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Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and here is your other host.



Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to rehydrate myself after taking a long bike ride yesterday in the Utah desert in July, basically.




KK: That’s not cool. Okay, so here's my here's my story. So the last 10 days of June, my wife and I were out in Vancouver visiting our son. And it was lovely. It was, you know, 65 degrees every day, and we took a side trip to Banff. And which, if you've never been, I cannot recommend enough expect ACULA really beautiful. We had a wonderful time. We took the redeye Wednesday night back from Seattle to Orlando. Thursday morning. I had a sore throat for a couple of days.




EL: Uh oh.




KK: Yeah. I took a COVID test and it was negative. Okay, but I still don't feel great. Thursday I didn’t feel good. Friday morning I'm feeling worse. Take another COVID test. Guess what?




EL: It got you?




KK: It got me. I had a good run. It was two and a half years. But anyway, so




EL: This is a podcast from quarantine, although it's exactly the same as all our other podcasts because we’re always on Zoom anyway.




KK: Yeah, so anyway, here I am. So if I sound a little froggy, that’s why. I'm feeling a lot better and so is Ellen, but yeah, it's been a rough few days in the Knudson house. And it's 100 degrees here and miserable.




EL: Right? Yeah. A little less conducive to fun.




KK: Yeah, yeah. But enough of my petty problem problems, which — look, you know, everybody, if you're not vaccinated, get vaccinated, right? I'm of an age where I can be double boosted. And you know, I just, I got a bad cold. That's it. So get your shots, people. There, enough political — anyway. It shouldn't be political, but somehow it is. So today, we are pleased to welcome Philip Ording to the show. Phillip, why don't you introduce yourself?




Philip Ording: Hi, thank you, Kevin. Thanks, Evelyn, for having me on the show. Yeah, I am a mathematician and a writer and I teach at Sarah Lawrence College in Westchester, New York state. It's not as hot here right now. It was over the weekend.




EL: You’ve got your moments up there, I’m sure. It can be very oppressive.




KK: Yep.




PO: And if you hear some some child background noise, that's because COVID got the summer camp up here. My son came back from upstate Catskills camp, because they had to shut it down after a week.



KK: That’s a bummer.




EL: Oh, man. Yeah, that's rough. Well, we have invited you on the show to talk about your favorite theorem. But first, I wanted to sort of digress to a theorem that you have written about quite extensively.




KK: A lot!




EL: That apparently is not your favorite theorem. But I wanted to invite you on here because of this amazing book 99 Variations on a Proof that came out a few years ago, and I read it, you know, last year, or something, and I kept thinking, “Oh, I should invite him on here.” And it's 99 proofs of a theorem — maybe we might not even call it a theorem, a statement.




PO: It’s generous to call it a theorem.




EL: Yeah, that about the roots of a cubic polynomial, one particular cubic polynomial, and you just talk about it, you have 99 different ways to prove that the roots of this polynomial are 1 or 4. Sorry, a little minor spoiler for this book. I think you’ll still be able to enjoy it. So yeah, can you talk about that? Like the how you had the idea to write this book and kind of maybe tell us about some of the styles of proof or styles of presentation that you've included in here?




PO: Yeah, sure. Thanks for bringing that up. And the, the book, yeah, it’s not my favorite theorem. I chose it almost at random. And the book is really about everything around it. So I was interested in whether or not you could fill a book by thinking about the expressive material of mathematics outside of the content, or almost parallel to the content. I had a friend in grad school who said that he — he said this, I think, over drinks, but with gravitas — that he thought that the thing that mathematics had over other subjects was that it has so much content; you know, if you make one statement in mathematics, it's the kind of thing that not only is very condensed, and is probably the result of a long, long track to study, but it's also something you can return to a lot. So I was interested whether or not even a very humble equation and solution, something that anybody who has been exposed to math would recognize as mathematics, would be able to support that kind of an investigation of something — mathematicians don't talk about style that much. I think philosophers maybe are starting to talk about it more recently — and just carry it through the things that I like about math, the things I don't like, the history, and some of the folklore as well. So the titles are kind of the style for each of those chapters. And they range from things like “Psychedelic” to “Medieval” to “A proof that's found in a book.” And everything in between. So there are proofs from school, from graduate school, from college. There are person different languages. There are proofs that are linguistic, I guess, you could say, that draw attention to the particular notations, or the sound, that the proof reads as. Yeah. And it was a lot of fun. It kind of was a project that once it started, it took over and had a life of its own, which was probably what what got me to the very end of it, even though it was a long project.




EL: Yeah, a long time to be thinking about one cubic equation. I was flipping through today, and I did you know, towards the back, you have a mondegreen, which is one of those kind of misheard lyrics sort of things. “Their omelet: eggs, beer, eel” is the the first line of it. So you know, “their omelet” instead of “theorem: Let”




KK: Yeah. Yeah.




EL: And I read through it. This is one, you just have to concentrate so hard to read it and try to figure out the math version of it, but yeah, so you got, yeah, so many different ways to roll over this equation. So yeah, I hope people will check that out. It's a lot of fun.




PO: That was a that particular proof was a lot of fun to work on. I had some students helping me in the summer, and we just turned over the language of one of the simplest proofs in the book from a mathematical point of view. I think it comes from a kind of sleight of hand that could easily be misunderstood if somebody wasn't paying attention. And I remember when I was in college, I had a friend who said that she liked math, and she'd taken some courses but gave up after a calculus course in which she couldn't understand what the professor was saying. And all she remembers is this professor would get very excited and say “Knees the baby, knees the baby.” And she didn't have any idea what that meant, but she knew it was important. And so I tried to think of things that sounded very similar.




KK: Yeah.




PO: And I think it's an experience that everybody has, at some point, you're sitting in a talk and you kind of are reading the person's emotions as much as you're reading, you're listening, to that particular details of the techniques that are used, and there's often things that are lost in that channel. So it was fun to make fun of that phenomenon that I think most people who have studied mathematics at a certain level have experienced.




KK: What’s knees the baby? I can't figure it out.




PO: I still don't know. If anyone figures it out.




EL: Listener submissions.




PO: Multivariable calculus




KK: Okay, well, anyway, that’s, okay. We’ll try to figure it out offline.




EL: We’ll try not try to be thinking about that the whole time we’re recording this.




KK: That’s right. Okay, so you've told us what isn't your favorite theorem. You do have an actual favorite theorem. Why don’t you tell us about it?




PO: I do. And I love this question, because it’s, to me, it's very appealing. It's also very challenging. It's not the first time, actually, somebody asked me for my favorite theorem. The first time it wasn't for a podcast, it was for bathroom. I had a friend, some family friends, that had remodeled their apartment and they thought that this bathroom they had designed, it was like black paint or wallpaper inside. And they thought it would be fun to have their mathematician friend make a theorem or some kind of statement of gravitas in the bathroom. Or maybe they just thought it would go well with the marble sink or something. I'm not sure. But I thought about it for a long time. And I thought, okay, you know, is this going to be like something that is, I think, the most important piece of mathematics? Or is it going to be something really personally meaningful? Or maybe, like, I was in a bathroom at a bar once downtown and some, I think it was a grad student at NYU maybe, had done, like, the de Rham cohomology sequences, and I thought that looked cool. Maybe it should just be graffiti. But yeah, I sort of never got around to it, because I felt like I didn't really attach that much meaning to particular theorems. But anyway, what I came up with is something that's instead of a theorem, it's an idea. So it's called the Erlangen program.




KK: Okay.




PO: And it's credited to Felix Klein, German mathemati