To Infinity and Beyond With Mathematician Eugenia Cheng
One of the first numbers you learned as a kid was infinity. Perhaps you’ve also heard the typical six-year-old ask for infinity-plus-one cookies or minutes before bedtime. But what is the mathematical definition of infinity? Is it a number or just a boundless limit? Why are some infinities bigger than others? And what does the concept tell us about how the world works? Mathematician Eugenia Cheng guides us through the conundrums of this complex concept.
Dr. Eugenia Cheng is Scientist in Residence at the School of the Art Institute of Chicago and author of several books, including The Art of Logic in an Illogical World (Basic Books, 2018) and Is Math Real?: How Simple Questions Lead Us to Mathematics’ Deepest Truths (Basic Books, 2023).
IRA FLATOW: This is Science Friday. I’m Ira Flatow. You know as a kid, you talk about numbers. Isn’t one of the first numbers you like to talk about as a kid, make fun of it, is infinity, you know? Typical six-year-old conversation, here it goes. I have 10 cookies. I have 100 cookies. Well, I have infinity plus one cookies, right?
You’ve been through that conversation. You hate it when that kid comes up. The entire playground goes silent. Well, kids might use infinity to mean really big, beyond our imaginations. But what does infinity really mean? Is it an actual number? Is it just a boundary? Does infinity plus one actually exist? And what can baby carrots tell us about this complex concept?
My next guest talks about all of this stuff in her new book and is here to help us wrap our brain around this big idea. Eugenia Cheng is a mathematician and scientist in residence at the School of the Art Institute in Chicago. And her new book is called Beyond Infinity: An Expedition to the Outer Limits of Mathematics. Welcome back.
EUGENIA CHENG: Hello, thank you for having me back.
IRA FLATOW: Oh, it’s great. This is such a great book and so easy to read. It’s filled with–
EUGENIA CHENG: Oh thank you. I’m so glad you think so.
IRA FLATOW: Well, it’s filled with such vernacular, like little phrases like when you go through a problem and you say wait for it. You’re going to come up with the answer. It’s great. Why tackle infinity now?
EUGENIA CHENG: Well, as you say, it’s something that even little kids love to think about and play around with in their heads. But what does it actually mean? And it’s amazing, because it took mathematicians thousands of years to figure out how to think about it in such a way that it would make sense logically and not just in our imaginations.
IRA FLATOW: Why do we even need the concept of infinity?
EUGENIA CHENG: Well, what does need really mean, you know? There’s a difference between want and need that my little nephew has been studying in school. And the thing is that math, some people think that math is all about getting the right answers. And one of the big messages that I want to get across is that actually it’s more than that. It’s not always just about getting the right answers.
But it’s about asking the right questions and exploring possible answers and seeing what possible answers are available in different types of situations. And infinity doesn’t have just one answer. It doesn’t just have one way of thinking about it. There are so many fascinatingly different ways of thinking about it that produce different weird scenarios, like can you eat infinite cookies? Do you ever actually travel anywhere? What is time? And are there baby carrots?
IRA FLATOW: Let’s start backwards here. What do you mean are there baby carrots? How does that fit in?
EUGENIA CHENG: Well, this is a real and very important argument I once had with a mathematician, of course, about whether baby carrots really exist or whether they’re just cut, you know, like those baby cut carrots that you can get that are unnervingly cylindrical and don’t look like they really grew like that. But when I was little, we grew carrots in the garden. So I knew perfectly well that if we pulled them out of the ground before they were big, then, well, they would be small.
And this is actually something to do with a theorem in mathematics called the intermediate value theorem, which is something that students usually only study when they get to some quite high level of calculus. But it comes from understanding not just infinitely big things but infinitely small things. And the infinitely big and the infinitely small are, well, infinitely related.
IRA FLATOW: You know, getting back to the kids’ conversation, a kid talks about infinity plus one, as if infinity is a number. And you–
EUGENIA CHENG: Yes.
IRA FLATOW: –you about how it can’t be a number in your book, and with very easy, simple explanations about zero.
EUGENIA CHENG: Right, it depends what you mean by number. So if you’re going to say infinity is or isn’t a number, then first of all you have to say, what are numbers? And so I start by talking about the fact that numbers are things we can add up and multiply and subtract. And if you just take those basic operations and you pretend that infinity is a number for a second, then you get to deduce logically that everything equals zero, which is not that great. If everything equaled zero, well, there wouldn’t really be anything, would there?
IRA FLATOW: No, no. I want to quote you from your book, some really interesting things that you say throughout. Let me read one. “One of the things that is tantalizing about infinity is that it’s so easy to stumble upon the idea, so easy to stumble upon the strange, apparently magical behavior that surrounds it, yet so difficult to work out what on earth is really going on.”
EUGENIA CHENG: That’s right, and I like to think of the exploration of mathematics as being just like wandering around a weird jungle or a new landscape, where you can see things and you can feel things about them, but you don’t really know what they are. And so if you find yourself in that weird, mysterious world, well, if I find myself there, I really want to know what’s going on. I’m curious about it.
And that’s what drives my love of mathematics. I’m a bit like that two-year-old child who keeps saying why? And then you answer and then they go why and why and why? So when you are faced with the fact that infinity causes strange things to happen, you can either run away, and unfortunately that’s what too many people do with mathematics. And I have sympathy for that because it can be a little unnerving. Or you can delve into it and try and figure out what’s going on.
IRA FLATOW: We’re going to delve into it a little bit more with some calls from listeners because they want to talk about infinity. Peter in Iowa City, hi. Welcome to Science Friday.
PETER: Oh, hello, thanks for taking my call. I really appreciate everything you guys do.
IRA FLATOW: Thank you.
PETER: And here’s my favorite question about infinity. How many different infinities are there? And then which one is it?
IRA FLATOW: OK, Peter. Well thank– we’ll try to answer that. Eugenia, what do you think?
EUGENIA CHENG: There are infinitely many infinities–
IRA FLATOW: There you have it.
EUGENIA CHENG: –as it turns out. There’s a hierarchy of infinities because as soon as you have one infinity, then you can create a bigger one by saying 2 to the power of infinity. And then you can say 2 to the power of that. And then you can say 2 to the power of that. And that never stops.
And so if you’re having an argument with a child and they’re saying, well, I’m right times infinity. Then you can say, well, I’m right times 2 to the power of infinity. But unfortunately they can then say, well, I’m right times 2 to the power of 2 to the power of infinity. And then you’ll never really get anywhere, so perhaps we shouldn’t have arguments like that.
IRA FLATOW: Well you know, you just saying this, it just reminded me. You know, physicists use mathematics to prove all kinds of stuff, and one of the things they talk about is the infinite number of universes that might be, the multiverses.
EUGENIA CHENG: Yes.
IRA FLATOW: Is this an idea they got from math about infinities?
EUGENIA CHENG: I wouldn’t like to speak for physicists, but physics does often get a lot of ideas from mathematics. And I like to think that math is there to serve everyone else and to try and clarify the way that we can think about the world around us. The great thing about mathematics is that we can think about possible worlds that only exist in our imagination.
But just like fiction helps us understand real life, those fictional worlds help us understand real life as well. But we’re not constrained by reality in the way that physicists are trying to understand actual reality. I get to wander around in the depths of my imagination all the time, and then some of it ends up helping physicists understand the real world.
IRA FLATOW: But if you’re really making up stuff in mathematics, which is what I hear you saying– and tell me if I’m wrong– you talk about your book, about mathematical rigor and that this stuff has to really work out.
EUGENIA CHENG: Yes, we do make stuff up, but there are a few rules and guidelines that we have to stick to. And that is that we have to use the rules of logic. And that’s what mathematics is all about. Different subjects and disciplines get at information by different methods. And so science uses evidence, and mathematics uses logic. So we can make anything up that we want, as long as it obeys the rules of logic. And if you have a contradiction in what you try to make up, then the whole world implodes. But oh well.
IRA FLATOW: Oh well.
EUGENIA CHENG: It was only an imaginary world, so you can just try again and do a different one.
IRA FLATOW: Gosh, you can’t be wrong? Does that mean you can’t be wrong?
EUGENIA CHENG: When I’m teaching my art students, I have discovered that my art students at the School of the Art Institute have often been put off math in the past by the fact that they could be wrong. And no one really likes being wrong, I don’t think. So what I say to them, I don’t really say that they’re wrong. I just say that they caused a contradiction, and their whole world imploded.
IRA FLATOW: Well, that’s what I was going to ask you about. Why is a mathematician in residence, scientist in residence, at the School of Art Institute in Chicago?
EUGENIA CHENG: Isn’t it fantastic? It’s my dream job, actually, because the School of the Art Institute has an amazing vision that there shouldn’t really be boundaries between subjects. And I agree with that. And so we’re trying to erase the boundary between art and science and show that science and scientific thinking and mathematical thinking is relevant to everybody, and there isn’t really a cut-off between there and art and artistic practice.
And in fact, since I’ve been there, amazingly I’ve started making mathematical art. I was commissioned by the new hotel EMC2, so I have become an artist making mathematical art installations to show also that my mathematics is also a bit like art.
IRA FLATOW: You’re a musician too, and we’ve heard over the years that music and math go together, and you seem to prove that point.
EUGENIA CHENG: Let’s be careful here, because that’s a sample size of one. So we shouldn’t say that that actually–
IRA FLATOW: It’s more than one.
EUGENIA CHENG: –proves the point.
IRA FLATOW: I can find 10 people.
EUGENIA CHENG: So it’s an interesting question. And I think about it a lot because people ask me. And I think that for me, because math isn’t just about numbers, the relationship between math and music is about something deeper than that. It’s about structures. And there’s so much interesting structure in the way that music is built up. And mathematics, for me, really looks at the way things work and the way things are built up, the way that ideas fit together.
Just like you build a building by fitting maybe bricks and mortar together– probably not so much bricks anymore, but steel pieces. You fit pieces together, and that makes a physical structure. In mathematics, we fit ideas together, and we make an abstract structure. And the aim of my work in the field called category theory is to look at what it is that holds those ideas together and what it is that might make ideas collapse and fall down.
IRA FLATOW: Now you talk about math and ideas. No one in school has taught that way that I know of. You taught the count and whatever and never to really appreciate math. Do you think we need to have more math art classes or to teach kids the beauty behind the math?
EUGENIA CHENG: I do think we need to teach kids the beauty, the fun, and the fascination behind the math. And there are some really great math teachers out there, and I’ve met them. And I talked to them, and they’re trying really hard. Unfortunately, they’re often constrained by the tests and the standards that they’re required to meet. And the tests are often the exact opposite of beauty.
They’re finding the answers, the formulae, doing the things over and over again to try and jump past an arbitrary hurdle. Whereas I’ve discovered by teaching art students that if I show students what fascinates me and what’s beautiful and to show that it’s an exploration of possibilities rather than a journey with just one specific aim in mind, then they’re much more interested by it, and they’re drawn into it, especially art students who are fundamentally creative people.
They think visually. They’re not interested in just memorizing formulae and solving specific problems. They’re interested in uncovering ideas. And that’s what math is about, really. So I think that we should be telling that, teaching that to students of all ages at school.
IRA FLATOW: Talking with Eugenia Cheng, the author of Beyond Infinity: An Expedition to the Outer Limits of Mathematics, a really great book to have a read, on Science Friday from PRI, Public Radio International. And we don’t usually get a full board of folks who want to talk about math, but we have maxed out our math here.
EUGENIA CHENG: Oh, that’s great.
IRA FLATOW: Let’s go to Cedar Rapids, Iowa. I lost his name or her name. Hi, Cedar Rapids. Welcome to Science Friday.
DOUGLAS: Hi, this is Douglas.
IRA FLATOW: Thank you, Douglas.
DOUGLAS: And I had a quick question. I was hoping you could maybe help me solve a longstanding dispute I had with my mother.
EUGENIA CHENG: Uh-oh.
DOUGLAS: We were born on the same day 24 years apart. And when I was taking calculus in high school, I told her that eventually if we both lived to infinity that we’d be the same age. And she vehemently disagreed with me. And so I was just hoping maybe I could get some clarity on this.
EUGENIA CHENG: That’s a great question. And I love the if at the beginning of that. Because if you could both live to infinity, well, it depends how we take logic. Because you’re not both going to live to– well, you’re probably not both going to live to infinity. And so if you did, then logically, kind of anything would be true. So you would both be the same age, and you would also both not be the same age. And you’d also both be elephants.
But I think if we take it a little bit more literally than that, then I think you would both be infinity years old if you both lived to infinity. And because– now here’s the thing, because the difference in your ages happened at the beginning and not at the end, I think it would be the same. Whereas infinity plus one, now that could be really something different.
And I think that we learned this from Shakespeare. Because in The Taming of the Shrew, he talks about forever and a day. So I think he knew that forever and a day is longer than forever. Otherwise he’d have just said forever.
IRA FLATOW: Is infinity a constant?
EUGENIA CHENG: Is infinity a constant? Well, it depends what kind of infinity you’re talking about. I think that we have pinned down infinity to be very different possible types of number. It can be thought of as an ordinal number, which means if you lined lots of people up in a queue, how long is that queue? It’s that sort of idea.
It’s also a cardinal number, where it’s just the size of a certain number of things. So if you think about the number 10, what is the number 10? Well, one way we tell children is that it’s the number of fingers that you have. And so infinity, the first infinity, is the number of whole numbers that there are. And that is constant, because I’m pretty sure we’re not going to spontaneously discover a new whole number that we didn’t know about. So I would say that yes, infinity is constant.
IRA FLATOW: Well, my last question is about pi.
EUGENIA CHENG: Ah.
IRA FLATOW: Pi, it’s a real number, but it can go on infinitely long, right?
EUGENIA CHENG: Yes, that is to say that the decimal expansion of it goes on forever and doesn’t repeat itself because it’s an irrational number. And this is related to infinitely small things. Because what mathematicians realized, really 2,000 years ago, is that if you just look at the fractions, things like 2 over 3, 3 over 5, 6 over 7, things like that, they’re very, very close together, but not totally close together. There are always teeny, weeny little gaps in between them.
And so to fill in those gaps, if we want to fill in those gaps, then we need the irrational numbers. That is the decimal numbers that go on forever. And if we don’t have those gaps, well, then every line we draw would have– I mean, if we don’t fill in those gaps, then every line we draw will have tiny gaps in it, which is a bit odd. Then when we walk from a to b, we might skip some places along the way, which is also quite odd. And it also means that baby carrots might not exist.
IRA FLATOW: Wow, and a rational way to think about the irrational, and only as Eugenia Cheng can do it. She’s author of the new book Beyond Infinity: An Expedition to the Outer Limits of Mathematics. She’s a mathematician and scientist in residence of, of all places, the School of Art Institute of Chicago.
And you can read an excerpt from this terrific, this really great book for the weekend. sciencefriday.com/infinity, we have an excerpt. Thank you, Dr. Cheng, for taking time to be with us today, great book.
EUGENIA CHENG: Thank you for having me.
IRA FLATOW: We’ll have you back.