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Such as reporting overseas on who owns the rights to knowledge. Or thousands of baby crabs. Other thought provoking stories on... I want to talk about the war in Ukraine. Do we need to get people more scared about climate change? About a black hole? Or do we need to get people more hopeful about climate change? Actually hitting Earth. Exactly. Your no-strings-attached, non-committal gift will allow us to continue to make this type of work into the new year and beyond. Now, of course, you could already be dating into the program, joining the thousands of people who are already donors and members that help us survive economic downturns and headwinds like the ones we're going through right now. But you're not. No, you're sitting there listening to this right now without a care in the world, consuming free content, eating up at your disposal. Everything is just here for you, for you to consume. Gobble, gobble. Gobble, gobble, gobble, you little turkey. Gobble, gobble, gobble. Radiolab. Org/donate, a place where forever can just be a moment. So the next time you're feeling terrified by the prospect of commitment, take a moment and come over to radiolab. Org/donate. Any amount is appreciated.

Radiolab. Org/donate. Come on, join the fun. That was from our genius, deranged and slightly commitment-apobic producer, Matt Kielty. I'm Lou Miller. This is Radiolab. To remind you of the 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 roller coaster ride through all different kinds of numbers, these things which can sometimes seem cold, but in Radio Labs, Love and Care are shown to contain real warmth. Without further ado, our episode numbers enjoy the ride. Again, if it makes you chuckle, if it 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/donate. Thanks for thinking of us. Here we go. Wait, you're listening. Okay. All right. Okay. All right. You're listening to RadioLab. -to RadioLab. From- W-N-Y- C. C? Yeah. Rewind. Chad? Yes. Listen to this just for a second. Well, 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 beams for my last- There's no math here. What are you talking about? I know 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 Calgary. 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 listening to this song for three hours already. The numbers are making it teddy. 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. Let me just ask you something very simple. You go to buy some M&Ms and you have $5 bill in your hand and you give it to the vendor and the vendor gives you back the M&Ms 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 asked 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, 10 minutes late. I usually just wait for the call before I leave. I know that. Which you know is true. I know it's true. So yeah, don't need 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 card yet? I wait for the light to come on. You're leaving. Suppose you want to call me, right? And you can't remember my phone number. Two words, speed dial. How many words? Oh, God. God. You see you, golly used numbers. I got two more minutes to go. I can see the buzzards. I can hear the crows. One more minute to go. Now I'm swinging and here I go.

Is that how it ends? Yeah. That's a great ending. Ending made possible, all thanks to the disciplined use of numbers. Yeah. Then that's going to be our hour. What do numbers do to us and for us? Don't do for us. Get a… 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 Krowitch. This is RadioLab. Stay with us. So, Chad, do you want to introduce this person? This is little, Emile. Hi, Emile. So how old is he now? He's hungry right now. He's about 30. Carla, how old is he? Thirty-six. 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 numbers or quantities or anything? What do you mean like thing? Can he count? I'm not asking you 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.

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 Emile. Hi, Emile. Emile, this is our producer, Lulu Miller. By the way, Jed, while you were on fraternity leave, we sent Lulu on a little mission to ask, where does a number of cents come from and how soon does it arrive in a person? Oh, there we go. Hello? This is the first guy I spoke to. His name is Stani Slas-Dahen. Yes, speaking. Who is he? He is a neuroscientist in Paris. 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 of Cent that talks all about what babies understand. 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, it'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 spongees, which contain a very small electrode that you can place on the head of the baby. It's the 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 in front of a computer screen and on the screen are a bunch of little pictures. Like 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, 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 it slowly the firing just 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 the what we call the temporal lobe. Meaning the baby can notice that change. Yeah, but that's not numbers. No, no, no, I know. He's just getting started. Because Stan runs the whole thing again, starting out the same way.

Eight dogs, eight dogs, eight dogs. But then instead of changing to trucks, he just changes the number. Eight dogs, eight dogs, 16 dogs. 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. The suggestion is, according to Stan, that they're noticing that this is a different 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 eight versus 16. Or say, 10 and 20. 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 nine and 10. These numbers are too close together. But it's not quite as simple as you might think. According to Stan- What is most extraordinary, I think- 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 logarism of the number. The what? The logarism of the number. You mean's logarism? Yeah. Sorry, my English is getting really bad. No, logarisms. 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. Imagine in your head the distance between one and two. What is that? One. Right. Now imagine the distance between eight and nine. One. Also. They feel like the same distance from each other. 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 one and two is huge. It's this vast space. The distance between eight and nine? Oh, tiny. Why is that? Well, because one to two is doubling. But eight to nine... It's a ratio of close to 1, only one point something. 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. . 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. We've done this very funny experiment in 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 reside these numbers. 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? . And he asked them, What number is exactly between one and nine? Okay. You'd say- Five. -exactly, but-What they put in the middle is three. Three. Wait, help me here a little bit. The property of the logarism 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 one, then you multiply by three to get to three, and then hey, hey, you multiply by three again to get to nine.

I see. Those are equal jumps on either side. Three is to one as nine is to three. Get it? Yeah. 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. 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, six, seven is wrong? I wouldn't go too far. But then I talked to Susan Carey. I'm a professor of psychology at Harvard University. She said that numbers, as we think of them today, are certainly made up. Those are human constructions. And even someone at odds with how we feel numbers intuitively. That's right, they are. 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.

Microphone? Yeah. Over a couple of years. Can you count? Yeah. Let's hear it. One more quick introduction that is... Nina. Who you might remember from The Laster show. Yes, you've met Nina before. And her mother, producer Amanda Ronchik. She will be two in a week. Two. Yes, it's her birthday. We've called them in today because of an experiment. It's 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. Honey, we're going to play a game, okay? 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. 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.

I think you have more than two pennies. You have 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, 1, 2, 3, 4. And you say, Is that two? They say, Yes. Oh, okay. And sometimes they count- How many pennies is that? One, two, two, two. It's like they somehow note 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. 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. Good job. She's probably still living in the land of that baby man.

But there does come a moment when they finally step away. Can you sing that song? 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 this on your towel? 1, 2, 3, 4, 5, 6. You do four, five? Seven, eight, and nine. The last one's 10. 1, 2, 3, 4, 5, 6. Even though the kid is baffled by these numbers and they don't know what five or six or sevens, four, five, six, seven, eight, and nine, the last one's 10. One, two, three. At some point, after enough pressure, count, they just count, throw up their hands, count, count, and believe the song. That's in a very bold leap that children must make. And so now what five means for the child is one more than four. And then what six means is one more than five. But now you've got integers. We're all relying on this song. Yeah, one, two, three.

We just have one day decided, okay, that means something. That's right. So this is a trick. What does she mean by that? Sounds almost like a dirty word. Well, she doesn't use it like a dirty word. She says it's a wonderful trip. The point is, once you have that trip, you build on that. That opens up the whole world of mathematics to you and you can build buildings and launch rockets into space. No other animal has invented that trip. 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. Which you were born with? Yeah. But look what you get on the other side, though. You get to play and have remarkably interesting… If you like math, you get to play with deeply abstract and beautiful thoughts. Yes, and that's great. 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 different talents, that's all. But, Robert, I think I know what Lulu is talking about.

I mean, it's refreshing somehow to know that the numbers that we use day to day are somehow made up. 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, toddling was unusual. Then toddling became an adventure, and then that became usual. Then you learn 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. I think there's a lot of people listening right now who probably feel that way about numbers. So maybe we're just - What does that mean, though? 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. Well, we'll be right back. This week on the New Yorker Radio Hour, with immigration policy front and center in Washington, staff writer Dexter Philkins traveled along the Southern border looking for answers. I think it's difficult to appreciate the scale and the magnitude of what's happening there unless you see it by yourself up close.

The dilemma is at the border. That's the New Yorker Radio Hour from W. N. Y. C. Studios, listen wherever you get your podcasts. Hello, I'm J. D. Avomrod. And I'm Robert Krollwitch. This is RadioLab. We're still talking about numbers, and now we're going to switch. 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 Negrinny. 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 accent was that? That was 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 the 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 a $1.40, then you know, Oh, teenager with no money. Why? Wait, explain that. Because that's all the kid can afford. Quite right. Sometimes I'll see $10.04 and I'll say, Oh, you're meant to do $10, but you were a bit slow today. You 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. Imagine an office in Schenectody, New York, at the GE Research Laboratories. In the 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 is 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? Log tables were a very convenient way of doing multiplication in the early part of the last century. Remember, this is before there were calculators.

If you wanted to multiply something like 145 times 3,564, you could just go to this book and look it up. It starts with numbers you might want to multiply by 100 on the first pages, then up to 200 and 300. The back of the book is like 900. The further you go, the higher and higher the numbers you use to multiply. That's right. Our Bedford fellow, he's sitting there doing his calculations and he's looking at the numbers, flipping through the book. He's staring at the pages and- He notices something 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? 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 seven, eight, or nine.

Just because his book is more? That's what started him thinking. Here's what he did. He compiled some tens of thousands of statistics. That's Steve Strogat's mathematician at Cornell University. Just anything he could think of that was numerical. Molecular weights of different chemical, baseball statistics, census data. They were the revenues of all the companies listed on the main stock exchange is in America. And everywhere he looked in all these different categories, it seemed, yes, there were more numbers beginning with one and twos than eights and nines. 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 a number of deaths in a war, areas of counties. Streamflow data. What if you were to say get all the people in New York together and look at their bank accounts? Oh, bank account balances follow Brentford's Law, nearly perfectly. Meaning that if you just look in at the amount of money that people have, matter of fact, being all the bank accounts, you'll find they begin with one more often than they begin with two. Perfectly, yes. Actually they begin with one, 30.1% of the time.

They'll begin with a two, 17.6% of the time. They'll begin with a 3.12.5 % of the time. That's a big difference. Why would three be two? I'm sorry, keep going. And the poor nine would only occur as a first digit 4.6 % of the time, which actually would make the one approximately six times as likely as the nine. 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 a 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 ones and then twos and then threes. Robert? Mm-hmm. So what? Well, this is not just a mathematical curiosity, Judd. No, no, no. There is something you can do with this information. What? Well, when Mark Negreeney 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, Negreeney 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 moving 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. So he started giving lectures on the idea that Bedford 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, Darryl de Dorel. Darryl de Dorel. Darryl de Dorel. Darryl de Dorel. It's an alliterative heaven. It's like palandromic. Yeah, it's Darryl de Rale. Oh, Darrell. Darrell. Oh. But I should say Darrell is, what does he call it? He calls 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 Bedford's law... A little bit skeptical... One day... I happened to talk to one of my neighbors who was a retired statistics professor, and he said, Oh, Benfords. I have my students do that proof every year. He actually wrote out the proof for me, and it's immutable. It's a mathematical law. And now it's one of his favorite tools of his trade. 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 know you can't tell us what this business is doing, but is it a- Let's say it's a regulated business. It's a business that each of you purchase on a regular basis through your local governmental authority, like a trash collection or a sewage. Cycling. Sure. Anyway, this one woman thought that she was being cheated. She've got an attorney involved. The attorney requested data. We have seven years of income tax return. That's all he had, just tax returns? , yes. He entered them all into the computer.

Aggregate them, run Benfords, and 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 Benfords. Very suspicious. Yes. Blue out the Benfords 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- They don't know about it. -as a practical tool has probably been around maybe 10 years, maybe 15 at the outset. Please welcome Darryl Carell. 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 five % of the people. Can you just look at Penney? Just a couple of observations, Darryl. To me, it doesn't make sense to exclude the Penney.

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 Darryl tells them, Oh, yeah. Yes. 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 and 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 mess dealer. The then has suspected her of having dipped into the till. Run Benford and boom. Oh, busted. She eventually pledged. Or the guy with a $40 million Ponzi scheme. Run Benford's and boom. Well, almost boom. I mean, Benford's was an element in all these cases. It wasn't the clincher. But still, it is a very compelling argument. And then 10 years from now, it'll be the equivalent of a fingerprint. That's it. You still haven't addressed the central mystery here. Why in the world would there be more ones than nines?

Shouldn't they be equ- 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. 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, 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 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. Yeah. Let's go with our reporter, Ben Calhoun, and meet a crowd of mathematicians. Ben? Yep. You decided to, I don't know, it was some 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 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, CAMP. I had heard that if I went to this room, there was going to be a bunch of mathematicians from all over the place. I'm a mathematician from Sweden, attending CAMP. They would be able to tell me where they taught, what their name was. But they would have this other way of identifying themselves. 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 Erdis number. What's an AirDish? Airdish is a guy. Oh. Your AirDish number is how many steps away you are from this guy, Paul AirDish. 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 Erdis. So we start out in Budapest, Hungary, in 1913. At 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. Oh, my God. 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 here's a kid whose whole life is mathematics from the beginning. But let's fast forward. Paul Erdish gets his PhD in his early 20s. This is in the early 1930s. Paul Erdish is Jewish, which means he knows that he's got to get out of Hungary.

Then 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 hurt in 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 Erdits was at Princeton University. He was just 27 years old, completely cut off from his family. He was 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 life experience might have ended up in a mental institution or worse, but he didn't. He turned this inwardness into making mathematics a joyous and social occasion. He started connecting with people. I don't get this. 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. Erdish didn't know how to tie his own shoes until he was 11. He had some 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 bang pots and pans around in the kitchen at 4:00 AM 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 are 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 Erdish 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 Erdish on his 80th birthday. I should add that the only good wish for an old man you can say is the easy cure of the incurable disease of life. Surrounded by-A lot of people.

-the mathematicians who loved him and put him up in their house. We wanted to express all our deep, warm feeling to you, and I want to raise your distoste for you. He was a saint. A saint? A saint. That's Joel Spencer. He's a mathematician. He was also friends with Paul Erdish. Now that he is gone, I think of him sometimes in a religious context because he gave this faith to those 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 while you put this enormous effort into finding these statements. Mathematicians will spend years of their lives trying to prove these things that 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 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 Ernest was the embodiment of that feeling. Somewhere along the way, mathematicians started keeping track of their connection to Paul Erdish.

And that's what Erdish numbers actually are. If you published a paper with Paul Erdish, your Erdish number is one. If you published a paper with someone else and they published a paper with Paul Erdish, then your Erdish number is two and so on and so on. So this is like all the people that Paul Erdish in some way has touched. All the people who are connected to him through their ideas. There are about 500 people with Erdish number one and about 8,000 people with Erdish number two. This is Professor Jerry Grothman. He's at the University of Oakland in Michigan. And what he did was he took each ring of Erdish numbers and he charted it out. Erdish number three has about 34,000 people in it, about 84,000 with Erdish 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. Two hundred thousand? Two hundred thousand? Picture that for a second. It's like a solar system with more than 200,000 mathematicians all orbiting around Paul Erdish.

And your Erdish number is one. One? What? You want me to say it here? Dennis Icorn is a number two. Your Erdish 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 Erdish. My Erdish number is three. Your Erdish number is one. And everybody with Erdish number one knows that they've got that most- Everybody in this room knows their number. I would be very surprised if there are people you don't know. Ben Calhoun's erotic number is 0.5, 7, 7, 8-B minus 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. It's all about mathematicians, but this is a very unusual friendship. I'm Chad Abomrad. I'm Robert Krollwitz. Stick around. Okay, I'm Chad Abomrad. I'm Robert Krollwitz. This is RadioLab. Our topic today is- Mathematics, mathematics, and mathematics. I suppose that is our topic. But actually, we do have a gripping story for you coming up now from our producer, Sauron Wheeler. Hey, Sauron. Hey. This is about math, right?

Yeah. Well, math and friendship, really. I heard it from Steve Strogats. He's a mathematician at Cornell University, and he's been on the show once or twice. 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. 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 space out. He'd get a look in his eye, a faraway look, and then he'd say, Oh, this reminds me with the hushed tone. This reminds me of the time Jamie Williams calculated the formula for the nth term in the Fibonachi sequence.

Who is Jamie Williams? Jamie Williams was a student. He was just a couple of years ahead of Steve in Mr. Jofrey's class. That was part of the mystique. 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 chain that we were now becoming part of. Yeah. So then I'm off to college. 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 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 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 showoff 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. 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. 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. Man, I loved it. I couldn't wait for the next question. 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 ecstasy and seeing this answer. It was 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. I feel embarrassed. It feels mean, but I remember not liking those parts of the letters. I didn't write about that. 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. Guess what? In my letter back to him, I didn't say anything. Do you remember 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. Like in that time, the letters were a refuge from all that. That is, 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. I've got another math problem for you.

Look at this. Yeah. Then more than 20 years into this relationship of letter writing, Mr. Joffrey retired. 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 how much spherical area can the Hawks see if it's at such and such altitude. What is happening at this time is that now I have just gotten married and we've started having kids. I'm not answering his letters anymore. They're sitting in an 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. -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 in the first sentences, Eak, 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 of months later. My brother died very suddenly, and he heard about it from someone else and immediately wrote to me how that he and his wife had heard and they were very sorry to hear that my brother had died. That to me was... I still had never said I'm sorry about Marshall all those years ago, and it kept nagging at me. Why won't you talk to him? Beckoning and obviously care about him. It's 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. I thought I got to go talk to him and ask him, Can I come to your house? He seemed a little reluctant about it, but, okay, fine. 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. He seems to be recording. 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. And here's a picture of me doing an Eagle Watch out in the Connecticut River. And there's a lot of stuff about-Hawk in his typical way- People I don't know- -and this is a bird that's moved up from the south too. You never saw these, well, what some people call buzzards. 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. He produced. 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.

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. And paying benefits and all the other stuff. There was a pause. My heart was beating fast. Then I thought I'm going to ask him now. I don't think we ever talked about Marshall, but I wanted to- I did. I asked what? I didn't really know him either, but I know that he died very young. And what happened? What happened to Marshall? Well, we had something we don't really- Do you want to talk about that? Okay. I think he was going to say, That's something we don't talk about. Well, he had a- I remember him as a star. And he... He did. He had a wonderful 27 years. Music was going to be- -and it was so beautiful and 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-to-page. It's something that he'd never seen. He could sight, read it, play it, and sing it all in one time with us.

And I thought, God, this guy's got a multichannel mind that I wish I had. He talked about what a great life he had in his 27 years. Even in his waning moments, he'd stay up all night long playing the piano. The house was just filled with beautiful music. 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. I mean, in that moment, did it change the way you see him? Or did it change you? Well, 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. That was a good thing that I think he went peacefully. Then actually conversation drifted to easier things like calculus problems. And we talked to him 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?

Yeah, I'd like to do something where I get outdoors a little before- 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. Rofrey 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 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 infinity than Don had thought about before. Thanks to Sauron 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 Abomrod. Three seconds to go.

You are? Robert Rowland. Bye. Radiolab was created by Jad Abomrod and is edited by Sauron Wheeler. Lulu, Miller, and Latif Nassar are our co-hosts. Dylan Keith is our Director of Sound Design. Our staff includes Simon Adler, Jeremy Bloom, Becka Bresler, Iketi, Foster-Kies, W. Harry Fortuna, David Gable, MariaVaz Guetierez, Sandu Nihana-Sambandum, Matt Kielte, Annie McEwen, Alex Neeson, Sara Kari, Alyssa Jiang-Perry, Sarah Sandback, Arianne Wack, Pat Walters, and Molly Webster. Our fact checkers are Diane Kelly, Emily Krieger, and Natalie Middleton. I was on to do this. Hi, my name is Michael Smith. I'm calling from Pennington, New Jersey. Leadership support for RadioLab's science programming is provided by the Gordon and Betty Moore Foundation, Science Sandbox, the Simmons Foundation Initiative, and the John Templeton Foundation. Foundational support for RadioLab was provided by the Alfred P. Sloan Foundation.