Numbers
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That was from our genius, deranged, and slightly commitment-phobic producer, Matt Kilty. I'm Lou Miller. This is Radiolab. And to remind you of the kind of fun we sometimes get up to over here, I thought I would play one of my all-time favorite episodes. It's called Numbers, and it is a rollercoaster ride through all different kinds of numbers. These things which, you know, can sometimes seem kind of cold. But in Radiolab's loving care...

are shown to contain real warmth. So without further ado, our episode numbers. Enjoy the ride. And again, if it makes you chuckle, makes you feel warm and you want to support the work we do, if you feel like tossing a few quarters into our proverbial bin, you can do that over at Radiolab.org slash donate. Thanks for thinking of us. Here we go. Wait, you're listening? Okay. All right. Okay. Okay.

You're listening to Radiolab from WNYC. Rewind. Chad? Yes? Listen to this, just for a second. They're building a gallows outside my cell. I've got 25 minutes to go.

Is that Johnny Cash? Yes, it's Johnny Cash, and he's singing a song about the deep importance of mathematics in your life. I got 24 minutes to go. Well, they gave me some beans for my last meal. There's no math here. What are you talking about? No, there's a lot of math here, because you see, what he's doing is he's moving to his extinction, it seems, but he's being very careful to calibrate. I got 22 minutes to go.

Well, I sent for the governor and the whole darn bunch with 21 minutes to go

And I sent for the mayor, but he's out to lunch. I got 20 more minutes to go. Oh my God, we're going to go all the way to one? I feel like I've been listening to this song for three hours already. The numbers are making a titty. If I were him, I'd lose the numbers. You'd lose the numbers? Yeah. You can't lose the numbers. You cannot lose the numbers because numbers create order in your life. I could lose the numbers. I could survive an entire, well, my whole life without them.

That's just completely ridiculous. Easily. Try me. Try me. Let me just ask you something very simple. You go to buy some M&M's and you have a $5 bill in your hand and you give it to the vendor and the vendor gives you back the M&M's and...

What? No numbers required. I hand him the bill, he hands me some change. I just go by trust. You go by trust. Yeah. He asks you how old you are. What do you say? I'm middle-aged, I tell him. Listen to that. You hear that? Suppose that you're late for an appointment or something like that. Yeah? So you call up and you say, I'm going to be three minutes late, five minutes late, ten minutes late. I usually just wait for the call before I leave. Yeah.

I know that. Which you know is true. I know it's true. So yeah, don't eat them. Don't eat them. You're a test. You're taking a test in school. You get a 98. You get a 52. You don't care? Pass, fail. How much gas is in your car, Jed? I wait for the light to come on. You wait for the light. Suppose you want to call me. And you can't remember my phone number. Two words. Speed dial. How many words?

Oh crap! You see you gotta use numbers! Is that how it ends?

That's a great ending. An ending made possible all thanks to the disciplined use of numbers. Yeah. And that's going to be our hour. What do numbers do to us and for us? Or don't do for us. We've got, what do we have? We have a... We're going to have a detective story, a love story, some Nazis, and lots of numbers. I'm Chad Abumrad. I'm Robert Krolwich. This is Radiolab. Stay with us.

So, Chad, do you want to introduce this person? This is little Emil. Hi, Emil. So how old is he now? He's hungry right now. He's about 30. Carla, how old is he? 36? At the time of this recording, he is 36 days old. Well, I mean, you must have wondered, do you think he has any sense at all of...

or quantities or anything. What do you mean? Can he count? I'm not asking if he can count, but do you think he has a, I don't know, a numeric sense at all? Do I think he has a numeric? No. No, I don't think he knows that that is his hand that he's chewing. So I don't think there are any numbers in there. In fact, I'm pretty sure there aren't. Well,

Actually... Lulu, you should introduce yourself to Emil. Hi, Emil. Emil, this is our producer, Lulu Miller. And by the way, Jed, while you were on paternity leave, we sent Lulu on a little mission to ask, where does a number sense come from and how soon does it arrive in a person? Uh-oh.

Hello? So this is the first guy I spoke to. His name is Stanislas Dahen. Yes, speaking. Who's he? He is a neuroscientist in Paris. So we've been brushing up my English for a few minutes. Currently, he's like the godfather of this research. Really? He wrote a whole book called The Number Sense that talks all about what babies understand. Mm-hmm, yeah. And he said that for a long time, people thought that babies came into the world just empty.

Piaget and many other thinkers thought that there is what people have called the blank slate. That we could only learn numbers if we were taught them. Yeah, that's what I think. But now we know it's just completely wrong. And how do they know this? Well, experiments. Lots and lots of baby experiments. The equipment we have is a set of little sponges, which contain a very small electrode that you can place on the head of the baby. It's a little net.

And these babies are how old? In this case, it was babies of two or three months. So he plunks the baby down. Grrrr!

in front of a computer screen, and on the screen are a bunch of little pictures. Like, you know, little ducks, for instance. It's always a set of eight of the same object. So you do eight ducks, eight ducks, eight ducks, eight ducks, eight ducks. And what he sees is that at first the baby's brain is a little excited about getting to see ducks, and then slowly the firing just kind of fizzles out. Another eight ducks, another eight ducks. And then at some point, suddenly... He changes it to... Eight trucks...

And he sees a spike in brain activity. In what we call the temporal lobe. Meaning the baby can notice that change. Yeah, but that's not numbered. No, no, no, I know. He's just getting started. Because Stan runs the whole thing again, starting out the same way. Eight ducks, eight ducks, eight ducks, eight ducks. But then, instead of changing to trucks, he just changes the number. Eight ducks, eight ducks, sixteen ducks.

And once again, the baby notices the change. But now, it's in a different part of the brain. What we call the parietal lobe. So the suggestion is, according to Stan, that they're noticing that this is a different kind of change. That in some sense, they're noticing this is a change in quantity. Which is very important because it means that even in newborns, they have in their minds and in their brains an intuition of numbers.

Is he sure that they're seeing numbers or maybe they're just seeing a change in the pattern? Some, some, some, some more. Yeah. Well, sure. What they're good at is making these gross distinctions like 8 versus 16. Or say 10 and 20. You know, and as the difference in number gets smaller and smaller, then they're not so good. There is no baby that will ever know the difference between 9 and 10. These numbers are too close together. But it's not quite as simple as you might think.

According to Stan, the way that they're actually experiencing quantities is not just a dumbed-down version of what adults do. It's a completely different version of what adults do. They seem to care about the logarithm of the number.

The what? The logarithm of the number. He means logarithm. Yeah. Sorry, my English is getting really bad. No, logarithms. I don't know if this will scare the people who listen to this show. It scares me a little, but it's actually not that bad. You can think of it in terms of ratios. First think about you. Meaning? Us, how we think about numbers. Okay. Imagine in your head the distance between 1 and 2. Okay. What is that?

One. Right. Now imagine the distance between 8 and 9. One, also. They feel like the same distance from each other. Yeah. But that's because we think of numbers in these discrete ordered chunks. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. But now if you were to think about it logarithmically... Like the baby. The distance between 1 and 2 is huge! It's this vast space.

And the distance between 8 and 9? Oh, tiny. Why is that? Well, because 1 to 2 is doubling. But 8 to 9... It's a ratio of close to 1, only 1 point something. Huh.

Now, here's the spooky thing about this. You might think what must happen is that eventually as we grow up, we just naturally switch from logarithmic thinking to the numbers we all know now. Uh-huh. But this is not true. According to Stan, if left to your own devices, you'd never switch. What do you mean? You would stay in this logarithmic world forever.

So we've done this very funny experiments in the Amazon with people from the Amazon who do not count. Basically, in their culture, they do not have number words beyond five and they don't recite these numbers. So what we found is that these people still think of numbers in a logarithmic way, even the adults.

What that means is that if you give them a line and on the left you place one object and on the right you place nine objects. You got that? Uh-huh. And he asked them, what number is exactly between one and nine? Okay. So you'd say... Five. Exactly, but... What they put in the middle is three. Okay.

Wait, help me here a little bit. The property of the logarithm is that each time you multiply the number, you move by a constant displacement. Okay, so this is a bit tricky, but the gist is, if you're thinking in ratios, and you're starting at 1, then you multiply by 3 to get to 3, and then, hey, hey, you multiply by 3 again to get to 9. I see, I see.

So those are equal jumps on either side. 3 is to 1 as 9 is to 3.

Get it? Yeah, well, it's such a sophisticated way to go about thinking about it. Yeah, to us, but not to them. That feels intuitively simply like the middle. Dozens of people did this without hesitation. I mean, this experiment gives me chills. These are the numbers that we all, for want of a better word, naturally feel. At least that has been my theoretical claim for many years. Hmm.

And I don't quite know how to phrase this question, but is there some, is it almost like the way we think about numbers with an equal distance between 1, 2, 3, 4, 5, 6, 7 is wrong?

You know, I wouldn't go too far. But then I talked to Susan Carey. I'm professor of psychology at Harvard University. And she said that numbers as we think of them today are certainly made up. Those are human constructions. And even somewhat at odds with how we feel numbers intuitively. That's right, they are. So there is the problem. Then how do we ever come to understand the numbers we know now? That's a $64,000 question. She says it happens gradually. Okay, don't touch the microphone. Bye.

Over a couple of years. Can you count? Yeah. Let's see it. One more quick introduction. That is... Mina. Who you might remember from The Laughter Show. Yes, you've met Mina before. Me.

And her mother, producer Amanda Aronchik. She will be two in a week. Yes, it's her birthday. And we've called them in today because of an experiment. An incredibly simple set of tasks that Susan told me about. If you have a two-year-old at home, you can do these tasks. So we're going to play a game, okay? So you put a bunch of pennies on the table. I'm going to give you some pennies, okay? Just a second. Let mommy get them for you.

And you say to the child, can you give me one penny? Can I have one penny?

and the child very carefully picks up one and hands it to you. That's right, that's one penny. Thank you. Young two-year-olds, almost all, can do that. Then you ask for two pennies. Now, can I have two pennies? No. No? Please, can I have two? It doesn't matter what you ask for. They just pick up a handful and hand them to you. If you have more than two pennies, you have like one, two, three, four. And so they've given you four pennies, and you say, is that two? And they say, yeah.

Right? And then you say, can you count how many pennies you have? Or can you count and make sure? How many pennies is that? Two.

So they go one, two, three, four, and you say, is that two? They say yes. And sometimes they count. How many pennies is that? One, two, two, two. So it's like they somehow know that all of their other words contrast with one in meaning. That is, they're giving you a number, and they're giving you a number more than one, but they haven't the slightest idea.

what two is or three is or four is or five is. And they don't know what two means for nine months. And they're in that stage for several months, and then they become three-knowers, and then they become four-knowers. That process takes a year and a half. In other words, even though it sounds like Nina understands numbers like we do, she's probably still living in the land of that baby man. One, two,

But there does come a moment when they finally step away. And it happens right when the kid's about three and a half years old. What they do, I think, this is speculative, but... After years of everyone around them saying... Count. Can you count how many pennies you have? This is something parents do. One, two... They practice counting with children. Can you count how many pennies you have?

One, two, three, four, five, six. Can you do four, five? Seven, eight, and nine. The last one's ten. One, two, three, four, five, six. Even though the kid is baffled by these numbers, they don't know what five or six or seven means. Four, five, six, seven, nine. The last one's ten.

At some point, after enough pressure, count, count, they just sort of count. Throw up their hands, count, count, and believe the song. That's a very bold leap that children must make. And so now what five means for the child is one more than four. And what six means is one more than five.

And now you've got integers. We're all sort of like relying on this song. Yeah, one, two, three. We just that one day decided, okay, that means something. That's right. So this is a trick. What does she mean by that? Trick. Sounds almost like a dirty word. Well, she doesn't use it like a dirty word. She says it's a wonderful trick. The point is, once you have that trick,

You build on that. And that opens up the whole world of mathematics to you, and we can build buildings and launch rockets into space. And no other animal has invented that trick. But I can't help feeling there's something about this that's a little bit sad. Why? Well, just the idea that to step into this world of numbers, we all had to leave something behind. What you were born with. Yeah. But no, look what you get on the other side, though. You get...

you get to play and have remarkably interesting, if you like math, you get to play with deeply abstract and beautiful thoughts. Yes, yes, and that's great. So do you feel sad when somebody's good at trapeze work? No, that's just something that they're good at, and they practice it and they learn it. Just like different talents, that's all. But Robert, I think I know what Lulu's talking about. I mean, it's refreshing somehow to know that the numbers that we use day to day are somehow made up. Yeah.

Because sometimes the numbers for me at least feel like these hard, fussy, foreign things that don't feel real. They feel actually the opposite of real. But are you sure that real isn't just unfamiliar or a little strange? Foreign, yeah, sure. Because before you could walk, when you were just a crawler, you know, toddling was kind of unusual. And then toddling became kind of an adventure, and then that became kind of usual, and then you learned how to walk. Yeah, but eventually you do walk. But there's something about numbers where I feel like...

Personally, I never learned how to walk. And I think there's a lot of people listening right now who probably feel that way about numbers. So maybe we're just logarithmic people. Come on. Lulu? Yes? Thank you very much for that lesson. Lulu, stay strong in your opposition to integers. Yeah. We'll be right back.

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♪♪

Hello, I'm Jad Abumrad. And I'm Robert Krelwich. This is Radiolab. We're still talking about numbers. Now we're going to switch, though it may fatigue some of us. If you think about them a little differently, if you learn to embrace them, give them a bit of a hug, wonderful things can happen. I'm going to introduce you to a...

Well, a nosy man named Mark Negreene. I'm an associate professor at the School of Business at the College of New Jersey. Has a really heavy New Jersey accent. But what he really likes to do. What kind of accent was that? That was kind of a... I originally grew up in Cape Town, South Africa. South African, yes. He likes to play detective and the clues he looks for are numbers. I can't walk past a number without just wondering about it.

What went into that number? How did it get there? For example, after I finished filling up at a gas station, sometimes I would just walk around and look at the dollar amounts on the pump. So he peeks in at the pump right next door. And it's rather amazing. You can almost tell who's been there before. If you see a number like $1.40, then you know, oh, teenager with no money. Why? Wait, explain that. Because that's all the kid can afford. Quite right. So.

Sometimes I'll see $10.04 and I'll say, ah, you meant to do $10, but you were a bit slow today. So you'd go to the gas pumps and they tell you all little short stories. Yes. And his favorite story that numbers tell actually starts back in 1938. So imagine an office in Schenectady, New York, at the GE Research Laboratories.

And in that office is a man and he's sitting at his desk. Mr. Frank Benford. And Mr. Frank Benford is a physicist, so he's doing some difficult calculations and he's hunched over a book.

Probably, actually, one of the most boring books you could imagine. This is a book of logarithmic tables. What are logarithmic tables? So, log tables were a very convenient way of doing multiplication in the early part of the last century. So, remember, this is before there were calculators. So, if you wanted to multiply something like 145 times 3,564, you could just go to this book and look it up.

So it starts with numbers you might want to multiply by. One to a hundred on the first page, then a hundred and one, a hundred and two, up to two hundred and three hundred. And the back of the book is like nine hundred. The further you go, the higher and higher the numbers you use to multiply. That's right. So our Benford fellow, he's sitting there doing his calculations, and he's looking up the numbers.

Flipping through the book. He's staring at the pages and... He notices something kind of weird. He noticed that the first few pages were more worn than the last few pages. Meaning more smudgy and darker and oily, as if he was using the front of the book more than the last few pages. And he wondered, why is this happening? Strange. I'm not aware of favoring one part of the book over the other. Am I doing something a little odd? Or maybe it's something bigger.

And that's when it hit him. He thought maybe in this world there are more numbers with low first digits than with high first digits. What? More numbers that start with one or two

The numbers that start with 7, 8, or 9. Just because his book is more? Well, that's what started him thinking, so here's what he did. He compiled some tens of thousands of statistics. That's Steve Strogatz, mathematician at Cornell University. Just anything he could think of that was numerical. Molecular weights of different chemicals, baseball statistics, census data. The revenues of all the companies listed on the main stock exchanges in America, and every

Everywhere he looked in all these different categories, it seemed, yes, there were more numbers beginning with 1s and 2s than 8s and 9s. Wait, really? Oh, yeah. This has been checked out again and again and again, and it's true. Size of rivers. Earthquakes and things like that. Populations or number of deaths in a war. Areas of counties. Stream flow data. What if you were to say, get all the people in New York together and look at their bank accounts. Bank.

Bank account balances follow Benford's Law nearly perfectly. Meaning that if you just look in at the amount of money that people have, matter of fact, in all the bank accounts, you'll find they begin with one more often than they begin with two? Perfectly, yes. So actually, they begin with one 30.1% of the time. They'll begin with a

two 17.6 percent of the time they begin with a three 12.5 percent of the time that's a that's a big difference why would three be i'm sorry keep going and the poor nine would only occur as a first digit 4.6 percent of the time

which actually would make the 1 approximately 6 times as likely as the 9, and it is quite amazing. That is more than quite amazing. That's deeply suspicious. I mean, this is crazy what I'm telling you, and I can't give you good intuition why it's true. But Steve and Mark and many, many, many mathematicians will tell you, despite what you may think, there is a preference, a deep preference in the world, it seems, for number sequences that start with 1s and then 2s and then 3s.

Um, Robert? Mm-hmm? So what? Well, this is not just a mathematical curiosity, Jed. No, no, no. There is something you can do with this information. What? Well, when Mark Negreani first ran across Benford's Law, he thought, maybe I can use this law.

To bust people. For payroll fraud, tax return fraud. You thought, hey, we can use this to catch a thief? That's right. Huh?

How? Well, Negreani figured if you look at a bunch of numbers, say bank statements or expense reports and so on, and you see that the numbers in the business do not match the natural pattern of Benford's Law, so the numbers don't begin with ones more than with twos and will begin with twos more than with threes and so on, then you could say, hey, this is not natural. This may not be true. This may be fraud. Fraud.

So he started giving lectures on the idea that Benford is a way to catch thieves. The only problem was... They didn't quite believe Benford's law, which means the rest of my talk isn't going to go anywhere. It is now my great pleasure to introduce you to one of the most fabulous people I've ever had the name to say.

Daryl D. Dorrell. Daryl D. Dorrell. Daryl D. Dorrell. Daryl D. Dorrell. Daryl D. Dorrell. Daryl D. Dorrell. Daryl D. Dorrell. Daryl D. Dorrell. It's an alliterative heaven. It's like palindromic. Yeah. It's Daryl D. Dorrell. Ah.

Darrell. Darrell. Darrell. But I should say, Darrell is, what does he call it? I'm a forensic accountant. Forensic accountant. Which means his job is to examine numbers and figures to see if someone is stealing. It's an investigative process. And while at first he was unsure about Benford's Law, one day... I happened to talk to one of my neighbors who was a...

retired statistics professor. And he said, "Oh, Benford's? I have my students do that proof every year." He actually wrote out the proof for me. And it's just, it's immutable. It's a mathematical law. And now it's one of his favorite tools of history. We have a case right now underway, relatively small company, family shareholders, there are four of them,

One of them feels like she has been misrepresented as a shareholder. Meaning she thinks these other three guys might be stealing? Yes. I see. And I know you can't tell us what this business is doing, but is it a business? Let's say it's a regulated business company.

It's a business that each of you purchase on a regular basis through your local governmental authority. Trash collection or suing. Sure. Anyway, this one woman thought she was being cheated. So she got an attorney involved. The attorney requested data. So we have seven years of income tax return. And that's all he had.

Just tax returns. Uh-huh. Yes. So he entered them all into the computer. Aggregate them, run Benford's, and boom. Clicked on the graph. We instantly saw, bingo, for a couple of the years, coincident with when the dispute began, the way they've reported their taxes violates Benford's. Hmm.

Very suspicious. Yes, blew out the Benford's pattern. You mean like there are too many nines on the tax returns? Meaning if you looked at the tax returns of this company, you will see a pattern that isn't natural exactly. Not enough ones and too many sevens, eights, and nines.

But now you have to convince detectives and then lawyers and then judges that this is real evidence of wrongdoing, but they've not heard of this thing. Benford's... You don't know about it. As a practical tool, has probably been around maybe 10 years, maybe 15 at the outset. Please welcome Daryl Terrell. I'm at a conference now with about 700 people.

Nice to see all of you here. I've spoken four times, and each time I've asked about Benford's, who's heard of it? Who's familiar with Benford's Law? Maybe.

Maybe 5% of the people. Can you just look at pennies? Just a couple observations. To me, it doesn't make sense to exclude the pennies. Can you use Benford's? And they're asking, do judges allow Benford's in as evidence that suggests that someone's committed a crime? Is there case law out there that actually cite the use of Benford's law? And Darrell tells them, oh yeah, you can use this evidence in court. Yes, federal, state, and local, from the experiences we've had. And then he tells them stories.

Like the case of the CEO stealing money to buy... Automobiles. Firearms. Artwork. Jewelry. Run Benford's and... Boom. The CEO is still in federal prison. Or the case of the dentist and his wife. She began having an affair with a guy who turned out to be a meth dealer. The dentist suspected her of having dipped into the till. Run Benford's and... Boom. Oh, busted. She eventually pled. Or the guy with a $40 million Ponzi scheme. Run Benford's and... Boom. Well, almost boom. I mean...

Benviz was an element in all these cases. It wasn't the clincher. But still... It is a very compelling argument. And ten years from now, it'll be the equivalent of a fingerprint.

That's it. Huh. You still haven't addressed the central mystery here. Why in the world would there be more 1s than 9s? Shouldn't they be equi... Equi-coincident? Yes. Well, the answer is actually very complicated and deeply mathematical. The simple answer is... Is there an answer, though? Yes, there is an answer, and it has to do... Do you understand the answer? No. No.

I mean, I understand that it has to do with logarithms and the business of doubling and the culture of numbers. But if you were to sit me down and say, explain it to me carefully and well, I mean, no, it's just too numeric for me to explain it to you.

Okay, all right. But I will now take a little sidestep to a group of people who would be able to explain it to us if they were in this room, but we didn't find them in this room. We found them in another room. We're rarely in the same room that they are in. So let's go with our reporter Ben Calhoun and meet a

- Ben? - Yep? - You decided to, I don't know, it was some kind of a busman's holiday. You wanted to go to a math conference? - I did, badly. - And what happened? - Well, I went to CUNY, which is the City University of New York, and I went to a math conference

And it was a math conference called, it was on combinatorial and additive number theory. Oh, a good time had by all. Yeah, it goes by the optimistic acronym CAN'T. So I had heard that if I went to this room, there was going to be a bunch of mathematicians from all over the place. And they would be able to tell me where they taught, what their name was.

But they would have this other way of identifying themselves. Why? They had this number. My number is two. Three. Two. Yeah. Three. Three. Two. Mine is three, actually. Oh, nice. Yeah, I'm really excited about it. What does that mean, I'm a two, I'm a three? Well, it's an Erdős number. What's an Erdős? Erdős is a guy. Oh. So your Erdős number is how many steps away you are from this guy, right?

Paul Erdős. So you're going to tell me his story? Yep. Are you ready? Okay. Let me turn off my cell phone so we don't ruin the best take. That's Paul Hoffman. He wrote a book about Paul Erdős. So we start out in Budapest, Hungary in 1913. It's spring. Two math teachers have a son named Paul. And he had two sisters. They were three and five, and they had scarlet fever, and they died the day he was born. I mean, imagine that. His mother loses her two daughters and gains a son.

Yeah, she was so terrified after that that Paul would get a fatal disease and die that she didn't let him leave the house pretty much for the first 10 years of his life. She didn't let him play with other kids, really. Didn't let him go to school. Didn't let him go outside. Also, when he was one and a half, his dad was captured and put in a Soviet prisoner of war camp for six years of his life.

So here's this kid at home without other children around. His mother is out teaching mathematics. All the books in the house were math, and he taught himself basically to read by looking at these math books. And he also said to me that numbers became my best friends. So I mean, here's a kid whose whole life is mathematics from the beginning. But let's fast forward.

Paul Erdős gets his PhD in his early 20s. This is in the early 1930s. Paul Erdős is Jewish, which means he knows that he's got to get out of Hungary. And he managed to get to the United States. But he has to leave his family behind.

When the Nazis moved into Budapest, four of his mother's five siblings were killed. His father died as they were herding Jews and trying to move them into the ghetto, and he only had his mother left. But she was in Hungary. And in 1941, Paul Erdős was at Princeton University. He was just 27 years old, completely cut off from his family.

He was lonely and he was homesick. I mean, this guy had no conventional friendships. He had no sexual relationships. His only contact with the world was the people he worked with. I mean, what's remarkable to me is other people who had been through this kind of life experience might have ended up in a mental institution or worse. But he didn't. He turned this

this sort of inwardness into making mathematics a joyous and social occasion. He started connecting with people. I don't get this. Like, what do you mean? Well, he started traveling. He would hear about somebody who was working on something interesting, and he would find a way to get there, show up at their door, and he had this phrase that he would say, my brain is open.

And he was there to work with them on whatever it was that they were working on. And he just kept moving. He made a circuit of 25 different countries. Eventually, he gave up almost all of his possessions and he became essentially homeless. He had no home? He had no home. So everywhere he went, people had to put him up. And as a house guest...

The man was an acquired taste. He didn't know how to do basic things. He couldn't cook, he couldn't even boil water for tea, he could barely change his clothes. Erdős didn't know how to tie his own shoes until he was 11. He had some kind of skin condition, so he only wore silk. Silk clothes you had to wash. I mean, he went through life this way. And

There was the schedule. He did mathematics 20 to 22 hours a day. He'd bang pots and pans around in the kitchen at 4 a.m. because he wanted you to come downstairs and do more math. So why would anybody want to visit from this guy? This sounds like a walking nightmare. You have to cook for him and stay with him and wash his clothes and tie his shoes. Regardless of all of it, you wanted him to come see you. Why? Because he was just that good. This was like God coming to visit you.

He knew your strengths. He knew how you thought. And it was fascinating to watch him. I mean, there were times like I went with him to a math conference. He was there in his hotel room. And at one point, there were like 10 or 12 mathematicians in the room. Some were sprawled on his bed, some were sitting on the floor. And he'd be working with one for a few minutes.

And then he would turn to another. And then he'd go back to a third. And he was working simultaneously with all these people on different problems. Paul Erdős wrote more papers and collaborated with more people than any other mathematician who's ever lived. He did mathematics with anybody, even if the person was a dim bulb in the world of mathematics.

What's this? This is Paul Erdős on his 80th birthday. The only good wish for an old man you can say is an easy cure of the incurable disease of life. Surrounded by the mathematicians who loved him and put him up in their house. We want to express all our deep, warm feelings to you, and I want to raise this toast for you. Thank you.

He was a saint. A saint? A saint. That's Joel Spencer. He's a mathematician. He was also friends with Paul Erdős. Now that he is gone, I think of him sometimes in a religious context because he gave this faith to those of us that are doing mathematics, which after all is, if you look at it from the outside, it's a little bit of a strange activity why you

put this enormous effort into finding these statements. Mathematicians will spend years of their lives trying to prove these things that, you know, from the outside look totally obscure and pointless. And yet it was clear working with him that what we were doing was we were trying to find truth with a capital T. A truth that

that transcends our physical universe. I think that's the reason why we like to talk about our connection to Paul because our feeling of mathematics, the feeling for what we want mathematics to be, Paul Erdős was the embodiment of that feeling. Somewhere along the way, mathematicians started keeping track of their connection to Paul Erdős.

And that's what Erdős numbers actually are. If you published a paper with Paul Erdős, your Erdős number is one. If you published a paper with someone else and they published a paper with Paul Erdős, then your Erdős number is two, and so on and so on. So this is like all the people that Paul Erdős in some way has touched. All the people who are connected to him through their ideas. There are...

about 500 people with Erdos number one and about 8,000 people with Erdos number two. This is Professor Jerry Grossman. He's at the University of Oakland in Michigan. And what he did was he took each ring of Erdos numbers and he charted it out. Erdos number three has about 34,000 people in it, about 84,000 with Erdos number four. Then they start decreasing

84,000. That's a lot of people. So if you go ring upon ring upon ring and you do the whole deal, how many people did this man in the end influence? I think it's about 200,000 mathematicians. 200,000? 200,000. Picture that for a second. It's like a solar system with more than 200,000 mathematicians all orbiting around Paul Erdős.

And your Erdős number is? One. One. Dennis Eichhorn is number two. Your Erdős number is? Two. I wrote a paper with my advisor and the other students, and she had written a paper with a mathematician who had written a paper with Erdős. My Erdős number is three. Your Erdős number is? One. Everybody with Erdős number one knows that they've got that. Everybody in this room knows their number.

I would be very surprised if there are people who don't know.

Ben Calhoun's Erdős number is 00.5778-b-1 radical 6. Coming up, a story from our friend Steve Strogatz, the mathematician from Cornell, who tells about a friendship he has, a very precious friendship with his math teacher. So it's all about mathematicians, but this is a very unusual friendship. I'm Chad Abumrad. I'm Robert Krulwich. Stick around.

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Okay, I'm Jad Abumrad. I'm Robert Krolwit. This is Radiolab. Our topic today is... Mathematics, mathematics, and mathematics. I suppose that is our topic. Okay.

But actually, we do have a gripping story for you coming up now from our producer, Soren Wheeler. Hey, Soren. Hey. And this is about math, right? Yeah. Well, math and friendship, really. And I heard it from Steve Strogatz. He's a mathematician at Cornell University, and he's been on the show once or twice. Okay. And we sat down in the studio, and he told me about... Why don't you back up and tell me a little bit about high school and about his high school math teacher?

Don Joffrey. Well, there were several striking and peculiar things about him. I mean, probably the first thing is that he was physically incredibly impressive. When he would hold the chalk between his enormous fingers and write on the board, the chalk would pulverize with each stroke so that there would be this cloud of chalk dust all over him and his big sweater and everything.

Another thing that was very unusual about him, he'd be in the middle of a calculation, standing at the board, chalk dust all over him as usual. Then he would space out, and he'd get a look in his eye, a kind of faraway look, and then he'd say, oh, this reminds me, with a hushed tone, oh, this reminds me of the time Jamie Williams calculated the formula for the nth term in the Fibonacci sequence. Who's Jamie Williams? Jamie Williams was a student. Ha ha ha ha ha ha ha.

He was just a couple of years ahead of Steve in Mr. Joffrey's class. And that was part of the mystique, you know, that now he was graduated and it was as if the secret was lost to the ages. But the point was that he would talk about a student. With reverence. With reverence. What was very thrilling about that is that there was this kind of chain that we were now becoming part of. ♪

Yeah, so then I'm off to college and it started very early. I started to write to him. It was like an annual tidbit. Dear Mr. Joffrey, here's the gem that I learned this year in math. So Steve would write to him, Mr. Joffrey would write back, add something, ask him a new question, and it went on like that for a while, with Steve kind of still being like a student and Mr. Joffrey still like a teacher. There was one moment, though, where...

Something new happened where he wrote to me asking for help. He said a question came up in his class about an elliptical swimming pool.

So, you know, picture a swimming pool. Often there's a little border on the edge of the swimming pool, like a piece of concrete that lines the pool you stand on that part before jumping in. And so the question was, if you had an elliptical swimming pool with a one-foot border around it, is the outer edge of the border also an ellipse? Something about that really appealed to me. It was a very nice math problem. Probably there was a little bit of a show-off in me, like I thought if I could do this, he's going to

say something nice. You'll become part of the pantheon. Yeah, maybe I'll enter the pantheon. They'll start talking about me like they used to talk about Jamie Williams. So I stopped whatever I was doing and I worked hard on that ellipse problem and I figured out two or three different ways to... It turns out it's never an ellipse. It cannot be an ellipse. So Steve sat down and wrote back to Mr. Joffrey about this puzzle, but... I didn't just show him the answer.

I wrote the answer in a very loving and gentle way that was meant to be empathetic. That is, I know where you're coming from, and I'm going to just start from scratch to lead you from where you are to where you need to be to solve this problem. In other words, Steve acted like he was the teacher.

And Mr. Joffrey played along. And this was such a generous thing in retrospect. The humility, the modesty, the kindness in playing the role of a student. It's like he knew that that's what I needed. And man, I loved it. I couldn't wait for the next question. And as Steve went off to graduate school to become a math professor himself, he and Mr. Joffrey kept writing to each other. In fact, they were writing to each other quite a lot. There was one sequence in March of 1989 where we wrote to each other almost every day.

He sent me a puzzle. I worked on it. I showed him a really beautiful answer. He expressed kind of ecstasy in seeing this answer. It was kind of a mathematician's dream correspondence of puzzles and equations, and Steve loved it. But every so often, Mr. Joffrey would...

Break the routine. A little bit. He would say things about that he was doing some jazz piano gig. He would sometimes write about he had three sons. He would talk about them a little bit. And, you know, I feel embarrassed. It feels mean. But I remember not liking those parts of the letters. And I didn't write about that. I mean, I would say.

Maybe I was playing some tennis. But I have lines in some of my letters that say, after a few of those sentences, okay, enough stalling. Here's the math problem. But then in later years, he would almost pointedly ask me things. Like there was a time when he said, that rumor has it that you're engaged. We wish you the best if this is true. And guess what? In my letter back to him, I didn't say anything. Do you remember like...

Thinking not to respond or just... Well, I can tell you what was going on, which is that I was already in couple therapy with my fiancé. You know, like in that time, the letters were a kind of refuge from all that. That as we could go into this pristine world of math where things are simple and logical and well-ordered, there may have been part of me that felt like, oh, come on, this is the one place where it's all perfect.

But over the years, that perfect world got a little less perfect. Because his oldest son died. Marshall died. Marshall died when he was only 27. And I didn't ask about it. Can you believe this? I feel so sick about this when I think about it now. So you would just write back, oh, I've got another puzzle. Got another math problem for you. Look at this. Yeah.

And then more than 20 years into this relationship of letter writing, Mr. Joffrey retired. And now that he couldn't teach anymore... He'd write to me. He'd show me these beautiful math problems that he would make up for himself, usually about hawks flying over the earth and, you know, how much spherical area can the hawk see if it's at such and such altitude.

And what is happening at this time is that now I have just gotten married and we've started having kids. And I'm not answering his letters anymore. They're sitting in their envelope stacking up. He's writing them faster than I can answer them, a lot faster. And then at one point I got one more letter from Mr. Joffrey. Except as soon as I looked at the envelope, I could see that something was really very wrong.

His handwriting didn't look normal. My address, my name, was written in a craggy... Like shaky kind of... Shaky. And I knew what that looked like because my dad wrote like that when he had Parkinson's. So I thought, what's this? And I opened the letter and the first sentence is, eek, I just had a mild stroke. I didn't write back to him right away. I didn't call him. And then just a couple months later... My brother died very suddenly and...

He heard about it from someone else and immediately wrote to me how, you know, that he and his wife had heard and they were very sorry to hear that my brother had died. That, to me, was, you know, I still had never said I'm sorry about Marshall all those years ago. And it kept going.

nagging at me, "Why won't you talk to him?" Beckoning, and you obviously care about him. It's sort of like in math, there's this concept of bifurcation, which really means a fork in the road, a splitting. When the forces on a system get too large, there can be a moment when the dynamics of that system change abruptly and qualitatively. This was a moment of bifurcation. I should have just said how sorry I was to hear about Marshall.

So I thought, I've got to go talk to him and ask him, can I come to your house? You know, he seemed a little reluctant about it. But, you know, okay, fine. So I bought a little pocket tape recorder.

Just a cheap thing. Drove up Route 95 to his house in Connecticut on the shore. Knock on the door, hear the piano that was playing inside stop. He comes and rushes to see me. We give each other hugs, take out a big plate of cold cuts and say, let's sit out on the porch. Does that work? Hello. And so we're eating. Seems to be recording. Okay.

And then he takes out his journal. I decided that I would keep a journal since I was retired. Where he's drawn pictures of all kinds of birds. Here's a picture of me doing an eagle watch out in the Connecticut River. And there's a lot of stuff about people I don't know. This is a bird that's moved up from the south, too. You never saw these, well, what some people call buzzards. Yeah. They've moved up here and...

This is one of my favorite birds. It's a marsh hawk and it flies low over the meadows. More about that. Hank says, I'm going to take you over to see a rough-legged hawk. Now, he didn't say, we're going to see if we can see a rough-legged hawk. Yeah. He produced. Uh-huh.

And I'm thinking to myself, I'm not really interested in this. I want to talk about him, about all these things that we never talked about, that are emotional, hard things. Like, what happened? How did your son die? They have a lot of work. They're just trying to make guys put in extra hours to pay guys extra hours. There was a fidgeting feeling inside me. Right, sure. And there was a pause. My heart was beating fast.

And I thought, I'm going to ask him now. So I don't think we ever talked about Marshall, but I wanted... And I did. I asked what... I didn't really know him either, but I know that he died very young. And I... What happened? You know, what happened to Marshall? Well, we, you know, that's something we don't really... Do you want to talk about that? Okay. And I think he was going to say, that's something we don't talk about. Well, it was... I remember him as a star. And he... He did. He had a wonderful 27 years.

music was going to be. It was so beautiful. And so, um, and so he went to man, so uplifting and sweet. He'd be at home and we'd sit around the piano and I'll get out the Cole Porter songbook and just turn to a page, something that he'd never seen. He could cite, read it, play it and sing it all in one time with us. And I thought, golly, this guy's got a multi-channel mind that I wish I had. You know, he talked about what a great, what a great life he had.

in his 27 years. In his waning moments, he'd stay up all night long playing the piano. Just the house was just filled with beautiful music. And he had made plans to get a job at the New England Conservatory and things like that, but the fates were wrong for him. Oh yeah, we miss him.

Was was that I mean In that moment did it change the way you see him well change? I have to tell you how that day ended so we talked more and I asked him at one point. Do you think Marshall had a religious feeling and he said oh, yeah, I think he felt close to Having to come to terms with somebody out there. Yeah, that was a good thing that that I think he went peacefully and

Then actually conversation drifted to easier things like calculus problems. And we talked some more about math and then he said, "How about a swim or let's go to the beach?" How would you like to go out to the beach and relax? I'd like to do something where I get outdoors a little bit. So we did go to the beach and it was a beautiful evening and there were waves coming in from Long Island Sound.

In fact, we were talking about a math problem about waves, about Fourier analysis. Which is really about, well, infinity and the fact that if you take an infinite number of simple waves, you can create any shape of wave you want. As long as it's a wave that repeats. But then Mr. Joffrey asked, how do you create waves that don't repeat, waves that change? Sometimes waves don't exactly repeat. They can grow or die out.

And Steve told him that to deal with those kinds of waves, you need a different kind of infinity. Not the kind where you just keep adding and adding and adding numbers, but the kind that sits in the space between two numbers. This higher kind of infinity than Don had thought about before. Thanks to Soren Wheeler, our producer, who interviewed Steve and produced that story. Thanks to Steve Strogatz, who has a book out now, which tells this very story called The Calculus of Friendship.

I'm Jad Abumrad. Three seconds to go. You are? Robert Rovich. Bye. Radiolab was created by Jad Abumrad and is edited by Soren Wheeler. Lulu Miller and Latef Nasser are our co-hosts.

Dylan Keefe is our Director of Sound Design. Our staff includes Simon Adler, Jeremy Bloom, Becca Bressler, Ikedi Foster-Keys, W. Harry Fortuna, David Gable, Maria Paz Gutierrez, Sindhu Niyana Sambandhan, Matt Kielty, Annie McEwen, Alex Neeson, Sara Khari, Alyssa Jung Perry, Sarah Sandback, Ariane Wack,

Yeah. I always want to do this.

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