Ideas - How geometry is embedded in daily life

Episode Date: July 15, 2026

"Geometry is the cilantro of math. People are not neutral about it," says Jordan Ellenberg. The mathematician is aware not everyone loves math like he does. But his book is not about the kind of geometry some people feel intimidated by. It's about geometry as a way of thinking. As a method of reasoning and argument. And as a system for making sense of the world. *This episode originally aired on May 11, 2022.Guest in this episode:Jordan Ellenberg is a mathematician is a child prodigy. He scored a perfect 800 on the math portion of his college entrance exams — the SAT's — at the age of 12. He's a Guggenheim Fellow, a Harvard graduate, and a professor of mathematics at the University of Wisconsin-Madison. He's also the author of Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else.

Hi, it's Thomas LeBlanc, host of Celine Understood. A deep dive into the cultural, political, and economic alchemy that created Celine Dion. And there are more seasons have understood. Follow our feet to learn more about other big stories in the news like the rise and fall of Cryptoaking Sam Bankman-Fried, or the controversy behind another notable Quebec institution, Pornhub. And if you're already a follower, thanks. Please tell your friends about us. This is a CBC podcast.

How many holes is in a straw? One hole. It's two holes. One hole. Welcome to ideas. I'm Nala Ayyad. The reason why I say two, because if you got two exits, that's two holes. You got a back door and the front door of the house.

That's two doors. That's two exits. That's two holes. So you're saying your house got one hole? No. What? What? You're listening to a long-running argument on TikTok.

How many holes are in a straw? Is it one, two, zero? It's one, it's a tube. It's one continuous hole. It's all one thing. That makes no sense and it's completely illogical. The thing is like your intestines, one long continuous tube. I got into a heated argument with my best friend's dad who kept telling me it's two holes.

The holes in a straw argument took off on TikTok. in early 2021. It had already gone a few rounds on Twitter, Reddit, and Snapchat before that. And on the surface, it seems to be a debate about semantics. What exactly qualifies as a whole? My first goal is to convince you you're confused about the holes. But that confusion is also fun. It is always a pleasure for those of us in the mathematical professions.

When the internet spends a day or two tying itself in a knot over a math problem, we get to watch other people discovering and enjoying the mode of thought we spend our whole lives taking pleasure in. When you have a really nice house, you like it when people unexpectedly come over. Jordan Ellenberg was a child prodigy. He scored a perfect 800 on the math portion of his SATs or college entrance exams at the age of, wait for it, 12. He's a Guggenheim fellow, a Harvard graduate, and a professor of mathematics at the University of Wisconsin-Madison. Oh, he's also the Great Square Rudio. You say a number? He'll tell you it's Square Root. Pretty closely, anyway.

Mental computation of Square Roots was a party trick I learned in college. Its social utility in that context was not as great as I'd anticipated. His latest book is called Shape. the hidden geometry of information, biology, strategy, democracy, and everything else. But it's not the geometry of Euclid, that simple set of axioms that lets you prove that this angle is equal to that one. It's about geometry as a way of thinking, as a method of reasoning and argument, and a system for making sense of the world. I spoke with Jordan Ellenberg from his home in Madison, Wisconsin, Wisconsin, About geometry, sure, but also about democracy, education, and yes, how many holes there are in a straw.

Hey, Siri, does a straw have one hole or two holes? A straw has zero holes. It is a tube. I am a mathematician who talks about math in public. And this seems to unlock something in people. They tell me things. They tell me stories I sense they haven't told. anyone in a long time, maybe ever. Stories about math.

Sometimes sad stories. A math teacher rubbing a kid's ego in the mud for no reason but meanness. Sometimes the story is happier. An experience of abrupt elimination that burst open a child's mind, an experience the grown-up wanted to find a path back to but never quite could. Actually, this one is kind of sad, too. Often, these stories are about geometry. It seems to stand out in people's high school memories,

like a weird, loud, out-of-scale note in a chorus. There are people who hate it, who tell me geometry was the moment math stopped making sense to them. Others tell me it was the only part of math that made sense to them. Somehow, it's primal, built into our bodies. From the second we exit hollering from the womb, were reckoning where things are and what they look like. When South American mystics and their non-South American imitators

drink ayahuasca, the sacred hallucinogenic tea, the first thing that happens, okay, the first thing that happens after the uncontrollable vomiting, is the perception of pure geometric form. Repeating two-dimensional patterns like the latticework in a classical mosque or full three-dimensional visions of hexahedral cells clustered into pulsating honeycombs, geometry is still there when the rest of our reasoning mind is stripped away.

You describe geometry not just as a branch of math, but an actual way of thinking. What is that way of thinking entail? I think people don't appreciate how much it's woven into our ordinary way of thinking that we do when we're not in the classroom, when we're not working problems with a pencil and paper. But to make that claim, I suppose, I should say that my interpretation of our way of what counts as geometry is very broad. You might not be surprised to hear. I mean, the word literally means like the measurement of the world, right?

Like, okay, I'm not going to try to speak Greek to you guys, but like Jios and Metros, right, earth and measure. And I would say whenever we talk about distance, whenever we talk about sort of two things being close to each other, like, oh, a close relative. Okay, if you think about your family tree already when you talk about a close relative versus a distant relative, That's a geometric way of thinking about your family. When you describe it as a tree, when you draw a picture, you're drawing a fundamentally geometric diagram that takes these familial relationships that you have and geometrises them, sort of makes a picture, like draws a shape that your family forms. When you talk about something you do on the radio a lot, right, the two sides of an argument, already you're sort of imposing a geometric metaphor.

And I actually think being conscious of that is very useful because often you say to yourself, wait, like, why does it have two sides? Like lots of figures have different numbers of size, not just two. If you sort of interrogate these metaphors a little bit and think more seriously about what they say, I think that can be a very valuable practice. Just walk me through that utility. Why, you know, help me as an interviewer if I'm talking about a two-way argument. Why is why is that useful? So I would say that often, nobody ever asked me this before.

Everybody always just accepts as his face value. You're going to make me go through my paces, which is awesome. I'll try. So let me spin it out a little bit. Let me improvise on that. What I would say is that if you sort of somewhat thoughtlessly say the two sides of the argument, what are you doing? First of all, as I said, you're saying there are only two sides. So you may be missing something, right? You may be missing an approach that you haven't thought of. But thing two is, I think,

if you're metaphor is, oh, it's like the two sides of a stop sign, the front and the back, you're saying something very specific about the relationship of those two positions, that they are opposite of each other, that whatever is true of one is false of the other and whatever is true of the other is false of the one. And most interesting social or political or intellectual or literary or philosophical arguments are not actually like that, right? The two sides of the argument, if you like, maybe in opposition, but they're not literally mirror images of one another, which is what you might think of, which is what the metaphor applied to carelessly might lead you to think. In your book, you write that geometry is not just a way of measuring ourselves or

measuring, but it's a form of honesty. What's so honest about geometry? Yeah, and that's another feature. And if I, you know, I often, I often say, and I put this in the book that geometry is the cilantro of math. Like, people are not neutral about it. There's people who love it and people who hate it. And if you talk to people about it, which, you know, going around giving talks and talking to people about math, people, I mean, you know, they come to me with their math stories. It's like, I'm like the therapist. And like, they've been waiting like 20 years to like unload this trauma that they have of there. I mean that very seriously. But what I find, and this is sort of part of the impetus of writing this particular book about this particular subject, that there are two kinds of people. There's people

who are like, I loved math in high school, except geometry. Like, what was up with that? Then there was this weird interlude where we just like drew pictures and proved obvious things. Like, why do we do that? And then there's other people, I would actually say maybe a little more numerous who are like, I hated math, the entire thing in high school, except geometry. Why wasn't it all like that? That was the part that made sense. But people are not neutral. People recognize that it's different. And I am coming to answer your question, by the way. I'm making my way towards it. You know, if the case I was making was, oh, it really organizes the way we think about the world, what we started with, I could say that about geometry, but I could say it about algebra too. I could say it about probability too. I could

sort of argue, boy, we've got to think about things probabilistically. We've got to think about things algebraically. Of course, I believe all those things. I'm a mathematician. But I do think the geometry has a special role that makes it very different from everything else we learn in school math and frankly, everything else we learn in school. The honesty piece is this. The geometry is where we prove things, where we write proofs of statements. And that is very, very special. It's the place in school where we can make our own knowledge. We're not reliant on the authority of the teacher, we're not reliant on the authority of a book. There's certain rules, and we can construct a fact, even if it's a sort of somewhat abstract fact about a triangle or a

circle, we can make it from scratch and nobody can tell us that we're wrong. That's extremely powerful. And I think people respond to that. There's a certain electricity to it. People could respond to it positively or negatively, but I think that's what sets it apart. So, honesty, you know, when I write a book, it's never about what I think it's going to be about. I always sort of start researching and then like find super exciting stories that I didn't know were there. So when I was writing this book, I discovered that Abraham Lincoln was a huge geometry enthusiast. Who knew?

I did not know any of this, but I sort of found out about it. I don't know it's something people have written about. In 1864, the Reverend J.P. Gulliver of Norwich, Connecticut, recalled a conversation with Abraham Lincoln about how the president had acquired his famously persuasive rhetorical skill. The source, Lincoln said, was geometry. He has this kind of crazy crisis of faith when his political career is sort of in the toilet and he's going around being a country lawyer.

And he says to an interviewer, you know, every day I was going into court and being asked to prove things and I was like, what does that mean? In the course of my law reading, I constantly came upon the word, demonstrate. I thought at first that I understood its meaning. meaning, but soon became satisfied that I did not. I consulted Webster's dictionary, that told of certain proof, proof beyond the possibility of doubt, but I could form no idea what sort of proof that was. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. At last I said, Lincoln, you can never

make a lawyer if you did not understand what demonstrate means. And I left my situation in Springfield, went home to my father's house, and stayed there till I could give any propositions in the six books of Euclid at sight. I then found out what demonstrate means and went back to my law studies. He's kind of made of different stuff in a way, right? So he's... Well, he is honest Abe. Right, exactly, honest Abe. And he became a kind of lifelong enthusiast. What is it saying? about Lincoln that in your estimate that he went to geometry and not these other more traditional sources? I think, okay, I'm just going to keep on doing my like amateur psychology of Lincoln. Remember this thing I said that geometry doesn't rest on authority. It doesn't rest on it's in the book,

therefore it's correct. It rests on here's the rules. You build it up yourself. That's very much Lincoln's personality. Like you build it yourself, the self-made person, right? I mean, he's not from a patrician background, like the earlier American leaders we talk about, like Jefferson, right? So I do think the idea of I built this myself is like very much in keeping with Lincoln's character. And I think that's what geometry offers you. It offers you that ability to sort of build it up from scratch and say, I know this is right and I know this is sound because I made it. And I trust myself. You're very covetous of the word, therefore. Talk to me a bit what but why geometrists are, in your words, kind of the only ones who can actually...

Okay, but you struck on another of my pet peeves because we're geometers, not geometers. Sorry, sorry, sorry. I said a geometer. No, no, it's okay. Yes, and it's because, I mean, you do see that, right? I mean, definitely this is something that a problem that mathematicians have. You read an article in the newspaper that will say it proves something or we'll say, especially it's a tell, right, this word.

Therefore, it must be that, oh, if you believe this, then therefore you must believe that. And for those of us who live in the world of proofs, who live in the world of rigor, who live in the world of deduction, we're like, I know what therefore means. And it doesn't mean that. Like, you're saying, therefore, because you mean there's something I really want you to believe and there's something I really want to say. And if I sort of say, therefore, I'm saying it was sort of a more ringing force. But it's not a proof. And by the way, when I say that, I have to always hurry to point out that I am not saying that I wish the stuff that was in the opinion page of the newspaper were proofs of a Euclidean nature. I mean, that would be horrible.

First of all, most of the issues that we care about are not subject to proofs like that. And second of all, it would be like, they would be very long. Bit cumbersome to read. So, you know, so I think it's not so much. This is always a big question because people will say, why are students learning to do proofs? Why do we spend this year in America, it's in the ninth grade. I don't know when it is in Canada. Why do we spend this year training students to do proofs. Are there proofs in real life? Mostly no, I'm going to be honest. I hope we won't get in trouble for saying that. But what there are is a lot of non-proofs wearing the clothes of proofs. A lot of non-proofs that have words like therefore in them or words like must or

words like prove. And I think once you really know what a proof is, once you've really felt one clicked together, you become kind of impervious to the fake proofs. You're able to distinguish a non-proof from a real proof. And I think that's kind of what Lincoln wanted. He He wanted to have that ability. I promise you, it wasn't like he went into court and was giving Euclidean deductions and drawing diagrams on an easel, like in the middle of the court. But it meant that once you really know what it is, you're much less likely to be tricked. By others or by yourself.

What Lincoln took from Euclid was the idea that, if you were careful, you could erect a tall, rock-solid building of belief and agreement by rigorous deductible. steps, story by story, on a foundation of axioms no one could doubt, or, if you like, truths one holds to be self-evident. I hear the echoes of Euclid in Lincoln's most famous speech, the Gettysburg Address, where he characterizes the United States as dedicated to the proposition that all men are created equal. A proposition is the term Euclid uses for a fact that follows logically from the self-evident axioms, one you simply cannot rather than.

rationally deny. Lincoln wasn't the first American president to look for a basis of democratic politics in Euclidean terms. That was the math-loving Thomas Jefferson. Jefferson had studied Euclid at William and Mary as a young man, an esteemed geometry highly ever afterward. In 1812, retired from politics, Jefferson wrote to his predecessor in the presidency, John Adams.

I have given up newspapers in exchange for Tacitus and Thucydides for Newton and and Euclid, and I find myself much the happier. Here we see a real difference between the two geometer presidents. For Jefferson, Euclid was part of the classical education required of a cultivated patrician of a piece with the Greek and Roman historians and the scientists of the Enlightenment. Not so for Lincoln, the self-educated rustic. For Lincoln, unlike Jefferson, the Euclidean style isn't something belonging to the gentleman or the possessor of a formal education

because Lincoln was neither. It's a hand-hewn log cabin of the mine. Built properly, it can withstand any challenge and anybody in the country Lincoln conceived can have one. Here's an easier question. If Lincoln was drawn to geometry because of propositions, proofs, persuasion, what about you?

Do you remember a moment or a memory that something kind of just clicked, so to speak, and brought you to that place. Here's one. And it sort of comes back to this idea of making knowledge yourself. This is what I was a pretty small kid, but I knew my multiplication tables or I knew like some of them. And I was just kind of like line on the floor, like looking at whatever around the house and like, you know, my parents had a stereo. This is the 70s. I'm dating myself here. So I was a kid in the 70s. So everybody had like wood paneled stuff in their living rooms. And so we had a stereo with like a wood panel in front of it with like a bunch of holes in it, you know, so like the sound can get out.

And the holes were in kind of a rectangular array, like a six by eight array of holes. And it's kind of like line there contemplating it. You look at this rectangle and you're like, oh, okay, so there's like six columns and each one of those columns has eight holes. But then on the other hand, there's eight rows and each of those rows has six holes. And then suddenly you're like, oh, crap. like six eights is the same thing as eight sixes. They're the same because I'm looking at them. There they are in the rectangle.

The six rows of eight are the same thing as the eight columns of six. Now, of course, because I knew the tables, I knew that and I observed the symmetry. I knew that if you knew six times eight, you also know eight times six because that's the way it looks in the table and anybody who learns their multiplication table notices that. I knew it, but I didn't know it, right? I didn't really know it until that moment. And the mechanism by which I suddenly came to know it was a geometric one, right? it's because they're in a shape.

Then once you know that, it's not something that somebody told you. Because if it's just something that somebody tells you, right, then how do you know that 11 times 12 is the same as 12 times 11, unless you know it in the table? I mean, you might sort of infer it. You might be like, oh, it seems like that's the way it works. But once you see the rectangle, once you do the geometry, you really know it, the click.

And that's incredibly satisfying and incredibly empowering. You're listening to Ideas on CBC Radio 1 in Canada, across North America on Sirius XM, in Australia, on ABC Radio National and around the world at cbc.ca.ca. You can also hear ideas on the CBC Listen app or wherever you get your podcasts. Uncover from CBC Podcasts brings you award-winning investigations year round. Infiltrate an international network of neo-Nazi extremists. Ranted with racist language. Discover the true story of the CIA's attempts at mind control.

Their objective was to wipe my memory. Or dig into a Crypto King's mysterious death and a quarter billion dollars missing. There are deep oddities in this case. With episodes weekly, Uncover is your home for in-depth reporting and exceptional storytelling. Find Uncover wherever you get your podcasts. Jordan Ellenberg is a mathematician. professor and author of the book Shape, the hidden geometry of information, biology, strategy, democracy,

and everything else. He's not just a mathematician. He plays one. Who can tell me? What? Four. Seven thousand and six hundred and ninety-five. In the movie, gifted, about an exasperated child prodigy,

Ellenberg briefly appears, as, you guessed it, a math professor. What I want to notice is that when you compute P of N for N congruent of four or mod five, the answer is a multiple of five. Which means he's now one of a select group of people in the world with what's known as an Erdosch Bacon Number. Maybe you've heard of a Bacon Number. That's the number of steps between any given movie star and prolific actor,

Kevin Bacon. For example, Diane Keaton has a bacon number of two. She was in Marvin's room with Robert De Niro. Do you mind if I call you Augustina? Well, my name is Bessie. Bessie, of course, I'm sorry. And Robert De Niro was in sleepers with Kevin Bacon. I may not be in your division, but I do weigh more than 85 pounds. Yeah, that's right. So how you been? Mathematicians have their own version of the bacon number. It's the aird number. The number of steps away from co-authoring a paper with the prolific Hungarian mathematician, Paul Erdos. For example, Albert Einstein has an Erdos number of two since he co-authored with Ernst Strauss,

who co-authored with Erdos. But only a very select few have an Erdoche Bacon number, linked both to a paper published by Erdoch and a movie starring Kevin Bacon. Jordan Ellenberg's Erdos number is three. I wrote a paper in 2001 about modular forms with Chris Skinner, who as a Bell Labs intern in 1993, wrote a paper about Zeta Functions with Andrew Adlisco, who wrote three papers with Erdisch between 1979 and 1987. My Bacon number is two, thanks to being engifted with Octavia Spencer,

who played big customer opposite Kevin Bacon's Jorge in the 2005 Queen Latifah vehicle Beauty Shop. So my Erdisch Bacon number is 3 plus 2, or 5. With that 5, Jordan Ellenberg joins Stephen Hawking, Colin Firth, Elon Musk, and just a handful of others. Besides the prestige, Ellenberg appreciates the geometry of his Erdos Bacon number, that when you think of two movie stars or two mathematicians and the distance from one to another, you're thinking geometrically.

And otherwise scattered and meaningless relationships can also be ordered into shape. Here's the second part of my conversation with Jordan Ellenberg. Rita Dove is a Pulitzer Prize winning poet, a former poet laureate of the United States, and the Commonwealth Professor of English at the University of Virginia, where Thomas Jefferson and James Joseph Sylvester both thought deep mathematical thoughts in their day. But in the early 1960s, she was a nerdy kid in Akron, Ohio. Her father was an industrial chemist, the first black research chemist a good year tire. Most great poets never write even one math poem, but Dove wrote two. Geometry. I prove a theorem and the house expands. The windows

jerk free to hover near the ceiling, the ceiling floats away with a sigh. As the walls clear themselves of everything but transparency, the scent of carnations leaves with them. I am out in the open. And above, the windows have hinged into butterflies, sunlight glinting where they've intersected. They are going to some point, true, and unproven. There's something special about geometry, something that makes it worth writing poems about. Everywhere else in the school curriculum, you must, in the end, defer to the teacher's authority or a textbook's. when it comes to who fought in the French and Indian wars or what the principal products of Portugal are.

In geometry, you make your own knowledge. The power is in your hands. The Pythagorean theorem isn't true because Pythagoras said it was. It's true because we can, ourselves, prove that it's true. Behold! But truth and proof are not the same thing. That's where Dove's poem ends with the point true but unproven.

Proof is an essential tool for us, the measure of our certainty just as it was for Lincoln. But it is not the point. The point is to understand things. We want not just the facts, but the souls of the facts. It's at the moment of understanding

that the walls go transparent, the ceiling flies off, and we're doing geometry. I want to read you part of an article from the Washington Post way back in 1987. It was about you winning a competition called the USA Math Olympiad. This is what you had to say about math way back in 1987. I always think of it as a zoo. There are a million different mathematical objects. They are like animals. Some are like each other and some are unalike. The

amazing thing is it all connects. Something or anything you prove with trigonometry is just as true if you do it with algebra. I think it's kind of amazing, actually, if you think of it from an emotional point of view. I don't think a lot, or perhaps most people listening to our conversation, would have an emotional point of view about mathematics. So what is it that's emotional about it for you? So first of all, I am curious because what 16-year-old me wrote makes perfect sense to 50-year-old me, but does it make sense to you? I'm curious. Or do you read that and you're like, what the hell is that kid talking about? Is he drunk? I'm not curious how that reads. No, it reads in a wondrous way, actually.

To talk about math in an emotional point of view at 16. Wow. Like, yeah, I want to know more. I'd like to interview that kid. I have a 16-year-olds. 16-year-olds are filled with emotions. Trust me, I know this very well. I mean, it's, and we can forget this when we emphasize the formal and the rigorous

side of the subject. Although it's incredibly important, and it makes, it's one of the things that sort of makes math special and makes it different from other endeavors. We must never forget. math is made of people. I mean, it's a human activity that human beings do for human reasons and everything that people do, they have feelings about. Right? Like the desire to prove something is a desire. Like you feel frustration. You feel eagerness. Like you feel, okay, maybe it's more like 95% frustration, 5% eagerness in your actual life as a researcher. That's life in science. But, you know, as a teacher,

let me say this. Okay, so I teach math, right? I teach the University of Wisconsin. I'm in front of students all the time. The biggest human challenge we have as teachers, and this is going to sound sort of silly, but it really is important, is that we know math very well. Okay, that sounds like that should be good, right? And it is good, but we know math very well and our students do not. And in order to be an effective teacher, you have to kind of imagine yourself into the mind of not knowing it. That can be very hard, actually, if you know it very well and you've known it for a long time. Actually, I'm going through this right now because I'm teaching my, I said I have a 16 year old. I'm teaching him how to drive.

It's hard to explain how to drive, right? Because you just know. That's so intuitive. Right. It's actually hard to put yourself into the mindset of not knowing how to drive. And in the same sense, it's hard for me to put myself into the mindset of not knowing geometry or not knowing calculus or something like that. Do you still think of math as a zoo? Yes.

Absolutely. I mean, it's, and I think it's actually, maybe I was only just dimly grasping it then. but, you know, Henri Puancares, who's one of the great geometers and is a central figure in my book, and also just an incredible creator of aphorisms. I mean, if this guy could be on Twitter, it would be, like, incredible because he was, like, so good with these, like, little pithy sayings that he would come up with. So he said, so he has this remark that is sort of resounded with the history of mathematics ever since, where he says mathematics is the art of calling different things by the same name.

Beautiful, beautiful because it has that ironic tang that we love, right? It's like, wait, I thought mathematicians were supposed to be super precise and call everything exactly what it is. Well, yes, we do. But at the same time, it's also incredibly important to know what distinctions not to make. And in a way, what we do at the zoo is like that. Right? On the one hand, if you need to know exactly what subspecies of tiger that is, you can find that information. It's there on the little placard.

But you also, when you think about animals, you can say, okay, there's this tiger and there's that tiger. and sometimes you just need to know that it's a tiger. In any given moment, there's certain distinctions that are important to make and certain distinctions that are unimportant is kind of how we think about animals and other kinds of menageries. And it's how we think about math do. Well, it's funny. Actually, it's a good way to get into this next topic I want to tackle with you,

which you foreshadowed, which is how we teach kids about math. Because I think one of the points you were making about math is that we make the mistake of saying if kids don't understand it, it's probably their fault. Okay, well, I do have a little bit of a heterodox view about this. So I'll say, so I'll say this. I don't think that the goal of math teaching should be for every school kid to love math, which I think plenty of my colleagues would say, like, that's the goal. Like, we have to find a way for them to experience the love of math that people like me do, people in my profession and people who teach math at the K-12 level do. And I guess I don't believe in that

because I think it's authentically like kind of weird how much I love math. Like I think that that's It's rare. In the same way that there's people who like, you know, I also like super, super love like ultra salty like Swedish like herring rose spread like on a cracker. And like I know the people who love it, love it and like not everybody loves it. And that's okay. But what I what depresses me about about what's happening in math education is not if kids. are like, I don't care about math that much. I don't love it. Okay, you don't love it.

Fine. What is sad is if kids are like, I can't do this. I'm like congenitally incapable of do this. Kids who are trained to be like, this is just impossible for me. I think that's just almost never the case. And I do think we've trained a lot of children to believe that about themselves. And that's pretty bad. And yet you point out that one of the biggest problems in how we educate children in math is that we oftentimes,

often tells them that math is easy. What's wrong with doing that? Because it's not. That's the reason. Math is actually very hard. And there is a fine line. I am definitely not going to sit here and tell people how to do it because I am not a K-12 teacher. But even in college, we have the same issue. I think we need to find a way and it takes a lot of emotional intelligence, like it's not an easy thing, to send the message that it's hard, but it's a hard thing that you can do. that people in general can do. Like, we do hard things. We do hard things all the time.

Just like learning to, like, be an adult and live life is hard. But people do it. Raising a child is incredibly hard, right? Nobody would say it's easy to, like, somehow have a kid, like, survive from, like, infancy to, like, you know, through five years old. And yet, look around you. Like, everybody's doing it. And we don't tell people it's going to be easy. We don't.

We tell people it's hard, but there's going to be something in it for you. it's worthwhile and it's possible for you. And I think with work, we can send that message. But if we say it's easy, this is, I do you think it's really dangerous? Because if, and I had to unlearn this, by the way, it comes naturally to us, partly because if you know it really well, it is easy for you. Like, what if I said to my kid, like, okay, driving's easy. You just get on the highway and go.

Just, you know, I do this every day. Like, it's easy. Just like, okay, I might be like a smear on the pavement right now if that's what I told him. No, no. And I think that if you tell a child, this is simple, this is easy, and then they encounter the fact that it is manifestly not simple or easy, what are they going to think? Are they going to think my teacher lied to me? No, because kids trust teachers. They're going to think, oh, I'm stupid. The problems with me. This is easy and I couldn't immediately do it. So therefore, this whole enterprise, I'm out. I'm out. And we do send that message and I think it's incredibly

bad. Here's something that we heard about recently. The Florida Department of Education recently declared that it had rejected several math textbooks because they included what they called, quote unquote, critical race theory. The department didn't offer any details about what was in these math textbooks, but here's Florida's lieutenant governor appearing on Fox and Friends. Q tape. I knew it was being taught in school, but you know that there's an ideology here that they're trying to indoctrinate our children when it's tucked into math books. Indeed, Rachel, and good morning. And you're right. What we've seen is obviously a systematic attempt by these publishers to infiltrate our children's education by embedding topics such as

critical race theory, things that have nothing to do with math. Right. So they're seeing like math problems that, you know, have gender issues involved and CRT in it. What's been the reaction of parents from Florida on this? Especially in those youngest grades when kids are most impressionable. We have to make sure that our kids are free from indoctrination, that we're not embedding this ideology. And I think that parents really want to make sure that in Florida, their kids are going to be taught, not indoctrinated. So of course, we have no way of knowing what's actually in those books because the department didn't give any examples. But this is not the first political controversy over math education. There was a conflict over the so-called common core math a few years ago.

and before that, it was a debate between liberals and conservatives over new math in the mid-20th century. Why is teaching math so politically contentious? First of all, let me say this. I'm not going to like to say anything about the textbooks Florida is using or not using because I have not seen them. I don't know what's in them. Like, you know, I have my own biases, but I'm not, I mean, but I will say this. Okay, we have an amazing collection at the University of Wisconsin of historical math textbooks, of basically like books that were used in Wisconsin from about 1910 until the present.

And like three full bookcases of math textbooks. And looking at these books just standing there and picking them off the shelf and looking at them, what you realize is that every single controversy about math education that exists, we have been having for a hundred years or more. Every single approach. You know, so for instance, okay, I'll give you an example. You know, Common Core, which is very controversial. One of the things that people really didn't like is like, okay, I mean, there's a lot of different stuff that makes up Common Core.

But one thing is like, okay, we're going to sort of have a part of the classroom time be students trying to develop ideas about what's the case. Like trying to sort of rather than being told, here's the method, do it. Here's another method, do it. Here's another method do it. Being told like, here's a bunch of answers. Can you sort of figure out, can you sort of try to understand like what. relates them, like what the commonalities are, can you sort of like, you know, the so-called discovery teaching. I'm not even going to weigh in on whether or not that's like a good idea and for which

grades, et cetera, et cetera. I'm just going to say that, you know, while researching this book, I found a geometry textbook from 1860 that uses the exact same approach where there's no instruction at all, just a list of problems. And it's like, okay, try to do these like 180 problems in this order. Think about each one in the light of the ones you did before and discover for yourself, the principles of geometry. Nothing is new. Nothing is new. If you want to say, like, what is this, like, new thing where math textbooks try to have questions that touch directly on questions of social import, you can find this in, like, textbooks from the 1930s. I've seen them. I mean, it's just, we go around, around and around in circles, like, are having the same arguments

again and again. And there's probably, like, no way to stop doing that. But you asked, why do we have this argument about specifically about math? And I would say, I mean, first of all, to some extent, I can promise you, there's a huge curricular issues about how we teach about the Civil War and history class, what books, which authors we read in English class, et cetera, et cetera. But I do think math is special because of this feature, I'm going to sort of bring us back to where we started. It has a certain prestige and it has this certain freedom from authority.

The idea that math is kind of threatening because it enables students to draw their own conclusions that authorities cannot overrule. That's very old. In fact, are you ready? There's like, you go back to the 17th century. I feel like you want to interrupt the train, but the train is rolling and then I'm going to let you talk. Please keep going.

Okay. You can go back to the 17th century and there's an amazing book written in defense of Newton called Geometry, No Friend to Infidelity. So people were like writing whole books being like, I know you think that if people get into this calculus stuff, they're going to be like, okay, forget Jesus. Now it's all calculus. Like that's, I mean, like, you don't have to write a whole book about that unless there's a controversy, unless some lieutenant governor at the time was like, I don't, you know, I don't like this Newton thing.

This seems dangerous. It seems like it's some locus of authority that's not the one I'm comfortable with. So that's old. And it's not just the 20th or 21st century. It's not just America or Florida. It's like a pretty common thing that because mathematics, and even especially geometry has this property of kind of, I can make the knowledge myself, it's dangerous. It's seen as dangerous because it is kind of dangerous. Maybe the purest example of this point of view is the short novel Flatland, written by English schoolmaster Edwin Abbott in 1884.

It is a story told by a square. I call our world Flatland. Not because we call it so. know, but to make its nature clearer to you, my happy readers who are privileged to live in three dimensions. The book takes place in a two-dimensional world whose inhabitants are unable to conceive any direction not spanned by the four compass points they know. The people in the plane are geometric figures whose shape determines their position in society.

The more sides a person has, the higher their station. professional men and gentlemen are squares, to which class I myself belong. The most exalted of all, polygons so multi-sided as to be indistinguishable from circles. Isosceles triangles constitute the masses. The only people below them are women who are mere line segments, and who are presented in a novel as terrifying creatures, nearly mindless, leafily pointy and invisible when viewed head on. It was the first day of a long vacation. Having amused myself till a late hour with my favorite recreation, geometry,

I had retired to rest with an unsolved problem in my mind. Upon awakening, the square is startled by a disembodied voice, which reveals itself as coming from a tiny circle that has somehow gotten inside his house. I am indeed in a certain sense a circle and a more perfect circle than any in Flatland. But to speak more accurately, I am many circles in one. The circle grows and shrinks inexplicably. But of course, that's because the circle is not a circle, but a sphere, whose cross-section inside our narrator's universe grows and shrinks as the sphere moves up and down in the third dimension.

The sphere tries to explain itself to the square. Your country of two dimensions is not spacious enough to represent me, a being of three dimensions, but can only exhibit a slice or section of me, which is what you call a circle. After words fail, the sphere lifts the narrator out of his home plane and tilts him, so that he can see for himself the shape of the world he had previously only inferred. Awestruck at the site. of the mysteries of the three-dimensional earth,

thus unveiled before my unworthy eye, I said to my companion, the matter is now so clear to me, the nature of real space so palpable that me thinks I could make a child understand it. Permit me to descend at once and enlighten the world. Returning to his plane after this revelation, the square tries to spread the news about what he's seen.

Predictably, he's imprisoned. And that's where the number, novel leaves him, locked up, his revelation ignored. I exist in the hope that these memoirs in some manner may find their way to the minds of humanity in this dimension and stir up rebels who shall refuse to be confined to just two dimensions. Flatland was received at the time of publication with a mixture of bemusement and dismissal. The New York Times said, it's a very puzzling book and a very distressing one, and to be enjoyed by about six or at the outside seven persons in the whole of the United States and Canada.

But it's become a favorite of young people with a taste for geometry, adapted for films several times and continually in print. I read it again and again as a kid. But I didn't understand as a child that the book was a satire, lampooning rather than embracing the already old-fashioned view of social hierarchy that holds sway in flatland. Far from seeing women as empty-headed death needles, Abbot was an advocate of equality in education. He served on the Council of the Girls Public Day School Company, which financed secondary education for women. The power of geometry in this telling

is that a two-dimensional being can infer by pure thought the properties of a higher world he can't directly observe. The principles of geometry, far from enforcing an oppressive social order, are a way out of it for those able to accept the reality of the world beyond. The geometry we know can be used to endorse conventional ways, but the geometry we don't yet know is a threat.

In 17th century Italy, the Jesuits stamped out mathematicians' attempts to develop a rigorous theory of infinitesimals and compute the areas and volumes of previously inaccessible figures. If it went beyond Euclid, it was suspect. Geometry, especially new geometry, offers a locus of authority that rivals the established order. In this way, it can be a destabilizing force and a radical measure.

Hey, babe, does a straw have one or two holes? Technically, there's one there and there's one there. So that's two, but always at one hole that continues all the way through. Because you want to know, I have the unpopular opinion. I think it's one hole. I think it's one hole. I think it's one hole. Zero.

One hole's two holes. Look, you're telling me, by cut off. a hole in the bottom of this cup, right? And I dropped this through there. It went through two holes, or is it one hole? Two is one. There's two holes, the bottom of the straw on the top. Top hole, bottom hole. You tackle the problem very early in your book of how many holes there are in a straw. I have my own opinions about this, which I feel best not to share. But I'm curious first, if we could begin with kind of an overview. What is that problem? What is the question? What is the question we're really asking there. I mean, the question is just what it says. How many holes are there in a

straw? And then amazingly, I learned that you can find a huge amount of dispute over that question, including some immensely entertaining YouTube of like just people arguing about how many holes there are and getting like more and more irate and upset and like really into it. And I love this. As a teacher, I love this because to me, I'm like, why are people so upset? Like, why do they get so into it? And I would claim that it's because on some level they understand that there's some actually mathematical deep issue here. It's because there is an issue that people are like not willing to like let go of it and find themselves really, really arguing about it. When you say issue, an issue of ambiguity or something beyond that? Yeah, an issue of realizing, I mean, maybe maybe let me put it this way.

When you see an argument between two people, one of whom says and the two most popular positions are, look, there's this one hole in the straw and it goes all the way through and somebody else who says, well, there's obviously two holes. There's the top hole and the bottom hole. It makes you realize that even a word as simple as whole, you don't really know what it means. We're sort of in the position I'm going to dignify these like random dudes who argue on YouTube. They're like Abraham Lincoln, right? They're like people who just suddenly realized that a word they've been using all their life. They don't actually know what it means. And like Lincoln, they have that drive to actually address it and actually try to like work out what they mean by it. And that is salutary. I

love that about people. It's our best, it's our best selves when we're doing that. How do you as a child pro, you know, a child prodigy and a Harvard educated mathematician, how do you approach the question? Well, I would approach it. I always say, and this is something I even like tell my grad students, I tell like people I teach. I'm like, you know, when you're faced with like a math problem that you don't understand, one thing you can do is try to make the problem easier and see if you can understand it then. But another thing you can do, which is like a little paradoxical, but very useful, is make the problem harder and see how your understanding of it changes. So how do you do that? I like to say, okay, how many holes are there

in a pair of pants? That's more complicated. That's a more complicated shape than a straw. But I think sometimes making the problem harder and making it more general allows you to sort of see more general features that you couldn't see from the original problem. So for a pair of pants, there's actually still some people who will say there's just one hole. There's sort of the whole interior of the pants. A lot of people will say there's three holes. There's the waist and there's the two legs. But this is like me being the dad and being like, oh, look at my clever kids. This is what my daughter said when I asked her this. She's like, well, there's two holes because the waist hole is just like the combination of the two leg holes. That is clever. And that's actually kind of a deep insight.

If you think the straw has one hole, I think you kind of have to think the pants have two holes. But that brings you to this sort of very amazing revolutionary thing that happened in geometry in the beginning of the 20th century and Puancares, who we mentioned earlier, was involved, and Emmy Nurture was involved, this development of the idea of what's called in math homology, this idea that the waste hole is the combination of the two leg holes, all of a sudden it's like you're doing arithmetic, but with holes, with geometric things. You're sort of saying like waste equals leg plus leg. And this insight, like, created an entire new field of mathematics. The idea the geometric things could be added and subtracted as if they were numbers.

And that it made sense to talk about, yeah, doing this kind of something that looks like algebra, but with geometric entities. And so in the end, I'll tell you, I encourage people to watch these videos, by the way. But if you want my actual answer to the question about the straw. I do want your answer. Yeah. So how many, how many holes are in the straw? I would say it has the top hole and the bottom hole.

One is the negative of the other. in this kind of arithmetic sense that Pocherea Nurtur introduced. They're not identical, but they're also related. It's kind of like asking, like, are you and your reflection in the mirror, like two different people are the same?

Like, you know, one is reversed left right so that you could say like, oh, the me in the mirror is not the same because they're left-handed and I'm right-handed. And yet you wouldn't really say there's two different people, right? There's sort of like one person and their reflection. And that's the status of the two of the two holes in the straw. One is the reflection of the other.

What's the lesson? What can we learn from arguing about holes in a straw or that we can kind of apply to education and democracy and knowledge and everything else? I think the mental process you go through when you have this argument, which is kind of different, by the way, from the mental process that you have when you do school geometry, is just learning to interrogate your own reasoning and ask yourself how it works. Like to say it's one thing to sort of have an opinion. God knows the world is full of people with opinions. I'm a one-hole or I'm a two-s-hole or it just seems to me like it has two holes. But what I think is valuable is to be like there must be some reason I think that. And to actually sort of be open-minded enough to like reflect on your own. I was going to say reasoning, but really it's a combination of reasoning and intuition. I mean, this is another one of like Poong-Gare's great insights that geometry, contrary to what we might present it in school, is not an austere and arid exercise of pure reason.

It's reason married to intuition. If you don't have intuition, you can't start. Okay, I'll tell you something that my PhD advisor told me, but it's actually kind of mathematical folk wisdom, but I learned from my advisory. And he said it in the context of doing mathematics research, but I think it applies more generally. He said, like, you know, you try to prove by day and you try to disprove by night. Like, whatever it is that you think is the case,

whatever opinion it is that you have, of course, you sort of spend a certain amount of time trying to bolster it and like sort of arguing for it. But you also got to spend a certain amount of your day, whatever it is that you think is the case, however it is you think the world works. I'd better be able to inhabit the possibility, since I haven't proved it yet, I'd better be able to inhabit the mental realm of disbelieving it and being the person who's trying to prove the opposite so that I can see whether I can do that or not. And if I can, maybe there's a reason. Is it too far a stretch to say we could be more empathetic and better have better arguments if we thought that way more often? Yeah, absolutely. And I think, you know, this is a bigger conversation

and math is only one strand of it. But I think it's definitely true that we've sort of accidentally built a kind of information sphere that doesn't reward that at all. Like it rewards winning. It rewards the so-called dunk, right? If you're like on Twitter all day, as I like to be, because math Twitter is like cool in its way, you can definitely see that somehow you don't get points for being able to empathetically imagine yourself into the position of holding a position opposite to the one you hold. You get points for kind of tenaciously being a bulldog and being like, here is where I am and I will never leave. And I think that math teaches you to sort of not be in this mindset, at least as regards like questions about straws and pants.

Thank you so much for your time. and your insights. Thank you. This is really fun talking. Jordan Ellenberg teaches mathematics at the University of Wisconsin-Madison. He's also the author of Shape, the hidden geometry of information, biology, strategy, democracy, and everything else. For more on the wonders of geometry, head to our website, cbc.ca.ca. slash ideas, where you can find our episode on the power of Euclid, peace, order, and good geometry.

This episode was produced by Matthew Lazen Ryder. Technical production, Danielle Duvau. Web producer Lisa Ayuso. Senior producer Nicola Luxchich. Greg Kelly is the executive producer of ideas, and I'm Nala Ayyid. For more CBCPy. podcasts, go to cBC.ca.ca slash podcasts.