you don’t do well on those tests, then everyone is upset: the teacher, your parents, the school, society, government, the world—everybody. One defense against everyone being upset at you is to decide that you didn’t like it anyway and to declare that it is pointless and not relevant to life. And then, you have adults who have grown up using that as a defense mechanism and it is passed on to the next generation. Parents and teachers say that can’t do math and children look at the think this is OK. And so it goes on.
The other thing is that it is one of the few subjects where you can be called wrong. When that starts in elementary school, it can be quite shocking to some children. In school, you learn and play and make things, but then suddenly there is this thing where you are wrong. For people who don’t like being wrong or told they are wrong, that can be upsetting. Children then can become afraid. Unfortunately, there are some societies where you can be randomly arrested by walking out onto the street. That is going to cause fear, and that’s similar to what happens with math.
We need to address all the issues at once as there are so many interrelated issues. Presenting math in a way where there isn’t so much emphasis on right and wrong would be a good start. Although math does have right and wrong, what math really is about is a way of discovering things (whether right or wrong) and a way of finding the truth. It is also a way of building more truth. If you put it that way, then it becomes less scary and more about discovery and creation.
Many of my art students at the School of the Art Institute of Chicago tell me they were put off math because it was about right and wrong and they wanted to do something creative. It made me sad because math is creative. What that shows me is that it has been presented in a way that removes the creativity. I think creativity is there in kindergarten, where there is math learning through play. I also think creativity is there at university, and somehow in the middle it gets lost.
Adding creativity back into the education system is easier said than done. We can’t just ask teachers to add creativity to math. We first have to release them from the pressures of standardized testing and curriculum. You have to make sure they are valued for that extra work. If teachers are not valued either through respect or money, then they won’t do it. We need to find fantastic math teachers who can put their skills towards doing that. Typically, great mathematicians are encouraged to teach math at higher levels rather than at an elementary school level. Having specialist math teachers teach at elementary schools would help change things.
AB: What is the earliest memory of being excited by mathematics?
EC: It was definitely when my mother showed me some really cool things that had nothing to do with homework or school. Neither of us can remember why she did it—I think it was just part of her consciousness. She told me fun things about how to draw graphs. I was really young so it was about how you can turn a thought process into a picture. I thought it was amazing and I loved having my brain stretching in strange directions. That feeling of not understanding something and enjoying not understanding something has stayed with me. It would be great if we could nurture that in people. To me, not understanding something is a whole possibility for discovering something.
AB: How did you come to Cambridge to pursue your university education, and what was the environment there like completing your doctorate?
EC: I was there for ten years as an undergraduate, a post-graduate, and then as a post-doctoral fellow. I wanted to go there because it was, and maybe is, the “best” place to do math in the UK. In undergraduate degrees in the UK we specialize, and I had been dying to specialize in mathematics all through my high school career. I felt that math was the only subject rigorous enough to warrant formal study. I am interested in everything, but I thought I could read the other stuff in books. With math, I knew I wanted to pursue a structured and rigorous form of education in it.
It was extremely competitive to get in and I had to work really hard to get the grades in extra exams. It was also extremely difficult when I got in. I did well in it, but I don’t like exams. I think exams test your ability to do exams and it’s not an extremely transferable skill. I can do well in exams, but I hate them and I have sympathy for those who don’t do well in them. I realized I had to sweat blood to do well in those exams. When I started doing research, I did much better than how I did at exams. Many of the people who did better than me in exams with less work turned out to be less good at research. Those students dropped out by either getting their Ph.D. and then stopping, or burning out in the middle of their first post-doc. I think this is interesting as it shows we are using the wrong selection criteria for going into research. I think this is applicable to many aspects of life where we use particular selection criteria as it is the easiest or most quantifiable. We use the wrong criteria to assess people for things, whether it is jobs, research, or leadership. I came into my own when I started doing abstract research. I felt like I came home mathematically. Before that, I took classes and wished they weren’t so boring.
AB: For non-mathematicians, can you describe the field of category theory?
EC: I like describing it as the mathematics of mathematics. The idea is that mathematics is what helps us understand the structures within science. And science helps us understand the structures in the world. Mathematics helps to explain how science works. It takes things common to all science and studies them all at once and save brain effort. There is math inside physics, chemistry, biology, and all of the sciences.
Math as a subject itself has become so broad that we do the same thing to branches of math. We can ask what do different parts of math have in common so we can save ourselves brain effort. That is category theory: it’s how math works and what is holding it together. I like to think of it as conservation of brain energy or front-loading the effort, like when we build machines to do things to save us time or energy. I love conserving brain energy. You can call it being lazy, but I call it being efficient.
AB: What research are you working on right now?
EC: I’ve been working on different definitions of higher dimensional category theory, and the relationship between them have never been well understood. The subject is relatively new, and I’ve spent most of my research career in finding relationships between those definitions.
An important test case of how higher dimensional category theory works is to do degenerate versions of them, where you collapse all the lower dimensions to a single point. There is a kind of dimension shift so you can go back to the low dimensional case, but there is a bit more structure floating around in the footprint or trace of the lower dimensions you collapsed. Comparing that to bona fide lower dimensional cases can be a good way to get to grips with how the higher dimensional cases are working.
I am doing that on a definition that I really like and seems intuitive. It is so intuitive it is not clear where all the content has gone. I love that intellectual optical illusion. The content is there, but where? I am looking for it.
AB: You are the author of the highly successful popular math books How to Bake Pi and Beyond Infinity. How did you come to write these books and what were you trying to accomplish with them?
EC: I always wanted to share my love of math very widely. Because I think fear and dislike math for the wrong reasons, I think it is something that can be addressed. They think math is all about right or wrong, but it isn’t about that. They think they can’t do it, but maybe they can. Maybe the thing they were looking at was arithmetic or school math.
I got to the point where I felt like I wasn’t reaching enough people by teaching undergraduate courses in a university. I decided I wanted to reach more people. I started making YouTube videos, beginning with things on category theory. And then I made videos aimed at high school students—especially the ones interested in going to university—to show them the more interesting part of math are in my opinion. Then I started making videos for a completely general audience.
It struck me that I wanted to write a book about it so everyone could see the parts of math I thought were the most interesting. I am not trying to turn everyone into a research mathematician. But I wanted to explain things that people don’t learn about in school like category theory or abstract thinking.
I wrote the first book How to Bake Pi to explain what math really is, not the subject that people come to dislike at school. I wanted it to have a really wide audience, especially for people who wouldn’t normally read a popular math book. There are many popular math books, but I feel they are often aimed at people who already love math. The book is aimed at adults who wished they knew more about math. I met many people who said that. Ten years ago, if I met someone and said I was a mathematician, they would say “I can’t do math.” And then I noticed a slight shift where people starting saying “I wish I knew more about what math is.”
It is also for keen children and teenagers who are fascinated by math but may be a bit bored at school. Or maybe their parents want to encourage them in math but don’t know which books to give them. Or maybe it is for those who like math in school but don’t fit in with the normal problem-solving geeks, who do competitions and Olypmiads—those people will always be fine. I wanted to help people left out by that. So my book is aimed at them, but it also aimed at serious math students and graduate students and all mathematicians to explain category theory. I think all pure mathematicians need to know category theory. Most modern mathematicians would agree with me. There weren’t many books that explained the point of category theory rather than just the theory. There is also the extreme niche audience of families of category theorists who can discover what their spouse or parent has been doing all these years.
AB: You appeared on the Colbert Show and you gave the awesome quote “I am a mathematician, not a calculator”. What was it like as a mathematician on a comedy show?
EC: It was hilarious. I feel like I am many things, not just a mathematician. In a way it didn’t feel extraordinary to me because I don’t really understand how famous things are. What made it me realise it was out of the ordinary was the extraordinary excitement of my friends.
Whenever I’m giving talks or teaching my students, I feel like I am part mathematician and part stand-up comedian as I am trying to keep everyone entertained. The good thing about being a part stand-up comedian during math classes is that the bar is very low, as no one is expecting it to be funny. You can say something ever so slightly amusing and people will have hysterics as it is so unexpected in the math environment.
I enjoy that. I’ve always been a performer and like being the center of attention. I’d like to use that attention to convey some math profundity by stealth. You get everyone laughing and tell them something profound about math. That seems to be the way I do things. The Colbert Show was a great opportunity to do that. I love it.
AB: You are Scientist In Residence at the School of the Art Institute of Chicago. Would you tell us how you came to that position and what you do there?
EC: It came about through a series of flukes, although I like to think of it as all the meticulous preparation throughout my life so I was ready for it when it arrived! It is a liberal arts school so they need people to teach everything, including math. I had no idea that happened there. They needed someone to do maternity cover for a semester. I got this mass e-mail about it, and I thought teaching math to Art students was the best thing in the entire universe. I replied immediately, and they wrote back and said I was rather over qualified, and could I instead be Scientist in Residence? When I asked what that was, they said anything I wanted to make of it really.
It sounded perfect, as they are all on the same page that the boundaries between science and art are artificial. It’s all part about thinking clearly about the world. They believe in teaching art students how to think. In universities, math is often taught as a tool for doing something else rather than a way of thinking. That is my least favorite thing about mathematics. I am glad it is a useful tool, but if you just take it as a tool then it is so boring. I used to teach many students who had to take math because they needed it for engineering or wanted to go into finance. Art students aren’t going to become engineers or go into finance. They just want to think about the world in different ways.
I also have a general platform to explore ways of ridding the boundaries around math and be in an extremely supportive environment where the institution as a whole appreciates and values the work I do in bringing math to more people.
AB: What advice would you give to students in university, especially women, about a career in mathematics?
EC: Especially for women, in my experience, statistically speaking, men tend to overrate their mathematical ability while women underrate it. Male students think they are doing fine when they are not, while female students think they are doing terribly but they are doing very well. Female students are more likely to doubt themselves and think they can’t go into research. They often perceive the male students to be doing better, but it may be that the male students are posturing and over valuing themselves. I’m not saying this is always true, but it’s been a noticeable trend in my years of teaching.
I would say—especially to female math people—that they are probably better than they think they are. Everything has exceptions and it may not be true. But for example, if they think they don’t understand something, it may be that they are seeking a deeper level of understanding than other people are, in which case they might be extremely suited to doing research.
AB: What do you think are the trends in mathematics education?
EC: I am very optimistic about the future of math education. I am generally an optimistic person, which is why I do all these things as I think things can get better. Everyone realizes we need to do something about math education—that is the first step. I see a lot of support for the changing the way we think about math. Changing it from just right or wrong to a mindset where everyone can get better at it. It’s not simply something you are born with.
Also, I want to get rid of the boundaries around math so there are more project based learning opportunities rather than subject based. I read about some amazing projects where a class will do a project for a whole year that encompasses everything. For example, in one class students made bread by first growing wheat, grinding it into flour, baking it, then the business of a bakery, and investment and stocks and shares. That was an amazing overarching approach.
One of the great thing that has happened since I have been doing a lot of public math work is that I get invited to give talks all over the world and I get to meet people who care about improving math education. I found that there are so many wonderful teachers, organizers, and educators who are working towards the same goals that I am. I’m optimistic that within the next generation we are going to change things for the better.