Into the Impossible With Brian Keating - Category Theory: Exploring Mathematics’ Deepest Truths With Eugenia Cheng (#362)

Episode Date: October 30, 2023

Is math real? How to bake pi? And how much is x+y, really?  Many people don't like math because they find it too complicated or boring. But math can actually be a lot of fun, and we can find it everywhere in life, even in the most mundane things like baking. And it is through baking that today's guest, Eugenia Cheng, decided to rid the world of math phobia. Dr. Cheng is a renowned mathematician, educator, author, and concert pianist. She's a scientist in residence at the School of the Art Institute of Chicago, where she teaches mathematics to art students. She is an expert in category theory and has recently published a book, Is Math Real?, which we will discuss in detail today! Join Eugenia and me as we explore mathematics’ deepest truths. Key Takeaways:  Intro (00:00) Judging a book by its cover: Is Math Real? (01:10) On the unreasonable power of mathematics in the physical sciences (04:05) If there were no physical world, would math exist? (08:14) The number zero (10:30) Is our brain a massive computer? (17:04) How to Bake Pi (22:38) Category theory (27:07) How to revitalize and modernize education (39:21) Is math created or discovered? (45:12) Outro (49:33) — Additional resources:  🥗 Thanks, HelloFresh! Go to HelloFresh.com/50impossible and use code 50impossible for 50% off plus 15% off the next 2 months. 📝 With a MasterClass annual membership, you can take one-on-one classes from the world’s best for $10 a month with your annual membership, get unlimited access to every class — and even better, right now, as an Into The Impossible listener, you can get 15% off when you go to MASTERCLASS.com/impossible. 🧑‍💻 Visit LinkedIn.com/IMPOSSIBLE to post your job for free! ➡️ Check out Eugenia Cheng:  📚 Is Math Real? By Eugenia Cheng: https://a.co/d/j68vZuU   ✖️ Twitter: https://twitter.com/DrEugeniaCheng/  💻 Website: https://eugeniacheng.com/  ➡️ Follow me on your fav platforms: ✖️ Twitter: https://twitter.com/DrBrianKeating  🔔 YouTube: https://www.youtube.com/DrBrianKeating?sub_confirmation=1  📝 Join my mailing list: https://briankeating.com/mailing_list  ✍️ Check out my blog: https://briankeating.com/blog.php  🎙️ Follow my podcast: https://briankeating.com/podcast  — Into the Impossible with Brian Keating is a podcast dedicated to all those who want to explore the universe within and beyond the known. Make sure to follow so you never miss an episode! Learn more about your ad choices. Visit megaphone.fm/adchoices

Dr. Eugenia Cheng is a renowned mathematician, educator, author, and concert pianist. She's a scientist in residence at the School of the Art Institute of Chicago, or she teaches mathematics to art students. She is best known for trying to rid the world of mathematics phobia by using analogies from baking. Eugenia has written many wonderful books exploring the world of numbers. Is math real? How to bake pie? And how much is X plus Y really?

Tune in to find out once and for all why math is so much fun. Any sufficiently advanced technology is indistinguishable from magic. Open the pod bay doors, Hal. Hello, everybody. We are joined today by a renowned author and mathematical muse, and this is Dr. Eugenia Chang of the Art Institute of Chicago. What a mathematician be doing at an art institute? We'll find out in today's episode.

Eugenia, how are you today? I'm well, thank you. Thanks very much for having me. You've done so many things. You've written so many wonderful and extremely popular books like How to Bake Pie and X Plus Y. And today's book is, is math real? So we're going to be discussing the reality of math. And I know that math is not always real because there are imaginary numbers.

And we'll get into those maybe later on. But, Eugenia, as you know, we have a Bayesian framework on this podcast, how to evaluate a book other than to judge it by its cover. So today, I'm going to ask you to play our favorite game, judging books by their covers. Please explain to my audience how you came up with the title, the subtitle, and especially the delightful artwork on the cover as you are an artist and scholar in residence at an art institute. Take it away. Well, the title is math real? a provocative question. It does provoke certain people to just answer yes or no and think that's going to be the end of the book. But that's why there's a subtitle because the whole book isn't

just about whether or not math is real. And the subtitle is how simple questions lead us to mathematics deepest truths. Because it really is about how simple and possibly naive sounding questions can turn out to be really profound. And so each chapter of the book starts with a simple, or naive sounding question, like, is math real? Why does one plus one equal two? Why can't I divide by zero? And things like that. And then weaves a story that shows how that pushes mathematicians into deeper and deeper research. So that's where the subtitle comes from. As for the cover art, well, I have to give credit to the art department at my wonderful publisher, Basic Books, because it's, it is the art department who comes up with the cover. And if you hold it up again, perhaps we can

see that it's got a kind of digital feel to it that maybe the colors are reminiscent of, I don't know, the kind of green and blue text that you might see on a black background if you did old style computer programming, maybe. I definitely grew up with kind of green digitized text on black backgrounds in the old monitors. So maybe that's calling to mind those early forays into the computer world before we all used them for everything all the time. And the idea that those early things led us into this whole new world that we're in now. In the background, there's kind of generic looking mathy things,

symbols floating around just a kind of big mysterious forest or jungle of mathematical symbols that might seem confusing. And then in the middle of it, there's this big question on top of it that may be the question that really taxed many people, not all the answers to the little equations and the integrals and the differentials, but this other looming question is math real. Indeed. And I like that you, although I believe that you spent some time in England, maybe you're from there, you can clarify that. I like that you didn't use the word maths, because I always find that so, I don't know, I find insulting. But I want to get your reaction, first of all, first and foremost, to this following quote.

by a famous physicist, the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. And this is, of course, Eugene Vigner, who spoke about the unreasonable effect and the unreasonable power of mathematics in the physical sciences. How do you react to Vigner's quotes? Do we deserve it? Do we understand it? We definitely don't understand it. And I think that's really important because I think many people are put off math because they feel like they don't understand it. And that they've been given the impression that that means they're not good at it. But actually none of us understands it.

And that's the whole point. We accept that we will never fully understand it, but we still want to understand more. And so that drives us to want to understand more. We don't just go, oh, I don't understand it. It's over. We go, OK, well, I don't understand it yet. I'm going to understand more all the time. As for the unreasonable effectiveness, I sort of think.

think it's reasonable. I think it's a reasonable effectiveness because it's the use of reason that's at the core of everything. And as for whether we deserve it, well, that's a complicated question. What do humans deserve? In one sense, we behave badly, we destroy the environment, we're horrible to people, we don't deserve

much. On the other hand, we all deserve wonderful things because we're all just trying. And so maybe we do deserve it. I don't know that deserving is something that is easy to work out. Is that a gift? know, I mean, I think if you talk to a lot of, you know, fifth graders doing, you know,

rational and irrational numbers, they'll say, if it's a gift, you can have it back. Vigner called it a gift. What do you make of that, of that kind of notion that such tools? And I believe somebody once said that math is the machine or the tool that has saved the most human labor throughout history. So why is there such an outsized kind of differential between the appreciations, you know, of math and its power. I mean, nobody says I hate this iPhone. You know, it's too powerful. It seems like a magical gift. It's a piece of rock with electrons in it and it allows me to talk to you, right? So how can we

explain this discrepancy between the benefits and the public perception of mathematics? Well, first of all, I would like to point out that I think some people do hate iPhones and I think that some people are Android users, Eugene. Those are heathens. They are Android users. We do not discuss it. Personally, I personally have big trouble with touch screens. I can't do touch screens because I feel like my fingers in the way all the time. And it upsets my brain that my fingers in the way of what I'm trying to do. So I'll just put that out there for a second. I think that the trouble is that too many people have been put off.

They may well appreciate the fact that someone else has done it. So I think most people are glad that technology, not all, but many people are glad that it has been done and that technology exists and that all of these developments have happened. There are problems with those developments, but many people think, well, that doesn't mean I have to do it myself. And I think the problem is that what happens in mainstream math education is so much about testing people and separating out people into those who are good and those who are not good. And we don't really focus very much on just the sheer appreciation of it. And I think it would be wonderful to have math appreciation as a really,

conscious part of math education and to acknowledge that it's really important to preserve the appreciation of math all the way through rather than just trying to drum facts and algorithms and techniques into people's brains. If in the process of doing that, you kill off the appreciation because the process was so painful, then I think we've really done a disservice to everybody and to mathematics. You have many wonderful illustrations in the book. It's such a rich and enjoyable book. One of the parts I enjoy the most is you show like a square at one point and you just have the numbers, you know, one and one on the two sides of the square. And it's to introduce the concept of the irrational length of the hypotenuse is square root a two. And then you say, well,

what about the units? I haven't put the units in. And then you say, never mind. It's irrelevant what the units are. And I'm an experimental physicist. So, you know, for me, the notion of, you know, just shut up and calculate. Don't think about, you know, is this real or is this not real? But I want to ask you a philosophical question. If there were no physical world, you know, would math exist? In other words, we can conceive of things like a triangle, but you can't hand me a triangle. And we can conceive of infinity, but there's nothing possibly that's infinite, right? So or else it would crowd out most of everything else.

So I wonder if you can explain, in your opinion, does math exist independent of a physical world and the quote unquote real world? Well, first of all, I mean, this is how I address the question, is math real world? in the book. What does real mean? And I'm glad you put quote-unquote around it, because if I think about it too hard, I can come to the conclusion that nothing is real. I'm not real, you are not real, is anything real? What's going on? Nothing's real. It's all the figment of my imagination. I think that this is what Descartes was probably taxing himself about. Do we exist? All this kind of thing. I think that if we accept that anything is real, then I think ideas are real. They are real concepts. They just happen to be abstract. We can't touch them. But then,

are plenty of real things that we can't touch. I think love is real, but we can't touch it. Hunger is real. We can't touch that either. I think that the planets are like the sun is real. We can't touch the sun. The center of the earth is real. We can't touch that either. And so I think that the idea of realness is confusing and possibly a red herring. I think what's more important is does it help us. And I think that it does help us just in the same way that, for example, fiction and works of literature. The stories in those works of literature aren't real, they didn't happen, but it's still real literature. And that real literature can give us real insights to the part of the world that we do consider to be real. I think a lot of, you know, what really amused me

about the book was, you know, the different various categories, and maybe we're going to get into a discussion of what category theory is, because despite my father being a mathematician, I've never really understood it, but the various examples from geometry to algebra, set theory, imaginary numbers, complex numbers, real numbers. But the number that kind of speaks loudest in this book is zero for a variety of reasons. And I always think zero is sort of, it seems almost like a magical number in that it's maybe not real. It's, for example, in cosmology, the universe can have a variety of different shapes. It can have spatial geometry that are positively curved, negatively curved, or it can have zero curvature. But there's an infinite number of radii of curvature, you know, corresponding to different size, positive curvature spheres.

There's an infinite number of negatively curved numbers that could have been chosen for the universe's curvature. But we live in a universe that's precisely zero. It's precisely flat. There's something strange about that. Can you talk about zero? Is it invented? Was it discovered? Would we have had to come upon it by some other means?

Maybe an alien brought it here. What about zero? Why is it so fascinating to you? Zero is fascinating because it is a thing that represents nothing. And that is a difficult thing to get our heads around. And if we think about teaching small children numbers, I love teaching small children about numbers because I love watching their heads get around things

before they've been infected by the fear that testing brings to so many people. And the thing about zero is you can't show, someone zero because it's not there. You can show them one and two and three and four things and you gradually see the pattern. But how do you show them zero? And I was reminded of this very strongly when I one day posted on social media that I realized I had one, one whole whole whole punch, two, two hole hole punches and three, three whole whole punches. And someone said to me, you also have zero, zero whole hole punches. And I sat there and I thought about it and I thought, I don't think I do actually. I think every object I own is a zero hole hole punch

because everything I own is sitting around punching zero holes all the time. And so you can look, once you realize that zero is everywhere, now I hope everyone can see that in their house now there are zero elephants unless you have an elephant in your house, which you might do. But I have zero elephants in my house. I have zero giraffes. Now I can see zero things everywhere. So now zero is even more confusing because it's everywhere.

One minute it was nowhere and the next minute it was everywhere. And it took humans a really long time between getting comfortable with the numbers, one, two, three, and so on, and then deciding that zero could also count as a number. And they went from just representing it as a gap. And then when you do, when you represent numbers using place value, we kind of need the zeros in order to say that there are none of those. So if you have places where you have one units and you have tens and you have hundreds, How do you represent the fact that there's nothing if you've got no tens and you're going straight to the hundreds?

You can leave a gap, but that's kind of confusing. So then you need a symbol to represent it. But that's still not regarding zero to be an actual number. It's just a symbol. And so then it took even more time. I'm not a historian of mathematics, but I just find it fascinating that there was so much suspicion about the idea of treating zero as a number. And I think it was very contentious for a long time. And it's actually still contentious among some particularly, I shouldn't be too judgmental,

I'll just say a certain kind of mouth person who gets really taxed about whether zero should count as a natural number or not. Because the natural numbers are the ones we use to count. And so some people still think zero shouldn't be one of those because we're not counting things. We're counting the absence of things. And that's a very confusing thing. And I think it's important for us to acknowledge how confusing it is. because if we just go around saying, oh, it's really obvious, then the people who don't find it obvious feel stupid. And I think that's something that has put a lot of people off math in the normal course of their math education.

Whereas if we acknowledge that it's actually quite strange and that it is quite weird and that in order to progress in mathematics, we accept it, but we still hold on to the idea that it's a bit strange. And so there's this thing that we do in our heads where we accept that it's strange and we also keep going with it. And that's a really wonderful thing to be able to do. And I think that's key for getting to grips or moving forward in mathematics. It's accepting that things are weird and also going ahead as well. Well, I want to ask you when it comes to building designs, which annoys you more? The fact that in the UK, they have a floor zero, which we call the first floor here in America, as you know.

And in America, if a building has more than 14 floors, there's no button. for the number 13 in the elevator. So my favorite thing to do, and when I go to high rise is ask the people in the elevator, could you push floor 13 for me? And then they look around and it's not there. Which annoys you more, the kind of superstition around numbers or this notion that, you know, there's a, you know, zero is this special number that's really not given the kind of attention or respect that it might desert.

Actually, neither of those things annoy me too much because I've lived, I grew up in the UK and I've lived in the US for 12 years now, and if I allowed the cultural differences to irritate me, then it would just be an ongoing irritation all the time. And there are so many things in the world to be irritated about that are more important than that, that I have accepted those things. What does irritate me more, I will say, about zero,

is when people insist that the millennium didn't start until 2001, that it didn't start at 2000 because there was no year. zero. And I think to myself, well, first of all, there wasn't a year one, two, three, or four, I understand, and that the current way of ordering years didn't start until about 500. Plus, when the digits change, when the first digit of a number changes, that is a big deal. Even if it doesn't, if you're going to be pedantic about it, and I think most of the people doing that, who are pointing out that there was no year zero, what they're really trying to do is assert some kind of superiority, as if they have some secret knowledge. And when mathematical concepts are

used to assert superiority, then I never like that. They shouldn't be there to assert superiority. They should be there to help us and shed light. Hey there, fellow Voyagers into the impossible Tizai, your fearful host. Professor Brian Keating here with a tiny little homework assignment before we get back to the episode. And that's to make sure that you're subscribed to the podcast, either following it or subscribing to it depending on your podcast catcher of choice. I did some research of my own and found out that I don't know about half of you are actually following or subscribing to the podcast. So please do that.

And for some extra credit, if you're looking to boost your position on the grading curve, please leave a rating or review. It really helps us out tremendously. Do it. Do it now. Before you forget, let's go back to the episode. One topic that's not directly addressed in the book, although it's rotten perhaps throughout all of mathematics nowadays

is artificial intelligence and computability of numbers. We've had on Sir Roger Penrose many times. We've had on Stephen Wolfram and many other Stephen Strogetz, many mathematicians. I wonder if you could speculate on this. One of my arguments for the kind of non-concern that I have with artificial intelligence is that there's things that are very good for computers to do that computers are very good at doing, but the human brain can't do. And conversely, the converse is true as well.

There are things the human brain can do like represent infinity. A computer represents infinity by the largest number and its possible registers that it can represent. It doesn't really represent it the way that we think of infinity, which is maybe abstract, but we can at least manipulate infinities, which a computer will, you know, throw up at. So I wonder if you have any thoughts about the, you know, the issue of what is unique in the mind and the human brain about the ability that we have to be to compute, to compute numbers, to work with concepts that a computer cannot. Do you think of the brain as a massive computer or is there something inherently advantageous about the brain that allows it to manipulate numbers as if they were real, as your book speculates? I do think of the brain as a massive computer, actually, but I also know that it's an organic thing. And so it physically changes as it goes along. And so that's what makes it so much more spectacularly complicated and hard for us to understand.

And it's really important for us to remember that it has enormous plasticity because that's how we learn to do things. And that when we're learning to do things, and I'm not a neuroscientist, but as I understand it, when we learn to do things, our brain is actually physically changing. And whereas when I don't know that a computer physically changes that much as it's doing things, whereas our brain really changes, it lays down layers, I think, and it makes new connections. And so our brain really changes as it goes along. And I think that's important because so many people are limited in mathematics by a belief that may have come from other people, that they are bad at math and that they can't do it.

And if they've been told that enough, then they start believing it. And once you believe you can't do something, you pretty much can't do it. And I don't know that that's a drawback. I didn't answer the question because I said a drawback. Instead, I don't think computers have that drawback. I don't think they get psyched out about the fact that they don't think they're going to be able to do something. But I think that there are, I would never say that AI will never be able to do things. First of all, I'm not that afraid of AI because I'm more afraid of natural stupidity than

artificial intelligence. Also, I feel like the AI is so bad at stalking me when they try and show me personalized adverts, they're so badly tailored to me that I can't believe, unless it's some kind of giant conspiracy to fool me into thinking I don't need to be afraid of them. They're so bad at understanding anything that I want. So there is also that. But I also, from a point of view of a mathematician, there are things I think are currently still so far beyond. I think that machines would be much better at doing things like having an immediate knowledge of all the literature. If only, we humans have told them what the literature is. So first of all, we have to find a way to explain to AI what is in the literature.

So that's a huge monumental undertaking in the first place. But then it's about making connections where you have some kind of idea. How do we come up with an idea? And this taxes me a lot because I often need to come up with an idea. And when I'm doing research, there is very rarely a deadline on when I need to come up with an idea. But there are other things I do where I have to come up with an idea to a deadline. And so, for example, if I'm writing a column, there's a deadline and I need an idea. How do you come up with an idea?

It's really interesting. And so I don't know if we understand that enough to teach AI how to do it. It can sift through huge quantities of information. But when we remember, when we're reminded of something, quite often when I have an idea, I'm just reminded of something. And I don't know even why I was reminded of it. My brain has made some connection that I'm not even conscious of. And I couldn't even put into words how it made that connection. And so I don't know how that happens.

Or if I come up with an idea for how to represent in mathematics, you often spot a pattern. And then you have to come up with an idea for how to represent it. How do we come up with that? I don't know. those are the really deep human things that I think are the furthest down the line from what AI could do. I'm not an expert in AI either. I've been sitting here saying all the things I'm not an expert in. But I would never say never because I know that we have a very poor understanding of what could happen technologically in the future.

But I think those are the things that are currently furthest away. You said this place was steps from the water. We just haven't found the steps yet. How much did we save? Enough. Enough to get lost. Or you could book a stay with Hilton. Welcome to your ocean front room.

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I'm a fan of finger puppets and my daughter kept this one kindly nearby. But this band said once he was asked, a woman came up and asked him allegedly to help her. You know, she has such trouble with mathematics. And he reportedly said, don't tell me about your troubles with mathematics. I have my own. And I always think back on computability and artificial intelligence, et cetera. And I always respond to people that do believe that the brain is essentially a big computer like Donald Hoffman, who's been a past guest,

or believes it's essentially a way of consciousness as a way of, you know, evolutionarily dealing with computability and information overload. But this man said that he had one of his happiest thoughts in life. And that was that if he was in free fall, he would experience no gravitational field. And that led him to the Einstein equivalence principle, which is the bedrock of general relativity, which is basically a set of 84 different partial differential equations that this defines. space-time curvature. But I always like to point out, like, how do I tell my iPhone, you know, what is it like to freefall? How do I tell it? What is a happiest thought? You know, how do I tell what a painful thought is? Do I, you know, pull out one of its circuit boards or what have you? So I like you, I am not too concerned about the kind of paperclip problem, taking AI takeover.

I think there are things the human brain is uniquely capable to do. And one of the things you're uniquely capable of doing, I think is explaining in terms that people can understand, starting with your first book, I believe, which was How to Bake Pie. So how did that idea come to you? How did you have this happiest thought, perhaps, which launched your brilliant career? What was the, what was sort of the genesis, the zero point of that particular adventure? It was very gradual. There wasn't one specific thing that started it, but for my whole life, I've enjoyed helping other people to understand things. And my upbringing and my education have always given me the belief in the principle that helping others is really important

in whatever way that you can. And for me, it turned out that helping other people understand mathematics was some way that I could contribute to that. And I always believed in education is a really important social good to help others who don't necessarily have as good as start in life as everyone else does. And so for me, wanting to go into education to help other people understand mathematics was a very obvious path. And then it gradually became that I wanted to help people who had been put off mathematics because I realized that when I was teaching math in normal universities to math students, I was not reaching as many people as I wanted and that those people could be taught by somebody else. That if I stopped teaching calculus to undergraduates,

somebody else would teach calculus. But the way I saw it, there weren't enough research-level mathematicians who wanted to talk about abstract mathematics to people who were fundamentally very afraid of math. And so I decided I wanted to do more of that. And I had been talking to my students who, some of my students at the university who was teaching at, weren't that keen on abstract math. They wanted to go into finance or accounting. And so they were really interested in the more applied things. But they were made to do the abstract part. They were reluctant. And so I told them all sorts of stories from my life to try and bring abstract math to life because the trouble with abstract math is it's really abstract which means it's very removed from daily life which is also the

powerful thing it gets its power from being removed from life but then it means that we can't relate to it in the same way until we've got used to the abstract world so I told them all sorts of stories from my life and they always perked up when I told them stories from my life they especially perked up when I told them stories involving food and so then that became a whole book because I realized there were so many stories I could tell about food and math and baking and cooking and creating things and building things, that that became an entire book. And that was how to bake pie came about. And each chapter kind of gives a recipe as well for a dessert, I believe I remember that book correctly. It was very cute.

It's not all desserts, but there is definitely a focus on desserts because that is my favorite thing to make. As I love to say, I love making dessert because it has absolutely no. nutritional value and so it exists just for my pleasure. Study and play come together on a Windows 11 PC and for a limited time college students get the best of both worlds. Get the unreal college deal everything you need to study and play with select Windows 11 PCs eligible students get a year of Microsoft 365 premium and a year of Xbox GamePass Ultimate with a custom color Xbox wireless controller learn more at windows.com slash student offer.

ends June 30th, terms at AKA.m.m.S. College PC. That's right. Exactly. Like we have the holiday of Labor Day, you know, which always comes after the long summer. And it's like, do you need a second dessert, you know, after? So that book had elements of category theory, which is your area of expertise. I wonder if you could explain category theory. Why didn't I never hear about it until, say, 10 years ago? What is responsible for its, you know, meteoric-like rise? Category theory is quite a recent branch of mathematics, and until recently, it was considered to be research level, and so it was thought that you couldn't possibly learn it unless you had already got through an entire undergraduate math degree. And it was also thought that it would only be useful to pure mathematicians, because what it does is it makes a theory of math for math. Just like math is kind of a theory at the root of science, category theory is like the theory at the root of mathematics. And then what happened is that it gradually expanded its reach and people realized it was really helpful to things way beyond pure mathematics.

And that started with theoretical physics and theoretical computer science. And then it moved into chemistry and biology and ecology and engineering. And then it expanded further into linguistics. And then once it got out of the realm of mathematics, there were people who wanted to learn about it who did not have a mathematics undergraduate degree anymore. And so then it became a thing that people wanted to learn, without having a math undergraduate degree, which is why I wanted to start talking about it.

But the other reason I wanted to start talking about it is because I think it's really great. And because it doesn't really have prerequisites. You don't have to know calculus in order to do category theory. You don't have to know any high school math, really. You just have to be curious. Because what category theory says, and I think it has really important principles at the root of it as well,

it's not about categorizing things. It's a very, it's a technical use of the word category. it's not just like, oh, we'll put this in this category and that category. What it's saying is that we can understand the world around us by looking at how things relate to each other rather than by looking at their intrinsic characteristics. And so in mathematics, often what we do is we're not just studying things. We're studying transformations between things and the maps between things.

And so in calculus, if you've ever done any calculus, then actually what's important in calculus is really the concept of a continuous map. it's all there to think about continuity and what it means for things to be continuous. And so category theory says, actually, it's those relationships between things that are really important. And it doesn't really matter what the intrinsic characteristics of something are. What matters is how it relates to the stuff around it. So you can write 25 books about what the number two is. But in the end, it doesn't matter what the number two is.

What matters is how the number two relates to the number one and the number three and the number before and all of that. And that was the big insight of category theory. And it turns out that loads of things get really illuminated by that, including, for example, whether you like it or not, large language models, which as I understand it, and again, I'm not an expert in that, but it's about you make a model of a language so that you don't have to understand what individual words mean, you understand where they fit into the rest of the language. And then you map the entire language onto another language and you find the same spot. So you find the point that relates to the things around it in the same way that that one related to the things around it. And that's how

you translate between things. And it can even translate between words and pictures. So you make a whole space of pictures and how they relate to each other. And then you can find the corresponding place according to how it relates to other things. And I think it's also really important for how we think about people in society because I think in the end what equality of people should mean is not that people are equal because we're not equal everyone is different but what matters is how we relate to other people it shouldn't matter what our intrinsic characteristics are you know it shouldn't matter what we look like and what all those things what matters is how we treat other people and I think that's fundamentally what category theory is about it's about thinking via relationships and putting everything

in a context according to how it relates to the stuff around it I wonder if you can comment a little bit some of the uses that I've seen in just a little bit of Google searching. Things where people are attempting to kind of categorize people, groups, individuals in terms of language of category theory. Is that some sort of math washing? Have you encountered this? I know you've thought a lot about gender and equality and privilege and so forth in the mathematical sciences. Can you comment on, you know, is there a dark side? Is there mathematical washing going on with category theory?

Or am I totally off? I think there sometimes is mathematical washing, but often what happens is that people invoke the word math to try and finish an argument. Although I don't like that because that's misuse of math, but in a way, to me it shows that at least there is awareness that math is a very powerful thing. But we shouldn't misuse it and we shouldn't just pay spurious homage to it

by saying, oh, well, mathematically, people say things like, oh well mathematically that can't happen. And I think, well, what do you mean? You just actually, you don't know whether it can happen or not, but you've invoked the word mathematically to try and finish the argument. But what I do think about people, and I've done this myself, because I've used some category theory, some very basic category theory, to understand the interactions between people.

What I do think it helps with is understanding the interactions of groups of people within structures, because unfortunately our society is made up of power hierarchies. And if we can understand people according to where they fit inside different power hierarchies and the fact that there isn't just one, but there are many along different dimensions and they might not agree with each other, that is something that I find very helpful. And the point about learning category theory and mathematics is something you refer to before about using our brain to basically deal with a complex world. I'm paraphrasing what you said.

But to me it's to do with being able to understand more complex structures so that we don't have to simplify so much. Because we do have to simplify the world in order to understand it. That's inevitable because the world is really complicated and our brains are tiny. But if we simplify it by just ignoring important details, that's not a very good way of simplifying it. If we simplifying it by packaging things up into structures that we can then understand in a single unit, that's a really great way to simplify it because what in a way we've done is we've become more intelligent so that the world has become simpler relative to our ability to understand it. So I wonder if you could wait in a little bit. Something that's bothered me

quite significantly is this notion that saying that there are mathematical truths is redolent of white supremacy. Recently, in the last year or so, there was article published somewhere in Toronto, I think, and it was speaking to the fact that, you know, if you say that there's one right answer, two plus two equals four, that you're somehow enforcing, you know, some sort of ideological or racist notion. And I did a search. I went to the University of Ibadan, which is in Nigeria and Covenant University and the University of Lagos. And these are all, you know, top thousand universities in the world. Nowhere in their math department do they teach anything other than two plus two equals four.

So I wonder, what do you attribute? First of all, what do you make of this notion that having a correct answer somehow, you know, reminiscent or reinforcing of the idea of something to do with color or race? I find it sort of abhorrent, but what is your, what's your impression? I think there's a lot of nuance here because I think the first thing I want to challenge is the idea that nobody teaches anything other than 2 plus 2 equals 4. I teach that 2 plus 2 can be anything because 2 plus 2 can be anything. And actually the idea that there's only one right answer is a very narrow view of mathematics, which is not the way research mathematicians see it. The way research mathematicians, and because we're talking about this, I should say,

research mathematicians as defined by the framework of mathematics that was built by European white men in the 18th or 19th century, according to that framework, there are many different possible answers, and what we do is we find the contexts in which different things are true. So 2 plus 2 equals 4 in the natural. numbers and the real numbers and various other number systems. But in other number systems, two plus two can equal different things. And we study those contexts as different mathematical worlds. So I don't think mathematics has absolute truth. I think that mathematics provides contexts in which different things are true. And I think it's a very subtle difference.

And the thing is that many people get upset when I say that because they want to cling to the idea that math does have absolute right and wrong. And I just think that's an unrued. realistic view of what research mathematics is. And I think that it is definitely exclusionary, because what it's saying is that where I am going to, I have decided that I'm going to gatekeep this form of truth and I'm not going to let anything else be true. Now, while I don't think that that is directly a form of white supremacy, I do think it's related. And here's why, and it's a very long story that I find very uncomfortable to think about. So the history of mathematics is very, very long. Mathematics has been developed by human civilizations across the world since the

beginning of human civilizations. And it has really got all its starts in non-white cultures, in ancient civilizations. And again, I'm not a historian of mathematics, but including ancient Egypt, Mesopotamia, Maya, native cultures, Arab cultures, India, China, Africa. But it gradually developed, in various different ways. And then the academic mathematics that we know now had a framework that was defined by a bunch of white male Europeans who sat down and decided that mathematics needed a rigorous framework in order to be really, really sure that their foundations were completely rigorous.

And what you get out of rigorous foundations is that you can build more complicated structures. So if you just want to build a hut, you don't need deep foundations. If you want to build a skyscraper, you need really, really solid foundations in order to build higher. And that's what this rigorous form of mathematics has done. It means that we can really build really serious complexity. And that is what has enabled spectacular advances in science and technology in the last couple of hundred years, which is amazing. And I love it. But I also recognize that it's very, it's, it makes me

uncomfortable because those extraordinary developments are also what have enabled some people in the world to exploit other people in the world because technology, whether it is in the form of weapons or in the form of computer technology, as what has enabled a lot of imperialism and colonialism and exploitation. And that makes me very uncomfortable, especially because I know that another side effect of us doing that is that we're destroying the environment that we depend on, whereas native cultures who didn't make those extraordinary developments, perhaps probably understood much better how to live in harmony with the environment and not destroy it. So what is it that we've done exactly?

I don't know. I think it makes me uncomfortable, but I do think that those things are related. I don't think it is white supremacy to say that one plus one equals two, even if you acknowledge that one plus one equals two in the real numbers. I think if you think that one plus one always equals two, you're actually just wrong. And that's a whole different question from whether it's white supremacy. But I always say, you know, if basic books decides, you know, they're going to give pay you in two installments, you know, equal amounts, it better add up to the advance that you were promised, right? You're not going to say, well. Well, that's because that's the context. That is the context of natural numbers. But that is a very specific context.

For a lay person, right? They'll say, you know, you're adding, you know, comparing apples and oranges, right? So you can add one apple to one orange and that's two fruit, but then that's the category of fruit, as I've learned, you know, from you, from you, you know, if I've learned nothing else. But, but you can't add, you don't get two apples and you don't get two oranges, right? So, so I think colloquially there is a lay person should not, you know, just as you said before that people make math out to be, you know, say you're stupid if you don't understand this or they give you this impression of math anxiety. because you don't understand our math doesn't come as naturally. So, too, we shouldn't also, you know, kind of obfuscate what we do and say in no circumstance. And you didn't say this.

I'm just, I'm just saying what I've heard. And that's where I would push back and say, look, if it was white supremacy, I wouldn't see Africans in Nigeria that would teach in the, in the appropriate category, as you point out, these are facts or these are truths or these are proofs. And that's what I think, I think the lay person might react to. And I think that's why I got attention. I don't want to dwell too much on this, but I did want to ask you because you did touch upon education and you did touch upon, you know, the history of teaching, you know, as being generated from Europeans. And I want to ask in one of your chapters, you have this illustration. And I called it like in my notes, I call like the stupid question resonance cycle. And let me see if I can find.

Yeah, here it is. It starts in the following way. students ask a disingenuous question. Teachers give a dismissive answer, and there's a sort of negative flywheel. And I'm wondering, education really hasn't changed. You pointed out it's European. Yes, that's true. The first major Western university was in Bologna in the year 1080.

Almost nothing has changed. They're still like some person standing up with a piece of rock, scraping on another piece of rock. The only thing, Eugenie, I say, is better. Back then, they used to, the students could go on strike. and then we professors wouldn't get paid. So that's barbaric, and I'm glad that they've outlawed that, and we have tenure. Now, talk about education, and before you do, how we can revitalize and modernize education,

and then can you first speculate on that flywheel, that disingenuous and dismissive circle? What are the pernicious effects of that feedback cycle? Well, because my whole book is about how simple questions are really important, that the trouble is that some questions are disingenuous, that students are asking them to try and catch teachers out or to try and poke at them or rebel against the system, and then as a result, teachers may feel that they're being undermined, which they probably are,

and then they can't give a genuine answer to the question, or they feel that because the student thinks, the students think that they can catch a teacher out by asking a question the teacher can't answer. And so then the teacher feels they have. have to be able to answer every question, otherwise they'll be caught out. And so if they can't answer a question, then they have to do something to protect themselves, which may be to say that the question is a bad question. Whereas actually, what we should do, I think, is to create

an environment in which it's fine if a teacher can't answer a question, because then that's a good, it's a great question. Then what we can do is we can all sit down and learn how to figure out the answer to a question using the resources around us. There's an entire internet. We can look anything up and then we can learn how to discover things. And then it won't matter if a student asks a question and teacher can't answer. So then hopefully the disingenuous questions will go away because there will be no social cachet left to being able to ask a question and teacher can't answer. So then students will only ask things that they generally want to know. So the other type of disingenuous question is when students ask things just to show how clever they are. And I always talk

to my students at the beginning of the semester saying, I don't want to know how clever anyone is. I'm not interested. And unfortunately, mainstream education is all about that. It's all there to test how clever people are. But luckily I teach at art school and it's nothing to do with how clever anyone is. What I want to know is how curious people are and how deeply they are prepared to think about something and how open they are prepared to be in thinking about different points of view. And so in that case, the questions where they're posturing or showing how clever they are, those questions will go away because they're not rewarded. While we carry on rewarding behavior, humans are just at some level basic animals.

If you reward behavior, then the behavior continues. And if you stop rewarding the behavior, the behavior will probably stop. And so I always want to nurture a classroom environment where I try to convince everyone that I will not reward the posturing behavior, that I think it's wonderful if they ask me a question I can't answer. and where any questions stemming from curiosity is a good question and that I will take it seriously and I won't let anyone else in the room make people feel stupid for it because the thing that then stops the genuine questions

is if you're worried that someone's going to say that you're stupid and that I know this happens because I felt it. People have told me at all sorts of levels that my questions are stupid and so it's not something that people have made up and you can't, I don't think it's right to just say or stop being so scared of asking questions, because the danger is real. If you have been humiliated by other people,

no one, I might say no one, but most of us don't like being humiliated in front of other people. And so we will try to avoid it. So the other thing is to stop humiliating people if they are perhaps slower at understanding something. And I always say to my students, if you're slower at understanding something, it might be because you're seeking deeper understanding.

And mathematicians seek really deep. understanding and so we spend ages thinking about things that might seem simple because we don't feel like we've really understood it yet okay ambition comes in all shapes and sizes at first citizens bank we roll with your goals because we're built for what you're building fit for your ambition for citizens back a dr. Chang we have reached the segment where I like to take audience questions and we have a few too many to actually take in the remaining five minutes or so that we allotted for the podcast, but I'm going to ask one for my good, good friend,

Alex Bilzerian, who follows us on Twitter, and you can follow Dr. Chang at DR. Eugenia Chang, on Twitter or X now, they call it now. You must like that, right? The joy of X, right? Alex asks the following, or X plus Y, I should say. So we need to find an app called Y. You need to, you need to build one. Yeah, right. Then you could be mega wealthy. I'll say, I knew you went. A question from Alex is, how do you resolve the ontological tension between mathematics as a human constructed formalism and as a domain of discoverable entities? And he has a follow-up question after that. So I did see that question on Twitter. So I looked some words up because those are big words.

And that I thought maybe everyone else would like to know what those words mean as well. So I looked up ontology because it's one of those philosophy words that I can never remember what it means because I don't think about it very much. I think it means something to do with the state of being or something like what it means to be. And so I think what this question is asking in simpler terms is, isn't it about whether math is created or discovered? That's what it's asking. Invented or, yeah, invented or discovered. Right. And so I think it's both.

And I think that very often we ask questions that have a premise of a false dichotomy inside them. And instead of saying, is it this or this? I prefer to say in what sense is it this and in what sense is it that. And I think that there is a sense in which it is created because we come up with, we create ways to talk about the ideas that we see around us. And there is a sense in which it is discovered because we're just wandering around, looking at things around us. And then they start tying into each other because once you've created a way of thinking about

something or you have said, it's like inventing the rules for a game, you can invent the rules for a game and then explore what, happens in that game. And at that point, you're discovering what happens in the game. So whoever invented the rules for chess invented them. But now there's all sorts of discoveries about how chess works. And then you're discovering the consequences. So it's like discovering the consequences of some idea you came up with. And I think it's really both of those things in different ways. Well, we spoke about the prospects of you becoming a billionaire. I should mention that none other than Jim Simons, who is a very good friend of mine, who was the first mathematicians I

had on the podcast. I asked him that same question, as math discovered or invented, and he said yes. He said both. So you're in very good company with another billionaire, Eugenia. Congratulations. So the last thing I want to ask you about is related to the name of this podcast. So this podcast is called The Into the Impossible podcast because I am the associate director of the Arthur C. Clark Center for human imagination. And Sir Arthur C. Clark had all these wonderful things. And also he would have liked you very much because he was very much interested in art and science. And we didn't talk about your piano playing and your leader later. I assume that goes with the hosen.

I'm not sure you tell me if that's if that's wrong. No, no, later is leber and leader is songs. Was that a dad joke? Yes, it was. Okay, just joking. Thank you. I'm glad you got it. I'm glad you got it.

It's in my Twitter profile. So that's all I can be held responsible for. But I want to ask you a question related to sort of advice to your younger self. And it's the following. So Arthur C. Clark said, the only way of determining the limits of the possible is to go beyond them into the impossible. And I like to phrase this in terms of a question that's predicated on you going back in time, you know, teleporting back in time. You've got 30 seconds with pre-Dr. Chang. And you're going to tell her some piece of advice to give her the courage to do as she will later do to go into the

possible. What advice do you give to your younger, pre-doctoral self? I think I'd say that the people who have more confidence in themselves, it's not necessarily because they're wiser or better. They've just got more confidence in themselves, and it might be because they're deluded. Wonderful, Eugenia. Well, can I ask you to summarize where should people find you? What would you like to point people towards? Give yourself, this is the plug zone. So please feel free to plug away. Well, my website is my name, www. And I am on the place formerly known as Twitter as Dr. Eugenia Chen. Very good. And you can get this book now. It's out. By the time this episode, which probably

take a couple weeks to process, comes out, it'll be available. And hopefully it'll be number one or, you know, as close to number one on the best-selling charts as possible. That's my, that's my second favorite number after zero, right? You don't want to be zero, but hopefully, may you be one. Eugenia, thank you so much for joining us today. so much. Yamava Resort and Casino at San Manuel is California's number one entertainment destination for today's superstars. Catch the Jonas Brothers return to the Yamava Theater stage on April 30th, the powerful vocals of Demi Lovato on May 17th and the signature Southern Country Rock of Eric Church on July 19th. Tickets on sale now at Yamavah Theater.com, only at Yamava Resort and

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