CLEARER THINKING

with Spencer Greenberg
the podcast about ideas that matter

Episode 185: Rethinking what it means to learn math (with Eugenia Cheng)

November 23, 2023

What should the goals of math education be? What does it mean to "think well"? Is math real? Why are feelings of bewilderment or confusion so common in math classes but not as common in other subjects? Schools now generally offer reading and writing instruction separately — even though both are important for language use — because the skill sets they require can differ so widely; so how might math education benefit from drawing a similar distinction? What should math classes impart to students that will enable them to engage as citizens with complex or controversial issues? What does it mean to ask good questions in math? Can math teach empathy? What is category theory? Can most people learn most things if they just have the right teacher and/or educational materials?

Eugenia Cheng is a mathematician, educator, author, public speaker, columnist, concert pianist, composer, and artist. She is Scientist In Residence at the School of the Art Institute of Chicago. She won tenure in Pure Mathematics at the University of Sheffield, UK, and is now Honorary Visiting Fellow at City, University of London. She has previously taught at the Universities of Cambridge, Chicago, and Nice, and holds a PhD in pure mathematics from the University of Cambridge. Alongside her research in Category Theory and undergraduate teaching, her aim is to rid the world of "math phobia". Eugenia was an early pioneer of math on YouTube, and her videos have been viewed around 15 million times to date. She has also written several books, including: How to Bake Pi (2015); Beyond Infinity (2017); The Art of Logic (2018); x + y : A Mathematician's Manifesto for Rethinking Gender (2020); The Joy of Abstraction: A Exploration of Math, Category Theory, and Life (2022); Is Math Real? How Simple Questions Lead Us to Mathematics' Deepest Truths (2023); and two children's books: Molly and the Mathematical Mysteries and Bake Infinite Pie with x + y. She also writes the "Everyday Math" column for the Wall Street Journal and has completed mathematical art commissions for Hotel EMC2, 6018 North, the Lubeznik Center, and the Cultural Center, Chicago. She is the founder of the Liederstube, an intimate oasis for art song based in Chicago. As a composer she has been commissioned by GRAMMY-nominated soprano Laura Strickling and is one of the composers for the LYNX Amplify series, setting work by autistic poets who are primarily non-speaking. Learn more about her at her website, eugeniacheng.com.

SPENCER: Eugenia, welcome.

EUGENIA: Thanks.

SPENCER: You and I are both lovers of math, so I'm excited to talk about one of my favorite subjects with you today. And I think you bring a really interesting perspective to math and math education. So why don't we start with the question: what should math education be about in your view?

EUGENIA: Thanks. Well, if I may, I'm going to answer what it should not be about, first of all. It should not be about forcing children to memorize things that they're really never going to use.

SPENCER: And why do you say that? Would you say that most of math education today is really about memorizing things that people are not going to use?

EUGENIA: I think, unfortunately, that's what it can come across as. And for too many people, that's what their main memory of math education is.

SPENCER: What about things that we can use? Let's say, memorizing times table. Presumably, those will come up in everyday life. Would you say that it's fine for math education to be about memorization in cases where someone is likely to encounter it again?

EUGENIA: Here's the thing. I never use my times tables. And I think that many people never use their time tables. And that's why they get the idea that it's all completely useless. And this is a controversial thing for me to say. Lots of people get mad at me when I say we don't need timetables, but we're all carrying around calculators in our pockets. There are much more important things that math can do for us than stuff that our phones could do instead.

SPENCER: Okay, so what are those important things? What can math do for us?

EUGENIA: I think that math should not just be about numbers and equations and solving specific problems. Math is about using our brains to think well. And it does so happen that numbers and equations and solving specific problems is one form of thinking well. But if we only learn to solve specific problems, then all we learned is, well, to solve those specific problems. It's not very transferable. Whereas if we learn how to use our brains in a clear, logical way, then that can be used for absolutely anything.

SPENCER: So let's unpack this idea of thinking well. What to you does that mean that math helps us think well?

EUGENIA: I think that every academic discipline has a point of view on a way to think better. And the point of view of math is to use logic and logical frameworks to decide what counts as good information. And I think this is so important in the world at the moment, because we don't have a lack of information. In the olden days, you know, hundreds of years ago, it was hard to get ahold of information. So having knowledge was important. Whereas now we've got the opposite problem. There's way too much information everywhere. And so it's much more important to have ways to decide what is good information, and what is bad information. And also to have ways to communicate that to other people.

SPENCER: When you think about the kind of thinking that math teaches, it's a very deductive form of thinking a lot of the time. Would you agree with that?

EUGENIA: Yeah, that's what logic is. Logic says, you start with something that is known to be true. And then you see what you can deduce from it using just logic. So you take small steps to make sure that your argument is really secure.

SPENCER: In that type of thinking — and I think it's clearly very useful — but I also think it's quite limited. Because so often in the real world, we don't have a set of premises that are completely unattackable, where we can just say, "Okay, take this set of premises, let me deduce all the conclusions from it." So while I do think it's a useful, powerful set of tools, it also feels to me limited in its real world applicability. What do you think about that?

EUGENIA: I think it's limited in the sense that it can't do everything, and nothing can do everything, right? And I'm definitely not trying to claim that math can do everything. But what it really can do is be used anywhere to help us get started in understanding what counts as a good argument. Because even though we don't know what necessarily is a really, really secure starting point, logic enables us to say what we can deduce from that if it does turn out to be true. And it also enables us to understand why people think things. So it's a forwards and backwards process. And I think that what isn't limited about it is its scope, the fact that we can then use it anywhere.

SPENCER: Yeah, it's pretty cool that in math, you can get a whole bunch of mathematicians in a room together, and they can kind of agree, "Oh, yeah, if we start with these premises, we come to these conclusions," which feels like it almost never happens in other areas of life, or at least much less often, right? It's kind of an almost unique aspect of math.

EUGENIA: Yeah, that is a really cool thing that mathematicians are able to agree on what counts as true. And if you want to say to another mathematician that they're wrong, according to math, you have to point out a flaw in their logic. And once you've done it, then everyone agrees that they were wrong. It's a very weirdly agreeable situation. And I think one of the consequences of that is that some people think that it's just a conspiracy, because there's a level of agreement that you don't see anywhere else in the world. And they think that the only way that could possibly happen is if there's some kind of conspiracy or brainwashing going on. And that is, unfortunately, what happens when people haven't been shown what math really is enough. And that's why I think it's so important for us to show what math really is so that people can understand, "Oh, it's because there's a really secure framework for deciding things. It's not because there's a giant conspiracy."

SPENCER: It's funny, because then when you get all the way down to the origins of math, and you start saying, "Well, are the premises really true? How do we know?" Then it all goes to shit. [laughs]

EUGENIA: Well, because the thing is that mathematicians are not claiming the premises are true. We're just saying, if they are true, there are a whole load of things that follow from that, so that then we can look around us in the world. And anytime we see that being true, we can conclude some things. So we're not actually saying that these things are true.

SPENCER: Yeah, it's funny, because I feel like in the general world, people could benefit a lot more from that way of thinking about things like, "Okay, well, you say this thing is true, what do you actually mean by that? Let's agree on what terminology we're using and what set of premises we're working with. Then, we can agree on what's true from that." But it just so rarely happens.

EUGENIA: Yeah, agreeing on your starting point is so crucial. And there are so many times I look at divisive arguments in the world. And I think, "Oh but wait, everyone's just starting from a different definition." And it's really frustrating. But then often, when mathematicians get really precise about definitions, everyone goes, "Oh, mathematicians are so pedantic." And it can seem a little bit pedantic to really pin down what definitions are. But how can you even have a sensible argument, if you're all talking about different things? And unfortunately, I think that's what goes wrong in so many arguments in the world that everyone's actually talking about something different.

SPENCER: Yeah. And you see this even in really hot button topics like abortion, where someone will say, "Well, a fetus is alive, so aborting it is murder." And it's like, you just get into these semantic debates where it's like, "Well, you're not even using the words the same as your opponent. You're literally having two different conversations, and it happened to be directed at each other.

EUGENIA: Right. And I think it's funny that you said even in hot button topics, I think it's especially in hot button topics. Everyone picks the definition that serves their argument the most.

SPENCER: Yeah, exactly. It's interesting to me that you point to logical thinking, like deduction from premises, as sort of the key math thinking skill, which absolutely is. There's no question about that. But when I think about the way that math made me a better thinker, the thing I come to first is actually something different than that, which is, I first think about how math is a language for patterns. And so, you kind of look at the world, and then you notice a pattern, then you turn that into math that represents that pattern. And then you can use the math to think about the thing in a very rigorous way.

EUGENIA: Yeah, that's a starting point for math. So what I always say about math is that we're trying to use logic. And unfortunately, most things don't actually behave according to logic. So before we can use logic carefully, we have to do abstraction, which is where we forget certain details about a situation in order to turn it into a kind of idealized situation where things do behave according to logic. And pattern spotting is really a form of abstraction. Because what you're saying is that something going on over there is somehow the same as something going on over there, if you just ignore a few of the details.

SPENCER: Yeah, I think sometimes people get this backwards. Like, sometimes people think, "Well, we know that one plus one equals two, because if I have one ball and another ball, and I put the balls in a bag, now there's two balls in the bag." To me that's almost exactly the reverse. It's like, "No, what you're doing is you're looking at the world, and you're noticing a pattern that behaves like addition. And then you turn that into addition, so you can think about the thing in the world by generalizing and subtracting away the fact that it's balls in a bag. So it's not a proof of math, it's how you actually use math in the world.

EUGENIA: Right. And in fact, that is a pattern that has been spotted, where if you take one ball and another ball, you get two, if you take one banana and another banana, you get two, if you take one apple and another apple, you get two. And so someone said, "Okay, let's abstract from that situation, there's a pattern going on. And we'll call that one plus one equals two." But then you start noticing limits to where that works. And there may be some situations where you can't say that. So for example, if you put one color of paint and another color of paint in a bag, you do not get two colors of paint. First of all, you get a leaky bag. But secondly, you get one color of paint, because they are mixed together. And so there are scenarios in which that doesn't work. And so then math says, "Okay, well, what can we say about those situations? And actually, maybe we should have put some limitations on our one plus one equals two thing, and said, Oh, only in certain circumstances does one plus one equal two actually. It's not an absolute truth. It depends on what context you're in."

SPENCER: I guess I would put it slightly differently. I guess I would say that if you define plus as the standard operator over the real numbers, then one plus one always does equal two. It's just that when you're looking at the world and mapping it onto sort of adding things together, that's not always the right representation. Sometimes you want to use a different operator to reflect adding these together. Like when you're adding paint together, you're not using the standard plus operator that you would usually think about, you may be using a different kind of addition operator.

EUGENIA: Right, but what abstract mathematics does is it says that we're doing things in different contexts. And that the plus operator can work in different contexts, and it has different results. And so it's a different point of view, but it's a slight shift that gives us a bit more flexibility, because we're not so stuck on one thing being attached to one concept. Because the concept of addition is quite broad. And so, we like to see it as something we can think about in different contexts producing different results, rather than just fixing it into one world.

SPENCER: Would you say that there's some aspect of addition that makes it addition as opposed to just calling it something else?

EUGENIA: That's a really great question. And that's one of the things I talk about in my book, "Is Math Real?: How Simple Questions Lead Us to Mathematics' Deepest Truths" So it's right there. There's a simple question: Can anything count as addition? And it leads us into some of the deepest mathematics in my research field, which is category theory, a very, very abstract form of math. But it comes from thinking about what should count as addition, and why it counts as addition?

SPENCER: Do you want to just give us a taste of that? I know it's a very difficult field to talk about.

EUGENIA: Sure, I'd love to. Well, when you first do addition with small children, what you typically do is you take objects, and you put them together in front of their eyes. So if you want to talk about two plus three, you take two things and three things and you put them together. And actually, that's what abstract mathematics does. It says any operation that can consist of taking two collections of things and putting them side by side, without combining anything, and without anything spontaneously combusting, then that is a possible form of addition. And so for example, if you pour paint into another paint, then it has kind of combined, and so that wouldn't count as that form of addition. Whereas if you take, say, a shape, like a triangle, and you put it next to a circle, but you don't make any of it touch, because that would be like combining it, then that is another possible form of addition. And that is quite a profound concept in my research field category theory, which is called taking a coproduct. And we can do that with multiplication as well. And it may be a bit more vivid when we think about multiplication. Because if we think about how we multiply with objects, if we want to do two times three, for a small child, we might take three objects, and then put them in a grid with another three objects so that we have a two by three grid of six things. And if we think about multiplying shapes, you might go, "What? How can you possibly multiply shapes? Why should anything count as multiplying shapes?" But what you can do to say multiply a circle by a straight line is you sort of wave the circle through the air in a straight line. And if you try and imagine doing that, what you get is a cylinder. And if you wave a circle through the air in a circle, then you get a doughnut shape, which is technically called a torus. And this is how we do something that resembles multiplication of shapes, inspired by how we do multiplication of numbers, using objects or counting blocks, or the things that seem very childish or childlike when we first start doing arithmetic with children. But those childish ways of doing arithmetic lead us into those deep truths of mathematics, in a way that we wouldn't do if we were just the kind of person who went, "Oh, two times three is obviously six. Duh?"

SPENCER: Yeah, that's really cool to think about. So the idea is that you can take these two shapes, and you can define this addition-like operation, you're going to find this multiplication-like operation. And then you can say, "Well, this is sort of the deep thing that makes addition addition and makes multiplication multiplication, and then we can prove things about this very general form of addition and multiplication that will apply to the usual form, but also apply to all these other versions of it as well. Is that right?

EUGENIA: Yeah, exactly. And so it's all about, when we prove something in a more general setting, it's all about saving ourselves some effort because mathematicians are fundamentally very lazy. We don't want to do the same thing over and over again. So if we see something similar going on in different contexts, we'll go, "Okay, I want to prove this one time, so I can then use it in all those different contexts instead of having to do it over and over again." I always like to think it's a bit like building a dishwasher so that you don't have to keep washing your dishes. You can just get the dishwasher to do it.

SPENCER: How much of it is driven out of laziness versus love of generalization and abstraction?

EUGENIA: I think it goes together. I might avoid doing something because I feel like it's so boring and I don't want to do it. Whereas someone else might find it really fun. So for example, I do know that people exist who really, really love vacuum cleaning. I cannot stand vacuum cleaning. And if I never had to vacuum clean ever again in my whole life, I would be delighted. But some people find it therapeutic. And I was talking to someone the other day who does long multiplication as a form of therapeutic meditation. I absolutely do not do long multiplication as a form of therapeutic meditation. And also, by the way, it wasn't a mathematician who said this, it was just somebody who said, "Oh, I love doing long multiplication."

SPENCER: I don't think I've ever met a mathematician who loved multiplication.

EUGENIA: No, exactly. Mathematicians mostly hate doing arithmetic, because it seems like a sort of futile use of our brain power when something else can do it. But I admit that there are some futile things that I love doing in math. So there are some proofs that I find so beautiful, I'll just do them again, even though I've done them before, they're not new, all mathematicians in the world know how to do them, I still find them so satisfying to just do. It's a bit like going for a walk somewhere that's very beautiful that you've been before, you wouldn't — well, most people probably won't go, "Oh, I don't want to go on that walk again, I've seen it before." — you might go, "Oh, I really want to see that view again, I'm going to go again."

SPENCER: That reminds me of this feeling I sometimes have in math that we're kind of exploring the space of stuff that sort of already there in a sense, like every time you return to the proof, it comes out the same way. But there's also sort of this infinite variety of things we've never looked at before, but they're sort of waiting there for us to look at them.

EUGENIA: Right. And every time we come up with a new way of thinking about something that generates more things. And so it's a bit like climbing a mountain and then discovering that actually beyond the mountain, there's a whole 'nother mountain range, and that somehow in math, miraculously, every single time you climb up the next mountain, there is a whole 'nother mountain range behind it. And I think it's because math is so abstract. So every time we come up with a way of thinking about math, that is actually a new piece of math. Whereas, if you're studying birds, and you think of a new way of studying birds, all you've done is come up with a new way of studying birds. It's not an actual new bird. Whereas in math, it's really a new piece of math. And so then that generates a whole new world for us to look at.

SPENCER: Yeah, it's an interesting idea how, at first, math is about things in the world. Like, you might want to figure out how to add up money; that's really useful. Or figure out how to divide up a plot of land. But then, you start asking questions about the math itself, and then you ask questions about the math that's about the math, you eventually are in category theory.

EUGENIA: Yeah, right. And so, I often think that mathematicians, or at least me (or maybe I should just speak for myself), I'm like those toddlers who just never stop asking 'why' about everything, because you're never quite satisfied, because there's always something beyond it that you haven't quite understood yet. And for some people that might, unfortunately, make some people feel stupid, and feel because they don't understand it, that means that they're bad at math. And that's a real shame. Because actually, if you feel it, you don't understand it, but you want to understand it, that's a really, seriously mathematical urge. The thing that is not mathematical is go, "Oh, I understand everything now. I'm done." Mathematicians never do that.

SPENCER: I feel like math is very humbling to study because no matter how good you are, no matter how much you feel like you understand, you know that there's just an infinite amount of complexity beyond your reach.

EUGENIA: Yeah, and actually, all the brilliant mathematicians I know, all think that they're stupid and slow, they don't understand anything, they've been struggling to understand the same things for years. And this is why it makes me so sad that so many people put off math at school because they felt stupid and because they felt they weren't fast enough. Because being fast to get to the end of something is not really anything to do with what I think math is really about. Math is really about having endless curiosity, and being able to see how much stuff there is left for us to understand. And we want to understand it, and we feel the impulse to understand it. And so we keep trying.

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SPENCER: So there's two related concepts. There's math phobia, and then there's math hatred, and I have a theory about math hatred, and I want to run it by you and see what you think. I've met a bunch of smart people who just sort of like they thought math was the worst. Like, they just tried to do it. And they're like, "I can't do this." And then they decide that it's not for them. And I suspect part of what's happening is in most of their other classes, they expect to understand things right away, like in history, they're never confused about what the teacher is saying fundamentally. It never feels like they're just totally lost in history class, right? In math class though. I think the normal experience is that you just don't understand a lot of what's happening. And I think that's totally normal. But I think people don't realize that that's normal. And so it seems like there's something really wrong with them when they're actually studying something difficult math the first time.

EUGENIA: Yeah, I think that's right. And I think that it's made worse by the fact that there's always someone in the class who claims they find it really obvious and make everyone else feel stupid as a result.

SPENCER: Usually, someone who studied before.

EUGENIA: Well, yeah, or somebody who just isn't seeking such a deep level of understanding, I've seen this happen a lot with students that the people who claim to understand it and find it easier, it's not that they actually do understand it better, it may just be that they're not as curious. And so they're not seeking to understand it more.

SPENCER: Right, because with math, there's these infinite levels of understanding.

EUGENIA: Yeah.

SPENCER: Imagine something like the exponential function. What does it mean to understand it? Well, you could draw a curve of it on a piece of paper, okay, that's sort of one way to understand it. You could know that it's its own derivative. That's another way to understand it. You could know how to write it in a series expansion. It's like, it never ends. There's this deeper and deeper.

EUGENIA: Right, right. And then there's the category theory. There's the category theory explanation of it, which goes even deeper, and I don't think I understand it. In fact, I'm grappling with it right now.

SPENCER: Well, I definitely don't understand that one. But it's like, yeah, what does it mean to understand it? Well, the answer is no, you can't fully understand it, all you can do is kind of get deepening approximations in different ways.

EUGENIA: Right. And so it's a real shame that so many people, like you say, they kind of come to be so afraid of it and almost hate it. And I think that sometimes the hatred part comes out of the fear part, it started as fear. And then because fear is such an uncomfortable feeling, it becomes hatred. And the thing that's really sad to me about that is that fear should be for things that are dangerous. You know, I'm afraid of skydiving because I think if I jump out of a plane, then I'm quite likely to die. And so when people are afraid of heights, I can understand that because even if you are behind a glass window, your body might not fully realize that and so think you're about to fall over the edge. So if you're really standing on the edge of a cliff, it's quite right to be scared because it's genuinely dangerous. Or if you wander into a forest and there's a bear (bears don't live in the forest, do they?) If you wander into the forest and there's a snake, (I don't know what lives in the forest that's really scary) but there's something that really might attack you. Whereas math is just ideas. It's not going to threaten your life. What's made it scary, I think, is the humiliation that people have experienced early in their life in math class. And that's really a shame because it's so unnecessary and unhelpful.

SPENCER: Right. I think there's two sides of that. There's the "Oh, other people are gonna judge me or think I'm dumb, or make fun of me because I don't understand." And then there's the self image thing. Like, "Maybe I'm not as good as I thought, Maybe I am truly dumb. It's not just the people who perceive me that way."

EUGENIA: But you know, I don't think it's just other people will think I'm stupid. I think it's that other people have called them stupid, especially for women and people of color, who suffer much more social threat in situations. I've had people laugh at me countless times for asking a really stupid question. And I'm an actual mathematician.

SPENCER: I think that attitude of like, "Don't ask a stupid question because you're going to look bad" is so detrimental to people's learning, especially in math, where if the math is at the right level, you will still be confused a considerable amount of time. And if you don't feel comfortable asking questions to clarify, it's just such an impediment to your ability to progress.

EUGENIA: Right. But I would like to turn that around and put it the other way, because I don't want to blame the people who are afraid. I want to blame the people who made them afraid. Because I am afraid of asking questions. And I'm absolutely not going to blame myself. It's because people made fun of me and made me afraid. And they were horrible to me for it. And I don't think it's my fault that I got put off asking questions because other people were mean to me.

SPENCER: Totally fair. What was your experience like? What do you think motivated the people that made fun of you?

EUGENIA: I think it's usually insecurity, isn't it, when people make fun of other people? And there's definitely a sense in which I make people insecure, because I sort of don't look like the expected look of a mathematician, perhaps. Or because I have more degrees than normal people have or because I am a female person, and I'm not white, and I'm working in a field that's very, very dominated by white male people, or because I got my degrees from a very prestigious university. And so then some people, unfortunately, feel the need to try and put me down and belittle me to get some of their self esteem back. And unfortunately, at an early point in my career, I sort of didn't understand that that's why people were doing it because I didn't see myself as threatening. I still don't really see myself as threatening to everybody. But I've seen other people's reactions to me enough that I do know that it happens.

SPENCER: What is the kind of context in which this happens? Is it in front of a group of people?

EUGENIA: Yeah, it's in front of a group of people. It might be at a research seminar. It might be just in some kind of workshop or a group setting. It might be in the kind of, "Okay, now we're going to discuss something in a group" type situation. And it's so deeply ingrained in me to believe what other people say, because I don't want to be the kind of person who just ignores what other people say and carry on believing in myself. I want to be responsive to what other people say. And so, I take all those things really to heart. And I do understand that if I were a bit more arrogant perhaps, then those things wouldn't bother me as much. And as a result, if I extrapolate that to other people's experience in math class, people who are more arrogant about their own abilities are kind of impervious to those kinds of comments. But people who are sensitive and listen to people around them will really be hurt by that and take it to heart.

SPENCER: What do you think could be done to help that? Because, obviously, the best thing would be if people would stop engaging in those kinds of behaviors. But if we can't reach those people and get them to stop, what can be done?

EUGENIA: I try and put this message out as far and wide as I can, in many of the books I write, including this one, "Is Math Real?" because I want to reach all the people who have been made to feel stupid in a way that I think is unfair, and try to validate them and reassure them that, first of all, that doesn't mean they're stupid, it means that the other people are mean. And second of all, that I have those experiences as well. And I am a very successful research mathematician, and so that shows that being made to feel stupid doesn't prove that you're a bad mathematician, and being slow to understand things doesn't prove you're a bad mathematician. And that thinking you don't understand anything also doesn't make you a bad mathematician.

SPENCER: That's really interesting. So it's about changing their view of what means you're good or not, right? Because people might have an idea that certain things mean they're not good at it, which are in fact completely untrue, because a lot of great mathematicians might have that same trait.

EUGENIA: Right, and so if you think back to normal school math class, most people think that the people who are good at math are the ones who are really fast, who get the right answer quickly, who finished more questions on the test and who score very highly on the test. But that's nothing to do with what I consider to be the actual skills of mathematics, because mathematics is a very, very complex pursuit. And it's really about being able to spot patterns between different situations, and come up with ways of thinking about those patterns. And it's about being able to hold ideas in your head and move them around. And it's about being able to understand logical steps both forwards and backwards, and to be able to pivot and think of things as being the same and different at the same time. Because every time we have an equation, there's a sense in which the two sides are the same, and the sense in which the two sides are different. And that's not even the end of it. But just to start with, that's a very complicated set of skills that's nothing to do with being faster at getting the right answer.

SPENCER: Right, and so different from so much of what we do in the rest of our lives. It's a really, really different skill set. On the point of speed, something that really stuck with me, in college, when I was studying math, one of my favorite professors, one day in class said, "I want you to remember, the people who do the best in the exams are never the people that walk out of the room first." And that really stuck with me. The people that do the best are the ones that are taking their time, carefully checking what they did, not trying to prove that they're smartest so everyone can see them leave the room for people to say, "Oh, they're so smart. She's so smart."

EUGENIA: But also, the people who do best in the exam aren't even necessarily the best mathematicians.

SPENCER: Oh, absolutely.

EUGENIA: Because why should math be done under timed conditions, when you don't have anything to look at? Math is never like that in the rest of life. You never have a two-hour limit in which you have to — okay, there are some very, very slight exceptions. So for example, if you are flying a plane and something goes wrong with it, then you have a very short space of time in which to fix it. Or if you're on a spacecraft and you have to fix it before you run out of oxygen. — There are some very, very specific situations in which there is a time pressure. But I would say that at that point, it's more you're doing engineering at that point, more than just abstract math. Basically, math research is never done under time pressure. And so, testing who can do it right in a fixed time setting is extremely contrived. And it's not a good indicator for very much at all.

SPENCER: Right, and even if we're talking about not research math, but just using math for things that are valuable, helping you with whatever you're doing, that's rarely in a time circumstance either.

EUGENIA: Right. So there is a theory, and I do understand this and I want to address it, that people say, "We should save our cognitive load." So the idea is that every time we do something with our brain, our brain gets tired. And then if we have to do it consciously, our brain gets more tired than if we can do it automatically. So if we don't have to engage our conscious brain, then we won't get so tired. And people use this as an argument for why we should memorize our times tables, because they say, "Oh, well, if you have to think every time you do six times eight, you're gonna get really tired. Whereas if you can do it subconsciously, then you can get further because you haven't got tired by just doing six times eight." And what I say to that is, while that is true, something else is true as well, which is that there are different ways to get something into your subconscious, and memorizing it might not get it into your subconscious as well as internalizing it by learning to recognize patterns. And so, for example, if you see a four by six grid on a muffin tray very often, then you'll start really associating that with 24. And you don't have to memorize it. So the other thing I say about getting things into your subconscious so that you reduce your cognitive load, is that if you reduce your cognitive load, but you increase your hatred of mathematics, then I don't think that's a very good trade off. Because what that means is that — and I've met many people like this, many of them are my students — their hatred of mathematics becomes so strong that as soon as they see anything like a number, they run away. So I don't think it really achieved the thing it was trying to achieve.

SPENCER: Those are good points. It seems like that people just have a lot of misconceptions about what math is really about and what makes people good at it. I do wonder, though, about the usefulness point. Realistically, most people are never going to become a research mathematician, right? You already talked about how math can help us to think logically. But I'm curious, do you see other benefits why everyone should learn math even if they're not gonna become a research mathematician, beyond the sort of logic benefits?

EUGENIA: Yeah, I think there's a couple of things. One is that, I think it's for the good of society. Because if we have many people in society, who are so afraid of math, that they run away from it every time they see it coming, or that they think it's a conspiracy, then we can't communicate important things about the world that are based on science and mathematics. Things like, there's a pandemic coming, it's going to be really dangerous. Because math people who understood exponentials understood that by the behavior of exponentials, if things start multiplying even by a small number repeatedly, then the numbers grow really fast, even though they start slow at the beginning. And then other people said, "Oh, you're just scaremongering, you're fear mongering, you're just trying to predict the future. How can you say that?" or, "You're manipulating numbers," because they didn't understand some of the basic math behind it. And so then we get into real problems trying to communicate, or just things like understanding how probability works. So that, if you say that something is 60% likely, and then it doesn't happen, it doesn't mean you were wrong, it just means that the other 40% happened. And so there's that for the good of society for just being able to communicate science, I think is really important. But it doesn't mean that people need to be able to do it themselves. It just means that people need to be able to follow it, or at least appreciate it or understand that it's coming from a framework and not a conspiracy.

SPENCER: Yeah. So that seems to suggest to me that there are certain basic mathematical ideas that are very important for people to understand, so that they can understand what's happening in society, so that they can be communicated with. Take the mean, for example. It seems to me that without a notion of the mean, it's very hard to communicate a lot of different things like, what is the mean longevity that people live to? What is the mean age when people get cancer? All these kinds of ideas, without that concept of me, it's so hard to take them on board.

EUGENIA: Right. And what's even worse, I think, is that when people don't fully understand or have a very limited understanding of mean, they kind of forget that it doesn't mean that that's what happens to everybody. And that actually, there's a whole distribution, and that the mean is just some kind of abstract notion of a central point. And it's not even the central point, because that's really the median. But I think that the problem with that is that if we try and get people to understand it, by testing them on doing calculations, I don't think that's a good way of getting them to understand it. First of all, because you can learn algorithms for doing calculations without actually understanding anything. And secondly, because if you keep bashing them over the head with it, and then telling them off when they get it wrong, then again, what happens is that they just develop a hatred of it without any understanding. I think that there could be a way of conveying the understanding of it without having to force people to do endless repetitive tests about it.

SPENCER: So what would that look like?

EUGENIA: That's a great question. I think it would look more like showing people things rather than trying to make them do things. And one of the things I say is that when we teach reading and writing, like the most basic, even the really traditional reading, writing and arithmetic thing in education. For reading and writing, listen to that, it's been acknowledged that reading and writing are separate things. Reading is where you understand things that have already been written, and written is where you produce things. And those are separate. Whereas arithmetic is somehow only one thing. And so I think there should be a reading and a writing version of mathematics. There's the version where you can read it and understand it. And then there's a version where you try to produce it, and that we shouldn't cut everyone out of it if they can't produce it. It's a bit like what there's also that in learning foreign languages, right? There's Can you speak it, but also the comprehension, can you listen to other people doing it?

SPENCER: I've never heard anyone propose that before, that's really interesting. It seems to me that when I've come to understand math concepts, it's always almost always been through manipulating them in some way. Like, if someone teaches me the idea of a math concept, but then I don't actually put my hands on it and try to use it for something, I'm not sure that I really understand it at a deep level.

EUGENIA: That's you. And I think that perhaps everybody who's made it through the math education system who still likes math is somebody who understood things by manipulating them. Whereas there are other ways to understand them. And I think that watching other people manipulate them or watching a computer manipulate them. There are so many things that we can do now. We can take different datasets and draw pictures of them very quickly, without you having to do them yourself. And then you can see different shapes coming out without you having to go through the hard slog of doing it. And I think that there's a kind of survivor bias sometimes with mathematics, where those people who were successful at it were a certain kind of person, because that's what the system demanded. And so they're the ones saying, "Oh, well, this is what you have to do to be successful." But what if there was some other way of doing it as well? So I have come to a very deep understanding of a lot of kinds of mathematics, often not by manipulating them myself, but by doing something else, like thinking about them really hard, or listening to someone else do it. And I think that's a valid form of understanding as well, especially if you're not going to try and become a research mathematician, and it's just about being able to appreciate the math that someone else has done.

SPENCER: The word manipulate here may be a bit confusing, because I absolutely agree that you don't necessarily have to put your hands on a piece of paper with a pencil and write out math. But it seems to me there's something where your brain is like, "Okay, I'm going to take this concept, and I'm going to try to apply it, and make sure I applied it properly." And then if you're just doing passive reading, I think it's really hard to understand math, until you've actually tried to apply it to something,

EUGENIA: I would say play. I would say that we should let everybody play with it. Because I think it's when you play with it that you come to understand it. But playing with it shouldn't mean that you're supposed to solve particular questions, or that you're supposed to get certain answers, or do certain calculations, you just mess around with it and poke it and see what happens. And I think that is a really great way to understand how something behaves.

SPENCER: Yeah, that makes a lot of sense to me. I could definitely see someone, for example, interacting with interactive visualization about means that could really give them a deep understanding as they change things, and they see how the mean changes. And they're like, "Oh, okay, I get it now." They don't have to necessarily be solving a problem.

EUGENIA: You could make it something like, "What do you think the wealth distribution in a fair society would be like?" And then you could look at the salaries of people working for a company. And so you could go, "Oh, the CEO earns $100 million a year, and the cleaner earns $8,000 a year." And then you look at the whole distribution in between, and you see what the mean is, and then you try reducing the CEO salary by some amount and seeing how much the mean changes. And then you can compare that with what the median is. And think about your ideal society, and what you think the distribution should be like. And then everyone can do their own thing. So if some people think it's really fair for a CEO to earn hundreds of millions, then they can do that and see what happens to their mean. And if someone doesn't, then they can do that, and see what happens to their mean. So it's really not about trying to get a particular answer, but exploring scenarios to see how this mean thing behaves. And then to see how limited it is. So you could then play around and go, "Oh, what are the wildest distributions you could produce that would still have the same mean?" And so then you can see why it's a very limited way of summing up a whole load of data.

SPENCER: Yeah, I like that a lot. So I would put it on my list of concepts I think almost everyone should learn in math class: the mean of a distribution. But I'm curious, what other concepts do you feel are important so that people can be literate in, let's say like, reading the news, or understanding important societal concepts?

EUGENIA: That's an interesting question. I want to preface my answer with the fact that I don't think that having a list of concepts is the most important thing in math education. I think we can focus too much on trying to get concepts into people's brains, rather than preserving and nurturing a sense of curiosity and appreciation for what math does in general. Because if you appreciate what math does in general, you can learn any concept at any point you need to. What you need is to not hate math, not have a phobia of it, remain curious, and know how to be able to look things up. And so there may be all sorts of concepts that I didn't learn in school, but I believed that I could figure them out by looking them up now. And I think that that's much more important than actually conveying concepts to people.

SPENCER: It's like learning to use a dictionary rather than memorizing every word, right?

EUGENIA: Yeah, exactly.

SPENCER: But maybe in math, it's sort of less clear how to do that for most people. Everyone kind of knows, "Oh, if you don't know a word, you can go look it up." But people are much less familiar with how to look up a math concept."

EUGENIA: Right, and that's why it would be a really important thing to do in education.

SPENCER: The reality is I do this constantly. I'm like reading some paper, and I'm like, "I have no idea what that means. Let me go figure out what the heck that math is."

EUGENIA: Right. I look things up over and over again, even in my own research, because there are some definitions that don't stay in my brain because they don't make sense to me, or I don't use them enough, or there's something arbitrary about them, or it's a word named after some dude, and I can't remember which dude would did what exactly. And so in research seminars, now, I sometimes avoid having to ask what I'm afraid of stupid questions, because I get my computer on and I Google everything.

SPENCER: There's this other issue that comes up, which is that sometimes you don't even know that you're missing an important concept. A simple example would be: you're trying to do something with triangle shapes in your apartment, and you don't realize you need the Pythagorean theorem, right? It doesn't even occur to you that you would need the Pythagorean theorem. And so, in that instance, you don't need to know the formula of the Pythagorean theorem, you need to know that there is such a relationship that you can look up.

EUGENIA: Kind of, except I just want to say that I think there is a very rare situation where you're doing something in your house with triangles, and you'll need the Pythagorean theorem.

SPENCER: I agree, I agree. It's a silly contrived example. But I guess my point is just that, there are times when what you actually need is not a formula, you need a hook to say, "There's some math here that I could learn that will help me with the thing I'm doing."

EUGENIA: Right. And so one really important aspect of math education would be to expand everyone's awareness of what counts as math and where there probably is math going on.

SPENCER: Right. For me, a lot of times this comes up when I'm reading the news or reading about surveys, or I really like psychology papers, and they'll be talking about something. And really what they're referencing is something mathematical like, let's say you're reading a poll, and they say, "We surveyed two hundred people and 10% said this thing. And then, as a mathematician, my brain is immediately like, "Okay, 10%, what's the confidence interval on that?" Whereas, if you've never been trained in that way of thinking, then you might just say, "Oh, 10%, that's a fact. 10%, well, I guess that's just what it is," and not realize, "Oh, wait, there's maybe actually a reasonable size confidence interval around that. And also, wait, how did they actually sample this? Was it just a random sample or not?" It seems to me like the math training there gives you a bunch of questions to ask that you wouldn't even think to ask had you not been through the math training.

EUGENIA: Yeah, and that's a really great point that comes back to the main topic of my book, because the subtitle is: how simple questions lead us to mathematics' deepest truths. Because too often, I think people assume or have been told that math is about answering questions correctly. But a lot of math is about learning how to ask really good questions. And what good means there is questions that will uncover something. I don't mean good questions like you've caught somebody out, because they don't know the answer. Unfortunately, that's sometimes what counts as a good question. But I mean, questions that get to the heart of something that's going on. Like you say about what we should be asking if someone claims that something is true. And I also do that all the time when I'm reading articles about a poll, or about some supposedly scientific result where they say, doing this definitely results in that. You're, "Wait a minute, definitely, in the sense of what some percentage of people that there was some mean, and how confident are they? How significant was it and more to the point, for whom did this thing not work?" It's like one of my pet hates is when people say, "Oh, doing exercise is good for your mental health." Because doing exercise is often not good for my mental health. So maybe on average, it's true. But then I think a really interesting question is, first of all, how did they get this data? How did they control it? Did they control for other issues? For whom didn't it work? And what can we do for those people? And how can we figure out for whom it won't work so that we don't send them off doing loads of exercise that's going to be bad for them?

SPENCER: It's true. There's so much slippage in everyday language, where people will go from "Okay, this study showed that on average, this thing worked too. This thing works, too. This thing works for everybody." Right?

EUGENIA: Right. And unfortunately, there are so many problems causing that, including the way that news sites work now, and how everyone just wants clicks. And so you have to say something that's a really eye-catching headline to get people's clicks. And I suppose if everybody were better trained in not taking those things quite so black and white, then it wouldn't get quite so many clicks. Although I admit, I click on them just to figure out what's wrong with them, which I suppose is what clickbait is all about.

SPENCER: That's funny. I love clicking on scam advertisements for exactly the same reason. I'm just like, I wonder what kind of scam this is. It's entertaining. But obviously, for people without the training, they may not know the right question. So yeah, do you want to explain more about, what does it mean that you're learning to ask good questions through math?

EUGENIA: Asking good questions is often about probing around the limits of when something is true, rather than finding out what the answer is. So like we said about the limits of when one plus one equals two and when it doesn't, that leads to really deep math. Or why can't we do something? Why isn't infinity a number? That's something that children asked me quite a lot. And so then instead of saying, some people go, "Oh, infinity in a number." But instead, we can go, "What goes wrong if we try to consider infinity being a number?" And then you go, "Well, what even is a number?" And asking what a number is, is really profound. And then it turns out, actually, it kind of doesn't matter what a number is, what matters is what numbers do. And so then we ask ourselves, "Okay, what things get to count as numbers and what things don't get to count as numbers." And then they turn out to be loads more things that count as numbers than we originally thought. And so we go off on these winding journeys that don't have a particular end, but we get to discover all sorts of things along the way that we wouldn't otherwise have seen.

SPENCER: That reminds me of one of my own pet peeves, which is when someone says something that limits the scope of creativity or applicability, like, "Oh, you can't take one divided by zero," and they'll admonish people. I'm like, "Ah, no! You can. Try it."

EUGENIA: And so I think we should let people explore those things in math all the time. Unfortunately, math is a lot about play and exploration, maybe in kindergarten, when you're first just given things to play with blocks or counting beings or shapes that you can stick together to make pictures. And then suddenly, it turns into, "You have to get the right answer for this and the right answer for that." All the way, basically, until you're in grad school, and then you're allowed to play it again. And I think that's really unfortunate, and that we should be allowed to play all the way through. And I think that, unfortunately, there isn't time for that. I'm not blaming the teachers. But the system says, "Oh, you have to pass these standardized tests and these standardized tests, you can't play around with stuff, because then we won't be able to stuff all this information into your brain if you just spew out on this standardized test.

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SPENCER: There's another approach to math education, which is different, I think, than the standard approach, but also may be different than what you're suggesting. And I'm curious how you think about it, which is this idea that we should take what students are already interested in, and then use that to motivate teaching math, or whatever it is we're teaching. So let's say they're really interested in roblox. And it's like, okay, can we find ways that math ties into roblox that gets them excited to answer some question related to this thing that they're already interested in?

EUGENIA: Yeah, and I think that's a great approach. I think it's always really important to find out what motivates somebody in order to teach them, and then tap into that. And I actually have this backwards. I've got a version of this the other way around, where I really was put off sport when I was in high school, because I wasn't very good at it. And because people made fun of me, and they humiliated me for it, and I was always letting people down in a team. But only after I got out of those school situations and was doing it by myself, I started getting really interested in understanding the physics of it. So I got really interested in swimming and thinking about the fluid dynamics and getting the fluid dynamics to be better, so that I could swim, not necessarily faster, but more efficiently, with less water resistance. And I got more interested in running when I thought about how my center of gravity was moving. And I got more interested in tennis, when I understood that it was about using the physics of the motion of your body to put motion into the ball rather than just the sheer strength of your muscles. And so for me, if anyone had explained to me the physics behind those sports when I was younger, I would have been much more interested in them. And I think it goes the other way around for a lot of people that they think that math has nothing to do with them, or the questions that they get given have nothing to do with anything that they're interested in. And so they're not interested in it. And I do want to stress that unfortunately, when math gets made into a real life problem, it's often so awfully contrived. Like you buy 87 watermelons, and then you eat 16 of them, how many are left?

SPENCER: The pressing problems of our time. I remember there was some screenshot going around from a textbook that was like, this is Charlie's favorite triangle. Now he gets another triangle. [both laughing]

EUGENIA: That's hysterical because if I did of course start imagining what my favorite triangle would be. I saw some experiments somewhere where they showed people different rectangles to see which ones they liked the best. I thought that was kind of hilarious. I think they were trying to debunk something about the golden ratio, and they thoroughly debunked it. So that was very, very amusing.

SPENCER: Oh, yeah. So much for Golden Ratio bullshit.

EUGENIA: Yeah. So much.

SPENCER: So we've talked about a couple of different reasons why it might be important to teach math. We've talked about it as a way to train your brain, we've talked about it as a way to better be able to understand what's happening in society and be communicated with. What else would you point to? Are there any other big reasons you think math should be taught?

EUGENIA: Here's a reason I believe in teaching math that is quite counterintuitive, and something I only came to recently, which is, it actually helps me empathize with other people. And that's a really, I think, quite surprising answer. But let me explain a bit, which is that if there's someone I absolutely, completely vehemently disagree with, it can be so difficult to do anything except get angry and upset. But the discipline of abstract mathematics, for me, is a way to cut through those emotional responses, and try to follow their logic through to their starting point, according to their framework. And when I do that, then I can understand where they're coming from. It doesn't mean I'm validating their point of view, but it means that I understand where they're coming from, which means that I can always understand why people think the things they do, even if they're absolutely wildly opposed to what I think and really offensive.

SPENCER: That is so interesting, and it's something that I try to do a lot in my life, which is basically saying, "Okay, from this person's perspective, what they're saying makes sense. It makes no sense to me. But from their perspective, it makes sense. What set of premises must they be operating under, so that it makes sense?" Because everyone makes sense from their own worldview. But it's really cool that you feel like math has helped you get to that place.

EUGENIA: I think so because, like we were saying earlier, math is about starting from some premises. And you're not saying those are true, you're saying if I took these as true, what would follow from them. So we get really good at not claiming things are true, but just seeing what follows from certain starting points. And different starting points produce different mathematical worlds for us, just like different games, different rules for a game produce different games. And so understanding other people's thought processes is really like just going into a different mathematical world. You're not saying I agree with their starting point, you're saying, "Okay, I acknowledge that this is their starting point. And so this is how they get to that ending point." But crucially, actually, we usually have to do it backwards. Because often we go into another mathematical world, and we go, "Oh, this weird thing is going on? How can we understand this weird thing from some basic starting points?" And so for people, we can do that as well, "Whoa, how on earth does that person think that? Why do they think that? Oh, let's see, it's because they think this, and that's because they think this," and then you unpack it down to some really fundamental starting point, that might be extremely different from yours, or sometimes it's not, sometimes it can be just one small thing that they believe differently and then the whole thing diverges really dramatically.

SPENCER: Yeah, this seems especially important to me when I'm talking to people with a really different worldview. For example, if I'm talking to a very devoutly religious Christian person, I think it's very important for me to remind myself that that's where they're coming from, and that what I say has to make sense from the point of view of "the Bible is the Word of God, and Jesus died for our sins." And if it doesn't make sense from that point of view, they're just gonna miss the point, or we're gonna be talking past each other. But it's really interesting that you point out that even sometimes, these kinds of subtle differences can lead us off base as well. Even if we start with most of the same premises, maybe we just have one premise that's a little bit different, even just being able to acknowledge that could keep the conversation on track.

EUGENIA: Right. For example, it's something extremely divisive, which I hesitate to bring in, but say universal health care. So some people don't believe in universal healthcare for wildly opposing reasons. But I know there are some people who believe everybody should have good healthcare. It's just that they don't think that universal healthcare is the best way to achieve that, and that they genuinely think that a free market with private healthcare is a better way for everyone to have good healthcare. And that's a way closer position to mine, because at least we're all starting from the point of view that we want everybody to have good healthcare. And that's very different from people who think that no one should have healthcare, that that's a luxury, and that everyone should pay for their own or something.

SPENCER: Right. So then you may find yourself disagreeing with this person without realizing that your premises are actually pretty close together, and you're just disagreeing about some empirical question. So if you can realize, "Oh, wait, we actually kind of have similar goals here." And then identify, "Ah, the source of disagreement is just about which way to organize that gets those goals." That's a much more fruitful conversation, right?

EUGENIA: Yeah. Yeah.

SPENCER: When I think about what math has done for me, another totally different thing comes to mind, which is that I think there's certain sub-disciplines of math that have really influenced my thinking. The two that come up the most are probably probability theory, where I've just come to start to think of everything in terms of probabilities. And also kind of using sort of mental imagery around a lot of things, like if I'm thinking about, what's the chance this thing will go well or badly, I can imagine a distribution in my mind. And is it more like a normal distribution or a fat tail distribution? So that comes up for me a lot. And then another one is optimization theory of: you have a function, and you're trying to find what input to it leads to the highest output. And so you know, to put that in sort of the language of physical reality, you find yourself, you're on a bunch of hills, and you're trying to get to the top of the tallest mountain, how do you do that? How do you go from being somewhere on the hills to the top of this mountain, especially when there's fog all around, and you can't really see very far, and all you can do is kind of see what's right around you right now. Those mathematical ideas have just been really powerful for me, because I find that they apply to all over the kinds of things I'm trying to do. Probability theory, when you're making decisions about running a business or decision, even in our personal decisions. Optimization theory, when you're trying to think about how to make a system better? How do you make this thing as good as it could be? And so on. So I'm wondering, are there things like that for you, like ideas for math that you find you come to again and again in everyday life?

EUGENIA: Yeah, I do think that distributions is one of them. And that it ties into a more general mathematical concept for me, which is: if we can understand more complicated concepts as a single concept, then we can understand more things. So if we can only understand single pieces of data, then we're stuck reducing a whole population to a single number, like a mean or a median. And that's extremely...well, we've lost a lot of information. Whereas if we are able to comprehend a distribution, then we can understand a much larger amount of information as one concept. So, we can picture a distribution, and it doesn't take up very much space in our brain, because it's one concept. And for me, math is about packaging complicated things together so that they become one concept, instead of a whole load of concepts. Just like when we first learned to read, we have to read each letter individually, and it's really slow. And then we start learning to recognize whole words as packages, and then even whole sentences. And then we can get much further because our brains aren't taxed. And so a more general version for me of distributions is understanding contexts for things so that we don't just understand individuals floating around in space, but we put them inside a context, which is the entire experience of their life that has led up to this moment. And that is actually I feel like I've learned it from my field of research, which is category theory, which is all about putting things in context and making sure we're aware of what context we're thinking about in any given moment.

SPENCER: Can you give an example of how context ends up impacting things?

EUGENIA: Yeah, for example, when I have new students in my class at the beginning of the semester, they've all arrived to the same place. They're all sitting in my class at the same university. But some of them have had a huge struggle to get to that point, maybe because they're the first person in their family ever to go to university. And because everyone told them that university wasn't for them and that it was useless. Whereas other people have got to that point on a kind of conveyor belt, where everyone in their family goes to university. So it was just kind of vaguely expected of them that they would go, they don't really know what they're doing there, but they thought they'd better go because everyone else does. And those are two extremely different ways of having got to the same point.

SPENCER: And how does this come up in your math work? Could you maybe give an example of showing that as well?

EUGENIA: Yeah. So in category theory, the basic starting point is that we can understand things according to their relationships with each other, a lot. And that that is in a way more important than their intrinsic characteristics. Because, for example, the number two has certain characteristics in the context of all the whole numbers. But then if we look at it in the context of just the even numbers, it gets really different characteristics. And so, if we think about, say, when numbers are divisible by each other, it depends what we're talking about, right? We say that five isn't divisible by anything except one and itself, which makes it a prime number. But that's only true if we're thinking about whole numbers, right? If we're in the context of all the numbers, including fractions, then five is divisible by everything. And so those are some basic examples about how it's the relationships with what surrounds it that matters more than its intrinsic characteristic. And it's a very small change of a point of view. But category theory, which grew up in the middle of the last century, has really changed the way contemporary mathematics is done. It's quite extraordinary how much it has changed from just a slight shift in a point of view.

SPENCER: That's fascinating. So it's like the fundamental thing about two-ness is not so much the fact that it's two, but more about its relationship to the other elements, to the other numbers.

EUGENIA: Right. And then if we go into a different situation, then it will take on different characteristics. So if we go into a world where, say we're on a five-hour clock, so normally, we're on a 12 hour clock, so when we get around to 12, it's the same as zero, and then we go back to one. But in mathematics, there could abstractly be a five-hour clock, where we go round to five, and then five turns out to be the same as zero, and then we move to one again. And in that situation, two is really different. Two has different characteristics inside there, because we're on a circle that's got five spots on it, and two doesn't go into five. So suddenly, two becomes a bit more weird.

SPENCER: Ah, right, we're using a 12-hour clock, 12 divided by two, it's divisible. And so it ends up with different properties.

EUGENIA: Right.

SPENCER: You've touched a bit on category theory in this conversation. Can you tell us a little bit more about its impact on math? Because I hear more and more about category theory. It seems like a really exciting area, but I think most people would be like, "Category theory? What are you even talking about?"

EUGENIA: Mathematics periodically goes through a crisis where everyone goes, "Oh, what are we doing? We don't know what we're doing." And then they try and make it more secure. And so the foundations of mathematics, where mathematicians go, "Whoa, actually, we don't know what we're talking about. Let's try again." And they go back to deeper first principles. And one of the really first principal ways of doing it was set theory, which happened about 200 years ago, where people thought, "Okay, how can we make sure the math we're doing is really on firm foundations?" But set theory is very, very — I don't know, Well, I don't want to insult the set theorists. I don't know how many set theorists you have who listen to this podcast...

SPENCER: There's only one, but he's very belligerent. So be careful.

EUGENIA: Okay, great. Well, I'm gonna get hate mail. [chuckles] So, set theory, if you even look at how numbers are defined in set theory, it's really, really tedious. And I don't want to be judgmental, but it really is tedious. And so category theorists came along and said, "Okay, let's try this again." Because what we often do in math is we think about maps between things or relationships between things more than we think about the things themselves. And so when we have a set of two things, we often don't care what those two things are. It's like the number two, we don't care whether it was two bananas, or two cookies, or an apple and a banana. That's the whole point about the number two, it doesn't matter. And so it's a set of two things where we don't really care what the two things are. What matters is that there were two things in there. And we can express that in category theory by talking about a relationship between that and the set with one thing in it. And so it becomes more about how those things relate, rather than what's actually inside them. And I think this is true about people as well, that it should be more about how we relate to each other, rather than what's actually in us.

SPENCER: Hmmm. And so this shift in perspective, which sounds like it sounds like a very big shift in perspective, compared to the way that set theory thinks about things, what has that done for mathematics?

EUGENIA: It has enabled us to spot patterns between different subjects. So there are a lot of different fields of mathematics. And each one studies very, very different objects. And so there's things like number theory that studies numbers. And there's group theory that studies symmetries. And there's topology, which studies the shapes of things. And those objects are very, very different from each other. So you might think you have to use very different techniques to study them. But then you think about the relationships between things. And like we were talking about with the addition and the multiplication, we go, "Actually the relationships between numbers are, in some ways, similar to the relationships between shapes and the relationships between symmetries. And so if we study at the level of the relationships, rather than at the level of the things themselves, there are things that we can understand across the board all at once, instead of having to do it separately in each field."

SPENCER: So what's an example of something that has been proven in category theory, like just even a simplified level?

EUGENIA: Well, one thing is how we define things like multiplication and addition, and how we can identify the things that count as multiplication and addition. And category theory came up with a way of doing that, which happened at the level of relationships. But then it turned out to be something that people in different fields had been studying already. They just hadn't phrased it like that. And so sometimes mathematicians say that all category theory does is go around reproving things that people have already done. And it's kind of fair. It's sort of fair, but it's kind of like finding a way to do it that gives us more insight or makes more connections between different places. It's like if you introduce a person to another person, then you haven't discovered a new person, right? The other people knew those people already, but you've made a new connection, and that can be really helpful.

SPENCER: I see. So it's sort of like, there might already be a theory in group theory about something. And there might be another theory in topology about something. And then category theory might come in and say, "Hey, you know what, you're talking about the same thing here. And there's a more general version of the theory that actually shows that those two things you both proved are equivalent to each other, and you just never realized it." Is that kind of the concept?

EUGENIA: Yeah, that's kind of thing. And it's also about building bridges. So that's a really great example with group theory and topology, because a lot of modern fields of mathematics have two words in them. So back in the day, there was algebra, and there was topology. And now there's algebraic topology, because contemporary math has realized that if you can combine two fields of study, you can learn from both of them in both directions. But what you need in order to do it is a bridge from one field to the other. And category theory is a really good way of building those bridges, because it goes up to this more abstract level and says, "Oh, here's a way, in a sense, in which you can count these things as the same." And that constitutes a connection between those different fields that previously seemed to be different. Now, there's a way of getting backwards and forwards, so you can kind of transport things you know about topology into the field of algebra and use the algebra that's already been done. Or, you can do it the other way around. And so then you get to make more progress because you're not just building by yourself, you're kind of working together with another field.

SPENCER: I feel like those are some of the most exciting things in math where you suddenly realize, "Oh, wait, all the stuff we've been doing is equivalent to this other thing that we never thought of it as being related to, and then it means that we have a whole new way to look at all of our problems, because we can look at them through this other lens." One of my favorite examples of this is with the Fourier transform, where you take a problem and you're like, "I have no idea how to solve this." And you're like, "Well, what if we transformed it into thinking about frequencies?" And then suddenly your problem's easy, and you're like, "Oh, my gosh, I had no idea I was solving this easy problem secretly without realizing it."

EUGENIA: Yeah, sometimes it can really be like putting on a pair of glasses and you see better. And you suddenly realize, like when I go and get a new prescription, I don't realize I haven't been seeing quite clearly, because it happens very gradually, right? And then I get the new prescription, like, "Oh, my goodness, everything looks completely sharp now."

SPENCER: So before we wrap up, the last topic I just wanted to talk to you about is about continuing to learn as we get older. Because I think a lot of people have the sense that, "You know, I never study math, it's kind of too late for me." Is it hopeless? Once you get to 30 or 40, is it too late to learn math?

EUGENIA: What is definitely true is that if you believe you can't learn something, then you can't learn something. And the amazing thing is that there is a lot of recent research — I'm not an expert in this, but I've read some things — showing that the brain carries on being extraordinarily plastic all the way through our lives. Now, it may well be that it's the most plastic when we're tiny, and we have to learn so many things so quickly. But unfortunately, many people don't even believe it's plastic, even when we're young. And so many people still think that our brains are hardwired in a certain way when we're born, and that's it. But actually, there's a lot showing that it's extraordinarily plastic, and that the reason that people end up with such different abilities is largely just because of the environment that they've been in, and much less to do with whatever hard wiring means. Although there is some aspect of hardwiring, there's so much that we can train our brains to do. And I think there's even been some people training themselves into quite old age to do things like reverse some of the aspects of Alzheimer's or dementia. And so I think that it's really amazing what we can get our brains to do. And if we give up, then we definitely won't be able to. And this is true at all ages. So if you decide that you're bad at math, then you kind of will be bad at math. But if you decide that you can improve, then you can improve. And I'm not saying it means everyone can become Einstein or research mathematician. But why put limits on yourself? I've just answered my question for myself. But let's just throw that rhetorical question out there for a second. Why put limits on yourself?

SPENCER: Yeah, I think an attitude that I've always had that has worked really well for me is just assuming that if other people can learn things, I can learn it too. And so that just therefore not imposing sort of an artificial limit. But I think a lot of people think that they've proven the opposite. Like they've tried to learn something, they got frustrated, they gave up and so they come away thinking, "Oh, I can't learn that thing." And maybe I'm naive, but my attitude is like, "No, no, no. You just didn't get a good enough teacher or you just needed to put more time into it, but if it was a good enough teacher and more time, it's very, very likely you will learn the thing, unless it's something ridiculously complicated..."

EUGENIA: Like flying.

SPENCER: Yeah, but the vast majority of things, almost anyone could learn with the right teaching and enough effort. But I don't know, am I wrong about that?

EUGENIA: No, I agree. And I think the reason people often put limits on themselves, and I've done this myself, the reason people put limits on themselves is because they're afraid of being humiliated. They're afraid of being frustrated again. And I understand this, because I've done it with sport a lot. And it's because I was so humiliated before. And I just now have been thinking that, like you said, if I had a better teacher, I know all the ways in which people didn't help me in the best possible way. And if I did have some help, I do think I could get better. And actually, this happened to me in some talks I gave about my first book, "How to Bake Pi" where I asked someone to do a juggling demonstration for me to demonstrate the math of juggling. And I would start by saying, I'm terrible at juggling, so someone else do it for me. And someone came up to me at the end of one of my talks and said, "You know, when you say you're terrible at juggling, that's like people who say I'm terrible at math." And I thought about it. And I thought, "Gosh, it really is, isn't it?" So I started saying, "I am trying to get better at juggling." And as soon as I did that — and I know this sounds like a fake fairy tale, but I honestly went back to my hotel room that very day with my juggling balls, and thought, "Maybe I can get better at juggling," — and I was instantly better at juggling. I kid you not. I was immediately better and juggling. And I don't even know how that happened. Because I thought I had practiced really hard before. But I had convinced myself that I was just bad at it. And I've been doing it this summer. I've been trying to get better at tennis, because I really love playing tennis, but I'm really terrible at it. See, so I decided to stop saying "I'm really terrible at it." And I thought, "Maybe if I just watch loads of YouTube videos and find myself a teacher," because I believe I can learn anything as well. So I started watching tons of YouTube videos that did not exist when I was young, because there was no YouTube, and I've discovered that I can get better at it. And so, I think that I agree with you. Apart from things that are physically impossible, like flying or defying gravity, or being taller than you are. And certainly there, I don't believe that I will ever be able to run as fast as Usain Bolt. But I can definitely get faster at sprinting. I can get better at tennis. And certainly, intellectually, I believe that I can understand anything that anyone else can understand. And that is possibly naive. Maybe some people will say it's deluded. But I think that it is a kind of superpower that I have gained by being good at mathematics, by the training in mathematics I've had, has taught me how to understand things and how to use my brain, which has given me this amazing power of confidence to believe that I can understand anything. And I just want to give that to other people as well.

SPENCER: It's a lovely way of putting it. I think that it becomes more plausible that this is true when you start to realize everything can be broken down. Like if you had to learn a complex skill by just doing the whole thing, then, yeah, that does seem insurmountable. But, essentially, everything can be decomposed. Take the idea of a derivative and calculus. You don't have to start with writing out this complicated formula in terms of limits. That's ridiculous. Of course, people can't learn it like that. But there's many different ways to go bit by bit up to the idea. And then each of the bits, each of the little pieces is not that much more than what you already know. And then if you think of it as like you're building a bridge, then it is just about building a bridge made of pieces where each little piece you can understand and suddenly now it seems possible.

EUGENIA: Yeah, that's a great way of putting it. And I think that's such a fundamental principle in mathematics, that we're always trying to break things down into really basic pieces, and figure out how to put them together again. And, in a way, that's the fundamental principle of mathematics: break things down into small enough things that you can understand them. And then you just have to understand these tiny things and understand that way of putting them together. And then, you can understand anything.

SPENCER: Eugenia, thank you so much for coming on.

EUGENIA: Thank you so much. It's been really interesting.

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