The Hidden Geometry of Everything

Anne Strainchamps (00:00):

It's To The Best Of Our Knowledge. I'm Anne Strainchamps.

Speaker 2 (00:02):

Knowledge, power. Came full rip off of the taxpayer.

Anne Strainchamps (00:13):

In the spring of 2011 in a locked room at a law firm in Madison, Wisconsin, a group of Republican aides and consultants met to remap the state's voting districts.

Bill Whitford (00:24):

There are more Republicans that are affected by pairings than there are Democrats.

Anne Strainchamps (00:29):

Under a strict cone of silence, they devised a set of maps with a single overriding objective. Keep the Republican Party in power for a decade.

Bill Whitford (00:39):

We have once again fulfilled one of the promises that we made, and I'm not going to apologize for that. I'm not going to apologize for that.

Anne Strainchamps (00:49):

But meanwhile, another group of professionals was growing concerned about gerrymandering. Not just in Wisconsin, but across the country. Mathematicians.

Jordan Ellenberg (01:03):

We always thought it was a geometric story. We split the state up into districts here in Wisconsin. There's 99 assembly districts and the question of what those regions should look like is fundamentally a question about shape. Fundamentally a geometric question.

Anne Strainchamps (01:17):

This is Jordan Ellenberg. He's a world-class mathematician and a professor at the University of Wisconsin and he is about to convince me that geometry is crucial to democracy.

Jordan Ellenberg (01:32):

The classic idea of what gerrymandering looks like is it looks like these legislative districts that look ridiculous. They snake all over the place and have octopus tentacles, and you look at them and you're like, "That didn't happen naturally." I talked to somebody who was involved in the process in the '80s and the way it worked then is you sat across from each other at a big wooden table, kind of like the table we're sitting across from each other at right now and you rolled out this giant map at the state of Wisconsin on paper, and everybody would be there with their wax pens and like moving it a little bit here and moving it a little bit there and seeing how to eke out some advantage.

Jordan Ellenberg (02:04):

Well, we don't do it that way anymore. We can now look at 10,000 options in the space of a few seconds, right? So our ability to search the space of all possible ways to gerrymander has ramped up so much that people are increasing their own power and cementing their own hold on the majority in a way that just wasn't possible in the early days of this process.

Anne Strainchamps (02:29):

So what do you do if you're one of the few people in the world trained to understand the complex geometry that could push back against party hacks armed with advanced computing power.

Jordan Ellenberg (02:41):

Temperamentally, most of us are not rabble-rousers. When mathematicians get excited about something, it is because it's both of social importance and it's interesting to think about from a mathematical point of view, which this problem is.

Anne Strainchamps (02:52):

So is gerrymandering not actually all that interesting mathematically?

Jordan Ellenberg (02:55):

No, no, it is interesting. That's why I started thinking about it.

Anne Strainchamps (02:58):

Luckily there's a tried and true academic method for dealing with complex issues, convene a conference.

Jordan Ellenberg (03:06):

Yeah, it was cool. It was a real experiment. This was 2017 and I think people were just starting to grasp what a big political issue this was.

Anne Strainchamps (03:16):

The Geometry of Redistricting Conference ran for three days. Attending were mathematicians, political scientists, lawyers, engineers, geographers, plus the solicitor general of Wisconsin. The guy who was at the time defending the state's gerrymandered assembly districts in court, and also the plaintiff in that same case. The guy who was currently suing to have those maps overturned. So no pressure or anything.

Jordan Ellenberg (03:47):

So with a charged moment, the plaintiff in Gill versus Whitford, the Wisconsin researching case Bill Whitford is the one who stood up.

Bill Whitford (03:53):

Wisconsin is so far over any reasonable line. You don't have to say [inaudible 00:03:57].

Jordan Ellenberg (03:57):

In front of the man, [inaudible 00:03:59] who was arguing the case against him.

Speaker 2 (04:01):

So you would have federal courts deciding, well, what would it be under this map or that map? [inaudible 00:04:08].

Jordan Ellenberg (04:07):

And I was like, "Is something going to happen. Are these two old guys and suits going to fight?" Like, "No, it wasn't like that." They didn't fight. They didn't fight. In the end, there's two human beings with opposing goals who stand facing each other to sort of... How to put it? Something that's kind of abstract and I think there's a human part of the story. Maybe there's a stereotype that to think about things mathematically means neglecting that part of the story and I think just the opposite is true.

Anne Strainchamps (04:42):

So I wanted to begin with that story because it illustrates the fundamental point we want to dig into today. Geometry explains the world.

Jordan Ellenberg (04:52):

Yeah. I did one with a cellist who likes science and then we talked about [inaudible 00:04:57].

Anne Strainchamps (04:57):

Jordan Ellenberg, as you might have guessed, is not only a world famous mathematician. He's my neighbor.

Jordan Ellenberg (05:03):

Although actually, I [inaudible 00:05:04].

Anne Strainchamps (05:03):

We live a few blocks apart. We're always running into each other on campus or at the local coffee shop.

Jordan Ellenberg (05:08):

And then I [inaudible 00:05:10].

Anne Strainchamps (05:10):

He's one of those people who is curious about literally everything. He's also a great writer. His first book, How Not To Be Wrong was a huge hit and the most recent just out in paper is called Shape. That's about the hidden geometry of everything. See usually when I run into Jordan, I ask, "What's new?" Today, I asked a variation. What's new in geometry?

Jordan Ellenberg (05:36):

Oh, everything is new all the time. We are in a geometric era. We're in an era where geometry in all kinds of math are... More math is probably being created every day now than ever before in human history.

Anne Strainchamps (05:52):

Really?

Jordan Ellenberg (05:52):

By an incredible amount. One obvious place this is happening is of course in the development of machine learning and artificial intelligence, which is absolutely at its bottom, a geometric endeavor. You're trying to develop a program that can solve a problem. I want to write the first three paragraphs of my romance novel and I want to have the computer write the rest. Okay, so that's like, "How are you going to search that impossibly huge space of all possible textual strategies that you might use?" And that is a fundamentally geometric problem when the people are working on everywhere every day.

Anne Strainchamps (06:26):

Why is that geometric?

Jordan Ellenberg (06:27):

Because the space of all possible programs is a gigantic high dimensional space. And that's not just a metaphor, it really is. Okay, ready? We're going to get into it.

Anne Strainchamps (06:35):

Okay.

Jordan Ellenberg (06:36):

I feel like you implicitly asked me to, so I'm going to do it.

Anne Strainchamps (06:39):

Yep.

Jordan Ellenberg (06:41):

If you were trying to devise a program to do something, you can imagine what a computer program looks like. It looks like, I don't know some sort of long string of text and maybe in that text, there are some letters and there are some numbers. Maybe the program has 1,000 numbers in it and maybe if you chose 1,000 different numbers, the program would work better. Well, how the hell do you know? How are you going to choose? Go through each one and pick the very best value to make the program be as good at writing a romance novel as it can possibly be? Didn't sound like geometry, right? What I said.

Anne Strainchamps (07:16):

No.

Jordan Ellenberg (07:16):

I'm going to make it sound like geometry.

Anne Strainchamps (07:18):

Okay.

Jordan Ellenberg (07:19):

Because what if there was just two numbers? What if literally your program were incredibly simple and there was just two numbers in it? Well, two numbers to me is a point in the plane. This was the great insight of our hero, René Descartes, who had the idea that you can take a plane and you can describe every point by an X coordinate and a Y coordinate. In the context of geography, we call it a longitude and a latitude. You can describe any point on the earth by a pair of numbers.

Jordan Ellenberg (07:47):

And so there's this fundamental metaphor that's so profound it underlies everything in all of life that a list of numbers is the same thing as a point in space. A pair of numbers, like a longitude and a latitude is like a point on a surface, a two-dimensional space like the surface of the earth. Three numbers, an X, a Y, and a Z coordinate, that's like a point in three-dimensional space. If I tell you a longitude and a latitude, and then one more number for a height above the earth, I've specified some point in three-dimensional space.

Anne Strainchamps (08:16):

So you're suggesting that geometry touches all kinds of math, all numbers. That if you're working with numbers, there is a geometry to it no matter what?

Jordan Ellenberg (08:27):

Absolutely, and that's how we solve problems, right? We geometrize everything. So if this romance novel writing program is specified by a list of 1,000 numbers and you got to get each one of them right, I got to wander around 1,000 dimensional space until I find just the right spot, just the right spot that writes maximally arousing romance novels. And that problem is just finding your way to the best spot in some 1,000 dimensional room. When I put it that way, it's a geometry problem but all that I just told you, we knew 300 years ago and you were asking what's happening now.

Anne Strainchamps (09:03):

Oh my God. Okay, what's happening now?

Jordan Ellenberg (09:05):

What's happening now that's so interesting is basically all of these geometric algorithms for so-called artificial intelligence systems or machine learning systems that can carry out these tasks incredibly well, they can play Go better than any human. All of these methods we have, they work much better than they should. And nobody understands why. I think that's the fundamental question that people are wrestling with every day and I think it's fundamentally a geometric question because it's about searching this high dimensional space.

Anne Strainchamps (09:44):

What's it like to be a geometer today? And let's say-

Jordan Ellenberg (09:48):

Awesome. Oh, I'm sorry. Did you have a further question?

Anne Strainchamps (09:54):

And to try to imagine a shape or a thing that exists in, I don't know, eight dimensions.

Jordan Ellenberg (10:03):

Yes. So there's a wonderful quote of Geoff Hinton who is at the University of Toronto and it's one of the founders of the theory of neural nets. He's giving a talk and somebody asks him, "Professor Hinton, how do you visualize a 14 dimensional space?" And he said, "Well, you visualize a three-dimensional space and then very loudly say 14." And this is absolutely true. It's kind of a miracle, but we have this built-in geometric intuition. It's in our bodies, this intuition about what things look like and what it means when things are close to each other and far away and how things move, and what paths things take. All of that intuition is about the two-dimensional and three-dimensional geometry, and yet a lot of that stuff works really, really well in any number of dimensions

Anne Strainchamps (10:55):

In 14 dimensions, are you even talking about shapes then?

Jordan Ellenberg (11:00):

Oh, yeah. I would say.

Anne Strainchamps (11:02):

Because isn't the fourth dimension time?

Jordan Ellenberg (11:07):

So this is the kind of thing where if somebody asks this in my class, I'm like, "No, you're totally wrong, but in such a useful and good way."

Anne Strainchamps (11:19):

You must be a very good teacher, thank you. I just said something stupid. Okay, tell me why it's not-

Jordan Ellenberg (11:23):

No, no but it's not...

Anne Strainchamps (11:24):

Tell me why it's not so bad.

Jordan Ellenberg (11:25):

The point is it's not stupid. Do you know the phrase ontology recapitulates phylogeny. So teaching is like that, right? Over the course of the class, we're recapitulating the entire history of mathematics and our developing understanding, because it was the old fashion to be like, "What is the fourth dimension?" Now we just say, "Let's think in the abstract about what four-dimensional spaces can be."

Anne Strainchamps (11:51):

I'm still struggling to understand dimensions. I'm sorry, it fascinates me because part of me thinks, "Well, then if we happen to live in the universe with four dimensions, but we know all these other dimensions exist, then are you suggesting there are other universes possibly just moving through us and around us that are in different dimensions and so we just can't see them?" Except I think that's a complete misunderstanding of what a dimension actually is.

Jordan Ellenberg (12:20):

Well, that is a thing that people think may be the case, but in some sense, in math we care about what could be much more than we care about what actually is. Some people see that as a moral failing, but I see it as a great strength, probably our special and unique strength. If we can conceive it then for us, it exists because we can say something about it. Okay, I'll tell you a story. It's a story of Flatland. I don't know if you ever read this book when you were a kid.

Anne Strainchamps (12:46):

I haven't read it, but I know about it at it.

Jordan Ellenberg (12:57):

It's set in a two-dimensional world called Flatland and the narrator of this book is a square. His entire world is composed of polygons. Different social classes are different kinds of polygons, and one day he's in his house and he sees a little circle. It's like, "Ha, how'd the circle get into my house?" Circles are very noble in his world. And then the circle starts getting bigger. It starts growing. Now this is very science, fictional and weird to him, right? Just as if suddenly there was a little tiny miniature human in your house and then it started expanding and the square is like, "Who are you mystical size-changing circle?" And the circle says, "You misunderstand, I'm not a circle. I am a sphere. I'm a sphere that's slowly entering your world, crossing into your plane and my cross-section is getting bigger and bigger, but what you are seeing of me is only this two-dimensional section."

Jordan Ellenberg (13:52):

And of course, the square is like, "That is nonsensical. There's no such thing as the third dimension. Maybe you found some way, you're a circle that's found some way to make yourself bigger or smaller but what you're saying is ridiculous." And they have this long philosophical argument as one does in a 19th-century novel. And then finally the sphere gives up and just grabs the square by the corner and just jerks him up out of his world, turning him on his side so he can see his entire world globally that he was previously only able to see from the inside. So that he can see for himself from the third dimension, just how limited his perspective has been.

Jordan Ellenberg (14:31):

So here's where this story gets really subversive. The sphere explains to the square, "Oh, there's a thing like you in the third dimension called a cube." And they walk through the reasoning of how he could know that a cube has eight points. You have four vertices, you have four corners. A cube would have eight corners. And the square is very excited about this like our best math students and says, "Oh, so I get it. So the four-dimensional version of me would have to have 16 corners." And the sphere is like, "What the hell are you talking about? There's no fourth dimension. That's ridiculous." Three is how many dimensions there are.

Jordan Ellenberg (15:15):

So this is an incredibly powerful parable about learning mathematics because on the one hand, it shows you that can either the sphere or the square see or perceive the fourth dimension? No, they cannot but by the mechanism of our reason, the square is able to figure out things about it which are true and correct without actually experiencing it and that's what abstract math can do for us. We can reason past what we can see and feel. And of course, the other part of it, it really captures the way that math can be a locus of power, right? The square is figuring something out and this sphere who was moments ago in this position of authority is suddenly like, "Wait, wait, I don't like this." But there's nothing this sphere can do about it, right? The square figured it out and so there's a amazing fact about the history of geometry as the geometry is often seen as dangerous because it represents this other point of authority. Like knowledge that you can make yourself without taking it from some other source of received wisdom.

Anne Strainchamps (16:12):

I love that and I also love the connection to... I'm thinking about your chapter about Thomas Jefferson and Abraham Lincoln and the connections between geometry, in this case, Euclidean geometry and the foundation of the democracy.

Jordan Ellenberg (16:28):

Jefferson and Lincoln both were lovers of mathematics, but in very different ways. Jefferson is the patrician. He sees it as like hard and parcel, you learn Euclid because you also learn Thucydides.

Anne Strainchamps (16:40):

Euclid, just to be clear, that's the kind of geometry that we learned in school, right? Where you're forced to write theorems or prove that a triangle looks like a triangle.

Jordan Ellenberg (16:48):

I wish I could see the expression on Anne's face right now, which maybe is the expression on your face too as you recall your ninth grade geometry education. Yeah, so Euclid is this classical tradition of we are going to start from a set of axioms about points and lines and circles. And from that, build up everything step by step. People's feelings about it over the years have been very ambivalent. Many people find it dull. I'm going to be honest, I was one of them. That was not really my jam. Other people have found themselves incredibly inspired by it and Lincoln was one.

Anne Strainchamps (17:22):

It's the only kind of math I ever liked.

Jordan Ellenberg (17:25):

Yeah. Well, and that's where I started the book.

Anne Strainchamps (17:27):

I loved it because it was logical. It felt like you just moved from one thing to another very carefully and you... I loved it.

Jordan Ellenberg (17:34):

And you know, one of the Genesis, Genesisees? Genesisees? One of the origins of this book for me was having conversations just like this. I wrote another book about math about seven years ago so it meant I was going around giving a lot of talks to people about math, and then afterwards you give your talk and then afterwards people come and talk to you. It's like therapy. They come and talk to you about the math experience that they have been holding onto for like 30 years.

Anne Strainchamps (17:57):

And did you feel like you were a therapist to a generation of people who've been traumatized by math?

Jordan Ellenberg (18:01):

I'm not trained, but I did my best. But there's two things that you hear a lot. People who are like, "I really liked all of the stuff with algebra and trigonometry and calculus where there's a question and what's the answer? And it's a number and you get the answer, but geometry, what was going on there? Then suddenly there was all this stuff about two column proofs and triangles and saying some fact that I can completely see is obvious. Why are you asking me to prove it? Blah, blah, blah.

Anne Strainchamps (18:27):

No. I felt like finally we were talking about something real. I don't know why.

Jordan Ellenberg (18:30):

And then there's people like you, right? So this is why I say in the book that geometry is the cilantro of math. There's people who love it and there's people who hate it, but everybody recognizes that it's different from everything else. Now it's one of the things that maybe you want to make that the theme of the book in the first place. Like what's going on? Why does this one thing feel so different even from the rest of mathematics? What were we talking about? Lincoln? We were talking Abraham Lincoln?

Anne Strainchamps (18:53):

Lincoln and Jefferson.

Jordan Ellenberg (18:54):

Ah.

Speaker 4 (18:55):

Hello.

Anne Strainchamps (18:57):

Hi.

Speaker 4 (18:57):

Are you guys recording an episode?

Anne Strainchamps (18:59):

Yeah. We're recording an interview.

Speaker 4 (19:01):

I'm sorry. [inaudible 00:19:01].

Anne Strainchamps (19:01):

Oh, shoot. Did-

Speaker 4 (19:02):

[inaudible 00:19:02] in our yard.

Anne Strainchamps (19:03):

Oh, thank you very much.

Speaker 4 (19:05):

Push that open.

Speaker 5 (19:06):

The told me [inaudible 00:19:08]. Thank you.

Anne Strainchamps (19:11):

Okay. There's nothing like getting so absorbed in a conversation about math that you fail to notice the dog sneaking away.

Jordan Ellenberg (19:19):

That's how I know I'm good radio. You didn't even notice your own dog escaping.

Anne Strainchamps (19:24):

Hold on. We'll be right back onto The Best of our Knowledge. From Wisconsin public radio and PRX. You like going on tour?

Jordan Ellenberg (19:48):

Love it.

Anne Strainchamps (19:48):

Oh, really? Okay.

Jordan Ellenberg (19:48):

Love it.

Anne Strainchamps (19:50):

Hey, we're back with Jordan Ellenberg, author of Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else

Jordan Ellenberg (20:00):

I love going to some random store meeting like 12 people and like...

Anne Strainchamps (20:05):

Even though I know they all treat you like their math therapist. So we were talking about people who love and maybe don't like geometry. Well, it turns out some of America's founders not only loved geometry, they were so obsessed with it they even worked a little Euclid into the constitution. I think you say in the book, that if you read the constitution carefully, you can see how Euclidean geometry, at least the habit of thinking that's required when you do Euclidean geometry is in there.

Jordan Ellenberg (20:38):

And it's a famous argument. Of course, the very beginning, we hold these truths to be self-evident. That's an incredible Euclidean move, right? To say, "Let's start with some set of axioms and then build everything else up from that." Now it's actually very controversial about whether that line is there because of Euclid or whether it has anything to do with Euclid or even exactly who wrote it. We don't totally know but what's certainly-

Anne Strainchamps (21:05):

There was an earlier version. Somebody originally had said something like we hold these truths to be sacred.

Jordan Ellenberg (21:12):

Yeah. So I think that's right and I think-

Anne Strainchamps (21:13):

Yeah, and they switched self-evident, which does have a more democratic, "No, it's not sacred from on high. It's self-evident, as in you proved it or figured it out."

Jordan Ellenberg (21:23):

Oh yeah. I hadn't made that connection, but you're right. It's exactly the same distinction that I'm talking about at the end of the book. Well, missed opportunities because if you had this conversation before, because you're right. That that distinction is exactly the distinction that I'm making about where does the authority in the end come from? Does it come from yourself and your own reason or does it come from tradition, or does it come from on high?

Jordan Ellenberg (21:42):

But Lincoln is different, I want to say this. So Lincoln is self-educated, he learns Euclid not as a child in school because he didn't go to school but much later when he is already a lawyer and he is going around in court being asked to prove things and he is like, "What do they mean? Prove?" Well, he's said, "Demonstrate." He's like, "What does this word demonstrate mean that I'm allegedly doing in court?" He sets himself to read Euclid, to read it all and he is like, "If I'm going to be a real lawyer, I got to understand what it means to actually prove something."

Anne Strainchamps (22:09):

Does he actually read it with a pencil or pen in hand? Does he do the calculations?

Jordan Ellenberg (22:14):

Oh yeah. So he is law partner sort of just talked... They were just going around at the law circuit and these small towns in Illinois and he goes to sleep and Lincoln is sitting in bed-

Anne Strainchamps (22:22):

Scratching at...

Jordan Ellenberg (22:23):

... and working out his problems in Euclid. He tells this amazing story of Lincoln trying to square the circle, which is this famously impossible classical problem of Euclidean and geometry. In Lincoln's time, it hadn't yet been proved to be impossible. It would be proved to be impossible in 1882 but people kind of knew. People kind of knew like, "Look, it's been 2000 years. Probably it's impossible because nobody's done it yet." But Lincoln tried as many other people have and he talks about... Lincoln basically didn't do anything for two days except sit at his desk in the law office drawing more and more circles and trying to crack this problem and then he gave up and his law partner is like, "We could tell he was upset about it. So we just never brought it up again."

Anne Strainchamps (22:57):

Yeah, they're all like, "Dude, let's get a drink. Come on."

Jordan Ellenberg (23:01):

But what they say about him and I found this quite moving was one of his contemporaries wrote about Lincoln. Now, this was not directly in the context of geometry, to me, it's clearly related that they said, "Look, what was special about Lincoln? Was it that he was so brilliant as a lawyer? Was it that his arguments were so clever and insight..." No. He said, "What's special about Lincoln is that he really was allergic to false argument. It wasn't in him to say something that he didn't feel like he had justified that he could say." Now that is the Euclidean habit right there to a T. And even now what do we call him? Honest Abe. The moral content of geometry is it is a enforced honesty.

Anne Strainchamps (23:44):

Oh, that's so interesting. Geometry inclines toward honesty. Huh.

Jordan Ellenberg (23:49):

By the way, I think that has always been understood to be part of the goal. There's a wonderful thing I found in the 1950s, there was a giant survey asking teachers of high school geometry like, "Hey, why are you doing this? What's the point of what you're doing?" And they offered a long list and one of the items was so that the students will know facts about geometry. Know how triangles and circles and lines, et cetera behave. And that did pretty well, it was number two but number one was to develop the habit of rigorous and logical thinking. That's why I think it has always been the consensus among educators that we're not teaching that subject because we need a population of people who know that the angles of a triangle, some to 180 degrees. It's because we need a population of people who can understand how you could convince yourself of that fact and other facts like it.

Anne Strainchamps (24:35):

What's it like to write about math? Because as a non-math person, the idea in my head is it must be like translating. Like you know one language, math, and then you're translating it into words.

Jordan Ellenberg (24:48):

Well, you know what writing about math is like? It's like writing about other things in than math. That's what it's mainly like. I got to be honest with you that I started out... I did start out seeing those two pursuits as separate tracks from each other, and I was really interested in writing. I took a lot of creative writing courses in college and then after I was done with college, I sort of knew I wanted to pursue math and get a PhD but for that first year I did something totally different. I went and did a master's degree in creative writing at John's Hopkins, this wonderful...

Anne Strainchamps (25:18):

And you published a novel.

Jordan Ellenberg (25:18):

I did... Well, I wrote a novel that sat in my filing cabinet for 10 years until someone wanted to publish it. So that comes a little later but at that stage of my life, these were two different things. I'm going to stop doing the math thing and spend a year doing the writing thing. And it's been incredibly important to me to have this moment at which I was given permission to call myself a writer. It's not a hobby, it's one of the things that I am. Nevertheless, I miss math every day. And knowing that-

Anne Strainchamps (25:45):

What did you miss about it?

Jordan Ellenberg (25:48):

What can I say except that it's fun? In the doing of mathematics and in the making of mathematics there's always some kind of spirit of play. Even in the way we talk about it, we say we're messing around, we're playing around. Maybe I'm trying to make some argument where there is no argument and it's a purely temperamental distinction. It may just be that I have more fun doing that.

Anne Strainchamps (26:06):

Well, that could be, but I am also thinking the emphasis in writing, it's very much about your individual voice. Whereas math, it seems, I'm sure there's that in math also, but it also feels like you're taking part in some larger profession that you share with a lot of other people and you're all making contributions.

Jordan Ellenberg (26:26):

Actually, that is a really great point and I think that does speak to a true difference. That math is a fundamentally communal activity that you're creating something that's your own, but it's part of a gigantic project that's 1,000s of years long and that, knock on wood, is going to continue for thousands of years after us. If there was some theorem I try to prove, and I don't prove, if it's true, somebody else is going to prove it. I'd like to do it. I would feel proud and I'd feel like I contributed something, but if I don't somebody else will. And maybe when you're younger, you have a little bit more of a hunger for recognition, and maybe if you try to do something and you don't and then somebody else does it, you're like, "Oh, man. I wish I'd gotten that." Now I love it. Now nothing makes me happier when somebody proves something that I tried to do 10 or 20 years ago and I couldn't do.

Anne Strainchamps (27:22):

Do you remember when you first recognized numbers?

Jordan Ellenberg (27:27):

That's a great question. I was very interested in math and numbers from a young age. A lot of people discover they're excited about math later in life and some early. So I was one of the early ones. So I guess in a way that's my roundabout way of saying no, I don't really remember the moment. I can tell you, it's a story I tell a lot because I think realizing what mathematics is a little bit different from realizing what numbers are. And I think when I realized what mathematics was, and this is going to be a geometry story in its way, was just this moment when I was like zoning out. I was probably six, I don't know, looking at my parents' stereo system.

Jordan Ellenberg (28:11):

And to set the scene, this is the '70s, right? So I'm dating myself. So in the '70s, you got to have dark wood panels on everything and so there's a dark wood panel over the stereo and there's a bunch of holes in it so the sound can come out, and the holes are arranged in a rectangular array, like a six by eight rectangle of holes. And there I am just lying on the thick brown shag carpet, because remember it's the '70s, and looking at these holes, six rows of eight holes each, right? Which are also eight columns of six holes each. Flip your attention back and forth and organize them in your mind, seeing the rows, then seeing the columns, then seeing the rows, and then seeing the columns.

Jordan Ellenberg (28:53):

And then in that moment you're like, "Whoa. So six eights is the same thing as eight sixes, right? Six rows of eight is the same thing as eight columns of six." And this was an amazing fact. This is an amazing moment because if you start to learn your multiplication tables, whenever you learn them, you might say, "Well, I know that six times eight is eight times six because that's in the table." But there's a difference between knowing it because somebody told you and knowing it because you know it. Knowing it in this way where you're like, "Oh, there's no other way it could possibly be. It's not the case because somebody told me it's the case because it just is. It has to be." That's mathematics.

Jordan Ellenberg (29:30):

That's what separates it from everything else that we learn in school. That most things that happen in school, in some sense, you don't really know it's so except on authority of the teacher or on authority of a book that you read, right? Math is different and especially geometry is different. Ideally, you're giving the student the capacity to make knowledge on their own from scratch. So you want to talk about dangerous? There's a reason people saw geometry is dangerous. Exactly because it represented some other locus of authority.

Anne Strainchamps (30:05):

Last ending question. You came up with a playlist recently for shape, which I had fun listening to and the last song you chose, it's one of my old favorites, Talking Heads. Why'd you choose that one for the end?

Jordan Ellenberg (30:19):

Yeah. The song Once in a Lifetime, which it feels like math to me, partly because the incredible spirited-ness and forcefulness and joyfulness in it. But also because literally when I was in high school in training for the math Olympiad, the cassette of Stop Making Sense is what I'd play every night. So it feels like math to me because it was when I was doing math that I was listening to it all the time and drinking Mandarin orange slice, my math drink. That was everything. The '80s, young people, that's what the '80s was like. Mandarin orange slice and talking heads. That's why I closed the playlist.

Anne Strainchamps (30:54):

Oh, that's so interesting because the playlist includes a lot of songs that are very definitely about math, and this one is less obviously and so I was trying to think-

Speaker 6 (31:02):

And you may ask yourself.

Anne Strainchamps (31:05):

... Why this song? And I was thinking about the line that keeps getting repeated.

Speaker 6 (31:12):

Water at the bottom of the ocean.

Anne Strainchamps (31:14):

Because he's conjuring up our ordinary boring lives with our houses and our suburban driveways and stuff.

Speaker 6 (31:21):

Large automobile.

Anne Strainchamps (31:22):

And then you let it all go down into the water flowing underground and I thought maybe that's how you feel about math and geometry, that it is the hidden substrate of everything.

Jordan Ellenberg (31:35):

Yeah, and this is a great place to come back to you because it comes back to the beginning.

Speaker 6 (31:38):

And you may find your soul. Remove the...

Jordan Ellenberg (31:44):

Under the rocks and stones, water flowing underground, right? This is this famous motto of the 68 revolutionaries in Paris. Under the stones, the beach, which Greil Marcus writes about as the founding document of punk rock. That's where he says, "Even though this is this built-up city around us, these structures that appear impregnable, six inches under it-

Speaker 6 (32:03):

Under the water.

Jordan Ellenberg (32:03):

There's the living earth.

Speaker 6 (32:05):

Carry the [inaudible 00:32:07] with you.

Jordan Ellenberg (32:07):

Right? It's just a shell under which is the real living thing and maybe we'll finish with [inaudible 00:32:13] because one thing I learned is a that [inaudible 00:32:15], besides creating modern topology and geometry also was incredibly quotable. Maybe one of the most quotable mathematicians who ever lived and he talks about geometry, this kind of formal rigid structure that many people find oppressive when they learn it in school. Strict logic. He says, "You got to think of it like the skeleton of a sponge."

Speaker 6 (32:40):

[inaudible 00:32:40].

Jordan Ellenberg (32:42):

You sort of come across it and you see this very beautiful structure and you're like...

Speaker 6 (32:46):

This is not my beautiful house.

Jordan Ellenberg (32:48):

Wow. What an interesting structure. I can really see all kinds of things about it and it's perfectly formed and perfectly rigid. And he says, "But if you don't know that it was created by a living being in order to support it, you're missing something." And the living being, the sponge in the skeleton, that's the intuition.

Speaker 6 (33:07):

How did I get here?

Jordan Ellenberg (33:10):

And this is [inaudible 00:33:10] vision of geometry. That there's the structure, there's the rigidity, but it's there to support something that's under it that's alive.

Speaker 6 (33:19):

Same as it ever was.

Jordan Ellenberg (33:20):

I hope one thing I can do in the book is always to have the sponge be there. Put you in contact with the water flowing underground, the living thing that it's all there to support.

Speaker 6 (33:30):

Same as it ever was.

Anne Strainchamps (33:32):

Wow. Thank you so much.

Jordan Ellenberg (33:34):

Thank you. This was fantastic.

Speaker 6 (33:37):

Here the twister come. Here comes the twist.

Anne Strainchamps (33:43):

Jordan Ellenberg is Professor of Mathematics at the University of Wisconsin, Madison. And now that you've learned to love geometry, check out his new book. And it's called Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else.

Anne Strainchamps (34:04):

Coming up, French neuroscientist Stanislas Dehaene says it's not language that makes us human. It's geometry. The story of how your brain makes shape. Next onto The Best of Our Knowledge from Wisconsin Public Radio and PRX.

Jordan Ellenberg (34:21):

How'd we do? I feel like I got into it at the end. I feel like it was like...

Anne Strainchamps (34:27):

You were fantastic. You're always just...

Jordan Ellenberg (34:31):

Well, you know [inaudible 00:34:31].

Speaker 6 (34:32):

Same as it ever was.

Anne Strainchamps (34:39):

If you give a four year old a crayon and ask her to draw a house, she'll probably come up with a wobbly square or a rectangle, maybe a triangle for a roof. And this might not seem all that remarkable, but take a minute and think about what it means. From an incredibly early age, humans see the world in terms of its underlying geometry. We recognize shapes before we even know their names. And recent neuroscience suggests that intuitive understanding of geometry might just be what makes us human. Steve Paulson is here to explain.

Steve Paulson (35:19):

There's a long history of theories about what sets humans apart from every other animal. Once, it was thought to be agriculture. Then tool making. Then the ability to deceive the people around you, but scientists kept finding other species that do all these things and today, language is often seen as the one unique human talent. Or at least language with grammar and syntax, but a prominent French neuroscientist has another theory based on his research using FMRI neuroimaging and sophisticated computational models. Stanislas Dehaene believes there's an even more basic cognitive skill that might have evolved before language, geometry.

Steve Paulson (36:02):

Now, if you've ever watched a dog race after a ball that's been thrown in the air and then in one single acrobatic motion leap up to catch it, you know animals have a basic sense of geometry, but that's not the kind Dehaene is talking about.

Stanislas Dehaene (36:18):

Well, actually we have to distinguish two different forms of geometry. All animals, because they have to move around, they have to have a sense of their environment and there is a lot of beautiful research showing how they map space around them and you can call that a first sense of geometry. But my claim is that we humans have something else. We have a sense of discrete symbolic perception of the world or around us and we've done experiments, not just speculation, we've done experiments suggesting that our perception of something like a square, for instance, is quite peculiar compared to all other animals.

Steve Paulson (36:55):

We perceive right angles and squares and parallel lines, which are geometric symbols that don't actually exist in the natural world. At least not as perfectly formed as we can see them in our minds. Yeah, there are lines that look almost parallel on a slab of striated rock. Or if you look into the honeycomb of a beehive, all those tiny cells look kind of hexagonal but when you look closely, they're not actually perfect angles or exact parallel lines. Now you might be thinking, "What's the big deal?" Well, for some reason, our ancient ancestors were obsessed with geometric shapes and patterns. They carved them into rocks and drew them on cave walls.

Stanislas Dehaene (37:34):

For instance, if you go to the Lascaux Cave in the south of France, you see not just these beautiful depictions of animals, but you see a rectangle. And whenever you see rectangle, well, you know it's not an animal that has been sketching the walls. It's an intentional geometrical drawing. We think that this is unique to humans.

Steve Paulson (37:56):

Well, and if you go way back, even much further than the cave paintings in Lascaux, go back in human history, 10s of 1,000s of years, there are symbols engraved in rocks of spirals and zigzags and parallel lines. Symbols that I think we would call geometric.

Stanislas Dehaene (38:12):

Absolutely. In fact, this is not so well known, but the earliest forms are not paintings, they are not real depictions of animals or life forms. They are really symbolic geometrical forms.

Steve Paulson (38:31):

There's a freshwater shell that was found on the island of Java that has a zigzag engraved in it. Two sets of alternating parallel lines. Scientists say it's half a million years old. So that even predates homo sapiens. It goes back to homo erectus. And just to draw this simple pattern, Dehaene says, you need a basic sense of angles and parallel lines.

Stanislas Dehaene (38:54):

And if you go even earlier than that, everybody knows about the tools that are made of stone, which are called bifaces. They're very common in pre-history and date perhaps as early as 1.5, 1.8 million years ago and they are very geometrical. They are symmetrical in two planes. That's why they're called bifaces. So we see that in order to design these tools, the ancient humans must have had some idea of concepts such as perpendicularity and parallelism.

Steve Paulson (39:28):

So why do you think these, I don't even know if we would call them our ancient ancestors. We're talking species that predate homo sapiens. Why do you think they made those engravings? Why would they be compelled to carve these symbols into rock?

Stanislas Dehaene (39:43):

Well, you see, we don't really know the meanings that they could have attributed to these signs, but I'm not really interested in this question of the meaning. I'm interested in the existence of the signs themselves and the shapes that they take because I came to this trying to understand the human singularity, what is special to humans? And I think that this existence of the sciences is enough to say there is a sort of language of geometry. It looks like an organized mathematical language. So my own research has been trying to prove that in addition to the language that we use to communicate meanings, there is also an internal language in the brain, a language that allows us to formulate shapes.

Steve Paulson (40:21):

In other words, to see shapes as a kind of language that's actually part of how we think.

Stanislas Dehaene (40:26):

We are not just talking about zigzags, of course, but as you say it's spirals, squares. Freezes that we see for instance, in ancient Greece and of course much before. All of the decorative arts are based on a sort of language of shape.

Steve Paulson (40:39):

So this may be a stupid question, but what exactly is geometry? What makes shape or a collection of lines geometric?

Stanislas Dehaene (40:49):

That's a great question. It was well defined by Euclid, and of course, we talk about Euclidean geometry as perhaps the first formulation of what is geometry. So it's a system made of a very small set of elements, which are points and lines, angles, and planes basically in their combinations. It's amazing that humans accept to build such a complicated, wonderful system out of such puny foundations. But of course, it's the same in all areas of mathematics, we start with a very small set of concepts and axioms and from that, we build up a whole system. Mathematics is a construction based on a very small sets of elementary foundation.

Steve Paulson (41:34):

Do you think geometric thinking, this ability to see shape and think in that way, do you think that came before language in the evolution of the human brain?

Stanislas Dehaene (41:45):

This is now speculation, but yes, I think there's a strong possibility that the first thing that evolved during hominization is the capacity to think better about the external world. And geometry is a way to perceive abstract structures in the external world. It's a way to represent information in a more compact form. In neuroscience, we speak of the brain as a system for compressing information and symbolic representation is the ultimate compression, it's down to just a few symbols that express a zigzag, for instance.

Stanislas Dehaene (42:19):

So yes, this could have come before the ability to speak about these objects and to share them with each other. And this is, I think, the key function of language is to save time. We don't have to re-invent or rediscover what others have discovered. By sharing information inside a culture, we narrow down the search. It's been called the ratchet effect of culture. So to summarize, perhaps first we have the ability to think very abstract ideas and this could date back perhaps as long as 2 million years ago. And then perhaps more recently, we have an explosion of culture with homo sapiens because we're able to share these ideas.

Steve Paulson (43:02):

There's one other thing that's kind of surprising. You might think doing geometry, seeing these patterns of shapes and lines would be an inherently visual process. But Dehaene says there are plenty of blind people who are gifted mathematicians.

Stanislas Dehaene (43:20):

It's actually extraordinary that you can be blind and be an excellent mathematician and be an excellent geometer. So surely vision helped, but it's a very common observation that many blind people love mathematics and are very good in mathematics. This is essentially not just a visual ability. It's essentially an internal construction and the construction can start with data from any of the sensors. The brain generates ideas and projects them onto the outside world. So we think of squares, we think of spirals, even though we never see a real one in external world, it's never perfect. In our mind, it's perfect and that's where the ideas are generated. Before we ever see a triangle, we can think about one.

Steve Paulson (44:13):

So there is an age-old debate about whether mathematics is discovered or invented. Is it out there in the world or is it created by the human mind? What do you think?

Stanislas Dehaene (44:25):

I think I've made my position clear, right? It's a creation of the human mind. It starts in the brain and that's why you can be blind and still develop mathematics. However, we select our mathematics in part because it's useful to the external world, and of course, the foundations of our mathematics, the sense of space, the sense of number were selected through evolution because they were useful in many more species.

Steve Paulson (44:51):

I'm actually surprised that you say that it comes from the mind. If you think about the laws of physics, the structure of the universe, isn't that fundamentally mathematical in nature regardless of whether humans are around or not?

Stanislas Dehaene (45:04):

Well, it's a big debate. I think that we pick out, at the scale we believe in, we pick out structures and we simplify them. This is the whole purpose of the brain, is to discretize, simplify, compress information. So mathematics are useful obstructions, but they are arising in our brains as we try to describe the extraordinary complexity of the universe.

Steve Paulson (45:28):

Stan, you have been studying these questions for many years and doing incredible research. Why are you so interested, particularly in the neuroscience of mathematics?

Stanislas Dehaene (45:39):

There are many reasons. One of them is that I loved mathematics as a student. I was a very good student of mathematics. I actually boarded the [foreign language 00:45:49], which is very famous in France for its mathematicians and I also have a very strong interest in education. So as I was learning mathematics, I became fascinated in how we change. I remember very well for instance, when I learned about complex numbers and I don't know if you know about complex numbers, but there is this very funny idea that you can take the square root of a negative number. And the first time I met this, I was flabbergasted. I was told for my entire young life that a square is a positive number. Two times two, it has to be positive. Three times three, four times four, it's always positive. Even if you do minus two times minus two, you get a positive number, right?

Stanislas Dehaene (46:28):

And what's very funny in mathematics is that after a few weeks, you totally accept these concepts. You find them not only natural, but very useful and you know how to compute with them and then you develop very strong intuitions of them. So I remain completely fascinated by how we learn, and that's the title of my book. But it's also very concrete motivation, which is that there are some children that find it very difficult. They are just discouraged. They never cross this threshold where you find mathematics fascinated and like a mind teaser, which is exciting, like a puzzle. So I work now for the French government also to try to help design pedagogical tools or better pedagogies that can convey to all children, because I think all children of this potential, the power and the beauty of mathematics.

Steve Paulson (47:18):

So why do so many people, and I'll include myself in here, when you get older you become a little afraid of mathematics in any kind of a complicated way. We seem to do something wrong in school in terms of how we usually teach mathematics.

Stanislas Dehaene (47:35):

Well, you say that, it has a lot to do with anxiety and also convincing yourself that you are not good at it. You go into math class and you start to sweat because you might be cold to the front and you will not be able to answer, and you have a feeling of being completely lost. This is a situation where the brain's learning algorithm is blocked. So I think this is what we are doing wrong and here in France, we're trying to introduce pedagogies that are based on games or challenges, puzzles, all of mathematics was constructed because there were interesting problems to solve and instructions to develop.

Steve Paulson (48:12):

So suppose the French government were to appoint you czar of all education in the country, and it's up to you to design schools that tap into this curiosity, this innate drive that we have to learn, what would you recommend? How should schools work differently?

Stanislas Dehaene (48:35):

I'm not a czar or I'm not able to predict the future, but what we are trying to do here, I started the Scientific Council for Education with the spirit of experimenting, first of all. So for the moment, we are running a giant experiment with 67,000 children where we see whether the introduction of card games and board games in first grade make a difference in terms of learning mathematics. And at the end of the year, we're going to know whether the kids who received the games were doing better. It seems that for instance, playing with board games where you move by a certain number as a function of the number that you draw on a dice, for instance, this is very useful to understand and develop the sense of number. The mapping of number to space is really at the heart of geometry and mathematics. So this is what I would propose. The introduction of games, but the very careful monitoring and measurement of whether this is having a positive effects on children or not, because we need to learn much more before we can apply this science to the schools.

Anne Strainchamps (49:49):

Stanislas Dehaene is a cognitive neuroscientist at the [foreign language 00:49:52]. He's the author of The Number Sense: How the Mind Creates Mathematics and most recently, How We Learn. Steve Paulson brought us that story.

Anne Strainchamps (50:08):

Thanks for joining us today. Whether you love math or generally try to avoid it, I hope we helped you find some wonder in the underlying geometry of the world and of your own mind.

Anne Strainchamps (50:22):

To The Best of Our Knowledge is created by a small team of audio producers who use math every week to make a 59 minute show. Thanks to Angelo Bautista, Shannon Henry Kleiber, Charles Monroe-Kane, Mark Riechers, Joe Hardtke, Steve Paulson, and me, Anne Strainchamps. And thanks also to Sarah Hopefl for extra sound design this week. Until next time.

Anne Strainchamps (50:46):

What's it like to be a geometer today? Let's say-

Jordan Ellenberg (50:51):

Awesome. Sorry, did you have other questions?

Speaker 7 (51:00):

PRX.