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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