Is there more to mathematics than the equal sign? A pair of shoes is different from a pair of gloves, yet we recognize a sameness in them because both meet our definition of a pair. Category theory is a branch of mathematics that examines how things can be essentially the same without being exactly equal. It’s what the mathematician Eugenia Cheng uses to make connections between abstract concepts and to identify patterns across many topics and scales. Cheng is a scientist in residence at the School of the Art Institute of Chicago and the author of The Joy of Abstraction: An Exploration of Math, Category Theory and Life. In this episode, she and host Steve Strogatz unpack category theory and show how it can apply to everyday matters — including recognizing that irritating person who keeps resurfacing in your life.
Steven Strogatz (00:00): Hi. I’m Steve Strogatz, and this is The Joy of Why, a podcast from Quanta Magazine that takes you into some of the biggest unanswered questions in math and science today. In this episode, we’re going to ask: Is there more to math than the equal sign? In other words, can we use the logical power of math to talk about what it means to be essentially the same, without having to be exactly and strictly equal?
(00:28) Which brings us to category theory. It’s often described as the mathematics of mathematics — a kind of bird’s eye view of the entire subject. But what does that mean? If mathematics relies on abstractions, then proponents of category theory say it provides a sound scaffolding for those abstractions, a way to make connections and recognize patterns and relationships across a broad range of topics and scales, such as the various shapes of space, or different types of symmetry.
(00:59) But some mathematicians, especially some grouchy old-timers, question what category theory is really telling us about math that is new. What’s the point of it? Even some of its most ardent practitioners have affectionately described it as “abstract nonsense.”
(01:16) Lately, though, category theory is all the rage. It’s not just hot in the world of math. It’s now being used in computer science, physics, engineering, chemistry, linguistics, and more. Though it began as an abstract branch of pure math, it’s turning out to be a whole new way of thinking about lots of things, including situations in everyday life.
(01:37) Our guest today, Dr. Eugenia Cheng, will help us understand category theory better. She’s a scientist-in-residence at the School of the Art Institute of Chicago, and the author of several books, including her most recent one, The Joy of Abstraction: An Exploration of Math, Category Theory, and Life. Welcome, Dr. Eugenia Cheng,
Eugenia Cheng: Thank you so much for having me.
Strogatz (02:01): I’m really excited to have a chance to talk to you. I’ve been a big fan of yours for many years. Category theory is such an interesting subject in our world of math. And I honestly, until I started looking at your book, didn’t appreciate how it’s begun to reach out into so many other disciplines — and even everyday life, as I just said. So how about if you get us started with — Imagine you were making a TikTok video to define category theory. What would you say? And just spit it out very quickly.
Cheng (02:32): Like you say, I would say that the category theory is the mathematics of mathematics. Remember that mathematics is not just about numbers and equations. It’s about how we build arguments and how we find patterns between things. So it’s about finding patterns in patterns and making arguments about arguments.
Strogatz (02:50): I like it. Is it something that you’ve been interested in your whole career? When did you start getting interested in it?
Cheng (02:56): I’m going to go out on a limb and say I’ve been interested in it my whole life without realizing that’s what I was interested in. Because ever since I was little, I was interested in patterns and arguments — not in the sense of people yelling at each other, but in the sense of building justifications for how we know things. And although I didn’t realize category theory existed at all, I was always interested in deep explanations. The kind of child who asks why repeatedly not just to annoy the grown-ups but actually because I wanted to know. And math was always the thing that gave me the most satisfying answers to why things are true.
(03:35) And then within math, it was pure math that gave me the most satisfying answers. And then within pure math, it was algebra. And then within algebra, it was category theory. And when I finally met category theory for the first time, it was like coming home or finding the home that I had sort of been dreaming of all along. I realized that that was what I had been really looking for without realizing it. And even now, when I go back to some of the things that intrigued me when I was little — and these are some of the examples I give, when I give talks about category theory — I realized that the thing that intrigued me was really something to do with category theory, deep down.
Strogatz (04:11): I love this sense of you kind of coming home. It’s like you have this long-lost paradise or a place that you were seeking, not subconsciously, but —
Cheng (04:21): It really did feel like that.
Strogatz: I’m happy you could be reunited with or —
Strogatz (04:27): Yes. Which may be very fitting, actually, since the whole subject seems to be about unification of so many different things. Is that, is that right?
Cheng (04:35): Indeed. Yes, it’s really about spotting similarities between different situations, and then finding a unified way to think about them — essentially so that we can use our brains better. Because our poor, finite brains are really very small compared with the complexity of the world that we are trying to understand. And one way to deal with that complexity is just to willfully ignore parts of it. And unfortunately, that is quite a common way of doing it.
(05:05) But I believe that a better way to deal with the complexity of the world is to take a broader view of things and to find similarities between different situations, so that you can study many things at the same time at a certain level. And I think that’s what you mean by unification. We’re not trying to declare things are the same when they’re not. We’re trying to find some deep essence about them that they have in common, so that we can at least study that part as the same thing before then zooming back in on the individual situations to look at the details.
Strogatz (05:40): You’ve given a beautiful manifesto there of what it is to think like a mathematician. I think it’s something that all of us in the field — well, maybe that’s an exaggeration, to say all of us. But certainly, it’s — I should be careful with such statements. You even say something like that in your book. I want to be careful not to exaggerate. But still, I think it’s a fair approximation to the truth to say that the appeal to abstraction, the shedding of details that seem irrelevant in a particular context, can give us great insight. So maybe, maybe this is a place for you to tell us a little about your views about abstraction, the pros and cons of it.
Cheng (06:16): Yes, I love abstraction because I enjoy it. And I think we don’t talk about that enough with math. There’s a big push to tell everyone how useful math is and how important it is because it’s going to help you solve problems in life. And I think that that’s doing math a little bit of a disservice. Yes, it is very useful. And yes, it does help me in various ways. But I also just love doing it. And that’s why I gave the title of my book, the title that I gave it, which is The Joy of Abstraction. It’s not the usefulness of abstraction — it’s the joy of abstraction. Because to me, it really is a joyful process. It’s like shining light on things. You know, I just went outside today, and the sun was shining so brightly, and that just gave me joy, not because I’m looking for something, and I can see it better. You know, if I dropped something on the street and I was looking for it, then it, yes, it would be helpful that the sun was shining. But it’s just nice, isn’t it, to be able to see things clearly. And that’s what I love about abstraction. What it feels like to me is that there’s kind of fog everywhere. And then when you perform an abstraction, you’re clearing away irrelevant details, as you say, and then you can see things more clearly. And that, to me is joyful.
Strogatz (07:29): So this part could be a little tricky to discuss, but let’s try it. So you know, I have come up in a little bit of a different subculture in math. I’m from the very applied end of math. But I did have exposure to category theory in the final year of my undergraduate training. I was taking a topology course. And I guess where I want to go with this is that my only exposure to it before your book is, I think, a kind of old-fashioned point of view that category theory relies on knowing a lot of advanced math. And then its great utility, as at least it was argued to us at the time, is that it helps us see connections between — even if these words don’t make sense to some of our listeners, I think they’ll get the idea — that we were doing some hard problems in topology about shapes, and we were converting them into problems about algebra, about things called groups. And that was the big virtue, we were told: That with category theory, you could do a kind of translation from one part of math to another, and maybe make a problem easier. But what do you think about that? That was the received wisdom for a long time about category theory, right? That it’s a sort of very sophisticated thing.
Cheng (08:43): Yes. And that’s one of the reasons I call it the mathematics of mathematics. Because where mathematics goes around looking at the world and finding similarities between different things in the world, you could say that category theory does that for mathematics. So it goes around, like finding similarities between different things in mathematics. So in mathematics, you might say, oh — for example in topology, you sort of say… Well, for example, there’s this thing that is like a coffee cup, and there’s this thing that’s like a bagel or a donut to pick a famous example. And there’s some sense in which they are the same, which is not that you can eat them both (because you can’t) but it’s to do with how you can transform one into the other. And so topology comes up with a way of talking about those things being the same.
(09:23) And then category theory goes a level up and says, oh, there’s some way that we can make a connection, as you say, between the whole field of topology and the entire field of algebra, so that we can then make a transformation not between a coffee cup and a doughnut, but between the whole idea of a shape and the entire idea of algebra. And so it did grow up from that idea. And it grew from wanting a framework for making those connections in a way that is rigorous, because math is all about doing things in a rigorous, logical way — not just saying, “I sense somewhere in my gut that those things are connected.” We often start with a sense in our gut, but then we need to give it a logical framework because we believe in building our arguments on logic, not just gut instinct. It can seem that you need to know those advanced fields in mathematics in order to see what category theory is doing.
(10:17) But honestly, I think that’s a converse error. It’s not necessary. It is one way of getting at it. But another way of getting at it is just to think about abstraction. And I came to it from that point of view because, honestly, I don’t think I had a very good grip on mathematics when I finished my undergraduate degree. And one of the reasons I didn’t, I like to think, is because I hadn’t done category theory. And when I did category theory, I came at it just from the point of view of enjoying the algebra in its own right. I wasn’t trying to unite different areas of mathematics, because I was honestly quite confused by all the different areas of mathematics.
(10:58) And when category theory came along, category theory made so much sense to me just from itself. You don’t need to know any other mathematics to understand the definition of a category. It doesn’t depend on anything. It doesn’t involve words from other parts of math, it just goes right in.
(11:12) But then what happened was that it helped me understand all the parts of math that I had done before in my undergraduate degree. And so I remember some friends and I sitting down and going, “Well, really, this was a prerequisite for all the things we did before.” And I felt personally that it would have helped me understand all the other things if I had done category theory first, rather than using the other things as a jumping point to understand category theory. And the thing is that — radical thought — everyone’s different. And so some people understand things via the other parts of mathematics. And some people understand other parts of mathematics via category theory.
(11:52) And I think one of the big problems with the way that mathematics education is at the moment is that there’s a belief that there’s a certain order that you have to do mathematics. And so I’ve described it in my book at the beginning as a series of hurdles. If you think that math is a series of hurdles, and that they get higher and higher as you go along, then yes, indeed, there’s not much point trying to get over a higher hurdle if you can’t get over the lower hurdle.
(12:18) But the thing is, math is not actually a series of hurdles. It’s an interconnected network of ideas. And so there are many different paths around that. Because everything is connected. You can go in all sorts of different routes around that network. And here’s the radical thought, again: Different routes will suit different people in different ways. And it’s like just how you present mathematics in the first place. Some people like seeing specific examples first, and then looking at the general theory, based off their understanding of their specific examples.
(12:50) But I prefer seeing general theory, and then using the specific examples afterwards, to help me while using the general theory to help me understand the specific examples. And when I go to research seminars, when the seminar starts with the examples and does the theory afterwards, I always feel like I want to watch the seminar backwards. So then I have to, I have to hold the examples in my brain and ignore them, listen to the general theory, and then rapidly try and rewind and go over the examples afterwards.
Strogatz (13:18): It’s wonderful, all this psychology that you’re bringing in about human diversity, because I can hear this is not just being a research mathematician. It may even be no accident that you’re at an art institute, you know. You have a view of math and math education that is so refreshing and so egalitarian and different. You know, like even just rejecting this hurdle metaphor, or the gatekeeping that we so often hear about — that, really — look, we know something is not working, right? The way we’re doing it. There are so many people with strong aversion to math, verging in some cases on math-phobia, or math-anxiety, and it’s so unnecessary. I really think you may have, you have found a way in for at least some of these people.
(14:03) But what’s really surprising to me is this idea of “abstraction first” or “theory first,” because we’re so often taught — like, in a different domain: of writing. And both of us are interested in writing, I know. Like if you read Strunk and White or a lot of the old timers, they’ll say “examples first” or “prefer the concrete to the general.” And yet, that’s apparently not true — it wasn’t true of your brain. It may be true for some people’s brain.
Cheng (14:26): Exactly. It’s true for some people. And I have been showing category theory to my art students at the School of the Art Institute for seven or eight years now. And what I have found is that they are art students. And they, many of them, have been really put off math and the whole of mainstream education in the past. And the concrete form of math or the examples form of math has not resonated with them. And the thing is that abstraction — so my class is called “The Elegance of Abstraction” — it might seem that abstraction is even less relevant to life than other forms of math. And I used to think this myself, actually. I used to think that the math I did, category theory, was only useful to other parts of pure math. And that pure math was helpful to parts of applied math, and applied math is useful to engineering, science. And that’s useful to the human world. And so I thought, I just accepted that my form of math was going to be useful to normal life in a very, very long series of knock-on effects.
(15:28) And then I realized that, actually, because abstraction is about how to think better, it can be helpful directly to all of human life — and in fact, more of it than applied math and engineering. And so if you say that math is useful because, oh, it means that we can make phones that use GPS, and that we can fly planes and build bridges. Yeah, that’s all great. But it means that anyone who isn’t going to go into science and engineering doesn’t need to do it. And it gives those people a genuine and, I think, valid reason to say, “I’m very glad that some people do math, but I don’t see why I need to do it myself.” The kind of thing where students go, “Oh, why do we have to learn trigonometry? I’m never going to use trigonometry in life.” And I think that’s valid. And I want to say that’s true. I very rarely use any trigonometry in my daily life. The only times I use trigonometry is when I’m drawing math diagrams for a math paper, and I have to figure out some coordinates for something. And that’s not a very mainstream part of daily life.
Strogatz (16:27): I’m sorry, if you don’t mind my interrupting. I’d like to give you a little setup here, like in the style of volleyball. I think this should be an easy one for you to spike. But let’s see. Here’s what I have in mind: that the standard argument that people will make — I’m sure there are some listeners who are shaking their head, “Oh, no.” They think trigonometry is sacrosanct. “You better not get rid of trigonometry!” Because even though we could all agree, say these people, that trigonometry is worthless for 90% of people or 99% of people, it teaches you to think, we’re told. “It’s good exercise for your mind.” So go ahead, spike that down.
Cheng (17:02): That’s exactly what I was about to say. This is wonderful. We’re agreeing. You can interrupt and agree with me, it’s great! That’s the thing, that when we push the usefulness of math for the applications — no offense to applied mathematicians. But when we push that, that’s not the be-all and end-all. Because as you say, it’s about learning how to think. And so when we learn about trigonometry, it’s not because trigonometry is useful. It’s because it’s a training for our brains. And once you present math as a way of thinking, then it becomes important for anyone who cares about thinking. And I would hope that everyone — sometimes it seems like not everyone is thinking. But then anyone who cares about thinking, and my art students really care about thinking, they then become interested in it in a different way, when they realize it’s not about direct applications. And it’s not about directly solving problems. It’s about learning new ways of using your brain, and learning how to use your brain to get many different points of view on the same thing.
(18:02) And I think that’s what abstraction is about for me. It’s that when you go up to a higher level where you — I don’t mean higher in the sense of more difficult, I just mean a kind of bird’s eye view where you ignore some of the small details about things. You can then turn around better, you can see different points of view. And I think it’s a shame that math is so often presented as something that’s fixed and rigid, because I truly believe that it’s about flexibility, and about binding different points of view on the same thing. And even trigonometry is really deep down about switching points of view between circular points of view and square points of view. And I don’t think that that is presented enough when we talk about trigonometry. It’s usually about memorizing all those dumb formulae or trig functions. But it’s actually about if we understand the world via a circular point of view, as opposed to a square grid, how do we move between those two points of view?
(18:59) And I think that’s what all of math is about, even the, those dreaded equations and the equality sign, which as you say, we’re moving away from. Equations are really about moving points of view. What’s an equation saying? It’s saying that something is equal to something else. But it says more than that. It’s saying that some things that were not obviously the same as each other in one sense are actually the same as each other in another sense, which enables us to change our point of view from one to the other. Even something simple, like 5 + 1 = 1 + 5, is telling us that one point of view is we could take five things and then add another one to it. And another point of view is that we could take one and add five afterwards. And that might not seem like a radically different point of view. But if you teach a small child that kind of idea, one way you can do it is you can put one thing and five things on a plate, and then you can rotate the plate or you can get them to walk around to the other side of the plate, at which point the one and the five things have swapped places. So then you’ve literally changed your point of view. And that’s what abstract math is about.
Strogatz (20:01): Okay, this may be a little counter to the spirit of what you’ve been telling us about abstraction and theory building, as opposed to examples and details. But I think maybe it would help if you gave us some example, if you could, of either what a category is or how you could use a higher level of thinking to shed light on something by ignoring certain details — some example of something in the categorical way of thinking. Could we start with a category? Or is that too technical to start with?
Cheng (20:29): No, I think we can start with that, because the audience may be interested to hear. So a category is, it’s a piece of algebra. But the idea is that we could, on the one hand, look at sets of things. And a set of things is just a bunch of objects. We don’t have any extra information, except there’s a bunch of objects. And so what a category does is it says, well, if we think about the relationships between those objects as well, then we get a huge amount more that we can think about. And so a category consists of a bunch of objects, and some chosen relationships between them satisfying some, some mild axioms. But the idea is that if we study objects via their relationships rather than via their intrinsic characteristics, then we get to understand a lot and a lot more that’s relevant. It’s just like when we study people — it really makes a lot of sense to look at how they interact with other people. If you’re writing a biography of a person, it would be very odd to just write a whole book describing their intrinsic characteristics rather than looking at their relationships with other people, their relationships with their family, their relationships with their friends, their partners, their children, the people they work with. And from there, we build up a view of their character through their interactions with others.
(21:45) And that’s what category theory is. It’s saying, we put everything into a context. And the flexibility is in saying that we can put the same objects into different contexts by thinking about different relationships between them. So we could think about, for example, age relationships between people, or we could think about education relationships between them. So we could think about how old people are, or we can think about how many levels of education they have. And we can think about how they interact with each other at work, or we could think about which mathematicians have collaborated with others — which is really nothing much to do with how old they are. I suppose they probably have to be alive at the same time. And so then that puts people in different contexts.
(22:26) And category theory is all about saying, we should not think about anything outside of a context, because things really change character depending on what context they’re in. And even numbers — I don’t mean “even numbers” — numbers themselves, change context depending on what character they’re in. For example, ordinary numbers — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13… They go on forever, they get bigger and bigger and bigger and bigger forever. Whereas numbers on a clock go round and round in circles. And that’s really important, because if we didn’t go around in circles, we’d be stuck saying things like, “Oh, I’ll meet you at 1,000,233 o’clock.” And so the context really matters.
(23:06) And, in fact, one of the things I’ve been doing with my students recently is thinking about a particular kind of category called a partial ordering. And so a total ordering is where you can put everything in a straight line, like the number. You can line them all up. There’s a perfectly sensible way of putting them all in a direct hierarchy. And the thing is, there aren’t very many things in life that fit into a direct hierarchy. Unfortunately, our human tendency is to try and shove everything into a direct hierarchy anyway. And I think that’s because we’re not sufficiently used to the complexity of something that’s got more dimensions in it. And so I believe in getting our brains better at dealing with more dimensions so that we can be more nuanced about the situation.
Strogatz (23:50): Maybe we should pick up on your mention of the partial ordering, since I’m not so sure I know what you mean by that.
Cheng (23:55): Right. So a total ordering is where you can say clearly that everything is in a straight-line order. So 1 is less than 2; 2 is less than 3; 3 is less than 4. There’s no ambiguity about it. But then we get into situations — and one of the ones I talk about a lot is to do with different kinds of privilege. And so if we accept that some people’s identities give them more privileges in life, and that doesn’t mean that they are better off than other people, it just means that that particular aspect of them is not causing them problems. So for example, white people hold structural privilege over non-white people. And male people hold structural privilege over non-male people. So just white people over non-white people is a total ordering. We’ve put them in order. And a male people over non-male people is also a total ordering because we’ve put them in an order.
(24:48) But then when we combine the two, we can’t put them in an order anymore. Because definitely out of those groups of people, white male people have the most privilege and then non-white non-male people have the least but then if we look at non-white male people, and white non-male people, we can’t put those in an order because they’ve got one type each. And so then if we try, then we can get into potential situations of antagonism. And so often what happens is that white women are so focused on the ways in which they’re oppressed as women, that they then end up being negligent about the issues of racism, because they’re focusing too much on their own problems as women. And if they just see the whole world as men versus women, then they will be neglecting the problems of racism. At the same time, people who are not white may be so focused on the issues that they have as non-white people, that they then neglect the issues that that women have.
(25:41) And so this, in other areas of sociology perhaps, is called intersectionality. Of course, intersections in math means something slightly different. But I like to think of it as a higher-dimensional situation, because instead of putting everyone on a one-dimensional straight line, that has now become a two-dimensional situation, and that’s where it’s not a total order, but it’s a partial order. Because we can’t completely put everyone in line, we can only partially do it. And if you draw a diagram of it, it comes out looking like a square, instead of a straight line. And then if you add another type of privilege, you get a cube, and then you keep adding more dimensions for each type that you add in.
Strogatz (26:21): That’s a rich, interesting example. I mean, it will also — with lots of historical significance, if we started to talk, let’s say, about the history of feminism. I suppose everyone listening realizes these are very real-world things that you just brought up. These are not abstractions that are irrelevant to daily life; in fact, they’re monumentally relevant. We’ve been talking about matters of — I don’t know what we should call them — social justice, or concerns about real people in real life. That’s not inherent to category theory, right? That’s one of many possible ways that we could use it. But we could also be using it in in chemistry or physics.
Cheng (26:57): Oh, yes. So category theory is inherently just a piece of algebra. That’s all it is. It’s a definition that looks exactly like a piece of algebra. You start with some objects and some arrow structure. (That’s the data: You put in some structure, and you have some axioms, and then you manipulate it using logic). So the way I’m describing it sounds somewhat philosophical. But that’s because I’m not giving a lecture to graduate students who need to know the full details. Yes, we should not give the impression that it’s just some vague piece of philosophy. It really is a very, very precise piece of mathematics. And it’s a very technical piece of algebra. I have found that I can apply it to these questions, these very difficult questions, of social justice.
(27:38) And there are two things. One is that I was just doing it anyway. I just realized, that’s how I think and that that’s how I understand the world around me. And that that’s why I feel able to understand people, even when I completely disagree with them. You know, I quite often see people on the internet who I completely disagree with, and then other people are like “Oh, my goodness, I can’t believe anyone would think that, or how can anyone vote for such and such a thing?” And I always think I absolutely can understand why anyone would think that. And if I can’t, I’m jolly well going to sit down and try harder to understand it. And it’s not that other people are irrational. Some people go, “Oh, well, how can you deal with it if other people are just being irrational?” I think that’s absolutely the wrong approach. It is patronizing, and wrong of us to declare other people to be irrational, because also, they will declare us to be irrational. And then all we do is have a yelling match. What’s much more important is to find what their logic is, and to analyze their logic from their point of view, rather than imposing our logic on them.
(28:38) And that’s what we do in mathematics. We don’t study one part of math using the axioms of a different part of math — that wouldn’t make any sense. But what we do try and do is, as you say, map from one part to another, using some kind of transformation or map. And that’s one thing that category theory does. It gives us a way of finding an abstract structure in one world, and then mapping it to another world in a way such that the abstract structure is still relevant. And that’s how I understand people I completely disagree with, via a form of empathy that comes from abstraction. That’s how I then talk about abstraction with my art students. Because they don’t really care about how to build a plane or build a bridge. They really care about questions of social justice and other people and the people that they disagree with, and how to get the world to become a better place. And so in showing them abstraction as applied to those questions, it not only helps them understand those questions, but also piques their interest in abstraction and convinces them that abstract math can be relevant to them as well.
Strogatz (29:47): So at the risk of going too much into algebra, there was one passage in your book that really struck me, where you talk about a class of relationships called equivalence relations. And the interesting thing that you said about them— I mean, I’ve certainly heard about them my whole life and in math — is that they might be too well behaved to be what you call broadly expressive. And I feel like this is a good topic for us because we said, you know, in my intro, I said that, is there math beyond the equal sign? And it feels like this gets us into what’s sort of wrong — if I can put it that bluntly — what’s a bit inflexible or somehow too rigid or something about the concept of equivalence. But I’d like to hear what you have to say about it. I mean, maybe if you could just remind us or teach us what is what are equivalence relations? And then what’s wrong with them from this point of view? What does it mean to be too well behaved, to be broadly expressive?
Cheng (30:43): Yeah. An equivalence relation is, is something that we study properties of relationships abstractly. And so we think about types of relations such as “has the same birthday as,” that’s something that you can ask about any two people. Does Person A have the same birthday as Person B? Or we can say, “is the same age as,” that’s another type of relation, or “is older than.” All of those are examples of relations, and that we can then study the properties of those relations, not among individual people, but the type of relation itself.
(31:15) So an equivalence relation is one that satisfies certain properties. One is that everyone is related to themselves. So, does everyone have the same birthday as themselves? Yes, they do. But is everyone older than themselves? No, they are not. In fact, no one’s older than themselves. So that’s the first property, which is called reflexivity.
(31:33) The next property is called symmetry, which is if A has this relationship with B is it necessarily true that B has that relationship with A? So if Person A has the same birthday as Person B, yes, it is necessarily true that Person B has the same birthday as Person A. But if Person A is older than Person B, then Person B is not older than Person A; it’s not a symmetric relation, actually, it’s an anti-symmetric relation, because the exact opposite is true backwards.
(32:00) And then the third one is transitivity, which is, can you transfer across a person in the middle? So if Person A has that relationship with Person B, and Person B has it with Person C, does that mean Person A has it with Person C? So if he has the same birthday, as B and B has the same birthday as C, you can deduce that A has the same birthday as C. And if Person A is older than Person B, and Person B is older than Person C, then yes, you can also deduce that person A is older than Person C. But for example, if A is the mother of B, and B is the mother of C, that does not transfer: A is not the mother of C. But there is a sensible relationship between them, which is that A is the grandmother of C. And so there are many interesting relations that don’t satisfy these properties at all. And that’s because those properties are, as, as you said, they are much too restrictive to allow for a great expressivity.
(32:51) And so it turns out that having an equivalence relation is just like putting people in pigeonholes. It’s the same as just partitioning your whole world into completely separate boxes and shoving people in those boxes, and you don’t allow for any gray areas, you don’t allow for any people to straddle boundaries or move between places. So it is very, very restrictive. And when you have a lot of laws, I mean, it’s a bit like in life, really — laws are necessary to keep a little bit of order. But if you have too many laws, then you’re so restrictive, that no one can express themselves. And that’s true in society as well. But if you have too few, then you may end up with anarchy.
(33:31) And so this is also true in math. And that the kind of relations that we want to study in math go way beyond equivalence relations and the ones in life also as well. I discussed this with my students, and we talked about “is in love with,” and so I go okay, is that reflexive? Is everyone in love with themselves? Or even just “is a friend of” — is everyone a friend of themselves? Well, sadly, some people are not friends with themselves. And then we talk about whether friendship is symmetric. You know, if A is a friend of B, does that mean B is a friend of A? “In love with” is definitely not because that gives us unrequited love. But is it possible to have a one-way friendship? Are you really even friends with someone if they’re not nice to you back? It’s a kind of an interesting question.
(34:10) And then transitivity is what social media wants to impose on us. They want us to be friends with all the friends of our friends. But that’s not necessarily true. And so category theory enables us to study more expansive versions of relationships where the relations don’t have to transfer literally across people in the middle. There’s just something that we can say that’s sensible, so that if we have a relationship between A and B, and one between B and C, there is then something we can say about A and C, like “grandmother” or “uncle,” so that if you go to the brother of your father, we call that an uncle. And so in normal life, in family relationships, we have words for compiling those relations up. And we want to do that in category theory, as well.
(34:57) And that slight shift enables us to express way more situations. So many situations that the ones that the original founders of category theory had in mind are just a tiny proportion of what it’s now used for.
Strogatz (35:16): There are people who tell me that in another part of mathematics known as algebraic geometry — which sounds like it’s combining something about algebra and shapes, as it does — that algebraic geometry was revolutionized by an influx of ideas from category theory and the people like [Alexander] Grothendieck and so on. Would you be able to tell us anything about that? I mean, that seems to be one of the great success stories of the field as a part of mathematics.
Cheng (35:41): Yes, category theory, as you said before, is often about making connections between different things. And that can be individual things, or entire branches of mathematics. And I think one of the powerful aspects of abstraction that I love is how it enables us to zoom in and out. It works at many scales, because it’s not dependent on the scale. So you can use it to study individual objects, but you can also use it to study whole classes of objects.
(36:08) And one of the big insights from the beginning of category theory was the idea that totalities of objects — worlds of objects, mathematical objects — are mathematical objects in their own right. And this only works for abstract things. Because if you think about, say, birds — OK, so you can study a bird. And then if you look at a flock of birds, that is not a bird. A flock of birds is a different thing. Whereas if you look at a mathematical object, like a topological space, and then you look at the totality of that — the whole community of topological spaces. That is itself a mathematical object. So a single topological space can be expressed as a category, but also, the totality of topological spaces can also be expressed as a category. And then the totality of groups can also be expressed as a category. And then category theory gives us a way not only to move between topological spaces, but to move between the whole world of topology and the whole world of groups.
(37:06) And then for algebraic geometry, what we’re thinking about is not topological spaces, but sort of geometrical spaces. And the difference between geometry and topology is really what we count as the same. And so we’re coming back to the idea of equality versus equivalence, because a doughnut is manifestly not equal to a coffee cup. It’s just that there is a point of view in which they are the same. And that is the topological point of view. But it is not the geometrical point of view. The geometrical point of view says that things are the same; we do need to take curvature into account. Topology only really takes connectedness into account. (This is a, this is a simplification. But I think that’s the essence.) Whereas geometry takes curvature into account, and that means that we count different things as the same.
(37:52) And the thing is that in life, we are very flexible about what we count as the same in some sense. And my wonderful Ph.D. advisor, Martin Hyland, often says there is a sense in which — it’s a kind of catchphrase, but it’s not just a catchphrase, it’s also a point of view, it’s a way of thinking — where we always remind ourselves that there is a sense in which one thing is true, and there is also a sense in which another thing is true. So there is a sense in which we can count a coffee cup and a doughnut as being the same. Of course, there’s also another sense in which they’re not the same. But we get to make those choices in math. And I think there’s definitely not enough emphasis in mainstream math education on our ability to make choices. And the fact that we get to do that, and that impulse, that imperative, is usually not given to students of math. The choices have been made for them. They are told, “We’re going to do these things. Do it like that or you’ll be wrong, and you’ll fail this test.” Whereas in abstract math, we go, “Let’s choose! What are we going to count as the same?”
(38:51) And that’s what equivalence is about — it’s a more flexible notion than equality, because it’s saying we’re going to make a choice in this context of which things we want to count as the same just for now to give us a point of view, and we are going to study what happens in the world if we treat those things as the same. And I think that that has huge lessons to teach us about the world as well, because we talk about equality and equity in the world. And then we get into very divisive arguments, because some people say, “Well, men and women aren’t the same.” That’s true. Men and women aren’t the same. Because if they were the same, we wouldn’t need the words “men” and “women” at all. And maybe at some point in Utopia, we can get away from talking about men and women, but at the moment, we can’t. And the thing is that there are differences. The question is, when should we treat them as the same?
(39:37) And we get to make that choice. And it’s really important that we make a good choice, if we care about those things. And I think the point is, like in category theory, usually not about their intrinsic characteristics, but about the roles that they can play in society, and the roles that they can play in context. And in almost all contexts, they can play exactly the same rules, and so we should treat them as the same.
(40:02) And in category theory, when you have equivalent objects, or in a basic category, they’re called isomorphisms. And in higher dimensional categories, they’re called equivalences, which is, as we add more dimensions, we get even more nuance. And the whole point is that the category does not see those things as different. The category actually can’t tell those objects apart, because they can play the same role in that category.
(40:26) And I think that we should be like that toward humans as well. If people can play the same role, there isn’t a difference that we need to make between them. And in fact, I’ve taken this so far that I like to say this is why I can’t remember what people look like, because all I remember is the role that they play in my life. And so once we start interacting, then I can remember who they are by their interaction, rather than by what they look like. And there’s even been a situation where there was someone who was quite obnoxious towards me. And I recognized that particular brand of obnoxiousness. And it reminded me of someone else who had been really obnoxious towards me. And then when I met a third person who was obnoxious to me in exactly the same way, I sat down, I went, “Wait a minute.” I trolled through years of emails and realized they were all the same person. And it was actually just their form of interacting with me that I had remembered, not their name.
Strogatz (41:26): That’s an amazing story. That’s great. Well, I’m sorry about the obnoxiousness part. But I love the insight that came from it.
(41:35) But now, you mentioned higher category theory, just in passing, and that’s a phrase I’ve been hearing lately. I wondered about that. Is it related to when you mentioned earlier, the category theory is itself a mathematical object? And so you could use category theory to study category theory? Is it connected to that?
Cheng (41:54): Yes, yes, bingo! It’s connected to many of the things we’ve been talking about. And yes, if you do a theory of categories themselves, then you need another dimension, because categories already had another dimension. So there are many impulses that push us into higher dimensions. And that’s one of them — that if you want to study the categories themselves, you kind of need another dimension to deal with it. And that extra dimension comes from thinking about relationships, because that really is a higher-dimensional thing from just thinking about objects. If you think about objects in isolation, we can think of that as a zero-dimensional. It has no dimension, it’s just blobs. Whereas if you then think about relationships between them, it’s like making paths between different things. And that’s a one-dimensional relationship.
(42:38) So then you might say to yourself, well, shouldn’t we then think about the relationships between the relationships? If our point of view on life is that we should put everything in context, shouldn’t we put the relationships themselves in context as well? And that urge is an urge that pushes us into higher dimensions, where we also want to think about relationships between relationships. And then guess what? We go, “Well, what about relationships between those?” and then we go, “Oh, what about relationships between this?” And then if we never stop, then we get to infinity. And that’s how we get infinite dimensional categories — where we go, “We should never stop thinking about relationships between things!” And each of those dimensions adds nuance to the situation, just like — in arguments, we could do this as well. Where if we’re, say, comparing books, so you might go, “Oh, I think this book is better than that book.” And I go, “Oh, I think this book is better than that book,” and then we could, if we weren’t very nuanced about it, we would just yell at each other and go, “Oh, you’re so dumb, I can’t believe you like that book.”
(43:31) But if we were a little bit — if we were one-dimensionally nuanced, we could, we could talk about senses in which we could say, “Well, this book has a more interesting plot, but that book has more interesting characters.” And then we could acknowledge that one of us is more interested in character development, and the other of us is more interested in plot. And then we could compare those, and we could go “OK, well, what do we like about plot as opposed to character?” And then we could keep going as many times as we can, to get more and more nuance into our situation.
(43:59) The trouble is that it also adds a lot of complexity. So it’s very difficult to deal with infinite dimensions. And so we try and stop. So we try and make lower-dimensional approximations of the higher-dimensional things, which is why one-dimensional categories are still helpful and they’re illuminating. But there has been more of a push to go into higher dimensions, first of all, because we are studying many higher-dimensional things in other parts of math now. And secondly, because once the whole theory of one-dimensional categories is developed, then it becomes easier to do the higher dimensions because we’re better at the lower dimensions. So two-dimensional category theory got quite well developed to deal with one-dimensional categories. And now that we’re more comfortable with two categories, we’re kind of going into the higher dimensions, because everything gets easier when you’re comfortable with it.
Strogatz (44:49): Very interesting. It’s true that it does work that way. I mean, when you were just riffing about the idea that when you get comfortable with something, even if… Like, you mentioned in the book, that even the concept of, like, the number three. Or maybe your example was two, but I could probably generalize. I think you’d say something — you know, two bananas and two apples, you know, when you’re a little kid and you’re counting my two fingers, or these two sounds. At some point, you just start to inherently understand the concept, the abstract concept of two — not in reference to any particular thing, just two. And we tend to forget that two, the number two, is an abstraction.
Cheng (45:30): Right. And people who say that they don’t like abstraction, or that category theory is too abstract— well, maybe everyone has some kind of limit that they have reached so far about what abstraction that they are comfortable with, or what they want to do. But we can all expand the abstraction that we’re comfortable with. And we may also expand what abstraction we want to do, if we realize that it can help us in more ways. Just like my students — they don’t really like numbers. I mean, honestly, I’m not that keen on numbers, either. Numbers are really boring; that’s the whole point of numbers is that they’re boring. We boil something down to a really boring essence. And I’m interested in things that are richer than that. And my students didn’t feel interested in those abstract concepts.
(46:14) But they’re really interested when we get to talk about social structures, or to think harder about interactions between people. And then that gives them the urge to think about more abstractions. Some of us like abstraction just for the sake of it, because I love abstraction. But some people need to be drawn into it more by what it can do for us.
(46:36) And I would just like to add, since you mentioned it at the beginning, that I personally don’t like calling category theory “abstract nonsense.” That phrase was coined, I think, as an insult at the beginning, because some mathematicians thought it wasn’t doing anything. And in a way, it’s not doing anything. But that’s kind of the whole point — what it’s doing is shining light so that other people can do more things. And then I think there was a move to reclaim the term abstract nonsense. As is often the case when, when people feel insulted, one thing you can do is reclaim the insulting term for yourself. And so people have reclaimed it, but I don’t like that because it’s not nonsense. It’s abstract wonderfulness. It’s abstract deliciousness. It’s not nonsense at all, because it helps us with things.
Strogatz (47:16): Oh good for you, bravo. I know, I realize it’s a very provocative phrase. And I wanted to provoke you a little. And I love how you rose to the occasion. But you’re right, “abstract wonderfulness.” Let’s go with that.
Cheng (47:29): I think that if we get comfortable with abstraction, then we don’t need to use funny words to describe what we’re doing. We do some abstract calculations. And then we come out with this insight. And I think that understanding that numbers were already an abstraction can be very helpful for anyone who is skeptical about abstraction or thinks they can’t do it, because everyone can do it. Because we all do it the whole time around us in life. I think it’s a very human urge, actually, where math can seem like it’s a very contrived thing, and it sort of is. But it’s also, I think it’s based on a very human urge.
Strogatz (48:05): That’s a beautiful place to end. Thank you for helping us see, as you call it, the joy of abstraction. I think we can all celebrate it now. Thank you, Dr. Eugenia Cheng.
Cheng (48:16): Thank you for a very interesting and in-depth conversation.
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Strogatz (48:40): The Joy of Why is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons foundation have no influence on the selection of topics, guests or other editorial decisions in this podcast or in Quanta Magazine. The Joy of Why is produced by Susan Valot and Polly Stryker. Our editors are John Rennie and Thomas Lin, with support by Matt Carlstrom, Annie Melchor and Zack Savitsky. Our theme music was composed by Richie Johnson. Julian Lin came up with the podcast name. The episode art is by Peter Greenwood, and our logo is by Jaki King. Special thanks to Burt Odom-Reed at the Cornell Broadcast Studios. I’m your host, Steve Strogatz. If you have any questions or comments for us, please email us at [email protected]. Thanks for listening.