Abstract Mathematology (UH, IS MATH REAL?) with Eugenia Cheng
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Oh, hey, it's quite frankly the episode that you never thought you'd click on. And it's the one that I never thought I would do because mathematics, it's an X, it's not an ology. But over the years, y'all have begged me to cover this, I swear. And I found an ology suitable and an ologist who's going to get us emotional about numbers. Believe it. And also non-numbers. Come on a journey. Take this VIP pass.

Come slip behind the curtain backstage for a more intimate chill hang with a topic that maybe has seemed a little out of reach, a little intimidating. You're about to meet the really charming and artsy side of math. So this ologist, of course, is a professional mather, an author, and a speaker who holds a PhD in pure mathematics from the University of Cambridge and taught at the University of Cambridge, Chicago, and Nice. She won tenure in pure mathematics at the University of Sheffield in the UK and

has appeared on late night shows, hyping people up about math, has given TED Talks and written several books on the topic, such as How to Bake Pie. And her latest one is Is Math Real? How Simple Questions Lead Us to Mathematics' Deepest Truths. She's one of us and the best person ever.

to cover this topic. So she jumped on a mic from the Midwest where she's currently the scientist in residence at the School of the Art Institute of Chicago. And she's also a concert pianist. You're just going to love her.

But first, thank you to patrons at patreon.com slash ologies for submitting your questions. You can join for a bug a month. And thank you to everyone wearing Ologies merch from ologiesmerch.com. Heads up, if you are hoping to play this episode for your kids, this is not an all-ages episode, but we do have many of those. They're called Smologies, and you'll find them at the link in the show notes. Thank you also to everyone who leaves a review because, you know, I look at them with my eyes so I can...

say one with my mouth. And this one was left this week by Minecraft Girl 215837, who wrote, I listen to you every day and you are so good, but can you make a cookie one? Minecraft Girl 215837, first off, that's an appropriate number of numbers in your name. And also we do discuss cookies. So you have no idea how appropriate that is. So let's get into it. Mathematology. Nobody write me and balk at this, okay? So many people have used this word. It's just, it's in the air and the

Also, fun that the word math stems from a root meaning simply to learn. So come along with me. Let me hold your hand and guide you to the wings of academia, the behind the scenes of a topic you did not know you could adore like this. As I hear about Fibonacci sequences, golden ratios, common core, loving thy neighbor, slide rules versus calculators, imaginary numbers.

The Nature of Zero, Infinite Cookies, Fingers, Toes, Knuckles, How Math Will Change Your Relationships, How Math Is Not Just Numbers, But Dare I Say a Lifestyle, and How Math Is Weirder and More Artsy Than You Think, and more with author and mathematician and mathematologist, Dr. Eugenia Cheng.

I'm Eugenia Chen and I use she, her. And you are a pure mathematician? That's right. And an abstract mathematician? Yes, it's kind of the same thing. I mean, maybe one is more specific than the other. Which one is more specific?

That's a great question. Do you know, I haven't thought about this. I think that pure mathematics is more the name that it gets given to distinguish it from applied mathematics in formal contexts like in university departments and in funding bodies.

I call it abstract mathematics because I think pure sounds a bit judgmental. It's pure. It kind of does. It kind of implies that there's an impure mathematics. Exactly. Which seems like illogical, right? Exactly. And I think that there's been enough people looking down their noses at other people over mathematics that we can do with getting rid of any of those things wherever we can. Yeah, it does seem antithetical to your whole mission, which is to make people appreciate and get rid of those things.

get over their math phobia, right? Or invite them to. Invite them. Yes. It's a good distinction. Okay, quick aside. What is the difference between the types of math? What makes one kind of math abstract versus applied if you apply abstract to a

Applied math. How abstract is abstract? It's one of those things where the boundary isn't very clear cut, but the general idea is that abstract mathematics is trying to understand things from the point of view of uncovering logical structures and just exploring how things fit together for their own sake. Whereas applied mathematics is much more about looking at problems in the concrete world around us and

coming up with theories to help us understand those specific problems. So applied mathematics is closer to physics because physics is really about understanding the physical world around us. But applied mathematics uses the techniques that pure mathematicians come up with often, but they are really

specifically trying to solve things in the concrete world. Sometimes people say real world, but I don't like that either because you know, what's real anyway? Are we real? Is anything real? And I suppose that's what the title of my book is.

I don't know. Is math real? Is math real? Am I real? Nobody knows. And so one thing that I love about your book is just that it's one of those questions that we're all afraid to ask. I think some very intrepid TikToker was doing a Get Ready With Me video and was like, how do we

even know math is real. And I feel like a year or two ago, it shook the world. I was just doing my makeup for work, and I just wanted to tell you guys about how I don't think math is real. And I know that it's real because we all learn it in school or whatever, but who came up with this concept? And you're like, Pythagoras. But Pythagoras

Like, he didn't even have plumbing. And he was like, let me worry about y equals mx plus b. Which, first of all, how would you even figure that out? How would you, like, start on the concept of algebra? Like, what did you need it for?

Do you remember that video? Yeah. And you know, that did what, that was one of the things that, that inspired me to write this book because I replied to her. I wrote a document and I just stuck it on my webpage and I just replied to each of her questions. And then I did a few radio interviews and people, because people picked up on the fact that I'd replied to all her questions and people emailed me from all over the world saying, Oh my goodness, I've wanted to ask those questions for my life. And it

I was stupid and this was the first time I thought maybe I'm not stupid or I thought I was intelligent in every other way except this and I'm sitting in my car crying because you finally validated me after all these years. And I thought, wow, this is really, really tapped into something that has hit people all around the world and it's really affected all these grown adults, including grown men, emailing me to say that they were weeping in their cars because...

Because I had finally validated their questions after all these years. People who thought of themselves as otherwise intelligent. And so I thought, okay, maybe I'll write a whole book about this.

And so did that spark the book? Did you email your book agent and say, listen, I already started on a doc here. We got something here. Yeah, I mean, I already had thoughts about the questions that people have always wanted to ask because I've been teaching art students at the School of the Art Institute for some years now, and they quite often ask,

Quite often what we do is like therapy for their past math trauma. And they tell me that all the things that went wrong and all the ways in which they were made to feel stupid and the burning questions they had that never got answered because they were told, oh, that's a stupid question. Or, no, you're not supposed to ask that question here. You're just supposed to answer the questions that I tell you. And so I'd already been building up a kind of catalog of questions.

that people have really wanted to ask and that they never get answered. And then when that happened, I thought, okay, maybe there is scope for this as a whole book. Was part of that math trauma, Math Barbie? Do you remember? Math class is tough. Do you remember that one? Oh, no. I mean, Barbie's very topical at the moment, isn't it? I know. I know.

Boy, howdy. Okay, so this Teen Talk Barbie came out in 1993 and rightly enraged female scientists and mathematicians and really any human with a brain who thought sexism was weird and that children get brainwashed into gender stereotypes.

So there's one guerrilla group calling themselves the Barbie Liberation Organization, and they bought a bunch of Teen Talk Barbies and Talking G.I. Joe dolls, and they swapped their voice boxes so that Barbie said things like, troops, attack, and made machine gun noises. And then G.I. Joe said, let's go shopping. They did this every

in December of that year. And a lot of kids and parents were surprised on Christmas morning to have a tiny man in combat fatigues lamenting that. I do not remember that one, but there was a spoof that appeared

a few years ago in my research field. So my research field is category theory, which is considered to be one of the most abstract parts of mathematics. And even abstract mathematicians sometimes think it's too abstract. And so there was this whole series of Barbie cartoons where Ken would say things like, oh my goodness, what even is a monad? And then Barbie would just go, well, obviously it's just a monoid in the category of endofunctus.

Okay, P.S. For context, this comic was part of a meme known as Feminist Hacker Barbie, which arose in 2014 like a phoenix from the firebombed ashes of a Mattel book called Barbie, I Can Be a Computer Engineer, which was marketed toward children and featured a teen Barbie looking like a baby.

Learning to Computer. And it involved illustrations of Barbie at a laptop and in class. And it was captioned with passages such as, I'm only creating the design ideas, Barbie says, laughing. I'll need Stephen's and Ryan's help to turn it into a real game. Naturally, computer people, some of whom were known as women...

did not enjoy the existence of this book. And so one of them, Kathleen Tweet, created a meme generator that spawned so many realistic and really sardonic takes on feminist hacker Barbie. And you can read more about it in an article titled, Barbie fucks it up again. So when Barbie explains to her male classmates that a monad is just a monoid in the category of endofunctors, it's

a definition so legendarily opaque that it's become like a math and programming joke itself. But if you'd like to know, a monad in math is an algebraic structure, and in programming, it's used to structure computations as a sequence of steps. And a monad can be a thing that describes how something is supposed to be modified, but it isn't really a thing. And the etymology of monad even means everything or...

or nothing at all. So if it's confusing, that's the joke. And so now we get the joke. If you're seated next to someone at a dinner party and they were to turn to you and say, is math real? Who invented it? And how do we trust it? How do you begin in a short interaction to get them to trust that math is real or at least look at it in

in a different way? Well, first I would validate their question because often when they're asking those questions, they're already full of doubt, skepticism and past trauma. Sometimes they're trying to say, none of this is real. It's all a load of

Oops, can't say any of those words out loud. Yes, you can. It's all a load of codswallop. It's all a load of codswallop. Codswallop. Okay. That's often the subtext of their question. And sometimes it's because, especially in a dinner party situation, they may feel intimidated by finding that I'm a mathematician. And unfortunately, that happens a lot, especially because I'm

not a male person, I'm not a white person, and I'm not necessarily how people expect a mathematician to be, whatever that is. So quite often people feel intimidated. And so it depends where those questions are coming from. I also say, well, what does real even mean? And spoiler alert from my book, I don't actually say math is real, nor do I say math is not real. What's really more the point is how does it help us? It doesn't matter whether something is real or not, really.

if it helps us in some way, that's what I think in the end. So then the question is, does math help us? And does it help us in some way that you, the person asking me the question, cares about? Because people might say, oh, well, sure, people can use math to fly planes and to make computers work, but that doesn't mean that I have to care about it. And so then I say, try and find out what they care about themselves and whether they care about thinking clearly about

about the world around them. Do you surf or eat or paint or eat paint or golf or wear clothes?

Math is in the tides, it's in the temperature gauges in your oven, putting angles, Rubik's cubes, hair braids, fabric knits, and lasagna gilds. So we all have stuff that gives us butterflies, just nauseated with happiness. And there is a shitload of friendly and helpful and benign and dazzling math involved. And sometimes, depending on where the person is coming from, when I say that math helps me, and

empathize with other people. That can be really mind blowing because that is not something that is often presented in math class, I think. How does it help you empathize? There's two ways, I think. First of all, it's a technique for understanding how arguments are structured. If I want to try and understand somebody who has a completely opposed point of view from mine, then I

can do it by understanding where their argument is coming from. Because it's always coming from somewhere. And it's never going to help if we just sit there and go, oh, that person's just not being logical. Their point of view has come from somewhere. And one way to empathize with other people is to understand where their point of view has come from. So being in a math mindset primes your brain for taking something complex and

and going a step back and breaking it down through logic. How does one get from there to here? It doesn't mean that we're supporting it, and it doesn't mean that we are claiming it's good, but we are just understanding where it came from, from their point of view. Abstract mathematics also helps because it's a process of

seeing patterns and making analogies between different situations. And so at a basic level, if you say two plus three equals five, what you're really saying is that anytime you take two objects and another three objects, as long as they don't kind of spontaneously combust or coalesce, you will end up with five objects. And that's a pattern that when we teach...

arithmetic to small children, we show them doing that with objects, physical objects, over and over again until they see the pattern forming in front of their eyes. And that's really a way, I think, to access empathy with people who have differing opinions from us, because it's about finding an analogy between our situation and theirs. So between you and the guy who just flipped you off in the Trader Joe's parking lot, there must be a common denominator. And

And so it can be quite a far-fetched analogy, but because abstract mathematics is really about seeing deep patterns where the surface looks completely different, but if you strip away enough details, you get down to something that's the same underneath. I can do that with people who I really disagree with. By just stripping away so many details and finding some kind of

Yeah.

getting angry with me for thinking something. And it's not going to help me change my mind. So for example, if we think about people who don't believe in vaccines and all the ways that people kind of get angry with them to try and get them to believe in vaccines. And then if I think about a situation where a lot of people are getting angry with me and want me to believe something, then I can see that that isn't going to help me.

And so that kind of abstraction and analogy helps me understand why people take the positions that they take. On that note, and stay with me, there was something about this chat and Eugenia's book that reminded me of therapy, of cognitive behavioral therapy. It was funny because I was reading your book thinking about how much math must give you kind of...

an edge into understanding your own brain and other people's brains by saying, okay, everybody hates me. No one's texting me back today. Is that true? Does everyone hate you? What's the logic? Right. What is my reason for thinking this? Exactly. And the thing is that I absolutely don't always use logic over feelings. I actually do the opposite. I always observe that feelings are correct. That's it. Feelings are always true.

as a basic starting point. But sometimes there are other things going on as well. And so I've noticed for myself, for example, sometimes when I'm feeling terrible because some bad things have happened to me and then I think about them too much, they go around my head and then I feel terrible. And sometimes I think to myself,

I know intellectually, because I have done cognitive behavioral therapy, I know that if I just go to bed, I will feel better in the morning. And you know what happens after that is that I don't want to do it because I don't want to just go to sleep and feel better in the morning. So then I might say, well, that's not logical. But no, I think, no, that is a true feeling. I truly feel this. Okay, now why am I feeling this? And then I realize it's because I'm

it seems like cheating. I don't want... Then it will feel like my feelings aren't being validated if I can just go to sleep and they'll go away. So true. You're also a musician too. Yes. I think...

We know that there's a lot of math in music, but is there a lot of emotion in math? There's tons of emotion in math. That's such an interesting question. I feel things so deeply when I'm doing and seeing and experiencing math. And I try to write that into my books because on the one hand, the power and the strength of math comes from the fact that it doesn't have emotions in the actual argument of it. So it doesn't depend on...

emotions, it shouldn't depend on emotions at all to build the argument. But humans are emotional creatures. And so when we're communicating math, if we don't communicate it with emotions, then I don't think anything gets through or at least much less gets through. And I think that's one of the big problems with teaching and learning math is that there's the idea that math shouldn't have any emotions in it, which is true. But at the same time,

all human experiences have emotions in. And if we try and teach someone something without showing the emotional side or without giving them an emotional connection to it, then I just don't think it goes in as deeply. So there are many papers on this, but a fresh 2023 study called Emotions and Motivation in Mathematics Education, Where We Are Today and Where We Need to Go, stated that

Female students' enjoyment of and interest in fashion was found to result in lower engagement in mathematics and prevent them from solving word problems. And I was like, hold up, what? That seems weird. So I checked out the 1994 study they cited titled, When Do Girls Prefer Football to Fashion? An Analysis of Female Underachievement in Relation to Realistic Mathematic Contexts.

And what that study was actually looking at was the tendency for contextual problems in math lessons to make no fucking practical sense at all. Like you're given this long fictional scenario, which you'd approach from an entirely different perspective in real life. Like the fashion problem in the actual 1994 study was that girls scored lower on the fashion design math calculation than one about footballs.

or just an abstract question, because the fashion problem made no sense. Like in order to divide hours of labor in a mathematically sound way, which is what the problem was about, you'd have to deliver the finished dresses before you sewed them. And this 1994 study concluded that two-thirds of girls used their common sense as well as their mathematical knowledge and then were penalized for doing so.

Anyway, there was actually more engagement in the problem, but the problem only made sense on a math test, not in the real world. And research has also shown that figuring out solutions to hands-on actual scenarios gets us more engaged in finding the solution. So it's actually a two-pronged approach to make people actually care. And yes, folks are quick to penalize others without realizing that, hey, they're actually asking really good, important questions. And the problem is more complex

And that the emperor, wait a second, is very naked. Why am I seeing his bare buttocks? Let me put it this way. If you do have an emotional experience when you're doing something, you will remember it more deeply. And so I try to talk about things. First of all, talk about how I feel emotionally.

about mathematics. And secondly, I encourage my students and my readers to have feelings about it. And I try to link it to topics that they already have feelings about. Because if I'm talking about why one plus one equals two, someone may have no feelings about that apart from horror when they remember their math lessons from school. But then if I can find something else, like my wonderful students who came up with thinking about

when you mix paints together, when you mix one color of paint with one color of paint, you actually don't get two because you get a new color of paint. And maybe mixing paint is something that some people have much more feeling about. Or if you're,

making cookies and you've got balls of dough and you decide you're going to make one bigger one instead of two small ones, you take two bits of dough and you stick them together and you make a bigger cookie. That may be something that someone can have a feeling about rather than just it being some abstract concept. And so the power of abstract mathematics is that it does not involve emotions, but that also makes it difficult to learn and understand. Well, how does an abstract mathematician think

deal with those balls of cookie dough what's the answer that's gonna trouble me my whole life

Well, it depends because in one way, if you take two balls of cookie dough, you get two balls of cookie dough. So you take one ball, you take one ball, and that's two. But there's another thing you can do, which is smush them together and get one bigger one, in which case you've kind of done one plus one equals one bigger one. And those are two scenarios that are both real. And so it's not that one plus one always equals two. It's under what circumstances does one plus one equal two? And that's where we get to the whole, oh, as long as we don't smush things together and we don't

eat the cookie dough because cookie dough is delicious. So in her book, Beyond Infinity, An Expedition to the Outer Limits of Mathematics, Eugenia also just stares into

infinity right in the mouth while discussing dividing cookies for all eternity. So many cookies, man. Hell yeah. Yum, yum, yum, yum. Right on. Where in the brain is math coming from? And why, when I am stoned, do I think I understand math better? I cannot possibly address that last question. I have absolutely no experience of that whatsoever. And that is for real. However, there's a popular...

that math is on one side of the brain. And the whole left-right brain thing, I think, has been mostly completely debunked. But I remember it was actually my piano teacher who introduced me to the book, Drawing with the Right Side of the Brain. And so I think off the top of my head, the idea is that the right side is the logical side and the left side is the creative side. Is that it?

Is that what the myth was? Right. So I would just like to stress that I'm pretty sure it's all been debunked and that both sides of the brain work together and are very highly connected. So while the hypothesis was flipped and the left side of the brain is supposedly logical, the right is supposedly illogical,

creative, it's been flimflammed by medicine itself. And for more on this, you can read an evaluation of the left brain versus right brain hypothesis with resting state functional connectivity magnetic resonance imaging. And you got to use your whole brain for that study because it concluded from over a thousand scans that there's no evidence that the people nestled into their brain imaging chambers use one side of their pumpkin to

more than the other. Rather, your whole shebang is interconnected and quote, the two hemispheres support each other in its processes and functions, which is tender and kind. So if you feel like you need to get it together,

don't worry, you have, you are together. And there are people who have either never had one side of their brain or lose the use of it. And then the other side is able to compensate. And of course, there's tons of stuff about the plasticity of the brain learning to do things. But even if one side of the brain were logical and the other side's creative, that just panders to the idea that math is only logical and not creative. And it's really both together. It's just that when

When you're doing arithmetic in elementary school or wherever you first do it, that might not be extremely creative. But in that case, people who came up with arithmetic in the first place

That's creative. I have questions. Yeah. I have so many questions. And I don't even remember how old I was when I realized, oh, everything's multiplied by 10 because of our fingers. What? It didn't occur to me. The base 10 thing didn't occur to me for so long. But where did math come from and how many...

generations has been passed down. And if we didn't learn it from someone older than us who raised us, would we even have any capacity to just come up with

theorems and proofs and calculus and everything out of thin air? Or does it really just keep building on itself? That's a fascinating question. And I think those are really great questions. And so humans did come up with those things, but it took them thousands of years. I mean, people came up with the idea of numbers thousands of years ago, but it was ancient cultures. While the numbers that we use today are mostly based on 10 fingers, different cultures base things on different things. And so there are some cultures that base things on

eight because of knuckles. And then there are some people who use the spaces in between their fingers as well. And there are some cultures who based on 20, 20 fingers and 20 toes. And the number system in French has traces of that where 80 is

is quatre-vingt, four twenties. And then ninety isn't ninety, it's four twenties plus ten, because it's as if you're counting up in twenties. So there are different things, but then this base ten thing has really taken over. So base ten systems have been used for probably as long as we've had ten fingers. And though the written records go back to 3000 BCE in Egypt,

It wasn't until between 100 and 400 years into the Common Era that the Hindu-Arabic numeral system of 0 to 10 kind of won out. And this number system has only been in use in Europe for like the last 1,000 years. But yeah, it goes far, far beyond that in different forms. But staring at your hands for a while, that's math, people. And...

There were some ancient cultures, I think it might have been Mayan culture, that used base 60, which is why there are 60 minutes in an hour and 60 seconds in

in a minute. No. Right. So many of our things are intense. The lovely American system still uses Fahrenheit, but it doesn't fit well with hundreds of things. No, it does not. But we have this thing with 60 seconds in a minute and 60 minutes in an hour. And 60 is a great number to use because it has a lot of factors. And so you can divide an hour into a lot of really nice chunks. Whereas if we went decimal on...

I think there's some fantastic society somewhere that thinks we should do a metric on time and have 100 minutes in an hour. We would actually not be able to divide it up into quite so many groups.

handy units because we wouldn't be able to do a third in quite a handy way. So we can divide an hour into half, thirds, quarters, fifths, tenths, twelfths. I mean, we can do all sorts of things. And so 60 is a pretty good number for that. But the whole 10 finger thing eventually took over. And to answer your question about

if we grew up and nobody older than us taught us math, would we be able to come up with all of it? Well, here's what I think, and this is just pure speculation.

I think it would be hard to do that in a single lifetime. It did take humans thousands of years to get to the point we are. And I think it's amazing how fast an individual human is now able to learn all of those things that it took humans thousands of years to learn. And it's because we communicate with each other. And so it's really dependent on people who already know it, passing it down to the next generation. If we each had to develop it from scratch,

It would take a really long time. And I don't know how far an individual would get.

Well, do you think there's any math systems that have been completely forgotten that some people cultivated for a couple thousand years and then just wasn't a record of it and we just have no idea that there's a whole math system based on pi or the Fibonacci sequence or like seven or something? Almost certainly, especially because so many cultures pass things down from generation to generation and then maybe...

died out or got killed off by white European people or are living uncontacted somewhere. And I think that there are certainly pieces of math that are obsolete now because we've developed more technology. So, for example, there's the whole math of the slide rule. The what?

My parents' generation had to learn how to use a slide rule, which is a really clever device for multiplying large numbers together using logarithms. And the thing is, we just really don't need that anymore because we've got calculators. So that slide rule system was invented by a dude named Edmund Gunther in 1620. And that was the same year that the Mayflower crashed the party that's now known as North America.

And the slide rule was technically an analog computer. And it had this ability to glide to different positions to, like, reveal these complex math solutions. And this is how folks conquered big multiplication and division of numbers until about the 1970s when electronic calculators just beep-booped their way onto office desks next to ashtrays and Diapepsi.

and became accessible and commonplace. Now, as for this so-called slipstick, it became obvious

obsolete, partly because you could not write boobies upside down with it. And so the slide rule has become obsolete. I don't know how to use one myself. I expect my parents had it drummed into them so hard that they could still do it. I'm sure there's somebody out there who still loves using their slide rule. I mean, I remember the anxiety of how to afford a

Texas Instruments TI-87. Oh, the graphical calculator. Yeah, like needing one of those in high school being like, I need $109 for a calculator size of a brick. Maybe that's not used anymore, but maybe it is in testing situations. But I know you get asked this all the time, but who is good at

at math. How much is it aptitude? How much is it attitude? How much is it access? And why are we so afraid of it? Why do so many people just throw their hands up and say, nope, I suck, bye. Like you split the bill. I'm not dealing with it. So first of all, I think it's mostly access and how much help you've had.

and how much that help was specifically helpful to you. All the scientific research at the moment points to brains being spectacularly plastic. Neuroplasticity, it's extraordinary how much brains can change according to how they are used and how they are stimulated. So you can coax your brain toward a better life, which means maybe one day I will be good at dancing. And so there is...

almost no evidence. There's basically no evidence to show that there's some kind of hard wiring at birth that means that some people

are destined to be better at math than others. And in my previous book, "X+Y: A Mathematician's Manifesto for Rethinking Gender," I talked about this a bit because there are still some people who think that maybe there's just some biological reason for men to be better than women at math and therefore there's nothing we can do about it. And the scientific evidence for that is just so thin that it's actually

I mean, I laugh at it. That means it's laughable, right? And so, because how can you even tell that something is hardwired at birth except by testing newborn babies? But how do you even test newborn babies? They don't do anything. You can't get them to do anything. They're just hanging around. And so the idea that anything you can get a newborn baby to do is going to be indicative of their future math ability is just ludicrous to me because math ability is a really complicated concept

combination of things. It's not just about being how fast you can do arithmetic. And so I always say that the things that you can test in lab controlled situations are necessarily very restrictive. Just like you can test how far people can sprint the hundred meters. And I don't think anybody really argues with the fact that men can, the fastest men can run a hundred meters faster than the fastest women. It's to do with body strength and stuff like that. But ultra marathons,

Women have been beating men at ultramarathons. I can't remember how long an ult counts as an ultramarathon, but it might be 250 miles or something. Okay, so technically everything over 26.2 is an ultramarathon, but the longest and the most grueling is the Hong Kong 298K, which

with no stopping or sleep or support on the trails. Just two to three days of continuous running. Why? Why, why, why, why? But take a gander at this one article from 2021 titled, Why Women Are Faster Than Men in Long Runs. And you'll learn that the men among us tend to have

larger hearts to pump oxygen for powerful sprints and more muscle mass to power those bursts of energy. However, that's the ganders. The geese in this situation, lady athletes, excel at endurance due to a multitude of factors, such as a higher body fat percentage that helps when they hit a wall, figuratively, more slow twitch muscle fibers, and yes,

emotional resilience in general. Because some of these bodies have thrust a whole person out of their more sensitive aperture. So maybe an ultra marathon is like a walk in the park, but just without ever stopping until you've collapsed at the end. And that's a really complicated combination of skills, much more complicated than walking.

than running 100 meters, which isn't to say running 100 meters isn't hard. It's just it's a much more focused thing. Whereas an ultramarathon involves planning, strategy, self-knowledge, pacing. And math is like that because math isn't just about memorizing things or manipulating large numbers. Math is about spotting patterns and being able to perform abstractions in order to see patterns that previously weren't visible. And how

having ideas for how to group things

objects together to make structures that will be useful to us. It's a bit like designing a useful tool for building a house, except that it's an abstract tool for building ideas. Okay, so it's like a hammer to drive nails versus the idea of a hammer to drive the idea of a nail, which can be applied to the mathematics of the physics of the fabrication of the tool to drive the nails, which becomes the real hammer. Woodworkers, you love math.

So how on earth do you measure a biological predisposition for doing that? They're not skills that are just something that you can be born with. How are we going to test whether a newborn baby can do that? We can't.

Well, what do you think about Common Core? Because I don't have kids, but all I know is that my friends with kids seem horrified or confused by Common Core. I don't quite understand what it is, but why has there been this shift in the way that we're teaching math and I guess like the last 10 or so years?

The reason there's been a shift in the way we're teaching math is because most people have acknowledged that it wasn't going very well. It's just that people haven't really agreed on what to do about it. And I think one of the problems with changing the system is that

a lot of teachers don't get autonomy over what they're allowed to teach. They get it plonked on them. Now you have to teach like this and that's that. And then you have to prepare people for standardized tests and then you get judged on how they do in those standardized tests. And I think that, that any time that we're aiming to teach people content, like can you do this thing at the end of it, then that is going to be less successful than teaching them, um,

appreciation and appreciation

allowing teachers more autonomy to teach the things that they care about in the way that they care about. And as the world changes, some basic things become less important, just like the slide rule is not important anymore and controversial idea, but I don't think that long multiplication is important anymore either because we've got calculators. And it's just like learning to ride a horse isn't important anymore, which doesn't mean no one should learn to ride a horse. If someone loves riding a horse, great, but it's not

It's not a crucial skill for most people's daily lives. And I think that sometimes parents get really upset if they don't understand the work that their children are being asked to do. And I've talked to many math teachers who say that the parents complain to them that it's not like it was when they were young. But then the teachers say to them, well, did you like math? And they go, no, I hated it. And they go, well, why do you want me to do that to your children? Oh, yeah.

It's empathetic. That's nice. And then sometimes people say, oh, it's terrible. I can't help my children with their homework. And sometimes I want to go, well, don't help them with their homework then. It's their homework. Okay, this just in. It turns out that I had no idea what Common Core really meant because I don't have kids and I don't listen well. So Common Core was adopted in U.S. states around 2010. And it's actually the set of standardized assessments measuring where kids should be for every school year. But

But that term common core often mistakenly is applied to just this new way of teaching math that's more intuitive and is actually a very old way of doing math. So instead of going and fetching a piece of paper to figure out what's 63 minus 42 and then borrowing some numbers and stacking figures on top of each other, kids learn, well, if you add 1 to 42, then 63 minus 43 would be 20, so it's 20 plus 1, which

which is 21. Because we all have pocket computers for the bigger arithmetic issues, teaching these quicker and less fussy ways of handling numerical concepts is more valuable. And future generations will probably thank us when it comes to adding tips to dinner bills. That is if American restaurant workers are still making minimum wage and at the mercy of grouchy customers to actually pay for the rent. We thought we'd have flying cars, but really just want health care.

Well, I wonder, how do you think the world will change with a new generation learning math in a different way? Do you think we'll have more mathematicians, more statisticians, more scientists, or do you think we'll progress even faster?

through mathematics? Is that a good thing? I'm much more worried about the people who fall off and get put off and get traumatized by it. What I would like to see is not more mathematicians and more scientists. It's fewer people who hate it. That's what I really want to see. And partly because when there's half or more of adult humans who hate

either hate math or are traumatized by it, or who are actively hostile towards math and science. We've got problems with persuading people that things like vaccines are real, that the pandemic is real and that we need to do something about it, that climate change is definitely something that we need to worry about. I think that that's a big problem for the way that society is going. If people are ready to just

believe those kinds of lies. And I do think that my background and training in abstract mathematics really helps me to not be manipulated by people who are lying to me. And it helps me always to be aware of the frameworks for finding out whether information is good or not, and deciding whether something that someone is saying is probably true or not. And I think that that is all part of

what I think education should be aiming for, not can you multiply these large numbers together and can you calculate this thing and get the right answer and can you solve this equation? But can you think clearly about the world around you? Can you make sure that people don't manipulate you and can't lie to you about things just so that they can get your vote or your money? And can we make a contribution to the world that benefits more people and not just ourselves? The way that social media is

I feel like we're going toward kind of a quantitative rather than a qualitative assessment of our lives. And maybe there creates a little bit of an anxiety around math there too. And we look at how many followers, how many likes something has. We've really started to introduce numbers to things that are much more qualitative numbers.

That's a really interesting point. Yeah. Right. I think it's because society is trying to rank everything all the time. Yeah. And it starts in school because the system tries to rank students, but then we get into this, this frame of mind where everything has to be ranked by a number and it's,

It might sound funny because I'm a mathematician. I'm saying we shouldn't rank things by numbers anymore. But I mean, a case in point is that at the School of the Art Institute where I teach, we don't have grades. And so there is an understanding that we are not trying to rank everybody by a GPA because there are no grades.

And I just think that's wonderful. So we can focus on educating rather than ranking. For more on this, you can see the 2021 paper, Gradeless Learning, The Effect of Eliminating Traditional Grading Practices on Student Engagement and Learning, which notes that throughout their study, it became clear that students want to learn. Accurate feedback is a vital part of the learning process. It says, but grades are not.

Their traditional grading system pits students and teachers against one another, often leading to either side bickering over fractions of percentage points, which I guess is applied mathematics, but I don't think that's the lesson here. And I wanted to ask a little bit about some basics for people who maybe are not math majors, but when things go from numerical to letters, where in the learning process of math from like

arithmetic to pre-algebra to algebra to pre-calculus to calculus, where do things start turning from numbers into letters? Why does that happen? Oh, thank you. That is one of the questions that I address in the book because it is something people say to me a lot. Like, why does math and global numbers turn into letters? Why do we do that? Yeah, exactly. And so we're thinking about the idea of a person. And so we don't name them because we don't know who they are. And so that's why we do that with numbers as well. It's

possible that I also introduced it like a murder mystery where someone is a murderer and you're trying to find out who they are, but you can't refer to them by name yet because you don't know who they are. So you gather a whole load of evidence about them and then you pin them down and go, ah, it was actually James. How could you? Often in math, we're trying to find out what something is.

But we don't know what it is yet. So how can we refer to it? So we use something like a pronoun, but it's a letter because that's what we do in math instead of a pronoun. And then we gather evidence about it. And so we say, oh, well, it's related to this other thing like this. And when we do this to it, it behaves like that. And when we do this to it, so it's like 20 questions. You say, what happens when you do this? What happens when you do this to it? And then you find all these relationships. And so that's what solving equations is about. It's about putting that thing you don't know what it is yet.

into a relationship where you found out various aspects of its behavior and now you find out, now you can pin down what that thing actually is. And that is the point of using letters and that is the point of solving equations. It really is like a murder mystery. So anyone who enjoys any kind of murder mystery, I think that that's, I think it's math. One thing that does not embarrass me is asking really basic questions for me and for the good of all of us, such as what is

What is the difference between a logarithm and an algorithm? They sound alike. Are they friends? A logarithm is very different from an algorithm. Okay. A logarithm is a particular function and it's

I'm now doing hand motions. That doesn't help on an audio. The graph of a logarithm, it starts down at infinity and it goes up really fast and then it tails off. So it's kind of like the opposite of an exponential, so much so that it actually is the opposite of an exponential. So it's the inverse of an exponential function. And so that tails off as it goes along. That's completely different from an algorithm. So an algorithm is a method

for doing something. And it's a particular form of method, like a really, really step-by-step recipe, the step-by-step process that you could tell a computer to do. So for example, the dreaded long multiplication is an algorithm. It's a step-by-step process whereby you can follow these steps and multiply large numbers together by following the steps

all the way through. But the algorithm says, if this happens to do that, now if this happens to do that, now it's like a flow chart. And so that's not really math. And that's why I say that we don't really need long-mouthed creation anymore because it's an algorithm. It's a handy algorithm for doing something, but it's not really math. And when people say the algorithm, are you talking about the internet?

Yeah. The algorithm is showing my videos or the algorithm wants me to listen to the new. Oh, yes. Right. So that's a particular, really, it's a particular algorithm that the internet or most places we interact with on the internet use some process for deciding how to show you something next. And it's a bit murky exactly what their process is. And that

that many people think that there's something nefarious going on behind it, which there probably is. And it's probably based on money. So they have applied some step-by-step algorithm. And that one really is implemented by computer to say, based on this person's previous activity, follow these steps and show them this next thing. That's kind of capital T, capital A, the algorithm. So this is the math that expertly hooks us with engaging, but ultimately never satisfying and

infinity-like scroll of content, just harvesting our tastes and churning out more like it back with this boggling speed. Now, Twitter looks at half a billion tweets every day and then decides exactly what to show you. And one article titled, The TikTok Algorithm Knew My Sexuality Better Than I Did, pretty much says it all.

So we cover more of this in, yes, the TikTokology episode with your favorite psych homer on there, Mr. Hank Green. And yeah, we will link that in the show notes. Great.

There are many other algorithms. And sometimes algorithms are great because they help us conserve our brain energy. And I think that's really important because our brains are finite and very puny. And so if we can conserve that energy as much as possible, that's really helpful. So I personally have all sorts of algorithms for helping me run my life

and this may be me as a mathematician speaking, but it's because I don't want to use up my brain energy on something that's kind of irrelevant. What do you have? Yeah, tell me, tell me, tell me. Oh, for example, I have algorithms for when I buy more coffee, for example. Okay, tell me everything. So I buy five pound bags of beans, but then I keep a pot,

of them on the counter. So I refill the pot on the counter from the bag of beans. And then once the last bit of the bag of beans has filled the pot, then I immediately buy another bag of beans so that it's ready by the time I get to the end of the pot. And I don't have to think about that. Now, maybe one day I can have a smart coffee bean pot, which will automatically order it for me. But that is my algorithm for coffee beans. I love that those are algorithms. Can you tell me some other places where we don't realize

There's math just everywhere. I know a video going viral. I know you've seen sine waves in your wraps and burritos. There's math in braids and challah. Where are some of the unexpected places where math is really making our lives amazing? Well...

One thing that I do when I'm walking across Chicago, which is on a grid, is that thing where I walk in one direction until I hit a stop sign. And so if I don't get a crosswalk, the light on the crosswalk, then I turn and take the crosswalk going the other way. And then I keep going in that direction. So that's my algorithm for walking across Chicago. And there's some math in there that's telling us that wherever we turn, it doesn't matter. So if I need to go five blocks east and 10 blocks north,

then it doesn't matter where I turn, it will still be the same distance of walking as long as I consider that turning doesn't exhaust me. And that's actually a part of

of metric spaces. And so a metric is a more general form of distance. And we think about distance usually as distance as the crow flies, but you don't fly like a crow when you're walking across Chicago because there are buildings in the way. And so the distance we actually need to go to get somewhere is not the distance of the crow flies. It's the distance along this grid. And

So we need to know that it doesn't matter where we turn. And that's intuitively kind of clear. Right. And then you can do things like say, what's a circle if we're using this form of distance? Because a circle doesn't quite look like a circle anymore. It looks more like a diamond shape. But that counts as a circle because it's...

It's all the points that are four blocks away from you. And I think that's kind of hilarious because I like things that,

challenge the received wisdom, which makes me sound like a pointless rebel. But I like to think I'm a point-full rebel, because math is about not making assumptions that you don't need to make. And I think that that's really important in life as well, because a lot of the problems with society come from people making assumptions about other people that you don't need to make, which doesn't mean we shouldn't make any assumptions about people, because

actually that's impossible. As soon as we meet someone, we have to start somewhere. And so we have to have an instinctive, intuitive gut response to them. But the really important thing is to be ready to change it when we get new information. And math is really about that. You can be aware you made them so that if they turn out not to be right, you can change them and then get a different result. It seems like

Once you embrace math, you understand that it can be very empowering and that could offer a lot of clarity. That is certainly what I think. So if that's the impression I've given you, that's fantastic. And I just realized, I don't think I really addressed the part of your question earlier where you said, why are people so put off it and afraid of it? And this is what makes me sad because I think it's there to help us. And I think that it's not presented like that enough. Mm-hmm.

Okay, I have some questions from listeners. They know that you specifically are coming on. But first, let's raise the bottom line of a cause of theologist choosing. And Eugenia pointed us toward mathcirclesofchicago.org, whose mission statement says, we offer engaging, flexible, and free math programs to students in grades 3 through 12 and focus on reaching Black and Latino communities and other communities where most children live in low-income households.

You can find out more at mathcirclesofchicago.org, and that will be linked in the show notes. And that donation was made possible by sponsors of Ologies. When U.S. Bank says they're in it with you.

They mean it, not just for the good stuff, the grand openings and celebrations, although those are pretty great, but for all the hard work it took to get there, the fine tuning of goals, the managing of cash and workflows and decision making. They're in to help you through all of it.

because together they're proving day in and day out that there is nothing as powerful as the power of us. Visit usbank.com to get started today. Equal housing lender, member FDIC, copyright 2024, U.S. Bank.

This podcast and my life is brought to you by Squarespace. Do you know that I didn't have a website for forever because I was putting it off because I was scared and then I heard another podcast talk about Squarespace. I was like, I'm going to give it a shot. I had a website up that day. They have beautiful templates. They host. Squarespace is the all-in-one website platform for entrepreneurs to stand out and succeed online. Look at me. Even I did it.

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Okay, so think about your childhood and think about some highlights. I bet they were probably out essentially tinkering. This is why I love KiwiCo. Each month, they send a kid a crate. It's packed with these engaging hands-on activities. They introduce them to science and technology and art concepts.

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All right, let's divide and conquer your queries. One wonderful one that was asked by Mallory Skinner, Doug Pace, Maddie S., Talia Dunyak, Emily Stauffer, Devin Naples. Devin asked- They all asked separately. All asked this similar question. Oh, wow. Want to know, please explain infinity in a way that my brain can really comprehend. Oh, well-

You know, infinity is really difficult. And so if your brain, if you feel like your brain isn't comprehending it, then maybe it is. Oh, wow. Go on. It's supposed to be mind blowing. If people who think they understand infinity are kind of just deluded. And so I think that the correct feeling is to feel like it's...

it's too mind-boggling to get your head around. That's the whole point of it. It's bigger than anything that we can think of. It behaves in really weird ways. And I think what often makes some people mathematicians and some people mathophobic is embracing the feeling of not understanding something. Rather than taking it as a sign you're bad, taking it as a sign that there's something really interesting going on. If you go to the edge of a cliff...

I don't like looking over the edge of a cliff because I'm worried I will fall and die. Same. And I think that there's a similar kind of vertigo that sometimes people get when thinking about math concepts. And people have been so humiliated and traumatized by their past math experience that it feels like they're going to get hurt if they look over that cliff. And what I want to say is that you're not going to die anyway by looking over the mathematical cliff. You might get

some emotions dug up from some past negative experiences. But the good thing is that you won't physically die by looking over the math cliff. And the thing that's over the math cliff is something that maybe kind of gives you intellectual vertigo. I don't run away from those things. And I think that's the only difference. It's not that I'm better at it or something. It's just that I have

decided that I'm interested in it and that it's not a sign that I'm bad at it if I don't understand it. I don't understand anything if I think about it hard enough. The number one, what even is that? I can't understand the number one. If I don't understand the number one, how am I going to understand infinity? But that's what drives us to do more research. It's the feeling that there is always more there to understand and we want to understand it. We don't go, oh, I can't understand this. We go, oh, I want to understand more. I

I'm never going to understand all of it, but that doesn't mean I'm going to give up. I'm just going to keep trying to understand more all the time I can. Some people call this a growth versus a fixed mindset. And fixed mindsets tend to be afraid to try and fail. But growth is like, you know, whatever happens, I want to try.

I learned some. And from personal experience, two of the smartest people I've ever encountered, my friends, doctors, Casey Hanmer and Christine Corbett, have worked at NASA. Casey now heads Terraform Industries, working on these new ways to capture atmospheric carbon and turn it into natural gas. And one thing about the two of them and their marriage is that they just try stuff.

They want to learn something. They just dive right in. As a result, Dr. Corbett has an extra master's in creative writing and just got a black belt in Kung Fu. They are smart people who ask smart people sometimes basic questions. And Casey once told me the key to how much they get done is that they just learn shamelessly from any failure. They get it.

They keep moving. So they are heroes. They are not zeros. Oh, which reminds me that Brenna had a question. On that note, Jenny Lowe Rhodes, Maggie Morgan, star Lillian Wright, Christine Pickstein, Meg C, Lily Hagedon, and Brenna Pixley all have questions about zero. Jenny wants to know, how long has zero been a thing?

Maggie's like, is zero a number? Great questions. I do know that it has troubled people for thousands of years about whether it really ought to count as a number or not. My recollection is that maybe the Greeks...

or the Romans, one or the other, or maybe both, that were really unconvinced that zero should count as a number. Philosophically, it really bothered them. And that really held them back. Because here's the thing, you can decide whether you want zero to count as a number or not. But

The question is, how does it help us? There are often no right and wrong answers to these things. Mathematicians say, okay, here's a world in which this thing is true. And here's a different world in which something else is true. Which one is going to help us? So we can make a world in which zero isn't a number. We're not going to be able to do very much there. Zero is a really helpful number to have around because if we're going to represent nothing, what are we going to represent it with? If we subtract five from five, what do we get?

If we don't have a number called zero, we can't do that. And then we can't really make negative numbers either. And if we can't make negative numbers, there are all sorts of things we can't do. And so if we do call zero a number, then we do get to do all these things. And so that's why mathematicians generally have decided that zero should get counted as a number just so that we can do all of those things. It's just language. So

As long as we can manipulate zero in some way, we can get to do those things. If you feel like you don't want to call it a number, then that's fine. Don't call it a number. As long as you can still manipulate it in those ways, because it's really helpful to do it. So zero, we can wrap our brains around that. Some of you had a weirder concept plaguing you, such as Anna Thompson, Jennifer Lemon, Ariel Van Sant, Christina Kuhn's

Eleonora, Renaud Banville, CJ Wyatt, Tony Vessels, Sarah Val McKelvey, Felicia Chandler, Valerie Bertha, and first-time question asker, Bill LaBranch, and Christina Kuntz asks, imaginary numbers, what the fuck? What is an imaginary number?

What's going on? Yeah, this is great. And this is a great thing to say after is zero a number because sometimes people go, are imaginary numbers a number? So mathematicians really don't like having rules imposed on them. And it makes me sad that one of the things that puts people off math is that there seem to be all of these rules that you have to follow. Whereas mathematicians go, wait, okay, those rules, we

We had to follow those rules in this world, but I don't want to follow them anymore. Can I build a different world in which I don't have to follow those rules? And so one of the rules mathematicians don't like having to follow, and I do talk about this in the book, is you can't take the square root of a negative number.

Well, why can't you take the square root of a negative number? Because if you try squaring numbers, square rooting is sort of the opposite of squaring, right? So you're saying, is there any number such that when I multiply it by itself, I get negative one? Well, let's think about that. If I multiply a positive number by itself, I get a positive number.

If I multiply a negative number by itself, I also get a positive number. You are employing a double negative. So we seem to be out of options for things we could multiply by themselves and get a negative number. But mathematicians go, never mind, I'll just make something up and see what happens. And so it's called an imaginary number because we sort of just imagined it. And the wonderful thing about abstract concepts is as soon as you imagine them,

They become something that exists because it's an abstract concept. And so I would love to imagine my dinner and for it just to exist. But it doesn't work like that. Or imagine some dollars in my bank account. Oh, there they are. But I love that about math. You can just imagine something into existence. So you just imagine that there is something and you take it square and it is negative one. So what is it? Well, it doesn't really matter because in math, it doesn't matter what something is. It only matters what it does.

And I think this is a wonderful thing to think about in life as well, because really, it shouldn't matter what somebody is. No, it shouldn't matter what the color of their skin is or what they look like or how large they are or what gender they are or anything. It should only really matter what they do. Are they a nice person? Are they helpful? Are they kind? Are they generous? Those kinds of things. And so in math,

We kind of put aside the question of what that imaginary number is. We just say, what does it do? And it's like when children make up a game, they will make up a whole world and they will play that game. And that's what math is. But then it goes one step further because it goes,

oh, is this helpful? Crikey, it actually is. And so it starts off as being just some kind of ludicrous game that we're playing in our head. But it turned out to be really helpful for solving problems in physics. Isn't that extraordinary? And I

I just think it's extraordinary that this thing that we made up that doesn't have any physical reality to it is helpful in physics. Because what happens is that you draw pictures of it. And when you have ordinary numbers, you draw them on a number line, right? We put numbers on a line and it's one line. A line is very flat. It's one dimensional. But if you add imaginary numbers into that mix...

Where do your imagining numbers go? Well, they're not on the line. So they just have to go in a different direction. So you can just make them sort of go up the page. And then when you mix all those things up, you get a two-dimensional space instead of a one-dimensional space. And, you know, in two-dimensional space, you can see beautiful patterns that you can't see in one-dimensional space because it's so incredibly thin.

a line is just too thin. You can't see patterns. Whereas in the two-dimensional space, you can see gorgeous patterns. And that's how it's helpful in things like physics, because you see, you work out the patterns using the two-dimensional space. And then once you apply it to the real world again, you just take the part that's on the line, but it's just that you can recognize what the patterns are because you were exploring them in two-dimensional space. It's kind of like

If you're clearing out your closet, you kind of need to take everything off the rail and spread it out on your bed before you put it back on the rail again. So behind every perfectly Marie Kondo closet is the chaos and the pain and the discovery and the beauty of purging and whittling what's hiding in the universe's crevices and corners.

Well, kind of on that note, too, Felicia Chandler and James Dean Cotton want to know about the Fibonacci sequence. And Felicia asked, why do we see it so much in nature? I imagine that Dan Brown's The Da Vinci Code really...

put the Fibonacci sequence on the map. I'm into something here that I cannot understand. What does math have to say about that particular equation in nature? So perhaps I should remind everyone that the Fibonacci sequence starts 1, 1, and then you add up the two previous numbers to produce the next number. So you get 1 add 1, which is 2, and then you add that to the previous number, 1 add 2, which is 3, then you get 5, and then you get 8, and then you get 13, and so on. Okay.

Okay, so just a quick primer. Fibonacci Leonardo Pisano was an Italian guy who, around the year 1200, wrote a book about math and popularized the standard Hindu-Arabic numerals in Europe. Everyone loved him for it, thought he was dope at math, and then he died, and everyone forgot about him for like 400 years, until the 1800s. Two things. He didn't actually invent or discover the Fibonacci sequence.

That dates back to at least 300 BCE when this Indian poet, Bengala, was already down with it. Not Fibonacci's fault that we named this sequence after him. Another thing is that Fibonacci was from Pisa, which is where the Leaning Tower of Pisa is. And it was built around 1200, during his lifespan. So where was he when it came to applied mathematics? Yeah.

Yeah, I'm super busy these days. Also, I didn't intend this, but Worldwide Fibonacci Day is this week, occurring on the same day as the dog holiday Wolfenoot and American Thanksgiving. But every year you can celebrate Fibonacci by eating artichokes and Romanesco and pineapple because 1123 is Fibonacci-esque, even though, again, it wasn't him to discover the sequence.

Also, I need you to know that I only found out about Fibonacci Day because I googled, was Fibonacci hot? And I happened upon a blog that mentioned 1-1-2-3 and that there is little known of his physical appearance. I don't know. I just have a hunch. I bet he was hot.

Maybe like a 10. I think the first thing to say is that it is not quite as prevalent as its popularity may suggest. Okay. It's good to know. It's quite the golden ratio, which isn't nearly as prevalent as its popularity would indicate either. But I think the reason is that nature...

I don't want to anthropomorphize nature too much, but nature is trying to do as much as it can with the smallest starting point as possible because that's kind of efficient. And why that is the case is a whole deep philosophical or possibly biological question. But it may be to do with

of the fittest and evolution, the things that survived were the ones which did the most possible stuff with the smallest possible amount of information. And so sometimes it really is a sheer mathematical situation. So for example, the spirals on a pineapple are

typically Fibonacci numbers. And so if you count how many spirals there are in each of the different directions, there are three different directions that spirals on a pineapple can take. There's the kind of really vertical direction, there's the more obvious slanty direction, and then there's the really, really, really slanted direction that's less obvious. And the thing is that geometrically, the two

The two smaller numbers have to add up to the bigger one. You can work that out on any shape at all. Why they typically turn out to be Fibonacci numbers may be something to do with the way that the little fruitlets grow. And it's the same with the leaves on the stem of a plant, that they grow at different angles spiraling around the plant, possibly to get as much sunlight as possible.

And the angles, the way that the angles work in a Fibonacci sequence is sort of to try and make sure the next time a leaf lines up with the one beneath it, it's as far away as possible. But it's often only a very small part of the Fibonacci sequence. So it's just like two consecutive numbers or something. And so that's why I think it's not quite so prevalent. If you have leaves spiraling around and it happens to be in a pattern three and five,

That's only two numbers out of the Fibonacci sequence. So I wouldn't go, oh, wow, it's a Fibonacci sequence. And maybe it's just three and five and it's nothing to do with the Fibonacci sequence at all. And sometimes the reason that something that's constructed very simply in math pops up all over the place

is a beautiful and maybe slightly mysterious thing. And that's one of the things that's wonderful about it. And that is maybe not an answerable question. And this is why sometimes people believe in a higher being because it seems like some higher being created that. But I personally don't feel the need to say that it was a higher being, a specific kind of delineated higher being. I just think that math is a higher being. Math is my co-pilot.

This one was on the minds of Jim Pompeo, Milan Ilnicki, Mushroom Morgan, Rachel Gardner, Taylor, The Ren You Know, Karina Reagan, Felix Lassell, Claire Nurk, and Lizzie Lassner.

and listener Victoria Sauter, who wrote via patreon.com slash ologies, I'm so excited you're talking to Eugenia Chang. I have a huge math crush on her. I have two of her books. Victoria writes, as a teacher, I believe math is for everybody, and I try to make clear that math isn't an isolated subject when we use it to understand all parts of the world. Suggestions to bring math into the living room, so to speak, that's a little juicier, Victoria writes. Last question from listeners,

So many people wanted your expert advice. And Elsa Sparks wanted to know, any recommendations for teaching number sense to very young students? Blessed are the cheesemakers asked, math professor here, do you have any advice for college professors or any teachers who want to improve their teaching? So any advice to people teaching or learning math that you feel like has been empowering to people? Well, see her book.

This is why I write all my books to try and give people ideas and help people get over their past traumas. I think that one of the messages that I most want to say to everyone who's learning math, and therefore I want everyone who's teaching it to also pass that on, is that if you find it hard, that doesn't mean you're bad at it. You're probably just right. It is hard.

and that that's not a reason to be put off it. And if you look around you and it seems that other people are finding it easier, they might just be talking out of their armpit, so to speak, in order to intimidate other people. Because most mathematicians, in fact, every mathematician I know thinks they're stupid and finds everything very difficult. Is that true? Yes, that is the, I think that is a real, it's a correct impulse to think it's hard because it is hard. But

But the point is we can keep understanding more of it all the time. And I think that celebrating the questions that children ask and not being afraid if it's a question that you don't know how to answer, because then you can celebrate them for having asked a question you don't know how to answer. And then we can all learn about how you discover answers to questions. And then if it's a question that nobody knows the answer to, like why ask?

Does the Fibonacci sequence come from nature? Then you can say to a child, if you become an expert in that field, you could be the one who answers it. And then you could be the one going on this podcast, explaining it to other people. Please do. What about, I always have to end with these, but what's your least favorite thing about science?

what you do as a professional abstract mathematician? I think it is overcoming people's misconceptions. And it's the fact that the misconceptions are so deeply embedded culturally. And that's what I'm trying to overcome. I'm not trying to get everybody to love math because we don't all have to love the same things, right? I'm just trying to show it for what it really is. If everyone saw it for what I think it is, and they still didn't like it, then

then, you know, fine, we can all like different things. It's the fact that people see some really, really narrow, shallow side of it and then write it off from their lives when I think it can help everybody. I find that frustrating, but that's the challenge. It's to overcome those deeply embedded misconceptions. And

It's been so great talking to you about it because it doesn't sound like you have those things. But sometimes even getting something published can be difficult because the people who want to publish something have their own misconceptions and go, well, I want you to write about math, but this isn't math. And so then I have to persuade them that this is math. What about your favorite thing or your favorite number, your favorite theorem, your favorite moment? What has just made your heart sing the most in your career? My favorite thing is...

is being able to help people understand something they didn't previously understand and seeing their gasps of delight when some mathematical thing has got them to be as excited as when I see it. That is what gives me the most joy in math, I think, is being able to give that joy to other people.

I wonder if there's a word in any language for that moment when something clicks, when you are figuring it out, figuring out, figuring out, you don't get it. You don't get it. And I've sat in calculus classes before. I don't get it. And then suddenly something changes over and you're like, I get it. I get it. I get it. Whether it's a joke or whether it is a theorem or something, I wonder what is happening in the brain because that moment is just unlike anything else when you finally get it.

And it really feels like pieces falling physically into place, doesn't it? It does, yeah. I'm fascinated by the brain. I'd love to try and do brain scans while I'm doing things like that to try and see what's going on. This has just been such a joy. Oh, I've really enjoyed it too. Thank you so much. So you have it. Ask the simplest questions and you'll learn the deepest truths.

And for more on that, you can see Dr. Eugenia Chang's freaking book titled, Is Math Real? How Simple Questions Lead Us to Mathematics' Deepest Truths, which is linked in the show notes alongside her website and her social media. We are at ologies on Instagram and I guess Twitter.

But I'm at Allie Ward with one L on both. And Ologies merch is available at Ologiesmerch.com in case you need holiday gifts. Smologies are available for free. They are G-rated. They're at AllieWard.com slash Smologies. Those are kid-friendly versions of classic episodes. To join Patreon and submit your questions before recording, head to Patreon.com slash Ologies. Erin Talbert admins the Ologies podcast Facebook page.

group. Emily White of The Wordery makes our professional transcripts. Noelle Dilworth is scheduling producer. Susan Hale is our major managing director, who also assisted in research for this episode. Kelly R. Dwyer makes the website. Zeke Rodriguez-Thomas and Jera Sleeper of Mind Gem Media worked on Smologies alongside our lead editor of this episode and of Ologies, the infinitely talented Mercedes Maitland of Maitland Audio. Nick Thorburn wrote the theme music. And if you stick around after the show, I'll tell you a secret.

Okay, so this week it's that I used to make cocktail recipes for cooking sites. I used to write about nightlife way back when. So I'm pretty versed in making cocktails. Like at a party, people would be like, we got some peppermint schnapps and a cantaloupe and a pixie stick and some Tabasco. Make something out of it. And I could usually whip something up. But I'm not much of like a make cocktails at home kind of person these days. Pretty chill. But I just found myself making a beverage and I use like some lemon powder and some peach emergency and...

heaping scoop of Metamucil. And I had it in a Nalgene and I was just shaking that thing up because you have to. And I realized that the muscle memory of the cocktail shaking is still there. It's still strong, but just a very different contents and very different vibe. But hey, you got to hydrate. You got to get those electrolytes. Colon motility is important. And so is math. Math is real, but nothing's real. And life is very beautiful. And Metamucil can be tasty.

All right. Bye-bye.

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