Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcasts where there's no quiz at the end. I’m coming up with a new tagline for it.
Kevin Knudson: Good.
EL: I just thought I'd throw that in. Yeah, so I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer from Salt Lake City—or in Salt Lake City, Utah, not originally from here. And here's your other host.
KK: I’m Kevin Knudson, professor of mathematics at the University of Florida in Gainesville, but not from Gainesville. This is part of being a mathematician, right? No one lives where they're from.
EL: Yeah, I guess probably a lot of professions could say this, too.
KK: Yeah, I don’t know. It’s also a sort of a generational thing, right? I think people used to just tend to, you know, live where they grew up, but now not so much. But anyway.
KK: Oh, well, it's okay. I like it here.
EL: Yeah. I mean, it's great here right now it's spring, and I've been doing a ton of gardening, which always seems like such a chore and then I'm out smelling the dirt and looking at earthworms and stuff, and it's very nice.
KK: I’m bird watching like crazy these days. Yesterday, we went out and we saw the bobolinks were migrating through. They're not native here, they just come through for, like, a week, and then they're gone.
EL: The what?
KK: Bobolinks, B-O-B-O-L-I-N-K. They kind of fool you, they look a little bit like an oriole, but the orange is on the wrong side. It's on the back of the neck instead of underneath.
EL: Okay, I'll have to look up a picture of that later.
KK: And then this morning for the first time ever, we had a rose-breasted grosbeak at our feeder. Never seen one before and they're not native around here, they just migrate through. So this is
EL: Very nice. Yes.
KK: This is what I'm doing in my late middle age. This is what I do. I just took up bird watching, you know?
EL: Yeah. Well, I can see the appeal.
KK: Yeah, it's great.
EL: Yes. But we are excited today to be talking with Adriana Salerno. Do you want to introduce yourself?
Adriana Salerno: Hi. Yeah, I'm Adriana Salerno. Now I am an associate professor of math at Bates College in Maine. And I am also not from Maine. I live in Maine. I'm originally from Caracas, Venezuela, so quite a ways away.
AS: Again, you don't choose where you live, but maybe you get to choose where you work. So that's nice.
EL: Yeah. And you're you're not only a professor there, but you're also the department chair right now, right?
AS: Oh, yeah. Yeah, I'm trying to forget. No, I’m kidding.
KK: You know, speaking of, before we we started recording here, I spent my afternoon writing annual faculty evaluations. I’m in the first year as chair. I have 58 of them to write.
AS: Oh, I don't have to do those, which I'm very happy about. But we are hiring a staff position, and I'm in charge of that. And that's been a lot.
EL: And we actually met because both of us have done this mass media fellowship for people interested in math or science and writing. And so you've done a lot of writing not for mathematicians as well, throughout your career path.
AS: Yeah, yeah. I mean, I did the mass media fellowship in 2007. And since then, I've been trying to write more and more about mathematics for a general audience. These days, I mostly spend time writing for blogs for the AMS. And right now I'm editing and writing for inclusion/exclusion. I wish I had more time to write than I do. It's one of those things that I really like to do, and I don't think I do enough of, but these opportunities are great because I get to use those—or scratch that itch, I guess, by talking to you all.
KK: Well, so speaking of, we assume you have a favorite theorem that you want to tell us about. What is it?
AS: Well, so it's always hard to decide, right? But I guess I was inspired by a conversation I had with Evelyn at the Joint Math Meetings. So I've decided my favorite theorem is Cantor's diagonalization argument that the real numbers do not have the same cardinality as the natural numbers.
EL: Yes, and I’m so excited about this! Ever since we talked at the Joint Meetings, I’ve been very excited about getting you to talk about this.
AS: Good. Good.
EL: Because really, it’s such a great theorem.
AS: Yeah. Well, I was thinking about it today. And I'm like, how am I going to explain this? But I have chosen that, and I'm sticking with it. Yeah.
AS: So yeah, it’s—one of the coolest things about it is sort of it’s this first experience that you have, as a math student—at least it was for me—where you realize that there are different sizes of infinity. And so another way of saying that is that this theorem shows, without a doubt, I believe—although some students still doubt me after we go over it—it shows that you can have different sizes of infinity. And so the first step, even, is to say, “How do you decide if two things have the same size of infinity?” Right? And so it's a very, very lovely sort of succession of ideas. And so the first thing is, how do you decide that two things are the same size? Well, if they're finite, you count them, and you see that you have the same number of things. But even when things are finite—and say, you're a little kid, and you don't know how to count—another way of saying there's the same number of things is if you can match them up in pairs, right? So you know, if you want to say I have the same number of crayons as I have apples, you can match a crayon to an apple and see that you don't have anything left over, right?
AS: And so it's just a very natural idea. And so when you think about infinite sets—or not even infinite sets—but you can think of this idea of size by saying two things are the same size if I can match every element in one set to every element in another set, just one by one. And so I really like, I'm borrowing from Kelsey Houston-Edwards’ PBS show, but what I really like that she said that you have two sets, and every element has a buddy, right? And so then I love that language, and so I'm borrowing from from her. But then that works for finite sets, but you can extend it to an infinite set.You can say, for example, that two infinite sets are the same size if I can find a matching between every element in the first set and every element in the second set. It’s very hard to picture in your head, I think, but we're going to try to do this. So for example, you can say that the natural numbers, the counting numbers, 1, 2, 3, 4, etc, have the same size as the even numbers, because you can make a matching where you say, “Match the number 1 with the number 2 on the other side. And then the number 2, with the number 4 on the other side.” And you have all the counting numbers, and for every counting number, you have two times that number as the even buddy.
EL: Yeah. And I think this is, it's a simple example that you started with, but it even hints at the weirdness of infinity.
EL: You’ve got this matching, but the even numbers are also a subset of the natural numbers. Ooh, things are going to get a little weird here.
KK: Clearly, there aren’t as many even numbers, right?
KK: This is where you fight with your students all the time.
AS: That’s exactly—so when you're teaching this, the first thing you do is talk about things that have the same cardinality. And then everybody, it can take a while, you know, like, infinity is so weird that you can actually do these matching. So Hilbert’s infinite hotel is a really great way of doing this sort of more conceptually. So you have infinitely many rooms. And so for example, suppose that rooms numbers from 1, 2, 3, to infinity, mean, and so on. Yes, you have to be careful because infinity is not a number. You have to be careful with that. But say that all the rooms are occupied. And so then, you know, say someone shows up in the middle of the night, and they say, “I need a room.” And so what you do if you're the hotel manager is you tell everyone to move one room over. And so everyone moves one room over and you put this person in, and room number one. And so that's another way of seeing that. So the one-to-one pairing, or the matching here is every person has a room. And so the number of rooms and the number of people are the same—the word is cardinality because you don't want to say number because you can't count that.
AS: And so you you say cardinality instead. But it's really weird, right? Because the first time you think about this, you say, “Well, you know, there's infinity, and there's infinity plus one.” That's like the kind of thing that you would say as a kid, right? And they're the same! When you have the natural numbers and the natural numbers and one extra thing, or like with zero, for example—unless you're in the camp that says zero is an actual number—but we're not going to get to that discussion right now.
KK: I’m camp zero is a natural number.
AS: Okay. I feel like I know maybe half people who say zero is a natural number and the other half say it's not. And I don't think anyone has good arguments other than, ah, it must be true! And so then the cool thing is, once you start doing that, then you start seeing, for example—and these are, these are kind of tricky examples, it can get tricky. Like, you can say that the integers like the positive whole numbers, negative whole numbers and zero, that also has the same cardinality as the natural numbers. Because you can just start with zero—I mean, basically, when you want to say that something has the same cardinality as the natural numbers, what you're really trying to do is to find a buddy, so you're trying to pair someone with one or two, or three. But really, what you can do is just list them in order, right? Like you can have like the first one, the second one, the third one, the fourth one, and you know that that's a good matching. It's like the hotel. You can put everyone in a room. And then you know they're the same number. Everyone has a room. So with the integers, for example, the whole numbers, positive, negative and zero, then you can say, “Okay, put zero first, then one, then negative one, and two, then negative two then three, then negative three,” and then they're the same size, right? And so once you start thinking about this—I remember this pretty clearly from from college—once you start thinking about this, then you're like, “Well, obviously, because infinity is infinity.” That’s the next step. So the first step is like, well no, infinity plus one and infinity are different. But then you get convinced that there is a way of matching things that where you can get things that seem pretty different, or a subset of a set, and they have the same cardinality. And then you go the other direction, which is “Well, of course, anything infinite is going to be the same size as anything else that's infinite.” And so then it turns out that even the rationals are the same size as the natural numbers. And that's way more complicated than we have time for. But if you add real numbers, meaning irrationals as well, then you have a whole different situation.
KK: You do indeed.
AS: It’s mind blowing, right? And so if you just think about the real numbers between zero and one, so just get go real simple. I mean, small, relatively. So you're just looking at decimal expansions. And so if those numbers had the same cardinality as the natural numbers, then you should be able to have a first one and a second one and a third one, and a fourth one. Or you can pair one number was the number 1, one number with the number 2, etc. And that list should be complete, and in the words of Kelsey Houston-Edwards, everyone should have a buddy. And so then, here's the cool thing, this is a proof that these two sizes of infinity are not the same, and it's a proof by contradiction, which is, again, your favorite proof when you are learning how to prove things. I mean, when I was learning proofs, I wanted to do everything by contradiction. So proving something by contradiction means you want to assume, “Well, what if we can list all the all the real numbers?” There’s a first one, a second one, a third one, etc. So Cantor’s amazing insight was that you can always find a number that was not on that list. Every time you make this list: a first one, a second one, a third one… there is some missing element.
And so you line up all your decimals. So you have the first number in decimal. And so you have like, you know, 0.12345… or something like that. And then you have the next one. And the next one. And like, I mean, this is really hard to do verbally, but we're going to do it. And so you sort of line them up, and you have infinite decimals. So you have point, a whole bunch of decimals, point, a whole bunch of decimals. And so you can make a missing number by taking that first number in the first decimal place a just changing that number. Okay, so if it was a 1, you write down a 2. And so you know, because we’ve known how to compare decimals since we were little kids, that what you need to compare is decimal place by decimal place. So these are different because they're different in this one spot, right? And then you go to the second number, and the second decimal point. And then you say, “Well, whatever number I see there, I'm going to make the second decimal point of my new number different.” So if you had a 3, you change it to a 4, whatever it is, as long as it's not the original number. And and this is why it's called the diagonalization argument, or the diagonal argument, because you have lined all those numbers up, and you can go through the diagonal, and for each one of those decimal points, at each decimal place, you just change the value. And what you're going to get is a number, another real number, infinitely many decimals, and it's going to be different from every number on your list, just by virtue of how you made it. And so then, what that shows is that the answer to the “what if” is: you can’t. The “what if” is, if you have a list of all real numbers, it's not complete. So there is never going to be a way that you can make that list complete. And this is the part where every time I tell my students, at some point, they're like, “Wait, there are different sizes of infinity? What?” Then—and that’s sort of lovely, because it's just this this mind-blowing moment where you've convinced yourself, by the way, that you were to infinity is infinity, and then you realize that there's something bigger than the cardinality of the natural numbers. And and then it's really fun when you tell them, “Well, is there something in between?” They’re like, “Of course! There must be!” And then you're like, “Wait, no one knows.”
KK: Maybe not.
AS: So yeah, I just love that argument. And I love how simple it is. And at the same time, it's, simple, but it's very, very deep, right? You really have to understand how these numbers match up with each other. And it requires a big leap of imagination to just think of doing this and realizing that you could make a number that was not on this infinite list by just doing that simple trick.
AS: And so I just think it's a really, really beautiful theorem. And then I also have a really personal connection to this theorem. But it's one of my favorite things to teach. And I'm going to be teaching at this term, and I’m really looking forward to seeing how that how that lands. Sometimes it lands really well. Sometimes people are like, “Eh, you’re just making stuff up.” Yeah.
KK: Well, then you can really blow their minds then when you show them the Cantor set, right?
AS: Yeah, yeah.
KK: And say, “Well, look, I mean, here's this subset of the reels that has the same cardinality, but it's nothing.”
AS: Exactly. Yeah, there's nothing there. Yeah.
EL: Yeah. I remember, then, when I first saw this argument, really carefully talking myself through, “Like, okay, but what if I just added that number I just made to the end of the list? Why wouldn't that work?” And trying to go through, like, “Why can't I—Oh, and then there must be other numbers that don’t fit on the list either.” It's not like we got within 1 of being the right cardinality.
EL: For these infinite number. So yeah, it's a really cool idea. But you said you had some personal connections to this. So do you want to talk more about those?
AS Sure. So I am from Venezuela, and I went to college there. And I liked college, it was fine. I knew—Well, one thing that you do have to decide when you're a student in high school is, you don't really apply to college, you apply to a major within the college. And so then I knew I wanted to do math. And I signed up for math at a specific university. And so then the first year was very similar to what you would do in the States, which is sort of this general year where everybody's thinking calculus, or everybody's taking—you have some subset of things that everybody takes. And then your second year, you start really going into the math major. And so this was my first real analysis class. This was my first serious proof-y class in my university. And we learned Cantor’s diagonalization argument, which was pretty early. But I loved this argument. I felt so mind-blown. You know, I was like, “This is why I want to do math,” you know, I was just so excited. And I knew I understood everything. And so I took the exam, and I got horrible grade. And in particular, I got zero points on the “prove that the real and the natural numbers don't have the same cardinality.” And so I went to the professor, and I saw my exam, and I was really confused. And I went to the professor, and I said, “I really don't understand what's wrong with this problem. Could you help me understand?” Because I thought I understood this. And then—you know, that's a typical thing. I probably said it in a more obnoxious way than I remember now. But I felt like I was being pretty reasonable. I was not the kind of kid that would go up to my professors too often to ask for points. I really was like, “I don't know what I did wrong.” And especially because I felt like I really got it.
AS: And so then he just looked at me and said, “If you don't understand what's wrong with this problem, you should not be a math major.” And that was it. That was the end of that conversation. Well, I still don't know what's wrong with this problem, and now you just told me I need to do something else. Just go do something else at a different school. Right? And I mean, I don't know that that was particularly sexist. But I do know that I was the only woman in that class, and I know that I felt it a lot. I think he probably would have said that—I really do think that he in particular would have said that to any student. I don’t think it was just me being female that affected that at all. But I do think that if I had been less stubborn about my math identity, I might have taken him up on that. But I was just like, “No, I'm going to show you!” And eventually I got an A in his class. He taught real analysis every semester, so I had to take the class with him every time and at some point, I cracked his code. And he at some point respected me, and thought I deserved to be there. But he was just very old-fashioned. You know, I don't think it's even sexism. It's just very, very, like, this is how we do things. And then I went—eventually, I did talk to someone. I think it was a teaching assistant. And I was like, “I don't know what's wrong with this problem.” And he looked at it. And he said, “Well, here's the problem. When you were listing—so you needed to list all these generic numbers and their decimal expansion. And I did, “Okay, the first number is point A1, A2, A3, etc. The second number is point B1, B2, B3, etc. The third one is point C1, C2, C3, etc, dot dot dot, right? And he said, “You have listed 26 numbers. And that's not going to be an infinite list.” Right?
KK: That’s cheap.
AS: And I was just like, “Okay, but I got the idea, right?” I was like, “Okay, it's true.” He’s like, “The way you wrote it is incorrect.” And I'm like, sure.
EL: Sort of.
KK: I’ve written that same thing on a chalkboard.
AS: You know, this shows you—like, fine, you can be more careful, you can be more precise, but from this, you shouldn't be a math major? That’s pretty intense.
AS: And I knew the mechanics, I knew what was supposed to be happening, I knew how to make the missing number, right? Like you just need A1: you change it to some other number, B2: you change it to some other number, C3: you change it to some other number. And so, I just thought—I mean, that was a moment where I was just literally told I should not be in math because I made a silly mistake. And it was a moment where I realized that—now looking back, I realize my math identity was pretty strong, because I just said, “Well, ask someone else to see what was wrong, and I'm not going to ask this guy anymore because it's clear what he thinks.”
AS: And sort of the stubbornness of, “Well, I’ll show him that I do deserve to be here.” But I think of all the students who might have taken classes with him, who would have heard that and then been like, “Yeah, maybe I need to do something else.” I mean, it just makes me really sad to hear, especially now that I'm a professor, and teaching these kinds of things. It just makes me sad to see which people were just scared away by someone like that, you know?
AS: So that was a big moment for me. Yeah.
EL: Yeah. Quite a disproportionate response to, what’s basically a bookkeeping difficulty.
EL: So, you know, we like to get our mathematicians to pair their theorems, with something on this show. And what have you chosen as your pairing for Cantor's diagonalization argument?
AS: Well, now that you suggested, music and other things, I'm maybe changing my mind.
EL: You could pair more than one thing.
AS: I was trying to find something that was just like—I need to sort of express the sort of mind-blowing nature of this, right? And so I was like, a tequila shot! You know, really just strong. And like, “Whoa, what just happened?” And so that was one thing that I thought about. And then—I don't know, just mind-blowing experiences, like, when I saw the Himalayas from an airplane, or when—you know, there are some moments where you're just like, “I can't believe this exists.” I can't believe this is a thing that I get to experience. So I guess, you know, there's been—most of these have been with traveling, where you just see something that you're just like, “I can't believe that I get to experience this.” And so I think Cantor's diagonalization argument is something like that, like seeing this amazing landscape where you're just like, “How does this even exist?”
EL: Yeah, I like that. I mean, I've had that experience looking out of airplane windows too. One time we were just flying by the coast of Greenland. And these fjords there. Of course, an airplane window is tiny and it's not exactly high-definition picture quality out of the thick plastic there, but it just took my breath away.
EL: Yeah, I like that. And we can even invite our listeners to think of their own mind-blowing favorite experiences that they've they've had. Hopefully legal experiences in their jurisdiction.
KK: Well, oh wait, it's not 4/20 anymore. Oh, well. So we also like to invite our guests to plug anything they want to plug. So you write for the AMS, the inclusion/exclusion blog, are there other places where we might find your mathematical writing for the general public?
AS: Well, that's my main plug and outlet right now. But I I do write for the MAA Focus magazine sometimes. That's sort of my main, and sometimes the AWM newsletter. So you might find some of my writing there. And the blog. I mean, again, now that I'm chair and doing a lot of other things, I'm not writing as much, but I definitely like to—I’ve gotten really into maybe this is a weird plug, but I've gotten really into storytelling.
EL: Oh yeah, you’ve been on Story Collider?
AS: Yeah, I was on one Story Collider. I've done some of the local stuff. But you can find me on the internet telling stories about being a mathematician. Some of them about some pretty fantastic experiences, and some not so great experiences.
EL: Yeah. Okay. Yeah. Well, we'll link to your Twitter, and that can help people find you too.
AS: Oh, yeah. Cool.
EL: Thanks a lot for joining us.
AS: Yeah. Thanks for having me and listen to me ramble about infinity.
EL: Oh, I just love this theorem so much.
KK: Yeah, we could talk about infinity all day. Thanks, Adriana.
AS: Yeah. Thank you so much.
We were excited to have Bates College mathematician Adriana Salerno on the show. She is also the chair of the department at Bates and a former Mass Media Fellow (just like Evelyn). Here are some links you might enjoy along with this episode.
Salerno on Twitter
AAAS Mass Media Fellowship for graduate students in math and science who are interested in writing about math and science for non-experts
Hilbert’s Infinite Hotel
Evelyn’s blog post about the Cantor set
Salerno’s StoryCollider episode
The inclusion/exclusion blog, an AMS blog about diversity, inclusion, race, gender, biases, and all that fun stuff