Kevin Knudson: Welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, a professor of mathematics at the University of Florida. And I'm joined by my other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we just got a lovely few inches of snow last night. So I've developed a theory that podcasts cause snow here. Although it could be the other way. Maybe snow causes podcasts.

KK: Maybe.

EL: It’s hard to tell.

KK: I don't know. It's 85 degrees for today. Sorry.

EL: Yeah. I meant to say don't tell me what the weather is in Florida.

KK: It’s very nice.

EL: It’s too painful.

KK: It’s very nice. Yes, speaking of painful, we were having our pre-banter about I had a little hand surgery yesterday, and I have this ridiculous wrap on my right hand, and it's making me kind of useless today. So I have to do anything left handed, and to control everything on the computer left handed. But you know, it'll resolve my issue on my finger. That'll be good. So anyway, today, we are pleased to welcome Dave Kung. Dave, why don't you introduce yourself?

Dave Kung: Hi, there. I'm Dave Kung. I'm a mathematician by training. I spent 21 years at St. Mary's College of Maryland, and I've recently moved on from there, and I work at the Dana Center, the Charles A. Dana Center, at the University of Texas at Austin. I work with Uri Treisman down there on math ed policy.

KK: It’s very cool. So you got a really serious taco upgrade.

DK: Definitely. I'm still living in Maryland, but I get to visit to Texas every few months.

KK: Are you gonna relocate there? Or are you just gonna stay in Maryland?

DK: I’ll be here for now. Yeah.

KK: I mean, it seems like the sort of work that you could do remotely, it’s policy work mostly, right?

DK: A lot of policy work. There's going to be a fair amount of travel once that's more of a thing, but not all of it will be to Texas. Some of it will actually be in DC, in which case, I'm pretty close.

EL: Yeah, you’re right there.

KK: Yeah. Yeah. Well, they do really great work at the Dana center. I've been involved a little bit with the math pathways business, and it is really vital stuff. And of course, Uri is, like, well, he's the pied piper or something else. I don't know. But when you hear him talk about it, he's an evangelist, you really you can't help but like him.

DK: Yeah. And making sure that we know that students have the right math at the right time with the right supports, we're far from that goal right now, but we can get closer.

EL: Yeah, very important work for all mathematicians to care about.

KK: Yeah, yeah. Okay, but I think we're going to talk a little bit higher-level than math pathways today. So we asked you on to have a favorite theorem. What is it?

DK: My favorite theorem is the Banach-Tarski theorem, which is usually labeled the Banach-Tarski paradox.

KK: Yes. Yeah. So what is let's hear it. Well, let's let our listeners know.

DK: So the Banach-Tarski paradox says the following thing: that you can take a ball, think of a sort of a solid ball, and you can split it up into a finite number of pieces — we’ll come back to that word pieces in a bit — but you can split it up into a finite number of pieces, and then just move those pieces and end up with two balls the same size and the same shape as the original. It is incredibly paradoxical. And I remember hearing this theorem a long time ago, and it just sort of blew my mind.

EL: Yeah, it was one — I think I was an undergraduate, I don't think I'd even taken, like, a real analysis class. But I heard about this and read this book, there's this book about it that I think it's called The Pea and the Sun. Because another statement of — I mean, once you can make two of the same size things out of one, you can make kind of anything out of anything.

DK: Just repeat that process. That's the other statement, you take a pea and you do it enough times. And if you do it, well, you can reassemble them to form a sun. Absurd.

EL: Yeah, I just, I mean, reading it, there was a lot that went over my head at the time, because of what my mathematical background was, but at first, I was like, Okay, this means that math is irrevocably broken. And then after actually reading it, it’s like, okay, it doesn't mean math is broken. So maybe, maybe you should talk a little more about why it doesn't mean math is broken. If you have that perspective. Maybe you do think math is broken.

DK: It certainly feels like. I mean, I think my first reaction was: Cool. Let's do that with gold, right? We'll just take a small piece of gold and split it up and keep doing that. I think there's a lot in this theorem, right? And so I mean, it's one thing to understand it, sort of at a deep sense why it's not absurd. And I think it helps me to think about just sort of, you know, once you know a little bit about infinity and the fact that there are different sizes of infinity. And once you know that somehow the even numbers, the even integers, have the same size as all the integers, which is already sort of weird, this feels a little bit like that. I mean, you could sort of somehow take the odd integers and the even integers, and each one of those is the same size as the full integers. But some of the integers themselves, it's weird to be able to split the integers up into two things, which are the same size as itself.

EL: Right.

DK: And so fundamentally, this is about infinity. And the reason this is a little bit more than that, well, first of all, obviously, the ball here is not countable, right? We're not dealing with a countable number of points. This is uncountable. So now we're talking about the continuum in terms of the cardinality of the points. But I think then the surprising thing is that it works out geometrically. So it's not just about cardinality, but you can do this geometrically. And so you can actually define these sets. And, you know, the word pieces, when we say you can split it up into a number of finite number of pieces. I think the record is somewhere under 10 pieces.

EL: It might be just like five or something. It’s been a while.

DK: But the word pieces is doing a whole lot of work in that statement.

EL: Yes.

DK: And these pieces are not something you could ever do with like a knife and fork or, you know, even define easily. It requires the axiom of choice to define these pieces.

KK: Uh-oh!

DK: So it's really high-level mathematics, to understand how to do these pieces, but the fact that you could do these pieces, and then geometrically it works to just reassemble them, to just rotate and translate these pieces, and get back two balls the size of the origina, that’s just astonishing.

KK: Yeah, that's why I've always had a problem with this. I mean, I, I can read it and understand it and go, yes, you can follow every logical step. But you're right, it doesn't work visually, if you think about it. So can you describe these pieces at all?

DK: They are screwed up. So the analogy I like to make is, you know, if you're working on the interval from zero to one, you know, so first of all, the Banach Tarski paradox does not work in in one dimension. But in terms of these pieces, if you're thinking about the interval from zero to one, you could think of the rational numbers as a piece of that, right? And so it's a piece in the sense that it's part of the whole, it's not a piece in the sense that you could cut it out with a butter knife, or you could model this with a stick of butter or something like that, right? You have to hit 1/2, 1/3, 2/3, you have to hit, you know, 97/101 in there, right? All of those are rational numbers, but you can certainly think of all of the rational points in the interval from zero to one as a single piece. You can talk about them, you can define them, and then you can talk about moving them. And so you can think about it that way, right? So you have all those rational numbers, and you can think of that as one piece. And it certainly is a lot more complicated than that. You know, when you think about Banach, Tarski, one of the things I love to do is with students is to go back and think about what it would mean for a set to be non-measurable. So we can measure things like intervals, but it doesn't take too much mathematics to sort of dive into the fact that there have to be sets — if you have a sense of measure, which works on intervals and things like that, and you want it to have other properties, like when you take two disjoint sets, the measure of the two disjoint sets together should be the sum of the two measures, like really basic properties, it doesn't take too much to be able to prove that there are sets that are non measurable. And once you can prove that, like Oh, then then the world gets really screwed up. Because in the world, in our everyday living, everything seems measurable, in some sense. Like even if you have some screwed up sculpture, you could measure its volume. You dunk it in water and see how the water level rises, right? I mean, it is, it has a measure. And the idea that there is no way to measure something is just incredibly counterintuitive. But once you get that, then it's it's it's a little bit more of a leap, but to understand at a fundamental level that you can define pieces that are so screwed up that you could just rearrange them and get two copies of the original, it’s fantastic.

EL: It’s interesting that you, when you first introduced this, you said it's something about infinity and I remember what — so now I'm thinking I might have taken a real analysis class before I had read this book, because I remember when I read the book, thinking, Oh, this is telling me something about non-measurability and, like, really giving me a concept of what non-measurability does to things. And that's that's how I viewed it. It's a statement about how important measurability is.

DK: And just to be really clear, if these pieces were measurable, right, then we would just have some volume or something like that. And there's no way to double the volume. Right? So you can't you just can't do that. So clearly these pieces, at least some of them have to be non measurable.

KK: Right? So much for your alchemy idea, right?

DK: No more doing this with gold. Okay.

EL: It won’t work on atoms.

DK: There’s this fundamental idea that's permeates mathematics that we can continue to divide things, right? You can get things as small as you want. You see this — this is basically what calculus is all based on, infinitesimally small things. And it's just a reminder that the real world does not work like that. You take a gold atoms, you could keep splitting something up. But eventually, you’ve got one gold atom in each piece and you can't go any smaller than that.

KK: Well, you could, but then

DK: It wouldn't be gold anymore.

KK: Right. All right. So you sort of hinted at this, but why do you love this theorem so much?

DK: I love that it makes you question so many things. I mean, I love paradoxes in general, right? paradoxes are these moments when there's, there's so much cognitive conflict going on and cognitive dissonance, that it forces you, in order to resolve that cognitive dissonance, you have really have to question some other fundamental aspect of the world. So it's something that you were thinking before, is not true, or this paradox is like totally crap, right? So something like that. But in this case, the theorem is true, right? The Banach-Tarski paradox is true. And so it forces you to just go back and question some fundamental ideas about the world. And I love that there are statements like that, that can force you to go back and question so many things. I think we as humans need to do better at this. There are so many things that we just accept, as, as we take for granted, right, and we take them for granted as if they are true with a capital T. And in this case, it's about, like, you can measure all things. But of course, in our in our everyday lives, it's things about the world, it's things about people, it's things about politics, it's things about, you know, topical issues. And we grow up, and it's so normal that we think these things are just part of the world with a capital T on truth. And I love those moments that force us to go back and question those those fundamental “truths,” which all of a sudden turn out to be assumptions, some of which may not be right, or some of which we might want to reassess.

KK: So maybe you're arguing that we should study more mathematics to make the world better.

DK: We certainly should do that, Kevin.

KK: To build a better citizenry, right?

DK: Certainly if we all understood mathematics better, we would have been better off during this pandemic. That's certainly true.

EL: I have a question. So as you mentioned, this is called a paradox often. Do you think it is a paradox?

DK: Yeah, that's a great question. You know, paradox has a couple of different meanings. One is sort of this deep philosophical meaning, like, is it really something which is somehow both true and false simultaneously? And, and this is not in that sense, a paradox. It is a paradox in a sort of weaker sort of more everyday sense of that word where it really throws us into into some cognitive dissonance that forces us to question other things. We can't hold both true that like everything can be measurable and things have volume and you can take a ball, split it into six pieces, rearrange them and get two balls. Those are fundamentally in conflict, and one of them has to go.

KK: It does rely on choice, though, right? So there's something to argue about there. You know, there are those people who deny the axiom of choice.

DK: They're few and far between in the math community, but they're out there. Yeah, they're not measure zero, so to speak. So yeah, so it does, it does use the axiom of choice. So this idea that you can, you know, you can make infinitely many choices. And you can see in the proof where you have to do that. So you end up with uncountably many sets and you choose one point from each one of those uncountably many sets, and that's part of the way you get one of the sets that creates the the Banach-Tarski paradox.

KK: Yeah.

EL: I guess I'm a little surprised that the Banach-Tarski paradox hasn't made more people reject the axiom of choice, to say like, Okay, well, clearly, you can't do this. So, therefore, what is this relying on? Well, it's relying on the axiom of choice.

KK: Well, there's a lot of things about like that, like the Tychonoff product theorem. So that's equivalent to the axiom of choice. So do you want your product of compact spaces to be compact or not, you know, I mean, I don't know.

DK: Yes, I think Evelyn, you're hitting upon this fundamental thing: at a very deep level, math gets weird. Yeah. Right. And you can have things. I mean, you see that in, you know, in Gödel’s work, you see like, well, is that true? Well, you know, it can either be true or false. What's your pleasure today? Right? To the axioms or we can reject it. And, and fundamentally, it kind of doesn't matter. But you know, go ahead like, is there an infinity between the integers and the real line? Like, oh, you know, take your Take your pick?

EL: Yeah, what seems more more useful to you right now? Or what sounds like more fun to play with?

KK: Sure.

DK: And in some sense, that that is kind of the beauty of math, because so much of what mathematicians do is based on wherever you want to start. It’s theoretical, like, oh, well, if we start here, this is where we get. If we start at a different place, we get this other thing. And so you can hold those both in your head, despite the fact that maybe you can't have the hold them simultaneously. Like, either the axiom of choice, you take it to be true or you take it to be false, but you can't take it to really be both. Then things break down. But you can you can do a lot of mathematics either way.

KK: Yeah. So another thing we like to do on this podcast is invite our guests to pair their theorem with something. So I'm curious, what have you chosen to pair with this paradox?

DK: So there's this piece that I played, I think I first played this when I was in high school. I'm a violinist and Evelyn and I share this as string players, but it's called the Enigma Variations. It's by Edward Elgar. And it's a really interesting piece with sort of a fascinating story behind it. And the story tells, or explains the name, enigma. So the idea is that Edward Elgar was sitting there, and he played this little melody, and he was sitting at the keyboard — this is in, like, 1898, late 19th century. And he started riffing on this on this melody, right? And he's like, oh, okay, like, we could play it this way, or this way. And then he started putting names on it, like, Oh, my friend would play it this way. Or, you know, my wife, she would play it in this way. And then he sort of just kept taking that further and further, and he ended up writing a bunch of different movements based on how different people in his life, he imagined would play this particular melody. And one of the it's sort of ingenious if you want something to go viral, I guess. But he never, he never lays out what the what the melody was. And so that's the enigma part of it. Somehow out there somewhere is this melody, and all you're hearing is different people's take on it. He does tell you who the people are, and so some of these movements have initials telling you who some of them are. Some are explicit with names. But all you have is that, right? And it's beautiful music and when you listen to it, some of the some of the movements are really fast loud and really exciting. And others are just incredibly slow and languishing. And it's hard to imagine that all of that could sort of in some sense be based on a single melody, and somehow it is. That's what I'd like to pair it with, the Enigma Variations.

KK: Do you have a favorite so that we can insert a clip?

DK: Um, let's see. I do. So probably the most famous of these is called Nimrod. It's an incredibly slow movement. I've played it a bunch of times. It's the sort of thing that, you know, if you want to get people to cry in a movie you could play a little bit of Nimrod. I don't think I've ever successfully been on stage playing this without crying. It is that emotional come and it's also, for me it's more emotional if you play it slower, so it's incredibly evocative. [Clip of the Nimrod variation]

KK: Do you still play?

DK: I do. I do. Well, in theory. We’re in this pandemic where I haven’t been able to play much. But yeah, I play in a local community orchestra and play with my daughter and yeah, I've had some had some fun over during the pandemic, by myself playing, picking up some old pieces and playing them.

KK: Very cool.

EL: Yeah, I was thinking probably everyone has heard Nimrod without realizing it, because it's in the background. If it's not Adagio, for strings, it's Nimrod in the background of that, you know, swelling, emotional scene, your farewell or someone’s dying or whatever. You know, it’s there.

DK: And when you hear it, you can hear little hints of Pomp and Circumstance, which we hear all the time during graduations. And so you can see like, yeah, those are, those are similar sort of in structure, in the melodies and the harmonies and how they fit together,

KK: Right, plus it’s public domain by this point, so you can just throw it into movies pretty easily. Right? That's true, too.

EL: Yeah, I guess you're the practical one here.

DK: Yeah, that's important when you're doing things like podcasts. Yeah, yeah.

KK: Yeah. All right. So um, that's a really good pairing. I like that a lot. So we also like to give our guests a chance to promote anything they want to promote. So So we've talked about the Dana Center a little bit. Anything you want to pitch? Where can we find you online?

DK: You can find me at the at the date of the Charles Dana center, it's UT Dana Center. dot.i.com. I think maybe .edu I'm not sure actually sure. [Editor’s note: it’s neither! It’s actually https://www.utdanacenter.org/], but it’s a pretty easy web search to find us. You know, we're trying to make sure that mathematicians, that everybody has an opportunity to see themselves as a mathematician, and that everybody has access to the right math for them at the right time with the right supports. And so much of the math community, so much of our curriculum is grounded in things that are really old. T algebra, geometry, algebra II sequence was decided in 1892 by a group of 10 white men in the northeast who decided that that would be the right thing. And this focus on calculus comes out of the Cold War, it comes out of the need to produce a small number of engineers who are going to work pencil and paper, and then with massive computers that they could program, to win the Cold War, to beat the Soviets into space and to do all of that. That's no longer the world we live in. And we need to expand what we think of as mathematics. And it's not that calculus isn't important, it still is, but maybe not sending everybody in that direction would be better. And so some students would be much better off if they took statistics. Other students, Quantitative Literacy would be great. We're swimming in so much data, we don't know what to do with and the careers out there that deal with data are taking off like you wouldn't believe. What are the tools we need to give students so that they can deal with that?

KK: Yeah.

DK: All right. So all of these things are just updates that we need to do for their math curriculum. And and updating a system that's so complicated, with so many moving parts is difficult. And so that's part of what we work on at the Dana Center. And if anybody's out there who wants to help work on those things and at the same time, make sure all of these systems are equitable, because we know they haven't been in the past, and make sure all students have access to high-quality math instruction independent of your zip code, or who your parents are your economic circumstances. Those are the sorts of things that we work on at the Dana Center.

KK: Yeah, that's really important work. I mean, it's funny to hear you talk about this, because I've been saying the same thing. You know, the math degree that we still give out more or less is what I got 30 years ago, which isn't different than what they were handing out 30 years before that. I mean, things have changed. And we're still sort of stuck that way. It's unfortunate. And they're still doing it on the high schools. You know, my son when he was going through high school, they just marched him in lockstep through algebra, geometry. And he's a musician. I mean, it's great, you know, it's good for him, but I don’t know.

DK: And even some of these topics that are still important, like, I think algebraic thinking is still important. I think a lot of people do algebraic thinking out there all the time. It's just, it's not pencil and paper. It's not symbol manipulation. The most popular place that people use it, their algebraic thinking is when they're working with spreadsheets, right? Writing formulas in spreadsheets is incredibly algebraic. You have these placeholders that are really just variables, things change, some things stay the same. It's highly algebraic. And we could be teaching algebra using that so that everybody, every student coming out of algebra would have a basis for understanding how algebra is going to be used. You know, it's something like 98% of employers want their employees to be able to use spreadsheets. Well, there's a perfect example. We could we could tweak the curriculum to take something that right now is seen as kind of useless and make it very useful and in fact vital. And it's still teaching essentially the same core ideas, but it's really approaching them from a very different perspective.

KK: Yeah, yeah. All right.

DK: And as long as you give me the space Kevin, you know talking about mathematics and music, so I I have to take a little space. I did this set of 12 lectures for the Great Courses on math and music. And it's a sort of a tour through the listening of music. So like when when we record something, and that music eventually gets into your ear, like every step on that process involves mathematics. And over the course of 12 lectures, we talk about rhythm, we talk about harmonics, we talk about tuning, we talk about lots of, we even talk about the digital side of it. So there's a ton of mathematics that goes into CDs, there's a lot of mathematics that goes into compression that we're using now, so that we can, you know, actually hear each other from this far away. And all of that, there's just lots of mathematics embedded in that. Yeah, so it's really fun, it was just a real honor to be able to do that series. So you can find that out there in the Great Courses.

KK: Yeah. Do you talk about why you can't have perfect tuning on a piano or anything like that?

DK: Yeah, in fact, I got to demonstrate tuning. So they brought in a baby grand for me to do this on. And so we tuned an octave and pulled it out of tune a little bit and you could hear the beats. Again, like these things that I saw in high school and in a trig class, but I had no idea they were applicable, right? So I mean, you can actually hear — there are trig identities that tell you, if you play two notes that have really similar frequencies, that's kind of equivalent to something that goes a wah-wah-wah, it has beats in it. And you can actually do that. And so I played that and tuned it, so that was perfect. And then I took a fifth and we did the mathematics to figure out if you want everything to be sort of in tune, how out of tune do your fifths need to be right, which is, you know, fascinating thing. And I know, Evelyn, you've written about this, what was the column called, like, the saddest thing you know about the natural numbers?

EL: Yeah. Something like that.

DK: Like three is not a power of two or something like that.

EL: It is so sad.

DK: It is sad. And, you know, there are mathematical facts at the heart of that, and so much that we hear in music. And I get notes every week from somebody who has had these nagging questions in the back of their head about how music and math are related. And they watch these 12 lectures, and they're just so thrilled to sort of unpack some of that.

EL: Yeah, it's neat. And it's also it's not just, like, the sound waves and the math, which is definitely part of it. But there's also this extra perception in our ears and our brains that is involved and has some math, but that has some, basically, I don't know, wizardry that our brains do to be like, Oh, when I hear this kind of thing, I often hear this kind of thing with it, so it probably came together, I just missed something there.

DK: There’s a there's a whole section on auditory illusions, which are really fascinating, ways in which our brain can trick itself or, or like, because it's been useful in the past, like, our brain does certain things. And so you can intentionally use that to have these auditory illusions, which are really just fascinating. And my favorite fact about this is that is that essentially, your ear is doing a Fourier transform. When you are listening, your ear is doing this Fourier transform. And you know, just sort of physically doing it and breaking down the sound into its constituent frequencies. And that is just a phenomenally cool idea.

KK: That our brains are that sophisticated.

DK: That our brains somehow involved evolved to do something that we didn't describe mathematically until the 19th century, right? Like, oh, now we know what our brains did just sort of by, you know, random chance and little tweaks to random changes in a DNA sequence. And somehow we got to like, oh, oh, yeah. And that's now called Fourier transform.

EL: Yeah. All right. It's a lot of fun. I feel like we could almost do a whole nother podcast episode about one of these facts.

KK: Probably.

DK: That would be so much fun.

KK: Yeah. Okay. Yeah. Well, maybe we’ll have you back for part two sometime.

DK: That'd be great. Yeah.

KK: Well, this has been great fun, Dave. I'm surprised Banach-Tarski made it this long without being somebody's favorite. This is pretty good.

EL: Yeah. Yeah.

KK: Because we've had we've had repeats before, but but this one, I'm surprised. So thanks for joining us. And yeah.

DK: Thank you guys so much. I love the work that you do, and I really appreciate it.

On this episode of My Favorite Theorem, we welcomed Dave Kung from the Dana Center at the University of Texas at Austin to talk about the Banach-Tarski paradox/theorem. Here are some links you might enjoy:
Kung's website and Twitter account
The Dana Center website
Leonard Wapner's book The Pea and the Sun about the Banach-Tarski paradox
A shorter article by Max Levy explaining the theorem
A primer on the axiom of choice from the Stanford Encyclopedia of Philosophy

The Tychonoff product theorem
Kung's course How Music and Mathematics Relate from the Great Courses

Evelyn's article The Saddest Thing I Know about the Integers, mourning the fact that no power of 3 is also a power of 2