Short Wave - Is Math Real?

Episode Date: August 16, 2023

Kids ask, "Why?" all the time. Why does 1+1=2? Why do we memorize multiplication tables? Many of us eventually stop asking these questions. But mathematician Dr. Eugenia Cheng says they're key to uncovering the beauty behind math. So today, we celebrate endless curiosity and creativity — the driving forces of mathematicians. Regina G. Barber and Eugenia talk imaginary numbers, how to go beyond simply right and wrong and yes, Eugenia answers the question, "Is math real?"Eugenia's new book Is Math Real? is out now.Have a science story to share? Email us at [e-mail] pcm.adswizz.com for information about our collection and use of personal data for sponsorship and to manage your podcast sponsorship preferences.NPR Privacy Policy

You're listening to Shortwave from NPR. I think if we're lucky, we get to have a few big loves in our lifetime. The kind of love that makes you zone out in conversations and keeps you up at night. And if you're like me, when you love something or someone, it feels like there are reminders of them everywhere. For Dr. Eugenia Chang, one of those big loves was and still is math. Everywhere I look, I see math, everything I think about, I see math. And I think that once you've gone through a certain amount of training and practice of your brain thinking in certain ways, it just does that all the time. And so I think that I see patterns and structures.

And I think it's partly just having a very broad notion of what math is. Eugenia is a mathematician and author of the new book, Is Math Real? how simple questions lead us to mathematics deepest truths. Throughout the book, she invites us to see that math is more than just getting the right answer or arriving at a single conclusion. The beauty of math is that it's all about questions and curiosity. It's about wondering why things are happening. I am really like one of those toddlers who just keeps asking why forever and never stops.

Fundamentally, I just want to understand why things are going on. And every explanation I find or come up with pushes me to ask why that is true. Eugenia, like me, likes to test the limits of things. See what the rules reveal. But suddenly she's down a rabbit hole and asking, well, why does a certain rule even have to exist in the first place? You sort of feel hemmed in by those rules and you go, well, what if I don't want to follow that particular rule, but I want to do something else? Where can I do that? And that's how math grows and develops.

we don't want to stay put. An easy way to understand this is with a really basic example from Eugenia's book. It's a word problem, like one's from grade school. Okay, if Joe had seven apples and uses five of them to make a pie, how many apples does he have left? The answer the asker is probably looking for is two. Joe has two unused apples left over, but Eugenia was delighted when she heard that a kid didn't answer to.

Instead, they asked, well, has Joe eaten the pie yet? And I just think it's brilliant. And I remember my friend who posted that. And I just think it's fantastic when children do that. It's so subversive, but in a really logical way. And it's getting an additional truth beyond the question being asked. And besides, Joe still has seven apples. So two isn't the only answer.

It's not about right and wrong answers. It's about there's always some sense in which something else could be true. So today on the show, we break the rules with Dr. Eugenia. Chang. She helps us understand how the abstract can remind us that math isn't cut and dry. It's creative. I'm Regina Barber. You're listening to Shortwave from NPR. So, Eugenia, let's stay on the breaking rules path here and building like new worlds. It seems like we actually get to do that, you know, push boundaries, get creative with the help of a math concept called abstraction. What is abstraction? Abstraction is where you.

you forget certain details about situations so that you get into a kind of idealized version of a situation. So it's like if you look at a window and you declare that it is a rectangle, then you've forgotten quite a lot about the window apart from its basic shape. And then you can go around going, oh wait, this table, if I forget the legs and stuff, that's also a rectangle. And my laptop is also a rectangle. And so you're looking at some abstract version of it that is more. ideal, more idealized, and is something that you can then study using logic. And that's where

numbers come from in the first place, because numbers come from looking at a collection of things and forgetting what they are and just remembering the quantity of them. And then what mathematicians do is that we don't just make an analogy. We take the analogy really seriously as a new concept. And so that's what the number is, or the shape called a rectangle, where you don't just say this table and this window have something in common. You say, I'm going to call that a rectangle. And once you've named it, you can kind of think about it and study it. Right. And so in chapter four, you take this basic premise and you use it to explain kind of more advanced math, imaginary and complex numbers. I'd love to talk about these, starting with what is an imaginary

and complex number? An imaginary number is what happens when they tell you at school, you're not allowed to take the square root of a negative number. And then the next year they go, now we're going to take the square root of a negative number. And so it looks like that we're just contradicting ourselves. But what we're really doing is we're starting in one world and then we're saying, okay, I'm done with this world. I don't want to follow the rules of that world anymore.

I want to play a different game. I want to expand the world and do something more. Well, let's walk through this slowly. What is a square root? It is a number such that when you multiply it by itself, you get something. So the square root of four, we're looking. for a number such that when you multiply it by itself, you get four. So two times two is four.

Great. Negative two times negative two is also four. It's still four. So how are we going to multiply something by itself and get a negative number? Because if you square a positive number, you get a positive number. If you square a negative number, you get a positive number as well. So there aren't any options left. So I'm going to imagine that I have a number and when I take it square, I get negative one. So then you can go, well, what is that number? And mathematicians go, well, I don't really know, but let's pretend it exists. And so let's give it a name. Let's call it I for imaginary. And then see what happens. And then you build a whole world around it, just like, really just like children making an imaginary world with imaginary friends. And that's what we're doing.

And then the curious thing is that it turns out to be actually useful. Because if you, make a mathematical plane with an axis of real numbers, you know like the one, two, three we know and love, and then an axis of imaginary numbers, it lets you move from 1D using the normal numbers to 2D, which is made of complex numbers, one part real, one part imaginary. And I would like to reassure everyone that if you feel like you don't understand what's going on there, you're on the right track. All the good mathematicians I know think nothing is obvious and think that feel confused all the time and that's what drives us. The confusion doesn't make us stop and go, I give up. The key thing is

that the confusion drives us to try and understand more all the time. You can't see me, but I'm like swooning because I love complex numbers. Because I told you I was a physicist, they help explain waves, right, and like how waves move and like through space or through a rope you wiggle. But along all these lines, you write about the tension between our desire to build and develop these new concepts in mathematics and some of the history of the field. So like you acknowledge that there's a lot of current definitions of math that come from like European white men in the last few hundred years. While the field has this deep history beginning with like non-white cultures, like how do you reconcile these things? I don't know. How do you reconcile these things? So I

fully fess up in the book that I don't know, but I think we should still think about it. And the thing is that math, as we currently do it today in the research world, is all about building up extremely logical arguments using just logic and being extremely clear what assumptions you're making. And so mathematicians as a community are able to agree on what counts as right and what counts as wrong and then build on it, which is quite radical, given how little anyone in the world is able to agree about what's right and wrong anywhere. And so some European white male mathematicians decided that all the math that was being done, like you say, by mostly non-white cultures, many ancient cultures for thousands of years, there was something kind of hazy about it because it wasn't on totally secure foundations. And so they basically sat down and made up rules for mathematics.

And it's pretty much like they made up some gates and appointed themselves as gatekeepers. And the advantage was that mathematics was able to develop on really secure foundations. But another consequence is that while it has enabled us to make extraordinary technological and scientific advances, that has then enabled us to kind of destroy the environment that we depend on. And so I'm personally very conflicted about it because I love that kind of complexity that builds up on complexity. But it has also enabled humans to do pretty destructive things. and people were excluded. Because, of course, women didn't have access to the education

that would teach them those rules. People from other countries that weren't part of the European academic elite didn't have access to those kinds of rules. And so I do not know how to reconcile those things. Yeah, that tension feels like it's all over math and science. It's all over physics, a world I was in for a long time. And I think that sometimes I've wondered, what is this all for?

and I can also get existential, like in the vein of your book title, Is Math Real? And so, Eugenia, my final question is, is math real? And is that even the right question? I'm not trying to answer whether math is real or not. I'm trying to show that considering the question at all leads us to interesting thoughts. And in the end, what I say is that with all these questions, I don't think there are yes or no answers. And we shouldn't claim that there are. what we should do instead is say there is a sense in which, you know,

what is the sense in which math is real and what is the sense in which math isn't real? And the thing is, I think a lot of people who say math isn't real are using that to say, oh, so it's irrelevant and stupid, why should we study it? It's made up. And what I want to say is that just because it is made up doesn't mean that it's irrelevant. And actually, the fact that it's kind of made up makes it really powerful because, well, it really in a way more accessible because you don't need a lot of money to get it. All you need is an imagination. And I think that's a really amazing thing about it. And just like fiction isn't real, but fiction can give us insights about the world around us to highlight much more

specifically things about society. And that's what I think is powerful about abstract math as well because we're not constrained by reality. I really like fantasy. I think the more abstract you get, the more people can relate. Yes, that's a great point. That abstraction seems like it takes us far away, but actually it means that more people can join. Yeah, you can bring in more examples,

and more people can be part of it. Thank you. That was so wonderful. Thank you so much. It's always great coming on this show. I love it. Before we head out, we want to take a minute to talk about Shortwave Plus. Plus subscribers help make shows like this one possible.

And they also get to listen to all of our shows without any sponsor breaks. Find out more at plus.npr.org slash shortwave. And to everyone who's already subscribed, we see you, we appreciate you. Thank you so much. This episode was produced and fact-checked by Rachel Carlson. It was edited by managing producer Rebecca Ramirez and Patrick Murray was the audio engineer. Beth Donovan is our senior director and Anya Grenman is our senior vice president of programming. I'm Regina Barber.

Thank you for listening to Shortwave from NPR.