[RADIOLAB INTRO]

ROBERT KRULWICH: Jad?

JAD ABUMRAD: Yes.

ROBERT: Listen to this just for a second.

[ARCHIVE CLIP, Johnny Cash: Well, they're building a gallows outside my cell. I got 25 minutes to go. And the whole town's waiting ...]

JAD: Is that Johnny Cash?

ROBERT: Yes, it's Johnny Cash and he's singing a song about the deep importance of mathematics in your life.

[ARCHIVE CLIP, Johnny Cash: I got 24 minutes to go. Well, they gave me some beans for my last meal. I got 23 minutes to go]

JAD: There's no math here. What are you talking about?

ROBERT: No, there's a lot of math here because, you see, what he's doing is he's moving to his extinction it seems, but he's being very careful to calibrate.

[ARCHIVE CLIP, Johnny Cash: 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.]

JAD: Oh my God. We're gonna go all the way to one?

ROBERT: [laughs]

JAD: I feel like I've listened to the song for three hours already. The numbers are making it tedi—if I were him, I'd lose the numbers.

ROBERT: You'd lose the numbers?

JAD: Yes, this is stress.

ROBERT: You can't lose the numbers. You cannot lose the numbers because numbers create order in your life.

JAD: I could lose the numbers. I could survive an entire—well, my whole life without them.

ROBERT: That's just completely ridiculous.

JAD: Easily.

ROBERT: No.

JAD: Please, try me. Try me.

ROBERT: All right, let me just ask you something very simple.

JAD: Yeah.

ROBERT: 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?

JAD: No numbers required. If I hand him the bill, he hands me some change. I just go by trust.

ROBERT: You go by trust?

JAD: Yeah.

ROBERT: He asks you how old you are? What do you say?

JAD: I'm middle-aged, I tell him.

ROBERT: [laughs]

[ARCHIVE CLIP, Johnny Cash: With 11 more minutes to go.]

ROBERT: Listen to that. You hear that? Suppose that you're late for an appointment or something like that.

JAD: Yeah. Yeah.

ROBERT: So you call up and you say, "I'm going to be three minutes late, five minutes late, 10 minutes late."

JAD: I usually just wait for the call before I leave.

ROBERT: [laughs]

ROBERT: I know that!

JAD: Which you know it's true.

ROBERT: I know it's true.

JAD: So yeah, don't need them. Don't need them.

ROBERT: Your test. You're taking a test in school. You get a 98, you get a 52. You don't care?

JAD: Pass, fail.

ROBERT: How much gas is in your car, Jad?

JAD: I wait for the light to come on.

ROBERT: [laughs] Suppose you want to call me, right?

JAD: Yeah.

ROBERT: And you can't remember my phone number.

JAD: Two words.

ROBERT: What?

JAD: Speed dial.

ROBERT: How many words?

JAD: Oh, crap! Crap! Damn it! [laughs]

ROBERT: You see? You gotta use numbers.

[ARCHIVE CLIP, Johnny Cash: I got two more minutes to go. I can see the buzzards, I can hear the crows. One more minute to go. And now I'm swinging and here I go ...]

JAD: Is that how it ends?

ROBERT: Yeah.

JAD: That's a great ending!

ROBERT: An ending made possible all thanks to the disciplined use of numbers.

JAD: Death!

ROBERT: And that's gonna be our hour: what do numbers do to us and for us?

JAD: Or don't do for us. Forget the—what do we have? We have a ...

ROBERT: We're gonna have a detective story, a love story, some Nazis and lots of numbers.

JAD: I'm Jad Abumrad.

ROBERT: And I'm Robert Krulwich.

JAD: This is Radiolab. Stay with us.

[infant fussing]

ROBERT: So Jad, do you want to introduce this—this person?

JAD: This is—this is a little Amil.

ROBERT: Hi, Amil. So how old is he?

JAD: He's hungry right now. He's about 30—Karla, how old is he? 36?

JAD: At the time of this recording he is 36 days old.

ROBERT: [laughs] Well, I mean, you must have wondered: do you think he has any sense at all of numbers or quantities or anything?

JAD: What do you mean, like, can he count? Is that what you're asking?

ROBERT: I'm not asking if he can count, but do you think he has a—I don't know, a numeric sense at all?

JAD: Do I think he has a numeric—no. No, I don't think he knows that that is his hand that he's chewing.

ROBERT: [laughs]

JAD: I don't think there are any numbers in there. In fact, I'm pretty sure there aren't.

LULU MILLER: Well ...

JAD: [laughs]

LULU: ... actually ...

ROBERT: Lulu, you should introduce yourself to Amil.

LULU: Hi, Amil.

JAD: Amil, this is our producer, Lulu Miller.

ROBERT: And by the way Jad, while you were on paternity leave, we sent Lulu on a little mission to ask where does a number sense come from, and how soon does it arrive in a person?

STANISLAS DEHAENE: Hello?

LULU: So this is the first guy I spoke to. His name is Stanislas Dehaene.

STANISLAS DEHAENE: Yeah, speaking.

JAD: Who's he?

LULU: He is a neuroscientist in Paris.

STANISLAS DEHAENE: We've been brushing up my English for a few minutes.

LULU: Currently, he's like the godfather of this research.

JAD: Really?

LULU: He wrote a whole book called The Number Sense that talks all about what babies understand.

STANISLAS DEHAENE: Mm-hmm. Yeah.

LULU: And he said that for a long time, people thought that babies came into the world just empty.

STANISLAS DEHAENE: Piaget and many other thinkers thought that there is what people have called the blank slate.

LULU: That we could only learn numbers if we were taught them.

JAD: Yeah, that's what I think.

STANISLAS DEHAENE: But now we know it's just completely wrong. [laughs]

JAD: And how do they know this?

LULU: Well, experiments.

STANISLAS DEHAENE: Right.

LULU: Lots and lots of baby experiments.

STANISLAS DEHAENE: The equipment we have is a set of little sponges which contain a very small electrode that you can place on the head of the baby. It's a little net.

LULU: And these babies are how old?

STANISLAS DEHAENE: In this case it was babies of two or three months.

LULU: So he plunks the baby down in front of a computer screen, and on the screen are a bunch of little pictures.

STANISLAS DEHAENE: Like, you know, little ducks, for instance. It's always a set of eight of the same object, so you do eight ducks, eight ducks, eight ducks, eight ducks.

LULU: And what he sees is that at first, the baby's brain is a little excited about getting to see ducks.

ROBERT: Mm-hmm.

LULU: And then slowly the firing just kind of fizzles out.

ROBERT: Another eight ducks. Another eight ducks. [snores]

STANISLAS DEHAENE: And then at some point, suddenly ...

LULU: He changes it to ...

STANISLAS DEHAENE: Eight trucks.

LULU: And he sees a spike in brain activity.

STANISLAS DEHAENE: In the what we call the temporal lobe.

LULU: Meaning the baby can notice that change.

ROBERT: Yeah, but that's not numbers.

LULU: No, no, no, I know. He's just getting started, because Stan runs the whole thing again, starting out the same way.

STANISLAS DEHAENE: Eight ducks, eight ducks, eight ducks, eight ducks.

LULU: But then instead of changing to trucks, he just changes the number.

STANISLAS DEHAENE: Eight ducks, eight ducks, eight ducks, 16 ducks.

LULU: And once again, the baby notices the change, but now it's in a different part of the brain.

STANISLAS DEHAENE: What we call the parietal lobe.

LULU: So the suggestion is, according to Stan, that they're noticing that this is a different kind of change, that in some sense, they're noticing this is a change in quantity.

STANISLAS DEHAENE: Which is very important because it means that even in newborns, they have in their minds and in their brains an intuition of numbers.

JAD: Is he sure that they're seeing numbers, or maybe they're just seeing a change in the pattern?

ROBERT: Some, some, some, some, more.

JAD: Yeah.

LULU: Well, sure. What they're good at is making these gross distinctions like eight versus 16.

STANISLAS DEHAENE: Or, say, 10 and 20.

LULU: You know, and as the difference in number gets smaller and smaller then they're not so good.

STANISLAS DEHAENE: There is no baby that will ever know the difference between nine and 10. These numbers are too close together.

LULU: But it's not quite as simple as you might think. According to Stan ...

STANISLAS DEHAENE: What is most extraordinary, I think ...

LULU: ... 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.

STANISLAS DEHAENE: Mm-hmm. They seem to care about the logarithm of the number.

JAD: The what?

STANISLAS DEHAENE: The logarithm of the number.

ROBERT: He means logarithm.

LULU: Yeah.

STANISLAS DEHAENE: Sorry. My English is getting really bad. [laughs]

LULU: No, logarithms.

STANISLAS DEHAENE: I don't know if this will scare your listeners to the show ...

LULU: It scares me a little, but it's actually not that bad.

STANISLAS DEHAENE: You can think of it in terms of ratios.

LULU: First think about you.

ROBERT: Meaning?

LULU: Us. How we think about numbers.

ROBERT: Okay.

LULU: Imagine in your head the distance between one and two.

ROBERT: Okay.

LULU: What is that?

ROBERT: One.

LULU: Right. Now imagine the distance between eight and nine.

ROBERT: One also.

LULU: They feel like the same distance from each other.

ROBERT: Yeah.

LULU: Well, that's because we think of numbers in these discrete, ordered chunks.

STANISLAS DEHAENE: One, two, three, four.

LULU: But now if you were to think about it logarithmically …

STANISLAS DEHAENE: Like the baby.

LULU: ... the distance between one and two is huge! It's this vast space. And the distance between eight and nine? Tiny.

ROBERT: Why is that?

LULU: Well, because one to two is doubling.

ROBERT: Ah, interesting.

LULU: But eight to nine ...

STANISLAS DEHAENE: It's a ratio of close to one. Like, only one point something.

ROBERT: Huh.

LULU: 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.

ROBERT: Uh-huh?

STANISLAS DEHAENE: But this is not true.

LULU: According to Stan, if left to your own devices, you'd never switch.

ROBERT: What do you mean?

LULU: You would stay in this logarithmic world forever.

STANISLAS DEHAENE: So we've done these very funny experiments in the Amazon with people from the Amazon who do not count. Basically, in their culture they do not have number words beyond five and they don't recite these numbers, so what we found is that these people still think of numbers in a logarithmic way—even the adults. What that means is that if you give them a line, and on the left you place one object and on the right you place nine objects ...

LULU: You got that?

ROBERT: Uh-huh.

LULU: And he asked them, "What number is exactly between one and nine?"

ROBERT: Okay.

LULU: So you'd say?

ROBERT: Five.

LULU: Exactly. But ...

STANISLAS DEHAENE: What they put in the middle is three.

ROBERT: Three?

LULU: Wait, help me here a little bit. So the ...

STANISLAS DEHAENE: The property of the logarithm is that each time you multiply the number, you move by a constant displacement ...

LULU: Okay, so this is a bit tricky, but the gist is if you're thinking in ratios ...

ROBERT: Uh-huh?

LULU: And you're starting at one, then you multiply by three to get to three.

ROBERT: Right.

LULU: And then—hey, hey!—we multiply by three again to get to nine.

ROBERT: I see. Oh!

LULU: So those are equal jumps on either side.

ROBERT: Huh.

STANISLAS DEHAENE: 3 is to 1 as 9 is to 3.

LULU: Get it?

ROBERT: Yeah. Well, it's such a sophisticated way to go about thinking about it.

LULU: Yeah, to us but not to them. That feels intuitively simply like the middle. Dozens of people did this without hesitation.

ROBERT: Huh.

LULU: I mean, this experiment gives me chills. These are the numbers that we all, for want of a better word, naturally feel.

STANISLAS DEHAENE: Or at least that has been my theoretical claim for many years.

LULU: Hmm. And I don't quite know how to phrase this question but is there some—is it almost like the way we think about numbers with an equal distance between one, two, three, four, five, six, seven is wrong?

STANISLAS DEHAENE: Mmm, you know, I wouldn't go too far.

LULU: [laughs]

LULU: But then I talked to Susan Carey.

SUSAN CAREY: I'm a professor of psychology at Harvard University.

LULU: And she said that numbers as we think of them today are certainly made up.

SUSAN CAREY: Those are human constructions.

LULU: And even somewhat at odds with how we feel numbers intuitively.

SUSAN CAREY: That's right they are. [laughs] So there is the problem.

LULU: Then how do we ever come to understand the numbers we know now?

SUSAN CAREY: That's a $64,000 question.

LULU: She says it happens gradually.

AMANDA ARONCZYK: Okay, don't touch the microphone.

MINA: Microphone?

AMANDA: Yes.

LULU: Over a couple of years.

AMANDA: Can you count?

MINA: Yeah.

AMANDA: Let's hear it.

MINA: One, two, three.

LULU: One more quick introduction. That is ...

MINA: Mina.

LULU: Who you might remember from the Laughter show.

AMANDA: You've met Mina before.

LULU: And her mother, producer Amanda Aronczyk.

AMANDA: She will be two in a week.

MINA: Two.

AMANDA: Yes. It's her birthday.

LULU: And we've called them in today because of an experiment ...

SUSAN CAREY: An incredibly simple set of tasks.

LULU: ... that Susan told me about.

SUSAN CAREY: If you have a two-year-old at home, you can do these tasks.

AMANDA: Honey, so we're gonna play a game, okay?

MINA: Money?

SUSAN CAREY: So you put a bunch of pennies on the table.

AMANDA: I'm gonna give you some pennies, okay?

MINA: Money.

AMANDA: Just a second. Let mommy get them for you.

SUSAN CAREY: And you say to the child, "Can you give me one penny?"

AMANDA: Can I have one penny?

SUSAN CAREY: And the child very carefully picks up one and hands it to you.

MINA: One.

AMANDA: That's right, that's one penny. Thank you!

SUSAN CAREY: Young two-year-olds almost all can do that. Then you ask for two pennies.

AMANDA: Now can I have two pennies?

MINA: No.

AMANDA: No? Please can I have two?

SUSAN CAREY: It doesn't matter what you ask for, they just pick up a handful and hand them to you.

ROBERT: [laughs]

AMANDA: That's more than two pennies. You have like one, two, three, four.

SUSAN CAREY: And so they've given you four pennies, and you say, "Is that two?" And they say ...

MINA: Yeah.

SUSAN CAREY: [laughs] Right? Then you say ...

AMANDA: Can you count how many pennies you have?

SUSAN CAREY: "Well, can you count to make sure?"

AMANDA: How many pennies is that?

MINA: Two.

SUSAN CAREY: So they go one, two, three, four, and you say, "Is that two?" They say, "Yes."

MINA: Yes. More pennies.

AMANDA: Okay, okay.

SUSAN CAREY: And sometimes they count ...

AMANDA: How many pennies is that?

SUSAN CAREY: One, two, two, two ...

MINA: One, two, two.

SUSAN CAREY: So it's like they somehow know that all of their other words contrast with one in meaning. That is they're giving you a number and they're giving you a number more than one, but they haven't the slightest idea what two is or three is or four is or five is. And they don't know what two means for nine months.

LULU: Wow!

SUSAN CAREY: And they're in that stage for several months, and then they become three knowers and then they become four knowers. That process takes a year and a half.

LULU: In other words, even though it sounds like ...

MINA: Seven, eight, nine, ten.

LULU: ... Mina understands numbers like we do ...

AMANDA: Good job!

LULU: ... she's probably still living in the land of that baby math.

MINA: One, two, ten.

ROBERT: [laughs]

LULU: But, there does come a moment when they finally step away.

AMANDA: Can you sing that song?

LULU: And it happens right when the kid's about three and a half years old.

SUSAN CAREY: What they do, I think—this is speculative, but ...

LULU: After years of everyone around them saying ...

SUSAN CAREY: Count.

AMANDA: Can you count how many pennies you have?

SUSAN CAREY: This is something parents do.

AMANDA: One, two, three, four

SUSAN CAREY: They practice counting with children.

AMANDA: Can you count to three?

CHILD: One, two, three.

AMANDA: Can you do four, five?

CHILD: Two, one.

LULU: Even though the kid is baffled by these numbers, and they don't know what five or six or seven means, at some point, after enough pressure, they just sort of throw up their hands and believe the song.

SUSAN CAREY: That's a very bold leap that children must make. And so now what five means for the child is: one more than four.

CHILD: Five, six, seven.

SUSAN CAREY: And what six means is one more than five. And now you've got integers.

CHILD: Eight, nine, ten.

LULU: We're all sort of like relying on this song.

SUSAN CAREY: Yeah. One, two, three.

LULU: We just one day decided, "Okay, that means something."

SUSAN CAREY: That's right. So this is a trick.

ROBERT: What does she mean by that, "trick?" Sounds almost like a dirty word.

LULU: [laughs] Well, she doesn't use it like a dirty word. She says it's a wonderful trick.

SUSAN CAREY: The point is once you have that trick, you build on that.

LULU: And that opens up the whole world of mathematics to you, and we can build buildings and launch rockets into space.

SUSAN CAREY: And no other animal has invented that trick.

LULU: But I can't help feeling there's something about this that's a little bit sad.

ROBERT: Why?

LULU: Well, just the idea that to step into this world of numbers, we all had to leave something behind.

ROBERT: That you were born with.

LULU: Yeah.

ROBERT: But look what you get on the other side, though. You get—you get to play and have remarkably interesting—if you like math, you get to play with deeply abstract and beautiful thoughts.

LULU: Yes, and that's great, but ...

ROBERT: 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. It's just like different talents, that's all.

JAD: But Robert, I think I know what Lulu's talking about. I mean, it's refreshing somehow to know that the numbers that we use day to day are somehow made up, 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.

ROBERT: But you're sure that real isn't just unfamiliar, a little strange? I mean ...

JAD: Foreign. Yeah, sure.

ROBERT: Because before you could walk, when you were just a crawler, you know, toddling was kind of unusual. And then toddling became kind of an adventure, and then that became kind of usual.

JAD: Yes, but eventually you do walk. But there's something about numbers where I feel like personally, I never learned how to walk. And I think there's a lot of people listening right now who probably feel that way about numbers.

ROBERT: What does that mean though?

JAD: We're just logarithmic people.

ROBERT: Come on! Lulu?

LULU: Yes?

ROBERT: Thank you very much for that lesson.

JAD: Lulu, stay strong in your opposition to integers.

ROBERT: Yeah. [laughs] We'll be right back.

AMANDA: Hello there. This is Amanda and Mina, and we are gonna read the credits. Okay Mina, you ready? Can you say "Radiolab"?

MINA: Radiolab.

AMANDA: Radiolab is funded in part by the Alfred P. Sloan Foundation, the Corporation for Public Broadcasting and the National Science Foundation. Mina, can you say "National Science Foundation?"

AMANDA: Radiolab is produced by WNYC and distributed by National Public Radio. Good job. Okay.

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