February 20, 2018
‘Is it opposite day?’ and other mind-bending paradoxes
Subscribe
Think about it: the answer to the question “Is it opposite day?” will always be no. It’s a head-scratcher. So how do you figure out if it is, in fact, opposite day?
We talk to two philosophers who walk us through how questions like these can bend and twist the truth — and our minds. We learn about the sinister-sounding “Liar Paradox.” And we find out that it’s not only our brains that use logic, it’s used by the machines all around us too.
Plus: A brand new mystery sound and an answer to the question: How do erasers erase?
Audio Transcript
Download transcript (PDF)
MARY MOORE: You're listening to Brains On, where we're serious about being curious.
Brains On is supported in part by a grant from the National Science Foundation.
MOLLY BLOOM: So Mary, what would the opening of Brains On sound like if it were opposite day right now?
MARY MOORE: Maybe something like this. [PLAYS PIANO]
MAN: You are not listening to Brains On. This is not Molly Bloom. And we are not curious about anything. That tree over there, boring.
My phone? I don't care how this thing works. That dog, it is not super cute. And I'm definitely not interested in finding out why it wants to smell all the things.
Nope. You want a mystery sound? Too bad. Here's the sound of me turning off this microphone because there are no questions to answer and no one interesting to talk to. Bye.
MEN: Ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, brains off.
[COW MOOING]
MOLLY BLOOM: I'm not sure I'd want to listen to that show.
MARY MOORE: Well, at least the episodes would be really short.
[MUSIC PLAYING]
MOLLY BLOOM: Luckily, it's not opposite day here on Brains On, but opposite day is on our minds.
MARY MOORE: We're going to talk all about logic and paradoxes today.
MOLLY BLOOM: And we're going to have some fun tying our minds in knots.
MARY MOORE: Keep listening.
[MUSIC PLAYING]
MOLLY BLOOM: You're listening to Brains On for American Public Media. And here with me today is 15-year-old Mary Moore from Lake Shore, Minnesota. Hi, Mary.
MARY MOORE: Hi, Molly.
MOLLY BLOOM: And this episode was inspired by a question that you sent in to us. Let's hear it.
MARY MOORE: I'm wondering how I can determine whether today is opposite day or not. When I asked if today is opposite day, my teacher said no. If it is opposite day and he says no, then it really must be opposite day.
But if it isn't opposite day, his answer would be the same. Please help me figure out this paradox. Sincerely, Mary Moore.
MOLLY BLOOM: That is a great question. So Mary, what made you think of this question?
MARY MOORE: Well, we play opposite day a lot at our house. We just like-- everything's opposite.
MOLLY BLOOM: So give me an example of what happens when it's opposite day at your house.
MARY MOORE: So like, if I asked, do you want a glass of water? and you say yes, that really means no.
MOLLY BLOOM: So it's kind of just responding to questions with the opposite of what you really mean? You don't like wear your clothes like your pants on your head or something?
MARY MOORE: Yeah, or like sometimes you like walk backwards or whatever.
MOLLY BLOOM: Excellent. So is it a thing that you ever do at school?
MARY MOORE: Sometimes. Like, we played at recess or whatever.
MOLLY BLOOM: What is it like when you do it at recess?
MARY MOORE: Especially on the playground, it's hard to like use the monkey bars backwards.
MOLLY BLOOM: Yeah, I don't think your arms run that way.
MARY MOORE: Yeah.
MOLLY BLOOM: So is it hard to understand what you and your family are talking about when it's opposite day at your house?
MARY MOORE: Well, sometimes because if you ask the question backwards of what you really mean to ask and then they answer it backwards, it gets all confusing.
MOLLY BLOOM: Like, what are we talking about here? So how would you describe what a paradox is?
MARY MOORE: It's a puzzle that's confusing and doesn't really have an answer.
MOLLY BLOOM: Excellent. Well, we're going to dive more into paradoxes in a little bit. But first, we're going to look at opposite day, and we're going to help you untangle the question that you sent us.
How do you find out whether it's opposite day or not? And here to help us is Justin Khoo, who's a professor of philosophy at MIT.
JUSTIN KHOO: This is a great question.
[MUSIC PLAYING]
Mary wants to know whether it's opposite day. Opposite day is the day when you say the opposite of what you really mean. Suppose Mary in wanting to know this asks her teacher who does know whether it's opposite day--
MARY MOORE: Is it opposite day?
JUSTIN KHOO: OK. So now we can think about whether the answer to this question could really tell Mary whether it's opposite day or not. So let's suppose that it is opposite day. Well, then Mary's teacher should say no.
TEACHER: No, it's not opposite day.
JUSTIN KHOO: And that's because since it is, in fact, opposite day, Mary's teacher should say the opposite of what she really means. But let's suppose that it's not opposite day. Then Mary's teacher should also say no because she should just say what she means and it's not opposite day. So no would be the right answer.
And so it seems like the question is it opposite day can only have the answer no, in which case Mary could never find out whether it's opposite day by asking someone who knows.
MOLLY BLOOM: Does that clear things up?
MARY MOORE: No.
JUSTIN KHOO: One way to try to find out whether it's opposite day is to ask someone who knows a different question. So one kind of question that you could do to find out whether it's opposite day is you can ask someone a question that you know the answer to and that you know-- that they also know the answer to. Let's suppose for a second that Mary knows that her name is Mary and that she also knows that her teacher knows that her name is Mary. Then Mary could ask her teacher--
MARY MOORE: Ms. Brown, is my name Mary?
JUSTIN KHOO: OK. So now we can sort of see how the answer to this question might allow Mary to infer whether it's opposite day. So suppose that the teacher says--
TEACHER: Well, yes, your name is Mary.
JUSTIN KHOO: To the question, is my name Mary? Well, then Mary can infer that it is not opposite day. Because if it were opposite day, the teacher would have had to say no because then she would have had to say the opposite of what she meant.
But if the teacher says--
TEACHER: No, your name is not Mary.
JUSTIN KHOO: Then Mary can infer it really is opposite day.
TEACHER: Your name is not Mary. This isn't school. And class can throw away their homework. We're going to study on the playground and climb on our desks.
Oh, and the cafeteria food is actually delicious.
STUDENTS: Yay.
JUSTIN KHOO: Mary, to me, seems like a natural philosopher. The opposite day question and puzzle is exactly the kind of issue that animates philosophers and gets us thinking and excited.
[MUSIC PLAYING]
MOLLY BLOOM: So Mary, what do you think of Justin's solution?
MARY MOORE: I think that's a pretty good solution.
MOLLY BLOOM: So you can just say, is my name Mary, and you'll have your answer. So now, in his answer, he talked about inferring. Do you know what inferring is?
MARY MOORE: Yeah, it's kind of like using the stuff to make a good guess of what it means.
MOLLY BLOOM: Exactly. So this is how Justin describes it.
JUSTIN KHOO: Inferring is something when what you do when you move from one piece of information to another using the information that you have to gain new information. So one type of inference is inductive inference.
MOLLY BLOOM: This kind is where you gather specific examples or data and then make a conclusion based on it. So let's say every morning your cat wakes up at 5:30, wanting to be fed.
[MEOWING]
[ALARM]
So an inductive inference would be, my cat will wake me up at 5:30 tomorrow.
[MEOWING]
[ALARM]
This is the kind of inference that scientists make when they collect data, and then they make a conclusion.
JUSTIN KHOO: There's another kind of inference which is called a deductive inference.
MOLLY BLOOM: This is when the conclusion is directly contained in the information that you have in front of you. If the starting point is true, then the conclusion is guaranteed to be true. Here's an example,
You know your cat only eats food that he likes. Your cat eats the food that you give him. So you can infer that your cat likes that food.
This kind of way of looking at situations is called logic.
JUSTIN KHOO: One answer about what logic is is it's an attempt to find the fundamental laws of reality.
MOLLY BLOOM: We're going to get real with reality in just a little bit. But first, it's time for the mystery sound.
GIRL: Shh. Mystery sound.
MOLLY BLOOM: Are you ready?
MARY MOORE: Yes.
MOLLY BLOOM: All right. Here it is
[MYSTERY SOUND]
Any guesses?
MARY MOORE: Sort of sounds like gears turning. Maybe paper shuffling.
MOLLY BLOOM: OK. Something turning or shuffling. Excellent. Well, we are going to hear it again in just a little bit.
And while you puzzle through that, we have a special guest here.
MARY MOORE: Agustin Rayo is a philosophy professor at MIT who specializes in paradoxes. Thanks for being here today.
AGUSTIN RAYO: Oh, I'm so delighted to be here.
MARY MOORE: What makes a paradox a paradox?
AGUSTIN RAYO: It's about having some principles which all seem true and a bit of reasoning which seems like it's got to be right and the conclusion that has got to be wrong. And then what's interesting is that because a valid piece of argument can't take you from something true to something false, you know that you've made a mistake somewhere. And the challenge is to find the mistake.
MARY MOORE: What's your favorite paradox or puzzle?
AGUSTIN RAYO: There's a paradox which I think is the hardest paradox ever. And that's the liar paradox. And it's really, really easy to state.
So you take a sentence that says this very sentence is false.
MOLLY BLOOM: OK, this is Molly interrupting for one minute because this is a little confusing. A liar's paradox is just one sentence. And that one sentence is, this sentence is false.
That's the sentence. This sentence is false. So the sentence is saying that itself is false, OK? OK.
AGUSTIN RAYO: And then you ask, OK. So is that sentence true, or is it false? And it's very hard to know what to say because if it's true, then what it says is true, but it says it's false.
And if it's false, then what it says is not true, but it says it's false. So it's not false. So it's true.
MOLLY BLOOM: So no one's ever been able to find like a definitive answer to that.
AGUSTIN RAYO: That's right. So many, many people, myself included, have suggested ideas about how one might go about it. But very often, the only person who thinks that's the right solution is the person who wrote it. And that's definitely true of me. I'm the only person in the world who thinks that my solution is the right one.
MOLLY BLOOM: What was the solution you came up with?
AGUSTIN RAYO: Well, my solution is really radical. What I think is that language doesn't come with ready-made meanings. So it's not like we use words. In each word is like a little box that has its meaning. Instead, what happens is that as we communicate, we build meanings on the fly.
And what goes on in the liar paradox is that the reasons that-- the right meaning cannot be built. So it's a little bit like saying, OK, let's play this game. Let's play the game of doing something that Agustin is not doing right now.
So Mary can win that game. Maybe she'll just scratch her head. And I'm not scratching my head. So she'll win. But I can never win the game because if I try scratching my head, then I'll be doing something that I am doing.
And if I try jumping up and down, then again, I'll do something that I am doing. So I can never win. And that's kind of what happens in the liar paradox. We try to build meanings, but it's built into the game we're trying to play that I'll never get to build this table meaning.
MOLLY BLOOM: And so how does your brain feel when you think about these things, Mary?
MARY MOORE: It's just like-- it's like it messes it up. It's like twisting and pulling it.
MOLLY BLOOM: And you really like math?
MARY MOORE: Yeah, I do.
MOLLY BLOOM: So how does math make your brain feel?
MARY MOORE: I don't know. Math is kind of simple for me. It's just like it all makes sense.
MOLLY BLOOM: That's very clean, and this is like the opposite.
MARY MOORE: Yeah, it's messy.
AGUSTIN RAYO: Well, I just wanted to say to Mary that sometimes when you feel like your mind is being stretched and turned, that's only making it stronger. It's like exercising a muscle. Then at first, it hurts a little bit. Then you have a bigger muscle. And you can do it.
You can use it to do really cool things. So I say keep working that mind.
MARY MOORE: Thanks for answering my questions.
AGUSTIN RAYO: Oh, of course. Any time.
MARY MOORE: Yeah, I had lots of fun. Thank you.
MOLLY BLOOM: Do you have questions or paradoxes you'd like us to tackle on Brains On--
MARY MOORE: A mystery sound to share--
MOLLY BLOOM: Or maybe you want to send us a drawing or a high five?
MARY MOORE: Then email us. The address is hello@brainson.org.
MOLLY BLOOM: And if you share your ideas with us, you'll join Ben on the Brains Honor Roll. He sent us this question.
BOY: How do erasers erase pencil marks?
MOLLY BLOOM: Oh, erasers? That's easy. I mean, everybody knows that it's--
MARY MOORE: Listen for the answer to that question during our moment of um at the end of the show.
MOLLY BLOOM: And the latest group to be added to the Brains Honor Roll.
MARY MOORE: Keep listening.
[MUSIC PLAYING]
MAN: You are listening to Brains Off, where we half-heartedly sit around staring at stuff. I'm not Molly Bloom. So, yeah.
Mm-hmm. Yep. Have you ever wondered about what the universe is made of and what time is and how weather patterns form and how animals communicate and why we dream?
Yeah, me neither. I'm just going to sit here and stare at this spider's web and not wonder how that eight-legged little creep managed to make something so beautiful. I mean, boring.
Well, that's it for this episode. Don't tune in next week. Seriously, don't. It's going to be really boring.
Whatever. Bye.
MEN: Ba, ba ba, ba, ba, ba, ba, ba, ba, Brains Off.
[COW MOOING]
MARY MOORE: You're back listening to Brains On from American Public Media. I'm Mary.
MOLLY BLOOM: And I'm Molly. OK, Mary. Are you ready to hear the mystery sound again?
MARY MOORE: Yes.
MOLLY BLOOM: All right. Here it is.
[MYSTERY SOUND]
Any new thoughts?
MARY MOORE: It kind of sounds like a marble going down like a marble track.
MOLLY BLOOM: Hmm, excellent guess. Well, here is the answer.
CHILD: Hello, my name is [INAUDIBLE] [? Weaver ?] and I'm from Havertown, Pennsylvania. That mystery sound that you just heard was the sound of my salad spinner. Because vegetables are the best.
MOLLY BLOOM: Vegetables are the best. Have you seen a salad spinner before?
MARY MOORE: No.
MOLLY BLOOM: So it's basically a bowl. You put wet leaves and lettuce in it. And then it has a little thing you press on the top that spins it around. There's a top and a spinner.
And then it dries your lettuce. So you can eat all your vegetables. What's your favorite vegetable?
MARY MOORE: I like carrots and peas.
MOLLY BLOOM: Those are classic. I think I like brussels sprouts.
MEN: Ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, Brains On.
MARY MOORE: We've learned a bit about paradoxes and logic and the way our brains can work through these puzzles.
MOLLY BLOOM: But there's another kind of logic at play. And that's the logic found in artificial intelligence.
MARY MOORE: Here to tell us more is Brains On producer Sanden Totten.
SANDEN TOTTEN: It's hard to go a single day without relying on logic in some form. I mean, it's running in every smart device you own. And it even runs in some of the not so smart devices.
RANDI WILLIAMS: If you use a computer, if you use a calculator, if you use a microwave or a vending machine, then logic is sort of working behind all of that.
SANDEN TOTTEN: That's Randi Williams. She's a master's student at MIT, and she works with robots and AI, which stands for artificial intelligence. She says computers of all kinds think in a series of ones and zeros. Basically, these ones and zeros act sort of like switches. One for on, the other for off.
You can learn more about how they work in our episode on how the internet travels to your home. By using enough ones and zeros, you can program complex tasks based on simple logic.
RANDI WILLIAMS: So let's say I'm writing a program to run a vending machine. So I'm like, OK. What I want to happen is someone puts in a number, the computer thinks about something, and then it drops me out food.
SANDEN TOTTEN: For the user, you just give the machine some change, hit a few buttons, and voila, snack time. Simple enough, right?
RANDI WILLIAMS: What that means to me in my head is very different than it means to the computer. So first, you have to break it down into all of its itty-bitty, bitty steps. So someone is putting in a number. So you have to register a button push.
ROBOT: Button push.
RANDI WILLIAMS: Then you have to take that button push and map it to some combination.
ROBOT: Button for A and button for 2 have been pushed.
RANDI WILLIAMS: So maybe A2 is the donuts. That's the one I would go for. And then after you realize that you want the donuts, then you have to check the money.
ROBOT: If button for A and button for 2 have been pushed, then check to see if customer put in $1.
RANDI WILLIAMS: Does the money match the amount of money that I should have?
ROBOT: $1 confirmed.
RANDI WILLIAMS: When it finally puts out the donut, you have to tell the motors to start moving and to stop.
ROBOT: If $1 is confirmed, then start motor to release one item from location A2.
SANDEN TOTTEN: The whole process relies on logic statements that go, if this, then this. Randi says it's crucial to think this way if you want to program machines. Starting with simple logic and building from there, we can create artificial intelligences capable of lots of complicated tasks, like recognizing faces or even answering simple questions. But Randi says even these highly sophisticated machines still think very differently from humans.
RANDI WILLIAMS: When you build robots, people sometimes ask, does the robot know who I am? And I would say yes but not the same way. And the same thing is kind of true of logic.
So if your idea of knowing someone is you know their name and you know what they like and when you're around them, maybe you feel happy or sad, then no. Maybe a robot doesn't know those things. But it has a memory in its head, so it knows your name as zeros and ones.
And when you are around it, maybe it feels a different combination of zeros and ones. So maybe the two things can kind of map together. But it's this wholly different representation of those kinds of emotions.
[MUSIC PLAYING]
SANDEN TOTTEN: Randi Williams says you don't have to work in computers to appreciate logic, though. She says it can help you solve all kinds of problems, and it's a great tool for your cognitive toolbox.
[MUSIC PLAYING]
MARY MOORE: The answer to the question, is it opposite A? will always be no.
MOLLY BLOOM: But you can find the answer by asking another question and inferring the answer.
MARY MOORE: Paradoxes could seem absurd or contradictory but may actually be true.
MOLLY BLOOM: And logic is an important part of how we program everything from vending machines to artificially intelligent robots.
MARY MOORE: And now before we go, it's time for our moment of umm.
[VOICES SAYING "UM"]
BEN: Hello. My name is Ben, and I live in Lee's Summit, Missouri. My question is, how do erasers erase pencil marks?
GREGORY WEISS: Hi, this is Greg Weiss, a professor of chemistry at University of California at Irvine. My lab uses a type of carbon called carbon nanotubes to study how proteins work. Carbon is an element found on the periodic table, and it can be found in different forms.
In this pencil that I'm holding, the carbon atoms are in the form of sheets called graphite. And when you write with a pencil as I'm doing now, the paper turns color. It turns dark. But what's happening is a sheet of graphite is being left behind on the paper.
So now the pencil eraser reverses the process. What it's doing is actually wearing away that layer of paper. And let's take a close look at this, OK?
So imagine with me that you're using the eraser as I'm doing right now. And as you can imagine, my desk here is filling up with little grains of dust. It's actually the rubber.
And if I look closely at these grains-- because that's what we do as scientists. We look really, really closely at things. These grains of dust are actually-- they're dark-colored now, and the paper is white. That dark color is actually the graphite stuck to the rubber, and it's no longer stuck to the layer of the paper.
So now, I'm going to do what scientists like to do. So now we're getting even closer to the sample. And now I'm going to start feeling it.
And when I feel it, the surface texture has changed. It feels rough, where I did the erasing. And what's happened is the eraser has worn away the top layer of the paper material.
And the surrounding area-- that part that hasn't been erased-- is nice and smooth. So the way the eraser works is it scrapes off the graphite, puts it on this rubber that then turns into these little sandy grains, and then also scrapes off a top layer of paper in doing so.
Rubber is a hydrophobic material, meaning it doesn't mix well with water. And graphite, it turns out, is also hydrophobic. So these two types of material really love each other.
And because they have very similar anti-water tendencies, they'll stick to each other very well. So rubber is kind of ideally suited for this purpose.
This is a great experiment to do at home because you can actually look really closely at the piece of paper and the little grains. And so you can try to track where the graphite is going at the end of this. And that's how erasers work.
[VOICES SAYING "UM"]
MOLLY BLOOM: These names are going down in the permanent record and not erasable. It's the latest group to be added to the Brains Honor Roll. These are the listeners who share their ideas, questions, and mystery sounds with us. Here they are.
[MUSIC PLAYING]
[LISTING HONOR ROLL]
ROBOT: Brains Honor Roll [INAUDIBLE]
[MUSIC PLAYING]
MOLLY BLOOM: That's it for this episode of Brains On.
MARY MOORE: Brains On is produced by Marc Sanchez, Sanden Totten, and Molly Bloom.
MOLLY BLOOM: We had engineering help this week from Randy Johnson, Johnny Vince Evans, Matt Russell, and Veronica Rodriguez and production help from Lauren Dy, Jon Lambert, and Jacqueline Kim.
MARY MOORE: Brains On is supported in part by a grant from the National Science Foundation.
MOLLY BLOOM: Many thanks to Kieran Moore, Brandon Santos, Elizabeth Dunbar, and Senora Sandoval's third grade class.
MARY MOORE: If you want to hear more Brains On, you can find all our episodes at brainson.org or listen on your favorite podcast app. We're on Apple Podcasts, Google Play, Spotify, RadioPublic, basically wherever you get your podcasts.
MOLLY BLOOM: And if you're a fan of the show, consider leaving a review in Apple Podcasts.
MARY MOORE: It helps other people find out about the show and bring smiles to our faces and joy to our hearts.
MOLLY BLOOM: You can also find us on Instagram and Twitter.
MARY MOORE: We're Brains_On.
MOLLY BLOOM: And Facebook too. We'll be back soon with more answers to your questions.
MARY MOORE: Thanks for listening.
[MUSIC PLAYING]
Brains On is supported in part by a grant from the National Science Foundation.
MOLLY BLOOM: So Mary, what would the opening of Brains On sound like if it were opposite day right now?
MARY MOORE: Maybe something like this. [PLAYS PIANO]
MAN: You are not listening to Brains On. This is not Molly Bloom. And we are not curious about anything. That tree over there, boring.
My phone? I don't care how this thing works. That dog, it is not super cute. And I'm definitely not interested in finding out why it wants to smell all the things.
Nope. You want a mystery sound? Too bad. Here's the sound of me turning off this microphone because there are no questions to answer and no one interesting to talk to. Bye.
MEN: Ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, brains off.
[COW MOOING]
MOLLY BLOOM: I'm not sure I'd want to listen to that show.
MARY MOORE: Well, at least the episodes would be really short.
[MUSIC PLAYING]
MOLLY BLOOM: Luckily, it's not opposite day here on Brains On, but opposite day is on our minds.
MARY MOORE: We're going to talk all about logic and paradoxes today.
MOLLY BLOOM: And we're going to have some fun tying our minds in knots.
MARY MOORE: Keep listening.
[MUSIC PLAYING]
MOLLY BLOOM: You're listening to Brains On for American Public Media. And here with me today is 15-year-old Mary Moore from Lake Shore, Minnesota. Hi, Mary.
MARY MOORE: Hi, Molly.
MOLLY BLOOM: And this episode was inspired by a question that you sent in to us. Let's hear it.
MARY MOORE: I'm wondering how I can determine whether today is opposite day or not. When I asked if today is opposite day, my teacher said no. If it is opposite day and he says no, then it really must be opposite day.
But if it isn't opposite day, his answer would be the same. Please help me figure out this paradox. Sincerely, Mary Moore.
MOLLY BLOOM: That is a great question. So Mary, what made you think of this question?
MARY MOORE: Well, we play opposite day a lot at our house. We just like-- everything's opposite.
MOLLY BLOOM: So give me an example of what happens when it's opposite day at your house.
MARY MOORE: So like, if I asked, do you want a glass of water? and you say yes, that really means no.
MOLLY BLOOM: So it's kind of just responding to questions with the opposite of what you really mean? You don't like wear your clothes like your pants on your head or something?
MARY MOORE: Yeah, or like sometimes you like walk backwards or whatever.
MOLLY BLOOM: Excellent. So is it a thing that you ever do at school?
MARY MOORE: Sometimes. Like, we played at recess or whatever.
MOLLY BLOOM: What is it like when you do it at recess?
MARY MOORE: Especially on the playground, it's hard to like use the monkey bars backwards.
MOLLY BLOOM: Yeah, I don't think your arms run that way.
MARY MOORE: Yeah.
MOLLY BLOOM: So is it hard to understand what you and your family are talking about when it's opposite day at your house?
MARY MOORE: Well, sometimes because if you ask the question backwards of what you really mean to ask and then they answer it backwards, it gets all confusing.
MOLLY BLOOM: Like, what are we talking about here? So how would you describe what a paradox is?
MARY MOORE: It's a puzzle that's confusing and doesn't really have an answer.
MOLLY BLOOM: Excellent. Well, we're going to dive more into paradoxes in a little bit. But first, we're going to look at opposite day, and we're going to help you untangle the question that you sent us.
How do you find out whether it's opposite day or not? And here to help us is Justin Khoo, who's a professor of philosophy at MIT.
JUSTIN KHOO: This is a great question.
[MUSIC PLAYING]
Mary wants to know whether it's opposite day. Opposite day is the day when you say the opposite of what you really mean. Suppose Mary in wanting to know this asks her teacher who does know whether it's opposite day--
MARY MOORE: Is it opposite day?
JUSTIN KHOO: OK. So now we can think about whether the answer to this question could really tell Mary whether it's opposite day or not. So let's suppose that it is opposite day. Well, then Mary's teacher should say no.
TEACHER: No, it's not opposite day.
JUSTIN KHOO: And that's because since it is, in fact, opposite day, Mary's teacher should say the opposite of what she really means. But let's suppose that it's not opposite day. Then Mary's teacher should also say no because she should just say what she means and it's not opposite day. So no would be the right answer.
And so it seems like the question is it opposite day can only have the answer no, in which case Mary could never find out whether it's opposite day by asking someone who knows.
MOLLY BLOOM: Does that clear things up?
MARY MOORE: No.
JUSTIN KHOO: One way to try to find out whether it's opposite day is to ask someone who knows a different question. So one kind of question that you could do to find out whether it's opposite day is you can ask someone a question that you know the answer to and that you know-- that they also know the answer to. Let's suppose for a second that Mary knows that her name is Mary and that she also knows that her teacher knows that her name is Mary. Then Mary could ask her teacher--
MARY MOORE: Ms. Brown, is my name Mary?
JUSTIN KHOO: OK. So now we can sort of see how the answer to this question might allow Mary to infer whether it's opposite day. So suppose that the teacher says--
TEACHER: Well, yes, your name is Mary.
JUSTIN KHOO: To the question, is my name Mary? Well, then Mary can infer that it is not opposite day. Because if it were opposite day, the teacher would have had to say no because then she would have had to say the opposite of what she meant.
But if the teacher says--
TEACHER: No, your name is not Mary.
JUSTIN KHOO: Then Mary can infer it really is opposite day.
TEACHER: Your name is not Mary. This isn't school. And class can throw away their homework. We're going to study on the playground and climb on our desks.
Oh, and the cafeteria food is actually delicious.
STUDENTS: Yay.
JUSTIN KHOO: Mary, to me, seems like a natural philosopher. The opposite day question and puzzle is exactly the kind of issue that animates philosophers and gets us thinking and excited.
[MUSIC PLAYING]
MOLLY BLOOM: So Mary, what do you think of Justin's solution?
MARY MOORE: I think that's a pretty good solution.
MOLLY BLOOM: So you can just say, is my name Mary, and you'll have your answer. So now, in his answer, he talked about inferring. Do you know what inferring is?
MARY MOORE: Yeah, it's kind of like using the stuff to make a good guess of what it means.
MOLLY BLOOM: Exactly. So this is how Justin describes it.
JUSTIN KHOO: Inferring is something when what you do when you move from one piece of information to another using the information that you have to gain new information. So one type of inference is inductive inference.
MOLLY BLOOM: This kind is where you gather specific examples or data and then make a conclusion based on it. So let's say every morning your cat wakes up at 5:30, wanting to be fed.
[MEOWING]
[ALARM]
So an inductive inference would be, my cat will wake me up at 5:30 tomorrow.
[MEOWING]
[ALARM]
This is the kind of inference that scientists make when they collect data, and then they make a conclusion.
JUSTIN KHOO: There's another kind of inference which is called a deductive inference.
MOLLY BLOOM: This is when the conclusion is directly contained in the information that you have in front of you. If the starting point is true, then the conclusion is guaranteed to be true. Here's an example,
You know your cat only eats food that he likes. Your cat eats the food that you give him. So you can infer that your cat likes that food.
This kind of way of looking at situations is called logic.
JUSTIN KHOO: One answer about what logic is is it's an attempt to find the fundamental laws of reality.
MOLLY BLOOM: We're going to get real with reality in just a little bit. But first, it's time for the mystery sound.
GIRL: Shh. Mystery sound.
MOLLY BLOOM: Are you ready?
MARY MOORE: Yes.
MOLLY BLOOM: All right. Here it is
[MYSTERY SOUND]
Any guesses?
MARY MOORE: Sort of sounds like gears turning. Maybe paper shuffling.
MOLLY BLOOM: OK. Something turning or shuffling. Excellent. Well, we are going to hear it again in just a little bit.
And while you puzzle through that, we have a special guest here.
MARY MOORE: Agustin Rayo is a philosophy professor at MIT who specializes in paradoxes. Thanks for being here today.
AGUSTIN RAYO: Oh, I'm so delighted to be here.
MARY MOORE: What makes a paradox a paradox?
AGUSTIN RAYO: It's about having some principles which all seem true and a bit of reasoning which seems like it's got to be right and the conclusion that has got to be wrong. And then what's interesting is that because a valid piece of argument can't take you from something true to something false, you know that you've made a mistake somewhere. And the challenge is to find the mistake.
MARY MOORE: What's your favorite paradox or puzzle?
AGUSTIN RAYO: There's a paradox which I think is the hardest paradox ever. And that's the liar paradox. And it's really, really easy to state.
So you take a sentence that says this very sentence is false.
MOLLY BLOOM: OK, this is Molly interrupting for one minute because this is a little confusing. A liar's paradox is just one sentence. And that one sentence is, this sentence is false.
That's the sentence. This sentence is false. So the sentence is saying that itself is false, OK? OK.
AGUSTIN RAYO: And then you ask, OK. So is that sentence true, or is it false? And it's very hard to know what to say because if it's true, then what it says is true, but it says it's false.
And if it's false, then what it says is not true, but it says it's false. So it's not false. So it's true.
MOLLY BLOOM: So no one's ever been able to find like a definitive answer to that.
AGUSTIN RAYO: That's right. So many, many people, myself included, have suggested ideas about how one might go about it. But very often, the only person who thinks that's the right solution is the person who wrote it. And that's definitely true of me. I'm the only person in the world who thinks that my solution is the right one.
MOLLY BLOOM: What was the solution you came up with?
AGUSTIN RAYO: Well, my solution is really radical. What I think is that language doesn't come with ready-made meanings. So it's not like we use words. In each word is like a little box that has its meaning. Instead, what happens is that as we communicate, we build meanings on the fly.
And what goes on in the liar paradox is that the reasons that-- the right meaning cannot be built. So it's a little bit like saying, OK, let's play this game. Let's play the game of doing something that Agustin is not doing right now.
So Mary can win that game. Maybe she'll just scratch her head. And I'm not scratching my head. So she'll win. But I can never win the game because if I try scratching my head, then I'll be doing something that I am doing.
And if I try jumping up and down, then again, I'll do something that I am doing. So I can never win. And that's kind of what happens in the liar paradox. We try to build meanings, but it's built into the game we're trying to play that I'll never get to build this table meaning.
MOLLY BLOOM: And so how does your brain feel when you think about these things, Mary?
MARY MOORE: It's just like-- it's like it messes it up. It's like twisting and pulling it.
MOLLY BLOOM: And you really like math?
MARY MOORE: Yeah, I do.
MOLLY BLOOM: So how does math make your brain feel?
MARY MOORE: I don't know. Math is kind of simple for me. It's just like it all makes sense.
MOLLY BLOOM: That's very clean, and this is like the opposite.
MARY MOORE: Yeah, it's messy.
AGUSTIN RAYO: Well, I just wanted to say to Mary that sometimes when you feel like your mind is being stretched and turned, that's only making it stronger. It's like exercising a muscle. Then at first, it hurts a little bit. Then you have a bigger muscle. And you can do it.
You can use it to do really cool things. So I say keep working that mind.
MARY MOORE: Thanks for answering my questions.
AGUSTIN RAYO: Oh, of course. Any time.
MARY MOORE: Yeah, I had lots of fun. Thank you.
MOLLY BLOOM: Do you have questions or paradoxes you'd like us to tackle on Brains On--
MARY MOORE: A mystery sound to share--
MOLLY BLOOM: Or maybe you want to send us a drawing or a high five?
MARY MOORE: Then email us. The address is hello@brainson.org.
MOLLY BLOOM: And if you share your ideas with us, you'll join Ben on the Brains Honor Roll. He sent us this question.
BOY: How do erasers erase pencil marks?
MOLLY BLOOM: Oh, erasers? That's easy. I mean, everybody knows that it's--
MARY MOORE: Listen for the answer to that question during our moment of um at the end of the show.
MOLLY BLOOM: And the latest group to be added to the Brains Honor Roll.
MARY MOORE: Keep listening.
[MUSIC PLAYING]
MAN: You are listening to Brains Off, where we half-heartedly sit around staring at stuff. I'm not Molly Bloom. So, yeah.
Mm-hmm. Yep. Have you ever wondered about what the universe is made of and what time is and how weather patterns form and how animals communicate and why we dream?
Yeah, me neither. I'm just going to sit here and stare at this spider's web and not wonder how that eight-legged little creep managed to make something so beautiful. I mean, boring.
Well, that's it for this episode. Don't tune in next week. Seriously, don't. It's going to be really boring.
Whatever. Bye.
MEN: Ba, ba ba, ba, ba, ba, ba, ba, ba, Brains Off.
[COW MOOING]
MARY MOORE: You're back listening to Brains On from American Public Media. I'm Mary.
MOLLY BLOOM: And I'm Molly. OK, Mary. Are you ready to hear the mystery sound again?
MARY MOORE: Yes.
MOLLY BLOOM: All right. Here it is.
[MYSTERY SOUND]
Any new thoughts?
MARY MOORE: It kind of sounds like a marble going down like a marble track.
MOLLY BLOOM: Hmm, excellent guess. Well, here is the answer.
CHILD: Hello, my name is [INAUDIBLE] [? Weaver ?] and I'm from Havertown, Pennsylvania. That mystery sound that you just heard was the sound of my salad spinner. Because vegetables are the best.
MOLLY BLOOM: Vegetables are the best. Have you seen a salad spinner before?
MARY MOORE: No.
MOLLY BLOOM: So it's basically a bowl. You put wet leaves and lettuce in it. And then it has a little thing you press on the top that spins it around. There's a top and a spinner.
And then it dries your lettuce. So you can eat all your vegetables. What's your favorite vegetable?
MARY MOORE: I like carrots and peas.
MOLLY BLOOM: Those are classic. I think I like brussels sprouts.
MEN: Ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, ba, Brains On.
MARY MOORE: We've learned a bit about paradoxes and logic and the way our brains can work through these puzzles.
MOLLY BLOOM: But there's another kind of logic at play. And that's the logic found in artificial intelligence.
MARY MOORE: Here to tell us more is Brains On producer Sanden Totten.
SANDEN TOTTEN: It's hard to go a single day without relying on logic in some form. I mean, it's running in every smart device you own. And it even runs in some of the not so smart devices.
RANDI WILLIAMS: If you use a computer, if you use a calculator, if you use a microwave or a vending machine, then logic is sort of working behind all of that.
SANDEN TOTTEN: That's Randi Williams. She's a master's student at MIT, and she works with robots and AI, which stands for artificial intelligence. She says computers of all kinds think in a series of ones and zeros. Basically, these ones and zeros act sort of like switches. One for on, the other for off.
You can learn more about how they work in our episode on how the internet travels to your home. By using enough ones and zeros, you can program complex tasks based on simple logic.
RANDI WILLIAMS: So let's say I'm writing a program to run a vending machine. So I'm like, OK. What I want to happen is someone puts in a number, the computer thinks about something, and then it drops me out food.
SANDEN TOTTEN: For the user, you just give the machine some change, hit a few buttons, and voila, snack time. Simple enough, right?
RANDI WILLIAMS: What that means to me in my head is very different than it means to the computer. So first, you have to break it down into all of its itty-bitty, bitty steps. So someone is putting in a number. So you have to register a button push.
ROBOT: Button push.
RANDI WILLIAMS: Then you have to take that button push and map it to some combination.
ROBOT: Button for A and button for 2 have been pushed.
RANDI WILLIAMS: So maybe A2 is the donuts. That's the one I would go for. And then after you realize that you want the donuts, then you have to check the money.
ROBOT: If button for A and button for 2 have been pushed, then check to see if customer put in $1.
RANDI WILLIAMS: Does the money match the amount of money that I should have?
ROBOT: $1 confirmed.
RANDI WILLIAMS: When it finally puts out the donut, you have to tell the motors to start moving and to stop.
ROBOT: If $1 is confirmed, then start motor to release one item from location A2.
SANDEN TOTTEN: The whole process relies on logic statements that go, if this, then this. Randi says it's crucial to think this way if you want to program machines. Starting with simple logic and building from there, we can create artificial intelligences capable of lots of complicated tasks, like recognizing faces or even answering simple questions. But Randi says even these highly sophisticated machines still think very differently from humans.
RANDI WILLIAMS: When you build robots, people sometimes ask, does the robot know who I am? And I would say yes but not the same way. And the same thing is kind of true of logic.
So if your idea of knowing someone is you know their name and you know what they like and when you're around them, maybe you feel happy or sad, then no. Maybe a robot doesn't know those things. But it has a memory in its head, so it knows your name as zeros and ones.
And when you are around it, maybe it feels a different combination of zeros and ones. So maybe the two things can kind of map together. But it's this wholly different representation of those kinds of emotions.
[MUSIC PLAYING]
SANDEN TOTTEN: Randi Williams says you don't have to work in computers to appreciate logic, though. She says it can help you solve all kinds of problems, and it's a great tool for your cognitive toolbox.
[MUSIC PLAYING]
MARY MOORE: The answer to the question, is it opposite A? will always be no.
MOLLY BLOOM: But you can find the answer by asking another question and inferring the answer.
MARY MOORE: Paradoxes could seem absurd or contradictory but may actually be true.
MOLLY BLOOM: And logic is an important part of how we program everything from vending machines to artificially intelligent robots.
MARY MOORE: And now before we go, it's time for our moment of umm.
[VOICES SAYING "UM"]
BEN: Hello. My name is Ben, and I live in Lee's Summit, Missouri. My question is, how do erasers erase pencil marks?
GREGORY WEISS: Hi, this is Greg Weiss, a professor of chemistry at University of California at Irvine. My lab uses a type of carbon called carbon nanotubes to study how proteins work. Carbon is an element found on the periodic table, and it can be found in different forms.
In this pencil that I'm holding, the carbon atoms are in the form of sheets called graphite. And when you write with a pencil as I'm doing now, the paper turns color. It turns dark. But what's happening is a sheet of graphite is being left behind on the paper.
So now the pencil eraser reverses the process. What it's doing is actually wearing away that layer of paper. And let's take a close look at this, OK?
So imagine with me that you're using the eraser as I'm doing right now. And as you can imagine, my desk here is filling up with little grains of dust. It's actually the rubber.
And if I look closely at these grains-- because that's what we do as scientists. We look really, really closely at things. These grains of dust are actually-- they're dark-colored now, and the paper is white. That dark color is actually the graphite stuck to the rubber, and it's no longer stuck to the layer of the paper.
So now, I'm going to do what scientists like to do. So now we're getting even closer to the sample. And now I'm going to start feeling it.
And when I feel it, the surface texture has changed. It feels rough, where I did the erasing. And what's happened is the eraser has worn away the top layer of the paper material.
And the surrounding area-- that part that hasn't been erased-- is nice and smooth. So the way the eraser works is it scrapes off the graphite, puts it on this rubber that then turns into these little sandy grains, and then also scrapes off a top layer of paper in doing so.
Rubber is a hydrophobic material, meaning it doesn't mix well with water. And graphite, it turns out, is also hydrophobic. So these two types of material really love each other.
And because they have very similar anti-water tendencies, they'll stick to each other very well. So rubber is kind of ideally suited for this purpose.
This is a great experiment to do at home because you can actually look really closely at the piece of paper and the little grains. And so you can try to track where the graphite is going at the end of this. And that's how erasers work.
[VOICES SAYING "UM"]
MOLLY BLOOM: These names are going down in the permanent record and not erasable. It's the latest group to be added to the Brains Honor Roll. These are the listeners who share their ideas, questions, and mystery sounds with us. Here they are.
[MUSIC PLAYING]
[LISTING HONOR ROLL]
ROBOT: Brains Honor Roll [INAUDIBLE]
[MUSIC PLAYING]
MOLLY BLOOM: That's it for this episode of Brains On.
MARY MOORE: Brains On is produced by Marc Sanchez, Sanden Totten, and Molly Bloom.
MOLLY BLOOM: We had engineering help this week from Randy Johnson, Johnny Vince Evans, Matt Russell, and Veronica Rodriguez and production help from Lauren Dy, Jon Lambert, and Jacqueline Kim.
MARY MOORE: Brains On is supported in part by a grant from the National Science Foundation.
MOLLY BLOOM: Many thanks to Kieran Moore, Brandon Santos, Elizabeth Dunbar, and Senora Sandoval's third grade class.
MARY MOORE: If you want to hear more Brains On, you can find all our episodes at brainson.org or listen on your favorite podcast app. We're on Apple Podcasts, Google Play, Spotify, RadioPublic, basically wherever you get your podcasts.
MOLLY BLOOM: And if you're a fan of the show, consider leaving a review in Apple Podcasts.
MARY MOORE: It helps other people find out about the show and bring smiles to our faces and joy to our hearts.
MOLLY BLOOM: You can also find us on Instagram and Twitter.
MARY MOORE: We're Brains_On.
MOLLY BLOOM: And Facebook too. We'll be back soon with more answers to your questions.
MARY MOORE: Thanks for listening.
[MUSIC PLAYING]
Transcription services provided by 3Play Media.
