Numbers

SPEAKER_09: This week on the New Yorker Radio Hour, staff writer Dexter Filkins traveled to the southern border this year looking for answers to what seems like an impossible dilemma. That's the New Yorker Radio Hour, wherever you listen to podcasts. But what is it, Billy? What's holding you back? I don't know. I just can't commit. Do you find yourself feeling sweaty at the thought of making a recurring donation? SPEAKER_28: Quivering, yet frozen in terror when considering taking the big leap to being a monthly donor? Well, worry no more. Because here at Radiolab, from today until December 31st, you can make a one-time contribution to one of your favorite public radio programs. SPEAKER_18: When I'm online and someplace asks me to subscribe, I always do a free trial because anything more than seven days feels like just too much. That's okay. SPEAKER_22: I always feel like I'm getting asked to sustain things. Sustain this, sustain that. I just can't. I can't do that. No problem. 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No. You're sitting there, listening to this, right now, without a care in the world, consuming free content, eating it up at your disposal. SPEAKER_15: Everything is just here for you, for you to consume. Gobble, gobble. Gobble, gobble, gobble, you little turkey. Gobble, gobble, gobble. SPEAKER_22: Radiolab.org slash donate. SPEAKER_20: 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 slash donate. Any amount is appreciated. Radiolab.org slash donate. Come on. Join the fun. That was from our genius, deranged and slightly commitment-ophobic producer, Matt Kielty. 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 roller coaster 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 SPEAKER_09: 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. You're listening to Radiolab. Radiolab. From WNYC. 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've got 24 minutes to go. SPEAKER_14: Well, they gave me some beans for my last meal. There's no math here. SPEAKER_27: What are you talking about? SPEAKER_14: You 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 calibrate. SPEAKER_18: I've got 22 minutes to go. Well, I sent for the governor and the whole darn bunch with 21 minutes to go. SPEAKER_06: And I sent for the mayor, but he's out to lunch. I got 20 more minutes to go. SPEAKER_18: Oh my God, we're going to go all the way to one? I feel like listening to this song for three hours already. SPEAKER_06: The numbers are making it tedious. If I were him, I'd lose the numbers. You'd lose the numbers? You can't lose the numbers. You cannot lose the numbers because numbers create order in your life. I could lose the numbers. SPEAKER_18: I could survive an entire, well, my whole life without them. That's just completely ridiculous. Easily. No. Try me. SPEAKER_06: 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 buy trust. You go buy trust. He asks you how old you are. What do you say? I'm middle aged, I tell him. Listen to that. SPEAKER_19: You hear that? Suppose that you're late for an appointment or something like that. So you call up and you say, I'm going to be three minutes late, five minutes late, 10 SPEAKER_18: minutes late. I usually just wait for the call before I leave. I know that. SPEAKER_19: Which you know is true. I know it's true. So yeah, don't need them. Don't need them. SPEAKER_18: 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, Jen? I wait for the light to come on. You wait for the light? Suppose you want to call me. SPEAKER_18: Right? And you can't remember my phone number? SPEAKER_19: Two words. SPEAKER_19: Speed dial. SPEAKER_18: How many words? SPEAKER_18: Oh crap! Damn it! SPEAKER_06: You see, you gotta use numbers. SPEAKER_18: One more minute to go. And now I'm swinging and here I go. SPEAKER_18: Is that how it ends? Yeah. That's a great ending. SPEAKER_05: An ending made possible all thanks to the disciplined use of numbers. SPEAKER_18: Bleh. And that's going to be our hour. What do numbers do to us and for us? SPEAKER_05: SPEAKER_05: Don't do for us. SPEAKER_18: We've got, uh, what do we have? We have a... SPEAKER_18: We're going to have a detective story, a love story, some Nazis, and lots of numbers. SPEAKER_06: I'm Chad Abumrad. I'm Robert Kralwich. This is Radiolab. Stay with us. SPEAKER_18: Shhh. 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 numbers? SPEAKER_18: Or quantities? Or anything? What do you mean, like, can he count? I'm not asking you to 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... SPEAKER_19: Actually... Lulu, you should introduce yourself to Emil. Hi Emil. Emil, this is our producer Lulu Miller. SPEAKER_28: And by the way, Jed, while you were on paternity leave, we sent Lulu on a little mission to ask, SPEAKER_19: where does a number sense come from and how soon does it arrive in a person? SPEAKER_18: Hello? So this is the first guy I spoke to. His name is Stanislas D'Ahen. Yes, speaking. SPEAKER_19: Who's he? He is a neuroscientist in Paris. SPEAKER_18: So we've been brushing up my English for a few minutes. SPEAKER_19: 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. And he said that for a long time people thought that babies came into the world just empty. SPEAKER_09: Piaget and many other thinkers thought that there is what people have called the blank slate. SPEAKER_19: That we could only learn numbers if we were taught them. SPEAKER_18: 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 SPEAKER_09: that you can place on the head of the baby. It's a little net. SPEAKER_08: And these babies are how old? SPEAKER_09: In this case it was babies of two or three months. So he plunks the baby down in front of a computer screen. SPEAKER_08: SPEAKER_09: 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. SPEAKER_08: Eight ducks, eight ducks, eight ducks, eight ducks, eight ducks. SPEAKER_09: And what he sees is that at first the baby's brain is a little excited about getting to see ducks. SPEAKER_08: And then it's slowly the firing just kind of fizzles out. Another eight ducks, another eight ducks. SPEAKER_09: And then at some point suddenly... SPEAKER_08: 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. SPEAKER_09: Yeah, but that's not numbers. SPEAKER_08: No, I know. He's just getting started. Because Stan runs the whole thing again, starting out the same way. SPEAKER_09: Eight ducks, eight ducks, eight ducks, eight ducks. But then, instead of changing to trucks, he just changes the number. SPEAKER_09: Eight ducks, eight ducks, sixteen ducks. SPEAKER_08: And once again the baby notices the change. SPEAKER_09: But now, it's in a different part of the brain. What we call the parietal lobe. SPEAKER_09: So the suggestion is, according to Stan, that they're noticing that this is a different kind of change. SPEAKER_08: That in some sense they're noticing this is a change in quantity. SPEAKER_09: Which is very important because it means that even in newborns, SPEAKER_09: they have in their minds and in their brains an intuition of numbers. SPEAKER_09: Is he sure that they're seeing numbers or maybe they're just seeing a change in the pattern? Some, some, some, some. SPEAKER_08: More. Well, sure. What they're good at is making these gross distinctions, like eight versus sixteen. SPEAKER_09: Or say ten and twenty. SPEAKER_08: You know, and as the difference in number gets smaller and smaller, then they're not so good. SPEAKER_09: There is no baby that will ever know the difference between nine and ten. These numbers are too close together. SPEAKER_08: But it's not quite as simple as you might think. SPEAKER_09: According to Stan, the way that they're actually experiencing quantities is not just a dumbed down version of what adults do. SPEAKER_08: It's a completely different version of what adults do. They seem to care about the logarithm of the number. SPEAKER_19: The what? The logarithm of the number. You mean logarithm? Yeah. Sorry, my English is getting really bad. No, logarithms. SPEAKER_09: I don't know if this will scare the people who listen to this show. It scares me a little. SPEAKER_09: But it's actually not that bad. You can think of it in terms of ratios. First think about you. SPEAKER_08: Meaning? Us. How we think about numbers. Imagine in your head the distance between one and two. SPEAKER_09: Okay. What is that? One. SPEAKER_08: Right. SPEAKER_09: Now imagine the distance between eight and nine. 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. SPEAKER_08: One, two, three, four, five, six, seven, eight, nine, ten. But now if you were to think about it logarithmically. Like the baby. The distance between one and two is huge. SPEAKER_09: It's this vast space. SPEAKER_08: And the distance between eight and nine? SPEAKER_09: Tiny. SPEAKER_08: Why is that? Well, because one to two is doubling. SPEAKER_09: Ah. SPEAKER_08: But eight to nine. SPEAKER_09: It's a ratio of close to one. SPEAKER_18: Only one point something. SPEAKER_18: Now here's the spooky thing about this. SPEAKER_09: You might think what must happen is that eventually as we grow up SPEAKER_09: we just naturally switch from logarithmic thinking to the numbers we all know now. Uh-huh. SPEAKER_08: But this is not true. SPEAKER_09: According to Stan, if left to your own devices you'd never switch. SPEAKER_09: What do you mean? You would stay in this logarithmic world forever. So we've done this very funny experiment in the Amazon SPEAKER_08: with people from the Amazon who do not count. SPEAKER_08: Basically in their culture they do not have number words beyond five SPEAKER_09: 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 SPEAKER_08: and on the right you place nine objects. SPEAKER_09: You got that? Uh-huh. What number is exactly between one and nine? SPEAKER_09: Okay. So you'd say? Five. SPEAKER_08: Exactly. But what they put in the middle is three. Three. Wait. Help me here a little bit. So 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 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. So those are equal jumps on either side. Three is to one as nine is to three. Get it? Yeah. SPEAKER_09: Well, it's such a sophisticated way to go about thinking about it. SPEAKER_09: Yeah. To us but not to them. SPEAKER_08: That feels intuitively simply like the middle. SPEAKER_09: Dozens of people did this without hesitation. SPEAKER_08: I mean this experiment gives me chills. SPEAKER_09: These are the numbers that we all, for want of a better word, naturally feel. SPEAKER_08: At least that has been my theoretical claim for many years. And I don't quite know how to phrase this question but is there some, SPEAKER_09: is it almost like the way we think about numbers with an equal distance between one, two, three, four, five, six, seven is wrong? SPEAKER_08: SPEAKER_08: SPEAKER_09: 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. SPEAKER_09: Those are human constructions. And even someone at odds with how we feel numbers intuitively. That's right. They are. SPEAKER_08: So there is the problem. And how do we ever come to understand the numbers we know now? SPEAKER_09: That's a $64,000 question. She says it happens gradually. Don't touch the microphone. Over a couple of years. Can you count? Yeah. Let's hear it. One more quick introduction. That is Mina who you might remember from the Laughter Show. SPEAKER_08: Yes, you've met Mina before. Mina. And her mother, producer Amanda Aronchick. She will be two in a week. Two. SPEAKER_09: Yes, it's her birthday. And we've called them in today because of an experiment. SPEAKER_03: An incredibly simple set of tasks. SPEAKER_09: That Susan told me about. If you have a two year old at home, you can do these tasks. SPEAKER_09: So we're going to play a game. Money. So you put a bunch of pennies on the table. I'm going to give you some pennies, okay? Money. 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. SPEAKER_05: Mine. SPEAKER_09: That's right. That's one penny. And then you ask for two. Young two year olds, almost all can do that. Then you ask for two pennies. Now can I have two pennies? No. No? SPEAKER_03: Please can I have two? SPEAKER_09: 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. SPEAKER_03: 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. Two. So they go one, two, three, four. And you say, is that two? SPEAKER_03: They say yes. My pennies. Oh, OK. And sometimes they count. How many pennies is that? One, two, two, two. I mean, 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. SPEAKER_09: But they haven't the slightest idea what two is or three is or four is or five is. SPEAKER_03: They don't know what two means for nine months. Wow. And they're in that stage for several months. And then they become three-knowers, and then they become four-knowers. SPEAKER_03: That process takes a year and a half. In other words, even though it sounds like Mina understands numbers like we do, she's probably still living in the land of that baby man. SPEAKER_03: But there does come a moment when they finally step away. SPEAKER_03: 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. SPEAKER_03: One, two... They practice counting with children. Can you count for me? One, two, three. Four, five, six. Can you do four, five? SPEAKER_19: Seven, eight, and nine. SPEAKER_03: The last one's ten. SPEAKER_19: One, two, three. SPEAKER_09: Four, five, six. Seven, eight, and nine. Even though the kid is baffled by these numbers, they don't know what five or six or seven SPEAKER_16: SPEAKER_09: means. Four, five, six. Seven, eight, and nine. The last one's ten. One, two, three. SPEAKER_03: At some point, after enough pressure... SPEAKER_09: Count. Eight, nine, the last... Count. Ten. SPEAKER_03: They just sort of count. Throw up their hands. Count. Count. Count. SPEAKER_09: Count. Count. Count. Count. SPEAKER_09: Count. SPEAKER_03: Count. Count. SPEAKER_09: Count. SPEAKER_17: Count. Count. Count. Count. Count. Count. SPEAKER_11: Count. SPEAKER_17: Count. That bus is meant toapp playoffs. SPEAKER_09: Right, right right, right. That's one day for a mouse like everything. So when we placeEverything like that, the SPEAKER_05: She says it's a wonderful trick. SPEAKER_03: 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. SPEAKER_03: Why? Well, just the idea that to step into this world of numbers, we all had to leave something SPEAKER_09: behind. That you were born with. SPEAKER_09: But no, look what you get on the other side, though. SPEAKER_03: You get to play and have a remarkably interesting—if you like math, you get to play with deeply SPEAKER_18: abstract and beautiful thoughts. Yes, yes. And that's great. SPEAKER_09: But— 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. 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. Because sometimes the numbers for me at least feel like these hard, fussy, foreign things SPEAKER_18: that don't feel real. SPEAKER_09: They feel actually the opposite of real. But are you sure that real isn't just unfamiliar or a little strange? I mean— SPEAKER_18: Foreign, yeah, sure. Because before you could walk, when you were just a crawler, you know, toddling was kind SPEAKER_18: of unusual. And then toddling became kind of an adventure and then that became kind of usual. Yeah, but eventually you do walk. SPEAKER_09: But there's something about numbers where I feel like personally, I never learned how SPEAKER_18: 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. SPEAKER_19: Come on. Lulu? Yes. SPEAKER_19: Thank you very much for that lesson. Lulu, stay strong in your opposition to integers. Yeah. We'll be right back. SPEAKER_18: On this week's On the Media, 2023 was one of the deadliest years for journalists in SPEAKER_19: recent memory. They're like the canaries in the coal mine. All of our freedoms are at stake when so much violence is directed against journalists. Remembering the reporters we lost in Gaza, Ukraine, and all around the world on the next SPEAKER_19: On the Media. Find On the Media wherever you get your podcasts. Hello, I'm Jad Abumrad. SPEAKER_05: And I'm Robert Krelwich. This is Radiolab. SPEAKER_18: We're still talking about numbers. Now we're going to switch. SPEAKER_04: It may fatigue some of us. If you think about them a little differently, if you learn to embrace them, give them a SPEAKER_28: bit of a hug, wonderful things can happen. I'm going to introduce you to a, well, a nosy man named Mark Negrini. I'm an associate professor at the School of Business at the College of New Jersey. He 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. SPEAKER_01: South African, yes. Oh. 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. SPEAKER_18: 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 dollar forty, then you know, oh teenager with no money. Why? Wait, explain that? Because that's all the kid going for. Quite right. Sometimes I'll see ten dollars and four cents and I'll say, ah, you're meant to do ten dollars, but you were a bit slow today. SPEAKER_18: So you go to the gas pumps and they tell you all little short stories? SPEAKER_18: Yes. And his favorite story that numbers tell actually starts back in 1938. SPEAKER_25: So imagine an office in Schenectady, New York at the GE Research Laboratories. SPEAKER_25: And in that office is a man and he's sitting at his desk. Mr. Frank Benford. SPEAKER_18: And Mr. Frank Benford is a physicist, so he's doing some difficult calculations and he's SPEAKER_18: hunched over a book. SPEAKER_25: Probably actually the 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. SPEAKER_18: So if you wanted to multiply something like 145 times 3564, you could just go to this SPEAKER_25: book and look it up. So it starts with numbers you might want to multiply by one to 100 on the first pages SPEAKER_25: then 101, 102, up to 200 and 300. And the back of the book is like 900. The further you go, the higher and higher the numbers you use to multiply. SPEAKER_18: 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. SPEAKER_18: Meaning more smudgy and dark kernel, 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. SPEAKER_25: 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. SPEAKER_18: What? The more numbers that start with one or two, the numbers that start with seven, eight or nine. Just because his book is worn? 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. 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. SPEAKER_25: And everywhere he looked in all these different categories, it seemed, yes, there were more SPEAKER_18: numbers beginning with one and twos than eights and nines. SPEAKER_25: Wait, really? Oh yeah. SPEAKER_18: 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? And if you look at their account balances, follow Benford Sloan nearly perfectly. SPEAKER_25: 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 SPEAKER_18: with two? Perfectly, yes. So 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 three 12.5% of the time. SPEAKER_02: That's a big difference. Why would three be two? SPEAKER_18: I'm sorry, keep going. The poor nine would only occur as a first digit 4.6% of the time, which actually would SPEAKER_02: make the one approximately six times as likely as the nine. SPEAKER_25: And it is quite amazing. That is more than quite amazing. SPEAKER_18: 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 SPEAKER_02: that start with ones and then twos and then threes. Um, um, Robert? SPEAKER_19: So what? SPEAKER_18: Well this is not just a mathematical curiosity, Jad. No, no, no. There is something you can do with this information. SPEAKER_25: What? When Mark Negreany first ran across Benford's law, he thought, maybe I can use this law SPEAKER_19: to bust people. For payroll fraud, tax return fraud. SPEAKER_25: You thought, hey, we can use this to catch a thief? That's right. Huh? How? Well, Negreany 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 SPEAKER_02: of Benford's law, so the numbers don't begin with ones more than with twos and will begin SPEAKER_18: 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 Benford is a way to catch thieves. SPEAKER_19: The only problem was they didn't quite believe Benford's law, which means the rest of my SPEAKER_19: SPEAKER_18: 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'Doril. Daryl D'Doril. Daryl D'Doril. Daryl D'Doril. Daryl D'Doril. Daryl D'Doril. SPEAKER_18: Daryl D'Doril. Daryl D'Doril. It's an alliterative heaven. SPEAKER_17: It's like palindromic. Yeah, it's Daryl D'Doril. Oh. Daryl. Daryl. Daryl. Oh. And I should say, Daryl is, what does he call it? SPEAKER_18: He calls it... I'm a forensic accountant. I'm a 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... 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, 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. SPEAKER_25: 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. SPEAKER_18: Family shareholders, there are four of them. One of them feels like she has been misrepresented as a shareholder. SPEAKER_18: Meaning she thinks these other three guys might be stealing? Yes. Ah, I see. And I know you can't tell us what this business is doing, but is it a business? It's a regulated business. SPEAKER_11: SPEAKER_18: It's a business that each of you purchase on a regular basis through your local governmental authority. Like a trash collection or sewage. Cycling. Sure. Anyway, this one woman thought she was being cheated. SPEAKER_07: So she got an attorney involved. SPEAKER_18: The attorney requested data. SPEAKER_07: So we have seven years of income tax returns. And that's all he had. Just tax returns. Uh-huh. So he entered them all into the computer. Aggregated them, run Benfords, 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 Benfords. Mmm, very suspicious. Yes. Blew 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. SPEAKER_18: 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. SPEAKER_07: Benfords, I 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 Benfords, who's heard of them? SPEAKER_07: Who's familiar with Benfords law? SPEAKER_18: Maybe, maybe five percent of the people. SPEAKER_07: Can you just look at pennies? Just a couple observations. To me it doesn't make sense to exclude the pennies. Can you use Benfords? And they're asking, do judges allow Benfords in as evidence that suggests that someone's committed a crime? Is there case law out there that actually cite the use of Benfords law? And Daryl tells them, oh yeah. SPEAKER_18: You can use this evidence in court. SPEAKER_18: 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 Benfords 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, turned out to be a meth dealer. SPEAKER_07: The dentist suspected her of having dipped into the till, run Benfords and boom. Oh, busted. SPEAKER_29: She eventually pled. Or the guy with a 40 million dollar Ponzi scheme. SPEAKER_07: Run Benfords and boom. Well, almost boom. I mean, Benfords 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. SPEAKER_06: Pat Sam, huh. SPEAKER_18: You still haven't addressed the central mystery here. SPEAKER_01: Why in the world would there be more ones than nines? SPEAKER_18: Shouldn't they be equi... Equi-coincident? SPEAKER_07: Yes. Well, the answer is actually very complicated and deeply mathematical. SPEAKER_18: The simple answer is... Is there an answer, though? SPEAKER_07: Yes, there is an answer. Do you understand the answer? No. I mean, I understand that it has to do with logarithms and the business of doubling and SPEAKER_07: 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. SPEAKER_18: Okay, alright. But I will now take a little side step 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. SPEAKER_07: They were rarely in the same room that they are in. So let's go with our reporter Ben Calhoun and meet a crowd of mathematicians. SPEAKER_19: 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 it was a math conference called... It was on combinatorial and additive number theory. Oh, a good time had by all. SPEAKER_18: Yeah, it goes by the optimistic acronym CANT. SPEAKER_18: So I had heard that if I went to this room, there was going to be a bunch of mathematicians SPEAKER_18: from all over the place. I'm a mathematician from Sweden attending CUNY. 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. Like? They had this number. My number is two. Three. Two? Yeah. Mine is three. 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 Erdosch number. What's an Erdosch? Erdosch is a guy. So your Erdosch number is how many steps away you are from this guy, Paul Erdosch. So you're going to tell me his story? Yep. Are you ready? SPEAKER_19: Okay. SPEAKER_16: 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 Erdosch. 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. SPEAKER_16: Oh my God. Yeah, she was so terrified after that that Paul would get a fatal disease and die, that SPEAKER_16: 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 SPEAKER_16: go outside. SPEAKER_10: Also when he was one and a half, his dad was captured and put in a Soviet prisoner of war SPEAKER_10: camp for six years of his life. SPEAKER_18: So here's this kid at home without other children around. SPEAKER_16: 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. SPEAKER_18: And he also said to me that numbers became my best friends. SPEAKER_16: 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. SPEAKER_11: 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. He has to leave his family behind. SPEAKER_16: When the Nazis moved into Budapest, four of his mother's five siblings were killed. His father died as they were hurting Jews and trying to move them into the ghetto. And he only had his mother left. But she was in Hungary. SPEAKER_11: 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 SPEAKER_16: might have ended up in a mental institution or worse. But he didn't. He turned 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 SPEAKER_11: 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. SPEAKER_16: 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. SPEAKER_16: And as a houseguest, the man was an acquired taste. SPEAKER_11: 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. Erdos 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 SPEAKER_16: 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. SPEAKER_16: And it was fascinating to watch him. SPEAKER_11: I mean, there were times, like I went with him to a math conference. SPEAKER_16: He was there in his hotel room. And at one point, there were like 10 or 12 mathematicians in the room. SPEAKER_16: 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. SPEAKER_11: He was working simultaneously with all these people on different problems. Paul Erdosch wrote more papers and collaborated with more people than any other mathematician SPEAKER_11: who's ever lived. He did mathematics with anybody, even if the person was a dim bulb in the world of mathematics. SPEAKER_11: Now let me just thank you for speaking with us. SPEAKER_16: This is Paul Erdosch on his 80th birthday. The only good wish for an old man, you can say, is the easy cure of the incurable disease SPEAKER_16: of life. Surrounded by a lot of people. The mathematicians who loved him and put him up in their house. We want to express all our deep, warm feeling to you. SPEAKER_11: And I want to raise this dose for you. He was a saint. A saint? A saint. That's Joel Spencer. He's a mathematician. He was also friends with Paul Erdosch. Now that he is gone, I think of him sometimes in a religious context because he gave this SPEAKER_16: 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 while you put this enormous effort SPEAKER_11: into finding these statements. SPEAKER_16: The mathematicians will spend years of their lives trying to prove these things that, you SPEAKER_21: 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 SPEAKER_16: truth with a capital T. A truth that transcends our physical universe. SPEAKER_23: I think that's the reason why we like to talk about our connection to Paul, because SPEAKER_16: our feeling of mathematics, the feeling for what we want mathematics to be, Paul Erdosch was the embodiment of that feeling. SPEAKER_27: Somewhere along the way, mathematicians started keeping track of their connection to Paul Erdosch. And that's what Erdosch numbers actually are. If you published a paper with Paul Erdosch, your Erdosch number is one. If you published a paper with someone else and they published a paper with Paul Erdosch, then your Erdosch number is two, and so on and so on. SPEAKER_16: So this is like all the people that Paul Erdosch in some way has touched. All the people who are connected to him through their ideas. There are about 500 people with Erdosch number one and about 8,000 people with Erdosch number SPEAKER_27: two. Professor Jerry Grossman, he's at the University of Oakland in Michigan. And what he did was he took each ring of Erdosch numbers and he charted it out. Erdosch number three has about 34,000 people in it, about 84,000 with Erdosch 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, like how many people did this man in the end influence? SPEAKER_16: 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 Erdosch. And your Erdosch number is? One. SPEAKER_24: Dennis Eicorn is number two. Your Erdosch number is? Two. I wrote a paper with my advisor and the other students and she had written a paper with SPEAKER_16: a mathematician who had written a paper with Erdosch. My Erdosch number is three. Your Erdosch number is? SPEAKER_24: One. Everybody with Erdosch 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 who don't know. SPEAKER_18: SPEAKER_16: Ben Calhoun's Erdosch number is 00.5778-B-1-R-6. SPEAKER_21: Coming up, a story from our friend Steve Strogatz, the mathematician from Cornell, who tells SPEAKER_10: 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 Krolwich. Stick around. SPEAKER_11: Okay, I'm Chad Abumrad. SPEAKER_16: I'm Robert Krolwich. This is Radiolab. Our topic today is... Mathematics, mathematics, and mathematics. I suppose that is our topic. SPEAKER_11: But actually we do have a gripping story for you coming up now from our producer Soren SPEAKER_18: 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. 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. Another thing that was very unusual about him, he'd be in the middle of a calculation, SPEAKER_19: standing at the board, chalk dust all over him as usual. SPEAKER_28: Then he would space out and he'd get a look in his eye, a kind of far away 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 Fibonacci sequence. Who's Jamie Williams? Jamie Williams was a student. SPEAKER_19: 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 SPEAKER_14: now becoming part of. Yeah, so then I'm off to college and it was, it started very early. SPEAKER_14: 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. SPEAKER_02: 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 SPEAKER_14: 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. SPEAKER_02: 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. SPEAKER_14: They'll start talking about me like they used to talk about Jamie Williams. SPEAKER_02: 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. SPEAKER_14: 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. SPEAKER_02: 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. SPEAKER_02: 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. SPEAKER_14: He would talk about them a little bit. And I feel embarrassed. SPEAKER_02: 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 or, 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. SPEAKER_14: But then in later years, he would almost pointedly ask me things like there was a time when he SPEAKER_02: said that rumor has it that you're engaged. We wish you the best if this is true. And guess what? My letter back to him, I didn't say anything. SPEAKER_14: 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 SPEAKER_02: my fiance. You know, like in that time, the letters were kind of refuge from all that. That is, we could go into this pristine world of math where things are simple and logical and well ordered. SPEAKER_14: 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. SPEAKER_02: 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. 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. 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, shaky. SPEAKER_14: And I knew what that looked like because my dad wrote like that when he had Parkinson's. So I thought, what's this? SPEAKER_02: And I opened the letter and the first sentence is, eek, I just had a mild stroke. SPEAKER_02: 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. SPEAKER_14: He heard about it from someone else and immediately wrote to me how, you know, that he and his SPEAKER_14: 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 SPEAKER_02: ago and it kept nagging at me. SPEAKER_02: Why won't you talk to him? Beckoning and 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 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, OK, 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 SPEAKER_14: porch. SPEAKER_02: Does that work? Hello. And so we're eating and recording. 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 Hank in his typical way. 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. So Hank 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. Like what happened? How did your son die? They have a lot of work. SPEAKER_02: They're just trying to make guys put in extra hours to pay guys extra hours. There was a fidgeting feeling inside me. Paying benefits and all of the other stuff. I don't know. 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. SPEAKER_29: But I wanted. I did. SPEAKER_02: I asked what. SPEAKER_26: I didn't really know him either. SPEAKER_02: But I know that he died very young. SPEAKER_26: And I know what happened. You know what happened to Marshall? SPEAKER_02: Well. SPEAKER_26: We, you know. SPEAKER_02: That's something we don't really. SPEAKER_26: Do you want to talk about that? Ah. 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. SPEAKER_29: SPEAKER_26: And he. He did. He had a wonderful 27 years. Music was going to be. It was so beautiful. And so. 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 SPEAKER_02: and just turn to a page. 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. Golly. This guy's got a multi-channel mind that I wish I had. You know. He talked about what a great life he had in his 27 years. SPEAKER_26: Even in his waning moments he'd stay up all night long playing the piano. SPEAKER_26: Just the house was just filled with beautiful music. SPEAKER_02: And he had made plans to get a job at the New England Conservatory and things like that. The fates were wrong for him. SPEAKER_29: Oh yeah. SPEAKER_02: We miss him. SPEAKER_26: Was that. I mean. In that moment did it change the way you see him? SPEAKER_29: Well. SPEAKER_26: I have to tell you how that day ended. SPEAKER_13: So we talked more and I asked him at one point do you think Marshall had a religious feeling SPEAKER_26: and he said. Yeah. I think he felt close to having to come to terms with somebody out there. SPEAKER_02: That was a good thing. SPEAKER_26: That I think he went peacefully. SPEAKER_26: 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. I'd 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 SPEAKER_02: Long Island Sound. In fact we were talking about a math problem about waves about Fourier analysis. SPEAKER_26: Which is really about 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. SPEAKER_26: 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 SPEAKER_13: 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. SPEAKER_02: Thanks to Soren Wheeler our producer who interviewed Steve and produced that story. SPEAKER_02: Thanks to Steve Strogatz who has a book out now which tells this very story called the SPEAKER_26: calculus of friendship. I'm Jad Abumrad. Three seconds to go. SPEAKER_02: You are Robert Kowitch. Bye. Radiolab was created by Jad Abumrad and is edited by Soren Wheeler. Lulu Miller and Latif Nasser are our co-hosts. SPEAKER_14: Dylan Keefe is our director of sound design. Our staff includes Simon Adler, Jeremy Blum, Becca Bressler, Ekedi Foster-Keys, W. Harry Fortuna, David Gable, Maria Paz Gutierrez, Sindhu Niana Sambandhan, Matt Kielty, Annie SPEAKER_02: McEwen, Alex Neeson, Sara Khari, Alyssa Jung Perry, Sarah Sandback, Arianne Wack, Pat Walters, SPEAKER_14: and Molly Webster. Our fact checkers are Diane Kelly, Emily Krieger, and Natalie Middleton. Yeah. SPEAKER_02: I was on this. SPEAKER_02: Hi, my name is Michael Smith. SPEAKER_18: 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, Simons Foundation Initiative, and the John Templeton Foundation. Additional support for Radiolab was provided by the Alfred P. Sloan Foundation. SPEAKER_00: SPEAKER_12: