Kevin Knudson: Welcome to My Favorite Theorem, a podcast about mathematics, theorems, and, I don't know, just about anything else under the sun, apparently. I'm Kevin Knudson. I'm one of your hosts. I'm a professor of mathematics at the University of Florida. This is your other host.
Evelyn Lamb: Hi, I'm Evelyn lamb. I'm a freelance math and science writer based in Salt Lake City. So how are things going?
KK: It's homecoming weekend. We're recording this on a Friday, and for people who might not be familiar with Southeastern Conference football, it is an enormous thing here. And so today is is a university holiday. Campus is closed. In fact, the local schools are closed. There's a big parade that starts in about 20 minutes. My son marched in it for four years. So I've seen it. I don't need to go again.
KK: I had brunch at the president's house this morning, you know. It's a big festive time. I hope it doesn't get rained out, though. It's looking kind of gross outside. How are things for you?
EL: All right. Yeah, thankfully, no parades going on near me. Far too much of a misanthrope to enjoy that. Things are fight here. My alarm clock-- we're also recording in the the week in between the last Sunday of October and the first Sunday of November.
EL: In 2007, the US switched when it went away from Daylight Saving back to Standard Time to the first Sunday of November. But my alarm clock, which automatically adjusts, was manufactured before 2007.
KK: I have one of those too.
EL: Yeah, so it's constantly teasing me this week. Like, "Oh, wouldn't it be nice if it were only 7am now?" So yeah.
KK: All right. Well, yeah, first world problems, right?
EL: Yes. Very, very much.
KK: All right. So today, we are thrilled to have Skip Garibaldi join us. Skip, why don't you introduce yourself?
Skip Garibaldi: My name is Skip Garibaldi. I'm the director at the Center for Communications Research in La Jolla.
KK: You're from San Diego, aren't you?
SG: Well, I got my PhD there.
SG: Yeah, ish.
SG: So I actually grew up in Northern California. But once I went to San Diego to get my degree, I decided that that was really the place to be.
KK: Well, who can blame you, really?
EL: Yeah, a lot to love there.
KK: It's hard to argue with San Diego. Yeah. So you've been all over. For a while you're at the Institute for Pure and Applied Math at UCLA.
SG: Yeah, that was my job before I came to the Center for Communications Research. I was associate director there. That was an amazing experience. So their job is to host conferences and workshops which bring together mathematicians in areas where there's application, or maybe mathematicians with different kinds of mathematicians where the two groups don't really talk to each other. And so the fact that they have this vision of how to do that in an effective way is pretty amazing. So that was a great experience for me.
KK: Yeah, and you even got in the news for a while. Didn't you and a reporter, like, uncover some crime syndicate? What am I remembering?
SG: That's right. Somehow, I became known for writing things about the lottery. And so a reporter who was doing an investigative piece on lottery crime in Florida contacted me, and I worked closely with him and some other mathematicians, and some people got arrested. The FBI got involved and it was a big adventure.
KK: So Florida man got arrested. Never heard of that. That's so weird.
SG: There's a story about someone in Gainesville in the newspaper article. You could take a look.
KK: It wasn't me. It wasn't me, I promise.
EL: Whoever said math wasn't an exciting field?
KK: That's right.
Alright, so, you must have a favorite theorem, Skip, what is it?
SG: I do. So you know, I listened to some of your other podcasts. And I have to confess, my favorite theorem is a little bit different from what your other guests picked.
EL: Good. We like the the great range of things that we get on here.
SG: So my favorite theorem for this podcast answers a question that I had when I was young. It's not something that is part of my research today. It's never helped me prove another theorem. But it answers some question I had from being junior high. And so the way it goes, I'm going to call it the unknowability of irrational numbers.
So let me explain. When you're a kid, and you're in school, you probably had a number line on the wall in your classroom. And so it's just a line going left to right on the wall. And it's got some markings on it for your integers, your 0,1,2,3, your -1,-2,-3, maybe it has some rational numbers, like 1/2 and 3/4 marked, but there's all these other points on that number line. And we know some of them, like the square root of two or e. Those are irrational, they're decimals that when you write them down as a number-- like π is 3.14, we know that you can't really write it down that way because the decimal keeps on going, it never repeats. So wherever you stop writing, you still haven't quite captured π.
So what I wondered about was like, "Can we name all those points on the number line?
SG: Are π and e and the square root of two special? Or can we get all of them? And it comes up because your teacher assigns you these math problems. And it's like "x^2+3x+5=0. Tell me what x is." And then you name the answer. And it's something involving a square root and division and addition, and you use the quadratic formula, and you get the answer.
So that's the question. How many of those irrational can you actually name? And the answer is, well, it's hard.
KK: Like weirdly, like a lot of them, but not many.
SG: So if we just think about it, what would it mean to name one of those numbers? It would mean that, well, you'd have to write down some symbols into a coherent math problem, or a sentence or something, like π is the circumference of a circle over a diameter. And when you think about that, well, there's only finitely many choices for that first letter and finitely many choices for that second letter. So it doesn't matter how many teachers there are, and students, or planets with people on them, or alternate universes with extra students. There's only so many of those numbers you can name. And in fact, there's countably many.
KK: Right. Yeah. So are we talking about just the class of algebraic numbers? Or are we even thinking a little more expansively?
SG: Absolutely more expansive than that. So for your audience members with more sophisticated tastes, you know, maybe you want to talk about periods where you can talk about the value of any integral over some kind of geometric object.
KK: Oh, right. Okay.
SG: You still have to describe the object, and you have to describe the function that you're integrating. And you have to take the integral. So it's still a finite list of symbols. And once you end up in that realm, numbers that we can describe explicitly with our language, or with an alien language, you're stuck with only a countable number of things you can name precisely.
KK: Well, yeah, that makes sense, I suppose.
SG: Yeah. And so, Kevin, you asked about algebraic numbers. There are other classes of numbers you can think about, which, the ones I'm talking about include all of those. You can talk about something called closed form numbers, which means, like, you can take roots of polynomials and take exp and log.
SG: That doesn't change the setup. That doesn't give you anything more than what I'm talking about.
EL: Yeah. And just to back up a sec, algebraic numbers, basically, it's like roots of polynomials, and then doing, like, multiplication and division with them. That kind of thing. So, like, closed form, then you're expanding that a little bit, but still in a sort of countable way.
SG: Yes. Like, what kinds of numbers could you express precisely if you had a calculator with sort of infinite precision, right? You're going to start with an integer. You can take it square root, maybe you can take its sine, you know. You can think about those kinds of numbers. That's another notion, and you still end up with a countable list of numbers.
KK: Right. So this sounds like a logic problem.
SG: Yes, it does feel that way.
SG: So, Kevin and Evelyn, I can already imagine what you're thinking. But let me say it for the benefit of the people for whom the word "countable" is maybe a new thing thing. It means that you can imagine there's a way to order these in a list so that it makes sense to talk about the next one. And if you march down that list, you'll eventually reach all of them. That's what it means. But the interesting thing is, if you think about the numbers on the number line, we know going back to Cantor in the 1800s that those are not countable. You use the so-called diagonalization argument, if you happen to have seen that.
EL: Yeah. Which is just a beautiful, beautiful thing. Just, I have to put a plug in for diagonalization.
KK: Oh, it's wonderful.
SG: I've been thinking about it a lot in preparation for this podcast. I agree.
SG: So what that means is that that's the statement, these irrational numbers, you can't name all of them, because there are uncountably many of them, but only countably many numbers you can name.
It sort of has a hideous consequence that I want to mention. And it's why this is my favorite theorem. Because it says, it's not just that you can't name all of them. It's just much worse than that. So the reason I love this theorem is not just that it answers a question from my childhood. But it tells you something kind of shocking about the universe. So when you--if you could somehow magically pick a specific point on the number line, which you can't, because you know, there's--
SG: You have finite resolution when you pick points in the real world. But pretend you could, then the statement is the chance that the number you picked was a number you could name precisely is very low. Exactly. It's essentially zero.
SG: So the technical way to say this is that the countable subset of real numbers has Lebesgue measure zero.
SG: So I was feeling a little awkward about using this as my theorem for your podcast, because, you know, the proof is not much. If you know about countable and uncountable, I just told you the whole proof. And you might ask, "What else can I prove using this fact?" And the answer is, I don't know. But we've just learned something about irrational numbers that I think some of your listeners haven't known before. And I think it's a little shocking.
EL: Yeah, yeah. Well, it sounds like I was maybe more of a late bloomer on thinking about this than you, because I remember being in grad school, and just feeling really frustrated one day. I was like, you know, transcendental numbers, the non-algebraic numbers are, you know, 100% of the number line, Lebesgue measure one, and I know like, three of them, essentially. I know, like, e, π, and natural log two. And, you know, really, two of them are already kind of, in a relationship with each other. They're both related to e or the natural log idea. It's just like, okay, 2π. Oh, that's kind of a cheap transcendental number.
Like there's, there's really not that much difference. I mean, I guess then, in a sense, I only know, like, one irrational number, which is square root of 2, like, any other roots of things are non-transcendental, and then I know the rationals, but yeah, it's just like, there are all these numbers, and I know so few of them.
KK: Right. And these other these other things, of course, when you start dealing with infinite series, and you know, you realize that, say, the Sierpinski carpet has area zero, right? But it's uncountable, and you're like, wait a minute, this can't be right. I mean, this is, I think why Cantor was so ridiculed in his time, because it does just seem ridiculous. So you were sitting around in middle school just thinking about this, and your teacher led you down this path? Or was it much later that you figured this out?
SG: Well, I figured out the answer much later. But I worried about it a lot as a child. I used to worry about a lot of things like, your classic question is--if you really want to talk about things I worried about as a child--back in seventh grade, I was really troubled about .99999 with all the nines and whether or not that was one.
EL: Oh yeah.
SG: And I have a terrible story about my eighth grade education regarding that. But in the end, I discovered that they are they are actually equal.
KK: Well, if you make some assumptions, right? I mean, there are number systems, where they're not equal.
SG: Ah, yeah, I'd be happy--I'm not prepared to get into a detailed discussion of the hyperreals.
KK: Neither am I. But what's nice about that idea is that, of course, a lot depends on our assumptions. We we set up rules, and then with the rules that we're used to, .999 repeating is equal to one. But you know, mathematicians like sandboxes, right? Okay, let's go into this sandbox and throw out this rule and see what happens. And then you get non Euclidean geometry, right, or whatever.
KK: Really beautiful stuff.
SG: I have an analogy for this statement about real numbers that I don't know if your listeners will find compelling or not, but I do, so I'm going to say it unless you stop me.
EL: Go for it.
SG: Exactly. So one of the things I find totally amazing about geology is that, you know, we can see rocks that are on the surface of the earth and inspect them, and we can drill down in mines, and we can look at some rocks down there. But fundamentally, most of the geology of the earth, we can't see directly. We've never seen the mantle, we're never going to see the core. And that's most Earth. So nonetheless, there's a lot of great science you can do indirectly by analyzing as an aggregate, by studying the way, earthquake waves propagate and so on. But we're not able to look at things directly. And I think that has an analogy here with the number line, where the rocks can see on the surface are the integers and rationals. You drill down, and you can find some gems or something, and there's your irrational numbers you can name, and then all the ones you'll never be able to name, no matter how hard you try, how much time there is, how many alternate universes filled with people there are, you'll never be able to name, somehow that's like the core because you can't ever actually get directly at them.
EL: Yeah. I like this analogy a lot, because I was just reading about Inge Lehmann who is the Danish seismologist (who I think of as an applied mathematician) who was one of the people who found these different seismic waves that showed that the inner core had the liquid part--or I guess the core had the liquid part and then the solid inner core. She determined that it couldn't all be uniform, basically by doing inverse problems where, like, "Oh, these waves would not have come from this." So that's very relevant to something I just read. Christiane Rousseau actually wrote a really cool article about Inge Lehmann.
SG: Yes, that's a great article.
EL: So yeah, people should look that up.
KK: I'll have to find this.
EL: great analogy. Yeah.
KK: So, we know now that this, this is a long time there for you. So that's another question we've already answered. So, okay, what does one pair with this unknowability?
SG: Ah, so I I think I'm going to have to pair it with one of my favorite TV shows, which is Twin Peaks.
SG: So I watch the show, I really enjoy it. But there's a lot of stuff in there that just is impossible to understand.
And you can go read the stuff the people wrote about it on the side, and you can understand a little bit of it. But you know, most of it's clearly never meant to be understood. You're supposed to enjoy it as an aggregate.
KK: That's true. So you and I are the same age, roughly. We were in college when Twin Peaks was a thing. Did you did you watch it then?
SG: No, I just remember that personal ads in the school paper saying, "Anyone who has a video recording of Twin Peaks last week, please tell me. I'll bring doughnuts."
EL: You grew up in a dark time.
SG: Before DVRs, yeah.
KK: That's right. Well, yeah. Before Facebook or anything like that. You had to put an ad in the paper for stuff like this, yeah.
EL: Yeah, I'm really, really understanding the angst of your generation now.
KK: You know what, I kind of preferred it. I kind of like not being reached. Cell phones are kind of a nuisance that way. Although I don't miss paying for phone calls. Remember that, staying up till 11 to not have to pay long distance?
KK: Alright, so Twin Peaks. So you like pie.
SG: Yeah, clearly. And coffee.
KK: And coffee.
SG: And Snoqualmie.
KK: Very good.
SG: I don't know if you--
KK: Sure. I only sort of vaguely remember-- what I remember most about that show is just being frustrated by it, right? Sometimes you'd watch it and a lot would happen. It's like, "Wow, this is bizarre and weird, and David Lynch is a genius." And then there'd be other shows where nothing would happen.
KK: I mean, nothing! And, you know, also see Book II of Game of Thrones, for example, where nothing happens, right? Yeah. And David Lynch, of course, was sort of at his peak at that time.
KK: All right. So Twin Peaks. That's a good pairing because you're right, you'll never figure that out. I think a lot of it was meant to be unknowable.
SG: Yes. Yeah. Have you seen season three of Twin Peaks? The one that was out recently?
KK: No, I don't have cable anymore.
SG: About halfway through that season, there's an episode that is intensely hard to watch because so little happened on it. And if you look at the list the viewership ratings for each episode, there's a steep drop-off in the series at that episode. So this is like the most unknowable part of the number line if you if you follow the analogy.
KK: Okay. All right. That's interesting. So I assume that these these knowable numbers are probably fairly evenly distributed. I guess the rationals are pretty evenly distributed. So yeah.
So So our listeners might wonder if there's some sort of weird distribution to these things, like the ones that you can't name, do they live in certain parts? And the answer is no, they live everywhere.
SG: Yes. That's absolutely right.
EL: I wonder, though, if you can kind of--I'm thinking of continued fraction representations, where there is an explicit definition of number that's well-approximable versus badly-approximable numbers. I guess those are approximable by rationales, not by finite operations or closed form. So maybe that's a bad analogy.
KK: Mm hmm.
SG: Well, if you or your listeners are interested in, thinking about this question some more, then you can Google closed-form number. There's a Wikipedia entry to get people started. And there are a couple of references in there to some really well-written articles on the subject, one by my friend Tim Chow, that was an American Mathematical Monthly, and another one by Borwein and Crandall that was in the Notices of the AMS that's for free on the internet.
EL: Oh, great.
KK: Okay, great. We'll link to those.
EL: And actually, here's, this question, I'm not sure, so I'll just say, is this the same as computable or is closed form a different thing from computable numbers?
SG: Yeah, that's a good question. So there's not a widely-agreed upon definition of the term closed form number. So that's already a question. And then I'm not sure what your definition of computable is.
EL: Me neither.
EL: No, I've just heard of the term computable. But yeah, I guess the nice thing is no matter how you define it, your theorem will still be true.
SG: That's right. Exactly.
EL: There's still only countable.
KK: And now we've found something else unknowable: are these the same thing?
SG: Those are really hard questions in general. Yeah. That's the main question plumbed in those articles and referred to is: if you define them in these different ways, how different are they?
EL: Oh, cool.
SG: If you take a particular number, does it sit in which set? Those kinds of questions. Yeah, those are really hard usually, much like you said, what are the transcendental numbers that are--are certain numbers transcendental or not can be a hard question to answer.
EL: Yeah, yeah, even if you think, "Oh yeah, this certainly has to be transcendental, it takes a while to actually prove it."
KK: Or maybe you can't. I wonder if some of those statements are even actually undecidable, but again, we don't know. All right, we're going down weird rabbit holes here. Maybe David Lynch could just do a show.
SG: That would be great.
KK: Yeah, there would just be a lot of mathematicians, and nothing would happen
SG: And maybe owls.
KK: And maybe owls. Well, this has been great fun. Thanks for joining us before you head off to work, Skip. Our listeners don't know that it's you know, well, it's now nine in the morning where you are. So thanks for joining us, and I hope your traffic isn't so bad in La Jolla today.
SG: Every day's a great day here. Thank you so much for having me.
KK: Yeah. Thanks, Skip.