Science Friday - A Mathematician Asks ‘Is Math Real?’

Episode Date: January 1, 2024

The concept of math has been around for a long time, developing independently in many different cultures. In 1650 BC, the Egyptians were creating math textbooks on papyrus, with multiplication and division tables. Geometry, like the Pythagorean theorem, was used in ancient Greece. And negative numbers were invented in China around 200 BC.Some mathematical concepts are easier to understand than others. One apple plus one apple equals two apples, for example. But when it comes to complex equations, negative numbers, and calculus, concepts become abstract. All that abstraction prompts some to wonder: Is math even real?Mathematician Dr. Eugenia Cheng has heard this question many times over her career. The quandary is the basis of her latest book, Is Math Real?: How Simple Questions Lead Us to Mathematics’ Deepest Truths. She joins Ira from Chicago, Illinois.Transcripts for each segment will be available the week after the show airs on sciencefriday.com. Subscribe to this podcast. Plus, to stay updated on all things science, sign up for Science Friday's newsletters.

Sure, two plus two equals four, but is that all there is to math? It's not always just about getting the right answers, but it's about asking the right questions and exploring possible answers. It's Monday, January 1st. Happy New Year, but it's also Science Friday. I'm sci-fri producer Shoshana Buxbaum. Math is based on a lot of abstract concepts. For example, math has not one but many infinities. And scientists use it all the time to describe nature. But how do we know it describes, well, the real world?

Mathematician Eugenia Chang wrote a book grappling with the question, Is Math Real? Here's Ira Flato. The concept of math has been around for a long time. You know that 3,000 years ago. The Egyptians were using fractions. Geometry was used in ancient Greece. And negative numbers were invented in China around 200 b.C.

I've always found it fascinating that different cultures develop mathematics, and ultimately that knowledge was shared with the rest of the world. But you know what? There's an underlying question to all of this. If math is largely made up of abstract concepts, how do we know that it's real? I mean, what does real mean anyway? My next guest wrote a whole book grappling with this question, and she's here with us now. Dr. Eugenia Chang, mathematician and author of, is math?

Math real. How simple questions lead us to mathematics deepest truths. She's joining me now from Chicago. Welcome back to Science Friday. Thank you so much. It's great to be back. That's a weird question to ask is math real? Why do you even ask that question? Well, I've heard so many other people ask it and feel flummoxed by it. And often they're asking it because they really doubt whether it's real. And it's a way of saying, well, I don't think it's real. So why do I need to study it? And I think what's really coming out is many people's frustration with being forced to learn things that seem pointless at school. And so it comes out in this frustrated question. Well, is it even real? Right. I mean, if math wasn't real, how could we be using it to explain our world, right?

The laws of physics, the shape of the universe, how we get to the grocery store. So it must be real. Well, I think that the other question you posed is really, really on the nose, which is what does real mean anyway? And I think that the thing is that even if math isn't real, it can still help us with real things. And what I say at the end of the book, not to give it away the ending, spoiler alert, is that actually, maybe, as they say, that's a feature, not a bug. That the fact that it's made up is part of where its power comes from. And I think that it's a bit like fiction. I love reading fiction.

I know that some people don't like reading fiction and they only want to read nonfiction, but I love reading fiction. And that's made up. But often it can give us really deep insight into the actual world and our actual lives, even though it was made up. And I think sometimes it's because it was made up, a great novelist or great film writer or playwright can make up things that are particularly moving to us and relevant to us. Well, I want to go to the phones because so many people already have called in with their math questions. Yeah, isn't it? It's amazing. Anna in Birmingham, Alabama. Hi, welcome to Science Friday.

Hello. Friday's my favorite day because of science Friday. Well, thank you. I have wondered for a very long time whether we invented math or discovered math, because if you look at things like geometry, it's true. Whether we existed or not, geometry would still be true, but before we existed it wasn't a way to explain it. So did we invent math or did we discover math to explain our universe? What a great question. Thanks for calling. Well, what do you say to that? Did we discover it or invent it? I love that question. And I don't think we have to pick between those things. And I think the question, as posed, even led us to a really great answer, which is that there is a sense in which that we invented and a sense in which we discover it. I think that the concepts exist around us, whether or not we think about them. But what we invent is a way to think about them and a way to talk about. about them. And once we've done that, we can then, it kind of goes back and forth, we can then

discover more things about what we have invented, and then we can invent more things about what we've discovered. And so I really think it's both. And you know, when I'm doing research, I often feel like I'm just wandering around in a jungle and looking at what's already there. But then I invent a kind of language and methods to study what's there and to communicate it to other people. You know, I think we all have a concept of doing research if you're a biologist, you're looking at cells or DNA, if you're a physicist, you're looking at the universe. What kind of research does a mathematician do? Well, this research I do is very abstract, but it's really about spotting patterns. And I think that all of math deep down is about spotting patterns. It can seem like it's all about numbers

and equations, because that's what gets drummed into you at school maybe. But deeper than that, it's about finding similarities between different situations so that we can become more efficient in using our poor little brains to study a lot of things at the same time. So that instead of doing the same thing over and over and over again, we can go, oh, wait, there's something similar about these situations. If I do it once at a more abstract level, then I can go and apply it to all the different places without really having to do extra work. You say over and over in your book that there are no stupid questions when I'm going to. comes to math, why do you think it's so important to hammer this idea? I think too many people have been made to feel really stupid in math classes by asking questions that didn't get answered. And this makes me really sad because, as I say in the book, those questions are often very

profound questions that lead to wonderful, amazing, deep research math. But because they're not part of the curriculum and maybe because people don't have time to answer them because they're under so much pressure to cover the curriculum, they get fobbed off. The really deep questions like, where does math come from? Why does one plus one equal to, you know, why can't I divide by zero? Why isn't infinity a number? Whereas the people who get on really well with math in school just answer the questions, get full marks on every test and then move on. And so the people who have those deeper questions, start thinking like, they're stupid about math. And that makes me sad. And I want to try and change that. Do you think that may have something to do with the instructors, the teachers,

and how good they are at impressing upon their students' math? I had Mr. Cavalero. I'm going to shout out to him in my 10th grade geometry class. And we went in there like typical teenagers. And Mr. Cavalero said, by the time you're done here, you're going to love math. I love math. I love math. I love math, and you're going to love it too. And we all laughed at him. But when we left that class, we all did love math. It was just amazing. Oh, that's wonderful.

Because he was such a good instructor in instilling us his energy, right? Yes. It is absolutely true that some students get to university and they say to me, oh, I had this one really fantastic math teacher. But I don't want to put all the blame on the teachers because teachers are really in a bind. They're not given any freedom, really, to teach in a way that they want. in a way that's expansive and flexible and imaginative because they have so many constraints imposed on them by external bodies

or they have to train everyone to really boring standardized tests and then students get evaluated so in such a black and white manner according to their results on those tests and then the teachers in turn get evaluated according to how the students did on those tests and so there's really no incentive or reward for teaching in a way that is more imaginative and creative. Let's go to Jason in Santa Cruz, California. Hi, Jason. Hi, how you doing? Hi, go ahead. Thank you for taking my call. I have, I think, a couple better questions. Is math relevant versus real? And are statistics real? A statistic I don't think is real math,

so I don't want to go there. But math, we know is relevant. And when we talk about calculating the size of the universe and those kind of giant concepts, of theories using math, we can come up with definite answers every time. But until we get out into those spaces, is it relevant? Right? We know that we always have a definite answer when you do math. It always comes up with the same answer. Well, that's a good question. Do we always, is it relevant? He's attacking whether it's relevant or not, Dr. Chang? It's an interesting question. And I think that is a more relevant question than asking whether math is real, in fact. And I think that I think the point is that whether or not math is real, it is relevant.

Unfortunately, some of the math, the way it is taught to people in school, doesn't seem very relevant to their daily lives. Because let's face it, how often do we need to use solve a quadratic equation in our daily lives or calculate the angles of a triangle based on its sides or something? I don't think most of us do that in our daily lives. I definitely practically never have to do that. But the thing is, it's not just about solving specific problems, and it's not just about getting right answers all the time.

It's about learning how to use our brains well in a logical and rigorous way to build arguments that we can then defend without just saying, I think this is true, my opinion is valid. I never understood those upstream, downstream problems when I'm ever going to use that or one pipe filling a pool and the other pipe draining it. Oh, yeah. Who would ever need that? The laws of physics, as we know, have evolved over the centuries to explain how the universe works and is it built.

Has math evolved to keep up with it? Because math is supposed to explain nature, right? Math has evolved, yes, and it has evolved in two different ways. One, just by the sheer imagination of mathematicians. Sometimes mathematicians just dream things up based on the things that they already dreamt up, just like small children who invent a world and play a game in that world and then keep inventing more parts of that world. That's really how one kind of math research happens.

Another kind is driven by the things that are going on around us. So physics develops and then the new physics needs a new form of math to come with it. And that is relevant to how my field of research has developed, because one of the things it's done is gone higher dimensional. And it looks like we live in a three-dimensional world. But when we think about concepts and different ways of thinking about it, just like Einstein realized we could think of space time as four-dimensional. So suddenly now we need to understand four-dimensional things. And then we need math that can deal with four-dimensional things. So then the math is motivated to go home.

higher dimensional. And that's the kind of, those are two ways that math has continued to develop into the current times. But math, so that concept in physics may not be real, right? They're, they're thinking about what might exist. So the math might be describing something that doesn't exist. The math can get further in a way because math is not constrained by the physical world around us. And that's one of the things I love about it, because math is only constrained by logic and our imagination. So anything we can imagine, as long as it doesn't defy the laws of logic, we can do in math. And I don't think we put enough emphasis on that, because math seems to be about rules and rigidity. But really, it's endless because of our imagination. And as long as you

have an imagination, then there can be math that you can develop. You know, science is all about failure, right? People fail at experiments, and that's good. Is math about failure also? I prefer to think of all those failures as actual parts of the process rather than a failure because if you're still making progress, then I don't think it's really a failure. Good point. And math is about exploring and about asking questions, and every time you answer a question, it opens up more questions. And so I think what's really important is that it's not about fully understanding things. I think sometimes people get made to feel bad at math because they don't feel like they understand it.

it. But the difference with research mathematicians is we accept that we don't understand it, and we use that to drive us to try to understand more. We know we'll never understand all of it, but we just want to understand more and more all the time. Very interesting. This is Science Friday from WNYC Studios. In case you're just joining us, we're talking with Dr. Eugenia Chang, author of Is Math Real? She's based in Chicago, Illinois. And let's go to the phones. Another interesting question. Brian in Ozark, Missouri. Hi, Brian. Hi. I have a slightly stupid question. The more we study the universe and the more we study nature especially, the more we discover that nature has math built into it. My question is, do you think that there is life either on Earth that we

would consider not sentient that is better at math than we are? Can you give me an example of what you're talking about? You look at the way that tree branches form. That's math. The way that anything in nature uses math. Right. I think I'm getting what he's saying, the Fibonacci sequence of leaves and things like that.

What do you say about that, Eugenia? Nature's pretty good at math? It's a great question. Nature is pretty good at math, And I think we should be careful not to anthropomorphize nature, but the question definitely said, is there something that's not sentient? So it's not exactly a sentient being,

but there is something out there that is doing math really well in the world around us. And I think that that's an amazing thing. And when I am doing my research, I do often feel that math is this powerful force that is way bigger than any of us, because it all fits together so well. And the way that nature uses it,

I think it's important to remember that when it comes up with those structures, it's not that nature is a thing that's sitting there going, I think I'm going to use math to make this pineapple spiral. But because the math is often math that involves symmetry and efficiency, and maybe it's that nature is trying to, it just does things in the best way possible, just like gas just fills up the space around it. does the most efficient thing because it wouldn't it can't waste waste energy on extraneous things

it just grows in the way that it can fill things in efficiently and that is you that isn't it's not exactly using math because it's not consciously using math but there is math governing it and I think that is a little bit mysterious yes and I think that that's wonderful and it's not a stupid question it's something that is a mystery that we're still investigating and I think that's something that philosophers of mathematics really think about. Mathematicians often just are a bit more pragmatic about math. We just get on with it. We don't sit around going, oh, where does it come from? Does the nature know it's there? We just sort of do it, but we do marvel at it at the same time. So is it a coincidence that nature uses math? Well, it's not using math in a certain

sense. It's conforming. It's conforming to math, right? Yes. Or the math is there. It's just is inherent to it. I don't think it's a coincidence, unless you take coincidence very literally to mean it is happening at the same time. I think there is something wonderful, mysterious and powerful about it that we probably will never understand because we're just humans. But that's one of the things I love about math. I don't know really where its power comes from, but I do think it has great power. Wow. A mathematician thinking it's mysterious. Many things are mysterious. Yes. Even in the world of mathematics.

Oh, especially in the world of mathematics. It's what keeps me going. If it weren't mysterious, I would find it much less interesting. This is Science Friday. I'm Ira Flato. We're continuing our conversation about math. And my guest is Dr. Eugenia Chang, mathematician and author of Is Math Real, how simple questions lead us to mathematics deepest truths.

So many folks on the phone, I'm going to go right to the phone when we can and see what we can do. about it. Let's go to Eric in Oak Park, Illinois. Hi, welcome to Science Friday. Hi, Ira. So I have a senior in high school. She's taking a college algebra. I don't deign to understand it. So I can't help her. So we got her a tutor. And so just yesterday, she met with her tutor. And when I picked her up, she's like, mom, why do I, why do I have to, you know, I'm never going to use these equations. You know, I don't understand why math is so important. And in fact, she only has to take like seven semesters to graduate high school. So she's like, technically, I don't have to take math next semester,

but it looks good on, you know, when applying for college. So I'm asking, so Dr. Chang, are you saying that like math helps you think critically and argue better in other areas? Thank you for your question, and I have great sympathy for your daughter and anyone else who feels frustrated by things that they don't think they're ever going to use later in life. And the thing is that if you're taking math in high school to fulfill a requirement and then you're doing it because it's going to look good on paper in order to go to college or something, that can be really demoralizing. And the kind of math that you get trained to do in those situations really can. can feel like it is pointless. And honestly, when I look at it, I do agree sometimes that it really is. Because if it's being taught as just algorithms that you have to memorize

to churn out the right answer during tests, I think that has very limited use. I would love it if math could be taught so much more as a way of thinking. So that, yes, it does help you think more clearly and critically through any situation and so that it can help you understand other people's arguments, follow people's arguments, tell when somebody is trying to manipulate you, recognize when the media is biased, spot when there is fake news

or when scientific reporting is not showing what's really going on with the scientific research. All of those things are things I feel I am helped by my mathematical training so that I can get through the world in a much better way and also help other people to understand it better. So I feel every day that my mathematical training helps me with all of those things, even though it's not specifically any particular formula or mathematical method that I'm using.

Wow, Lericka, I hope you are writing all that down. Interesting. So basically what you're saying is math and like arguing politics or talking about philosophy are not mutually exclusive. Yeah, right. And I do think every intellectual, every academic discipline is about, or should be about,

learning how to use your brain well in a particular way and having certain frameworks for deciding what counts as good information and how to develop arguments. Math's point of view is about using logic really well. And that is at the core of, what should be at the core of all our thinking

and all the arguments and discussions we have about everything. And so in that sense, I think it's relevant to everyone. I'm sending you up to Capitol Hill very quickly. Let's go. Something more practical. Let's go to the phones. Cheyenne Pan and Winchester, Massachusetts. Hi, welcome to Science Friday.

Hi. Hi. So I just wanted to bring an Indian perspective to this philosophical argument of whether math is something. real or not. In that, our guest just opened up this discussion saying, you should ask questions first, not try to solve. And so from an Indian perspective, that is how ancient Indians who may or may not have been, I mean, this is getting more visibility these days because of the movie called Man Who Knew Infinity on Sunilwatra Ramunujan. But ancient Indian mathematician, have always, I mean, through trying to understand and ask questions, came up with the concept of

null and infinity. That is why these concepts came to them and they had to mathematically then tackle these things. What I meant to say is any of the numbering system or anything that we have is only for human benefits. And if you take humans out of the equation, all of this creation itself can be, you know, like not so concrete in that case. So for humans to realize the concept of null and infinity is totally unrealistic.

That is why math is just, for humans to figure out their surroundings. Thank you for the call. We have a lot of callers to get to. What do you think about that, Eugenia? Null and infinity. I think that math is there for humans to try to understand things and to try to communicate to other people.

But I do think it's there whether or not humans are trying to use it. I think that the concept of number is something that humans formalized, but I think that even if we had never come up with the concept of number. There still would have been a number of trees standing over there, you know, even if we didn't call it numbers. Dallas and Pensacola, welcome to Science Friday. Yes, thanks for taking my call. I'm just wondering, if we didn't have 10 fingers, would we still be using the base 10 system for counting? Good question. Oh, that's a lovely question. And you know, even though we do have 10 fingers, there were many cultures around the world who used different bases of numbering.

And so for those listeners who can't remember what a base is or never knew, it's the way that when we write our numbers, the last digit is units and then the next one is the number of tens and then the next one is the number of hundreds. So when we write 231, those digits 231 are 200s, 310s and 1. But there are other cultures who, for example, used base 12, and apparently it's because they used their knuckles and the space it way. what did they do? I can't remember.

Or they used base 20s. There are some people who use their knuckles and the spaces between their knuckles. There were some people who used base 20. And that is still evident in the way that the French language uses 20s when you count, you know, 10, 20, 30, 40, 50, 60. And then you go 6010. And then 80 is 420s in French, which is a remnant of using base 20. And so apparently there's another, there are other cultures.

I can't remember all the names, I'm afraid, which is bad of me. but they're in the book who use different body parts. And so you might count using your fingers and your wrists and your elbows and your knees and your toes. So 20 probably comes from using your fingers and your toes. If you use the spaces in between your knuckles, then you might be using base 8 instead. So even with 10 fingers, there were different ways of using them. I think if we had a different number other than 10 fingers, it seems fairly, fairly certain that we would have come up with that base as the main way for using writing numbers.

Question from Alex on Instagram. My question is whether irrational numbers like pie are truly irrational or merely an artifact of our enumeration system. Hmm. Hmm. Well, irrational in this case doesn't mean not logical. It means they aren't expressible as a ratio. So rational comes from the word ratio there, and it means you can't express it as a ratio of two integers. And those numbers are there because, for example, if you draw a square whose sides all have length one,

then that diagonal, the diagonal of that square exists, it's a thing. And then we can ask how long it is. And you just can't express that as a ratio of two things. So I don't think that it's an artifact of our numbering system because it's about what things are ratios and what things aren't ratio. Yeah, we try to do it as 22 over, what, seven, something like that. Right, that's an approximate, it's an approximation, and that's fine. It's good enough. That's an approximation of pie.

It's good enough. I mean, honestly, three is good enough for most purposes in daily life. There you go. Let's go to the phone. So many interesting questions. Larry and Kiddery, Maine. Hi, Larry. Hi, hi there.

I'll turn my radio down. The rule number one of talk radio. Yeah, yeah, I'm in the car. So I love this discussion because I've always loved math, and math came very easy to me in school as opposed to a lot of people I know. But I'm also a musician, and so I, from the beginning, have seen stringed instruments, and that's what I play. as I viewed music as math through stringed instruments and seeing I could figure out tunings and, you know, could understand how the frequency of certain things got either higher or lower, you know, due to limiting the size of the string or the, yeah, the size of the string.

And so I was wondering what our expert would say about that. Eugenia, the connection between math and music is sort of well established in some circles. Yeah, and I also am a mathematician and a musician. And so it's something that I think about. And actually the tuning of stringed instruments has pushed forward some mathematics across history. I believe there was a Chinese mathematician who, a really long time ago, calculated the square root of two to an extraordinary number of decimal places using an abacus for the purposes of tuning his stringed instrument.

And the square root of two is relevant to tuning string instruments because of the way that harmonics work. And harmonics work by ratios, so we're back to ratios again. And when Bach wrote his famous well-tempered clavier, it was to do with the fact that they had invented a way to tune keyboard instruments in a way that it wouldn't sound terrible in any one spot. Previously to that time, they couldn't figure out how to do it because you have to, for various reasons I won't go into it now,

you have to be able to take the 12th root of 2 in order to divide your octave up into 12 equal intervals. And that's a difficult math problem. And so it was only when they figured out how to do that that they could tune pianos so that the octave really would be split into 12 equal intervals. Before that, they had to shove the error into a corner and so they would sort of put it somewhere,

and then that part would sound terrible, and the rest of it would sound good. So you couldn't really write music in all the keys. And when they figured out how to do it better, Bach got really excited and wrote a piece in every key to celebrate. Wow, what a great story. This is Science Friday from WNIC Studios. Hearing great tales about math from our guest, Dr. Eugenie Cheng,

she's author of His Math Reel, how simple questions lead us to mathematics deepest truths. And we are really getting deeply into some really interesting, truths. And one of the truths you hear all the time from a lot of people is, I don't have a brain from math. Like it's some binary thing. You either got the brain from math or you don't. What's your take on this? It's absolutely not true. All the research is pointing to the extreme plasticity of our brains at the moment. Neuropasticity is a very fast developing field, as I understand it. Our brains develop according to how we use them and how we nurture, how we get nurtured to use them. And unfortunately,

thinking that there are math people and non-math people, it does a couple of things. One is it enables you to kind of give up and to have an excuse for not doing as well as you would like to. And the other is it gives educators an excuse not to help people be able to do better because they're just not math people. And I understand this sort of way of making excuses for oneself because I've been guilty of doing that about sport, because I was bad at sport at school and I felt I was made to feel stupid about it. So I just declared, oh, I'm just bad at sport. And then it can go even further where you actually denigrate the thing you're bad at so that it, you regain some of your self-esteem. So you don't just go, I'm bad at math. You go, and also math is pointless. So what's the

point? And I think that if we can encourage everyone to believe that it's not your fault if you have been unable to understand it, maybe you just didn't get the help that you really needed. and if you do get the help, especially early on the help that you need, then you can get better. And it's never too late. That's because people think they don't have them ahead for math, right? What's the sense of asking for help if I'm never going to understand this stuff? Well, what's the sense of asking for help if nobody helps you? And if someone makes you feel bad.

And that's what I really want to get rid of. So while we still make people feel bad for asking questions that we call stupid, then we'll get up in this vicious circle where they don't ask. questions because they get told they're stupid and then they feel that they're bad at it and then they don't get the help they need. And why did you, I'm asking the last question first, they usually ask first, why did you write this book? Because did you feel that people are not getting the understanding where the help they need? I do feel that in all the work I do, in the teaching I do, and in all the public work I do, it was catalyzed by an incident that some of your listeners may remember a few years ago when there was a teenager who went viral on Tick.

talk by asking questions about math and then people piled in and told her how stupid she was. And then mathematicians started saying, those are really great questions. And so I answered them and then people started writing to me from all over the world feeling so moved and validated by having questions answered for the first time and asking me questions that they had always wanted to know the answer to. And so I thought, well, I should turn this into a book so that I can compile all the questions that I've heard people ask many times that previously they have not had them answered and they've been made to feel stupid about it. And so I turned them into a book and I encourage everyone to ask the questions and or read the book and see if the questions get

answered in the book. And I thank you for your work, Dr. Cheng, and taking time to be with us today. It is a great book. Is math real? How simple questions lead us to mathematics deepest truth. Dr. Eugenia Chang. Welcome and thank you for taking time to be with us today. Thanks for having me back. And we have an excerpt from the book at ScienceFriday.com slash real math. That was a great book. And looking back over 2023, we featured a lot of great books on this program, over 35 different books. And if you're like me and you need another reminder of some of those great titles, go to Science Friday.com slash books of 2023. For a full list of this year,

year's best science books. That's Science Friday.com slash books of 2023. And that's it for today. A lot of folks help make this show happen, including Beth Rami, Santiago Flores, Diana Plasker, John Demcosky, Robin Kasmur, and many more. Tomorrow, looking at the science behind one of the most iconic science fiction franchises, Star Trek. We'll see you then. I'm SciFri producer Shoshana Bucksbaum. Thank you.