Steve Mirsky:Welcome to Scientific American's Science Talk, posted on June 2nd, 2015. I'm Steve Mirsky. On this episode—
Eugenia Cheng: So at a certain point in about the fifties, people realized that mathematics itself had become so complicated that it needed its own theory on how it worked.
Mirsky: That's mathematician Eugenia Cheng. She's tenured in the School of Mathematics and Statistics at the University of Sheffield in the U.K. She's currently scientist in residence at the School of Art Institute of Chicago. And she's the author of the new book, How to Bake Pi, spelled P-I, you know, the number pi, An Edible Exploration of the Mathematics of Mathematics, which means that she's in the branch of math called category theory, which you'll be hearing more about. In the book Cheng uses recipes and anecdotes from food and cooking to illustrate points about math. Appropriately, she recently met with a few reporters over lunch at a midtown New York restaurant. What follows is an edited version of that free ranging conversation. You'll hear Cheng, me, and Scientific American's Clara Moskowitz.
Cheng: I'm pretty excited about this book, which is why I wrote it, and it's the sum total of all the things that I've been talking to people about for all the years I've been thinking about mathematics, because I'm a very social person, and I meet a lot of people who aren't mathematicians, and so I'm always having conversations with them about what I do. And the responses are often things like, "Oh, I can't do math," or, "Oh, you must be really clever," or, "Oh, I'll never understand what you do."
And I think I've noticed that over the last ten years or so, the responses have just started slightly changing, and instead of being, "Oh, I couldn't ever do math," it started being, "I really wish I understood math more. I don't understand math, and I wish I did understand it more." And so I started sensing that maybe the time was right to reach out to people who actually regret a little bit that they don't understand math more. And so I had all this material in my head that I've been using either to teach reluctant students, or just to talk to people at parties, and one thing I noticed in my lectures is that whenever I use food analogies, my students all perk up, every single time, and everyone loves food.
And many more people love food than math, which is a shame, but I realized that if I brought the two together, then maybe I could show people some of what I love about math, and hope that people could find what they could love about math as well, and I think there are many things that are similar in an unlikely way between math and especially baking, because I love baking, because baking is kind of basically magic. You take some things, you take eggs and sugar and flour, and you do something with them, and then out comes a cake which looks like nothing like eggs or butter or sugar or flour. And it's amazing. And if you don't understand how it happens, it just looks like magic, mysterious magic.
And mathematics can look like mysterious magic as well, because you take some things, and something happens, and if you don't—if you don't see how it happens, then it just looks like it's a completely mystery, and you'll never be able to do it. And then it's much easier to go and do something else instead. And I would like to show people that in fact it's a very similar thing, that if someone explained—it just takes someone maybe showing you what it is and showing you how it actually can be quite fun, and that it doesn't—if you're not doing it just to score on the SAT or to get an A on your homework to keep your GPA up, once you're released from those terrible pressures, which I can't stand all of those, it's about how to think about things.
And another thing I've noticed is that people have become much more used to the idea of keeping physically fit, and that there's a—there's that great scene in The Secret of My Success, which I remember, and in England in those days, nobody went to the gym. And there's that scene where he wants to talk to his uncle, who's the big boss—I don't know if you remember—and the boss is on the treadmill. And so he's trying to go on a treadmill and keep up with him, and keeps falling off or running—he can't do it.
And I remember we all sat down in England and go, "That's hilarious. Imagine the idea of going to run on a treadmill. That's ridiculous." And now it's quite normal. You know, there's—I went to the gym in the hotel this morning, and there were tons of people there. It's just kind of normal that people see that it's good to work out, even if you're not an athlete or a sportsperson or going to run the Olympics.
And I like to think that math can play that role as well, people beginning to sense that it's good to keep mentally fit as well, and that the way I see math is a way of keeping mentally fit. And so it's about core training of your brain and your thought processes. And just like you train your core at the gym, even though you don't specifically—there isn't an Olympic medal for core use, which would be quite funny, but it wouldn't be very interesting to watch, I think, just ____, and math is the—my idea of it is it's the core of how we think, because it's all about logical thinking.
And it doesn't mean that that encompasses everything. It's just useful to know which things it can encompass, and to be able to think logically about things that should be considered logically. But I wanted to make it fun, because for me, it is fun. It's all—it's all fun. It's fun, and it's beautiful, and I want to share that with people, so that it can seem like it's a—it's something interesting, not a chore, because math has become a chore I think for many people because of their high school experiences. And that's – it's really a shame.
And also, people – I think other mathematicians too often, they don't really want to let people in. That's what I think. Because mathematicians especially, and I say this, and I am one, but mathematicians especially are very invested in feeling clever. They want to feel clever. And one very good way of feeling clever is to make everyone else feel stupid. And so if they can make everyone feel stupid, then they'll feel clever, and that's what their self-esteem is wrapped up in.
And so the sense that they can do something that other people can't do is something I think that they actually want to preserve a little bit, and I don't just – I just don't believe in that at all. I want everyone to be able – I don't believe in keeping people out. I believe in bringing people in, and that's what I want to do with this book.
Mirsky: I haven't had a chance to look at the book yet, but I know that there's a whole mathematical field of the best way to cut things, the best way to divide things.
Cheng: Mm-hmm. Mm-hmm.
Mirsky: Do you go into that?
Cheng: I do. Yes, a little bit. I mean, the theory is extremely complicated, and yet to solve it is one of those things where if you—to solve it for two people is completely easy, because it's the you cut, I choose thing. And then to solve it for three people is already phenomenally complicated. You can extend the you cut, I choose thing, but then you – first of all, you have to decide what it even means to be fair. What does it mean to share a cake fairly? And this is one of the things about mathematics. You have to decide what everything means really carefully, first, and once you've decided what it means, then you can think about it more clearly.
And one of processes of doing mathematics is first setting out your ground rules very carefully, and then following them carefully. And I've been teaching art students at the moment, because I'm teaching at the School of the Art Institute in Chicago, and in the first class, I explained to them that mathematics is a place where you can make up any rules you want, and then you can follow them.
And so—and they loved it, because they've known it as a place where someone gives them rules, and they don't like the rules. But then suddenly it was a place where they could make up any rules that they wanted, as long as they followed them. So they have to learn first of all what it means to make up rules, and then secondly, what it means to follow the rules, but once they've done that, they can do anything they want.
And I said, "The worst thing that will happen is that your rules will cause a contradiction, and then your world will implode."
Cheng: All the time I was in elementary school and high school, I really hated my math courses at school, really hated them. And it—I don't know why I knew that math was something better than that. Partly because of my mother, who is mathematical, had shown me some things, just a little bit. It doesn't take that much with children. She had just shown me a little bit of what was there. So I always knew there was more there.
And then when I got to university, I knew I wanted to do a math degree, because in England, you just do one subject when you're at university. So I knew I wanted to do a math degree, but then I knew that I only wanted to do pure math, because that was the essence inside math. And then I got to doing only pure math, and I – and then this thing—this thing called category theory showed up, and I just had one—I had one tutorial with the – Martin Hyland, to whom I've dedicated the book, and I just thought—his way of thinking, I just thought, "Oh, my goodness, I want to do my PhD with this person. I really want to."
And I didn't know what he did. And I went and looked it up, and it was category theory, and I read about what category theory was. I thought, "This is the thing I've been looking for for my whole life." And it's also a place where—well, another thing I said is that when you study—if you study birds, you need to come up with a method for studying birds, and that thing that you've come up with, the method for studying birds, isn't a bird, obviously. It's a method of studying birds.
Whereas with mathematics, if you come up with a method for studying mathematics, that's a new piece of mathematics. And so somehow the method that you are studying, you've reached a fixed point of where the method that you're using to study the thing you're studying is part of the subject itself. And so you can always—you keep generating more mathematics. That's why there's always more mathematics, because everything that you come up with as a way of studying that one becomes a new piece of mathematics that you can then study, and then you come up with a way of studying that, and you make a new – which sounds cyclic. It is a little bit cyclic.
But the particular field I do I think is the final – the final combination of that, because higher dimensional category theory is the place where it's really the final point where you're studying—you're studying mathematics.
Mirsky: So what is category theory?
Cheng: Category theory is the mathematics of mathematics. So just in the way that mathematics looks at how science works and it takes a certain aspect of science and kind of organizes it and looks at the structures inside it. Category theory does that for mathematics. So at a certain point in about the fifties, people realized that mathematics itself had become so complicated that it needed its own theory of how it works.
And so they had this amazing insight which said that instead of studying objects as objects, if you studied them via their relationships with other things, then you can understand a lot about anything just in terms of its relationships with other things rather than just by looking at the thing itself.
It's just like with people, if you write a biography of a person, you need to include all their relationships with all the people around them, their parents and their partners and their children and all their friends, because otherwise, it would be a very incomplete biography of them. And it's the same with mathematical objects, it turns out, that actually, the context of the—you can even ignore all the intrinsic characteristics of them and just look at how they relate to other things, and you – all this amazing structure appears. This was this extraordinary insight that the founders of category theory had that completely changed how mathematicians thought in the fifties.
And now modern mathematics is the stuff that uses that idea, and classical mathematics is the stuff that doesn't. So it really – it's the turning point.
Mirsky: Is that how it's possible to do category theory without understanding absolutely everything about all the other mathematical subfields? You can't know all of mathematics—
Cheng: No, you can't.
Mirsky: – so how can you do category theory unless a lot of the things that you don't know don't really matter that much for what you're doing?
Cheng: That's an interesting point, and people who—there are some mathematicians who say that category theory is too abstract, which is odd, because all of mathematics is abstract, and I want to say, well, why are you even a mathematician if you don't want to – it's like they don't want anyone to be more abstract than them. It's terribly competitive. They don't like the idea that someone has out-abstracted them.
But yes, because it's about ideas, this has two consequences. One is that it means that I can write a book about it for people who don't understand any mathematics, and it will be fine. The other is that you don't get some kind of nut cases who think that they can just go straight in to do category theory and do PhDs and research in it without having understood anything.
Mirsky: Other than obviously things like splitting a bill or figuring out the tip, what kind of ways do people use mathematics every day that maybe they don't realize? I mean, that's arithmetic, but maybe they don't realize they're actually employing mathematics on a daily basis.
Cheng: Any time you're evaluating something, and any time you're justifying a point of view, because logic is at the basis of that, and every—people should use logic. I mean, maybe most people don't use logic to justify points of view, but I wish they would. That is mathematics. And logic is what makes mathematics go, and it's what—I mean, it's what causes us to know that things are correct in mathematics, and it's what causes mathematics to progress in a way that other fields don't, because we don't—we don't disagree once something is done. In other fields, you have a theory, and you have another theory, and you can have an argument about it. In mathematics, if you haven't found a mistake in somebody's proof, then there is no argument.
And the process of constructing those arguments I think is the thing that is or at least should be everywhere, and be useful to everybody. And I would also like to say, though, that I don't think mathematics is just there to be useful. I think that—I feel sad that math has this burden on itself. It's got the burden of needing to be useful, whereas other things, like music, we don't say, well, what's music useful for? It's fun, and it gives people joy, and not all of mathematics—I mean, I think it is useful, and I say to people, "I don't do it because it's useful. I do it because I love it."
If it weren't useful, I would stop, so I always think there are two reasons for doing things. There's the reason you do it, and the reason you don't stop. So the reason I do it is because I love it. The reason I don't stop is because I think it's useful.
Clara Moskowitz: Can you tell us a bit more about the type of research that you do now? What are you studying?
Cheng: I study how higher dimensional structures fit together, basically, and it's related to spacetime, because spacetime is one of the most famous higher dimensional structures, because you have three dimensions of space, and then the fourth dimension, which you can think of as time. A lot of people say to me, "Oh, time is the fourth dimension." I say, "Well, it's one version of the fourth dimension."
And so the idea is that we need to understand what shape that is, and it's very, very complicated, understanding those shapes. And so one way you can understand it is to—is by breaking them down into smaller possible ones. But once you've broken them down, they're like Lego blocks, and you have to think about all the possible ways of fitting and sticking them back together again. And, I mean, you can build anything out of Lego blocks, despite the fact they're very—you have very simple pieces, and very few ways of putting them together. It's amazing, the complicated things you can build out of that.
And it's the same with higher dimensional—you can sort of imagine higher dimensional Lego. Imagine how complicated that would be. And so what we try to do is organize ways of expressing the way things fit together to make big structures, because mathematicians love reducing things to smaller parts so that you can then just understand—it's like going up a step. Then all you have to do is understand some small things and understand the process of sticking them together, and then you get everything.
But it's very hard, partly because the higher dimensions, they fit in our brain, but you can't write them down on a piece of paper. It's really difficult to even write down the research that you're doing, because you have to think of a way of squashing it onto this annoying two dimensional piece of paper. And I wish we had three dimensional pens that we could just write ____, but that would only give us one extra dimension.
Moskowitz: So does that relate—does your research relate to super symmetry or other theories that there are 11 dimensions in our universe? Does your research show how those 11 dimensions would fit together?
Cheng: It shows—it shows all the possible ways that 11 dimensions can fit together, and so it's something that theoretical physicists use, but we don't tell them how to use it. So they can—it's—we give them—we give them a language for talking about it. That's kind of what it is. It's like setting up the rules—the framework, and then people can use the framework—other mathematicians and physicists and computer scientists can use the framework.
Moskowitz: I'm still trying to wrap my head around category theory.
Moskowitz: So I liked your description of it, relationships between things, but like what are some examples? I'm trying to think of—
Cheng: You mean examples of—
Moskowitz: Of category theory, of the type of problem that you might solve in category theory.
Cheng: The type of problem you might solve is you—for example, you might—we look for things that are the biggest or the most extreme in a particular world. So it's like the fact that we look for the tallest person ever or the person who's won the most Olympic medals ever, and then—but then sometimes we turn it around and we say, okay, I want to find the world in which this person is the most extreme. And so then it's about changing the world that you're in in order to make the thing that you're looking at extreme, because then you can characterize it by it being an extreme.
And everyone likes to call themselves the best something, something, something. The example I give, there was this restaurant in Sheffield that declared itself to—on their—what was it that happened? That was the example I gave. No, it's that—it's the Sheffield Symphony Orchestra, which calls itself the largest amateur symphony orchestra in South Yorkshire.
Cheng: So they have found a world in which they are the—and not even the best, just the largest.
Moskowitz: So how is that about relationships between things?
Cheng: Because if you're the best, then that's your relationship with everything else. you've surpassed everything else.
Moskowitz: Okay. So is it about making groups of things, like categories of things for which some kind of law holds true?
Cheng: Exactly. You've understood category theory.
Cheng: So you decide what type of relationship you're interested in in any given moment, and you focus on that just for now. And it's quite mentally saving space, so you can just ignore all the other types of relationships between people, right at that moment.
Moskowitz: Okay. So it's different from set theory, which I don't know much about, but always thought about it as the mathematics of groups of things.
Cheng: Right. So set theory does not take relationships into account. And so it's like—I think I say this in the book. It's like building everything out of sand. And you can build things out of sand, but you can't build an awful lot of things out of sand. They kind of fall apart. And it's really tedious.
And so people have tried to construct a whole mathematics out of set theory, and it becomes incomprehensible very, very quickly, because actually, that's not how real mathematics is done. Real mathematics is really more about relationships between things. So if you start with that as a basis, set theory just says everything is a set of things. That's it.
Cheng: And I have to build everything up as a set of things. Whereas category says everything is a relationship—has a relationship with something else, and that's ____ to use to start. It's—so you can—you can move between the two. There's a way of moving between the two. It's just a question of one more – which one more closely fits how we really think?
Cheng: And I think it's category theory, because we don't think of things as just—in isolation with each other. We immediately—maybe—I mean, I know I do this more than other people. Whenever I see friends, I always have to—feel the need to figure out when I last saw them. And if I'm at a party, I always feel the need to figure out how everyone knows everyone else. Like how do you know the host? How did you meet? So I—just to understand how—rather than just sort of who are you and who are you and who are you.
Moskowitz: Yeah. So that's category ____.
Mirsky: That's it for this episode. Get your science news at our Web site, www.ScientificAmerican.com, where you can hear Christopher Intagliata's June 2nd "60-Second Science" podcast on how chimps would cook food if they could master the pesky fire issue. And follow us on Twitter, where you'll get a tweet whenever a new item hits the Web site. Our Twitter name is @sciam. For Scientific American Science Talk, I'm Steve Mirsky. Thanks for clicking on us.